diff --git a/-NFJT4oBgHgl3EQfpSw0/content/tmp_files/2301.11599v1.pdf.txt b/-NFJT4oBgHgl3EQfpSw0/content/tmp_files/2301.11599v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf171eddd51739364c584395c85f028f9a929abb --- /dev/null +++ b/-NFJT4oBgHgl3EQfpSw0/content/tmp_files/2301.11599v1.pdf.txt @@ -0,0 +1,5267 @@ +Unpolarized transverse-momentum dependent distribution functions of a quark in +a pion with Minkowskian dynamics +E. Ydrefors,1 W. de Paula,2 T. Frederico,2 and G. Salm`e3 +1Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China +2Instituto Tecnol´ogico de Aeron´autica, DCTA, 12228-900 S˜ao Jos´e dos Campos, Brazil +3INFN, Sezione di Roma, P.le A. Moro 2, 00185 Rome, Italy +(Dated: January 30, 2023) +The unpolarized twist-2 (leading) and twist-3 (subleading), T-even, transverse-momentum depen- +dent quark distributions in the pion are evaluated for the first time by using the actual solution +of a dynamical equation in Minkowski space. The adopted theoretical framework is based on the +homogeneous Bethe-Salpeter integral equation with an interaction kernel given by a one-gluon ex- +change, featuring an extended quark-gluon vertex. The masses of quark and gluon as well as the +interaction-vertex scale have been chosen in a range suggested by lattice-QCD calculations, and +calibrated to reproduce both pion mass and decay constant. The sum rules to be fulfilled by the +transverse-momentum dependent distributions are carefully investigated, particularly the leading- +twist one, that has to match the collinear parton distribution function, and hence can be scrutinized +in terms of existing data as well as theoretical predictions. Noteworthy, the joint use of the Fock +expansion of the pion state facilitates a more in-depth analysis of the content of the pion Bethe- +Salpeter amplitude, allowing for the first time to determine the gluon contribution to the quark +average longitudinal fraction, that results to be ∼ 6%. The current analysis highlights the role +of the gluon exchanges through quantitative analysis of collinear and transverse-momentum distri- +butions, showing, e.g. for both leading and subleading-twists, an early departure from the widely +adopted exponential fall-off, for |k⊥|2 > m2, with the quark mass ∼ ΛQCD. +I. +INTRODUCTION +Quark transverse-momentum dependent distribu- +tion functions (TMDs for short) are the basic ingredi- +ents for parametrizing the hadronic quark-quark cor- +relator (see the seminal Ref. [1] and for the complete +parametrization Ref. [2], while Refs. [3, 4] for corre- +lators involving gluons), and represent direct general- +ization of the parton distribution functions (PDFs), +so that both longitudinal and transverse degrees of +freedom (dof) can be addressed (see, e.g., Refs. [5, 6] +for an extensive introduction to the transverse dof +and related distribution functions). Clearly, the ac- +cess to the 3D imaging of hadrons allows us to +achieve a deeper and deeper understanding of the non- +perturbative regime of QCD, also exploiting the non- +trivial coupling to the spin dof (see, e.g., Refs. [7, 8] +and references therein). Hence, by means of TMDs, +one can gather unique information on QCD at work in +hard semi-inclusive reactions (both unpolarized and +polarized) at low transverse-momentum, like low-q⊥ +Drell-Yan (DY) processes, vector/scalar boson pro- +ductions or semi-inclusive deep inelastic scattering +(SIDIS) (see, e.g., Refs. [8–11] for a status-report on +the experimental measurements). Indeed, the extrac- +tion of TMDs from the experimental cross-section is +a highly challenging task, as shown by the intense +theoretical work on the factorization of the cross sec- +tions into transverse-momentum dependent matrix el- +ements (see, e.g., Refs. [12–16]) and the TMDs evo- +lution that becomes a two-scale problem, since the +rapidity ζ comes into play in addition to the renor- +malization scale µ (see, e.g., Ref. [12, 17–19] and +Ref. [20] for a recent review that covers also the factor- +ization). Noteworthy, one has to mention the efforts +for obtaining reliable global fits (see, e.g., Refs. [21– +24] and also Ref. [8] for a general discussion), early- +stage lattice calculations (see, e.g., Refs. [25–29] and +also Refs. [8, 30–32]) and, finally, the broad set of +phenomenological models, that we can only partially +list: +the bag model (see, e.g., Ref. [33] and refer- +ences therein), covariant model (see, e.g., Ref. [34] +and references therein), light-front (LF) constituent +quark models (see, e.g., Refs. [35, 36]) and the ba- +sis LF quantization framework [37], the approaches +based on the Nambu-Jona-Lasinio interaction (see, +e.g., Refs. [38, 39]), the holographic models (see,e.g., +Refs. [40, 41]), etc. In view of our study, one has to +separately mention the approaches developed within +the so-called continuum-QCD, that are based on solu- +tions (actually in Euclidean space) of dynamical equa- +tions like the homogeneous 4D Bethe-Salpeter equa- +tion (BSE) [42, 43] in combination or not with the +arXiv:2301.11599v1 [hep-ph] 27 Jan 2023 + +2 +quark gap-equation (see, e.g. Refs. [44–47]). +It should be recalled that the proton is the elective +target of much experimental (see, e.g., Refs. [9–11]) +and theoretical research (see, e.g., Refs. [48–50] and +references therein). While the pion, given the experi- +mental challenges its study poses, has surely attracted +less efforts in spite of its intriguing double-nature, be- +ing both a Goldstone boson (and hence fundamen- +tal for investigating the dynamical chiral-symmetry +breaking) and a quark-antiquark bound system (i.e. +the simplest bound system in QCD). In particular, a +first extraction of the pion unpolarized leading-twist +TMD from Drell-Yan data can be found in Ref. [51], +where the results of the E615 Collaboration [52] has +been used, and in Ref. [53], where both the previous +data and the E537 Collaboration cross-sections [54] +have been included. As to the phenomenological cal- +culations, a broad overview, embracing different ap- +proaches, can be gained from Refs. [35, 38, 40, 41, 44– +47, 55–58] (see also Ref. [59] for the generalized TMDs +in a spin-0 hadron). +As a conclusion to the above schematic introduc- +tion, it has to be emphasized that the vast amount of +nowadays theoretical studies on TMDs finds its strong +motivation in the very accurate measurements that +will come from forthcoming electron-ion colliders, that +promise to achieve greatly expected milestones in the +experimental investigation of non-perturbative QCD, +given the planned high energy and luminosity [60, 61]. +Our aim is to obtain, for the first time, T-even +leading- and subleading-twist unpolarized TMDs (uT- +MDs) of the pion, by solving a dynamical equation +directly in Minkowski space, namely relying on a gen- +uinely quantum-field theory framework based on the +4D homogeneous BSE [42, 43]. +The homogeneous +BSE is an integral equation and therefore suitable +for dealing with the fundamentally non-perturbative +nature of bound states. +One should not get con- +fused by the use of an interaction kernel expressed in +a perturbative series, since an integral equation has +a peculiar feature of infinitely many times iterating +the boson exchanges contained in each term of the +kernel, just what one needs for obtaining a pole in +the relevant Green’s function. In our approach (see +Ref. [62] for details and references therein), based +on the 4D homogeneous BSE in Minkowski space +and the Nakanishi integral representation (NIR) of +the BS-amplitude [63, 64], the interaction kernel is +given by the exchange of a massive vector boson in +the Feynman gauge, with three input parameters, in- +ferred from lattice QCD (LQCD) calculations (see, +e.g. +Refs. +[65–67]): (i) the constituent-quark and +gluon masses, and (ii) a scale parameter featuring the +extended quark-gluon vertex. +It should be pointed +out that the ladder kernel, i.e. +the first term in a +perturbative series, can be a reliable approximation +to evaluate the pion bound state, as suggested by the +suppression of the non-planar contributions for Nc = 3 +within the BS approach in a scalar QCD model [68], +and the presence of massive quarks and gluons, featur- +ing the confinement effects in a relatively large system +(rch ∼ 0.66 fm). There is another important conse- +quence stemming from the use of the BS-amplitude. +Although in the definition of the q¯q-pair BS-amplitude +there is a simple dependence upon two interacting +fermionic fields, one ends up dealing with an infinite +content of Fock states (the use of the Fock space al- +lows one to recover a probabilistic language within the +BS framework). In particular, by exploiting the Fock +expansion of the pion state, one can establish a for- +mal link between the LF-projected BS-amplitude (see, +e.g., Refs. [69–71]), and the amplitude of the Fock +component of the pion state with the lowest number +of constituents. Therefore, in our approach, it is natu- +ral to call the LF-projected BS-amplitude LF valence +wave function (LFWF), to be distinguished from the +valence wave function, when a SU(3)-flavor language +is adopted. In the latter case, the pion is composed +by only two fermionic constituents, suitably dressed. +One should keep in mind that within our framework, +the pion LFWF contributes only with 70% [62] of the +normalization, and consequently a significant role of +the higher Fock components has to be highlighted, and +possibly analyzed in-depth, as illustrated in what fol- +lows. Finally, we would emphasize that the first evalu- +ation of the uTMDs strengthens the reliability of our +approach and makes sound the ground for the next +step, already in progress, i.e. taking into account the +self-energy of the quarks (see Refs. [72] and [73, 74]). +Indeed, in spirit, our approach is similar to the one +developed in Ref. [45] for evaluating the leading-twist +uTMD, where it was also taken into account the self- +energy of the quark propagator (solving the gap equa- +tion) and a confining interaction, but in Euclidean +space. In this case, one resorts to a suitable method +(based on the moments and a parametrization of the +Euclidean BS-amplitude) to get the Minkowski-space +distribution function. Differently, in our approach the +NIR of the BS-amplitude allows one to successfully +deal with the analytic structure of the BS-amplitude +itself, obtaining an integral equation formally equiv- +alent to the initial 4D homogeneous BSE, but more +suitable for the numerical treatment. Many and rele- +vant applications of our approach to the pion, such as + +3 +the electromagnetic form factor [75], the PDF [76] and +the 3D imaging [62], have confirmed its reliability and +encouraged to broad the scope of our investigation. +It should be pointed out that (it will become clear in +what follows) the evaluation of quantities that depend +not only upon the longitudinal dof but also the trans- +verse ones leads to sharply increase the sensitivity to +the dynamical content of a given phenomenological +description of the pion, namely to increase its predic- +tive power. +Furthermore, the joint use of the Fock +expansion, meaningful in the Minkowski space, allows +one to resolve the gluonic content of the pion state. +The paper outline is as follows. In Sect. II, the gen- +eral formalism and the notations are introduced, high- +lighting the ingredients of our dynamical approach, +namely i) the Bethe-Salpeter amplitude, solution of +the 4D homogeneous Bethe-Salpeter equation, and +ii) the Nakanishi integral representation of the BS- +amplitude. +In Sect. III, the expressions of leading- +and subleading-twist uTMDs are given in terms of +the Bethe-Salpeter amplitude of the pion. In Sec. IV +and V, the leading and subleading-twist uTMDs are +shown and compared with outcomes from other ap- +proaches. +Finally, in Sect. IV, the conclusions are +drawn, and the perspectives of our approach are pre- +sented. +II. +GENERALITIES +For a pion with four-momentum P ≡ {P −, P +, P⊥} +(where P 2 = P +P − − |P⊥|2 = M 2 and the LF co- +ordinates are a± = a0 ± a3), and by adopting both +i) a frame where P⊥ = 0 and ii) the light-cone gauge +A+ +g = 0, the quark leading-twist uTMD, f q +1 (γ, ξ), is +defined as follows (for a general introduction see, e.g., +Ref. [1, 6]) +f q +1 (γ, ξ) = Nc +4 +� +dφˆk⊥ +� ∞ +−∞ +dy−dy⊥ +2(2π)3 +× ei[ξP + y− +2 −k⊥·y⊥]⟨P| ¯ψq(− y +2 )γ+ψq( y +2 )|P⟩ +�� +y+=0 , +(1) +where Nc is the number of colors, ψq is the fermionic +field, and the quark four-momentum is given in terms +of LF coordinate by pq ≡ {p− +q , ξP +, k⊥ + P⊥/2}, +with γ = |k⊥|2. +The antiquark uTMD is obtained +by using the proper four-momentum p¯q ≡ {p− +¯q , (1 − +ξ)P +, −k⊥ + P⊥/2}, recalling that P = pq + p¯q and +k = (pq − p¯q)/2. +The normalization of f q +1 (γ, ξ) is given by +� ∞ +−∞ +dξ +� ∞ +0 +dγ f q +1 (γ, ξ) = Nc +2 +� +dpq⊥ +� ∞ +−∞ +dp+ +q +P + +× +� ∞ +−∞ +dp− +q +2 +� ∞ +−∞ +d4y +(2π)4 ei pq·y⟨P| ¯ψq(− y +2)γ+ψq( y +2)|P⟩ += Nc +⟨P| ¯ψq(0)γ+ψq(0)|P⟩ +2P + += F q +π(0) = 1 , +(2) +where F q +π(t) is the quark contribution to the elec- +tromagnetic (em) form factor of the pion. The lat- +ter results to be equal to Fπ(t) = eqF q +π(t) + e¯qF ¯q +π(t), +with t = (P ′ − P)2, and is related to the matrix ele- +ment of the four-current by Nc ⟨P| ¯ψq(0)γµψq(0)|P⟩ = +2P µ Fπ(t = 0). Finally, it should be pointed that in- +serting a complete basis in Eq. +(1) and exploiting +the good and bad components of the fermionic field +one can easily demonstrate that f q +1 (γ, ξ) ≥ 0 (see Ref. +[77]). +In order to describe the pion by taking into ac- +count at some extent the QCD dynamics in the non- +perturbative regime, it is useful to resort to the Man- +delstam framework [78], where the interacting quark- +pion vertex is expressed in terms of the (reduced) BS- +amplitude, i.e. the solution of the 4D homogeneous +BSE, and defined by +Φ(k, P) = +� +d4x eik·x ⟨0|T +� +ψ( x +2) ¯ψ(− x +2) +� +|P⟩ , +(3) +where the fermionc field fulfills the Poincar´e trans- +lation ψ(x) = ei ˆ +P ·xψ(0)e−i ˆ +P ·x (recall that only the +component ˆP − is interacting in the LF dynamics, see, +e.g., Ref. [79]). +Thus, by using the Feynman-like diagrammatic pic- +ture inherent to the Mandelstam framework (see, e.g., +Ref. [75] for the application to the em form factor), +one can write the following expression for f q +1 (γ, ξ) +f q +1 (γ, ξ) = +Nc +4(2π)3 +� ∞ +−∞ +dk+ +2(2π)δ +� +k+ + P + +2 +− ξP +� +× +� ∞ +−∞ +dk− +� 2π +0 +dφˆk⊥Tr +� +S−1(−p¯q)¯Φ(k, P) γ+ Φ(k, P) +� +, +(4) +where +pq(¯q) = ± k + P +2 . +(5) +For the sake of completeness, let us write the BSE in +ladder approximation, i.e. the one we are adopting for + +4 +the numerical calculations, viz. +Φ(k; P) = S +� +pq +� � +d4k′ +(2π)4 Sµν(q)Γµ(q) +× Φ(k′; P)�Γν(q)S +� +−p¯q +� +, +(6) +where quark and antiquark momenta are off-shell, i.e. +p2 +q(¯q) = (±k + P +2 )2 ̸= m2, and q = k − k′ is the gluon +four-momentum. +In Eq. (6), the fermion propaga- +tor, the gluon propagator in the Feynman gauge and +the quark-gluon vertex, dressed through a simple form +factor, are +S(p) = +i +/p − m + iϵ , +Sµν(q) = −i +gµν +q2 − µ2 + iϵ , +Γµ = igγµ +µ2 − Λ2 +q2 − Λ2 + iϵ, +(7) +where g is the coupling constant, µ the mass of the +exchanged vector-boson and Λ is a scale parameter, +featuring the extension of the color distribution in the +interaction vertex of the dressed constituents. More- +over, in Eq. (6), one has �Γν(q) = C ΓT +ν (q) C−1, +where C = iγ2γ0 is the charge-conjugation opera- +tor. The normalization of the BS-amplitude reads (cf. +Refs. [80] and [62] for details) +Nc Tr +�� +d4k +(2π)4 +∂ +∂P ′µ +� +S−1� +k − P ′ +2 +� +¯Φ(k, P) +× S−1� +k + P ′ +2 +� +Φ(k, P) +�� +P ′=P += −2iPµ . +(8) +The antiquark uTMD is given by +f ¯q +1 (γ, 1 − ξ) = − +Nc +4(2π)3 +� ∞ +−∞ +dk+ +2(2π)δ(k+ + P + +2 − ξP +) +× +� ∞ +−∞ +dk− +� 2π +0 +dφˆk⊥Tr +� +S−1(pq)Φ(k, P)γ+ ¯Φ(k, P) +� +, +(9) +where the minus sign results from the property of the +normal-ordered em current to be odd under the action +of the charge conjugation operator. It is noteworthy +that in Appendix C 1, it is proven the identity of the +normalization condition, Eq. (8), and the half sum of +Eqs. (1) and (9). +Within a SU(3)-flavor symmetry framework, one +describes a pion as a bound system of a massive q¯q +pair. +This leads to introduce the so-called valence- +quark PDF in the pion, that is charge symmetric (once +the isospin breaking is disregarded [81]) as well as ful- +fills the charge conjugation. From those properties one +deduces that the SU(3)-valence PDFs in the charged +pions must verify: uv +π+(ξ) = dv +π−(ξ) = ¯dv +π+(ξ). In our +BS framework, in addition to the fermionic dof (still +massive) one introduces also gluonic dof, by adding +an explicit dynamical description of the binding. This +amounts to the ladder exchange of infinite number of +massive gluons. +Therefore, at the initial scale, the +quark and anti-quark longitudinal-momentum frac- +tion distributions are not expected to be symmetric +with respect to ξ = 1/2 (as it follows from the charge +symmetry), given the gluon-momentum flow in the +composite pion (see Sect. IV). The symmetric com- +bination of quark and anti-quark contribution allows +one to fulfill the charge symmetry, and hence it is rel- +evant in the comparison with experimental data (see +Ref. [76]). In what follows, in addition to the quark +distributions, symmetric and anti-symmetric combi- +nations are introduced for all the uTMDs we are going +to analyze. +The half sum (difference) of the quark and anti- +quark contributions, Eqs. (1) and (9), yields the fol- +lowing charge-symmetric (anti-symmetric) expression +for the leading-twist uTMD inside a π+ meson +f S(AS) +1 +(γ, ξ) = f q +1 (γ, ξ) ± f ¯q +1 (γ, 1 − ξ) +2 += +Nc +8(2π)3 +� ∞ +−∞ +dk+ +2(2π)δ +� +p+ +q − ξP +� � ∞ +−∞ +dk− +× +� 2π +0 +dφˆk⊥Tr +� +S−1(−p¯q)¯Φ(k, P) γ+ Φ(k, P) +∓ S−1(pq)Φ(k, P) γ+ ¯Φ(k, P) +� +. +(10) +Analogously to Eq. (1), one can define the T-even sub- +leading quark uTMDs, starting from the decomposi- +tion of the pion correlator [6, 82]. To be specific, one +has two twist-3 uTMDs (see, e.g., Ref. [35] for the +pion case) +M +P + eq(γ, ξ) = Nc +4 +� +dφˆk⊥ +� ∞ +−∞ +dy−dy⊥ +2(2π)3 +(11) +×ei[ξP + y− +2 −k⊥·y⊥]⟨P| ¯ψq(− y +2) 1 ψq( y +2)|P⟩ +�� +y+=0 , +M +P + f ⊥q(γ, ξ) = M +γ +Nc +4 +� +dφˆk⊥ +� ∞ +−∞ +dy−dy⊥ +2(2π)3 +(12) +×ei[ξP + y− +2 −k⊥·y⊥] ⟨P| ¯ψq(− y +2) k⊥ · γ⊥ ψq( y +2)|P⟩ +�� +y+=0 . +In analogy to Eq. (2), one gets for the twist-3 eq(ξ) +(see Refs. [77, 82] and Ref. [35] for the pion in phe- + +5 +nomenological models) +� ∞ +−∞ +dξ +� ∞ +0 +dγ eq(γ, ξ) = Nc +2 +� +dpq⊥ +� ∞ +−∞ +dp+ +q +P + +× +� ∞ +−∞ +dp− +q +2 +� ∞ +−∞ +d4y +(2π)4 ei pq·y ⟨P| ¯ψq(− y +2) 1 ψq( y +2)|P⟩ += Nc +⟨P| ¯ψq(0) 1 ψq(0)|P⟩ +2P + +, +(13) +where the matrix element ⟨P| ¯ψq(0) 1 ψq(0)|P⟩ has to +be proportional to the pion sigma term, once a QCD +framework is adopted. As a matter of fact, one gets +� 1 +0 +dξ +� ∞ +0 +dγ eq(γ, ξ) = +σπ +mcur +(14) +where mcur is the quark current mass and σπ is +the pion sigma term, that becomes σπ = M/2, in +the leading order of the chiral expansion, i.e. +the +Gell-Mann-Oakes-Renner relation [83]. It should be +pointed that recent LQCD calculations [84] confirm, +with high accuracy, the Gell-Mann-Oakes-Renner re- +lation in the range of the explored pion masses. In- +deed, the QCD equations of motion gives a decom- +position of the collinear PDF e(ξ) = +� +dγ e(γ, ξ) in +three terms. Among them, there is a singular term +proportional to the pion sigma term, that reads (see, +e.g., Ref. [85]) +esing(ξ) = δ(ξ) ⟨P| ¯ψq(0) 1 ψq(0)|P⟩/2P + , (15) +while the other two terms, one is due to quark- +antiquark-gluon correlations and the other is propor- +tional to the quark mass, do not contribute to Eq. (14) +(see Ref. [85], where the issue is analyzed, taking +the nucleon as actual case). In our phenomenologi- +cal model the strength is distributed over the whole +range of ξ (as in Ref. [35]), without the singularity at +ξ = 0, as it will be shown in Sect. V. Moreover, one +has for the first moment [85] +� 1 +0 +dξ +� ∞ +0 +dγ ξ eq(γ, ξ) = mcur +M +, +(16) +where the singular term and the gluonic contribution +vanish, and only the term proportional to the quark +mass contributes. +From the equations of motion of a free-quark model, +one deduces the following relations between the above +uTMDs (see,e.g., Ref [35, 85, 86]) +ξ eq +EoM(γ, ξ) = ξ ˜eq(γ, ξ) + m +M f q +1;EoM(γ, ξ) +ξ f q⊥ +EoM(γ, ξ) = ξ ˜f q⊥(γ, ξ) + f q +1;EoM(γ, ξ) , (17) +where the uTMDs with a tilde are the gluonic con- +tributions. The relevant point is the dependence of +all the subleading-twist uTMDs from only the leading +one, modulo the gluonic terms. In our fully interact- +ing framework, one can anticipate that the relations +are not recovered, and rather heavily broken. For a +derivation of the first line of Eq. (17), fully consistent +with QCD, one could apply the formalism presented +in Ref. [85]. +Following Eq. (10), one readily writes down charge- +symmetric and the anti-symmetric combinations for +the subleading TMDs. One has to take care how the +scalar and vector operators behave under the charge +conjugation that impose a different combination of +signs (cf. below Eq. (9)). Namely, one gets +M +P + eS(AS)(γ, ξ) = +Nc +8(2π)3 +� ∞ +−∞ +dk+ +2(2π)δ(p+ +q − ξP +) +× +� ∞ +−∞ +dk− +� 2π +0 +dφˆk⊥Tr +� +S−1(−p¯q)¯Φ(k, P) 1 Φ(k, P) +± S−1(pq)Φ(k, P) 1 ¯Φ(k, P) +� +. +(18) +M +P + f ⊥S(AS)(γ, ξ) = +NcM +8(2π)3γ +� ∞ +−∞ +dk+ +2(2π)δ(p+ +q −ξP +) +� ∞ +−∞ +dk− +� 2π +0 +dφˆk⊥Tr +� +S−1(−p¯q)¯Φ(k, P) γ⊥ Φ(k, P) +± S−1(pq)Φ(k, P) γ⊥ ¯Φ(k, P) +� +· k⊥ . +(19) +A. +The BS-amplitude and its Nakanishi integral +representation +It is useful to briefly recall some features of our ap- +proach for obtaining the actual solution of the ladder +BSE given in Eq. (6). +The basic ingredient is the +NIR of the BS-amplitude (see Ref. [64] for the general +introduction, and Refs. [62, 76, 87–89] for the appli- +cation to a two-fermion case), but let us first intro- +duce the general decomposition of the BS-amplitude, +Φ(k; P), for a 0− bound state, viz. [87, 90] +Φ(k; P) = S1(k; P)φ1(k; P) + S2(k; P)φ2(k; P) ++S3(k; P)φ3(k; P) + S4(k; P)φ4(k; P) , +(20) +where φi’s are unknown scalar functions, that depend +upon the kinematical scalars at disposal (k2, k ·P and + +6 +P 2), and Si’s are suitable Dirac structures, given by +S1(k; P) = γ5, S2(k; P) = /P +M γ5, +S3(k; P) = k · P +M 3 /Pγ5 − 1 +M /kγ5, +S4(k; P) = +i +M 2 σµνPµkνγ5 . +(21) +The functions φi must be even for i = 1, 2, 4 and odd +for i = 3, under the change k → −k, as dictated by +the anti-commutation rules of the fermionic fields, and +they can be written in terms of the NIR as follows +φi(k; P) = +� 1 +−1 +dz′ +� ∞ +0 +dγ′ +× +gi(γ′, z′; κ2) +[k2 + z′(P · k) − γ′ − κ2 + iϵ]3 , +(22) +where κ2 += +m2 − M 2/4. +The real functions +gi(γ′, z′; κ2), the unknowns of the problem under +scrutiny, are the Nakanishi weight functions (NWFs), +and assumed to be unique, following the uniqueness +theorem from Ref. [64]. The properties of the scalar +functions φi under the exchange k → −k translate +to properties of the NWFs, but under the exchange +z′ → −z′. +Finally, it should be mentioned that NWFs are de- +termined by solving a system of integral equation, so +that one is able to non-perturbatively embed dynam- +ical information that characterize the BS interaction +kernel. The system of integral equations is formally +deduced from the initial BSE, by exploiting the ana- +lytic structure of the scalar functions φi, made explicit +by means of the NIR. In fact, after inserting Eqs. (20) +and (22) in the BSE, Eq. (6), and performing both the +Dirac traces and a LF projection, i.e. the integration +over the k− = k0 − k3 component of the relative mo- +mentum, one gets a coupled system of integral equa- +tions for the NWFs (see details in Ref. [89]). Once +the NWFs are known, the BS-amplitude can be fully +reconstructed through an inverse path, i.e. Eqs. (22) +and (20). +III. +THE UNPOLARIZED TMDS AND THE +PION BS-AMPLITUDE +The evaluation of the leading- and subleading-twist +uTMDs, given in Eqs. (10), (18) and (19), can be +performed by inserting the decomposition of the BS- +amplitude in Eq. (20), obtaining +T S(AS) +i +(γ, ξ) = +Nc +8(2π)3 +� ∞ +−∞ +dk+ +2 δ(p+ +q − ξP +) +× +� ∞ +−∞ +dk− +2π +� 2π +0 +dφˆk⊥Tr +� +S−1(−p¯q) Ai(k, P) ++ ηS(AS) +i +S−1(pq) ¯ +Ai(k, P) +� += +i Nc +8(2π)2 +� +ℓj +� 1 +−1 +dz δ(z−(1 − 2ξ)) F i +ℓj(γ, z; S(AS)) , +(23) +where +Ai(k, P) = ¯Φ(k, P) Oi Φ(k, P) +¯ +Ai(k, P) = Φ(k, P) Oi ¯Φ(k, P) . +(24) +A new variable z is defined as z = −2k+/P + and +the three quantities: i) the functions Ti(γ, ξ), ii) the +operators Oi and iii) the phase ηS(AS) +i +are given by +T S(AS) +0 +(γ, ξ) ≡ f S(AS) +1 +(γ, ξ) , +O0 = γ+ , +ηS(AS) +0 += ∓1 , +T S(AS) +1 +(γ, ξ) ≡ +M +P + eS(AS)(γ, ξ) , +O1 = 1 , +ηS(AS) +1 += ±1 , +T S(AS) +2 +(γ, ξ) ≡ +M +P + f ⊥S(AS)(γ, ξ) , +O2 = +M +|k⊥|2 k⊥ · γ⊥ , +ηS(AS) +2 += ±1 . +(25) +Finally, the integrand F i +ℓj in Eq. (23) reads +F i +ℓj(γ, z; S(AS)) = +� ∞ +−∞ +dk− +2π +ai +ℓj(k−, γ, z; S(AS)) +× φℓ(k, P) φj(k, P) . +(26) +where ai +ℓj(k−, γ, ; S(AS)) are polynomial in k− (up to +the cubic power) and can be found in Appendix A for +each uTMDs, we are considering. +By exploiting the NIR, Eq. (22), one can perform + +7 +the integration on k−. This integration amounts to +restrict the LF-time to x+ = 0, and it is also known +as LF-projection (see, e.g., Refs. [70, 71, 91]). After +carrying out the k−-integration, the expression of each +T S(AS) +i +(γ, ξ) can be decomposed as follows (the details +of this formal step can be found in Appendix B) +T S(AS) +i +(γ, ξ) = 3Nc +(2π)2 +� +ℓj +� +Fi +0;ℓj(γ, z; S(AS)) ++ Fi +1;ℓj(γ, z; S(AS)) + Fi +2;ℓj(γ, z; S(AS)) ++ Fi +3;ℓj(γ, z; S(AS)) +� +, +(27) +where +ξ += +(1 − z)/2 +and +the +functions +Fi +n;ℓj(γ, z; S(AS)) +(n += +1, 2, 3, 4) +are +given +in +Eqs. (B19), (B20), (B21) and (B22), respectively. +IV. +THE LEADING-TWIST f S(AS) +1 +(γ, ξ) +The symmetric and anti-symmetric combinations of +the T-even leading-twist uTMD, f S(AS) +1 +(γ, ξ), allow us +to address the evaluation of both quark and anti-quark +contributions, f q(¯q) +1 +(γ, ξ), that in the BS framework +plus the Fock expansion of the pion state have inter- +esting features, distinct from the ones of f S(AS) +1 +(γ, ξ). +After integrating the leading-twist f q(¯q) +1 +(γ, ξ) on γ, +one gets the quark PDF uq(ξ), while the symmet- +ric combination provides the charge-symmetric PDF +uS(ξ), +i.e. the one is expected to have relevance at +the valence scale (see, e.g., Ref. +[81]). +Indeed, in +the Mandelstam approach the quark and antiquark +PDFs do not have in general a symmetry with re- +spect to ξ = 1/2, since each receives contributions +from states containing an infinite number of gluons, +as a consequence of the ladder-interaction kernel. But +if we restrict to the contribution from the first Fock +component in the expansion of the pion state, one +gets the LF-valence uLF +val(ξ), that is given by the BS- +amplitude projected onto the null plane [79] and is +fully compliant with the charge symmetry (see below +the discussion on the differences among uq(ξ), uS(ξ) +and uLF +val(ξ)). +To illustrate general features and relations, in this +Section we give some details, referring to Appendix C +for a more complete discussion. +The symmetric and anti-symmetric leading-twist +uTMDs, can be decomposed as follow +f S(AS) +1 +(γ, ξ) = IN(γ, ξ; S(AS)) + Id(γ, ξ; S(AS)) ++I2d(γ, ξ; S(AS)) + I3d(γ, ξ; S(AS)) , +(28) +where the non-vanishing symmetric contributions are +given by Eqs. (C2), (C3), (C4) and I3d(γ, ξ; S) = 0, +respectively. The anti-symmetric quantities are shown +in Eqs. (C5), (C6), (C7) and (C8), respectively. +Two comments are in order. The symmetry proper- +ties of the above quantities with respect to the trans- +formation z → −z are demonstrated in Appendix C, +and can be translated into the symmetry with respect +to ξ → 1 − ξ (that implements the charge-symmetry). +A relevant feature is given by the presence in the ex- +pressions of Id,2d,3d of the partial derivatives ∂n/∂zn, +that should be considered dual of the n-th moment in +k− of the relevant functions, generated by the formal +step of the LF-projection (cf Eq. (26)). This is not a +surprise since the variable z is proportional to k+. +A first consistency check of our formalism has been +carried out in Appendix C 1, where it is shown that, +within the Mandelstam approach, f S +1 (γ, ξ) and in turn +f q +1 (γ, ξ) are normalized to 1, as naturally follows from +the canonical BS-amplitude normalization [80, 92], +performed according to Eq. (8) (see also Ref. [62]). In +particular, the integral on γ and ξ of IN(γ, ξ; S) sat- +urates the normalization, while the other two terms +provide vanishing contributions. Hence, one gets +� 1 +0 +dξ +� ∞ +0 +dγ f S +1 (γ, ξ) += +� 1 +0 +dξ +� ∞ +0 +dγ IN(γ, ξ; S) += +� 1 +0 +dξ +� ∞ +0 +dγ f q +1 (γ, ξ) = 1. +(29) +It should be recalled that all the calculated uTMDs +vanish outside the interval 0 ≤ ξ ≤ 1, as dictated by +the conservation of the plus components of the four- +momenta of both pion and constituents (cf. Eq. (10)). +It is understood that the integral of f AS +1 +(γ, ξ) is van- +ishing, given the antisymmetry with respect to ξ → +1 − ξ. +A. +Longitudinal degree of freedom +The symmetric and the anti-symmetric PDFs, +uS(AS)(ξ) +(for +the +explicit +expressions +see +Ap- +pendix D) are defined by +uS(AS)(ξ) = +� ∞ +0 +dγ f S(AS) +1 +(γ, ξ) +(30) += uS(AS) +N +(ξ) + uS(AS) +d +(ξ) + uS(AS) +2d +(ξ) + uS(AS) +3d +(ξ) , + +8 +0 +0.25 +0.5 +0.75 +1 + ξ +-6 +-4 +-2 +0 +2 +4 +6 +u +S +( ξ) +0 +0.25 +0.5 +0.75 +1 + ξ +-0.5 +0 +0.5 +1 +1.5 +2 +u( ξ) +FIG. 1. (Color online). Left panel: the symmetric pion PDF, uS(ξ), with its contributions uS +N(ξ), uS +d (ξ) and uS +2d(ξ) (cf +Eq. (31)). Dash-dotted line: uS(ξ). Dashed line: uS +N(ξ). Dotted line: uS +d (ξ). Dash-double-dotted line: uS +2d(ξ). Right +panel: uq(ξ), uS(ξ), uAS(ξ) and the LF-valence PDF of the pion, uLF +val(ξ). Solid line: quark PDF, Eq. (31). Dashed line: +uS(ξ). Dotted line: uAS(ξ). Dash-dotted line: uLF +val(ξ) (see Ref. [76]), with normalization equal to Pval = 0.7 (see text). +with the normalization that follows from Eq. (29) +and the vanishing result of the double integration of +f AS +1 +(γ, ξ). +Finally, the quark and anti-quark PDFs +are evaluated through +uq(¯q)(ξ) = uS(ξ) ± uAS(ξ) , +(31) +with the normalization still given by Eq. (29). Within +the SU(3)-flavor symmetry, one has to implement the +charge symmetry (see, e.g. +Ref. [81]) at the initial +scale, and therefore uS(ξ) is the PDF to be com- +pared, after the proper evolution, with the experimen- +tal data, as it has been shown in Ref. [76]. +In the left panel of Fig. 1, uS(ξ) and its three +contributions (see Eqs. (D4), (D5) and (D6)) are +shown. +The calculation has been carried out by +adopting the BS-amplitude obtained by using the so- +lution of the BSE as described in Ref. [62], using +the following values of the three input parameters: +m = 255 MeV, µ = 637.5 MeV and Λ = 306 MeV, +able to reproduce the pion decay constant f P DG +π += +130.50(1)(3)(13) MeV [93] (recall that the pion charge +radius results to be rch = 0.663 fm [75], in excellent +agreement with rP DG +ch += 0.659 ± 0.004 fm [94]). A re- +markable cancellation among the contributions takes +place, and this represents a common feature for all +the integrated quantities generated by the uTMDs we +are considering. In the right panel, one can see the +comparison between the quark PDF, uS(AS)(ξ) and +the LF-valence PDF, resulting from the one-to-one +relation between the LF-projected BS amplitude and +the valence amplitude of the Fock expansion of the +pion state. +In particular, the LF-valence PDF (see +Refs. [62, 76]), is given by +uLF +val(ξ) = +� ∞ +0 +dγ +(4π)2 +� +|ψ↑↓(γ, z)|2+|ψ↑↑(γ, z)|2� +, (32) +where ξ = (1−z)/2, ψ↑↓(γ, z) is the anti-aligned com- +ponent of the LF-valence amplitude and ψ↑↑(γ, z) the +aligned one (of purely relativistic nature having an +orbital angular momentum equal to 1). These ampli- +tudes are suitable combinations of the LF-projected +scalar functions φi(k; P), Eq. (22). The integral on ξ +of LF-valence PDF gives the probability of the valence +state in the Fock expansion and amounts to +Pval = +� 1 +0 +dξ uLF +val(ξ) = 0.7 . +(33) +The striking feature shown in the left panel is the +shift toward low ξ of the quark PDF, so that for this +quantity the symmetry ξ → 1 − ξ is slightly violated. +B. +Analysing the shift and the gluon content +The PDF calculations based on the BS-amplitude +are able to capture an explicit gluonic effect, to be + +9 +taken distinct from the one responsible for the effec- +tive mass of the constituents. In particular, the dif- +ference between the two symmetric PDFs, i.e. uS(ξ) +and uLF +val(ξ) (recall that has Pval = 0.7), can be traced +back to the non negligible probability of the higher +Fock states (HFS), where a q¯q pair interacts by ex- +changing any number of gluons. Interestingly, the dif- +ference can be effectively described only by a factor, +since it turns out that uLF +val(ξ)/Pval largely overlaps +uS(ξ). Finally, also the small, but relevant, shift of +the quark PDF with respect to uS(ξ) has to be as- +cribed to the presence of HFS, as discussed in what +follows. +To get a qualitative view, we remind that the +pion state can be, in principle, decomposed in Fock- +components, which are schematically written in ladder +approximation as +|π⟩ = |q¯q⟩ + |q¯qg⟩ + |q¯q 2g⟩ + · · · +(34) +Due to the charge symmetry, each Fock-component is +invariant by q ↔ ¯q, and hence the valence state |q¯q⟩ +provides a symmetric contribution to uq(ξ), identified +with uLF +val(ξ). The following terms contain gluons up +to infinity. In our model, the gluon has an effective +mass about twice the quark mass, so that the HFS cu- +mulative effect results in a small shift of the uq(ξ) peak +at ξ < 1/2, as shown in the right panel of Fig. 1. Ac- +tually, a similar effect, related to the increasing mass +of the remnant, can be also recognized in the nucleon, +where one has a valence parton distribution with a +peak around 1/3 due to the presence of the other two +constituent quarks. In the case of the pion, the effect +is small since the valence component |q¯q⟩ has 70% of +probability (as generated by our dynamical calcula- +tion), and hence is largely dominant. +To become more quantitative and illustrate this ef- +fect, we schematically write the quark PDF by using +the Fock expansion of the pion state, Eq. (34), and +inserting LF variables [79], one has +uq(ξ) = +∞ +� +n=2 +� n +� +i +� d2ki⊥ +(2π)2 +� 1 +0 +dξi +� +×δ (ξ − ξ1) δ +� +1 − +n +� +i=1 +ξi +� +δ +� n +� +i=1 +ki⊥ +� +× +��Ψn(ξ1, k1⊥, ξ2, k2⊥, ...) +��2 , +(35) +where ξ1(2) is the longitudinal-momentum fraction +of the quark (antiquark) in each Fock state, com- +posed by a q¯q pair and n − 2 gluons, generated by +the iteration of the one-gluon exchange. +Moreover, +Ψn(ξ1, k1⊥, ξ2, k2⊥, ...) is the probability amplitude of +the corresponding Fock component and fulfills a nor- +malization condition that follows from the one of the +pion state. In the n-th state one has +ξ1 = 1 − ξ2 − +n +� +g=3 +ξg . +(36) +Since ξi > 0 for massive particles, the average value +of ξ1 starts to decreases while the number of gluons +increases, as quantitatively shown in what follows. +Looking at the right panel of Fig. 1, one can realize +that while the valence term, with probability Pval = +0.7, has a peak at ξ1 = ξ2 = 1/2, given the symmetry +of |Ψ2(ξ1, k1⊥, ξ2, k2⊥)|2 all the HFS shift the peak to +ξ1 < 1/2, and decrease the tail, due to the constraint +of the overall normalization. This is reflected in the +evaluation of the first moment (recall ξq ≡ ξ1) +⟨ξq⟩ = Pval ⟨ξq⟩val + +� +n>2 +Pn ⟨ξq⟩n += Pval ⟨ξq⟩val + (1 − Pval) ⟨ξq⟩HF S , +(37) +where Pn is the probability of the n-th Fock state +beyond the valence one. The first term in Eq. (37) +is equal to 0.35, since 1/2 is weighted by Pval, and +the rest is weighted by 0.3. Notice that for each HFS, +normalized to 1, one has +⟨ξq⟩n = 1 − ⟨ξ¯q⟩n − +n +� +i=3 +⟨ξgi⟩n += 1 − ⟨ξ¯q⟩n − (n − 2)⟨ξg⟩n , +(38) +where the gluon bosonic nature leads to the factor +n − 2. +The actual value of the first moment of uq(ξ) is +⟨ξq⟩ = +� 1 +0 +dξ +� ∞ +0 +dγ ξ f q +1 (γ, ξ) = 0.471 , +(39) +that amounts to an average of ⟨ξq⟩HF S equal to 0.40. +We can further analyse ⟨ξq⟩HF S, aiming at extract- +ing a quantitative estimate of the exchanged-gluon +contribution, ⟨ξg⟩. +From the momentum sum rule +Eq. (37), and recalling Eq. (36), we get +⟨ξq⟩HF S = +1 +1 − Pval +� +n>2 +Pn ⟨ξq⟩n += 1 − ⟨ξ¯q⟩HF S − ⟨ξg⟩ , +(40) +where +⟨ξg⟩ = +1 +1 − Pval +� +n≥3 +Pn(n − 2) ⟨ξg⟩n . +(41) + +10 +0 +0.2 +0.4 +0.6 +0.8 +1 + ξ +0 +0.2 +0.4 +0.6 +0.8 +ξ u( ξ) +0 +0.25 +0.5 +0.75 +1 + ξ +0 +0.1 +0.2 +0.3 +0.4 +0.5 + ξ u( ξ) +FIG. 2. +(Color online). Left panel. Pion longitudinal distributions, with different scales (see text for details). Dashed +line: ξ uS(ξ), with an assigned initial scale equal to 360 MeV, and first moment equal to 0.5. Solid line: ξ uq(ξ), with +a deduced scale equal 389 MeV, obtained by using a backward evolution of the first moment ⟨ξq⟩ = 0.471. Dot-dashed +line ξ uq(ξ) backward-evolved from 389 MeV to 360 MeV. Right panel. Comparison with the experimental data at the +scale 5.2 GeV. Solid line: evolved uS(ξ) starting from 360 MeV. Dot-dashed line: evolved uq(ξ), starting from 389 MeV. +Dashed line: DSE calculation from Fig. 5 of Ref. [95]. Dotted line: BLFQ result at 4.0 GeV [96]. Shaded area: LQCD +calculation extracted via Mellin moments from Ref. [97]. Full squares: reanalyzed data by using the ratio between the +fit 3 of Ref. [98], evolved to 5.2 GeV, and the experimental data [52], at each data point (see Ref. [76] for details). +Moreover, since each Fock component fulfills the +charge symmetry, i.e. +q ↔ ¯q, the corresponding +quark and antiquark momentum densities are equal +and hence for the Mellin moments one has ⟨ξk +q ⟩HF S = +⟨ξk +¯q ⟩HF S (this property does not imply the charge +symmetry of the total density, given the presence of +the gluon contribution, cf. Eq. (37)). From Eq. (40), +it follows that the gluon contribution reads +⟨ξg⟩ = 1 − 2 ⟨ξq⟩HF S . +(42) +Then, in our model one has ⟨ξg⟩ = 0.2. We should +note that i) ⟨ξq⟩ > ⟨ξq⟩HF S, as it should be, and ii) +the massive gluons carry 20% of the HFS momentum +fraction and contribute to the total longitudinal frac- +tion by 6% (recalling that PHF S = 0.3). This result +indicates that the exchanged gluons in the pion are not +soft (differently from the ones considered in Ref. [99] +where the subtraction of the effect due to soft gluons +is advocated for getting a symmetric PDF from the +LF projected BS amplitude). +It has to be emphasized that the above analysis, +made transparent by the adopted LF variables, is valid +in any gauge (both covariant gauges or the light-cone +one), and the only difference is the amount of the shift +one gets. +The possibility to regain the full gauge- +invariance by taking into account the additional gluon +exchanges that could affect the interaction between +the knocked-out quark and the spectator one (see, e.g., +the analysis of the gauge-invariance and the hand-bag +contribution in Ref. [14, 100]) will be explored else- +where. +The real test of the longitudinal dof is obviously +the comparison between the PDF and the experimen- +tal data [52]. As it is shown in Ref. [76], after evolving +uS(ξ) from an assigned initial scale of 360 MeV (sug- +gested by the inflection point of the effective running +charge αs(Q2)) to the scale of 5.2 GeV, as given by +the reanalysis in Ref. [101], the result compares very +satisfactorily with the experimental data extracted by +taking into account logarithmic resummation effects +in the hard part of the Drell-Yan cross-section, as per- +formed in Ref. [98]. Moreover, we have achieved a nice +agreement with other dynamical calculations, such +as the Dyson-Schwinger result of Ref. [95], the basis +light-front quantization calculation of Refs. [102, 103], +and also the recent LQCD outcomes of Ref. [97]. In +particular, both the overall shape and, importantly, +the tail for ξ → 1, gives great confidence in our for- +malism, and encourages the further steps we have un- +dertaken in this work. + +11 +0 +2 +4 +6 +8 +10 +γ/m +2 +10 +-4 +10 +-3 +10 +-2 +10 +-1 +10 +0 +10 +1 +D⊥(γ)/D⊥(0) +0 +2 +4 +6 +8 +10 +γ/m +2 +10 +-3 +10 +-2 +10 +-1 +10 +0 +10 +1 +m +2 f +S +1(γ,ξ=0.5) +FIG. 3. +(Color online) Left panel. +Normalized pion transverse distribution function, Eq. (43), vs γ/m2. +The +normalization is given by D⊥(0) = 22.945 GeV−2. +Thick solid line: full calculation. +Dashed line: the same as the +full line, but times (γ/m2)4. Dash-dotted line: the same as the full line, but times (γ/m2)2. Dash-double-dotted line: +exponential form e−γ/(m 0.42)2, with the parameter from Table 1 of Ref. [35], corresponding to a Gaussian Ansatz for +f1(γ, ξ) (see text). Right Panel. Pion unpolarized transverse-momentum distribution f S +1 (γ, ξ), Eq. (10), for ξ = 0.5. +Solid line: full calculation as in Fig. 4. Dashed line: LF constituent quark model [35, 56]. Dash-dotted line: LF wave +function from DSE calculations [45]. Dash-Double-dotted line: NJL model [38]. The adopted quark mass m = 255 MeV. +In Fig. 2, one can observe a further comparison, in- +volving the product ξ u(ξ), that sheds more light on +the link between the shift of the peak and the gluon +dynamics taken explicitly into account in the ladder +kernel of the BSE. In particular, we get a scale of 389 +MeV for uq(ξ), the solid line in the left panel of Fig. 1, +by backward-evolving its first moment, ⟨ξq⟩ = 0.471 +(cf. Eq. (39)), to 0.5, the first moment of uS(ξ), that +has an assigned hadronic scale of 360 MeV, as above +mentioned and thoroughly discussed in Ref. [76]. In +the right panel, the comparison at 360 MeV between +ξ uS(ξ) and the backward-evolved ξ uq(ξ) shows that +the effect of the interaction taken into account in the +ladder BSE is reproduced at large extent by applying +a leading-order DGLAP evolution with an effective +running charge as suggested in Ref. [104] and already +applied to our PDF in Ref. [76]. This is not surpris- +ing once we remind that the dressing of the quark- +gluon vertex, as expressed by the effective charge, is +governed by the same interaction kernel present in +the BSE (i.e. the q¯q amputated T-matrix). The left +panel shows the comparison at 5.2 GeV between the +evolved uS(ξ), starting from the scale of 360 MeV, and +the evolved uq(ξ), starting from the scale of 389 MeV. +Nicely, the difference is even smaller. +C. +Transverse degree of freedom +In the left panel of Fig. 3, it is shown the transverse +distribution defined by +D⊥(γ) = +� 1 +0 +dξ f S +1 (γ, ξ) = +� 1 +0 +dξ f q +1 (γ, ξ) . (43) +It has to be pointed out that the integration on ξ elim- +inates the anti-symmetric term f AS +1 +(γ, ξ), and there- +fore one gets the same transverse distribution also by +using f q +1 (γ, ξ). In order to emphasize the analysis of +the general pattern, we have presented D⊥(γ)/D⊥(0), +so that the widely adopted exponential or power-like +fall-off can be readily compared to our result. +In addition, in the left panel of Fig. 3 one can find: +i) an exponential form D⊥(γ)/D⊥(0) = e−γ/(m 0.42)2, +with the parameter given in Table 1 of Ref. [35], +corresponding to the so-called Gaussian Ans¨atz (re- +call γ = |k⊥|2), amounting to a factorized form for +f S +1 (γ, ξ) ∼ uS(ξ)e−γ/(0.422) very often adopted in phe- +nomenological studies; ii) our full results multiplied +by (γ/m2)2 and iii) our full results multiplied by +(γ/m2)4. +Not surprisingly, a power-like shape pro- +vides a better approximation to the dynamically cal- +culated D⊥(γ), but the proper power is different from + +12 +the one expected by the action of only a one-gluon +exchange, that should govern the ultraviolet behavior +and lead to a (γ/m2)2 (as suggested by a general- +ized counting rule in Ref. [105]). Indeed, the adopted +form-factor featuring the extension of the quark-gluon +interaction vertex (cf. Eq. (7)) generates a different +power-like fall-off, namely (γ/m2)4, as already pointed +out in Refs. [88, 89]. Finally, it is worth noticing that, +unlike the PDF, the two terms in f S +1 (γ, ξ) containing +derivatives of the delta-function do not contribute, as +it is discussed at the end of Appendix C 1. +In the right panel of Fig. 3, it is presented the +quantitative comparison between f S +1 (γ, ξ) at ξ = 0.5 +and some phenomenological outcomes from i) the ap- +proach based on the LF wave function obtained by +using the DSE calculation in Ref. [45]; ii) the LF con- +stituent quark-model of Ref. [35, 56]; iii) the NJL +model with Pauli-Villars regulator as given in Ref. +[38]. For γ/m2 → 0, there are remarkable differences +that, indeed, are present also on the tails. This last +feature impacts the value of ⟨γ/m2⟩ +1 +2 , as shown in Ta- +ble I, where, for the sake of completeness, the value +of uS(ξ = 0.5) and the pion charge radius are also +presented. As can be expected, the larger the average +transverse moment, the smaller the radius of charge. +The current model has the smaller ⟨γ/m2⟩ +1 +2 (of the +order of the infrared scale ΛQCD, effectively incorpo- +rated in the QCD-inspired choice of our parameters) +which in turn leads to a larger charge radius, in agree- +ment with the experimental value. +TABLE +I. +The +average +value +⟨γ/m2⟩ +(with +m += +0.255 MeV), uS(ξ = 0.5) and the pion charge radius are +presented for: i) f S +1 (γ, ξ = 0.5) from the present approach +(NIR+BSE); ii) the outcome from the LF wave function +obtained by using DSE calculation [45] (LFDSE); iii) the +LF constituent quark-model of Ref. [35, 56] (LFCQM) +and iv) the NJL with Pauli-Villars regulator [38]. +(Re- +call that the most recent PDG value of the charge radius +is rP DG +ch += 0.659 ± 0.004 fm [94]). +⟨γ/m2⟩ +1 +2 uS(ξ = 0.5) rch [fm] +NIR+BSE +1.25 +1.60 +0.663 +LFDSE +1.94 +1.36 +0.590 +LFCQM +1.65 +1.37 +0.572 +NJL +2.02 +1.01 +0.557 +In Fig. 4, the uTMD f S +1 (γ, ξ) is shown in full, in +order to appreciate the main features, i.e. i) the peak +at ξ = 0.5 for running γ/m2, ii) the vanishing val- +FIG. 4. +(Color online) Pion unpolarized transverse- +momentum distribution f S +1 (γ, ξ), Eq. (10), at the initial +scale. The normalization is +� 1 +0 dξ +� ∞ +0 +dγ f S +1 (γ, ξ) = 1. +ues at the end-points and iii) the order of magnitude +fall-off already for γ/m2 > 2. Comparing to other ap- +proaches, one can notice the sharp difference with the +results from the LF constituent model in Ref. [56] and +the LF holographic framework, like in Refs. [40, 57] +where a double-humped structure is found due to the +ξ-dependence in the holographic wave functions. Also +the value at ξ = 0.5 and small γ/m2 is substantially +lower than ours (almost an order of magnitude less). +Differently, the shape of our f S +1 (γ, ξ) is more similar, +i.e. without any double-humped structure, to the one +obtained in Ref. [45], where the pion LF-wave function +is determined from a beyond rainbow-ladder Dyson- +Schwinger equations (DSE) in Euclidean space, by ex- +ploiting the γ-dependent moments in ξ and a suitable +parametrization of the BS-amplitude. +V. +THE SUBLEADING-TWIST UTMDS +In this Section we present the numerical results for +(T-even) uTMDs beyond the leading-twist. The de- +tailed expressions can be found in the Appendix E, +but it is useful to recall that the decomposition in +symmetric and antisymmetric combinations adopted +for f1(γ, ξ) remains still valid, as well as the relations + +13 +0 +0.25 +0.5 +0.75 +1 + ξ +-2 +0 +2 +4 +6 +8 +e( ξ) +0 +0.25 +0.5 +0.75 +1 + ξ +0 +1 +2 +3 +ξe +q( ξ) +FIG. 5. +(Color online) Left panel. Pion unpolarized collinear PDFs: i) eq(ξ) (solid line), Eq. (46), ii) eS(ξ) (dashed +line) and eAS(ξ) (dotted line), Eqs. (45). It is also shown eq +EoM(ξ) (dash-dotted line), Eq. (47). Right panel. Quark +unpolarized collinear PDFs: ξ eq(ξ). Solid line: full calculation as in the left panel. Dashed line: m/M uq(ξ), with uq(ξ) +shown in the right panel of Fig. 1. Double-dot-dashed line: ξ eq +EoM(ξ), Eq. (47). +with the quark and anti-quark contributions. +As introduction to the outcomes of our dynamical +approach, it is worth anticipating that the comparison +between full calculations and naive estimates one can +infer from Eq. (17) by using a valence approximation +of the leading-twist f1(γ, ξ), highlights the inspiring +statement one can read in Ref. [77]: the higher-twist +distributions are naturally related to multiparton dis- +tributions. The role of the exchanged gluons becomes +definitely clear through a remarkable shift of the peak +in all the sub-leading uTMD we have analyzed, as +already discussed in the previous Section, as well as +through the sharp difference with the naive estimates, +which exclude the effect of the one-gluon exchange. +A. +Twist-3 uTMD: e(γ, ξ) +In the frame where P⊥ = 0 and hence P + = M, by +using Eq. (27), (B19), (B20), (B21) and (B22), with +i = 1 and the functions b1 +n;ℓj given in Table VII, one +gets the twist-3 uTMDs eS(AS)(γ, ξ), decomposed as +follows +eS(AS)(γ, ξ) = E0(γ, ξ; S(AS)) + Ed(γ, ξ; S(AS)) ++E2d(γ, ξ; S(AS)) + E3d(γ, ξ; S(AS)) , +(44) +where the functions in the rhs are given in Ap- +pendix E. +1. +Longitudinal degree of freedom +In the left panel of Fig. 5, the following collinear +PDFs are shown +e(S,AS)(ξ) = +� ∞ +0 +dγ e(S,AS)(γ, ξ) . +(45) +and +eq(ξ) = eS(ξ) + eAS(ξ) . +(46) +Moreover, in the spirit of Ref. [35], we also present +the collinear PDF, eq +EoM(ξ), obtained by integrating +the first line in Eq. (17), but disregarding the gluon +contribution, viz +eq +EoM(ξ) ∼ m +Mξ +� ∞ +0 +dγ f q +1;EoM(γ, ξ) +∼ m +Mξ +uLF +val(ξ) +Pval +, +(47) +where uLF +val(ξ)/Pval, normalized to 1 (cf. +Eq. (33)), +approximates the integral of f q +1;EoM(γ, ξ). The large +difference between our eq(ξ) and (m/Mξ)uLF +val(ξ)/Pval +indicates the sizable role of the gluon contribution +from the HFS generated by our dynamical model. In +addition, one should point out that the strength of +eq(ξ) is spread out on the whole range of ξ, and not +concentrated at the end-point ξ = 0 as QCD investiga- +tions indicate. The latter feature leads to the singular + +14 +0 +2 +4 +6 +8 +10 +γ/m +2 +10 +-4 +10 +-3 +10 +-2 +10 +-1 +10 +0 +E⊥(γ)/E⊥(0) +0 +2 +4 +6 +8 +10 +γ/m +2 +10 +-3 +10 +-2 +10 +-1 +10 +0 +10 +1 +m +2 e +S(γ,ξ=0.5) +FIG. 6. (Color online). Left panel. Normalized transverse distribution function E⊥(γ)/E⊥(0) (cf. Eq. (49)). Dotted +line: full calculation. +Solid line: D⊥(γ)/D⊥(0) for the sake of comparison. +Dash-double-dotted line: the same as +in the left panel of Fig. 3. +Right panel. +Pion unpolarized transverse-momentum distribution eS(γ, ξ), Eq. (44), for +ξ = 0.5. +Solid line: full calculation. +Dashed line: LF constituent quark model of Ref. [35, 56], but multiplied by +m/(M 0.5) (cf. Eq. (47)). Dash-dotted line: the same as the dashed line but with the LF wave function from DSE +calculations [45]. Dash-Double-dotted line: the same as the dashed line but with the NJL model [38]. The adopted +quark mass m = 255 MeV. +contribution given in Eq. (15) (see, e.g., Ref. [85], for +a detailed discussion, but notice that the focus is on +the nucleon). +In the right panel of Fig. 5, the comparison be- +tween ξ e(ξ) +and the other two approximations: +i) (m/M)f q +1 (ξ) and ii) (m/M)uLF +val(ξ) (cf. Eq. (47)) +is carried out. The relevance of such a comparison is +given by the possibility of more directly assessing the +gluon role, since the factor ξ eliminates the singular +term present in the QCD analysis of e(ξ), and one +remains with the mass contribution (m/M)f q +1 (ξ) and +the term from the quark-gluon-antiquark correlator. +Still +within +the +QCD +framework +(see, +e.g., +Ref. [85]), the moments ⟨ξn⟩eq, for n ≤ 2, read as +follow +� +dξ eq(ξ) = +σπ +mcur +� +dξ ξ eq(ξ) = mcur +M +� +dξ ξ2 eq(ξ) = mcur +M +� +dξ ξ f q +1 (ξ) , +(48) +and, for n > 2, they receive contributions not only +from the (n−1)-th moment of f q +1 (γ, ξ) , but also from +the n-th moment of the twist-3 contribution pertain- +ing to the quark-gluon-antiquark correlator. +Given +TABLE II. The moments ⟨ξn⟩eq +of the quark twist- +3 eq(γ, ξ) for n +< +4, and the ratio R(n, eq, f q +1 ) += +⟨ξn⟩eq/⟨ξn−1⟩fq +1 (it is assumed ⟨ξ−1⟩fq +1 = ⟨ξ0⟩fq +1 = 1, and +the values of ⟨ξ1⟩fq +1 = 0.471 and ⟨ξ2⟩fq +1 = 0.266 have been +numerically evaluated). +n +0 +1 +2 +3 +⟨ξn⟩eq +2.190 +0.814 +0.445 +0.292 +R(n, eq, f q +1 ) +2.190 +0.814 +0.943 +1.10 +the highly non trivial dynamical content of the eq(ξ) +moments, it is interesting to show the results obtained +with our dynamical model. +In Table II, both the moments up to n = 3 and the +ratio R(n, eq, f q +1 ) = ⟨ξn⟩eq/⟨ξn−1⟩f q +1 are presented. In +particular, as to the first two moments, to get rid of +the dependence upon mcur it is helpful to compare +the result obtained by multiplying the zero-th and +the first moment, (cf. Eq. (48)), with final outcome +σπ/M. The estimate of σπ at the leading order of the +chiral expansion leads to σπ/M = 1/2, as satisfacto- +rily confirmed by the lattice calculations in Ref. [84], +where σlat +π += 78.2±4.2 MeV, for M = 149.5±1.3 MeV + +15 +FIG. 7. +(Color online) Pion unpolarized transverse- +momentum distribution eS(γ, ξ), Eq. (45), at the initial +scale. +and mcur ∼ 4.9 MeV. Eliminating the current quark +mass, that is outside our approach, through the above +product, we get σπ/M = 1.78, instead of ∼ 0.5. +Such a conspicuous difference is surely influenced by +the different distribution of the eq(ξ) strength, as al- +ready mentioned, and points to a needed enrichment +of the gluon dynamics in our approach. However, it +is worth mentioning that for a simple non relativis- +tic constituent quark model one has σNR +π += 2m, so +that σNR +π +/M = 3.64 (with m = 255 MeV the con- +stituent mass), almost twice the result obtained in the +BS framework. +In QCD, the ratios R(1, eq, f q +1 ) and R(2, eq, f q +1 ) are +equal and amount to mcur/M (see Eq. (48)), while +R(3, eq, f q +1 ) = mcur/M + ∆3 +g, where ∆3 +g contains the +contribution from the twist-3 gluonic contribution. +In our calculation, the ratios for n = 1, 2 are al- +most equal, but different from the naive expectation +m/M = 1.82 with the adopted m = 255 MeV. The +difference with the third ratio indicates the onset of +the contribution from the twist-3 gluonic term. +2. +Transverse degree of freedom +The transverse dof can be analyzed globally by in- +troducing the following transverse distribution func- +tion, as already accomplished with the leading-twist +uTMD, viz. +E⊥(γ) = +� 1 +0 +dξ eS(γ, ξ) = +� 1 +0 +dξ eq(γ, ξ) . (49) +In the left panel of Fig. 6, it is presented our calcu- +lation and the ratio D⊥(γ)/D⊥(0) to show the simi- +lar fall-off, as generated from gluon dynamics and the +form-factor featuring the quark-gluon vertex. +A more close view of the decreasing as a function of +γ/m2 is provided by the right panel of Fig. 6, where +it is shown the comparison between our calculation of +e(γ, ξ = 0.5) and the outcomes obtained by using Eq. +(47) with i) the LF wave function from the constituent +quark model of Ref. [35, 56], ii) the LF wave function +from DSE calculations [45] and iii) the PDF from the +NJL model [38]. The differences again point to the +role of the interaction in the various approaches, and +highlight the relevance of an experimental investiga- +tion of the transverse dof. +In Fig. 7, the full dependence of eS(γ, ξ) is pre- +sented, displaying a double-hump shape that for larger +γ/m2 becomes smoother and smoother. +B. +Twist-3 uTMD: f ⊥(γ, ξ) +In an analogous way, for i = 2 and using Table VIII, +one gets the twist-3 f ⊥S(AS)(γ, ξ), with the following +decomposition +f ⊥S(AS)(γ, ξ) = P0(γ, ξ; S(AS)) + Pd(γ, ξ; S(AS)) ++P2d(γ, ξ; S(AS)) +(50) +where the above functions are given in Appendix E. +Notice that in this case P2d(γ, ξ; S(AS)) = 0. +1. +Longitudinal degree of freedom +In the left panel of Fig. 8, the following collinear +PDFs are shown +f ⊥S(AS)(ξ) = +� ∞ +0 +dγ f ⊥S(AS)(γ, ξ) , +(51) +and the corresponding quark combination. As a refer- +ence, it is also presented f ⊥q +EoM(ξ), obtained from the + +16 +0 +0.25 +0.5 +0.75 +1 + ξ +-2 +-1 +0 +1 +2 +3 +4 +5 +f +⊥( ξ) +0 +0.25 +0.5 +0.75 +1 + ξ +0 +1 +2 +ξ f +⊥( ξ) +FIG. 8. +(Color online) Left panel. The same as in Fig. 5, but for f ⊥q(ξ), f ⊥S(ξ) and f ⊥AS(ξ), Eq. (51), and f ⊥q +EoM(ξ) +as given in Eq. (52). Right panel. Quark unpolarized collinear PDFs ξ f q⊥(ξ). Solid line: full calculation as in left panel. +Dashed line: ξ f q⊥(ξ) obtained by using the second line in Eq. (17) and our f q +1 (ξ). Double-dot-dashed line: the same as +the dashed line but using the valence approximation of the PDF, uLF +val(ξ), with norm equal to 1. +0 +2 +4 +6 +8 +10 +γ/m +2 +10 +-4 +10 +-3 +10 +-2 +10 +-1 +10 +0 +P⊥(γ)/P⊥(0) +0 +2 +4 +6 +8 +10 +γ/m +2 +10 +-4 +10 +-3 +10 +-2 +10 +-1 +10 +0 +10 +1 +m +2 f +S⊥(γ,ξ=0.5) +FIG. 9. (Color online). Left panel. Normalized transverse distribution function P⊥(γ)/P⊥(0) (cf. Eq. (53)). Dotted +line: full calculation. Solid line: D⊥(γ)/D⊥(0) for the sake of comparison. Dash-double-dotted line: the same as in the +left panel of Fig. 3. Right panel. Pion unpolarized transverse-momentum distribution f S⊥(γ, ξ), Eq. (50), for ξ = 0.5. +Solid line: full calculation. Dashed line: by using f1(γ, ξ = 0.5) in Fig. 3 from the LF constituent quark model of +Ref. [35, 56] (cf. the second line in Eq. (17), without the gluonic term). Dash-dotted line: the LF wave function from +DSE calculations [45]. Dash-Double-dotted line: the NJL model [38]. The adopted quark mass m = 255 MeV. +second line of Eq. (17), without the gluon term, as +follows +f ⊥q +EoM(ξ) ∼ 1 +ξ +� ∞ +0 +dγf ⊥q +EoM(γ, ξ) ∼ uLF +val(ξ) +ξ +. (52) +For the sake of completeness, in the right panel of +Fig. 8, the product ξ f q⊥(ξ) is compared to f q +1 (ξ) and +uLF +val(ξ) that represents the approximation to f ⊥q +EoM(ξ) +as given in Eq. (52). Also for f q⊥(ξ), the full calcu- + +17 +lation substantially differs from approximated evalu- +ations, prompting further investigation of the gluon +contributions. +2. +Transverse degree of freedom +Also for f ⊥(γ, ξ), we introduce the transverse dis- +tribution function, viz. +P⊥(γ) = +� 1 +0 +dξ f S⊥(γ, ξ) = +� 1 +0 +dξ f q⊥(γ, ξ) .(53) +In the left panel of Fig. 9, a comparison, built with +the same spirit as in the left panel of Fig. 6, is shown +for the ratio P⊥(γ)/P⊥(0). +A more detailed view of the fall-off can be gained +from the right panel of Fig. 9, where f S⊥(γ, ξ = 0.5) +is compared with the results obtained by using i) the +LF constituent quark model of Ref. [35, 56] (cf. the +second line in Eq. (17), without the gluonic term). ii) +the LF wave function from DSE calculations [45] and +iii) the NJL model [38]. +FIG. 10. +(Color online) Pion unpolarized transverse- +momentum distribution f S⊥(γ, ξ), Eq. (50). +Finally in Fig. 10, the full dependence of f S⊥(γ, ξ) +is shown. Also in this uTMD, the double-hump shape +decreases when γ/m2 increases. +To summarize, a coherent view of the tail in γ is +plainly provided by Figs. 3, 6 and 9. Namely, the +interaction taken into account in the ladder kernel to- +gether with the extended structure of the quark-gluon +vertex governs the fall-off of both the leading and +subleading-twist uTMDS. Therefore, the quantitative +estimates obtained through our dynamical model, in +Minkowski space, is shown to be in a favorable po- +sition to provide insights into the interplay between +transverse dof and the role of gluons. +VI. +CONCLUSIONS +The twist-2 (leading) and twist-3 (subleading) un- +polarized (T-even) transverse-momentum dependent +parton distribution functions have been calculated for +the pion within an approach based on the solution +of the Bethe-Salpeter equation in Minkowski space, +namely, within a genuinely relativistic quantum-field +theory framework. We achieved this goal by exploit- +ing the Nakanishi integral representation of the BS- +amplitude in order to get actual solution of the homo- +geneous BSE, in ladder approximation, through a sys- +tem of integral equations that determine the Nakan- +ishi weight functions relevant for the problem under +scrutiny [62, 88, 89]. After obtaining the pion elec- +tromagnetic form factor [75], and the pion PDF [76], +we extended the yield of our approach by exploring +the dependence of the parton distributions upon the +transverse momentum. This additional step has its- +own importance in view of the planned experimental +efforts to achieve a fully three-dimensional investiga- +tion of hadrons (mainly of the nucleon and, more chal- +lenging, the pion). +The relevant message one gets from our calculations +is given by the essential role of the gluon exchange, +that cannot be captured by purely phenomenological +model. +The joint use of the Fock expansion of the +pion state, allows us to shed light on the gluonic con- +tent of the quark PDF obtained through the BS ampli- +tude, even determining a quantitative estimate, ∼ 6% +of the average longitudinal momentum fraction ⟨ξq⟩. +Moreover, the latter analysis explains also the source +of the small, but theoretically relevant, shift between +the uq(ξ) and the PDF that fulfills the charge symme- +try (an issue already investigated within the Dyson- +Schwinger approach, e.g., in Ref. [99], where a dif- +ferent interpretation was proposed). As to the trans- +verse degree of freedom, a power-like fall-off of the +transverse distributions, obtained by integrating on ξ +the uTMDs, is supported by the one-gluon exchange +interaction that governs the ultraviolet region, accord- +ing to our calculations. This outcome could suggest + +18 +to reconsider an exponential or Gaussian Ansatzes for +describing the high-momentum content (γ >> m2) of +the uTMDS. +Summarizing, our approach can be placed among +those in which the dynamics can be studied in +Minkowski space and in some detail. Moreover, the +additive construction of the interaction kernel allows +one to address step-by-step recognized effects, achiev- +ing an implementation of the dynamics in a controlled +way. +ACKNOWLEDGMENTS +E. Y. gratefully thanks INFN Sezione di Roma +for providing the computer resources to perform all +the calculations shown in this work. +W. d. +P. +acknowledges the partial support of CNPQ under +Grants No. 438562/2018-6 and No. 313236/2018-6, +and the partial support of CAPES under Grant No. +88881.309870/2018-01. T. F. thanks the financial sup- +port from the Brazilian Institutions: CNPq (Grant +No. 308486/2015-3), CAPES (Finance Code 001) and +FAPESP (Grants No. 2017/05660-0 and 2019/07767- +1). +E. Y. acknowledges the support of FAPESP +Grants No. +2016/25143-7 and No. +2018/21758-2. +This work is a part of the project Instituto Nacional +de Ciˆencia e Tecnologia - F´ısica Nuclear e Aplica¸c˜oes +Proc. No. 464898/2014-5. +Appendix A: Traces +In this Appendix the traces in Eqs. (10), (18) and (19), are explicitly evaluated, presenting the expressions of +the functions ai +ℓ,j(k−, γ, z; S(AS)) and bi +n;ℓ,j(γ, z; S(AS)). For the sake of convenience, let us rewrite the generic +trace entering Eq. (23) +TrS(AS) +i +(γ, ξ) = − i +2 +� +Tr +� +S−1(k − P +2 )¯Φ(k, P) Oi Φ(k, P) +� ++ ηS(AS) +i +Tr +� +S−1(k + P +2 )Φ(k, P) Oi ¯Φ(k, P) +�� += +� +ℓj +ai +ℓj(k−, γ, z; S(AS)) φℓ(k; P) φj(k; P) , +(A1) +with Oi and ηS(AS) +i +given by +O0 = γ+ , +ηS(AS) +0 += ∓1 , +O1 = 1 , +ηS(AS) +1 += ±1 , +O2 = +M +|k⊥|2 k⊥ · γ⊥ , +ηS(AS) +2 += ±1 . +(A2) +To proceed one has to insert the expression of the BS-amplitude, Eq. (20), and the definitions, Eqs. (7) and +(21), in Eq. (A1). Then one gets the results shown in Tables III, IV and V, for ai +ℓj(k−, γ, z; S(AS)). It is +also useful to organize the functions ai +ℓj in powers of k− for preparing the integration on such a variable (cfr +Appendix B), i.e. +ai +ℓj(k−, γ, z; S(AS)) = 2M +� +bi +0;ℓj(γ, z; S(AS)) + bi +1;ℓj(γ, z; S(AS)) k− +2M + bi +2;ℓj(γ, z; S(AS)) +� +k− +2M +�2 ++bi +3;ℓj(γ, z; S(AS)) +� +k− +2M +�3� +. +(A3) +In Tables VI, VII and VIII, one can find the expressions for bi +n;ℓj(γ, z; S(AS)) + +19 +TABLE III. Non vanishing coefficients a0 +ℓj(k−, γ, z; S(AS)). +ij +a0 +ℓj(S) +a0 +ℓj(AS) +11 +2M +2Mz +12 +−8m +13 +−2 m z − 4 m +M k− +14 +− 8 +M γ − M z2 − 2 z k− +−Mz − 2k− +22 +2M +−4k− +23 +M z + 2 k− +−8 γ +M − 2zk− − +4 +M (k−)2 +24 +2mz + 4 m +M k− +33 +M +2 +� +4 +γ +M2 + z2 +4 +� ++ z +2k− + +1 +2M (k−)2 +− +� +4 +γ +M2 + z2 +4 +� +k− − +z +M (k−)2 − +1 +M2 (k−)3 +34 +2m +� +4 +γ +M2 + z2 +4 +� ++ 2 m +M zk− + 2 m +M2 (k−)2 +44 +M +2 +� +4 +γ +M2 + z2 +4 +� ++ z +2k− + +1 +2M (k−)2 +M +2 z +� +4 +γ +M2 + z2 +4 +� ++ z2 +2 k− + +z +2M (k−)2 +TABLE IV. Non vanishing coefficients a1 +ℓj(k−, γ, z; S(AS)). +ij +a1 +ℓj(S) +a1 +ℓj(AS) +11 +−4m +12 +4M +2Mz − 4k− +13 +−2M +� +4 +γ +M2 + z2 +4 +� +− 2zk− − +2 +M (k−)2 +22 +−4m +24 +−2M +� +4 +γ +M2 + z2 +4 +� +− 2zk− − +2 +M (k−)2 +33 +m +� +4 +γ +M2 + z2 +4 +� ++ z m +M k− + +m +M2 (k−)2 +34 z M +2 +� +4 +γ +M2 + z2 +4 +� +− +� +4 +γ +M2 − z2 +4 +� +k− − +z +2M (k−)2 − +1 +M2 (k−)3 +M +� +4 +γ +M2 + z2 +4 +� ++ zk− + +1 +M (k−)2 +44 +m +� +4 +γ +M2 + z2 +4 +� ++ z m +M k− + +m +M2 (k−)2 +TABLE V. Non vanishing coefficients a2 +ℓj(k−, γ, z; S(AS)). +ij +a2 +ℓj(S) +a2 +ℓj(AS) +11 +−4M +13 +8m +14 +4M +2zM −4k− +22 +4M +23 +−2Mz + 4k− +−4 M +24 +−8m +33 +M +� +4 +γ +M2 + z2 +4 +� ++ zk− + +1 +M (k−)2 +44 −M +� +4 +γ +M2 + z2 +4 +� +− zk− − +1 +M (k−)2 + +20 +TABLE VI. Non vanishing coefficients b0 +n;ℓj(γ, z; S(AS)). +ij +b0 +0;ℓj(S) +b0 +1;ℓj(S) b0 +2;ℓj(S) +b0 +0;ℓj(AS) +b0 +1;ℓj(AS) +b0 +2;ℓj(AS) b0 +3;ℓj(AS) +11 +1 +0 +0 +z +0 +0 +0 +12 +−4m/M +0 +0 +0 +0 +0 +0 +13 +−zm/M +−4m/M +0 +0 +0 +0 +0 +14 −4γ/M 2 − z2/2 +−2 z +0 +−z/2 +−2 +0 +0 +22 +1 +0 +0 +0 +−4 +0 +0 +23 +z/2 +2 +0 +−4γ/M 2 +−2z +−8 +0 +24 +0 +0 +0 +zm/M +4m/M +0 +0 +33 +γ/M 2 + z2/16 +z/2 +1 +0 +− +� +4γ/M 2 + z2/4 +� +−2z +−4 +34 +0 +0 +0 +(m/M) +� +4γ/M 2 + z2/4 +� +2zm/M +4m/M +0 +44 +γ/M 2 + z2/16 +z/2 +1 +(z/4) +� +4γ/M 2 + z2/4 +� +z2/2 +z +0 +TABLE VII. Non vanishing coefficients b1 +n;ℓj(γ, z; S(AS)). +ij +b1 +0;ℓj(S) +b1 +1;ℓj(S) +b1 +2;ℓj(S) b1 +3;ℓj(S) +b1 +0;ℓj(AS) +b1 +1;ℓj(AS) b1 +2;ℓj(AS) +11 +−2m/M +0 +0 +0 +0 +0 +0 +12 +2 +0 +0 +0 +z +−4 +0 +13 +0 +0 +0 +0 +−4γ/M 2 − z2/4 +−2z +−4 +22 +−2m/M +0 +0 +0 +0 +0 +0 +24 +− +� +4γ/M 2 + z2/4 +� +−2z +−4 +0 +0 +0 +0 +33 (m/2M) +� +4γ/M 2 + z2/4 +� +zm/M +2m/M +0 +0 +0 +0 +34 +(z/4) +� +4γ/M 2 + z2/4 +� +− +� +4γ/M 2 − z2/4 +� +−z +−4 +2γ/M 2 + z2/8 +z +2 +44 (m/2M) +� +4γ/M 2 + z2/4 +� +zm/M +2m/M +0 +0 +0 +0 +TABLE VIII. Non vanishing coefficients b2 +n;ℓj(γ, z; S(AS)). +ij +b2 +0;ℓj(S) +b2 +1;ℓj(S) b2 +2;ℓj(S) b2 +0;ℓj(AS) b2 +1;ℓj(AS) +11 +−2 +0 +0 +0 +0 +13 +0 +0 +0 +4(m/M) +0 +14 +2 +0 +0 +z +−4 +22 +2 +0 +0 +0 +0 +23 +−z +4 +0 +−2 +0 +24 +−4m/M +0 +0 +0 +0 +33 +(1/2) +� +4γ/M 2 + z2/4 +� +z +2 +0 +0 +44 −(1/2) +� +4γ/M 2 + z2/4 +� +−z +−2 +0 +0 + +21 +Appendix B: The light-front projection +The Appendix is devoted to the integration over the variable k− in Eq. (26), that for the sake of clarity we +rewrite +F i +ℓj(γ, z; S(AS)) = +� ∞ +−∞ +dk− +2π +ai +ℓj(k−, γ, z; S(AS)) φℓ(k, P) φj(k, P) += 2M +� ∞ +−∞ +dk− +2π +φℓ(k, P) φj(k, P) × +� +bi +0;ℓj(γ, z; S(AS)) + bi +1;ℓj(γ, z; S(AS)) k− +2M + bi +2;ℓj(γ, z; S(AS)) +� +k− +2M +�2 ++bi +3;ℓj(γ, z; S(AS)) +� +k− +2M +�3� +, +(B1) +where the quantities ai +ℓj(k−, γ, z; S(AS)) and bi +n;ℓj(γ, z; S(AS)) are given in Appendix A. +The first step (see also Refs. [62, 75, 76]) is to introduce the NIR of φℓ(k, P), Eq. (22), and then apply the +Feynman parametrization as follows +φℓ(k, P)φj(k, P) = 30 +� 1 +0 +dv +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ v2(1 − v)2 gℓ(γ′, z′) gj(γ′′, z′′) +� +k−α − β + iϵ +�6 +, +(B2) +where +α = M +2 +� +λ(v) − z +� +, +β(zλ(v)) = γ + κ2 + M 2 +4 zλ(v) + vγ′ + (1 − v)γ′′ , +λ(v) = vz′ + (1 − v)z′′ . +(B3) +Hence, the following general expression, one can straightforwardly deduce from well-known result in Ref. [106] +(corresponding to the case m = 0) is useful for performing the relevant integrals +� ∞ +−∞ +dk− +2π +(k−)m +� +α k− − β + iϵ +�n = i (n − m − 2)! +(n − 1)! +(−1)m+1 +� +−β + iϵ +�n−m−1 δ(m)(α) +(B4) +where δ(m)(α) = ∂mδ(α)/∂αm. Finally, combining the results in Eqs. (B2) and (B4), one gets +F i +ℓj(γ, z; S(AS)) = −24i +� 1 +0 +dv v2(1 − v)2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Gℓj(γ′, z′; γ′′, z′′; κ2) +× +� +bi +0;ℓj(γ, z; S(AS)) δ(˜α) +[−β(zλ(v)) + iϵ]5 +− +1 +4M 2 +bi +1;ℓj(γ, z; S(AS)) δ′(˜α) +[−β(zλ(v)) + iϵ]4 ++ +1 +12M 4 +bi +2;ℓj(γ, z; S(AS)) δ +′′(˜α) +[−β(zλ(v)) + iϵ]3 +− +1 +24M 6 +bi +3;ℓj(γ, z; S(AS)) δ +′′′(˜α) +[−β(zλ(v)) + iϵ]2 +� +, +(B5) +where +Gℓj(γ′, z′; γ′′, z′′; κ2) = gℓ(γ′, z′; κ2) gj(γ′′, z′′; κ2) +˜α = 2 +M α = λ(v) − z , +(B6) + +22 +and the derivatives of the delta function is with respect to ˜α. Recalling that ∂ ˜α/∂z = −1, one can also write +F i +ℓj(γ, z; S(AS)) = −24i +� 1 +0 +dv v2(1 − v)2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Gℓj(γ′, z′; γ′′, z′′; κ2) +× +� +bi +0;ℓj(γ, z; S(AS)) δ(λ(v) − z) +[−β(zλ(v)) + iϵ]5 ++ +1 +4M 2 +bi +1;ℓj(γ, z; S(AS)) +[−β(zλ(v)) + iϵ]4 +∂ +∂z δ(λ(v) − z) ++ +1 +12M 4 +bi +2;ℓj(γ, z; S(AS)) +[−β(zλ(v)) + iϵ]3 +∂2 +∂z2 δ(λ(v) − z) + +1 +24M 4 +bi +3;ℓj(γ, z; S(AS)) +[−β(zλ(v)) + iϵ]2 +∂3 +∂z3 δ(λ(v) − z) +� +, +(B7) +Finally, by using +f(z) ∂m +∂zm δ(λ(v) − z) = +m +� +k=0 +cmk +∂k +∂zk +� +f (m−k)(z) δ(λ(v) − z) +� +, +(B8) +where the coefficient cmk can be obtained by repeatedly applying the Leibniz rule for the product of functions +and f (m−k) indicates the (m − k)-th derivative (with f (0)(z) ≡ f(z)), one recasts Eq. (B7) in a form more +suitable for the further elaboration. In practice, one trades derivatives on the delta functions with derivatives +on the functions bi +n;ij(γ, z; S(AS)), getting +F i +ℓj(γ, z; S(AS)) = −24i +� 1 +0 +dv v2(1 − v)2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Gℓj(γ′, z′; γ′′, z′′; κ2) +× +� +δ(λ(v) − z) +� +bi +0;ℓj(γ, z; S(AS)) +[−β(zλ(v)) + iϵ]5 − +1 +4M 2 +∂ +∂z +� bi +1;ℓj(γ, z; S(AS)) +[−β(zλ(v)) + iϵ]4 +� ++ +1 +12M 4 +∂2 +∂z2 +� bi +2;ℓj(γ, z; S(AS)) +[−β(zλ(v)) + iϵ]3 +� +− +1 +24M 6 +∂3 +∂z3 +� bi +3;ℓj(γ, z; S(AS)) +[−β(zλ(v)) + iϵ]2 +�� ++ +1 +4M 2 +∂ +∂z +� +bi +1;ℓj(γ, z; S(AS)) δ(λ(v) − z) +[−β(zλ(v)) + iϵ]4 +− +2 +3M 2 +∂ +∂z +� bi +2;ℓj(γ, z; S(AS)) +[−β(zλ(v)) + iϵ]3 +� +δ(λ(v) − z) ++ +1 +2M 4 +∂2 +∂z2 +� bi +3;ℓj(γ, z; S(AS) +[−β(zλ(v)) + iϵ]2 +� +δ(λ(v) − z) +� ++ +1 +12M 4 +∂2 +∂z2 +� +bi +2;ℓj(γ, z; S(AS)) δ(λ(v) − z) +[−β(zλ(v)) + iϵ]3 +− +3 +2M 2 +∂ +∂z +� bi +3;ℓj(γ, z; S(AS)) +[−β(zλ(v)) + iϵ]2 +� +δ(λ(v) − z) +� ++ +1 +24M 6 +∂3 +∂z3 +� +bi +3;ℓj(γ, z; S(AS)) δ(λ(v) − z) +[−β(zλ(v)) + iϵ]2 +�� +. +(B9) +Hence, by taking into account the expressions of bi +2;ℓj(γ, z; S(AS)) and bi +3;ℓj(γ, z; S(AS)), given in Tables VI, +VII and VIII, one can drop some derivatives. In particular the second derivative of bi +2;ℓj(γ, z; S(AS)) and all +the derivatives of bi +3;ℓj(γ, z; S(AS)), obtaining +F i +ℓj(γ, z; S(AS)) = −24i +� +Fi +0ℓj(γ, z; S(AS)) + Fi +1;ℓj(γ, z; S(AS)) + Fi +2;ℓj(γ, z; S(AS)) + Fi +3;ℓj(γ, z; S(AS)) +� +(B10) + +23 +where +Fi +0,ℓj(γ, z; S(AS)) = +� 1 +0 +dv v2(1 − v)2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Gℓj(γ′, z′; γ′′, z′′; κ2) +× +δ(λ(v) − z) +[−β(z2) + iϵ]5 +� +bi +0;ℓj(γ, z; S(AS)) + 1 +4 +� +β(z2) +M 2 +∂ +∂z bi +1;ℓj(γ, z; S(AS)) − z bi +1;ℓj(γ, z; S(AS)) +� ++ 1 +16 +� +z2 bi +2;ℓj(γ, z; S(AS)) − 2z β(z2) +M 2 +∂ +∂z bi +2;ℓj(γ, z; S(AS) +� +−z3 +64 bi +3;ℓj(S(AS)) +� +, +(B11) +Fi1 +1;ℓj(γ, z; S(AS)) = +1 +8M 2 +∂ +∂z +�� 1 +0 +dv v2(1 − v)2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Gℓj(γ′, z′; γ′′, z′′; κ2) +× +δ(λ(v) − z) +[−β(z2) + iϵ]4 +� +2bi +1;ℓj(γ, z; S(AS)) − z bi +2;ℓj(γ, z; S(AS)) + 4 +3 +β(z2) +M 2 +∂ +∂z bi +2;ℓj(γ, z; S(AS)) ++3 +8z2 bi +3;ℓj(S(AS)) +�� +, +(B12) +Fi +2;ℓj(γ, z; S(AS)) = +1 +12M 4 +∂2 +∂z2 +�� 1 +0 +dv v2(1 − v)2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Gℓj(γ′, z′; γ′′, z′′; κ2) +× +δ(λ(v) − z) +[−β(z2) + iϵ]3 +� +bi +2;ℓj(γ, z; S(AS)) − 3 +4z bi +3;ℓj(S(AS)) +�� +, +(B13) +and +Fi +3;ℓj(γ, z; S(AS)) = +bi +3;ℓj(S(AS)) +24M 6 +∂3 +∂z3 +�� 1 +0 +dv v2(1 − v)2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Gℓj(γ′, z′; γ′′, z′′; κ2) +× +δ(λ(v) − z) +[−β(z2) + iϵ]2 +� +. +(B14) +Collecting the above results, Eq. (23) becomes +T S(AS) +i +(γ, ξ) = iNc +8 +1 +(2π)2 × +� +ℓj +� 1 +−1 +dz δ(z − (1 − 2ξ)) F i +ℓj(γ, z; S(AS)) = += 3Nc +(2π)2 +� +ℓj +� +Fi +0;ℓj(γ, z; S(AS)) + Fi +1;ℓj(γ, z; S(AS)) + Fi +2;ℓj(γ, z; S(AS)) + Fi +3;ℓj(γ, z; S(AS)) +� +. (B15) +It is also useful for getting more explicit expressions to perform the integral on v in the Eqs. (B11), (B12) and +(B13). This can be accomplished by using the following result +� 1 +0 +dv v2(1 − v)2 δ[vz′ + (1 − v)z′′ − z] = v2 +0(1 − v0)2 Θ(v0) Θ(1 − v0) +|z′ − z′′| += v2 +0(1 − v0)2 ∆(z, z′, z′′) (B16) + +24 +with +∆(z, z′, z′′) = Θ(z′ − z)Θ(z − z′′) − Θ(z′′ − z)Θ(z − z′) +z′ − z′′ +, +v0 = z − z′′ +z′ − z′′ . +(B17) +The combination of the theta-functions implements the constraint 0 ≤ v0 ≤ 1. +Moreover, notice that i) +simultaneously changing the signs of z, z′ and z′′ the function ∆(z, z′, z′′) does not change sign, this reflects +the symmetry with respect ξ = 0.5, as implemented through the charge symmetry in Eq. (23); ii) ∆(z, z′, z′′) +is even under the exchange z′′ → z′ and in the limit z′′ − z′ = ϵ → 0 one has +lim +ϵ→0 ∆(z, z′, z′ + ϵ) = δ(z − z′) , +(B18) +so that the singularity can be addressed without particular problems. +By taking into account Eq. (B16), and the symmetries with respect to the transformation z′ → z′′ and +γ′ → γ′′, one gets +Fi +0;ℓj(γ, z; S(AS)) = 2 +� ∞ +0 +dγ′ +� ∞ +0 +dγ′′ +� +1 +−1 +dz′ +� +1 +−1 +dz′′ v2 +0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) +z′ − z′′ +× +¯Gℓj(γ′, z′; γ′′, z′′; κ2) +[−β0(z2) + iϵ]5 +� +bi +0;ℓj(γ, z; S(AS)) + 1 +4 +� +β(z2) +M 2 +∂ +∂z bi +1;ℓj(γ, z; S(AS)) − z bi +1;ℓj(γ, z; S(AS)) +� ++ 1 +16 +� +z2 bi +2;ℓj(γ, z; S(AS)) − 2z β(z2) +M 2 +∂ +∂z bi +2;ℓj(γ, z; S(AS) +� +−z3 +64 bi +3;ℓj(S(AS)) +�� +, +(B19) +Fi +1;ℓj(γ, z; S(AS)) = +1 +4M 2 +∂ +∂z +�� ∞ +0 +dγ′ +� ∞ +0 +dγ′′ +� +1 +−1 +dz′ +� +1 +−1 +dz′′v2 +0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) +z′ − z′′ +× +¯Gℓj(γ′, z′; γ′′, z′′; κ2) +[−β0(z2) + iϵ]4 +� +2bi +1;ℓj(γ, z; S(AS)) − z bi +2;ℓj(γ, z; S(AS)) + 4 +3 +β(z2) +M 2 +∂ +∂z bi +2;ℓj(γ, z; S(AS)) ++3 +8z2 bi +3;ℓj(S(AS)) +�� +, +(B20) +Fi +2;ℓj(γ, z; S(AS)) = +1 +6M 4 +∂2 +∂z2 +�� ∞ +0 +dγ′ +� ∞ +0 +dγ′′ +� +1 +−1 +dz′ +� +1 +−1 +dz′′ v2 +0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) +z′ − z′′ +× +¯Gℓj(γ′, z′; γ′′, z′′; κ2) +[−β0(z2) + iϵ]3 +� +bi +2;ℓj(γ, z; S(AS)) − 3 +4z bi +3;ℓj(S(AS)) +�� +, +(B21) +and +Fi +3;ℓj(γ, z; S(AS)) = +bi +3;ℓj(S(AS)) +12M 6 +∂3 +∂z3 +�� ∞ +0 +dγ′ +� ∞ +0 +dγ′′ +� +1 +−1 +dz′ +� +1 +−1 +dz′′ v2 +0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) +z′ − z′′ +× +¯Gℓj(γ′, z′; γ′′, z′′; κ2) +[−β0(z2) + iϵ]2 +� +(B22) + +25 +where the functions bi +n;ℓj(γ, z; S(AS)) are given in the Tables of Appendix A and +β0(z2) = γ + κ2 + z2 M 2 +4 ++ v0γ′ + (1 − v0)γ′′ , +¯Gℓj(γ′, z′; γ′′, z′′; κ2) = gℓ(γ′, z′; κ2)gj(γ′′, z′′; κ2) + gℓ(γ′′, z′′; κ2)gj(γ′, z′; κ2) +2 +, +v0 = z − z′′ +z′ − z′′ . +(B23) +Also notice that for a bound state one has +β0(z2) = γ + κ2 + M 2 +4 z2 + v0γ′ + (1 − v0)γ′′ ≥ m2 − M 2 +4 (1 − z2) ≥ κ2 > 0 , +(B24) +and therefore no poles are associated to such a quantity. It should be pointed out that the presence of the +theta-functions, that ensure 0 ≤ v0 ≤ 1, prevents singular behaviors, shrinking the area of integration in the +space z′ ⊗ z′′, when z′ → z′′. Interestingly, in the Appendix C 1, it is shown that only Fi0 +ℓj (γ, z), i.e. without +derivative of the delta-function, contributes to the norm of the twist-2 uTMD f1(γ, ξ). +Appendix C: The leading-twist uTMD f S(AS) +1 +(γ, ξ) +In this Appendix, the symmetric and anti-symmetric combinations of the quark and antiquark leading-twist +uTMDs are explicitly given and their relevant features discussed. +By specializing the expressions in Eq. (27), one can write +f S(AS) +1 +(γ, ξ) = IN(γ, ξ; S(AS)) + Id(γ, ξ; S(AS)) + I2d(γ, ξ; S(AS)) + I3d(γ, ξ; S(AS)) +(C1) +where the four contributions are obtained from Eqs. (B19), (B20), (B21) and (B22), respectively. Inserting the +functions b0 +n;ℓj(γ, z; S) listed in in the first three columns of Table VI in Appendix A, one gets the following non +vanishing symmetric contributions, viz. +IN(γ, ξ; S) = 3Nc +2π2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′v2 +0(1 − v0)2 +Θ(z′ − z) Θ(z − z′′) +(z′ − z′′) [−β0(z2) + iϵ]5 +× +�� +¯G11(γ′, z′; γ′′, z′′) + ¯G22(γ′, z′; γ′′, z′′) − 4 m +M +¯G12(γ′, z′; γ′′, z′′) +� ++β0(z2) + 8γ +8M 2 +� +¯G33(γ′, z′; γ′′, z′′) + ¯G44(γ′, z′; γ′′, z′′) − 4 ¯G14(γ′, z′; γ′′, z′′) +�� +, +(C2) +Id(γ, ξ; S) = − +3Nc +4π2M 2 +∂ +∂z +�� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ v2 +0(1 − v0)2 +Θ(z′ − z) Θ(z − z′′) +(z′ − z′′) [−β0(z2) + iϵ]4 +× +� +2 m +M +¯G13(γ′, z′; γ′′, z′′) + z ¯G14(γ′, z′; γ′′, z′′) − ¯G23(γ′, z′; γ′′, z′′) +�� +, +(C3) +and +I2d(γ, ξ; S) = +Nc +8π2M 4 +∂2 +∂z2 +�� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ v2 +0(1 − v0)2 +Θ(z′ − z) Θ(z − z′′) +(z′ − z′′) [−β0(z2) + iϵ]3 +× +� +¯G33(γ′, z′; γ′′, z′′) + ¯G44(γ′, z′; γ′′, z′′) +�� +, +(C4) + +26 +with β0(z2), ¯Gℓj and v0 given in Eq. (B23). The symmetry property under the transformation z → −z can be eas- +ily demonstrated, recalling also that under the exchange z′ → −z′′ and γ′ → γ′′ the functions ¯Gℓj(γ′, z′; γ′′, z′′) +do not change, since the NWFs gi(γ, z; κ2) are even for i = 1, 2, 4 and odd for i = 3. Moreover, under z → −z and +z′ → −z′′ one also has v0 → (1−v0), so that β0(z2) remains unchanged, as well as Θ(z′ −z) Θ(z −z′′)/(z′ −z′′). +The anti-symmetric combinations are +IN(γ, ξ; AS) = 3 Nc +2π2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Θ(z′ − z)Θ(z − z′′) +z′ − z′′ +× +v2 +0(1 − v0)2 +[−β0(z2) + iϵ]5 +� +z ¯G11(γ′, z′; γ′′, z′′) + z ¯G22(γ′, z′; γ′′, z′′) ++β0(z2) + 8γ +2M 2 +� +− ¯G23(γ′, z′; γ′′, z′′) + z +4 +¯G33(γ′, z′; γ′′, z′′) ++ m +M +¯G34(γ′, z; γ′′, z′′) + z +4 +¯G44(γ′, z′; γ′′, z′′) +�� +, +(C5) +Id(γ, ξ; AS) = +Nc +2M 2π2 +∂ +∂z +�� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Θ(z′ − z)Θ(z − z′′) +z′ − z′′ +v2 +0(1 − v0)2 +[−β0(z2) + iϵ]4 +× +� +−3 +2 +¯G14(γ′, z′; γ′′, z′′) − 3 ¯G22(γ′, z′; γ′′, z′′) + 3z +2 +¯G23(γ′, z′; γ′′, z′′) + 3 m +M +¯G24(γ′, z′; γ′′, z′′) +−β0(z2) + 3 γ +M 2 +¯G33(γ′, z′; γ′′, z′′) + β0(z2) +2 M 2 ¯G44(γ′, z′; γ′′, z′′) +�� +(C6) +I2d(γ, ξ; AS) = +Nc +8π2M 4 +∂2 +∂z2 +�� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Θ(z′ − z)Θ(z − z′′) +z′ − z′′ +v2 +0(1 − v0)2 +[−β0(z2) + iϵ]3 +× +� +−8 ¯G23(γ′, z′; γ′′, z′′) + z ¯G33(γ′, z′; γ′′, z′′) + 4 m +M +¯G34(γ′, z′; γ′′, z′′) + z ¯G44(γ′, z′; γ′′, z′′) +�� +(C7) +I3d(γ, ξ; AS) = − +Nc +4π2M 6 +∂3 +∂z3 +�� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Θ(z′ − z)Θ(z − z′′) +z′ − z′′ +× +v2 +0(1 − v0)2 +[−β0(z2) + iϵ]2 ¯G33(γ′, z′; γ′′, z′′) +� +(C8) +The anti-symmetry with respect to the transformation z → −z can be easily shown by using the properties +listed below Eq. (C4). +1. +The normalization of f S +1 (γ, ξ) +While the integration on ξ and γ of f AS +1 +(γ, ξ) triv- +ially yields zero, since the anti-symmetry in z trans- +lates in an anti-symmetry in ξ with respect to ξ = 1/2, +it is interesting to analyze how to recover the normal- +ization of f S +1 (γ, ξ), once the BS-amplitude is properly +normalized as in Eq. (8). To proceed in the most easy +way, let us perform a step backward, and reinsert the +dependence upon δ(z − λ(v)) in Eqs. (C2), (C3) and +(C4) by using Eq. (B16). Then one has + +27 +IN(γ, ξ; S) = 3Nc +4π2 +� 1 +0 +dv v2(1 − v)2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ +× +δ(λ(v) − z) +[−β0(z2) + iϵ]5 +�� +G11(γ′, z′; γ′′, z′′) + G22(γ′, z′; γ′′, z′′) − 4 m +M G12(γ′, z′; γ′′, z′′) +� ++β0(z2) + 8γ +8M 2 +� +G33(γ′, z′; γ′′, z′′) + G44(γ′, z′; γ′′, z′′) − 4G14(γ′, z′; γ′′, z′′) +�� +, +(C9) +Id(γ, ξ; S) = − +3Nc +8π2M 2 +∂ +∂z +�� 1 +0 +dv v2(1 − v)2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ +× +δ(λ(v) − z) +[−β0(z2)) + iϵ]4 +� +2 m +M G13(γ′, z′; γ′′, z′′) + z G14(γ′, z′; γ′′, z′′) − G23(γ′, z′; γ′′, z′′) +�� +(C10) +and +I2d(γ, ξ; S) = +Nc +16π2M 4 +∂2 +∂z2 +�� 1 +0 +dv v2(1 − v)2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ +× +δ(λ(v) − z) +[−β0(z2) + iϵ]3 +� +G33(γ′, z′; γ′′, z′′) + G44(γ′, z′; γ′′, z′′) +�� +. +(C11) +Performing the integration on γ and ξ = (1−z)/2, one gets the following results. From Eq. (C9), one recovers +the standard normalization of the BS-amplitude in ladder approximation (cf. Eq. (12) in Ref. [62]), viz. +� ∞ +−∞ +dξ +� ∞ +0 +dγ IN(γ, ξ; S) = − 3Nc +32π2 +� 1 +0 +dv v2(1 − v)2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ +× +1 +[κ2 + M 2 +4 z2 + vγ′ + (1 − v)γ′′]4 +�� +G11(γ′, z′; γ′′, z′′) + G22(γ′, z′; γ′′, z′′) − 4 m +M G12(γ′, z′; γ′′, z′′) +� ++κ2 + M 2 +4 z2 + vγ′ + (1 − v)γ′′ +2M 2 +� +G33(γ′, z′; γ′′, z′′) + G44(γ′, z′; γ′′, z′′) − 4G14(γ′, z′; γ′′, z′′) +�� +, +(C12) +while the other two terms do not contribute. In fact, let us first integrate on z and take into account that in +δ(λ(v) − z) one has 1 ≥ λ(v) ≥ −1. One gets for Eq. (C10) +� ∞ +−∞ +dξ +� ∞ +0 +dγ Id(γ, ξ; S) = − +3 +16π2M 2 +� ∞ +0 +dγ +�� 1 +0 +dv v2(1 − v)2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ +× +δ(λ(v) − z) +[−β(z2) + iϵ]4 +� +2 m +M G13(γ′, z′; γ′′, z′′) + z G14(γ′, z′; γ′′, z′′) − G23(γ′, z′; γ′′, z′′) +��z=+∞ +z=−∞ += 0 . +(C13) + +28 +For Eq (C11) one has +� ∞ +−∞ +dξ +� ∞ +0 +dγ I2d(γ, ξ; S) = +1 +32π2M 4 +� ∞ +0 +dγ +� +∂ +∂z +� 1 +0 +dv v2(1 − v)2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ +× +δ(λ(v) − z) +[−β(zλ(v)) + iϵ]3 +� +G33(γ′, z′; γ′′, z′′) + G44(γ′, z′; γ′′, z′′) +��z=+∞ +z=−∞ += += +1 +32π2M 4 +� ∞ +0 +dγ +� +∂ +∂z +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ v2 +0(1 − v0)2 +z′ − z′′ +× Θ(z′ − z)Θ(z − z′′) − Θ(z′′ − z)Θ(z − z′) +[−β0(z2) + iϵ]3 +� +G33(γ′, z′; γ′′, z′′) + G44(γ′, z′; γ′′, z′′) +��z=+∞ +z=−∞ +, +(C14) +where in the last step Eq. (B16) has been used. Moreover, by explicitly performing the derivative on z, given +by (recall β0(z2) = γ + κ2 + z2M 2/4 + v0γ′ + (1 − v0)γ′′) +∂ +∂z +� +v2 +0(1 − v0)2 +[−β0(z2) + iϵ]3 +� +Θ(z′ − z)Θ(z − z′′) − Θ(z′′ − z)Θ(z − z′) +�� += += ∂ +∂z +� +v2 +0(1 − v0)2 +[−β0(z2) + iϵ]3 +� � +Θ(z′ − z)Θ(z − z′′) − Θ(z′′ − z)Θ(z − z′) +� ++ +v2 +0(1 − v0)2 +[−β0(z2) + iϵ]3 +× +� +−δ(z′ − z)Θ(z − z′′) + Θ(z′ − z)δ(z − z′′) + δ(z′′ − z)Θ(z − z′) − Θ(z′′ − z)δ(z − z′) +�� += += ∂ +∂z +� +v2 +0(1 − v0)2 +[−β0(z2) + iϵ]3 +� � +Θ(z′ − z)Θ(z − z′′) − Θ(z′′ − z)Θ(z − z′) +� ++ +v2 +0(1 − v0)2 +[−β0(z2) + iϵ]3 +� +−δ(z′ − z) + δ(z − z′′) +�� +, +(C15) +one can straightforwardly see that the derivative vanishes for z = ±∞, being z′ and z′′ ∈ [−1, 1] +Hence +� ∞ +−∞ +dξ +� ∞ +0 +dγ I2d(γ, ξ; S) = 0 . +(C16) +Two comments are in order. First, the leading-twist +uTMD is vanishing outside the range ξ ∈ [0, 1], and +hence one can restrict the integration on z between +[−1, 1]. It is easy to prove that the same results can +be obtained also in this case, recalling that z′ and z′′ +are in the same range, and in the last line of Eq. (C15) +one has v0 = (z − z′)/(z′ − z′′) and 1 − v0 = (z′′ − +z)/(z′ − z′′). Second, the integrand in Eq. (C13) and +(C14) should lead to contributions to the transverse +distribution +D⊥(γ) = +� ∞ +0 +dξ f S +1 (γ, ξ) , +(C17) +but from the above results one can see that they are +vanishing. +Appendix D: The parton distribution function +and the leading-twist uTMD +By integrating f S +1 (γ, ξ) on γ one gets the symmetric +parton distribution function uS(ξ). In particular, one +has +uS(ξ) = +� ∞ +0 +dγ f S +1 (γ, ξ) += uS +N(ξ) + uS +d (ξ) + uS +2d(ξ) +(D1) +where the three contributions are obtained by in- +tegrating on γ of the three quantities IN(γ, ξ; S), + +29 +Id(γ, ξ; S) and I2d(γ, ξ; S) given in Eqs. (C9), (C10) +and (C11), respectively. By using the result in Eq. +(B16) and the integrals +� ∞ +0 +dγ +1 +[−β0(z2) + iϵ]n = (−1)n +n − 1 +1 +[D(z, v0, γ′, γ′′)]n−1 , +� ∞ +0 +dγ +γ +[−β0(z2) + iϵ]4 = 1 +6 +1 +[D(z, v0, γ′, γ′′)]2 , +� ∞ +0 +dγ +γ +[−β0(z2) + iϵ]5 = − 1 +12 +1 +[D(z, v0, γ′, γ′′)]3 , +(D2) +where +D(z, v0, γ′, γ′′) = κ2 + M 2 +4 z2 + v0γ′ + (1 − v0)γ′′ , +(D3) +one writes +uS +N(ξ) = +� ∞ +0 +dγ IN(γ, ξ; S) = − 3 +8π2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Θ(z′ − z)Θ(z − z′′) +z′ − z′′ +v2 +0(1 − v0)2 +[D(z, v0, γ′, γ′′)]4 +× +�� +¯G11(γ′, z′; γ′′, z′′) + ¯G22(γ′, z′; γ′′, z′′) − 4 m +M +¯G12(γ′, z′; γ′′, z′′) +� ++D(z, v0, γ′, γ′′) +2M 2 +� +¯G33(γ′, z′; γ′′, z′′) + ¯G44(γ′, z′; γ′′, z′′) − 4 ¯G14(γ′, z′; γ′′, z′′) +�� +, +(D4) +uS +d (ξ) = +� ∞ +0 +dγ Id(γ, ξ; S) = − +1 +4π2M 2 +∂ +∂z +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Θ(z′ − z)Θ(z − z′′) +z′ − z′′ +× +v2 +0(1 − v0)2 +[D(z, v0, γ′, γ′′)]3 +� +2 m +M +¯G13(γ′, z′; γ′′, z′′) + z ¯G14(γ′, z′; γ′′, z′′) − ¯G23(γ′, z′; γ′′, z′′) +� +, +(D5) +and +uS +2d(ξ) = +� ∞ +0 +dγ I2d(γ, ξ; S) = − +1 +16π2M 4 +∂2 +∂z2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Θ(z′ − z)Θ(z − z′′) +z′ − z′′ +× +v2 +0(1 − v0)2 +[D(z, v0, γ′, γ′′)]2 +� +¯G33(γ′, z′; γ′′, z′′) + ¯G44(γ′, z′; γ′′, z′′) +� +. +(D6) +If the BS-amplitude has the standard normalization +[80], after integrating uS +N(ξ) one gets +� +1 +0 +dξ uS(ξ) = +� +1 +0 +dξ uS +N(ξ) = 1 +(D7) +from i) Eq. (C12), (C13) and (C16) and ii) Eq. (12) +in Ref. [62]. +The anti-symmetric PDF uAS(ξ) is given by +uAS(ξ) = +� ∞ +0 +dγ f AS +1 +(γ, ξ) = uAS +N (ξ) + uAS +d (ξ) + uAS +2d (ξ) + uAS +3d (ξ) +(D8) + +30 +where +uAS +N (ξ) = +� ∞ +0 +dγ IN(γ, ξ; AS) = −3 Nc +8π2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Θ(z′ − z)Θ(z − z′′) +z′ − z′′ +× +v2 +0(1 − v0)2 +[D(z, v0, γ′, γ′′)]4 +� +z ¯G11(γ′, z′; γ′′, z′′) + z ¯G22(γ′, z′; γ′′, z′′) + 2D(z, v0, γ′, γ′′) +M 2 +× +�z +4 +¯G33(γ′, z′; γ′′, z′′) + z +4 +¯G44(γ′, z′; γ′′, z′′) − ¯G23(γ′, z′; γ′′, z′′) + m +M +¯G34(γ′, z; γ′′, z′′) +�� +, +(D9) +uAS +d (ξ) = +� ∞ +0 +dγ Id(γ, ξ; AS) = +Nc +6M 2π2 +∂ +∂z +�� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Θ(z′ − z)Θ(z − z′′) +z′ − z′′ +× +v2 +0(1 − v0)2 +[D(z, v0, γ′, γ′′)]3 +� +−3 +2 +¯G14(γ′, z′; γ′′, z′′) +−3 ¯G22(γ′, z′; γ′′, z′′) + 3z +2 +¯G23(γ′, z′; γ′′, z′′) + 3 m +M +¯G24(γ′, z′; γ′′, z′′) − 3D(z, v0, γ′, γ′′) +M 2 +¯G33(γ′, z′; γ′′, z′′) ++3D(z, v0, γ′, γ′′) +4 M 2 +¯G44(γ′, z′; γ′′, z′′) +�� +, +(D10) +uAS +2d (ξ) = +� ∞ +0 +dγ I2d(γ, ξ; AS) = − +Nc +16π2M 4 +∂2 +∂z2 +�� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Θ(z′ − z)Θ(z − z′′) +z′ − z′′ +× +v2 +0(1 − v0)2 +[D(z, v0, γ′, γ′′)]2 +� +−8 ¯G23(γ′, z′; γ′′, z′′) + z ¯G33(γ′, z′; γ′′, z′′) + 4 m +M +¯G34(γ′, z′; γ′′, z′′) ++z ¯G44(γ′, z′; γ′′, z′′) +�� +, +(D11) +and +uAS +3d (ξ) = +� ∞ +0 +dγ I3d(γ, ξ; AS) = − +Nc +4π2M 6 +∂3 +∂z3 +�� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ Θ(z′ − z)Θ(z − z′′) +z′ − z′′ +× +v2 +0(1 − v0)2 +D(z, v0, γ′, γ′′) +¯G33(γ′, z′; γ′′, z′′) +� +. +(D12) +Appendix E: Twist-3 unpolarized TMDs +The Appendix presents the explicit expressions of the twist-3 and twist-4 uTMDs, obtained from Eq. (27) and +Eqs. (B19), (B20), (B21) and (B22), by using the Tables VII and VIII. In particular for the twist-3 eS(AS)(γ, ξ), +i.e. for i = 1 in eq. (27), one has +eS(AS)(γ, ξ) = E0(γ, ξ; S(AS)) + Ed(γ, ξ; S(AS)) + E2d(γ, ξ; S(AS)) + E3d(γ, ξ; S(AS)) +(E1) + +31 +where the symmetric combinations are +E0(γ, ξ; S) = 3Nc +2π2 +� ∞ +0 +dγ′ +� ∞ +0 +dγ′′ +� +1 +−1 +dz′ +� +1 +−1 +dz′′ v2 +0(1 − v0)2 +Θ(z′ − z)Θ(z − z′′) +(z′ − z′′) [−β0(z2) + iϵ]5 +× {−2 m +M +¯G11(γ′, z′; γ′′, z′′) + 2 ¯G12(γ′, z′; γ′′, z′′) − 2 m +M +¯G22(γ′, z′; γ′′, z′′) ++8γ + β0(z2) +2M 2 +� +− ¯G24(γ′, z′; γ′′, z′′) + m +2M +¯G33(γ′, z′; γ′′, z′′) + z +2 +¯G34(γ′, z; γ′′, z′′) + m +2 M +¯G44(γ′, z′; γ′′, z′′) +�� +, +(E2) +Ed(γ, ξ; S) = − +Nc +4M 4π2 +∂ +∂z +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ v2 +0(1 − v0)2 +Θ(z′ − z)Θ(z − z′′) +(z′ − z′′)[−β0(z2) + iϵ]4 +× +� +6γ + β0(z2) +� +¯G34(γ′, z′; γ′′, z′′) , +(E3) +E2d(γ, ξ; S) = +Nc +4π2M 4 +∂2 +∂z2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ v2 +0(1 − v0)2 +Θ(z′ − z)Θ(z − z′′) +(z′ − z′′)[−β0(z2) + iϵ]3 +× +� +−2 ¯G24(γ′, z′; γ′′, z′′) + m +M +¯G33(γ′, z′; γ′′, z′′) + z ¯G34(γ′, z′; γ′′, z′′) + m +M +¯G44(γ′, z′; γ′′, z′′) +� +, +(E4) +and +E3d(γ, ξ; S) = − +Nc +4π2M 6 +∂3 +∂z3 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ v2 +0(1 − v0)2 +× +Θ(z′ − z)Θ(z − z′′) +(z′ − z′′)[−β0(z2) + iϵ]2 ¯G34(γ′, z′; γ′′, z′′) +(E5) +The anti-symmetric combinations are +E0(γ, ξ; AS) = 3Nc +2π2 +� ∞ +0 +dγ′ +� ∞ +0 +dγ′′ +� +1 +−1 +dz′ +� +1 +−1 +dz′′ v2 +0(1 − v0)2 +Θ(z′ − z)Θ(z − z′′) +(z′ − z′′) [−β0(z2) + iϵ]5 +× +� +2z ¯G12(γ′, z′; γ′′, z′′) − 8γ + β0(z2) +4M 2 +� +2 ¯G13(γ′, z′; γ′′, z′′) − ¯G34(γ′, z′; γ′′, z′′) +�� +, +(E6) +Ed(γ, ξ; AS) = − +3Nc +2π2M 2 +∂ +∂z +�� ∞ +0 +dγ′ +� ∞ +0 +dγ′′ +� +1 +−1 +dz′ +� +1 +−1 +dz′′v2 +0(1 − v0)2 +Θ(z′ − z)Θ(z − z′′) +(z′ − z′′) [−β0(z2) + iϵ]4 +× ¯G12(γ′, z′; γ′′, z′′) +� +, +(E7) +E2d(γ, ξ; AS) − +Nc +4π2M 4 +∂2 +∂z2 +�� ∞ +0 +dγ′ +� ∞ +0 +dγ′′ +� +1 +−1 +dz′ +� +1 +−1 +dz′′ v2 +0(1 − v0)2 +Θ(z′ − z)Θ(z − z′′) +(z′ − z′′) [−β0(z2) + iϵ]3 +× +� +2 ¯G13(γ′, z′; γ′′, z′′) − ¯G34(γ′, z′; γ′′, z′′) +�� +, +(E8) +and +E3d(γ, ξ; AS) = 0 . +(E9) + +32 +1. +The twist-3 uTMD f ⊥(γ, ξ) +For i = 2, one has the twist-3 uTMD f ⊥(γ, ξ), with the following decomposition +f ⊥S(AS)(γ, ξ) = P0(γ, ξ; S(AS)) + Pd(γ, ξ; S(AS)) + P2d(γ, ξ; S(AS)) + P3d(γ, ξ; S(AS)) . +(E10) +The symmetric contributions are given by +P0(γ, ξ; S) = −3 Nc +π2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ v2 +0(1 − v0)2 +Θ(z′ − z)Θ(z − z′′) +(z′ − z′′)[−β0(z2) + iϵ]5 +× +� +¯G11(γ′, z′; γ′′, z′′) − ¯G14(γ′, z′; γ′′, z′′) − ¯G22(γ′, z′; γ′′, z′′) + z ¯G23(γ′, z′; γ′′, z′′) ++2m +M +¯G24(γ′, z′; γ′′, z′′) − 8γ + β0(z2) +8M 2 +� +¯G33(γ′, z′; γ′′, z′′) − ¯G44(γ′, z′; γ′′, z′′) +�� +, +(E11) +Pd(γ, ξ; S) = +3 Nc +2π2 M 2 +∂ +∂z +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ v2 +0(1 − v0)2 +Θ(z′ − z)Θ(z − z′′) +(z′ − z′′)[−β0(z2) + iϵ]4 +× ¯G23(γ′, z′; γ′′, z′′) , +(E12) +P2d(γ, ξ; S) = +Nc +4π2M 4 +∂2 +∂z2 +� +1 +−1 +dz′ +� ∞ +0 +dγ′ +� +1 +−1 +dz′′ +� ∞ +0 +dγ′′ v2 +0(1 − v0)2 +Θ(z′ − z)Θ(z − z′′) +(z′ − z′′)[−β0(z2) + iϵ]3 +× +� +¯G33(γ′, z′; γ′′, z′′) − ¯G44(γ′, z′; γ′′, z′′) +� +, +(E13) +and +P3d(γ, ξ; S) = 0 +(E14) +The anti-symmetric contributions read +P0(γ, ξ; AS) = 3Nc +π2 +� ∞ +0 +dγ′ +� ∞ +0 +dγ′′ +� +1 +−1 +dz′ +� +1 +−1 +dz′′ v2 +0(1 − v0)2 +Θ(z′ − z)Θ(z − z′′) +(z′ − z′′) [−β0(z2) + iϵ]5 +× +� +2 m +M +¯G13(γ′, z′; γ′′, z′′) + z ¯G14(γ′, z′; γ′′, z′′) − ¯G23(γ′, z′; γ′′, z′′) +� +, +(E15) +and +Pd(γ, ξ; AS) = +− +3Nc +2π2M 2 +∂ +∂z +�� ∞ +0 +dγ′ +� ∞ +0 +dγ′′ +� +1 +−1 +dz′ +� +1 +−1 +dz′′v2 +0(1 − v0)2 +Θ(z′ − z)Θ(z − z′′) +(z′ − z′′) [−β0(z2) + iϵ]4 +× ¯G14(γ′, z′; γ′′, z′′) +� +, +(E16) +and +P2d(γ, ξ; AS) = P3d(γ, ξ; AS) = 0 . +(E17) +[1] R. 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D 7, 1780 (1973). + diff --git a/-NFJT4oBgHgl3EQfpSw0/content/tmp_files/load_file.txt b/-NFJT4oBgHgl3EQfpSw0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a9f8e9f77781f5734c1bf698ebffd494d0ee701 --- /dev/null +++ b/-NFJT4oBgHgl3EQfpSw0/content/tmp_files/load_file.txt @@ -0,0 +1,2353 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf,len=2352 +page_content='Unpolarized transverse-momentum dependent distribution functions of a quark in a pion with Minkowskian dynamics E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Ydrefors,1 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' de Paula,2 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Frederico,2 and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Salm`e3 1Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China 2Instituto Tecnol´ogico de Aeron´autica, DCTA, 12228-900 S˜ao Jos´e dos Campos, Brazil 3INFN, Sezione di Roma, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='le A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Moro 2, 00185 Rome, Italy (Dated: January 30, 2023) The unpolarized twist-2 (leading) and twist-3 (subleading), T-even, transverse-momentum depen- dent quark distributions in the pion are evaluated for the first time by using the actual solution of a dynamical equation in Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The adopted theoretical framework is based on the homogeneous Bethe-Salpeter integral equation with an interaction kernel given by a one-gluon ex- change, featuring an extended quark-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The masses of quark and gluon as well as the interaction-vertex scale have been chosen in a range suggested by lattice-QCD calculations, and calibrated to reproduce both pion mass and decay constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The sum rules to be fulfilled by the transverse-momentum dependent distributions are carefully investigated, particularly the leading- twist one, that has to match the collinear parton distribution function, and hence can be scrutinized in terms of existing data as well as theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Noteworthy, the joint use of the Fock expansion of the pion state facilitates a more in-depth analysis of the content of the pion Bethe- Salpeter amplitude, allowing for the first time to determine the gluon contribution to the quark average longitudinal fraction, that results to be ∼ 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The current analysis highlights the role of the gluon exchanges through quantitative analysis of collinear and transverse-momentum distri- butions, showing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' for both leading and subleading-twists, an early departure from the widely adopted exponential fall-off, for |k⊥|2 > m2, with the quark mass ∼ ΛQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' INTRODUCTION Quark transverse-momentum dependent distribu- tion functions (TMDs for short) are the basic ingredi- ents for parametrizing the hadronic quark-quark cor- relator (see the seminal Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [1] and for the complete parametrization Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [2], while Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [3, 4] for corre- lators involving gluons), and represent direct general- ization of the parton distribution functions (PDFs), so that both longitudinal and transverse degrees of freedom (dof) can be addressed (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [5, 6] for an extensive introduction to the transverse dof and related distribution functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Clearly, the ac- cess to the 3D imaging of hadrons allows us to achieve a deeper and deeper understanding of the non- perturbative regime of QCD, also exploiting the non- trivial coupling to the spin dof (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [7, 8] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Hence, by means of TMDs, one can gather unique information on QCD at work in hard semi-inclusive reactions (both unpolarized and polarized) at low transverse-momentum, like low-q⊥ Drell-Yan (DY) processes, vector/scalar boson pro- ductions or semi-inclusive deep inelastic scattering (SIDIS) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [8–11] for a status-report on the experimental measurements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Indeed, the extrac- tion of TMDs from the experimental cross-section is a highly challenging task, as shown by the intense theoretical work on the factorization of the cross sec- tions into transverse-momentum dependent matrix el- ements (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [12–16]) and the TMDs evo- lution that becomes a two-scale problem, since the rapidity ζ comes into play in addition to the renor- malization scale µ (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [12, 17–19] and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [20] for a recent review that covers also the factor- ization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Noteworthy, one has to mention the efforts for obtaining reliable global fits (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [21– 24] and also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [8] for a general discussion), early- stage lattice calculations (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [25–29] and also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [8, 30–32]) and, finally, the broad set of phenomenological models, that we can only partially list: the bag model (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [33] and refer- ences therein), covariant model (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [34] and references therein), light-front (LF) constituent quark models (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [35, 36]) and the ba- sis LF quantization framework [37], the approaches based on the Nambu-Jona-Lasinio interaction (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [38, 39]), the holographic models (see,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [40, 41]), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In view of our study, one has to separately mention the approaches developed within the so-called continuum-QCD, that are based on solu- tions (actually in Euclidean space) of dynamical equa- tions like the homogeneous 4D Bethe-Salpeter equa- tion (BSE) [42, 43] in combination or not with the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='11599v1 [hep-ph] 27 Jan 2023 2 quark gap-equation (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [44–47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' It should be recalled that the proton is the elective target of much experimental (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [9–11]) and theoretical research (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [48–50] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' While the pion, given the experi- mental challenges its study poses, has surely attracted less efforts in spite of its intriguing double-nature, be- ing both a Goldstone boson (and hence fundamen- tal for investigating the dynamical chiral-symmetry breaking) and a quark-antiquark bound system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' the simplest bound system in QCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In particular, a first extraction of the pion unpolarized leading-twist TMD from Drell-Yan data can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [51], where the results of the E615 Collaboration [52] has been used, and in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [53], where both the previous data and the E537 Collaboration cross-sections [54] have been included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' As to the phenomenological cal- culations, a broad overview, embracing different ap- proaches, can be gained from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [35, 38, 40, 41, 44– 47, 55–58] (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [59] for the generalized TMDs in a spin-0 hadron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' As a conclusion to the above schematic introduc- tion, it has to be emphasized that the vast amount of nowadays theoretical studies on TMDs finds its strong motivation in the very accurate measurements that will come from forthcoming electron-ion colliders, that promise to achieve greatly expected milestones in the experimental investigation of non-perturbative QCD, given the planned high energy and luminosity [60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Our aim is to obtain, for the first time, T-even leading- and subleading-twist unpolarized TMDs (uT- MDs) of the pion, by solving a dynamical equation directly in Minkowski space, namely relying on a gen- uinely quantum-field theory framework based on the 4D homogeneous BSE [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The homogeneous BSE is an integral equation and therefore suitable for dealing with the fundamentally non-perturbative nature of bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' One should not get con- fused by the use of an interaction kernel expressed in a perturbative series, since an integral equation has a peculiar feature of infinitely many times iterating the boson exchanges contained in each term of the kernel, just what one needs for obtaining a pole in the relevant Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In our approach (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [62] for details and references therein), based on the 4D homogeneous BSE in Minkowski space and the Nakanishi integral representation (NIR) of the BS-amplitude [63, 64], the interaction kernel is given by the exchange of a massive vector boson in the Feynman gauge, with three input parameters, in- ferred from lattice QCD (LQCD) calculations (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [65–67]): (i) the constituent-quark and gluon masses, and (ii) a scale parameter featuring the extended quark-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' It should be pointed out that the ladder kernel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' the first term in a perturbative series, can be a reliable approximation to evaluate the pion bound state, as suggested by the suppression of the non-planar contributions for Nc = 3 within the BS approach in a scalar QCD model [68], and the presence of massive quarks and gluons, featur- ing the confinement effects in a relatively large system (rch ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='66 fm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' There is another important conse- quence stemming from the use of the BS-amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Although in the definition of the q¯q-pair BS-amplitude there is a simple dependence upon two interacting fermionic fields, one ends up dealing with an infinite content of Fock states (the use of the Fock space al- lows one to recover a probabilistic language within the BS framework).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In particular, by exploiting the Fock expansion of the pion state, one can establish a for- mal link between the LF-projected BS-amplitude (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [69–71]), and the amplitude of the Fock component of the pion state with the lowest number of constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Therefore, in our approach, it is natu- ral to call the LF-projected BS-amplitude LF valence wave function (LFWF), to be distinguished from the valence wave function, when a SU(3)-flavor language is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In the latter case, the pion is composed by only two fermionic constituents, suitably dressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' One should keep in mind that within our framework, the pion LFWF contributes only with 70% [62] of the normalization, and consequently a significant role of the higher Fock components has to be highlighted, and possibly analyzed in-depth, as illustrated in what fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Finally, we would emphasize that the first evalu- ation of the uTMDs strengthens the reliability of our approach and makes sound the ground for the next step, already in progress, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' taking into account the self-energy of the quarks (see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [72] and [73, 74]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Indeed, in spirit, our approach is similar to the one developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [45] for evaluating the leading-twist uTMD, where it was also taken into account the self- energy of the quark propagator (solving the gap equa- tion) and a confining interaction, but in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In this case, one resorts to a suitable method (based on the moments and a parametrization of the Euclidean BS-amplitude) to get the Minkowski-space distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Differently, in our approach the NIR of the BS-amplitude allows one to successfully deal with the analytic structure of the BS-amplitude itself, obtaining an integral equation formally equiv- alent to the initial 4D homogeneous BSE, but more suitable for the numerical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Many and rele- vant applications of our approach to the pion, such as 3 the electromagnetic form factor [75], the PDF [76] and the 3D imaging [62], have confirmed its reliability and encouraged to broad the scope of our investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' It should be pointed out that (it will become clear in what follows) the evaluation of quantities that depend not only upon the longitudinal dof but also the trans- verse ones leads to sharply increase the sensitivity to the dynamical content of a given phenomenological description of the pion, namely to increase its predic- tive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Furthermore, the joint use of the Fock expansion, meaningful in the Minkowski space, allows one to resolve the gluonic content of the pion state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The paper outline is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' II, the gen- eral formalism and the notations are introduced, high- lighting the ingredients of our dynamical approach, namely i) the Bethe-Salpeter amplitude, solution of the 4D homogeneous Bethe-Salpeter equation, and ii) the Nakanishi integral representation of the BS- amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' III, the expressions of leading- and subleading-twist uTMDs are given in terms of the Bethe-Salpeter amplitude of the pion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' IV and V, the leading and subleading-twist uTMDs are shown and compared with outcomes from other ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Finally, in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' IV, the conclusions are drawn, and the perspectives of our approach are pre- sented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' GENERALITIES For a pion with four-momentum P ≡ {P −, P +, P⊥} (where P 2 = P +P − − |P⊥|2 = M 2 and the LF co- ordinates are a± = a0 ± a3), and by adopting both i) a frame where P⊥ = 0 and ii) the light-cone gauge A+ g = 0, the quark leading-twist uTMD, f q 1 (γ, ξ), is defined as follows (for a general introduction see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [1, 6]) f q 1 (γ, ξ) = Nc 4 � dφˆk⊥ � ∞ −∞ dy−dy⊥ 2(2π)3 × ei[ξP + y− 2 −k⊥·y⊥]⟨P| ¯ψq(− y 2 )γ+ψq( y 2 )|P⟩ �� y+=0 , (1) where Nc is the number of colors, ψq is the fermionic field, and the quark four-momentum is given in terms of LF coordinate by pq ≡ {p− q , ξP +, k⊥ + P⊥/2}, with γ = |k⊥|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The antiquark uTMD is obtained by using the proper four-momentum p¯q ≡ {p− ¯q , (1 − ξ)P +, −k⊥ + P⊥/2}, recalling that P = pq + p¯q and k = (pq − p¯q)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The normalization of f q 1 (γ, ξ) is given by � ∞ −∞ dξ � ∞ 0 dγ f q 1 (γ, ξ) = Nc 2 � dpq⊥ � ∞ −∞ dp+ q P + × � ∞ −∞ dp− q 2 � ∞ −∞ d4y (2π)4 ei pq·y⟨P| ¯ψq(− y 2)γ+ψq( y 2)|P⟩ = Nc ⟨P| ¯ψq(0)γ+ψq(0)|P⟩ 2P + = F q π(0) = 1 , (2) where F q π(t) is the quark contribution to the elec- tromagnetic (em) form factor of the pion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The lat- ter results to be equal to Fπ(t) = eqF q π(t) + e¯qF ¯q π(t), with t = (P ′ − P)2, and is related to the matrix ele- ment of the four-current by Nc ⟨P| ¯ψq(0)γµψq(0)|P⟩ = 2P µ Fπ(t = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Finally, it should be pointed that in- serting a complete basis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (1) and exploiting the good and bad components of the fermionic field one can easily demonstrate that f q 1 (γ, ξ) ≥ 0 (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [77]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In order to describe the pion by taking into ac- count at some extent the QCD dynamics in the non- perturbative regime, it is useful to resort to the Man- delstam framework [78], where the interacting quark- pion vertex is expressed in terms of the (reduced) BS- amplitude, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' the solution of the 4D homogeneous BSE, and defined by Φ(k, P) = � d4x eik·x ⟨0|T � ψ( x 2) ¯ψ(− x 2) � |P⟩ , (3) where the fermionc field fulfills the Poincar´e trans- lation ψ(x) = ei ˆ P ·xψ(0)e−i ˆ P ·x (recall that only the component ˆP − is interacting in the LF dynamics, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [79]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Thus, by using the Feynman-like diagrammatic pic- ture inherent to the Mandelstam framework (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [75] for the application to the em form factor), one can write the following expression for f q 1 (γ, ξ) f q 1 (γ, ξ) = Nc 4(2π)3 � ∞ −∞ dk+ 2(2π)δ � k+ + P + 2 − ξP +� × � ∞ −∞ dk− � 2π 0 dφˆk⊥Tr � S−1(−p¯q)¯Φ(k, P) γ+ Φ(k, P) � , (4) where pq(¯q) = ± k + P 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (5) For the sake of completeness, let us write the BSE in ladder approximation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' the one we are adopting for 4 the numerical calculations, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Φ(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P) = S � pq � � d4k′ (2π)4 Sµν(q)Γµ(q) × Φ(k′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P)�Γν(q)S � −p¯q � , (6) where quark and antiquark momenta are off-shell, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' p2 q(¯q) = (±k + P 2 )2 ̸= m2, and q = k − k′ is the gluon four-momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (6), the fermion propaga- tor, the gluon propagator in the Feynman gauge and the quark-gluon vertex, dressed through a simple form factor, are S(p) = i /p − m + iϵ , Sµν(q) = −i gµν q2 − µ2 + iϵ , Γµ = igγµ µ2 − Λ2 q2 − Λ2 + iϵ, (7) where g is the coupling constant, µ the mass of the exchanged vector-boson and Λ is a scale parameter, featuring the extension of the color distribution in the interaction vertex of the dressed constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' More- over, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (6), one has �Γν(q) = C ΓT ν (q) C−1, where C = iγ2γ0 is the charge-conjugation opera- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The normalization of the BS-amplitude reads (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [80] and [62] for details) Nc Tr �� d4k (2π)4 ∂ ∂P ′µ � S−1� k − P ′ 2 � ¯Φ(k, P) × S−1� k + P ′ 2 � Φ(k, P) �� P ′=P = −2iPµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (8) The antiquark uTMD is given by f ¯q 1 (γ, 1 − ξ) = − Nc 4(2π)3 � ∞ −∞ dk+ 2(2π)δ(k+ + P + 2 − ξP +) × � ∞ −∞ dk− � 2π 0 dφˆk⊥Tr � S−1(pq)Φ(k, P)γ+ ¯Φ(k, P) � , (9) where the minus sign results from the property of the normal-ordered em current to be odd under the action of the charge conjugation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' It is noteworthy that in Appendix C 1, it is proven the identity of the normalization condition, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (8), and the half sum of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (1) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Within a SU(3)-flavor symmetry framework, one describes a pion as a bound system of a massive q¯q pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' This leads to introduce the so-called valence- quark PDF in the pion, that is charge symmetric (once the isospin breaking is disregarded [81]) as well as ful- fills the charge conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' From those properties one deduces that the SU(3)-valence PDFs in the charged pions must verify: uv π+(ξ) = dv π−(ξ) = ¯dv π+(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In our BS framework, in addition to the fermionic dof (still massive) one introduces also gluonic dof, by adding an explicit dynamical description of the binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' This amounts to the ladder exchange of infinite number of massive gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Therefore, at the initial scale, the quark and anti-quark longitudinal-momentum frac- tion distributions are not expected to be symmetric with respect to ξ = 1/2 (as it follows from the charge symmetry), given the gluon-momentum flow in the composite pion (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The symmetric com- bination of quark and anti-quark contribution allows one to fulfill the charge symmetry, and hence it is rel- evant in the comparison with experimental data (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [76]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In what follows, in addition to the quark distributions, symmetric and anti-symmetric combi- nations are introduced for all the uTMDs we are going to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The half sum (difference) of the quark and anti- quark contributions, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (1) and (9), yields the fol- lowing charge-symmetric (anti-symmetric) expression for the leading-twist uTMD inside a π+ meson f S(AS) 1 (γ, ξ) = f q 1 (γ, ξ) ± f ¯q 1 (γ, 1 − ξ) 2 = Nc 8(2π)3 � ∞ −∞ dk+ 2(2π)δ � p+ q − ξP +� � ∞ −∞ dk− × � 2π 0 dφˆk⊥Tr � S−1(−p¯q)¯Φ(k, P) γ+ Φ(k, P) ∓ S−1(pq)Φ(k, P) γ+ ¯Φ(k, P) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (10) Analogously to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (1), one can define the T-even sub- leading quark uTMDs, starting from the decomposi- tion of the pion correlator [6, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' To be specific, one has two twist-3 uTMDs (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [35] for the pion case) M P + eq(γ, ξ) = Nc 4 � dφˆk⊥ � ∞ −∞ dy−dy⊥ 2(2π)3 (11) ×ei[ξP + y− 2 −k⊥·y⊥]⟨P| ¯ψq(− y 2) 1 ψq( y 2)|P⟩ �� y+=0 , M P + f ⊥q(γ, ξ) = M γ Nc 4 � dφˆk⊥ � ∞ −∞ dy−dy⊥ 2(2π)3 (12) ×ei[ξP + y− 2 −k⊥·y⊥] ⟨P| ¯ψq(− y 2) k⊥ · γ⊥ ψq( y 2)|P⟩ �� y+=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In analogy to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (2), one gets for the twist-3 eq(ξ) (see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [77, 82] and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [35] for the pion in phe- 5 nomenological models) � ∞ −∞ dξ � ∞ 0 dγ eq(γ, ξ) = Nc 2 � dpq⊥ � ∞ −∞ dp+ q P + × � ∞ −∞ dp− q 2 � ∞ −∞ d4y (2π)4 ei pq·y ⟨P| ¯ψq(− y 2) 1 ψq( y 2)|P⟩ = Nc ⟨P| ¯ψq(0) 1 ψq(0)|P⟩ 2P + , (13) where the matrix element ⟨P| ¯ψq(0) 1 ψq(0)|P⟩ has to be proportional to the pion sigma term, once a QCD framework is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' As a matter of fact, one gets � 1 0 dξ � ∞ 0 dγ eq(γ, ξ) = σπ mcur (14) where mcur is the quark current mass and σπ is the pion sigma term, that becomes σπ = M/2, in the leading order of the chiral expansion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' the Gell-Mann-Oakes-Renner relation [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' It should be pointed that recent LQCD calculations [84] confirm, with high accuracy, the Gell-Mann-Oakes-Renner re- lation in the range of the explored pion masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In- deed, the QCD equations of motion gives a decom- position of the collinear PDF e(ξ) = � dγ e(γ, ξ) in three terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Among them, there is a singular term proportional to the pion sigma term, that reads (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [85]) esing(ξ) = δ(ξ) ⟨P| ¯ψq(0) 1 ψq(0)|P⟩/2P + , (15) while the other two terms, one is due to quark- antiquark-gluon correlations and the other is propor- tional to the quark mass, do not contribute to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (14) (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [85], where the issue is analyzed, taking the nucleon as actual case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In our phenomenologi- cal model the strength is distributed over the whole range of ξ (as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [35]), without the singularity at ξ = 0, as it will be shown in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Moreover, one has for the first moment [85] � 1 0 dξ � ∞ 0 dγ ξ eq(γ, ξ) = mcur M , (16) where the singular term and the gluonic contribution vanish, and only the term proportional to the quark mass contributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' From the equations of motion of a free-quark model, one deduces the following relations between the above uTMDs (see,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Ref [35, 85, 86]) ξ eq EoM(γ, ξ) = ξ ˜eq(γ, ξ) + m M f q 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='EoM(γ, ξ) ξ f q⊥ EoM(γ, ξ) = ξ ˜f q⊥(γ, ξ) + f q 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='EoM(γ, ξ) , (17) where the uTMDs with a tilde are the gluonic con- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The relevant point is the dependence of all the subleading-twist uTMDs from only the leading one, modulo the gluonic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In our fully interact- ing framework, one can anticipate that the relations are not recovered, and rather heavily broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' For a derivation of the first line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (17), fully consistent with QCD, one could apply the formalism presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (10), one readily writes down charge- symmetric and the anti-symmetric combinations for the subleading TMDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' One has to take care how the scalar and vector operators behave under the charge conjugation that impose a different combination of signs (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Namely, one gets M P + eS(AS)(γ, ξ) = Nc 8(2π)3 � ∞ −∞ dk+ 2(2π)δ(p+ q − ξP +) × � ∞ −∞ dk− � 2π 0 dφˆk⊥Tr � S−1(−p¯q)¯Φ(k, P) 1 Φ(k, P) ± S−1(pq)Φ(k, P) 1 ¯Φ(k, P) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (18) M P + f ⊥S(AS)(γ, ξ) = NcM 8(2π)3γ � ∞ −∞ dk+ 2(2π)δ(p+ q −ξP +) � ∞ −∞ dk− � 2π 0 dφˆk⊥Tr � S−1(−p¯q)¯Φ(k, P) γ⊥ Φ(k, P) ± S−1(pq)Φ(k, P) γ⊥ ¯Φ(k, P) � k⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (19) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The BS-amplitude and its Nakanishi integral representation It is useful to briefly recall some features of our ap- proach for obtaining the actual solution of the ladder BSE given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The basic ingredient is the NIR of the BS-amplitude (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [64] for the general introduction, and Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [62, 76, 87–89] for the appli- cation to a two-fermion case), but let us first intro- duce the general decomposition of the BS-amplitude, Φ(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P), for a 0− bound state, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [87, 90] Φ(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P) = S1(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P)φ1(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P) + S2(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P)φ2(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P) +S3(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P)φ3(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P) + S4(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P)φ4(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P) , (20) where φi’s are unknown scalar functions, that depend upon the kinematical scalars at disposal (k2, k ·P and 6 P 2), and Si’s are suitable Dirac structures, given by S1(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P) = γ5, S2(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P) = /P M γ5, S3(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P) = k · P M 3 /Pγ5 − 1 M /kγ5, S4(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P) = i M 2 σµνPµkνγ5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (21) The functions φi must be even for i = 1, 2, 4 and odd for i = 3, under the change k → −k, as dictated by the anti-commutation rules of the fermionic fields, and they can be written in terms of the NIR as follows φi(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P) = � 1 −1 dz′ � ∞ 0 dγ′ × gi(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) [k2 + z′(P · k) − γ′ − κ2 + iϵ]3 , (22) where κ2 = m2 − M 2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The real functions gi(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2), the unknowns of the problem under scrutiny, are the Nakanishi weight functions (NWFs), and assumed to be unique, following the uniqueness theorem from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The properties of the scalar functions φi under the exchange k → −k translate to properties of the NWFs, but under the exchange z′ → −z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Finally, it should be mentioned that NWFs are de- termined by solving a system of integral equation, so that one is able to non-perturbatively embed dynam- ical information that characterize the BS interaction kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The system of integral equations is formally deduced from the initial BSE, by exploiting the ana- lytic structure of the scalar functions φi, made explicit by means of the NIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In fact, after inserting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (20) and (22) in the BSE, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (6), and performing both the Dirac traces and a LF projection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' the integration over the k− = k0 − k3 component of the relative mo- mentum, one gets a coupled system of integral equa- tions for the NWFs (see details in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [89]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Once the NWFs are known, the BS-amplitude can be fully reconstructed through an inverse path, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (22) and (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' THE UNPOLARIZED TMDS AND THE PION BS-AMPLITUDE The evaluation of the leading- and subleading-twist uTMDs, given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (10), (18) and (19), can be performed by inserting the decomposition of the BS- amplitude in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (20), obtaining T S(AS) i (γ, ξ) = Nc 8(2π)3 � ∞ −∞ dk+ 2 δ(p+ q − ξP +) × � ∞ −∞ dk− 2π � 2π 0 dφˆk⊥Tr � S−1(−p¯q) Ai(k, P) + ηS(AS) i S−1(pq) ¯ Ai(k, P) � = i Nc 8(2π)2 � ℓj � 1 −1 dz δ(z−(1 − 2ξ)) F i ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) , (23) where Ai(k, P) = ¯Φ(k, P) Oi Φ(k, P) ¯ Ai(k, P) = Φ(k, P) Oi ¯Φ(k, P) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (24) A new variable z is defined as z = −2k+/P + and the three quantities: i) the functions Ti(γ, ξ), ii) the operators Oi and iii) the phase ηS(AS) i are given by T S(AS) 0 (γ, ξ) ≡ f S(AS) 1 (γ, ξ) , O0 = γ+ , ηS(AS) 0 = ∓1 , T S(AS) 1 (γ, ξ) ≡ M P + eS(AS)(γ, ξ) , O1 = 1 , ηS(AS) 1 = ±1 , T S(AS) 2 (γ, ξ) ≡ M P + f ⊥S(AS)(γ, ξ) , O2 = M |k⊥|2 k⊥ · γ⊥ , ηS(AS) 2 = ±1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (25) Finally, the integrand F i ℓj in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (23) reads F i ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = � ∞ −∞ dk− 2π ai ℓj(k−, γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) × φℓ(k, P) φj(k, P) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (26) where ai ℓj(k−, γ, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) are polynomial in k− (up to the cubic power) and can be found in Appendix A for each uTMDs, we are considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' By exploiting the NIR, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (22), one can perform 7 the integration on k−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' This integration amounts to restrict the LF-time to x+ = 0, and it is also known as LF-projection (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [70, 71, 91]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' After carrying out the k−-integration, the expression of each T S(AS) i (γ, ξ) can be decomposed as follows (the details of this formal step can be found in Appendix B) T S(AS) i (γ, ξ) = 3Nc (2π)2 � ℓj � Fi 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + Fi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + Fi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + Fi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) � , (27) where ξ = (1 − z)/2 and the functions Fi n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) (n = 1, 2, 3, 4) are given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B19), (B20), (B21) and (B22), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' THE LEADING-TWIST f S(AS) 1 (γ, ξ) The symmetric and anti-symmetric combinations of the T-even leading-twist uTMD, f S(AS) 1 (γ, ξ), allow us to address the evaluation of both quark and anti-quark contributions, f q(¯q) 1 (γ, ξ), that in the BS framework plus the Fock expansion of the pion state have inter- esting features, distinct from the ones of f S(AS) 1 (γ, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' After integrating the leading-twist f q(¯q) 1 (γ, ξ) on γ, one gets the quark PDF uq(ξ), while the symmet- ric combination provides the charge-symmetric PDF uS(ξ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' the one is expected to have relevance at the valence scale (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [81]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Indeed, in the Mandelstam approach the quark and antiquark PDFs do not have in general a symmetry with re- spect to ξ = 1/2, since each receives contributions from states containing an infinite number of gluons, as a consequence of the ladder-interaction kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' But if we restrict to the contribution from the first Fock component in the expansion of the pion state, one gets the LF-valence uLF val(ξ), that is given by the BS- amplitude projected onto the null plane [79] and is fully compliant with the charge symmetry (see below the discussion on the differences among uq(ξ), uS(ξ) and uLF val(ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' To illustrate general features and relations, in this Section we give some details, referring to Appendix C for a more complete discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The symmetric and anti-symmetric leading-twist uTMDs, can be decomposed as follow f S(AS) 1 (γ, ξ) = IN(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + Id(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) +I2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + I3d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) , (28) where the non-vanishing symmetric contributions are given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (C2), (C3), (C4) and I3d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The anti-symmetric quantities are shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (C5), (C6), (C7) and (C8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Two comments are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The symmetry proper- ties of the above quantities with respect to the trans- formation z → −z are demonstrated in Appendix C, and can be translated into the symmetry with respect to ξ → 1 − ξ (that implements the charge-symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' A relevant feature is given by the presence in the ex- pressions of Id,2d,3d of the partial derivatives ∂n/∂zn, that should be considered dual of the n-th moment in k− of the relevant functions, generated by the formal step of the LF-projection (cf Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (26)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' This is not a surprise since the variable z is proportional to k+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' A first consistency check of our formalism has been carried out in Appendix C 1, where it is shown that, within the Mandelstam approach, f S 1 (γ, ξ) and in turn f q 1 (γ, ξ) are normalized to 1, as naturally follows from the canonical BS-amplitude normalization [80, 92], performed according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (8) (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [62]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In particular, the integral on γ and ξ of IN(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) sat- urates the normalization, while the other two terms provide vanishing contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Hence, one gets � 1 0 dξ � ∞ 0 dγ f S 1 (γ, ξ) = � 1 0 dξ � ∞ 0 dγ IN(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = � 1 0 dξ � ∞ 0 dγ f q 1 (γ, ξ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (29) It should be recalled that all the calculated uTMDs vanish outside the interval 0 ≤ ξ ≤ 1, as dictated by the conservation of the plus components of the four- momenta of both pion and constituents (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' It is understood that the integral of f AS 1 (γ, ξ) is van- ishing, given the antisymmetry with respect to ξ → 1 − ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Longitudinal degree of freedom The symmetric and the anti-symmetric PDFs, uS(AS)(ξ) (for the explicit expressions see Ap- pendix D) are defined by uS(AS)(ξ) = � ∞ 0 dγ f S(AS) 1 (γ, ξ) (30) = uS(AS) N (ξ) + uS(AS) d (ξ) + uS(AS) 2d (ξ) + uS(AS) 3d (ξ) , 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='75 1 ξ 6 4 2 0 2 4 6 u S ( ξ) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='75 1 ξ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5 2 u( ξ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (Color online).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Left panel: the symmetric pion PDF, uS(ξ), with its contributions uS N(ξ), uS d (ξ) and uS 2d(ξ) (cf Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (31)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dash-dotted line: uS(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dashed line: uS N(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dotted line: uS d (ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dash-double-dotted line: uS 2d(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Right panel: uq(ξ), uS(ξ), uAS(ξ) and the LF-valence PDF of the pion, uLF val(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Solid line: quark PDF, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dashed line: uS(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dotted line: uAS(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dash-dotted line: uLF val(ξ) (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [76]), with normalization equal to Pval = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='7 (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' with the normalization that follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (29) and the vanishing result of the double integration of f AS 1 (γ, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Finally, the quark and anti-quark PDFs are evaluated through uq(¯q)(ξ) = uS(ξ) ± uAS(ξ) , (31) with the normalization still given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Within the SU(3)-flavor symmetry, one has to implement the charge symmetry (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [81]) at the initial scale, and therefore uS(ξ) is the PDF to be com- pared, after the proper evolution, with the experimen- tal data, as it has been shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 1, uS(ξ) and its three contributions (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (D4), (D5) and (D6)) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The calculation has been carried out by adopting the BS-amplitude obtained by using the so- lution of the BSE as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [62], using the following values of the three input parameters: m = 255 MeV, µ = 637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5 MeV and Λ = 306 MeV, able to reproduce the pion decay constant f P DG π = 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='50(1)(3)(13) MeV [93] (recall that the pion charge radius results to be rch = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='663 fm [75], in excellent agreement with rP DG ch = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='659 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='004 fm [94]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' A re- markable cancellation among the contributions takes place, and this represents a common feature for all the integrated quantities generated by the uTMDs we are considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In the right panel, one can see the comparison between the quark PDF, uS(AS)(ξ) and the LF-valence PDF, resulting from the one-to-one relation between the LF-projected BS amplitude and the valence amplitude of the Fock expansion of the pion state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In particular, the LF-valence PDF (see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [62, 76]), is given by uLF val(ξ) = � ∞ 0 dγ (4π)2 � |ψ↑↓(γ, z)|2+|ψ↑↑(γ, z)|2� , (32) where ξ = (1−z)/2, ψ↑↓(γ, z) is the anti-aligned com- ponent of the LF-valence amplitude and ψ↑↑(γ, z) the aligned one (of purely relativistic nature having an orbital angular momentum equal to 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' These ampli- tudes are suitable combinations of the LF-projected scalar functions φi(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The integral on ξ of LF-valence PDF gives the probability of the valence state in the Fock expansion and amounts to Pval = � 1 0 dξ uLF val(ξ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (33) The striking feature shown in the left panel is the shift toward low ξ of the quark PDF, so that for this quantity the symmetry ξ → 1 − ξ is slightly violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Analysing the shift and the gluon content The PDF calculations based on the BS-amplitude are able to capture an explicit gluonic effect, to be 9 taken distinct from the one responsible for the effec- tive mass of the constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In particular, the dif- ference between the two symmetric PDFs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' uS(ξ) and uLF val(ξ) (recall that has Pval = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='7), can be traced back to the non negligible probability of the higher Fock states (HFS), where a q¯q pair interacts by ex- changing any number of gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Interestingly, the dif- ference can be effectively described only by a factor, since it turns out that uLF val(ξ)/Pval largely overlaps uS(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Finally, also the small, but relevant, shift of the quark PDF with respect to uS(ξ) has to be as- cribed to the presence of HFS, as discussed in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' To get a qualitative view, we remind that the pion state can be, in principle, decomposed in Fock- components, which are schematically written in ladder approximation as |π⟩ = |q¯q⟩ + |q¯qg⟩ + |q¯q 2g⟩ + · · · (34) Due to the charge symmetry, each Fock-component is invariant by q ↔ ¯q, and hence the valence state |q¯q⟩ provides a symmetric contribution to uq(ξ), identified with uLF val(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The following terms contain gluons up to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In our model, the gluon has an effective mass about twice the quark mass, so that the HFS cu- mulative effect results in a small shift of the uq(ξ) peak at ξ < 1/2, as shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Ac- tually, a similar effect, related to the increasing mass of the remnant, can be also recognized in the nucleon, where one has a valence parton distribution with a peak around 1/3 due to the presence of the other two constituent quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In the case of the pion, the effect is small since the valence component |q¯q⟩ has 70% of probability (as generated by our dynamical calcula- tion), and hence is largely dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' To become more quantitative and illustrate this ef- fect, we schematically write the quark PDF by using the Fock expansion of the pion state, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (34), and inserting LF variables [79], one has uq(ξ) = ∞ � n=2 � n � i � d2ki⊥ (2π)2 � 1 0 dξi � ×δ (ξ − ξ1) δ � 1 − n � i=1 ξi � δ � n � i=1 ki⊥ � × ��Ψn(ξ1, k1⊥, ξ2, k2⊥, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=') ��2 , (35) where ξ1(2) is the longitudinal-momentum fraction of the quark (antiquark) in each Fock state, com- posed by a q¯q pair and n − 2 gluons, generated by the iteration of the one-gluon exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Moreover, Ψn(ξ1, k1⊥, ξ2, k2⊥, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=') is the probability amplitude of the corresponding Fock component and fulfills a nor- malization condition that follows from the one of the pion state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In the n-th state one has ξ1 = 1 − ξ2 − n � g=3 ξg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (36) Since ξi > 0 for massive particles, the average value of ξ1 starts to decreases while the number of gluons increases, as quantitatively shown in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Looking at the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 1, one can realize that while the valence term, with probability Pval = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='7, has a peak at ξ1 = ξ2 = 1/2, given the symmetry of |Ψ2(ξ1, k1⊥, ξ2, k2⊥)|2 all the HFS shift the peak to ξ1 < 1/2, and decrease the tail, due to the constraint of the overall normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' This is reflected in the evaluation of the first moment (recall ξq ≡ ξ1) ⟨ξq⟩ = Pval ⟨ξq⟩val + � n>2 Pn ⟨ξq⟩n = Pval ⟨ξq⟩val + (1 − Pval) ⟨ξq⟩HF S , (37) where Pn is the probability of the n-th Fock state beyond the valence one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (37) is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='35, since 1/2 is weighted by Pval, and the rest is weighted by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Notice that for each HFS, normalized to 1, one has ⟨ξq⟩n = 1 − ⟨ξ¯q⟩n − n � i=3 ⟨ξgi⟩n = 1 − ⟨ξ¯q⟩n − (n − 2)⟨ξg⟩n , (38) where the gluon bosonic nature leads to the factor n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The actual value of the first moment of uq(ξ) is ⟨ξq⟩ = � 1 0 dξ � ∞ 0 dγ ξ f q 1 (γ, ξ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='471 , (39) that amounts to an average of ⟨ξq⟩HF S equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' We can further analyse ⟨ξq⟩HF S, aiming at extract- ing a quantitative estimate of the exchanged-gluon contribution, ⟨ξg⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' From the momentum sum rule Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (37), and recalling Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (36), we get ⟨ξq⟩HF S = 1 1 − Pval � n>2 Pn ⟨ξq⟩n = 1 − ⟨ξ¯q⟩HF S − ⟨ξg⟩ , (40) where ⟨ξg⟩ = 1 1 − Pval � n≥3 Pn(n − 2) ⟨ξg⟩n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (41) 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='8 1 ξ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='8 ξ u( ξ) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='75 1 ξ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5 ξ u( ξ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (Color online).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Pion longitudinal distributions, with different scales (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dashed line: ξ uS(ξ), with an assigned initial scale equal to 360 MeV, and first moment equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Solid line: ξ uq(ξ), with a deduced scale equal 389 MeV, obtained by using a backward evolution of the first moment ⟨ξq⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dot-dashed line ξ uq(ξ) backward-evolved from 389 MeV to 360 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Comparison with the experimental data at the scale 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Solid line: evolved uS(ξ) starting from 360 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dot-dashed line: evolved uq(ξ), starting from 389 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dashed line: DSE calculation from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 5 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dotted line: BLFQ result at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='0 GeV [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Shaded area: LQCD calculation extracted via Mellin moments from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Full squares: reanalyzed data by using the ratio between the fit 3 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [98], evolved to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2 GeV, and the experimental data [52], at each data point (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [76] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Moreover, since each Fock component fulfills the charge symmetry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' q ↔ ¯q, the corresponding quark and antiquark momentum densities are equal and hence for the Mellin moments one has ⟨ξk q ⟩HF S = ⟨ξk ¯q ⟩HF S (this property does not imply the charge symmetry of the total density, given the presence of the gluon contribution, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (37)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (40), it follows that the gluon contribution reads ⟨ξg⟩ = 1 − 2 ⟨ξq⟩HF S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (42) Then, in our model one has ⟨ξg⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' We should note that i) ⟨ξq⟩ > ⟨ξq⟩HF S, as it should be, and ii) the massive gluons carry 20% of the HFS momentum fraction and contribute to the total longitudinal frac- tion by 6% (recalling that PHF S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' This result indicates that the exchanged gluons in the pion are not soft (differently from the ones considered in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [99] where the subtraction of the effect due to soft gluons is advocated for getting a symmetric PDF from the LF projected BS amplitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' It has to be emphasized that the above analysis, made transparent by the adopted LF variables, is valid in any gauge (both covariant gauges or the light-cone one), and the only difference is the amount of the shift one gets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The possibility to regain the full gauge- invariance by taking into account the additional gluon exchanges that could affect the interaction between the knocked-out quark and the spectator one (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', the analysis of the gauge-invariance and the hand-bag contribution in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [14, 100]) will be explored else- where.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The real test of the longitudinal dof is obviously the comparison between the PDF and the experimen- tal data [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' As it is shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [76], after evolving uS(ξ) from an assigned initial scale of 360 MeV (sug- gested by the inflection point of the effective running charge αs(Q2)) to the scale of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2 GeV, as given by the reanalysis in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [101], the result compares very satisfactorily with the experimental data extracted by taking into account logarithmic resummation effects in the hard part of the Drell-Yan cross-section, as per- formed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Moreover, we have achieved a nice agreement with other dynamical calculations, such as the Dyson-Schwinger result of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [95], the basis light-front quantization calculation of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [102, 103], and also the recent LQCD outcomes of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In particular, both the overall shape and, importantly, the tail for ξ → 1, gives great confidence in our for- malism, and encourages the further steps we have un- dertaken in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 11 0 2 4 6 8 10 γ/m 2 10 4 10 3 10 2 10 1 10 0 10 1 D⊥(γ)/D⊥(0) 0 2 4 6 8 10 γ/m 2 10 3 10 2 10 1 10 0 10 1 m 2 f S 1(γ,ξ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (Color online) Left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Normalized pion transverse distribution function, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (43), vs γ/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The normalization is given by D⊥(0) = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='945 GeV−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Thick solid line: full calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dashed line: the same as the full line, but times (γ/m2)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dash-dotted line: the same as the full line, but times (γ/m2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dash-double-dotted line: exponential form e−γ/(m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='42)2, with the parameter from Table 1 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [35], corresponding to a Gaussian Ansatz for f1(γ, ξ) (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Right Panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Pion unpolarized transverse-momentum distribution f S 1 (γ, ξ), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (10), for ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Solid line: full calculation as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dashed line: LF constituent quark model [35, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dash-dotted line: LF wave function from DSE calculations [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dash-Double-dotted line: NJL model [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The adopted quark mass m = 255 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 2, one can observe a further comparison, in- volving the product ξ u(ξ), that sheds more light on the link between the shift of the peak and the gluon dynamics taken explicitly into account in the ladder kernel of the BSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In particular, we get a scale of 389 MeV for uq(ξ), the solid line in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 1, by backward-evolving its first moment, ⟨ξq⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='471 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (39)), to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5, the first moment of uS(ξ), that has an assigned hadronic scale of 360 MeV, as above mentioned and thoroughly discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In the right panel, the comparison at 360 MeV between ξ uS(ξ) and the backward-evolved ξ uq(ξ) shows that the effect of the interaction taken into account in the ladder BSE is reproduced at large extent by applying a leading-order DGLAP evolution with an effective running charge as suggested in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [104] and already applied to our PDF in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' This is not surpris- ing once we remind that the dressing of the quark- gluon vertex, as expressed by the effective charge, is governed by the same interaction kernel present in the BSE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' the q¯q amputated T-matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The left panel shows the comparison at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2 GeV between the evolved uS(ξ), starting from the scale of 360 MeV, and the evolved uq(ξ), starting from the scale of 389 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Nicely, the difference is even smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Transverse degree of freedom In the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 3, it is shown the transverse distribution defined by D⊥(γ) = � 1 0 dξ f S 1 (γ, ξ) = � 1 0 dξ f q 1 (γ, ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (43) It has to be pointed out that the integration on ξ elim- inates the anti-symmetric term f AS 1 (γ, ξ), and there- fore one gets the same transverse distribution also by using f q 1 (γ, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In order to emphasize the analysis of the general pattern, we have presented D⊥(γ)/D⊥(0), so that the widely adopted exponential or power-like fall-off can be readily compared to our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In addition, in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 3 one can find: i) an exponential form D⊥(γ)/D⊥(0) = e−γ/(m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='42)2, with the parameter given in Table 1 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [35], corresponding to the so-called Gaussian Ans¨atz (re- call γ = |k⊥|2), amounting to a factorized form for f S 1 (γ, ξ) ∼ uS(ξ)e−γ/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='422) very often adopted in phe- nomenological studies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ii) our full results multiplied by (γ/m2)2 and iii) our full results multiplied by (γ/m2)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Not surprisingly, a power-like shape pro- vides a better approximation to the dynamically cal- culated D⊥(γ), but the proper power is different from 12 the one expected by the action of only a one-gluon exchange, that should govern the ultraviolet behavior and lead to a (γ/m2)2 (as suggested by a general- ized counting rule in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [105]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Indeed, the adopted form-factor featuring the extension of the quark-gluon interaction vertex (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (7)) generates a different power-like fall-off, namely (γ/m2)4, as already pointed out in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [88, 89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Finally, it is worth noticing that, unlike the PDF, the two terms in f S 1 (γ, ξ) containing derivatives of the delta-function do not contribute, as it is discussed at the end of Appendix C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 3, it is presented the quantitative comparison between f S 1 (γ, ξ) at ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5 and some phenomenological outcomes from i) the ap- proach based on the LF wave function obtained by using the DSE calculation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [45];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ii) the LF con- stituent quark-model of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [35, 56];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' iii) the NJL model with Pauli-Villars regulator as given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' For γ/m2 → 0, there are remarkable differences that, indeed, are present also on the tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' This last feature impacts the value of ⟨γ/m2⟩ 1 2 , as shown in Ta- ble I, where, for the sake of completeness, the value of uS(ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5) and the pion charge radius are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' As can be expected, the larger the average transverse moment, the smaller the radius of charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The current model has the smaller ⟨γ/m2⟩ 1 2 (of the order of the infrared scale ΛQCD, effectively incorpo- rated in the QCD-inspired choice of our parameters) which in turn leads to a larger charge radius, in agree- ment with the experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The average value ⟨γ/m2⟩ (with m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='255 MeV), uS(ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5) and the pion charge radius are presented for: i) f S 1 (γ, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5) from the present approach (NIR+BSE);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ii) the outcome from the LF wave function obtained by using DSE calculation [45] (LFDSE);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' iii) the LF constituent quark-model of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [35, 56] (LFCQM) and iv) the NJL with Pauli-Villars regulator [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (Re- call that the most recent PDG value of the charge radius is rP DG ch = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='659 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='004 fm [94]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ⟨γ/m2⟩ 1 2 uS(ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5) rch [fm] NIR+BSE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='663 LFDSE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='590 LFCQM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='572 NJL 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='557 In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 4, the uTMD f S 1 (γ, ξ) is shown in full, in order to appreciate the main features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' i) the peak at ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5 for running γ/m2, ii) the vanishing val- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (Color online) Pion unpolarized transverse- momentum distribution f S 1 (γ, ξ), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (10), at the initial scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The normalization is � 1 0 dξ � ∞ 0 dγ f S 1 (γ, ξ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ues at the end-points and iii) the order of magnitude fall-off already for γ/m2 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Comparing to other ap- proaches, one can notice the sharp difference with the results from the LF constituent model in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [56] and the LF holographic framework, like in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [40, 57] where a double-humped structure is found due to the ξ-dependence in the holographic wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Also the value at ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5 and small γ/m2 is substantially lower than ours (almost an order of magnitude less).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Differently, the shape of our f S 1 (γ, ξ) is more similar, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' without any double-humped structure, to the one obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [45], where the pion LF-wave function is determined from a beyond rainbow-ladder Dyson- Schwinger equations (DSE) in Euclidean space, by ex- ploiting the γ-dependent moments in ξ and a suitable parametrization of the BS-amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' THE SUBLEADING-TWIST UTMDS In this Section we present the numerical results for (T-even) uTMDs beyond the leading-twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The de- tailed expressions can be found in the Appendix E, but it is useful to recall that the decomposition in symmetric and antisymmetric combinations adopted for f1(γ, ξ) remains still valid, as well as the relations 13 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='75 1 ξ 2 0 2 4 6 8 e( ξ) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='75 1 ξ 0 1 2 3 ξe q( ξ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (Color online) Left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Pion unpolarized collinear PDFs: i) eq(ξ) (solid line), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (46), ii) eS(ξ) (dashed line) and eAS(ξ) (dotted line), Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' It is also shown eq EoM(ξ) (dash-dotted line), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Quark unpolarized collinear PDFs: ξ eq(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Solid line: full calculation as in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dashed line: m/M uq(ξ), with uq(ξ) shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Double-dot-dashed line: ξ eq EoM(ξ), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' with the quark and anti-quark contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' As introduction to the outcomes of our dynamical approach, it is worth anticipating that the comparison between full calculations and naive estimates one can infer from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (17) by using a valence approximation of the leading-twist f1(γ, ξ), highlights the inspiring statement one can read in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [77]: the higher-twist distributions are naturally related to multiparton dis- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The role of the exchanged gluons becomes definitely clear through a remarkable shift of the peak in all the sub-leading uTMD we have analyzed, as already discussed in the previous Section, as well as through the sharp difference with the naive estimates, which exclude the effect of the one-gluon exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Twist-3 uTMD: e(γ, ξ) In the frame where P⊥ = 0 and hence P + = M, by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (27), (B19), (B20), (B21) and (B22), with i = 1 and the functions b1 n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj given in Table VII, one gets the twist-3 uTMDs eS(AS)(γ, ξ), decomposed as follows eS(AS)(γ, ξ) = E0(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + Ed(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) +E2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + E3d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) , (44) where the functions in the rhs are given in Ap- pendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Longitudinal degree of freedom In the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 5, the following collinear PDFs are shown e(S,AS)(ξ) = � ∞ 0 dγ e(S,AS)(γ, ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (45) and eq(ξ) = eS(ξ) + eAS(ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (46) Moreover, in the spirit of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [35], we also present the collinear PDF, eq EoM(ξ), obtained by integrating the first line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (17), but disregarding the gluon contribution, viz eq EoM(ξ) ∼ m Mξ � ∞ 0 dγ f q 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='EoM(γ, ξ) ∼ m Mξ uLF val(ξ) Pval , (47) where uLF val(ξ)/Pval, normalized to 1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (33)), approximates the integral of f q 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='EoM(γ, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The large difference between our eq(ξ) and (m/Mξ)uLF val(ξ)/Pval indicates the sizable role of the gluon contribution from the HFS generated by our dynamical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In addition, one should point out that the strength of eq(ξ) is spread out on the whole range of ξ, and not concentrated at the end-point ξ = 0 as QCD investiga- tions indicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The latter feature leads to the singular 14 0 2 4 6 8 10 γ/m 2 10 4 10 3 10 2 10 1 10 0 E⊥(γ)/E⊥(0) 0 2 4 6 8 10 γ/m 2 10 3 10 2 10 1 10 0 10 1 m 2 e S(γ,ξ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (Color online).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Normalized transverse distribution function E⊥(γ)/E⊥(0) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (49)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dotted line: full calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Solid line: D⊥(γ)/D⊥(0) for the sake of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dash-double-dotted line: the same as in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Pion unpolarized transverse-momentum distribution eS(γ, ξ), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (44), for ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Solid line: full calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dashed line: LF constituent quark model of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [35, 56], but multiplied by m/(M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (47)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dash-dotted line: the same as the dashed line but with the LF wave function from DSE calculations [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dash-Double-dotted line: the same as the dashed line but with the NJL model [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The adopted quark mass m = 255 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' contribution given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (15) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [85], for a detailed discussion, but notice that the focus is on the nucleon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 5, the comparison be- tween ξ e(ξ) and the other two approximations: i) (m/M)f q 1 (ξ) and ii) (m/M)uLF val(ξ) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (47)) is carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The relevance of such a comparison is given by the possibility of more directly assessing the gluon role, since the factor ξ eliminates the singular term present in the QCD analysis of e(ξ), and one remains with the mass contribution (m/M)f q 1 (ξ) and the term from the quark-gluon-antiquark correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Still within the QCD framework (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [85]), the moments ⟨ξn⟩eq, for n ≤ 2, read as follow � dξ eq(ξ) = σπ mcur � dξ ξ eq(ξ) = mcur M � dξ ξ2 eq(ξ) = mcur M � dξ ξ f q 1 (ξ) , (48) and, for n > 2, they receive contributions not only from the (n−1)-th moment of f q 1 (γ, ξ) , but also from the n-th moment of the twist-3 contribution pertain- ing to the quark-gluon-antiquark correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Given TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The moments ⟨ξn⟩eq of the quark twist- 3 eq(γ, ξ) for n < 4, and the ratio R(n, eq, f q 1 ) = ⟨ξn⟩eq/⟨ξn−1⟩fq 1 (it is assumed ⟨ξ−1⟩fq 1 = ⟨ξ0⟩fq 1 = 1, and the values of ⟨ξ1⟩fq 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='471 and ⟨ξ2⟩fq 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='266 have been numerically evaluated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' n 0 1 2 3 ⟨ξn⟩eq 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='814 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='445 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='292 R(n, eq, f q 1 ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='814 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='943 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='10 the highly non trivial dynamical content of the eq(ξ) moments, it is interesting to show the results obtained with our dynamical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In Table II, both the moments up to n = 3 and the ratio R(n, eq, f q 1 ) = ⟨ξn⟩eq/⟨ξn−1⟩f q 1 are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In particular, as to the first two moments, to get rid of the dependence upon mcur it is helpful to compare the result obtained by multiplying the zero-th and the first moment, (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (48)), with final outcome σπ/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The estimate of σπ at the leading order of the chiral expansion leads to σπ/M = 1/2, as satisfacto- rily confirmed by the lattice calculations in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [84], where σlat π = 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2 MeV, for M = 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='3 MeV 15 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (Color online) Pion unpolarized transverse- momentum distribution eS(γ, ξ), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (45), at the initial scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' and mcur ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='9 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Eliminating the current quark mass, that is outside our approach, through the above product, we get σπ/M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='78, instead of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Such a conspicuous difference is surely influenced by the different distribution of the eq(ξ) strength, as al- ready mentioned, and points to a needed enrichment of the gluon dynamics in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' However, it is worth mentioning that for a simple non relativis- tic constituent quark model one has σNR π = 2m, so that σNR π /M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='64 (with m = 255 MeV the con- stituent mass), almost twice the result obtained in the BS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In QCD, the ratios R(1, eq, f q 1 ) and R(2, eq, f q 1 ) are equal and amount to mcur/M (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (48)), while R(3, eq, f q 1 ) = mcur/M + ∆3 g, where ∆3 g contains the contribution from the twist-3 gluonic contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In our calculation, the ratios for n = 1, 2 are al- most equal, but different from the naive expectation m/M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='82 with the adopted m = 255 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The difference with the third ratio indicates the onset of the contribution from the twist-3 gluonic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Transverse degree of freedom The transverse dof can be analyzed globally by in- troducing the following transverse distribution func- tion, as already accomplished with the leading-twist uTMD, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' E⊥(γ) = � 1 0 dξ eS(γ, ξ) = � 1 0 dξ eq(γ, ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (49) In the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 6, it is presented our calcu- lation and the ratio D⊥(γ)/D⊥(0) to show the simi- lar fall-off, as generated from gluon dynamics and the form-factor featuring the quark-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' A more close view of the decreasing as a function of γ/m2 is provided by the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 6, where it is shown the comparison between our calculation of e(γ, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5) and the outcomes obtained by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (47) with i) the LF wave function from the constituent quark model of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [35, 56], ii) the LF wave function from DSE calculations [45] and iii) the PDF from the NJL model [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The differences again point to the role of the interaction in the various approaches, and highlight the relevance of an experimental investiga- tion of the transverse dof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 7, the full dependence of eS(γ, ξ) is pre- sented, displaying a double-hump shape that for larger γ/m2 becomes smoother and smoother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Twist-3 uTMD: f ⊥(γ, ξ) In an analogous way, for i = 2 and using Table VIII, one gets the twist-3 f ⊥S(AS)(γ, ξ), with the following decomposition f ⊥S(AS)(γ, ξ) = P0(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + Pd(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) +P2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) (50) where the above functions are given in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Notice that in this case P2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Longitudinal degree of freedom In the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 8, the following collinear PDFs are shown f ⊥S(AS)(ξ) = � ∞ 0 dγ f ⊥S(AS)(γ, ξ) , (51) and the corresponding quark combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' As a refer- ence, it is also presented f ⊥q EoM(ξ), obtained from the 16 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='75 1 ξ 2 1 0 1 2 3 4 5 f ⊥( ξ) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='75 1 ξ 0 1 2 ξ f ⊥( ξ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (Color online) Left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 5, but for f ⊥q(ξ), f ⊥S(ξ) and f ⊥AS(ξ), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (51), and f ⊥q EoM(ξ) as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Quark unpolarized collinear PDFs ξ f q⊥(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Solid line: full calculation as in left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dashed line: ξ f q⊥(ξ) obtained by using the second line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (17) and our f q 1 (ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Double-dot-dashed line: the same as the dashed line but using the valence approximation of the PDF, uLF val(ξ), with norm equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 0 2 4 6 8 10 γ/m 2 10 4 10 3 10 2 10 1 10 0 P⊥(γ)/P⊥(0) 0 2 4 6 8 10 γ/m 2 10 4 10 3 10 2 10 1 10 0 10 1 m 2 f S⊥(γ,ξ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (Color online).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Normalized transverse distribution function P⊥(γ)/P⊥(0) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (53)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dotted line: full calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Solid line: D⊥(γ)/D⊥(0) for the sake of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dash-double-dotted line: the same as in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Pion unpolarized transverse-momentum distribution f S⊥(γ, ξ), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (50), for ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Solid line: full calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dashed line: by using f1(γ, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 3 from the LF constituent quark model of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [35, 56] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' the second line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (17), without the gluonic term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dash-dotted line: the LF wave function from DSE calculations [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Dash-Double-dotted line: the NJL model [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The adopted quark mass m = 255 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (17), without the gluon term, as follows f ⊥q EoM(ξ) ∼ 1 ξ � ∞ 0 dγf ⊥q EoM(γ, ξ) ∼ uLF val(ξ) ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (52) For the sake of completeness, in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 8, the product ξ f q⊥(ξ) is compared to f q 1 (ξ) and uLF val(ξ) that represents the approximation to f ⊥q EoM(ξ) as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Also for f q⊥(ξ), the full calcu- 17 lation substantially differs from approximated evalu- ations, prompting further investigation of the gluon contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Transverse degree of freedom Also for f ⊥(γ, ξ), we introduce the transverse dis- tribution function, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P⊥(γ) = � 1 0 dξ f S⊥(γ, ξ) = � 1 0 dξ f q⊥(γ, ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (53) In the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 9, a comparison, built with the same spirit as in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 6, is shown for the ratio P⊥(γ)/P⊥(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' A more detailed view of the fall-off can be gained from the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 9, where f S⊥(γ, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5) is compared with the results obtained by using i) the LF constituent quark model of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [35, 56] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' the second line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (17), without the gluonic term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ii) the LF wave function from DSE calculations [45] and iii) the NJL model [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (Color online) Pion unpolarized transverse- momentum distribution f S⊥(γ, ξ), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Finally in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 10, the full dependence of f S⊥(γ, ξ) is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Also in this uTMD, the double-hump shape decreases when γ/m2 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' To summarize, a coherent view of the tail in γ is plainly provided by Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 3, 6 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Namely, the interaction taken into account in the ladder kernel to- gether with the extended structure of the quark-gluon vertex governs the fall-off of both the leading and subleading-twist uTMDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Therefore, the quantitative estimates obtained through our dynamical model, in Minkowski space, is shown to be in a favorable po- sition to provide insights into the interplay between transverse dof and the role of gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' CONCLUSIONS The twist-2 (leading) and twist-3 (subleading) un- polarized (T-even) transverse-momentum dependent parton distribution functions have been calculated for the pion within an approach based on the solution of the Bethe-Salpeter equation in Minkowski space, namely, within a genuinely relativistic quantum-field theory framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' We achieved this goal by exploit- ing the Nakanishi integral representation of the BS- amplitude in order to get actual solution of the homo- geneous BSE, in ladder approximation, through a sys- tem of integral equations that determine the Nakan- ishi weight functions relevant for the problem under scrutiny [62, 88, 89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' After obtaining the pion elec- tromagnetic form factor [75], and the pion PDF [76], we extended the yield of our approach by exploring the dependence of the parton distributions upon the transverse momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' This additional step has its- own importance in view of the planned experimental efforts to achieve a fully three-dimensional investiga- tion of hadrons (mainly of the nucleon and, more chal- lenging, the pion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The relevant message one gets from our calculations is given by the essential role of the gluon exchange, that cannot be captured by purely phenomenological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The joint use of the Fock expansion of the pion state, allows us to shed light on the gluonic con- tent of the quark PDF obtained through the BS ampli- tude, even determining a quantitative estimate, ∼ 6% of the average longitudinal momentum fraction ⟨ξq⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Moreover, the latter analysis explains also the source of the small, but theoretically relevant, shift between the uq(ξ) and the PDF that fulfills the charge symme- try (an issue already investigated within the Dyson- Schwinger approach, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=', in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [99], where a dif- ferent interpretation was proposed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' As to the trans- verse degree of freedom, a power-like fall-off of the transverse distributions, obtained by integrating on ξ the uTMDs, is supported by the one-gluon exchange interaction that governs the ultraviolet region, accord- ing to our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' This outcome could suggest 18 to reconsider an exponential or Gaussian Ansatzes for describing the high-momentum content (γ >> m2) of the uTMDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Summarizing, our approach can be placed among those in which the dynamics can be studied in Minkowski space and in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Moreover, the additive construction of the interaction kernel allows one to address step-by-step recognized effects, achiev- ing an implementation of the dynamics in a controlled way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ACKNOWLEDGMENTS E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' gratefully thanks INFN Sezione di Roma for providing the computer resources to perform all the calculations shown in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' acknowledges the partial support of CNPQ under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 438562/2018-6 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 313236/2018-6, and the partial support of CAPES under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 88881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='309870/2018-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' thanks the financial sup- port from the Brazilian Institutions: CNPq (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 308486/2015-3), CAPES (Finance Code 001) and FAPESP (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 2017/05660-0 and 2019/07767- 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' acknowledges the support of FAPESP Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 2016/25143-7 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 2018/21758-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' This work is a part of the project Instituto Nacional de Ciˆencia e Tecnologia - F´ısica Nuclear e Aplica¸c˜oes Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 464898/2014-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Appendix A: Traces In this Appendix the traces in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (10), (18) and (19), are explicitly evaluated, presenting the expressions of the functions ai ℓ,j(k−, γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) and bi n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓ,j(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' For the sake of convenience, let us rewrite the generic trace entering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (23) TrS(AS) i (γ, ξ) = − i 2 � Tr � S−1(k − P 2 )¯Φ(k, P) Oi Φ(k, P) � + ηS(AS) i Tr � S−1(k + P 2 )Φ(k, P) Oi ¯Φ(k, P) �� = � ℓj ai ℓj(k−, γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) φℓ(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P) φj(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' P) , (A1) with Oi and ηS(AS) i given by O0 = γ+ , ηS(AS) 0 = ∓1 , O1 = 1 , ηS(AS) 1 = ±1 , O2 = M |k⊥|2 k⊥ · γ⊥ , ηS(AS) 2 = ±1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (A2) To proceed one has to insert the expression of the BS-amplitude, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (20), and the definitions, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (7) and (21), in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Then one gets the results shown in Tables III, IV and V, for ai ℓj(k−, γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' It is also useful to organize the functions ai ℓj in powers of k− for preparing the integration on such a variable (cfr Appendix B), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ai ℓj(k−, γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = 2M � bi 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + bi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) k− 2M + bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) � k− 2M �2 +bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) � k− 2M �3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (A3) In Tables VI, VII and VIII, one can find the expressions for bi n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) 19 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Non vanishing coefficients a0 ℓj(k−, γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='a0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='a0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(AS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2Mz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='−8m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='−2 m z − 4 m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M k− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='− 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M γ − M z2 − 2 z k− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='−Mz − 2k− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='−4k− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M z + 2 k− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='−8 γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M − 2zk− − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M (k−)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2mz + 4 m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M k− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M2 + z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='+ z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2k− + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2M (k−)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M2 + z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='k− − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M (k−)2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M2 (k−)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M2 + z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='+ 2 m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M zk− + 2 m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M2 (k−)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M2 + z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='+ z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2k− + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2M (k−)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2 z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='M2 + z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='+ z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2 k− + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='2M (k−)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Non vanishing coefficients a1 ℓj(k−, γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ij a1 ℓj(S) a1 ℓj(AS) 11 −4m 12 4M 2Mz − 4k− 13 −2M � 4 γ M2 + z2 4 � − 2zk− − 2 M (k−)2 22 −4m 24 −2M � 4 γ M2 + z2 4 � − 2zk− − 2 M (k−)2 33 m � 4 γ M2 + z2 4 � + z m M k− + m M2 (k−)2 34 z M 2 � 4 γ M2 + z2 4 � − � 4 γ M2 − z2 4 � k− − z 2M (k−)2 − 1 M2 (k−)3 M � 4 γ M2 + z2 4 � + zk− + 1 M (k−)2 44 m � 4 γ M2 + z2 4 � + z m M k− + m M2 (k−)2 TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Non vanishing coefficients a2 ℓj(k−, γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ij a2 ℓj(S) a2 ℓj(AS) 11 −4M 13 8m 14 4M 2zM −4k− 22 4M 23 −2Mz + 4k− −4 M 24 −8m 33 M � 4 γ M2 + z2 4 � + zk− + 1 M (k−)2 44 −M � 4 γ M2 + z2 4 � − zk− − 1 M (k−)2 20 TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Non vanishing coefficients b0 n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ij b0 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S) b0 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S) b0 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S) b0 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(AS) b0 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(AS) b0 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(AS) b0 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(AS) 11 1 0 0 z 0 0 0 12 −4m/M 0 0 0 0 0 0 13 −zm/M −4m/M 0 0 0 0 0 14 −4γ/M 2 − z2/2 −2 z 0 −z/2 −2 0 0 22 1 0 0 0 −4 0 0 23 z/2 2 0 −4γ/M 2 −2z −8 0 24 0 0 0 zm/M 4m/M 0 0 33 γ/M 2 + z2/16 z/2 1 0 − � 4γ/M 2 + z2/4 � −2z −4 34 0 0 0 (m/M) � 4γ/M 2 + z2/4 � 2zm/M 4m/M 0 44 γ/M 2 + z2/16 z/2 1 (z/4) � 4γ/M 2 + z2/4 � z2/2 z 0 TABLE VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Non vanishing coefficients b1 n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ij b1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S) b1 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S) b1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S) b1 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S) b1 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(AS) b1 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(AS) b1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(AS) 11 −2m/M 0 0 0 0 0 0 12 2 0 0 0 z −4 0 13 0 0 0 0 −4γ/M 2 − z2/4 −2z −4 22 −2m/M 0 0 0 0 0 0 24 − � 4γ/M 2 + z2/4 � −2z −4 0 0 0 0 33 (m/2M) � 4γ/M 2 + z2/4 � zm/M 2m/M 0 0 0 0 34 (z/4) � 4γ/M 2 + z2/4 � − � 4γ/M 2 − z2/4 � −z −4 2γ/M 2 + z2/8 z 2 44 (m/2M) � 4γ/M 2 + z2/4 � zm/M 2m/M 0 0 0 0 TABLE VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Non vanishing coefficients b2 n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ij b2 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S) b2 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S) b2 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S) b2 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(AS) b2 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(AS) 11 −2 0 0 0 0 13 0 0 0 4(m/M) 0 14 2 0 0 z −4 22 2 0 0 0 0 23 −z 4 0 −2 0 24 −4m/M 0 0 0 0 33 (1/2) � 4γ/M 2 + z2/4 � z 2 0 0 44 −(1/2) � 4γ/M 2 + z2/4 � −z −2 0 0 21 Appendix B: The light-front projection The Appendix is devoted to the integration over the variable k− in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (26), that for the sake of clarity we rewrite F i ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = � ∞ −∞ dk− 2π ai ℓj(k−, γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) φℓ(k, P) φj(k, P) = 2M � ∞ −∞ dk− 2π φℓ(k, P) φj(k, P) × � bi 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + bi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) k− 2M + bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) � k− 2M �2 +bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) � k− 2M �3� , (B1) where the quantities ai ℓj(k−, γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) and bi n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The first step (see also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [62, 75, 76]) is to introduce the NIR of φℓ(k, P), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (22), and then apply the Feynman parametrization as follows φℓ(k, P)φj(k, P) = 30 � 1 0 dv � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ v2(1 − v)2 gℓ(γ′, z′) gj(γ′′, z′′) � k−α − β + iϵ �6 , (B2) where α = M 2 � λ(v) − z � , β(zλ(v)) = γ + κ2 + M 2 4 zλ(v) + vγ′ + (1 − v)γ′′ , λ(v) = vz′ + (1 − v)z′′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B3) Hence, the following general expression, one can straightforwardly deduce from well-known result in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [106] (corresponding to the case m = 0) is useful for performing the relevant integrals � ∞ −∞ dk− 2π (k−)m � α k− − β + iϵ �n = i (n − m − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (−1)m+1 � −β + iϵ �n−m−1 δ(m)(α) (B4) where δ(m)(α) = ∂mδ(α)/∂αm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Finally, combining the results in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B2) and (B4), one gets F i ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = −24i � 1 0 dv v2(1 − v)2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Gℓj(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) × � bi 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) δ(˜α) [−β(zλ(v)) + iϵ]5 − 1 4M 2 bi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) δ′(˜α) [−β(zλ(v)) + iϵ]4 + 1 12M 4 bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) δ ′′(˜α) [−β(zλ(v)) + iϵ]3 − 1 24M 6 bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) δ ′′′(˜α) [−β(zλ(v)) + iϵ]2 � , (B5) where Gℓj(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) = gℓ(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) gj(γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) ˜α = 2 M α = λ(v) − z , (B6) 22 and the derivatives of the delta function is with respect to ˜α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Recalling that ∂ ˜α/∂z = −1, one can also write F i ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = −24i � 1 0 dv v2(1 − v)2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Gℓj(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) × � bi 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) δ(λ(v) − z) [−β(zλ(v)) + iϵ]5 + 1 4M 2 bi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) [−β(zλ(v)) + iϵ]4 ∂ ∂z δ(λ(v) − z) + 1 12M 4 bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) [−β(zλ(v)) + iϵ]3 ∂2 ∂z2 δ(λ(v) − z) + 1 24M 4 bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) [−β(zλ(v)) + iϵ]2 ∂3 ∂z3 δ(λ(v) − z) � , (B7) Finally, by using f(z) ∂m ∂zm δ(λ(v) − z) = m � k=0 cmk ∂k ∂zk � f (m−k)(z) δ(λ(v) − z) � , (B8) where the coefficient cmk can be obtained by repeatedly applying the Leibniz rule for the product of functions and f (m−k) indicates the (m − k)-th derivative (with f (0)(z) ≡ f(z)), one recasts Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B7) in a form more suitable for the further elaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In practice, one trades derivatives on the delta functions with derivatives on the functions bi n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ij(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)), getting F i ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = −24i � 1 0 dv v2(1 − v)2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Gℓj(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) × � δ(λ(v) − z) � bi 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) [−β(zλ(v)) + iϵ]5 − 1 4M 2 ∂ ∂z � bi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) [−β(zλ(v)) + iϵ]4 � + 1 12M 4 ∂2 ∂z2 � bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) [−β(zλ(v)) + iϵ]3 � − 1 24M 6 ∂3 ∂z3 � bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) [−β(zλ(v)) + iϵ]2 �� + 1 4M 2 ∂ ∂z � bi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) δ(λ(v) − z) [−β(zλ(v)) + iϵ]4 − 2 3M 2 ∂ ∂z � bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) [−β(zλ(v)) + iϵ]3 � δ(λ(v) − z) + 1 2M 4 ∂2 ∂z2 � bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS) [−β(zλ(v)) + iϵ]2 � δ(λ(v) − z) � + 1 12M 4 ∂2 ∂z2 � bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) δ(λ(v) − z) [−β(zλ(v)) + iϵ]3 − 3 2M 2 ∂ ∂z � bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) [−β(zλ(v)) + iϵ]2 � δ(λ(v) − z) � + 1 24M 6 ∂3 ∂z3 � bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) δ(λ(v) − z) [−β(zλ(v)) + iϵ]2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B9) Hence, by taking into account the expressions of bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) and bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)), given in Tables VI, VII and VIII, one can drop some derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In particular the second derivative of bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) and all the derivatives of bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)), obtaining F i ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = −24i � Fi 0ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + Fi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + Fi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + Fi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) � (B10) 23 where Fi 0,ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = � 1 0 dv v2(1 − v)2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Gℓj(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) × δ(λ(v) − z) [−β(z2) + iϵ]5 � bi 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + 1 4 � β(z2) M 2 ∂ ∂z bi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) − z bi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) � + 1 16 � z2 bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) − 2z β(z2) M 2 ∂ ∂z bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS) � −z3 64 bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S(AS)) � , (B11) Fi1 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = 1 8M 2 ∂ ∂z �� 1 0 dv v2(1 − v)2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Gℓj(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) × δ(λ(v) − z) [−β(z2) + iϵ]4 � 2bi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) − z bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + 4 3 β(z2) M 2 ∂ ∂z bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) +3 8z2 bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S(AS)) �� , (B12) Fi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = 1 12M 4 ∂2 ∂z2 �� 1 0 dv v2(1 − v)2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Gℓj(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) × δ(λ(v) − z) [−β(z2) + iϵ]3 � bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) − 3 4z bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S(AS)) �� , (B13) and Fi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S(AS)) 24M 6 ∂3 ∂z3 �� 1 0 dv v2(1 − v)2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Gℓj(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) × δ(λ(v) − z) [−β(z2) + iϵ]2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B14) Collecting the above results, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (23) becomes T S(AS) i (γ, ξ) = iNc 8 1 (2π)2 × � ℓj � 1 −1 dz δ(z − (1 − 2ξ)) F i ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = = 3Nc (2π)2 � ℓj � Fi 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + Fi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + Fi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + Fi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B15) It is also useful for getting more explicit expressions to perform the integral on v in the Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B11), (B12) and (B13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' This can be accomplished by using the following result � 1 0 dv v2(1 − v)2 δ[vz′ + (1 − v)z′′ − z] = v2 0(1 − v0)2 Θ(v0) Θ(1 − v0) |z′ − z′′| = v2 0(1 − v0)2 ∆(z, z′, z′′) (B16) 24 with ∆(z, z′, z′′) = Θ(z′ − z)Θ(z − z′′) − Θ(z′′ − z)Θ(z − z′) z′ − z′′ , v0 = z − z′′ z′ − z′′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B17) The combination of the theta-functions implements the constraint 0 ≤ v0 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Moreover, notice that i) simultaneously changing the signs of z, z′ and z′′ the function ∆(z, z′, z′′) does not change sign, this reflects the symmetry with respect ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='5, as implemented through the charge symmetry in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (23);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ii) ∆(z, z′, z′′) is even under the exchange z′′ → z′ and in the limit z′′ − z′ = ϵ → 0 one has lim ϵ→0 ∆(z, z′, z′ + ϵ) = δ(z − z′) , (B18) so that the singularity can be addressed without particular problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' By taking into account Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B16), and the symmetries with respect to the transformation z′ → z′′ and γ′ → γ′′, one gets Fi 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = 2 � ∞ 0 dγ′ � ∞ 0 dγ′′ � +1 −1 dz′ � +1 −1 dz′′ v2 0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) z′ − z′′ × ¯Gℓj(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) [−β0(z2) + iϵ]5 � bi 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + 1 4 � β(z2) M 2 ∂ ∂z bi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) − z bi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) � + 1 16 � z2 bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) − 2z β(z2) M 2 ∂ ∂z bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS) � −z3 64 bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S(AS)) �� , (B19) Fi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = 1 4M 2 ∂ ∂z �� ∞ 0 dγ′ � ∞ 0 dγ′′ � +1 −1 dz′ � +1 −1 dz′′v2 0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) z′ − z′′ × ¯Gℓj(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) [−β0(z2) + iϵ]4 � 2bi 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) − z bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + 4 3 β(z2) M 2 ∂ ∂z bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) +3 8z2 bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S(AS)) �� , (B20) Fi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = 1 6M 4 ∂2 ∂z2 �� ∞ 0 dγ′ � ∞ 0 dγ′′ � +1 −1 dz′ � +1 −1 dz′′ v2 0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) z′ − z′′ × ¯Gℓj(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) [−β0(z2) + iϵ]3 � bi 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) − 3 4z bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S(AS)) �� , (B21) and Fi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) = bi 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(S(AS)) 12M 6 ∂3 ∂z3 �� ∞ 0 dγ′ � ∞ 0 dγ′′ � +1 −1 dz′ � +1 −1 dz′′ v2 0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) z′ − z′′ × ¯Gℓj(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) [−β0(z2) + iϵ]2 � (B22) 25 where the functions bi n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) are given in the Tables of Appendix A and β0(z2) = γ + κ2 + z2 M 2 4 + v0γ′ + (1 − v0)γ′′ , ¯Gℓj(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) = gℓ(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2)gj(γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) + gℓ(γ′′, z′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2)gj(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) 2 , v0 = z − z′′ z′ − z′′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B23) Also notice that for a bound state one has β0(z2) = γ + κ2 + M 2 4 z2 + v0γ′ + (1 − v0)γ′′ ≥ m2 − M 2 4 (1 − z2) ≥ κ2 > 0 , (B24) and therefore no poles are associated to such a quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' It should be pointed out that the presence of the theta-functions, that ensure 0 ≤ v0 ≤ 1, prevents singular behaviors, shrinking the area of integration in the space z′ ⊗ z′′, when z′ → z′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Interestingly, in the Appendix C 1, it is shown that only Fi0 ℓj (γ, z), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' without derivative of the delta-function, contributes to the norm of the twist-2 uTMD f1(γ, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Appendix C: The leading-twist uTMD f S(AS) 1 (γ, ξ) In this Appendix, the symmetric and anti-symmetric combinations of the quark and antiquark leading-twist uTMDs are explicitly given and their relevant features discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' By specializing the expressions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (27), one can write f S(AS) 1 (γ, ξ) = IN(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + Id(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + I2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + I3d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) (C1) where the four contributions are obtained from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B19), (B20), (B21) and (B22), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Inserting the functions b0 n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='ℓj(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) listed in in the first three columns of Table VI in Appendix A, one gets the following non vanishing symmetric contributions, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' IN(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = 3Nc 2π2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′v2 0(1 − v0)2 Θ(z′ − z) Θ(z − z′′) (z′ − z′′) [−β0(z2) + iϵ]5 × �� ¯G11(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + ¯G22(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − 4 m M ¯G12(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) � +β0(z2) + 8γ 8M 2 � ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + ¯G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − 4 ¯G14(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� , (C2) Id(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = − 3Nc 4π2M 2 ∂ ∂z �� +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ v2 0(1 − v0)2 Θ(z′ − z) Θ(z − z′′) (z′ − z′′) [−β0(z2) + iϵ]4 × � 2 m M ¯G13(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z ¯G14(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − ¯G23(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� , (C3) and I2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = Nc 8π2M 4 ∂2 ∂z2 �� +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ v2 0(1 − v0)2 Θ(z′ − z) Θ(z − z′′) (z′ − z′′) [−β0(z2) + iϵ]3 × � ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + ¯G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� , (C4) 26 with β0(z2), ¯Gℓj and v0 given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The symmetry property under the transformation z → −z can be eas- ily demonstrated, recalling also that under the exchange z′ → −z′′ and γ′ → γ′′ the functions ¯Gℓj(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) do not change, since the NWFs gi(γ, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' κ2) are even for i = 1, 2, 4 and odd for i = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Moreover, under z → −z and z′ → −z′′ one also has v0 → (1−v0), so that β0(z2) remains unchanged, as well as Θ(z′ −z) Θ(z −z′′)/(z′ −z′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The anti-symmetric combinations are IN(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' AS) = 3 Nc 2π2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Θ(z′ − z)Θ(z − z′′) z′ − z′′ × v2 0(1 − v0)2 [−β0(z2) + iϵ]5 � z ¯G11(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z ¯G22(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) +β0(z2) + 8γ 2M 2 � − ¯G23(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z 4 ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + m M ¯G34(γ′, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z 4 ¯G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� , (C5) Id(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' AS) = Nc 2M 2π2 ∂ ∂z �� +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Θ(z′ − z)Θ(z − z′′) z′ − z′′ v2 0(1 − v0)2 [−β0(z2) + iϵ]4 × � −3 2 ¯G14(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − 3 ¯G22(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + 3z 2 ¯G23(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + 3 m M ¯G24(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) −β0(z2) + 3 γ M 2 ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + β0(z2) 2 M 2 ¯G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� (C6) I2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' AS) = Nc 8π2M 4 ∂2 ∂z2 �� +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Θ(z′ − z)Θ(z − z′′) z′ − z′′ v2 0(1 − v0)2 [−β0(z2) + iϵ]3 × � −8 ¯G23(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + 4 m M ¯G34(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z ¯G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� (C7) I3d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' AS) = − Nc 4π2M 6 ∂3 ∂z3 �� +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Θ(z′ − z)Θ(z − z′′) z′ − z′′ × v2 0(1 − v0)2 [−β0(z2) + iϵ]2 ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) � (C8) The anti-symmetry with respect to the transformation z → −z can be easily shown by using the properties listed below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The normalization of f S 1 (γ, ξ) While the integration on ξ and γ of f AS 1 (γ, ξ) triv- ially yields zero, since the anti-symmetry in z trans- lates in an anti-symmetry in ξ with respect to ξ = 1/2, it is interesting to analyze how to recover the normal- ization of f S 1 (γ, ξ), once the BS-amplitude is properly normalized as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' To proceed in the most easy way, let us perform a step backward, and reinsert the dependence upon δ(z − λ(v)) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (C2), (C3) and (C4) by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Then one has 27 IN(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = 3Nc 4π2 � 1 0 dv v2(1 − v)2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ × δ(λ(v) − z) [−β0(z2) + iϵ]5 �� G11(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + G22(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − 4 m M G12(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) � +β0(z2) + 8γ 8M 2 � G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − 4G14(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� , (C9) Id(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = − 3Nc 8π2M 2 ∂ ∂z �� 1 0 dv v2(1 − v)2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ × δ(λ(v) − z) [−β0(z2)) + iϵ]4 � 2 m M G13(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z G14(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − G23(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� (C10) and I2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = Nc 16π2M 4 ∂2 ∂z2 �� 1 0 dv v2(1 − v)2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ × δ(λ(v) − z) [−β0(z2) + iϵ]3 � G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (C11) Performing the integration on γ and ξ = (1−z)/2, one gets the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (C9), one recovers the standard normalization of the BS-amplitude in ladder approximation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (12) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [62]), viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' � ∞ −∞ dξ � ∞ 0 dγ IN(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = − 3Nc 32π2 � 1 0 dv v2(1 − v)2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ × 1 [κ2 + M 2 4 z2 + vγ′ + (1 − v)γ′′]4 �� G11(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + G22(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − 4 m M G12(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) � +κ2 + M 2 4 z2 + vγ′ + (1 − v)γ′′ 2M 2 � G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − 4G14(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� , (C12) while the other two terms do not contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In fact, let us first integrate on z and take into account that in δ(λ(v) − z) one has 1 ≥ λ(v) ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' One gets for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (C10) � ∞ −∞ dξ � ∞ 0 dγ Id(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = − 3 16π2M 2 � ∞ 0 dγ �� 1 0 dv v2(1 − v)2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ × δ(λ(v) − z) [−β(z2) + iϵ]4 � 2 m M G13(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z G14(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − G23(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) ��z=+∞ z=−∞ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (C13) 28 For Eq (C11) one has � ∞ −∞ dξ � ∞ 0 dγ I2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = 1 32π2M 4 � ∞ 0 dγ � ∂ ∂z � 1 0 dv v2(1 − v)2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ × δ(λ(v) − z) [−β(zλ(v)) + iϵ]3 � G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) ��z=+∞ z=−∞ = = 1 32π2M 4 � ∞ 0 dγ � ∂ ∂z � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ v2 0(1 − v0)2 z′ − z′′ × Θ(z′ − z)Θ(z − z′′) − Θ(z′′ − z)Θ(z − z′) [−β0(z2) + iϵ]3 � G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) ��z=+∞ z=−∞ , (C14) where in the last step Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B16) has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' by explicitly performing the derivative on z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' given ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='by (recall β0(z2) = γ + κ2 + z2M 2/4 + v0γ′ + (1 − v0)γ′′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='∂z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='0(1 − v0)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='[−β0(z2) + iϵ]3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='Θ(z′ − z)Θ(z − z′′) − Θ(z′′ − z)Θ(z − z′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='= ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='∂z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='0(1 − v0)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='[−β0(z2) + iϵ]3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='Θ(z′ − z)Θ(z − z′′) − Θ(z′′ − z)Θ(z − z′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='0(1 − v0)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='[−β0(z2) + iϵ]3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='−δ(z′ − z)Θ(z − z′′) + Θ(z′ − z)δ(z − z′′) + δ(z′′ − z)Θ(z − z′) − Θ(z′′ − z)δ(z − z′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='= ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='∂z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='0(1 − v0)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='[−β0(z2) + iϵ]3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='Θ(z′ − z)Θ(z − z′′) − Θ(z′′ − z)Θ(z − z′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='0(1 − v0)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='[−β0(z2) + iϵ]3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='−δ(z′ − z) + δ(z − z′′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (C15) one can straightforwardly see that the derivative vanishes for z = ±∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' being z′ and z′′ ∈ [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' 1] Hence � ∞ −∞ dξ � ∞ 0 dγ I2d(γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (C16) Two comments are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' First, the leading-twist uTMD is vanishing outside the range ξ ∈ [0, 1], and hence one can restrict the integration on z between [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' It is easy to prove that the same results can be obtained also in this case, recalling that z′ and z′′ are in the same range, and in the last line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (C15) one has v0 = (z − z′)/(z′ − z′′) and 1 − v0 = (z′′ − z)/(z′ − z′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Second, the integrand in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (C13) and (C14) should lead to contributions to the transverse distribution D⊥(γ) = � ∞ 0 dξ f S 1 (γ, ξ) , (C17) but from the above results one can see that they are vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' Appendix D: The parton distribution function and the leading-twist uTMD By integrating f S 1 (γ, ξ) on γ one gets the symmetric parton distribution function uS(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In particular, one has uS(ξ) = � ∞ 0 dγ f S 1 (γ, ξ) = uS N(ξ) + uS d (ξ) + uS 2d(ξ) (D1) where the three contributions are obtained by in- tegrating on γ of the three quantities IN(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S), 29 Id(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) and I2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (C9), (C10) and (C11), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' By using the result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B16) and the integrals � ∞ 0 dγ 1 [−β0(z2) + iϵ]n = (−1)n n − 1 1 [D(z, v0, γ′, γ′′)]n−1 , � ∞ 0 dγ γ [−β0(z2) + iϵ]4 = 1 6 1 [D(z, v0, γ′, γ′′)]2 , � ∞ 0 dγ γ [−β0(z2) + iϵ]5 = − 1 12 1 [D(z, v0, γ′, γ′′)]3 , (D2) where D(z, v0, γ′, γ′′) = κ2 + M 2 4 z2 + v0γ′ + (1 − v0)γ′′ , (D3) one writes uS N(ξ) = � ∞ 0 dγ IN(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = − 3 8π2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Θ(z′ − z)Θ(z − z′′) z′ − z′′ v2 0(1 − v0)2 [D(z, v0, γ′, γ′′)]4 × �� ¯G11(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + ¯G22(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − 4 m M ¯G12(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) � +D(z, v0, γ′, γ′′) 2M 2 � ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + ¯G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − 4 ¯G14(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� , (D4) uS d (ξ) = � ∞ 0 dγ Id(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = − 1 4π2M 2 ∂ ∂z � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Θ(z′ − z)Θ(z − z′′) z′ − z′′ × v2 0(1 − v0)2 [D(z, v0, γ′, γ′′)]3 � 2 m M ¯G13(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z ¯G14(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − ¯G23(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) � , (D5) and uS 2d(ξ) = � ∞ 0 dγ I2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = − 1 16π2M 4 ∂2 ∂z2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Θ(z′ − z)Θ(z − z′′) z′ − z′′ × v2 0(1 − v0)2 [D(z, v0, γ′, γ′′)]2 � ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + ¯G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (D6) If the BS-amplitude has the standard normalization [80], after integrating uS N(ξ) one gets � +1 0 dξ uS(ξ) = � +1 0 dξ uS N(ξ) = 1 (D7) from i) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (C12), (C13) and (C16) and ii) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (12) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The anti-symmetric PDF uAS(ξ) is given by uAS(ξ) = � ∞ 0 dγ f AS 1 (γ, ξ) = uAS N (ξ) + uAS d (ξ) + uAS 2d (ξ) + uAS 3d (ξ) (D8) 30 where uAS N (ξ) = � ∞ 0 dγ IN(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' AS) = −3 Nc 8π2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Θ(z′ − z)Θ(z − z′′) z′ − z′′ × v2 0(1 − v0)2 [D(z, v0, γ′, γ′′)]4 � z ¯G11(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z ¯G22(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + 2D(z, v0, γ′, γ′′) M 2 × �z 4 ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z 4 ¯G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − ¯G23(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + m M ¯G34(γ′, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� , (D9) uAS d (ξ) = � ∞ 0 dγ Id(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' AS) = Nc 6M 2π2 ∂ ∂z �� +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Θ(z′ − z)Θ(z − z′′) z′ − z′′ × v2 0(1 − v0)2 [D(z, v0, γ′, γ′′)]3 � −3 2 ¯G14(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) −3 ¯G22(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + 3z 2 ¯G23(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + 3 m M ¯G24(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − 3D(z, v0, γ′, γ′′) M 2 ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) +3D(z, v0, γ′, γ′′) 4 M 2 ¯G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� , (D10) uAS 2d (ξ) = � ∞ 0 dγ I2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' AS) = − Nc 16π2M 4 ∂2 ∂z2 �� +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Θ(z′ − z)Θ(z − z′′) z′ − z′′ × v2 0(1 − v0)2 [D(z, v0, γ′, γ′′)]2 � −8 ¯G23(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + 4 m M ¯G34(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) +z ¯G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� , (D11) and uAS 3d (ξ) = � ∞ 0 dγ I3d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' AS) = − Nc 4π2M 6 ∂3 ∂z3 �� +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ Θ(z′ − z)Θ(z − z′′) z′ − z′′ × v2 0(1 − v0)2 D(z, v0, γ′, γ′′) ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (D12) Appendix E: Twist-3 unpolarized TMDs The Appendix presents the explicit expressions of the twist-3 and twist-4 uTMDs, obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (27) and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (B19), (B20), (B21) and (B22), by using the Tables VII and VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' In particular for the twist-3 eS(AS)(γ, ξ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' for i = 1 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (27), one has eS(AS)(γ, ξ) = E0(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + Ed(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + E2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + E3d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) (E1) 31 where the symmetric combinations are E0(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = 3Nc 2π2 � ∞ 0 dγ′ � ∞ 0 dγ′′ � +1 −1 dz′ � +1 −1 dz′′ v2 0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) (z′ − z′′) [−β0(z2) + iϵ]5 × {−2 m M ¯G11(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + 2 ¯G12(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − 2 m M ¯G22(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) +8γ + β0(z2) 2M 2 � − ¯G24(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + m 2M ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z 2 ¯G34(γ′, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + m 2 M ¯G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� , (E2) Ed(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = − Nc 4M 4π2 ∂ ∂z � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ v2 0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) (z′ − z′′)[−β0(z2) + iϵ]4 × � 6γ + β0(z2) � ¯G34(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) , (E3) E2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = Nc 4π2M 4 ∂2 ∂z2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ v2 0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) (z′ − z′′)[−β0(z2) + iϵ]3 × � −2 ¯G24(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + m M ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z ¯G34(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + m M ¯G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) � , (E4) and E3d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = − Nc 4π2M 6 ∂3 ∂z3 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ v2 0(1 − v0)2 × Θ(z′ − z)Θ(z − z′′) (z′ − z′′)[−β0(z2) + iϵ]2 ¯G34(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) (E5) The anti-symmetric combinations are E0(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' AS) = 3Nc 2π2 � ∞ 0 dγ′ � ∞ 0 dγ′′ � +1 −1 dz′ � +1 −1 dz′′ v2 0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) (z′ − z′′) [−β0(z2) + iϵ]5 × � 2z ¯G12(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − 8γ + β0(z2) 4M 2 � 2 ¯G13(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − ¯G34(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� , (E6) Ed(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' AS) = − 3Nc 2π2M 2 ∂ ∂z �� ∞ 0 dγ′ � ∞ 0 dγ′′ � +1 −1 dz′ � +1 −1 dz′′v2 0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) (z′ − z′′) [−β0(z2) + iϵ]4 × ¯G12(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) � , (E7) E2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' AS) − Nc 4π2M 4 ∂2 ∂z2 �� ∞ 0 dγ′ � ∞ 0 dγ′′ � +1 −1 dz′ � +1 −1 dz′′ v2 0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) (z′ − z′′) [−β0(z2) + iϵ]3 × � 2 ¯G13(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − ¯G34(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� , (E8) and E3d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' AS) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (E9) 32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' The twist-3 uTMD f ⊥(γ, ξ) For i = 2, one has the twist-3 uTMD f ⊥(γ, ξ), with the following decomposition f ⊥S(AS)(γ, ξ) = P0(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + Pd(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + P2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) + P3d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S(AS)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' (E10) The symmetric contributions are given by P0(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = −3 Nc π2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ v2 0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) (z′ − z′′)[−β0(z2) + iϵ]5 × � ¯G11(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − ¯G14(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − ¯G22(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z ¯G23(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) +2m M ¯G24(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − 8γ + β0(z2) 8M 2 � ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − ¯G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) �� , (E11) Pd(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = 3 Nc 2π2 M 2 ∂ ∂z � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ v2 0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) (z′ − z′′)[−β0(z2) + iϵ]4 × ¯G23(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) , (E12) P2d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = Nc 4π2M 4 ∂2 ∂z2 � +1 −1 dz′ � ∞ 0 dγ′ � +1 −1 dz′′ � ∞ 0 dγ′′ v2 0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) (z′ − z′′)[−β0(z2) + iϵ]3 × � ¯G33(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − ¯G44(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) � , (E13) and P3d(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' S) = 0 (E14) The anti-symmetric contributions read P0(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' AS) = 3Nc π2 � ∞ 0 dγ′ � ∞ 0 dγ′′ � +1 −1 dz′ � +1 −1 dz′′ v2 0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) (z′ − z′′) [−β0(z2) + iϵ]5 × � 2 m M ¯G13(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) + z ¯G14(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) − ¯G23(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) � , (E15) and Pd(γ, ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' AS) = − 3Nc 2π2M 2 ∂ ∂z �� ∞ 0 dγ′ � ∞ 0 dγ′′ � +1 −1 dz′ � +1 −1 dz′′v2 0(1 − v0)2 Θ(z′ − z)Θ(z − z′′) (z′ − z′′) [−β0(z2) + iϵ]4 × ¯G14(γ′, z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFJT4oBgHgl3EQfpSw0/content/2301.11599v1.pdf'} +page_content=' γ′′, z′′) � , (E16) and P2d(γ, ξ;' 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b/-tE5T4oBgHgl3EQfRw5F/content/tmp_files/2301.05523v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..19d26794439e7d0d48dc8de3def8d3463d3e02a7 --- /dev/null +++ b/-tE5T4oBgHgl3EQfRw5F/content/tmp_files/2301.05523v1.pdf.txt @@ -0,0 +1,720 @@ +arXiv:2301.05523v1 [cond-mat.str-el] 13 Jan 2023 +Bi layer properties in the Bi–FeNi GMR-type structures +probed by spectroscopic ellipsometry +Natalia Kovaleva,1, ∗ Dagmar Chvostova,2 Ladislav Fekete,2 and Alexandr Dejneka2, † +1Lebedev Physical Institute, Russian Academy of Sciences +Leninsky prospect 53, Moscow 119991, Russia +2Institute of Physics, Academy of Sciences of the Czech Republic +Na Slovance 2, Prague 18221, Czech Republic +(Dated: January 16, 2023) +Abstract +Bismuth (Bi) having a large atomic number is characterized by the strong spin-orbit coupling +(SOC) and is a parent compound of many 3D topological insulators (TIs). The ultrathin Bi films +are supposed to be 2D TIs possessing the nontrivial topology, which opens the possibility of devel- +oping new efficient technologies in the field of spintronics. Here we aimed at studying the dielectric +function properties of ultrathin Bi/FeNi periodic structures using spectroscopic ellipsometry. The +[Bi(d)–FeNi(1.8 nm)]N GMR-type structures were grown by rf sputtering deposition on Sitall-glass +(TiO2) substrates. The ellipsometric angles Ψ(ω) and ∆(ω) were measured for the grown series +(d=0.6, 1.4, 2.0, 2.5 nm, N = 16) of the multilayered film samples at room temperature for four an- +gles of incidence of 60◦, 65◦, 70◦, and 75◦ in a wide photon energy range of 0.5-6.5 eV. The measured +ellipsometric angles, Ψ(ω) and ∆(ω), were simulated in the framework of the corresponding mul- +tilayer model. The complex (pseudo)dielectric function spectra of the Bi layer were extracted. +The GMR effects relevant for the studied Bi–FeNi MLF systems are estimated from the optical +conductivity zero-limit (optical GMR effect). The obtained results demonstrate that the Bi layer +possesses the surface metallic conductivity induced by the SOC effects, which is strongly enhanced +on vanishing the semimetallic-like phase contribution on decreasing the layer thickness, indicating +its nontrivial 2D topology properties. +∗Electronic address: kovalevann@lebedev.ru +†Electronic address: dejneka@fzu.cz +1 + +I. +INTRODUCTION +The relativistic effect of spin–orbit (SOC) coupling is involved in the so-called Rashba +effect [1]. This phenomenon arises from the apparent loss of crystalline inversion symmetry +near the surface or heterojunction leading to the lifting of the spin degeneracy and generating +spin-polarized surface metallic states. In this respect, 3D (2D) topological insulators (TIs) +also exhibit spin-polarized surface metallic states due to SOC. However, contrary to the +Rashba effect, the surface metallic bands of a TI are determined by its bulk characteristics. +The TIs host metallic surface states in a bulk energy gap, which are topologically protected. +The surface (or interface) states of TIs can be topologically trivial or nontrivial. In the latter +case, for example, electrons cannot be backscattered by impurities. Bismuth (Bi) having a +large atomic number is characterized by the strong SOC and is a parent compound of many +3D TIs, such as Bi1−xSbx or Bi2Se3, even although 3D bulk Bi itself is topologically trivial. +The specific feature of the electronic band structure of bulk Bi having R¯3m rhombohedral +symmetry [2–4] is its inverted band gaps at both the Γ and M points of the Brillouin zone +due to the strong SOC. The uniqueness of Bi films associated with the surface metallic states +[5, 6] and the semiconductor-to-metal transition [7, 8] are well documented in the literature. +Theoretical analyses predict a 1-bilayer (BL) Bi(111) film to be a 2D TI [9, 10]. +If +there is no or weak inter-BL coupling, a stack of the odd-even 1-BL films will exhibit +nontrivial to trivial oscillations of topology (where the topological number ν [11] is equal +to 1 or 0, respectively). However, for the nontrivial topology in a stack of the 1-BL films, +the intermediate inter-BL coupling strength, which is, for example, higher than the van der +Waals strengths, is a mandatory condition. The direct (Γ point) and indirect band gap values +were calculated by Liu et al. as a function of the Bi film thickness [12]. It was established +that below 4BLs the film is a semiconductor with the direct gap open at the Γ point and +the positive indirect band gap leading to nontrivial topology peculiar for an intrinsic 2D TI. +Above 4BLs the indirect band gap becomes negative resulting in a semiconductor- semimetal +transition due to overlapping of two bands at the Fermi level around the Γ and M points. +This suggests that the Bi films from 5 to 8 BLs represent a 2D TI situated between two +trivial metallic surfaces [12]. +A comprehensive study of the associated SOC effects in ultrathin Bi layers opens the pos- +sibility of developing new efficient technologies in the field of spintronics. For this purpose, +2 + +here we aimed at studying the dielectric function properties of ultrathin periodic structures +Bi/Ni79Fe21, prepared by rf sputter deposition, which is one of the most common technologies +used to grow coatings and multilayered films (MLFs) exhibiting a giant magnetoresistance +(GMR) effect for various existing and modern nanotechnological applications. Earlier, we +have demonstrated that electronic band structure and surface electronic properties of ul- +trathin Bi layers in real GMR-type (Bi–FeNi)N MLF structures incorporating nanoisland +FeNi layers can be successfully studied by spectroscopic ellipsometry (SE) [13]. Here, by ap- +plying the elaborated SE approach, we investigate (Bi–FeNi) MLFs, where the thickness of +the FeNi layer was 1.8 nm, corresponding to the FeNi layer structural percolation threshold +[14, 15], and the Bi spacer layer was 0.6, 1.4, 2.0, and 2.5 nm thick, incorporating about 2, +4, 6, and 8 Bi(012)-type planes, respectively. We found that the Bi spacer layers have the +metallic surface conductivity, which demonstrates strongly enhanced metallicity properties +on vanishing the Bi semimetallic-like phase contribution on decreasing the layer thickness, +which can be constructive in finding new nontrivial 2D topology properties of the (Bi–FeNi) +GMR-type structures for their different nanotechnological applications. +II. +MATERIALS AND METHODS +The (Bi–FeNi)N MLFs were prepared in a sputter deposition system by cathode sput- +tering from 99.95% pure Bi and Fe21Ni79 targets in an alternative way. The base pressure +in a sputter deposition chamber was 2×10−6 Torr. The multilayers were deposited at ap- +proximately 80 ◦C in an argon atmosphere of 6× 10−4 Torr on to insulating glassy Sitall +(TiO2) substrates. We utilized the substrates having typical dimensions 15 × 5 × 0.6 mm3. +The nominal thicknesses of the FeNi and Bi layers were controlled by the layer deposition +times in accordance with the material deposition rates. A series consisting of four MLF +samples was prepared. In the series of the grown (Bi–FeNi)N samples, the nominal thickness +of the FeNi layer was 1.8 nm and the Bi layer thickness was of 0.6, 1.4, 2.0, and 2.5 nm, the +number N of the periodically repeated Bi/FeNi layers was 16. The thickness of the FeNi +layer was chosen to be 1.8 nm matching the structural percolation threshold [14, 15]. The +Bi layer thicknesses were chosen in such a way that the conditions for ferromagnetic (FM) +or antiFM coupling in the GMR-type structures would be optimized. To prevent degra- +dation, the deposited (Bi–FeNi)16/Sitall samples were covered with the 2.1 nm-thick Al2O3 +3 + +FIG. 1: AFM images (a–d) 5 × 5 µm2 and (e–h) 1 × 1 µm2 of the Al2O3/(Bi–FeNi)16/Sitall MLF +samples, where the nominal Al2O3 and FeNi layer thicknesses are 2.1 and 1.8 nm and the nominal Bi +layer thicknesses are 0.6, 1.4, 2.0, and 2.5 nm, respectively. The estimated surface RMS roughness +values are in (a-d) 3.6, 3.0, 3.1, and 5.2 nm and in (e-h) 3.2, 2.6, 2.7, and 5.3 nm, respectively. +(i,j) The typical height profiles for the MLF samples with the nominal Bi layer thicknesses of 0.6 +and 2.5 nm, respectively. +layer. The related [Bi–FeNi(0.8,1.2 nm)]N samples prepared by rf sputtering deposition onto +the Sitall substrates under similar conditions were investigated by X-ray diffraction (XRD) +as well as by X-ray reflectivity (XRR) experimental techniques in our previous study (see +Supplementary online information for the article [13]). The XRR spectra proved a good +periodicity and consistency with the corresponding nominal thicknesses of the FeNi and Bi +slices in the Bi/FeNi MLF structures, as well as relatively low interface roughness between +the constituent layers. The XRD characterization suggests (012)-type Bi plane orientation, +where the interlayer distance is 3.28 ˚A. It follows from this that in the studied MLF struc- +tures the Bi layers with a thickness corresponding to 0.6, 1.4, 2.0, and 2.5 nm incorporate +4 + +D!2SUce () +0 +500 +300 +400 +eoo +100 +008 +a0o +0001 +SO +10 +JO +(uw +SO +30D!2Sce (u) +0 +500 +300 +400 +ooa +100 +008 +a0o +0001 +2 +Heiapr (uw) +0 +2HGidpf 2GU20l +mn 0.00s +-34'2 UW +S4'0 UWHGidpf 2Gu20l +my o.r +mn 8.84- +401 UwHGidpf 26GU20l +mn 0.00s +mn 8.07- +mn 8.eHGidpf 26GU20l +my o.7 +mn c.er.HGidpf 26GU20l +mn 0.00s +n r.ar- +JI'SUwHeiauf 26u20l +my o.1 +mn &.rr- +mn c.srHGidpf 26u20l +mn 0.00s +mn r.rr- +mn f.crHeidpf 26u20k +my o.r +-531UWtwo, four, six, and eight Bi(012)-type planes, respectively. +In the present study, the surface morphology of the Bi–FeNi(1.8 nm) MLF samples, pre- +pared by rf sputtering deposition on the Sitall (TiO2) substrates, was studied at room +temperature using an ambient AFM (Bruker, Dimension Icon) in the Peak Force Tapping +mode with ScanAsyst Air tips (Bruker, k=0.4 N/m, nominal tip radius 2 nm). The SE mea- +surements for the investigated Al2O3/(Bi–FeNi)16/Sitall samples were performed at room +temperature in a wide photon energy range of 0.5 – 6.5 eV using a J.A. Woollam VUV-VASE +ellipsometer (see the scheme illustrating the SE study of the (Bi–FeNi)N MLFs in Fig. 1(a) +of Ref. [13]). The measured ellipsometry spectra are represented by real values of the angles +Ψ(ω) and ∆(ω), which are defined through the complex Fresnel reflection coefficients for +light-polarized parallel rp and perpendicular rs to the plane of incidence, tan Ψ ei∆ = rp +rs . +The ellipsometric angles, Ψ(ω) and ∆(ω), measured for the Bi–FeNi MLF samples were sim- +ulated using the multilayer model simulation available in the J.A. Woollam VASE software +[16]. From the multilayer model simulations, the (pseudo)dielectric function spectra of the +ultrathin 0.6, 1.4, 2.0, and 2.5 nm Bi layers and 1.8 nm FeNi layer inside the Bi–FeNi MLF +structures were extracted. The corresponding calculated optical conductivity spectra were +analyzed. +III. +RESULTS +A. +Atomic force microscopy study +The retrieved 5×5 µm2 and 1×1 µm2 AFM images of the Al2O3(2.1 nm)/[Bi(0.6, 1.4, 2.0, +2.5 nm)–FeNi(1.8 nm)]N/Sitall multilayered films (where the given layer thicknesses corre- +spond to their nominal values), presented in Figure 1a–h show discernable contrast because +of the available surface hight deviations. The surface roughness of the Sitall glass (TiO2) +substrates was investigated by us by AFM in our earlier publication [17]. The height profile +of the Sitall substrates (see Fig. 2a of Ref. [17]) demonstrated the height deviation within +the range 1-3 nm peculiar to the relatively large 0.3-1 µm lateral scale, which characterizes +the Sitall substrate surface roughness. From the AFM measurements on the areas 5×5 µm2 +and 1×1 µm2 the root-mean square (RMS) surface roughness values were evaluated, which +are presented in the caption to Figure 1. +The corresponding RMS roughness values are +5 + +notably higher for the Al2O3(2.1 nm)/[Bi(2.5 nm)–FeNi(1.8 nm)]16/Sitall MLF sample. The +smaller-scale (1 × 1 µm2) images clearly recognize a fine grainy-like structure of the surface +morphology, which seems to be characteristic for all studied film samples (see Figure 1e–h). +The typical grain size, being of about 50 nm, is notably larger for the FeNi(1.8 nm) – Bi MLF +sample incorporating the 2.5 nm-thick Bi layers, and, following the estimated RMS rough- +ness values, the average grain size decreases to about 20 nm with decreasing the Bi layer +thickness to 1.4 nm. As one can see from the typical height profiles presented in Figure 1i,j, +with decreasing the Bi layer thickness from 2.5 to about 0.6 nm, the surface morphology +becomes highly irregular due to the formation of conglomerates of nanoislands separated by +rather flat (relatively small roughness) areas of about 20 nm. +B. +Spectroscopic ellipsometry study of the ultrathin Bi–FeNi multilayer film samples +The ellipsometric angles Ψ(ω) and ∆(ω) were measured for the prepared Al2O3/(Bi– +FeNi)16/Sitall MLF samples at the angles of incidence of 60◦, 65◦, 70◦, and 75◦. Figure 2 +demonstrates the ellipsometric angles Ψ(ω) and ∆(ω) recorded at 65◦ and 70◦. To model +the contributions from free charge carriers and interband optical transitions, the complex +dielectric function ˜ε(ω) = ε1(ω) + iε2(ω) of the Bi and FeNi layers was interpreted in terms +of the Drude and Lorentz parts, respectively, +˜ε(E ≡ ℏω) = ǫ∞ − +AD +E2 + iEγD ++ +� +j +AjγjEj +E2 +j − E2 − iEγj +, +(1) +where ε∞ is the high-frequency dielectric constant, which takes into account the contribution +from the higher-energy interband transitions. The fitted Drude parameters were AD and free +charge carrier’s scattering rate γD. The fitted parameters of Lorentz bands were Ej, γj, and +Aj of the band maximum energy, the full width at half maximum, and the ε2 band height, +respectively. The obtained ellipsometric angles Ψ(ω) and ∆(ω) measured at different angles +of incidence of 60◦, 65◦, 70◦, and 75◦ were fitted for each sample simultaneously using the +J.A. Woollam VASE software [16] in the framework of the designed multilayer model. The +multilayer model for the studied Al2O3/(Bi–FeNi)/Sitall multiayers was constructed as it is +schematically presented in Figure 3, exactly so as the layers were deposited. In addition, we +attempted to take into account the roughness properties of the surface by using the conven- +tional approach of effective medium approximation (EMA) based on the (50% Al2O3–50% +6 + +FIG. 2: (a-d) Ellipsometric angles, Ψ(ω) and ∆(ω) (symbols), measured at the angles of incidence +of 65◦ and 70◦ for the Al2O3/[Bi(d)–NiFe(1.8 nm)]16/Sitall multilayered films where the Bi spacer +layer thicknesses d = 0.6, 1.4, 2.0, and 2.5 nm, respectively. The solid red, blue, green, and black +curves show the corresponding simulation results for the angle 65◦ by the dielectric function model +using Equation 1. +7 + +vacuum) Bruggeman model. The dispersion model for the Bi layers included three or four +Lorentz terms as well as the Drude part. The dispersion model for the 1.8 nm permalloy +layers incorporated in the studied MLF structures included the Drude term responsible for +the free charge carrier contribution and one Lorentz oscillator to account for the most pro- +nounced interband optical transition. In addition, the dielectric function spectra of the bare +Sitall substrate derived from our earlier SE studies [18, 19] were introduced to the elabo- +rated multilayer model. The dielectric response of the Al2O3 capping layer was represented +by the tabular complex dielectric function spectra [20]. The thicknesses of the Bi and FeNi +layers, as well as of the surface layers, were fitted. The unknown parameters were allowed to +vary until the minimum of the mean squared error (MSE) is reached. The best simulation +result for the studied [Bi(0.6, 1.4, 2.0, 2.5 nm)–FeNi(1.8 nm)]16 MLF samples corresponded +to the lowest obtained MSE values of 0.3843, 0.297, 0.2934, and 0.4508, respectively. The +good quality of the fit allowed us to estimate the actual Bi and FeNi layer thicknesses in the +MLFs under study. The quality of the fit is demonstrated by Figure 2, where we plotted the +measured ellipsometric angles along with the simulation results. The Drude and Lorentz +parameters resulting from the simulation of the Al2O3/[Bi(d)–FeNi(1.8 nm)]16/Sitall MLF +samples are given in Tables I and II, and the resulting ε1(ω) and ε2(ω) parts of the Bi and +FeNi (pseudo)dielectric function spectra are presented in Figure 4. +From Figure 4a,b one can see that the complex (pseudo)dielectric functions of the 0.6, 1.4, +2.0, and 2.5 nm thick Bi spacers inside the investigated Bi–FeNi MLFs demonstrate metal- +lic character. Moreover, the ε1(ω) function progressively decreases while the Bi thickness +decreases from 2.5–2.0 to 1.4 nm and the ε2(ω) increases at low photon energies, respec- +tively. According to our simulation results, we expect that the best metallicity properties +are demonstrated by the Bi layer in the [Bi(1.4 nm)–NiFe(1.8 nm)]16 structure. At the same +time, the complex (pseudo)dielectric functions of the thinnest 0.6 nm thick Bi layer look +somewhat different. Here, in addition to the low-energy metallic Drude response identified +by the characteristic behavior of the ε1(ω) and ε2(ω), the Lorentz band around 4–5 eV makes +an essential contribution to the dielectric function response (the corresponding Drude (AD +and γD) and Lorentz (Aj, Ej, and γj) parameters are listed in Table I). Next, being similar, +the dielectric functions of the 1.8 nm thick permalloy layers in the [FeNi–Bi(1.4, 2.0, 2.5 nm)] +MLFs are dominated by the ε2(ω) resonance and ε1(ω) antiresonance features, indicat- +ing the predominant contribution from the Lorentz oscillator peaking at around 3 eV (see +8 + +(a) +(b) +( ) +(d) +c +FIG. 3: +The multilayer model applied for the simulation of the Al2O3/[Bi(0.6, 1.4, 2.0, and +2.5 nm)–FeNi(1.8 nm)]16/Sitall samples. The Bi and FeNi thicknesses estimated from the model +simulations in (a) 0.684±0.037 nm and 2.082±0.116 nm, (b) 1.408±0.574 nm and 1.780±0.65 nm, +(c) 1.764±0.194 nm and 1.825±0.358 nm, and (d) 2.387±0.128 nm and 1.782±0.171 nm. Note good +agreement between the thicknesses of the FeNi and Bi layers estimated from the model simula- +tions and their respective nominal thickness values. The roughness and Al2O3 thicknesses esti- +mated from the model simulations in (a) 0.00±3.85 nm and 1.283±2.37 nm, (b) 0.000±4.97 nm and +4.967±2.17 nm, (c) 0.848±5.86 nm and 4.738±2.92 nm, and (d) 0.000±2.95 nm and 5.389±1.23 nm. +Figure 4c,d). +An upturn evident in the ε2(ω) at low photon energies indicates an addi- +tional Drude contribution, which is relatively less pronounced. Following our simulation +results, we expect the advanced metallicity properties of the FeNi layer in the [Bi(0.6 nm)– +NiFe(1.8 nm)]16 structure (see the corresponding Drude (AD and γD) and Lorentz (Aj, Ej, +and γj) parameters listed in Table II). +Figure 5a–d presents the evolution of the Bi intralayer optical conductivity, σ1(ω) = +ε2(ω)ω(cm−1)/60, upon decreasing the Bi spacer layer thickness in the [FeNi(1.8 nm) – +Bi(2.5, 2.0, 1.4, 0.6 nm)]16 structures, and Figure 5e–h shows the associated optical conduc- +tivity spectra of the 1.8 nm FeNi permalloy layer. Here, the contributions from the Drude +and Lorentz oscillators following the multilayer model simulations using Equation 1 are evi- +9 + +woge +0 +sti2(19VSl +mna.Oid) 19vsl +mn8.TingtS80.Sarmna.Oid +19vsl↑80.03 +91503 +Clε8S.T2lonap0'000wog6 +0 +sti219Vsl +mn8, Tingt +S1081.1arS +19Vsl +p!↓'4uw80↓.13 +91503 +Cl2lonap +40'000wogel +0 +lsti219Vsl +mn8, Tingt +328.1armno.Sid1043 +91503 +Cl881.2lonap +488.0woge +0 +ti219Vsl +mn8. Tingt +4S81.1arIS入Gl +mnc,.Sid188.S3 +91S03 +Cle88.2lonap +40'000FIG. 4: The complex (pseudo)dielectric function spectra, ε2(ω) and ε1(ω), of the (a,b) Bi layers and +(c,d) FeNi layers in the [Bi(d)–FeNi(1.8 nm)]16 structures shown for the Bi layer nominal thickness +values d = 0.6, 1.4, 2.0, and 2.5 nm by solid red, blue, green, and black curves, respectively. +TABLE I: Drude-Lorentz parameters for the Bi spacer layer in the [Bi(0.6, 1.4, 2.0, 2.5 nm)– +NiFe(1.8 nm)]16 multilayered films obtained from the model simulations of the dielectric functions +by using Equation 1. The values of Ej, γj, and γD are given in eV, and optical conductivity limit +σ1(ω→0) in Ω−1·cm−1. +Parameters 0.6 nm +1.4 nm +2.0 nm +2.5 nm +Drude +AD +46.(9)±4 +66.(7)±4 +24.(5)±4 +25.(1)±2 +γD +1.2(5)±0.09 1.51(0)±0.06 2.7(2)±0.4 +3.1(3)±0.2 +σ1(ω→0) +6300±540 +8970±540 +3290±540 +3370±270 +Lorentz +E1 +– +0.45(8)±0.05 0.35(9)±0.01 0.38(6)±0.004 +oscillator +A1 +– +15.(0)±6 +96.(0)±10 +70.(8)±2 +γ1 +– +0.52(6)±0.09 0.79(1)±0.02 0.67(6) +Lorentz +E2 +4.67 +5.31(5)±0.03 5.08(7)±0.04 4.77(5)±0.04 +oscillator +A2 +10.2(7)±0.6 2.53(2)±0.05 1.2(5)±0.1 +0.67(6)±0.08 +γ2 +4.2(1)±0.07 3.99(3)±0.07 3.4(7)±0.2 +2.5(5)±0.2 +Lorentz +E3 +11.1 +7.8 +7.7 +7.7 +oscillator +A3 +7.2 +4.1 +4.1 +4.1 +γ3 +8.9 +2.8 +2.8 +2.8 +10 + +TABLE +II: +Drude-Lorentz +parameters +for +the +1.8 nm +thick +NiFe +layer +in +the +[Bi(0.6, 1.4, 2.0, 2.5 nm)–NiFe]16 multilayered films obtained from the simulations of the model +dielectric function described by Equation 1. The values of E1, γ1, and γD are given in eV, and +optical conductivity limit σ1(ω→0) in Ω−1·cm−1. +Parameters 0.6 nm +1.4 nm +2.0 nm +2.5 nm +Drude +AD +33.(8)±2 +15.(0)±1 +21.(7)±2 +13.(1)±2 +γD +0.876(5)±0.04 2.8(2)±0.3 3.4(2)±0.4 3.1(3)±0.2 +σ1(ω→0) +4540±270 +2020±130 +2920±270 +1760±270 +Lorentz +E1 +1.87 +3.32 +3.32 +3.32 +oscillator +A1 +14.76 +14.28 +15.23 +14.74 +γ1 +3.62 +5.88 +5.65 +5.95 +dently demonstrated. The optical conductivity spectra of the Bi and FeNi layers follow the +main trends identified in their complex dielectric function spectra presented in Figure 4. +IV. +DISCUSSION +Initially, we would like to discuss GMR effects relevant for the studied MLF sys- +tems. +Our simulations of the dielectric functions for the 1.8 nm-thick NiFe layer inside +the [Bi(0.6,1.4,2.0,2.5 nm)–NiFe(1.8 nm)] MLFs show the presence of the Drude term com- +plemented with the pronounced Lorentz band located at around 2–3 eV (see Table II). From +the corresponding optical conductivity spectra presented in Figure 5e–h one can notice that +the associated Drude dc limit, σ1ω→0, displays an oscillating character (in agreement with the +results deduced for the corresponding Drude parameter AD, see Table II and Figure 6). We +can expect that the Bi spacer thicknesses for which the FeNi layers are preferentially antiFM +coupled in the studied MLFs are around 1.4 and 2.5 nm implying that the [Bi(1.4,2.5 nm)– +NiFe(1.8 nm)]16 film structures will exhibit a drop in the resistance (being negative magne- +toresistance) when exposed to an external magnetic field. It is well known from the literature +that the first antiFM maximum exhibits negative magnetoresistance of about 20%, while +the second antiFM maximum decreases to about 10%, and the presence of the third antiFM +maximum cannot confidently be retrieved (see, for example, Ref. [21] and references therein). +11 + +FIG. 5: The intralayer optical conductivity, σ1(ω) = ε2(ω)ω[cm−1]/60, for the (a-d) Bi layers and +(e-h) FeNi layers in the [Bi(d)–FeNi(1.8 nm)]16 structures shown for the Bi layer nominal thickness +values d = 2.5, 2.0, 1.4, and 0.6 nm by solid curves (a,e) black, (b,f) green, (c,g) blue, and (d,h) +red, respectively. The contributions from the Drude term and the Lorentz oscillator in (a-d) are +displayed by the yellow and cyan shaded area. In (e-h) the Drude term for the FeNi layers is +displayed by the magenta shaded area. Shown by the dotted curves are the summary of the Drude +and Lorentz contributions. +12 + +Using a simple model of a two-current series resistor [22], the magnetoresistance ∆R +R can be +estimated as +∆R +R = 100% +(α − β)2 +4 +� +α + +dBi +dF eNi +� � +β + +dBi +dF eNi +�, +(2) +where dBi and dF eNi are the thicknesses of Bi and FeNi layers, and α = +↓ρF eNi +ρBi +and β = +↑ρF eNi +ρBi +are the ratios of the resistivity in the FeNi layer to that in the Bi layer in the spin down +and spin up current channel, respectively. Exploiting values for ρ = σ−1 +1ω→0 estimated for +the 1.4 nm Bi and 1.8 nm FeNi layers from the current model simulations (see Table I and +II), namely, ρBi= +1 +8970Ω·cm, ↓ρF eNi= +1 +2020Ω·cm and ↑ρF eNi= +1 +4540Ω·cm (the latter estimate is +given by the FM coupling for the 0.6 nm Bi spacer), we obtain α=4.4 and β=2.0. Then, +using Equation (2) we have ∆R +R =10%. This means that the 1.4 nm Bi spacer corresponds +to the second antiFM maximum. Following the same approach for the 2.5 nm Bi spacer, +where ρBi= +1 +3370Ω·cm, ↓ρF eNi= +1 +1760Ω·cm and ↑ρF eNi= +1 +2920Ω·cm (corresponding to the FM cou- +pling for the 2.0 nm Bi spacer), we obtain α=1.9 and β=1.2. Using Equation (2), we have +∆R +R =1.4%, which may correspond to the very weakly pronounced third antiFM maximum. +From the analysis presented above, we may expect that the first antiFM maximum corre- +sponding to the magnetoresistance of about 20% occurs for the Bi spacer thickness of about +0.9 nm, which is in agreement with the results presented in Ref. [21]. +Further, +in +the +XRD +patterns +of +the +investigated +Al2O3/[Bi(1.4,2.0,2.5 nm)– +NiFe(1.8 nm)]16/Sitall film samples, the peak of the R¯3m crystalline Bi phase is identi- +fied at 2θ ≈ 26.2◦ suggesting (012) orientation of the Bi layers, which is characterized by +the interlayer distance of 3.28 ˚A. Using STM and reflection high-energy electron diffraction +(RHEED) techniques, it was shown that initial growth of Bi(012)-type films occurs in the +form of islands with the height increment of about 6.6 ˚A, indicating even-number layer sta- +bility leading to the laterally flat morphology of the Bi(012)-type islands [23]. Consequently, +we can expect that the 0.6, 1.4, 2.0, and 2.5 nm Bi spacer layers in the investigated MLFs +incorporate about 2, 4, 6, and 8 (012)-type Bi planes, respectively. +The model simulations for the [Bi(2.5, 2.0 nm)–FeNi(1.8 nm)]16 film samples reveal that +the low-energy dielectric function of the Bi intralayers has competing contributions from the +Drude term and from the intense Lorentz band around 0.36–0.39 eV with a ε2 maximum +height of 70–100 (see Table I). The Drude and Lorentz contributions are more clearly pro- +nounced in the corresponding optical conductivity spectra (see Figure 5a,b). The obtained +13 + +Drude and Lorentz parameters are in excellent agreement with those deduced in our pre- +vious study [13] for the Bi spacer layer incorporated in the [Bi(2.5, 2.0 nm)–NiFe(1.2 nm)]16 +structures under study. The pronounced Lorentz band found at low photon energies for +Bi single crystals (rhombohedral symmetry, space group R¯3m) [24, 25] and bulk Bi layers +[26, 27] is characteristic of the semimetallic-like electronic band structure due to the con- +tributions from the interband transitions near the Γ point, Γ+ +6 – Γ− +6 and Γ+ +45 – Γ− +6 [2], and +near the T point, T− +6 – T− +45 [4]. The estimated values (see Table I) of the Drude dc limit +σ1ω→0 (2750–3830 Ω−1·cm−1) as well as the free charge carrier’s γD (2.3–3.3 eV) are consis- +tent with those peculiar for the metallic surface states related to the Rashba SOC in Bi(111) +films, σ1ω→0 = 2300 Ω−1·cm−1 and γD = 2.0 eV) [6]. Meanwhile, the model simulation for +the [Bi(1.4 nm)–NiFe(1.8 nm)]16 structure indicates that for the 1.4 nm Bi layer the Drude +dc limit significantly increases to 8970±540 Ω−1·cm−1, while the γD essentially decreases to +1.50±0.06 eV. In this case, the Lorentz band is nearly suppressed. The associated found +Drude parameters for the ultrathin Bi layer inside the [Bi(0.6 nm)–NiFe(1.8 nm)]16 structure +are slightly different, namely, σ1ω→0 = 6300±540 Ω−1·cm−1 and γD = 1.2±0.1 eV, and the +Lorentz band is not present clearly (see Figure 5c,d and Table I). +Thus, we have discovered that, on the one hand, the optical conductivity spectra spectra +of the 2.0 and 2.5 nm thick Bi spacer layers in the (Bi–FeNi) MLFs incorporating 8 and 6 +Bi(012)-type monolayers, respectively, have contributions from the pronounced low-energy +Lorentz oscillator and from the free charge carrier Drude term (for details, see Figure 5a,b +and Table I). Here, the presence of the low-energy Lorentz band points on the Bi semimetallic +phase contribution, and the parameters obtained for the Drude conductivity indicate that +its origin can be associated with the surface metallic states [6]. +Therefore, the 2.0 and +2.5 nm Bi layers can be associated with the semimetallic Bi phase sandwiched between two +metallic layers on the top and bottom surfaces. On the other hand, the contribution from +the intrinsic Lorentz band is strongly suppressed for the 1.4 and 0.6 nm layers, where the +Drude conductivity displays notably improved metallicity properties, as one can see from +the optical conductivity spectra shown in Figure 5c,d (for details, see Table I). +From the above discussion of the obtained results, we can conclude that the Bi layer +consisting of 4 Bi(012)-type monolayers represents a kind of crossover regarding the contri- +butions from the semimetallic Bi phase and/or surface metallic-like states. Here we noticed +some similarity with the theory results presented for the ultrathin Bi(111) layers by Liu +14 + +et al. [12]. There, it was established that below 4 Bi(111) BLs the film is a semiconduc- +tor with the direct gap open at the Γ point and the positive indirect band gap, leading +to nontrivial Z2 topology (ν=1) peculiar for an intrinsic 2D TI. Hovewer, above 4 Bi(111) +BLs, the indirect band gap becomes negative resulting in a semiconductor-semimetal tran- +sition due to overlapping of two bands at the Fermi level around the Γ and M points. It +is argued by Liu et al. [12] that the Bi layers consisting of 5 to 8 Bi(111) BLs represent +a 2D TI suited between two “trivial” metallic surfaces [12]. This means that for the sur- +face considered as an individual 2D system its Z2 number is trivial (ν=0). The surface +bands have no contribution to the nontrivial Z2 topology and, therefore, these trivial metal- +lic surfaces are not robust and can easily be removed by surface defects or impurities. It +was found by us [13] that the Bi layers in the [Bi(2.0, 2.5 nm)–NiFe(0.8 nm)] multilayers, +incorporating the nanoisland permalloy layer, exhibit bulk-like semimetallic properties of +the electronic band structure, although the surface (Drude) metallic conductivity is absent +there (see Fig. 4(d) of Ref. [13]). Indeed, strong magnetic and spatial disorder induced by +magnetic FeNi nanoislands, as well as long-range many-body interactions between magnetic +moments of permalloy nanoislands [17], may lead to specific localization of free charge car- +riers [28]. However, the surface conductivity (or interface) states for the 1.4 nm layer in +the Bi–FeNi(1.8 nm) multilayers may be topologically nontrivial and, in this case, the elec- +trons cannot be backscattered by impurities. Here, the Drude dc limit is 8970±540 Ω·cm−1 +and the scattering rate γD=1.5±0.06 eV. We found that the 0.6 nm thick Bi layer exhibits +somewhat different Drude dc limit (6300±540 Ω·cm−1) and γD (1.2±0.1 eV), see Table I and +Figure 6, which can be attributed to the discontinuous nanoisland structure of this layer. +Finally, we would like to note that it will be challenging to investigate dc transport +and superconductivity properties of the ultrathin Bi films possessing 2D TI surface states +following the approach presented in Ref. [29], where the subkelvin superconductivity without +any external stimuli was discovered in 3D TI Cd3As2 films [30, 31]. +V. +CONCLUSIONS +In summary, using wide-band (0.5-6.5 eV) spectroscopic ellipsometry we studied the +optical properies of the [Bi(0.6, 1.4, 2.0, 2.5 nm)–NiFe(1.8˙nm)]16 MLFs prepared by rf +sputtering. The XRD analysis suggested that the 0.6, 1.4, 2.0, and 2.5 nm Bi layers in the +15 + +FIG. 6: (a,b) Parameters of the Drude term (AD and γD) for the Bi (filled symbols) and FeNi +(empty symbols) layers in the [Bi(0.6, 1.4, 2.0, 2.5 nm)–FeNi(1.8 nm)] MLF structures. +studied MLFs correspond to about two, four, six, and eight Bi(012)-type monolayers, +respectively. +From the multilayer model simulations of the measured ellipsometric data, +we extracted the Bi and FeNi layer dielectric functions. The dielectric function for the 2.0 +and 2.5 nm Bi spacer layers are represented by the Drude resonance due to the surface +states and the low-energy Lorentz band peaking at around 0.3-0.4 eV. The pronounced +Lorentz band is characteristic of the semimetallic bulk-like Bi electronic zone structure +due to the contributions from the interband transitions near the Γ point. We discovered +that the 2.0 and 2.5 nm Bi spacer layers can be associated with the semimetallic Bi phase +sandwiched between two trivial (where the topology number ν=0) metallic layers on the +top and bottom surfaces. The contribution from the low-photon-energy Lorentz band is +strongly suppressed for the 1.4 and 0.6 nm Bi layers, where the Drude conductivity displays +notably improved metallicity properties. This indicates that the Bi layer consisting of 4 +Bi(012)-type monolayers represents a kind of crossover regarding the contributions from +the semimetallic Bi phase and/or surface metallic-like states. +Therefore, the properties +of Bi layers below 4 monolayers may be associated with nontrivial topology (where the +topology number ν=1) peculiar for an intrinsic 2D TI. We expect that the Bi layers having +16 + +thickness of 0.9 nm will exhibit maximal GMR effect of about 20% in the (Bi-FeNi) MLFs, +where the Drude dc limit is about 8970±540 Ω·cm−1. These states may be protected from +backscattering, which makes them promising in spintronic devices and quantum computing. +Acknowledgement +We thank F.A. Pudonin for providing us with the Bi/FeNi multilayer film samples +and O. Pacherova for their XRD analysis. +We thank A. Muratov for participation in +the spectroscopic ellipsometry measurements. This work was supported by the European +Structural and Investment Funds and the Czech Ministry of Education, Youth, and Sports +(Project No. SOLID21, Cz.02.1.01/0.0/0.0/16−019/0000760). +Declaration of competing interest +The authors declare no conflict of interest. +[1] Bychkov, Y.A.; Rashba, E.I. JETP Lett., 1984, 39, 78. +[2] Golin, S. Phys. Rev. B, 1968, 166, 643. +[3] Gonze, X.; Michenaud, J.-P.; Vigneron, J.-P. Phys. Rev. B, 1990, 41, 11827. +[4] Liu, Y.; Allen, R.E. Phys. Rev. B, 1995, 52, 1566. +[5] Hofmann, Ph. Prog. Surf. 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Metals, 2020, 10, 1398. +18 + diff --git a/-tE5T4oBgHgl3EQfRw5F/content/tmp_files/load_file.txt b/-tE5T4oBgHgl3EQfRw5F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3514532ec3aba40d684ac66a9a91ecea12c9358 --- /dev/null +++ b/-tE5T4oBgHgl3EQfRw5F/content/tmp_files/load_file.txt @@ -0,0 +1,1026 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf,len=1025 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='05523v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='str-el] 13 Jan 2023 Bi layer properties in the Bi–FeNi GMR-type structures probed by spectroscopic ellipsometry Natalia Kovaleva,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ∗ Dagmar Chvostova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2 Ladislav Fekete,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2 and Alexandr Dejneka2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' † 1Lebedev Physical Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Russian Academy of Sciences Leninsky prospect 53,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Moscow 119991,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Russia 2Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Academy of Sciences of the Czech Republic Na Slovance 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Prague 18221,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Czech Republic (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' 2023) Abstract Bismuth (Bi) having a large atomic number is characterized by the strong spin-orbit coupling (SOC) and is a parent compound of many 3D topological insulators (TIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The ultrathin Bi films are supposed to be 2D TIs possessing the nontrivial topology, which opens the possibility of devel- oping new efficient technologies in the field of spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Here we aimed at studying the dielectric function properties of ultrathin Bi/FeNi periodic structures using spectroscopic ellipsometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The [Bi(d)–FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]N GMR-type structures were grown by rf sputtering deposition on Sitall-glass (TiO2) substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The ellipsometric angles Ψ(ω) and ∆(ω) were measured for the grown series (d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm, N = 16) of the multilayered film samples at room temperature for four an- gles of incidence of 60◦, 65◦, 70◦, and 75◦ in a wide photon energy range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The measured ellipsometric angles, Ψ(ω) and ∆(ω), were simulated in the framework of the corresponding mul- tilayer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The complex (pseudo)dielectric function spectra of the Bi layer were extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The GMR effects relevant for the studied Bi–FeNi MLF systems are estimated from the optical conductivity zero-limit (optical GMR effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The obtained results demonstrate that the Bi layer possesses the surface metallic conductivity induced by the SOC effects, which is strongly enhanced on vanishing the semimetallic-like phase contribution on decreasing the layer thickness, indicating its nontrivial 2D topology properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ∗Electronic address: kovalevann@lebedev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='ru †Electronic address: dejneka@fzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='cz 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' INTRODUCTION The relativistic effect of spin–orbit (SOC) coupling is involved in the so-called Rashba effect [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' This phenomenon arises from the apparent loss of crystalline inversion symmetry near the surface or heterojunction leading to the lifting of the spin degeneracy and generating spin-polarized surface metallic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' In this respect, 3D (2D) topological insulators (TIs) also exhibit spin-polarized surface metallic states due to SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' However, contrary to the Rashba effect, the surface metallic bands of a TI are determined by its bulk characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The TIs host metallic surface states in a bulk energy gap, which are topologically protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The surface (or interface) states of TIs can be topologically trivial or nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' In the latter case, for example, electrons cannot be backscattered by impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Bismuth (Bi) having a large atomic number is characterized by the strong SOC and is a parent compound of many 3D TIs, such as Bi1−xSbx or Bi2Se3, even although 3D bulk Bi itself is topologically trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The specific feature of the electronic band structure of bulk Bi having R¯3m rhombohedral symmetry [2–4] is its inverted band gaps at both the Γ and M points of the Brillouin zone due to the strong SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The uniqueness of Bi films associated with the surface metallic states [5, 6] and the semiconductor-to-metal transition [7, 8] are well documented in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Theoretical analyses predict a 1-bilayer (BL) Bi(111) film to be a 2D TI [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' If there is no or weak inter-BL coupling, a stack of the odd-even 1-BL films will exhibit nontrivial to trivial oscillations of topology (where the topological number ν [11] is equal to 1 or 0, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' However, for the nontrivial topology in a stack of the 1-BL films, the intermediate inter-BL coupling strength, which is, for example, higher than the van der Waals strengths, is a mandatory condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The direct (Γ point) and indirect band gap values were calculated by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' as a function of the Bi film thickness [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' It was established that below 4BLs the film is a semiconductor with the direct gap open at the Γ point and the positive indirect band gap leading to nontrivial topology peculiar for an intrinsic 2D TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Above 4BLs the indirect band gap becomes negative resulting in a semiconductor- semimetal transition due to overlapping of two bands at the Fermi level around the Γ and M points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' This suggests that the Bi films from 5 to 8 BLs represent a 2D TI situated between two trivial metallic surfaces [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' A comprehensive study of the associated SOC effects in ultrathin Bi layers opens the pos- sibility of developing new efficient technologies in the field of spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' For this purpose, 2 here we aimed at studying the dielectric function properties of ultrathin periodic structures Bi/Ni79Fe21, prepared by rf sputter deposition, which is one of the most common technologies used to grow coatings and multilayered films (MLFs) exhibiting a giant magnetoresistance (GMR) effect for various existing and modern nanotechnological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Earlier, we have demonstrated that electronic band structure and surface electronic properties of ul- trathin Bi layers in real GMR-type (Bi–FeNi)N MLF structures incorporating nanoisland FeNi layers can be successfully studied by spectroscopic ellipsometry (SE) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Here, by ap- plying the elaborated SE approach, we investigate (Bi–FeNi) MLFs, where the thickness of the FeNi layer was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm, corresponding to the FeNi layer structural percolation threshold [14, 15], and the Bi spacer layer was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm thick, incorporating about 2, 4, 6, and 8 Bi(012)-type planes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' We found that the Bi spacer layers have the metallic surface conductivity, which demonstrates strongly enhanced metallicity properties on vanishing the Bi semimetallic-like phase contribution on decreasing the layer thickness, which can be constructive in finding new nontrivial 2D topology properties of the (Bi–FeNi) GMR-type structures for their different nanotechnological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' MATERIALS AND METHODS The (Bi–FeNi)N MLFs were prepared in a sputter deposition system by cathode sput- tering from 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='95% pure Bi and Fe21Ni79 targets in an alternative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The base pressure in a sputter deposition chamber was 2×10−6 Torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The multilayers were deposited at ap- proximately 80 ◦C in an argon atmosphere of 6× 10−4 Torr on to insulating glassy Sitall (TiO2) substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' We utilized the substrates having typical dimensions 15 × 5 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 mm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The nominal thicknesses of the FeNi and Bi layers were controlled by the layer deposition times in accordance with the material deposition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' A series consisting of four MLF samples was prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' In the series of the grown (Bi–FeNi)N samples, the nominal thickness of the FeNi layer was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm and the Bi layer thickness was of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm, the number N of the periodically repeated Bi/FeNi layers was 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The thickness of the FeNi layer was chosen to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm matching the structural percolation threshold [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The Bi layer thicknesses were chosen in such a way that the conditions for ferromagnetic (FM) or antiFM coupling in the GMR-type structures would be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' To prevent degra- dation, the deposited (Bi–FeNi)16/Sitall samples were covered with the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1 nm-thick Al2O3 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' 1: AFM images (a–d) 5 × 5 µm2 and (e–h) 1 × 1 µm2 of the Al2O3/(Bi–FeNi)16/Sitall MLF samples, where the nominal Al2O3 and FeNi layer thicknesses are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm and the nominal Bi layer thicknesses are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The estimated surface RMS roughness values are in (a-d) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2 nm and in (e-h) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='7, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='3 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' (i,j) The typical height profiles for the MLF samples with the nominal Bi layer thicknesses of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The related [Bi–FeNi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2 nm)]N samples prepared by rf sputtering deposition onto the Sitall substrates under similar conditions were investigated by X-ray diffraction (XRD) as well as by X-ray reflectivity (XRR) experimental techniques in our previous study (see Supplementary online information for the article [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The XRR spectra proved a good periodicity and consistency with the corresponding nominal thicknesses of the FeNi and Bi slices in the Bi/FeNi MLF structures, as well as relatively low interface roughness between the constituent layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The XRD characterization suggests (012)-type Bi plane orientation, where the interlayer distance is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='28 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' It follows from this that in the studied MLF struc- tures the Bi layers with a thickness corresponding to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm incorporate 4 D!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2SUce () 0 500 300 400 eoo 100 008 a0o 0001 SO 10 JO (uw SO 30D!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2Sce (u) 0 500 300 400 ooa 100 008 a0o 0001 2 Heiapr (uw) 0 2HGidpf 2GU20l mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content="00s 34'2 UW S4'0 UWHGidpf 2Gu20l my o." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='r mn 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='84- 401 UwHGidpf 26GU20l mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='00s mn 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='07- mn 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='eHGidpf 26GU20l my o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='7 mn c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='HGidpf 26GU20l mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='00s n r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content="ar- JI'SUwHeiauf 26u20l my o." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1 mn &.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='rr- mn c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='srHGidpf 26u20l mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='00s mn r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='rr- mn f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='crHeidpf 26u20k my o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='r 531UWtwo, four, six, and eight Bi(012)-type planes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' In the present study, the surface morphology of the Bi–FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm) MLF samples, pre- pared by rf sputtering deposition on the Sitall (TiO2) substrates, was studied at room temperature using an ambient AFM (Bruker, Dimension Icon) in the Peak Force Tapping mode with ScanAsyst Air tips (Bruker, k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 N/m, nominal tip radius 2 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The SE mea- surements for the investigated Al2O3/(Bi–FeNi)16/Sitall samples were performed at room temperature in a wide photon energy range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 – 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 eV using a J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Woollam VUV-VASE ellipsometer (see the scheme illustrating the SE study of the (Bi–FeNi)N MLFs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' 1(a) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The measured ellipsometry spectra are represented by real values of the angles Ψ(ω) and ∆(ω), which are defined through the complex Fresnel reflection coefficients for light-polarized parallel rp and perpendicular rs to the plane of incidence, tan Ψ ei∆ = rp rs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The ellipsometric angles, Ψ(ω) and ∆(ω), measured for the Bi–FeNi MLF samples were sim- ulated using the multilayer model simulation available in the J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Woollam VASE software [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' From the multilayer model simulations, the (pseudo)dielectric function spectra of the ultrathin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm Bi layers and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm FeNi layer inside the Bi–FeNi MLF structures were extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The corresponding calculated optical conductivity spectra were analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Atomic force microscopy study The retrieved 5×5 µm2 and 1×1 µm2 AFM images of the Al2O3(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1 nm)/[Bi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm)–FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]N/Sitall multilayered films (where the given layer thicknesses corre- spond to their nominal values), presented in Figure 1a–h show discernable contrast because of the available surface hight deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The surface roughness of the Sitall glass (TiO2) substrates was investigated by us by AFM in our earlier publication [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The height profile of the Sitall substrates (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' 2a of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [17]) demonstrated the height deviation within the range 1-3 nm peculiar to the relatively large 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='3-1 µm lateral scale, which characterizes the Sitall substrate surface roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' From the AFM measurements on the areas 5×5 µm2 and 1×1 µm2 the root-mean square (RMS) surface roughness values were evaluated, which are presented in the caption to Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The corresponding RMS roughness values are 5 notably higher for the Al2O3(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1 nm)/[Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm)–FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]16/Sitall MLF sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The smaller-scale (1 × 1 µm2) images clearly recognize a fine grainy-like structure of the surface morphology, which seems to be characteristic for all studied film samples (see Figure 1e–h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The typical grain size, being of about 50 nm, is notably larger for the FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm) – Bi MLF sample incorporating the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm-thick Bi layers, and, following the estimated RMS rough- ness values, the average grain size decreases to about 20 nm with decreasing the Bi layer thickness to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' As one can see from the typical height profiles presented in Figure 1i,j, with decreasing the Bi layer thickness from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 to about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 nm, the surface morphology becomes highly irregular due to the formation of conglomerates of nanoislands separated by rather flat (relatively small roughness) areas of about 20 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Spectroscopic ellipsometry study of the ultrathin Bi–FeNi multilayer film samples The ellipsometric angles Ψ(ω) and ∆(ω) were measured for the prepared Al2O3/(Bi– FeNi)16/Sitall MLF samples at the angles of incidence of 60◦, 65◦, 70◦, and 75◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Figure 2 demonstrates the ellipsometric angles Ψ(ω) and ∆(ω) recorded at 65◦ and 70◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' To model the contributions from free charge carriers and interband optical transitions, the complex dielectric function ˜ε(ω) = ε1(ω) + iε2(ω) of the Bi and FeNi layers was interpreted in terms of the Drude and Lorentz parts, respectively, ˜ε(E ≡ ℏω) = ǫ∞ − AD E2 + iEγD + � j AjγjEj E2 j − E2 − iEγj , (1) where ε∞ is the high-frequency dielectric constant, which takes into account the contribution from the higher-energy interband transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The fitted Drude parameters were AD and free charge carrier’s scattering rate γD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The fitted parameters of Lorentz bands were Ej, γj, and Aj of the band maximum energy, the full width at half maximum, and the ε2 band height, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The obtained ellipsometric angles Ψ(ω) and ∆(ω) measured at different angles of incidence of 60◦, 65◦, 70◦, and 75◦ were fitted for each sample simultaneously using the J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Woollam VASE software [16] in the framework of the designed multilayer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The multilayer model for the studied Al2O3/(Bi–FeNi)/Sitall multiayers was constructed as it is schematically presented in Figure 3, exactly so as the layers were deposited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' In addition, we attempted to take into account the roughness properties of the surface by using the conven- tional approach of effective medium approximation (EMA) based on the (50% Al2O3–50% 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' 2: (a-d) Ellipsometric angles, Ψ(ω) and ∆(ω) (symbols), measured at the angles of incidence of 65◦ and 70◦ for the Al2O3/[Bi(d)–NiFe(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]16/Sitall multilayered films where the Bi spacer layer thicknesses d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The solid red, blue, green, and black curves show the corresponding simulation results for the angle 65◦ by the dielectric function model using Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' 7 vacuum) Bruggeman model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The dispersion model for the Bi layers included three or four Lorentz terms as well as the Drude part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The dispersion model for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm permalloy layers incorporated in the studied MLF structures included the Drude term responsible for the free charge carrier contribution and one Lorentz oscillator to account for the most pro- nounced interband optical transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' In addition, the dielectric function spectra of the bare Sitall substrate derived from our earlier SE studies [18, 19] were introduced to the elabo- rated multilayer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The dielectric response of the Al2O3 capping layer was represented by the tabular complex dielectric function spectra [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The thicknesses of the Bi and FeNi layers, as well as of the surface layers, were fitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The unknown parameters were allowed to vary until the minimum of the mean squared error (MSE) is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The best simulation result for the studied [Bi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm)–FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]16 MLF samples corresponded to the lowest obtained MSE values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='3843, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='297, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2934, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4508, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The good quality of the fit allowed us to estimate the actual Bi and FeNi layer thicknesses in the MLFs under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The quality of the fit is demonstrated by Figure 2, where we plotted the measured ellipsometric angles along with the simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The Drude and Lorentz parameters resulting from the simulation of the Al2O3/[Bi(d)–FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]16/Sitall MLF samples are given in Tables I and II, and the resulting ε1(ω) and ε2(ω) parts of the Bi and FeNi (pseudo)dielectric function spectra are presented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' From Figure 4a,b one can see that the complex (pseudo)dielectric functions of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm thick Bi spacers inside the investigated Bi–FeNi MLFs demonstrate metal- lic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Moreover, the ε1(ω) function progressively decreases while the Bi thickness decreases from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 nm and the ε2(ω) increases at low photon energies, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' According to our simulation results, we expect that the best metallicity properties are demonstrated by the Bi layer in the [Bi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 nm)–NiFe(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]16 structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' At the same time, the complex (pseudo)dielectric functions of the thinnest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 nm thick Bi layer look somewhat different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Here, in addition to the low-energy metallic Drude response identified by the characteristic behavior of the ε1(ω) and ε2(ω), the Lorentz band around 4–5 eV makes an essential contribution to the dielectric function response (the corresponding Drude (AD and γD) and Lorentz (Aj, Ej, and γj) parameters are listed in Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Next, being similar, the dielectric functions of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm thick permalloy layers in the [FeNi–Bi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm)] MLFs are dominated by the ε2(ω) resonance and ε1(ω) antiresonance features, indicat- ing the predominant contribution from the Lorentz oscillator peaking at around 3 eV (see 8 (a) (b) ( ) (d) c FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' 3: The multilayer model applied for the simulation of the Al2O3/[Bi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm)–FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]16/Sitall samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The Bi and FeNi thicknesses estimated from the model simulations in (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='684±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='037 nm and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='082±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='116 nm, (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='408±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='574 nm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='780±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='65 nm, (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='764±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='194 nm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='825±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='358 nm, and (d) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='387±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='128 nm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='782±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='171 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Note good agreement between the thicknesses of the FeNi and Bi layers estimated from the model simula- tions and their respective nominal thickness values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The roughness and Al2O3 thicknesses esti- mated from the model simulations in (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='00±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='85 nm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='283±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='37 nm, (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='000±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='97 nm and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='967±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='17 nm, (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='848±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='86 nm and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='738±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='92 nm, and (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='000±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='95 nm and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='389±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='23 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Figure 4c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' An upturn evident in the ε2(ω) at low photon energies indicates an addi- tional Drude contribution, which is relatively less pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Following our simulation results, we expect the advanced metallicity properties of the FeNi layer in the [Bi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 nm)– NiFe(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]16 structure (see the corresponding Drude (AD and γD) and Lorentz (Aj, Ej, and γj) parameters listed in Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Figure 5a–d presents the evolution of the Bi intralayer optical conductivity, σ1(ω) = ε2(ω)ω(cm−1)/60, upon decreasing the Bi spacer layer thickness in the [FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm) – Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 nm)]16 structures, and Figure 5e–h shows the associated optical conduc- tivity spectra of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm FeNi permalloy layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Here, the contributions from the Drude and Lorentz oscillators following the multilayer model simulations using Equation 1 are evi- 9 woge 0 sti2(19VSl mna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='Oid) 19vsl mn8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='TingtS80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='Sarmna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='Oid 19vsl↑80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='03 91503 Clε8S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content="T2lonap0'000wog6 0 sti219Vsl mn8, Tingt S1081." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1arS 19Vsl p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content="↓'4uw80↓." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content="13 91503 Cl2lonap 40'000wogel 0 lsti219Vsl mn8, Tingt 328." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1armno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='Sid1043 91503 Cl881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2lonap 488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0woge 0 ti219Vsl mn8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Tingt 4S81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1arIS入Gl mnc,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='Sid188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='S3 91S03 Cle88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content="2lonap 40'000FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' 4: The complex (pseudo)dielectric function spectra, ε2(ω) and ε1(ω), of the (a,b) Bi layers and (c,d) FeNi layers in the [Bi(d)–FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]16 structures shown for the Bi layer nominal thickness values d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm by solid red, blue, green, and black curves, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' TABLE I: Drude-Lorentz parameters for the Bi spacer layer in the [Bi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm)– NiFe(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]16 multilayered films obtained from the model simulations of the dielectric functions by using Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The values of Ej, γj, and γD are given in eV, and optical conductivity limit σ1(ω→0) in Ω−1·cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Parameters 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 nm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 nm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0 nm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm Drude AD 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' (9)±4 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' (7)±4 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' (5)±4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' (1)±2 γD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2(5)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='51(0)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='7(2)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1(3)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2 σ1(ω→0) 6300±540 8970±540 3290±540 3370±270 Lorentz E1 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='45(8)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='35(9)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='38(6)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='004 oscillator A1 – 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' (0)±6 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' (0)±10 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' (8)±2 γ1 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='52(6)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='79(1)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='67(6) Lorentz E2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='67 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='31(5)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='03 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='08(7)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='77(5)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='04 oscillator A2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2(7)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='53(2)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2(5)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='67(6)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='08 γ2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2(1)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='99(3)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4(7)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5(5)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2 Lorentz E3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='7 oscillator A3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1 γ3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 10 TABLE II: Drude-Lorentz parameters for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm thick NiFe layer in the [Bi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm)–NiFe]16 multilayered films obtained from the simulations of the model dielectric function described by Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The values of E1, γ1, and γD are given in eV, and optical conductivity limit σ1(ω→0) in Ω−1·cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Parameters 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 nm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 nm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0 nm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm Drude AD 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' (8)±2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' (0)±1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' (7)±2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' (1)±2 γD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='876(5)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8(2)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4(2)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1(3)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2 σ1(ω→0) 4540±270 2020±130 2920±270 1760±270 Lorentz E1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='87 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='32 oscillator A1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='76 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='28 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='23 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='74 γ1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='62 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='88 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='65 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='95 dently demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The optical conductivity spectra of the Bi and FeNi layers follow the main trends identified in their complex dielectric function spectra presented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' DISCUSSION Initially, we would like to discuss GMR effects relevant for the studied MLF sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Our simulations of the dielectric functions for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm-thick NiFe layer inside the [Bi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm)–NiFe(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)] MLFs show the presence of the Drude term com- plemented with the pronounced Lorentz band located at around 2–3 eV (see Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' From the corresponding optical conductivity spectra presented in Figure 5e–h one can notice that the associated Drude dc limit, σ1ω→0, displays an oscillating character (in agreement with the results deduced for the corresponding Drude parameter AD, see Table II and Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' We can expect that the Bi spacer thicknesses for which the FeNi layers are preferentially antiFM coupled in the studied MLFs are around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm implying that the [Bi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm)– NiFe(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]16 film structures will exhibit a drop in the resistance (being negative magne- toresistance) when exposed to an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' It is well known from the literature that the first antiFM maximum exhibits negative magnetoresistance of about 20%, while the second antiFM maximum decreases to about 10%, and the presence of the third antiFM maximum cannot confidently be retrieved (see, for example, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [21] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' 5: The intralayer optical conductivity, σ1(ω) = ε2(ω)ω[cm−1]/60, for the (a-d) Bi layers and (e-h) FeNi layers in the [Bi(d)–FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]16 structures shown for the Bi layer nominal thickness values d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 nm by solid curves (a,e) black, (b,f) green, (c,g) blue, and (d,h) red, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The contributions from the Drude term and the Lorentz oscillator in (a-d) are displayed by the yellow and cyan shaded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' In (e-h) the Drude term for the FeNi layers is displayed by the magenta shaded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Shown by the dotted curves are the summary of the Drude and Lorentz contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' 12 Using a simple model of a two-current series resistor [22], the magnetoresistance ∆R R can be estimated as ∆R R = 100% (α − β)2 4 � α + dBi dF eNi � � β + dBi dF eNi �, (2) where dBi and dF eNi are the thicknesses of Bi and FeNi layers, and α = ↓ρF eNi ρBi and β = ↑ρF eNi ρBi are the ratios of the resistivity in the FeNi layer to that in the Bi layer in the spin down and spin up current channel, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Exploiting values for ρ = σ−1 1ω→0 estimated for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 nm Bi and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm FeNi layers from the current model simulations (see Table I and II), namely, ρBi= 1 8970Ω·cm, ↓ρF eNi= 1 2020Ω·cm and ↑ρF eNi= 1 4540Ω·cm (the latter estimate is given by the FM coupling for the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 nm Bi spacer), we obtain α=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 and β=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Then, using Equation (2) we have ∆R R =10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' This means that the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 nm Bi spacer corresponds to the second antiFM maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Following the same approach for the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm Bi spacer, where ρBi= 1 3370Ω·cm, ↓ρF eNi= 1 1760Ω·cm and ↑ρF eNi= 1 2920Ω·cm (corresponding to the FM cou- pling for the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0 nm Bi spacer), we obtain α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='9 and β=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Using Equation (2), we have ∆R R =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4%, which may correspond to the very weakly pronounced third antiFM maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' From the analysis presented above, we may expect that the first antiFM maximum corre- sponding to the magnetoresistance of about 20% occurs for the Bi spacer thickness of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='9 nm, which is in agreement with the results presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Further, in the XRD patterns of the investigated Al2O3/[Bi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm)– NiFe(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]16/Sitall film samples, the peak of the R¯3m crystalline Bi phase is identi- fied at 2θ ≈ 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2◦ suggesting (012) orientation of the Bi layers, which is characterized by the interlayer distance of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='28 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Using STM and reflection high-energy electron diffraction (RHEED) techniques, it was shown that initial growth of Bi(012)-type films occurs in the form of islands with the height increment of about 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 ˚A, indicating even-number layer sta- bility leading to the laterally flat morphology of the Bi(012)-type islands [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Consequently, we can expect that the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm Bi spacer layers in the investigated MLFs incorporate about 2, 4, 6, and 8 (012)-type Bi planes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The model simulations for the [Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0 nm)–FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]16 film samples reveal that the low-energy dielectric function of the Bi intralayers has competing contributions from the Drude term and from the intense Lorentz band around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='36–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='39 eV with a ε2 maximum height of 70–100 (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The Drude and Lorentz contributions are more clearly pro- nounced in the corresponding optical conductivity spectra (see Figure 5a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The obtained 13 Drude and Lorentz parameters are in excellent agreement with those deduced in our pre- vious study [13] for the Bi spacer layer incorporated in the [Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0 nm)–NiFe(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2 nm)]16 structures under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The pronounced Lorentz band found at low photon energies for Bi single crystals (rhombohedral symmetry, space group R¯3m) [24, 25] and bulk Bi layers [26, 27] is characteristic of the semimetallic-like electronic band structure due to the con- tributions from the interband transitions near the Γ point, Γ+ 6 – Γ− 6 and Γ+ 45 – Γ− 6 [2], and near the T point, T− 6 – T− 45 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The estimated values (see Table I) of the Drude dc limit σ1ω→0 (2750–3830 Ω−1·cm−1) as well as the free charge carrier’s γD (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='3–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='3 eV) are consis- tent with those peculiar for the metallic surface states related to the Rashba SOC in Bi(111) films, σ1ω→0 = 2300 Ω−1·cm−1 and γD = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0 eV) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Meanwhile, the model simulation for the [Bi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 nm)–NiFe(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]16 structure indicates that for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 nm Bi layer the Drude dc limit significantly increases to 8970±540 Ω−1·cm−1, while the γD essentially decreases to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='50±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='06 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' In this case, the Lorentz band is nearly suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The associated found Drude parameters for the ultrathin Bi layer inside the [Bi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 nm)–NiFe(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)]16 structure are slightly different, namely, σ1ω→0 = 6300±540 Ω−1·cm−1 and γD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1 eV, and the Lorentz band is not present clearly (see Figure 5c,d and Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Thus, we have discovered that, on the one hand, the optical conductivity spectra spectra of the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm thick Bi spacer layers in the (Bi–FeNi) MLFs incorporating 8 and 6 Bi(012)-type monolayers, respectively, have contributions from the pronounced low-energy Lorentz oscillator and from the free charge carrier Drude term (for details, see Figure 5a,b and Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Here, the presence of the low-energy Lorentz band points on the Bi semimetallic phase contribution, and the parameters obtained for the Drude conductivity indicate that its origin can be associated with the surface metallic states [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Therefore, the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm Bi layers can be associated with the semimetallic Bi phase sandwiched between two metallic layers on the top and bottom surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' On the other hand, the contribution from the intrinsic Lorentz band is strongly suppressed for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 nm layers, where the Drude conductivity displays notably improved metallicity properties, as one can see from the optical conductivity spectra shown in Figure 5c,d (for details, see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' From the above discussion of the obtained results, we can conclude that the Bi layer consisting of 4 Bi(012)-type monolayers represents a kind of crossover regarding the contri- butions from the semimetallic Bi phase and/or surface metallic-like states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Here we noticed some similarity with the theory results presented for the ultrathin Bi(111) layers by Liu 14 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' There, it was established that below 4 Bi(111) BLs the film is a semiconduc- tor with the direct gap open at the Γ point and the positive indirect band gap, leading to nontrivial Z2 topology (ν=1) peculiar for an intrinsic 2D TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Hovewer, above 4 Bi(111) BLs, the indirect band gap becomes negative resulting in a semiconductor-semimetal tran- sition due to overlapping of two bands at the Fermi level around the Γ and M points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' It is argued by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [12] that the Bi layers consisting of 5 to 8 Bi(111) BLs represent a 2D TI suited between two “trivial” metallic surfaces [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' This means that for the sur- face considered as an individual 2D system its Z2 number is trivial (ν=0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The surface bands have no contribution to the nontrivial Z2 topology and, therefore, these trivial metal- lic surfaces are not robust and can easily be removed by surface defects or impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' It was found by us [13] that the Bi layers in the [Bi(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm)–NiFe(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)] multilayers, incorporating the nanoisland permalloy layer, exhibit bulk-like semimetallic properties of the electronic band structure, although the surface (Drude) metallic conductivity is absent there (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' 4(d) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Indeed, strong magnetic and spatial disorder induced by magnetic FeNi nanoislands, as well as long-range many-body interactions between magnetic moments of permalloy nanoislands [17], may lead to specific localization of free charge car- riers [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' However, the surface conductivity (or interface) states for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 nm layer in the Bi–FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm) multilayers may be topologically nontrivial and, in this case, the elec- trons cannot be backscattered by impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Here, the Drude dc limit is 8970±540 Ω·cm−1 and the scattering rate γD=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='06 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' We found that the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 nm thick Bi layer exhibits somewhat different Drude dc limit (6300±540 Ω·cm−1) and γD (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1 eV), see Table I and Figure 6, which can be attributed to the discontinuous nanoisland structure of this layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Finally, we would like to note that it will be challenging to investigate dc transport and superconductivity properties of the ultrathin Bi films possessing 2D TI surface states following the approach presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [29], where the subkelvin superconductivity without any external stimuli was discovered in 3D TI Cd3As2 films [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' CONCLUSIONS In summary, using wide-band (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 eV) spectroscopic ellipsometry we studied the optical properies of the [Bi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm)–NiFe(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8˙nm)]16 MLFs prepared by rf sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The XRD analysis suggested that the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm Bi layers in the 15 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' 6: (a,b) Parameters of the Drude term (AD and γD) for the Bi (filled symbols) and FeNi (empty symbols) layers in the [Bi(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm)–FeNi(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='8 nm)] MLF structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' studied MLFs correspond to about two, four, six, and eight Bi(012)-type monolayers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' From the multilayer model simulations of the measured ellipsometric data, we extracted the Bi and FeNi layer dielectric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The dielectric function for the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm Bi spacer layers are represented by the Drude resonance due to the surface states and the low-energy Lorentz band peaking at around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='3-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The pronounced Lorentz band is characteristic of the semimetallic bulk-like Bi electronic zone structure due to the contributions from the interband transitions near the Γ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' We discovered that the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='5 nm Bi spacer layers can be associated with the semimetallic Bi phase sandwiched between two trivial (where the topology number ν=0) metallic layers on the top and bottom surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' The contribution from the low-photon-energy Lorentz band is strongly suppressed for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='6 nm Bi layers, where the Drude conductivity displays notably improved metallicity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' This indicates that the Bi layer consisting of 4 Bi(012)-type monolayers represents a kind of crossover regarding the contributions from the semimetallic Bi phase and/or surface metallic-like states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Therefore, the properties of Bi layers below 4 monolayers may be associated with nontrivial topology (where the topology number ν=1) peculiar for an intrinsic 2D TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' We expect that the Bi layers having 16 thickness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='9 nm will exhibit maximal GMR effect of about 20% in the (Bi-FeNi) MLFs, where the Drude dc limit is about 8970±540 Ω·cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' These states may be protected from backscattering, which makes them promising in spintronic devices and quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Acknowledgement We thank F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Pudonin for providing us with the Bi/FeNi multilayer film samples and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Pacherova for their XRD analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' We thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Muratov for participation in the spectroscopic ellipsometry measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' This work was supported by the European Structural and Investment Funds and the Czech Ministry of Education, Youth, and Sports (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' SOLID21, Cz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='01/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='0/16−019/0000760).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Declaration of competing interest The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [1] Bychkov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Rashba, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' JETP Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=', 1984, 39, 78.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [4] Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Allen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' B, 1995, 52, 1566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [5] Hofmann, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=', 2006, 81, 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [6] Yokota, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Takeda, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Dang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Han, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' McCarthy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Nagao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Hishita, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Kitajima, K.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Meyer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Bartoli, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Chulkov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Blugel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Rev.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Pacherova O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Muratov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Fekete L.' metadata={'source': 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125, 465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [16] Woollam, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' VASE Spectroscopic Ellipsometry Data Analysis Software;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Woollam, Co.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Bagdinova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Demikhov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='I.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Saito, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Yaginuma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Fujikawa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Kogure, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Ohno, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Hasegawa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Sakurai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=', 2004, 93, 105501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [24] Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Jain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' B, 1970, 2, 2978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [25] Lenham, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Treherne, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Metcalfe, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=', 1965, 55, 1072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [26] Hunderi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' F, 1975, 5, 2214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [27] Toudert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Serna, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Express, 2017, 7, 2299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [28] Kovaleva, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Kusmartsev, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Mekhiya, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Trunkin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Chvostova, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Davydov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Oveshnikov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Pacherova, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Sherstnev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Kusmartseva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Kugel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' De- jneka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Pudonin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Luo,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Aronzon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' , 2020, 10, 21172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' [29] Suslov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Davydov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Oveshnikov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Morgun, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Kugel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Zakhvalinskii, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Pilyuk, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Kochura, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Kuzmenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Pudalov, V.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Chvostova, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Fekete, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Muratov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' Metals, 2020, 10, 1398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE5T4oBgHgl3EQfRw5F/content/2301.05523v1.pdf'} +page_content=' 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--git a/0NE3T4oBgHgl3EQfmwqR/content/tmp_files/2301.04619v1.pdf.txt b/0NE3T4oBgHgl3EQfmwqR/content/tmp_files/2301.04619v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b919a65cc88e28515411d856ac42426bcfbb6746 --- /dev/null +++ b/0NE3T4oBgHgl3EQfmwqR/content/tmp_files/2301.04619v1.pdf.txt @@ -0,0 +1,1165 @@ +TinyHD: Efficient Video Saliency Prediction with Heterogeneous Decoders using +Hierarchical Maps Distillation +Feiyan Hu1, Simone Palazzo2, Federica Proietto Salanitri2, Giovanni Bellitto2, Morteza Moradi2, +Concetto Spampinato2, Kevin McGuinness1 +1 Insight SFI Research Centre for Data Analytics, Dublin City University, Dublin, Ireland +{feiyan.hu, kevin.mcguinness}@dcu.ie +2 PeRCeiVe Lab, University of Catania, Catania, Italy +{simone.palazzo, concetto.spampinato}@unict.it +Abstract +Video saliency prediction has recently attracted atten- +tion of the research community, as it is an upstream task +for several practical applications. +However, current so- +lutions are particularly computationally demanding, espe- +cially due to the wide usage of spatio-temporal 3D convolu- +tions. We observe that, while different model architectures +achieve similar performance on benchmarks, visual varia- +tions between predicted saliency maps are still significant. +Inspired by this intuition, we propose a lightweight model +that employs multiple simple heterogeneous decoders and +adopts several practical approaches to improve accuracy +while keeping computational costs low, such as hierarchi- +cal multi-map knowledge distillation, multi-output saliency +prediction, unlabeled auxiliary datasets and channel re- +duction with teacher assistant supervision. Our approach +achieves saliency prediction accuracy on par or better than +state-of-the-art methods on DFH1K, UCF-Sports and Hol- +lywood2 benchmarks, while enhancing significantly the ef- +ficiency of the model. +1. Introduction +Video saliency prediction aims at estimating patterns of +human attention during free-viewing of dynamic scenes, +to emulate the capabilities of the human visual system of +quickly analyzing and interpreting the surrounding environ- +ment. Due to its several practical applications [7, 5, 28, 33, +9, 40, 12, 27], it is an active area of research in computer +vision. However, the solution to this problem is not triv- +ial, for several reasons. First, attention mechanisms in the +human visual system are not fully known, so it is not clear +how to emulate them. Also, it requires complex modeling +of both visual features and their motion and interaction: an +object with striking visual patterns may be shadowed by a +TASED +100% 84% +72% +ViNet +84% 100% 75% +HD2S +72% +TASED +75% +ViNet +100% +HD2S +CC Metric +TASED +100% 73% +57% +ViNet +73% 100% 58% +HD2S +57% +TASED +58% +ViNet +100% +HD2S +SIM Metric +Figure 1: Measuring prediction similarity among video +saliency prediction models TASED, ViNet and HD2S on +DHF1K validation set. +bland element of the scene that starts moving in a pecu- +liar way. Finally, modeling the temporal dimension may +become computationally expensive, especially with current +deep learning methods based on spatio-temporal 3D convo- +lutions, thus limiting the applicability to low-power devices. +Many solutions have been proposed, based on different +assumptions on how to capture video saliency. It is inter- +esting to note that, in spite of the remarkably different re- +search directions followed by the variety of works in the +literature, top results over video saliency prediction bench- +marks are very close [1, 3, 36], suggesting that predictions +of different models are similar. We assessed the validity +of this conclusion by comparing three of the best perform- +ing methods on the DHF1K dataset [36] — TASED [25], +HD2S [1] and ViNet [16] — not in terms of their scores +on summary metrics, but in terms of the relative similarity +of the predicted saliency maps. To illustrate our findings, +Fig. 1 shows pairwise similarities between predicted maps +over two common metrics, Linear Correlation Coefficient +(CC) and Similarity (SIM). Although the three approaches +achieve similar scores on both metrics on DHF1K (between +0.470 and 0.511 for CC, and between 0.361 and 0.406 for +SIM), the same metrics computed between each other are +relatively low, compared to what one would expect given +their similarity to the saliency ground truth. A visual in- +spection of the saliency maps generated by methods under +arXiv:2301.04619v1 [cs.CV] 11 Jan 2023 + +comparison confirms this behavior: Fig. 2 shows that it is +common to find cases where each approach produces re- +markably different saliency maps. +Notably, all three methods — TASED, HD2S and ViNet +— are encoder-decoder networks and share the same en- +coder, S3D [38], while employing different decoding strate- +gies (a U-Net–like approach for TASED and ViNet, a hi- +erarchical map aggregation for HD2S). This suggests that +a key factor underlying differences between current video +saliency prediction approaches lies in the way encoded fea- +tures are processed in the decoding path, leading to models +that learn specific (and often exclusive) representations. We +strengthen this hypothesis by experimenting with another +decoding strategy, exemplified by DLA [39], which com- +bines hierarchical decoding with complex feature interac- +tions: the results, also included in Fig. 2, show yet another +saliency prediction pattern, while using the same encoder +network, S3D. These results lead us to hypothesize that dif- +ferent model architectures introduce different inductive bi- +ases, which are more suitable to recognize certain patterns +more than others, thus requiring to increase model capacity +in order to generalize well to multiple saliency dynamics. +Indeed, the size of weights of models in our analysis range +between 82 MB and 116 MB, and the size of the top ten +models in the DHF1K leaderboard1 is on average 238 MB. +Given these premises, instead of increasing the complex- +ity of a single decoding strategy, it may be more efficient +to employ multiple simpler architectures with fewer param- +eters, relying on each architecture’s capability to attend to +different salient regions and combining their results. Hence, +we propose TinyHD, a lightweight, efficient and heteroge- +neous multi-decoder architecture for video saliency predic- +tion. The proposed method is inspired by encoder-decoder +architectures, but introduces the adoption of heterogeneous +decoding strategies in order to reduce the complexity of +each decoder, increasing efficiency (the weights of the re- +sulting model take only 16 MB) and improving the accu- +racy of predictions, as we show in our experiments. Fur- +thermore, along the direction of reducing computational +costs while retaining high accuracy, we also introduce a +novel knowledge distillation approach, based on exploiting +a teacher with multiple hierarchical predictions: this allows +the model to freely learn its own features, since no explicit +conditioning on representations is enforced, while at the +same time receiving a supervision signal that encodes in- +formation at different layers of abstraction. +Experiments confirm that our model can generate high- +quality predictions with low computational costs and model +size (only 16 MB). We assess the impact of our heteroge- +neous multi-decoder strategy by carrying out extensive ab- +lation studies and comparing alternative architectures. We +also demonstrate the effectiveness of our knowledge dis- +1https://mmcheng.net/videosal/ +tillation strategy, compared to the employment of a non- +hierarchical teacher. To summarize our contributions: +• We propose a decoding strategy for video saliency pre- +diction which combines heterogeneous decoders to ex- +ploit the specific pattern analysis capabilities, while re- +ducing the overall model complexity. To our knowl- +edge, we are first to propose multiple saliency maps +output using 3D CNN to improve model efficiency. +• We employ a knowledge distillation approach based +on a hierarchical teacher, providing saliency maps es- +timated from different abstraction layers. +• Extensive experiments show that our model achieves +state-of-the-art performance on the DHF1K bench- +mark, at lower computational costs of current meth- +ods. Ablation studies support the motivations for our +decoding and knowledge distillation strategies. +2. Related Work +The main contributions of the proposed approach con- +sist of a novel heterogeneous multi-decoder scheme, which +combines lightweight versions of common decoding strate- +gies, and a multi-objective knowledge distillation approach. +In this section, we briefly present the state-of-the-art on +these topics. +Decoding +strategies +for +video +saliency +prediction. +Among recent methods from the state-of-the-art for video +saliency prediction, leveraging encoder-decoder networks +can be considered a mainstream approach; however, sev- +eral architectural variations have been proposed for feature +sharing between encoder and decoder and for output recon- +struction. As shown in the taxonomy presented in Fig. 3, a +simpler class of approaches employs independent encoder +and decoder, with no feature sharing between the two paths. +Among these, approaches based on recurrent layers typ- +ically model temporal dynamics at the bottleneck of the +architecture [18, 37, 22]. Non-recurrent architectures, in- +stead, model time by means of 3D convolutions [42, 38, 4], +Other approaches employ architectures similar to U-Net by +introducing skip connections that encourage feature shar- +ing between encoder and decoder. +TASED [25] aggre- +gates spatio-temporal features through the use of auxiliary +pooling for reducing the temporal dimension. ViNet [16] +integrates S3D features from multiple hierarchical levels +by employing trilinear interpolation and 3D convolutions. +UNISAL [6] proposes a multi-objective unified framework +for both 2D and 3D saliency with domain-specific modules +and a lightweight recurrent architecture to handle temporal +dynamics; While single-decoder approaches are common, +multi-decoder output integration has recently attracted in- +terest. DVA [35] and HD2S [1] fuse maps predicted by +independent decoders operating at different abstraction lev- +els. RecSal [30] predicts multiple saliency maps in a multi- +objective training framework. Recent works introduce more + +(a) Frame +(b) GT +(c) TASED +(d) ViNet +(e) HD2S +(f) DLA +Figure 2: Examples of video saliency maps from state-of-the-art methods. Although they achieve very similar performance +on popular metrics, remarkable differences can be seem in the learned saliency patterns. +Figure 3: A taxonomy of decoding strategies commonly +employed in video saliency prediction. +Subfigures(top- +left, top-right, bottom-left, bottom-right): Independent en- +coder and decoder, with no feature sharing between the two +paths [4, 18, 22, 37, 42]; U-Net–like architecture, with fea- +tures sharing between encoder and decoder [6, 16, 19, 25]; +Deep Layer Aggregation [39]; Hierarchical intermediate +map aggregation [1, 30, 35]. +complex feature interactions among decoding paths, where +high-resolution features are affected by deeper high-level +features, as in DLA [39] and TSFP-Net [3]. All of the ap- +proaches presented employ either a single-decoder archi- +tecture or a homogeneous multi-decoder one, where differ- +ences between decoders lie in the number of layers rather +than in their structure. In our work, we propose an architec- +ture which combines heterogeneous decoder structures, in +order to better exploit their distinctive saliency prediction +properties and thus increase computational efficiency. +Knowledge distillation for visual saliency prediction. +Knowledge distillation [13, 11] is commonly employed +to train an efficient student model from a more complex +teacher model, with higher accuracy than when training +the student directly from dataset labels. +Several knowl- +edge distillation approaches have been recently proposed +for video saliency prediction. +SKD-DVA [20] proposes +spatio-temporal knowledge distillation with two teachers +and two students, with each pair focusing on either spatial +or temporal transfer. SV2T-SS [41] distills corresponding +features of teacher and student (implemented as encoder- +decoder networks), based on first- and second-order feature +statistics transfer. UVA-DVA [10] employs separate spatial +and temporal teachers, whose knowledge is transferred to +a single student model, which then fuses the resulting fea- +tures in the final saliency prediction, achieving reasonable +accuracy at impressive speed. Leveraging knowledge dis- +tillation for video salient object detection is the main theme +of the work in [34]. The knowledge distillation setting pro- +posed in our work differs from existing techniques in two +main aspects: 1) we define a multi-objective distillation tar- +get on saliency maps directly; 2) we employ a hierarchical +model as a teacher in order to further capture differences in +saliency patterns extracted at multiple scales. +3. Methodology +3.1. Overview +The overall architecture of the proposed saliency predic- +tion network with knowledge distillation is shown in Fig. 4. +Following the taxonomy introduced in Sect. 2, a shared en- +coder extracts multi-level features that are then processed +by three parallel decoding architectures: decoder 1 (D1) +implements hierarchical intermediate maps aggregation (in- +spired by HD2S); decoder 2 (D2) employs a U-Net–like ap- +proach; decoder 3 (D3) is based on deep layer aggregation +concepts (as in DLA [39]). +The hierarchical aggregation decoder (i.e., decoder 1 in +Fig. 4) produces four intermediate saliency maps from fea- +tures extracted at different encoder layers; then, the set of + +DVSS三三三三predictions from all decoders are fused into the final pre- +diction. At training time, we compute a supervised loss +by comparing the final prediction to the ground-truth map, +and a knowledge distillation loss on the final prediction and +the intermediate maps extracted by D1 (all losses are based +on Kullback-Leibler divergence between saliency maps; see +Sect. 3.3). In order to have a correspondence between inter- +mediate maps produced by D1 and teacher maps, we em- +ploy HD2S as a teacher, since it naturally and semantically +matches the decoder’s hierarchical structure. +3.2. Encoder structure +Depthwise separable convolutions are widely used for +efficient network design, as in MobileNetV2 [31], com- +monly pre-trained on ImageNet and used as a backbone for +lightweight models. In order to adapt it as a 3D video fea- +ture extractor, we follow the kernel inflation approach in- +troduced in [2] and already employed for static [14] and +dynamic [6] saliency prediction. A 2D convolutional ker- +nel of size Cin × Cout × H × W can be inflated into a 3D +kernel of size Cin × Cout × T × H × W by replicating +its weights along the temporal dimension T. This simple +trick provides a convenient initialization that responds to +common spatial patterns and can be gradually adapted to +temporal dynamics during training, eliminating the burden +of learning basic spatial structures from scratch. Given the +inflated MobileNetV2 encoder, we follow the approach in +FastSal [14] to extract four blocks of concatenated feature +from the whole set of layers. +3.2.1 +Decoder structure and multiple prediction +The set of heterogeneous decoders, employed in our model, +includes D1 (hierarchical map aggregation), D2(U-Net– +like) and D3 (deep layer aggregation). Our realizations of +each of these approaches are designed to process the four +input streams of features extracted by the encoder. +D1 +produces four intermediate saliency maps, while D2 and +D3 produce a map each. The fusion layer that computes +the final output map is implemented as a 1×1 convolu- +tion of the predicted maps. +Architectural details are re- +ported in the supplementary materials. As an additional ef- +ficiency consideration, we note that the high computation +cost of many state-of-the-art approaches due to multiple- +input/single-output (MISO) prediction, where a sequence +of frames is used to predict a single saliency map, usu- +ally referring to the last frame. This provides a full con- +text of previous frames to the model, but also means that, +in order to predict N saliency maps (without interpolation), +N forward passes are also required, with a proportional in- +crease of computational power. In order to further improve +efficiency, we implement a multiple-input/multiple-output +(MIMO) schema for output generation, by designing de- +coders that predict a number of saliency maps equal to the +number of frames provided to the encoder. MIMO decoders +can intrinsically make use of the similarity between con- +secutive saliency maps, and employ this information to re- +duce the computational power required to generate the same +number of saliency maps by MISO decoders. Of course, the +downside is that each frame has a different amount of sur- +rounding context; however, in our experiments this has little +impact on our model’s accuracy. +3.3. Knowledge distillation +Given the presence of multiple decoders in our model, +one of which also produces intermediate saliency maps, +choosing a distillation approach to supervise the student’s +training is not trivial. As illustrated in Fig. 4, we carry out +knowledge distillation by extracting intermediate and final +outputs from a hierarchical teacher to supervise intermedi- +ate of one student decoder and final student outputs. Our +design of the distillation process is guided by several ob- +servations. First and foremost, it is necessary to provide a +training signal at the very output of the model, in order to +train the final fusion layer. Second, carrying out distilla- +tion at the representation level, by enforcing similarity be- +tween teacher and student features, defeats the purpose of +having multiple decoders that are meant to recognize their +own distinctive saliency patterns and should therefore be +free to independently learn their own features. Also, us- +ing saliency maps directly ensures that the output and target +have the same size, so that the use of adaptation layers to +match feature size of student’s and teacher’s can be avoided. +We therefore choose to use intermediate saliency maps from +a hierarchical teacher, HD2S [1], since this makes it possi- +ble in a natural way to affect the model at different depths +of the encoder, without providing as strong a training signal +as internal features. +We formalize our knowledge distillation procedure as +follows. Let V be the space of video sequences and S be the +space of saliency maps (whether for the entire sequence or +for a single frame); let M be a family of models such that +each element in M is a function M : V → Sn+1, which +provides n intermediate and one output saliency maps. We +thus define a teacher T ∈ M and student S ∈ M. For +simplicity, the notations Si and Ti will indicate the i-th map +generated by, respectively, the student and teacher; indexes +from 1 to n will denote intermediate maps, while index n+1 +will refer to the final output. Saliency map distance is mea- +sured by Kullback-Leibler (KL) divergence: +LKL (x, y) = +� +i +yi log yi +xi +, +(1) +with i iterating over spatial locations of the saliency maps. +At each training iteration, we sample a video sequence +v ∈ V and its ground-truth saliency s ∈ S. +The em- + +CNN +Block 1 +CNN +Block 2 +CNN +Block 3 +CNN +Block 4 +Feature Aggregation +Decoder 1 +Decoder 2 +Decoder 3 +6 Intermediate +Saliency +predictions +CNN +Block 4 +Prediction +Teacher +Prediction +Ground +Truth +Teacher +Network +Student +Network +CNN +Block 1 +CNN +Block 2 +CNN +Block 3 +Input +Input +Blocks of Convolutional layers used for +encoders and decoders. Student network +structures are detailed in supplementary +material and Teacher network is the same as +HD2S. +Features from all layers of mobileNet V2 are +reorganized and aggregated to features from +4 abstraction level. +4 intermediate maps from student network +generated from decoder 1 (similar to HD2S ) +1 intermediate map from student network +generated from decoder 2 (U-Net like) +1 intermediate map from student network +generated from decoder 3 (DLA like) +Final predictions generated by fusing all 6 +intermediate maps +4 intermediate maps from teacher +network(HD2S ) +Final predictions generated by teacher +network (pseudo labels) +Ground truth saliency maps +KL divergence +losses are +computed and +back propagated +through student +network +Figure 4: Overview of the proposed multi-decoder architecture with hierarchical knowledge distillation. +ployed loss function aims at minimizing the KL divergence +between student and teacher maps (both intermediate and +final) and between (final) student and ground-truth maps: +L = +n+1 +� +i=1 +LKL +� +Si (v) , Ti (v) +� ++ LKL +� +Sn+1 (v) , s +� +. (2) +3.3.1 +Training with auxiliary dataset +The usage of unlabeled auxiliary datasets in a knowledge +distillation setting has been shown to help boost perfor- +mance [21, 32, 15]. Following this approach, we introduce +a new video distribution W, and extend the loss function +with a term that measures the distance between student’s +predicted saliency maps and “pseudo-labels” (which are, in +fact, also maps) provided by the teacher. As a result, given +a pair of input videos v ∈ V and w ∈ W, the new loss +function becomes: +L = +n+1 +� +i=1 +LKL +� +Si (v) , Ti (v) +� ++ +n+1 +� +i=1 +LKL +� +Si (w) , Ti (w) +� ++LKL +� +Sn+1 (v) , s +� +. +(3) +3.3.2 +Channel reduction with teacher assistant +Previous works have shown that, with a suitable network de- +sign, it is possible to decrease the number of channels in the +encoder’s layers, in order to reduce the computational cost, +without an excessive loss in accuracy [8]. Our channel re- +duction strategy applies multiple knowledge distillation it- +erations: at each of them, a new student is initialized by av- +eraging the weights of each pair of consecutive kernels into +a new kernel. Although kernel ordering is essentially ran- +dom, this approach has been shown to provide a meaningful +initialization to the new student. Additionally, we also ex- +plore the “teacher assistant” [26] distillation strategy: rather +than using the original teacher to perform knowledge dis- +tillation on reduced-channel students, we employ the full- +capacity student (i.e., before any channel reduction) as a +new teacher. As a result, by combining the channel reduc- +tion and teacher assistant, we encourage the model to distill +more information while reducing computational cost. +4. Experiments +4.1. Datasets and Metrics +We conduct experiments on DHF1K [36], +UCF- +Sports [29, 24] and Hollywood2 [23, 24] datasets, com- +monly employed to evaluate video saliency prediction. +DHF1K contains 1000 videos split into 600/100/300 for +training, validation, and test (unreleased). +Eye fixations +are collected from 17 participants in free-viewing experi- +ments. UCF-Sports is a task-driven dataset that includes +150 videos (103 for training, 47 for test) covering 9 sport +activities. Participants were asked to identify the activity in +each video sequence. Hollywood2 includes 1707 videos ex- +tracted from 69 movies and categorized between 12 action +classes. At data collection, 3 observers are free-viewing, 12 +observers are asked recognize the action, and 4 observers +are asked to recognize the scene. 823 videos are used for +training and 884 for test. +We also employ the Kinetic- +400 [17] action recognition benchmark as auxiliary dataset, +used by the teacher to generate additional training inputs +with pseudo-labels. For evaluation purposes, we report re- +sults in terms of the standard metrics for video saliency +prediction [35]: AUC-Judd (AUC-J), AUC-Borji (AUC-B), +Linear Correlation Coefficient (CC), Normalized Scanpath +Saliency (NSS), and Similarity Metric (SIM). + +Table 1: Comparison with SoA on the DHF1K and Hollywood2 test set in both the MISO and MIMO settings. +(a) Prediction accuracy and computational cost on the DHF1K test set. GMACs are +estimated for 16 frames, hence the ×16 multiplication for MISO models. Models +marked with a ∗ are image saliency models. +Models +AUC-J SIM sAUC +CC +NSS +GMACs +#params +Multi-input/single-output (MISO) prediction +SalGAN∗ +0.866 +0.262 0.709 0.370 2.043 +45.73×16 +31.92M +FastSal∗ +0.887 +0.293 0.712 0.426 2.330 +2.64×16 +2.47M +3DSal +0.850 +0.321 0.623 0.356 1.996 136.45×16 +46.15M +TASED +0.895 +0.361 0.712 0.470 2.667 +91.75×16 +21.26M +ViNet +0.908 +0.381 0.729 0.511 2.872 115.28×16 +31.1M +HD2S +0.908 +0.406 0.700 0.503 2.812 +11.08×16 +29.8M +TinyHD-S +0.909 +0.396 0.714 0.505 2.921 +5.57×16 +3.94M +Multi-input/multi-output (MIMO) prediction +SalEMA +0.890 +0.466 0.667 0.449 2.574 +640.16×1 +31.79M +STRA-Net +0.895 +0.355 0.663 0.458 2.558 +266.01×3 +168.02M +UNISAL +0.901 +0.390 0.691 0.490 2.776 +19.42×1 +3.71M +TinyHD-M +0.905 +0.387 0.707 0.493 2.819 +7.95×1 +3.92M +(b) Prediction accuracy on Hollywood2 +Models +AUC-J SIM +CC +NSS +Multi-input/single-output prediction +ACLNet +0.913 +0.757 0.623 3.086 +SalSAC +0.931 +0.529 0.670 3.356 +TASED +0.918 +0.507 0.646 3.302 +ViNet +0.930 +0.550 0.693 3.730 +HD2S +0.936 +0.551 0.670 3.352 +TinyHD-S +0.935 +0.561 0.690 3.815 +Multi-input/multi-output prediction +SalEMA +0.919 +0.487 0.613 3.186 +STRA-Net +0.923 +0.536 0.662 3.478 +UNISAL +0.934 +0.542 0.673 3.901 +TinyHD-M +0.934 +0.553 0.686 3.744 +Frame +GT +Output +D1 (1) +D1 (2) +D1 (3) +D1 (4) +D2 +D3 +Figure 5: Examples of video saliency maps predicted by the proposed model, as well as intermediate maps by multiple +decoders. Values between parentheses indicate one of the intermediate saliency maps by decoder D1. +4.2. Training procedure +Models are trained for 200 epochs using mini-batch +stochastic gradient descent, with a mini-batch size of 12. +The initial learning rate is 0.01, and it is reduced by a factor +of 0.1 at epochs 100, 150, and 180. Input sequence length +is 16 frames, spatially resized to 192×256. We carry out +data augmentation by means of random horizontal flips; in +our experiments, spatial resize and cropping do not lead to +significant benefits. When the teacher assistant strategy is +employed for channel reduction, we perform two additional +knowledge distillation, each time training a new student net- +work whose encoder contains, respectively, half and a quar- +ter of the original number of channel at each encoder layer. +4.3. Performance comparison with state-of-the-art +models +In these experiments, we report results of our model +in both the MISO and MIMO configurations (respectively, +TinyHD-S and TinyHD-M), trained with the auxiliary un- +labeled dataset but without channel reduction that using the +teacher assistant strategy (which introduces trade-offs be- +tween accuracy and computational costs that will be dis- +cussed later). +We also report the number of multiply- +accumulate operations (MAC) carried out by each method2 +to generate a 16-frame saliency sequence. +Results on +2Values are computed from official implementations when available +and from our own implementations otherwise. + +18-PAR4 +JASONDUFNER +SNDT +484 +LIVEDHF1K are shown in Table 1a. +In the MISO configu- +ration, our model is on par with state-of-the-art methods +(and even better on NSS), but only employs a fraction of +the their computational cost. In the MIMO configuration, +our method sets a new state of the art, outperforming (on +four metrics out of five) also UNISAL, which has a sim- +ilar number of parameters but is about twice as demand- +ing in terms of GMACs. Fig. 5 presents a few examples +of saliency predictions by our model3. For each example, +we also show the intermediate maps provided by each de- +coder. Qualitatively, our model predicts reasonable saliency +regions, sometimes identifying additional elements not in- +cluded in the ground truth (e.g., the third example). In- +termediate maps also exhibit a certain variability, although +similar patterns can be found in pairs (e.g., maps 1-2 and +maps 3-4 from D1, and maps from D2 and D3). In general, +the highest-level map from D1 (the fourth) mostly affects +the output prediction: this is expected, since the correspond- +ing architecture matches the teacher’s. However, the fusion +layer includes all information from intermediate maps, as +shown in the last example, where two salient areas identi- +fied by the highest-level map from D1 are discarded. +Table 1b and 2 report results on Hollywood2 and UCF- +Sports. While the model performs very well on the for- +mer, especially in the more efficient MIMO setting, ViNet +and UNISAL achieve higher accuracy on UCF-Sports. This +may be due to the lower performance of the HD2S teacher +on that specific dataset, and to the arguable suitability of +UCF-Sports as a video saliency prediction benchmark: the +vast majority of its videos has fewer than 100 frames, and +user fixations are driven by action classification, rather than +free-viewing saliency [1]. +4.4. Ablation studies +In order to experimentally substantiate our architectural +and methodological choices, we carry out a set of abla- +tion studies on each component of the model. the results +of these experiments are reported on the DHF1K validation +set, since testing set is not publicly available. First, we as- +sess the effect of our heterogeneous multi-decoder strategy, +evaluating the model’s performance under several decoder +configurations. We carry out this experiment in the MISO +configuration, which achieves higher accuracy, as shown in +Table 1a. In order to demonstrate the importance of com- +bining different decoder architectures, Table 3 reports re- +sults when using homogeneous decoders in our architec- +ture. Table 3 show that the heterogeneous approach gen- +erally performs better than configurations with a single de- +coder type, most remarkably in the NSS metric. For the +sake of completeness, we also show configurations where +a smaller number of homogeneous decoders are employed; +3More examples are provided in the supplementary materials, as well +as a visual comparison with state-of-the-art models. +Table 2: Performance comparison on UCF-Sports in both +the MISO and MIMO settings. +Models +AUC-J SIM +CC +NSS +Multi-input/single-output prediction +ACLNet +0.897 0.406 0.510 2.567 +3DSal +0.881 0.478 0.590 2.802 +TASED +0.899 0.469 0.582 2.920 +ViNet +0.924 0.522 0.673 3.620 +HD2S +0.904 0.507 0.604 3.114 +TinyHD-S +0.918 0.510 0.624 3.280 +Multi-input/multi-output prediction +SalEMA +0.906 0.431 0.544 2.638 +STRA-Net +0.910 0.479 0.593 3.018 +UNISAL +0.918 0.523 0.644 3.381 +TinyHD-M 0.911 0.499 0.609 3.234 +Table 3: Performance of our architecture with homoge- +neous decoders, on the DHF1K validation set. +Number +of parameters of models with homogeneous decoders are: +D1×1 (2.55M), D2×1 (3.57M). D3×1 (2.53M); D1×2 +(2.75M), D2×2 (4.78M). D3×2 (2.70M); D1×3 (2.95M), +D2×3 (6.00M). D3×3 (2.88M); TinyHD-S (3.94M). +Decoder +AUC-J AUC-B +CC +NSS +SIM GMACs +D1×1 +0.8993 0.8210 0.4881 2.8163 0.3939 3.55×16 +D2×1 +0.9040 0.8235 0.4837 2.7976 0.3820 2.88×16 +D3×1 +0.9034 0.8248 0.4836 2.7851 0.3794 2.45×16 +D1×2 +0.8998 0.8195 0.4882 2.8256 0.3928 5.45×16 +D2×2 +0.9046 0.8251 0.4855 2.8117 0.3806 4.11×16 +D3×2 +0.9046 0.8239 0.4864 2.8095 0.3819 3.24×16 +D1×3 +0.9013 0.8253 0.4922 2.8420 0.3924 7.35×16 +D2×3 +0.9049 0.8266 0.4847 2.8042 0.3774 5.33×16 +D3×3 +0.9047 0.8242 0.4845 2.7967 0.3799 4.03×16 +TinyHD-S 0.9075 0.8244 0.4945 2.8735 0.3887 5.57×16 +these setups are, of course, more computationally efficient, +but exhibit lower performance on average in the accuracy +metrics. +In the second part of our ablation study, we evaluate of +the impact of our knowledge distillation strategy. Table 4 re- +ports the results obtained by the proposed model, in MISO +configuration, when trained on ground-truth maps only, +and when gradually adding knowledge distillation terms on +DHF1K and on Kinetics-400, using HD2S as teacher. The +full loss setting achieves better performance on average — +as previously. This is most evident in the NSS metric. + +Table 4: Impact of loss terms on our model in the MISO +configuration, starting from training on ground-truth (GT) +maps only, and gradually adding knowledge distillation +terms on DHF1K (target dataset or TD) and on Kinetics- +400 (auxiliary dataset or AD), using HD2S as a teacher. +Loss term +AUC-J AUC-B +CC +NSS +SIM +GT maps +0.9033 0.8286 0.4864 2.7680 0.3765 ++ K.D. on TD +0.9058 0.8237 0.4875 2.8182 0.3846 ++ K.D. on AD 0.9075 0.8244 0.4945 2.8735 0.3887 +4.5. Channel reduction with teacher assistant +Finally, we investigate further reducing computational +costs by means of our channel reduction strategy: mul- +tiple distillation steps are carried out, with each student +progressively halving its number of encoding and decod- +ing features, as described in Sect. 3.3.2. We also evalu- +ate the performance of this approach when training on the +original teacher (HD2S) and when using the “teacher as- +sistant” technique, with the full-capacity student used as a +teacher. Table 5 reports results, on both MISO and MIMO +settings, after one and two reduction steps steps, respec- +tively resulting in models with half (marked as × 1 +2) and a +quarter (marked as × 1 +4) of the original number of convolu- +tional features (marked as ×1). Rows with “+TA” denote +the use of the full-capacity student as teacher for knowl- +edge distillation, rather than HD2S. As expected, channel +reduction introduces a trade-off between retaining the ac- +curacy of the original model and reducing computational +costs. As multiply-accumulate operations and model pa- +rameters are significantly reduced, accuracy also decreases, +most evidently in the NSS and, to a smaller extent, in the +SIM metrics. It is noteworthy that configurations employing +a teacher assistant outperform the counterpart using HD2S. +5. Conclusions +In this work, starting from the observation that differ- +ent encoder-decoder architectures recognize specific video +saliency patterns, we propose a heterogeneous multi- +decoder architecture that leverages simpler versions of +state-of-the-art decoding strategies to achieve high predic- +tion accuracy at a fraction of the computational cost. We +train our model in a multi-target knowledge distillation set- +ting, where a hierarchical decoder is used as a teacher to +supervise a matching internal decoder in our model and the +output prediction; additionally, we employ semi-supervised +learning on an unlabeled auxiliary dataset to further im- +prove model generalization. Our model sets new state-of- +the-art performance when employed in a multi-input/multi- +output setting, while being significantly more efficient in +terms of floating-point operations and number of parame- +Table 5: Performance of the proposed model when employ- +ing channel reduction and teacher assistant distillation. +(a) Number of parameters of models with reduced channels and +GMACs reported on generating 16 output saliency maps. +GMACs +#params +Models +×1 +× 1 +2 +× 1 +4 +×1 +× 1 +2 +× 1 +4 +TinyHD-S 89.12 59.52 37.44 3.94M 1.37M 513.1k +TinyHD-M 7.95 +6.92 +4.06 3.92M 1.37M 515.3k +(b) Performance of channel reduction reported on DHF1K valida- +tion set in both the MISO and MIMO settings. +Models +AUC-J AUC-B +CC +NSS +SIM +Multi-input/single-output prediction +TinyHD-S×1 +0.9075 0.8244 0.4945 2.8735 0.3887 +TinyHD-S× 1 +2 +0.9038 0.8331 0.4754 2.7194 0.3641 ++TA +0.9052 0.8330 0.4805 2.7317 0.3684 +TinyHD-S× 1 +4 +0.9005 0.8285 0.4560 2.5830 0.3514 ++TA +0.9018 0.8318 0.4667 2.6329 0.3569 +Multi-input/multi-output prediction +TinyHD-M×1 0.9050 0.8239 0.4880 2.8178 0.3844 +TinyHD-M× 1 +2 0.9016 0.8272 0.4687 2.6718 0.3612 ++TA +0.9021 0.8307 0.4718 2.6726 0.3630 +TinyHD-M× 1 +4 0.8980 0.8294 0.4487 2.5257 0.3438 ++TA +0.8999 0.8333 0.4564 2.5581 0.3478 +ters. We further push the limits of our model by applying +a channel reduction procedure through multiple distillation +steps and using the full-capacity student as a teacher, ac- +cording to the “teacher assistant” paradigm. In the resulting +model, the number of floating-point operations is approx- +imately halved compared to the full-capacity version, and +the number of parameters becomes as small as about 500k, +taking about 2.4 MB storage space without compression. +Acknowledgments +This publication has been financially supported by: +Science Foundation Ireland (SFI) under grant number +SFI/12/RC/2289 P2; +Regione Sicilia, +Italy, +RehaStart +project (grant identifier: +PO FESR 2014/2020, Azione +1.1.5, N. 08ME6201000222, CUP G79J18000610007); +University of Catania, Piano della Ricerca di Ateneo, +2020/2022, Linea 2D; MIUR, Italy, Azione 1.2 “Mobilit`a +dei Ricercatori” (grant identifier: Asse I, PON R&I 2014- +2020, id. 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Pattern Recognition Letters, +147:78–84, 2021. + diff --git a/0NE3T4oBgHgl3EQfmwqR/content/tmp_files/load_file.txt b/0NE3T4oBgHgl3EQfmwqR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..79b2232650e5e77aa7384eb52bc81be0d8dee415 --- /dev/null +++ b/0NE3T4oBgHgl3EQfmwqR/content/tmp_files/load_file.txt @@ -0,0 +1,743 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf,len=742 +page_content='TinyHD: Efficient Video Saliency Prediction with Heterogeneous Decoders using Hierarchical Maps Distillation Feiyan Hu1, Simone Palazzo2, Federica Proietto Salanitri2, Giovanni Bellitto2, Morteza Moradi2, Concetto Spampinato2, Kevin McGuinness1 1 Insight SFI Research Centre for Data Analytics, Dublin City University, Dublin, Ireland {feiyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='hu, kevin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='mcguinness}@dcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='ie 2 PeRCeiVe Lab, University of Catania, Catania, Italy {simone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='palazzo, concetto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='spampinato}@unict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='it Abstract Video saliency prediction has recently attracted atten- tion of the research community, as it is an upstream task for several practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' However, current so- lutions are particularly computationally demanding, espe- cially due to the wide usage of spatio-temporal 3D convolu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' We observe that, while different model architectures achieve similar performance on benchmarks, visual varia- tions between predicted saliency maps are still significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Inspired by this intuition, we propose a lightweight model that employs multiple simple heterogeneous decoders and adopts several practical approaches to improve accuracy while keeping computational costs low, such as hierarchi- cal multi-map knowledge distillation, multi-output saliency prediction, unlabeled auxiliary datasets and channel re- duction with teacher assistant supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Our approach achieves saliency prediction accuracy on par or better than state-of-the-art methods on DFH1K, UCF-Sports and Hol- lywood2 benchmarks, while enhancing significantly the ef- ficiency of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Introduction Video saliency prediction aims at estimating patterns of human attention during free-viewing of dynamic scenes, to emulate the capabilities of the human visual system of quickly analyzing and interpreting the surrounding environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Due to its several practical applications [7, 5, 28, 33, 9, 40, 12, 27], it is an active area of research in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' However, the solution to this problem is not triv- ial, for several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' First, attention mechanisms in the human visual system are not fully known, so it is not clear how to emulate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Also, it requires complex modeling of both visual features and their motion and interaction: an object with striking visual patterns may be shadowed by a TASED 100% 84% 72% ViNet 84% 100% 75% HD2S 72% TASED 75% ViNet 100% HD2S CC Metric TASED 100% 73% 57% ViNet 73% 100% 58% HD2S 57% TASED 58% ViNet 100% HD2S SIM Metric Figure 1: Measuring prediction similarity among video saliency prediction models TASED, ViNet and HD2S on DHF1K validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' bland element of the scene that starts moving in a pecu- liar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Finally, modeling the temporal dimension may become computationally expensive, especially with current deep learning methods based on spatio-temporal 3D convo- lutions, thus limiting the applicability to low-power devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Many solutions have been proposed, based on different assumptions on how to capture video saliency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' It is inter- esting to note that, in spite of the remarkably different re- search directions followed by the variety of works in the literature, top results over video saliency prediction bench- marks are very close [1, 3, 36], suggesting that predictions of different models are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' We assessed the validity of this conclusion by comparing three of the best perform- ing methods on the DHF1K dataset [36] — TASED [25], HD2S [1] and ViNet [16] — not in terms of their scores on summary metrics, but in terms of the relative similarity of the predicted saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' To illustrate our findings, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 1 shows pairwise similarities between predicted maps over two common metrics, Linear Correlation Coefficient (CC) and Similarity (SIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Although the three approaches achieve similar scores on both metrics on DHF1K (between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='470 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='511 for CC, and between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='361 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='406 for SIM), the same metrics computed between each other are relatively low, compared to what one would expect given their similarity to the saliency ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' A visual in- spection of the saliency maps generated by methods under arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='04619v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='CV] 11 Jan 2023 comparison confirms this behavior: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 2 shows that it is common to find cases where each approach produces re- markably different saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Notably, all three methods — TASED, HD2S and ViNet — are encoder-decoder networks and share the same en- coder, S3D [38], while employing different decoding strate- gies (a U-Net–like approach for TASED and ViNet, a hi- erarchical map aggregation for HD2S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' This suggests that a key factor underlying differences between current video saliency prediction approaches lies in the way encoded fea- tures are processed in the decoding path, leading to models that learn specific (and often exclusive) representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' We strengthen this hypothesis by experimenting with another decoding strategy, exemplified by DLA [39], which com- bines hierarchical decoding with complex feature interac- tions: the results, also included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 2, show yet another saliency prediction pattern, while using the same encoder network, S3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' These results lead us to hypothesize that dif- ferent model architectures introduce different inductive bi- ases, which are more suitable to recognize certain patterns more than others, thus requiring to increase model capacity in order to generalize well to multiple saliency dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Indeed, the size of weights of models in our analysis range between 82 MB and 116 MB, and the size of the top ten models in the DHF1K leaderboard1 is on average 238 MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Given these premises, instead of increasing the complex- ity of a single decoding strategy, it may be more efficient to employ multiple simpler architectures with fewer param- eters, relying on each architecture’s capability to attend to different salient regions and combining their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Hence, we propose TinyHD, a lightweight, efficient and heteroge- neous multi-decoder architecture for video saliency predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' The proposed method is inspired by encoder-decoder architectures, but introduces the adoption of heterogeneous decoding strategies in order to reduce the complexity of each decoder, increasing efficiency (the weights of the re- sulting model take only 16 MB) and improving the accu- racy of predictions, as we show in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Fur- thermore, along the direction of reducing computational costs while retaining high accuracy, we also introduce a novel knowledge distillation approach, based on exploiting a teacher with multiple hierarchical predictions: this allows the model to freely learn its own features, since no explicit conditioning on representations is enforced, while at the same time receiving a supervision signal that encodes in- formation at different layers of abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Experiments confirm that our model can generate high- quality predictions with low computational costs and model size (only 16 MB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' We assess the impact of our heteroge- neous multi-decoder strategy by carrying out extensive ab- lation studies and comparing alternative architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' We also demonstrate the effectiveness of our knowledge dis- 1https://mmcheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='net/videosal/ tillation strategy, compared to the employment of a non- hierarchical teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' To summarize our contributions: We propose a decoding strategy for video saliency pre- diction which combines heterogeneous decoders to ex- ploit the specific pattern analysis capabilities, while re- ducing the overall model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' To our knowl- edge, we are first to propose multiple saliency maps output using 3D CNN to improve model efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' We employ a knowledge distillation approach based on a hierarchical teacher, providing saliency maps es- timated from different abstraction layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Extensive experiments show that our model achieves state-of-the-art performance on the DHF1K bench- mark, at lower computational costs of current meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Ablation studies support the motivations for our decoding and knowledge distillation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Related Work The main contributions of the proposed approach con- sist of a novel heterogeneous multi-decoder scheme, which combines lightweight versions of common decoding strate- gies, and a multi-objective knowledge distillation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' In this section, we briefly present the state-of-the-art on these topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Decoding strategies for video saliency prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Among recent methods from the state-of-the-art for video saliency prediction, leveraging encoder-decoder networks can be considered a mainstream approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' however, sev- eral architectural variations have been proposed for feature sharing between encoder and decoder and for output recon- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' As shown in the taxonomy presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 3, a simpler class of approaches employs independent encoder and decoder, with no feature sharing between the two paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Among these, approaches based on recurrent layers typ- ically model temporal dynamics at the bottleneck of the architecture [18, 37, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Non-recurrent architectures, in- stead, model time by means of 3D convolutions [42, 38, 4], Other approaches employ architectures similar to U-Net by introducing skip connections that encourage feature shar- ing between encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' TASED [25] aggre- gates spatio-temporal features through the use of auxiliary pooling for reducing the temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' ViNet [16] integrates S3D features from multiple hierarchical levels by employing trilinear interpolation and 3D convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' UNISAL [6] proposes a multi-objective unified framework for both 2D and 3D saliency with domain-specific modules and a lightweight recurrent architecture to handle temporal dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' While single-decoder approaches are common, multi-decoder output integration has recently attracted in- terest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' DVA [35] and HD2S [1] fuse maps predicted by independent decoders operating at different abstraction lev- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' RecSal [30] predicts multiple saliency maps in a multi- objective training framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Recent works introduce more (a) Frame (b) GT (c) TASED (d) ViNet (e) HD2S (f) DLA Figure 2: Examples of video saliency maps from state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Although they achieve very similar performance on popular metrics, remarkable differences can be seem in the learned saliency patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Figure 3: A taxonomy of decoding strategies commonly employed in video saliency prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Subfigures(top- left, top-right, bottom-left, bottom-right): Independent en- coder and decoder, with no feature sharing between the two paths [4, 18, 22, 37, 42];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' U-Net–like architecture, with fea- tures sharing between encoder and decoder [6, 16, 19, 25];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Deep Layer Aggregation [39];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Hierarchical intermediate map aggregation [1, 30, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' complex feature interactions among decoding paths, where high-resolution features are affected by deeper high-level features, as in DLA [39] and TSFP-Net [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' All of the ap- proaches presented employ either a single-decoder archi- tecture or a homogeneous multi-decoder one, where differ- ences between decoders lie in the number of layers rather than in their structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' In our work, we propose an architec- ture which combines heterogeneous decoder structures, in order to better exploit their distinctive saliency prediction properties and thus increase computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Knowledge distillation for visual saliency prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Knowledge distillation [13, 11] is commonly employed to train an efficient student model from a more complex teacher model, with higher accuracy than when training the student directly from dataset labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Several knowl- edge distillation approaches have been recently proposed for video saliency prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' SKD-DVA [20] proposes spatio-temporal knowledge distillation with two teachers and two students, with each pair focusing on either spatial or temporal transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' SV2T-SS [41] distills corresponding features of teacher and student (implemented as encoder- decoder networks), based on first- and second-order feature statistics transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' UVA-DVA [10] employs separate spatial and temporal teachers, whose knowledge is transferred to a single student model, which then fuses the resulting fea- tures in the final saliency prediction, achieving reasonable accuracy at impressive speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Leveraging knowledge dis- tillation for video salient object detection is the main theme of the work in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' The knowledge distillation setting pro- posed in our work differs from existing techniques in two main aspects: 1) we define a multi-objective distillation tar- get on saliency maps directly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 2) we employ a hierarchical model as a teacher in order to further capture differences in saliency patterns extracted at multiple scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Overview The overall architecture of the proposed saliency predic- tion network with knowledge distillation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Following the taxonomy introduced in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 2, a shared en- coder extracts multi-level features that are then processed by three parallel decoding architectures: decoder 1 (D1) implements hierarchical intermediate maps aggregation (in- spired by HD2S);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' decoder 2 (D2) employs a U-Net–like ap- proach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' decoder 3 (D3) is based on deep layer aggregation concepts (as in DLA [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' The hierarchical aggregation decoder (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=', decoder 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 4) produces four intermediate saliency maps from fea- tures extracted at different encoder layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' then, the set of DVSS三三三三predictions from all decoders are fused into the final pre- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' At training time, we compute a supervised loss by comparing the final prediction to the ground-truth map, and a knowledge distillation loss on the final prediction and the intermediate maps extracted by D1 (all losses are based on Kullback-Leibler divergence between saliency maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' In order to have a correspondence between inter- mediate maps produced by D1 and teacher maps, we em- ploy HD2S as a teacher, since it naturally and semantically matches the decoder’s hierarchical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Encoder structure Depthwise separable convolutions are widely used for efficient network design, as in MobileNetV2 [31], com- monly pre-trained on ImageNet and used as a backbone for lightweight models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' In order to adapt it as a 3D video fea- ture extractor, we follow the kernel inflation approach in- troduced in [2] and already employed for static [14] and dynamic [6] saliency prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' A 2D convolutional ker- nel of size Cin × Cout × H × W can be inflated into a 3D kernel of size Cin × Cout × T × H × W by replicating its weights along the temporal dimension T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' This simple trick provides a convenient initialization that responds to common spatial patterns and can be gradually adapted to temporal dynamics during training, eliminating the burden of learning basic spatial structures from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Given the inflated MobileNetV2 encoder, we follow the approach in FastSal [14] to extract four blocks of concatenated feature from the whole set of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='1 Decoder structure and multiple prediction The set of heterogeneous decoders, employed in our model, includes D1 (hierarchical map aggregation), D2(U-Net– like) and D3 (deep layer aggregation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Our realizations of each of these approaches are designed to process the four input streams of features extracted by the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' D1 produces four intermediate saliency maps, while D2 and D3 produce a map each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' The fusion layer that computes the final output map is implemented as a 1×1 convolu- tion of the predicted maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Architectural details are re- ported in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' As an additional ef- ficiency consideration, we note that the high computation cost of many state-of-the-art approaches due to multiple- input/single-output (MISO) prediction, where a sequence of frames is used to predict a single saliency map, usu- ally referring to the last frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' This provides a full con- text of previous frames to the model, but also means that, in order to predict N saliency maps (without interpolation), N forward passes are also required, with a proportional in- crease of computational power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' In order to further improve efficiency, we implement a multiple-input/multiple-output (MIMO) schema for output generation, by designing de- coders that predict a number of saliency maps equal to the number of frames provided to the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' MIMO decoders can intrinsically make use of the similarity between con- secutive saliency maps, and employ this information to re- duce the computational power required to generate the same number of saliency maps by MISO decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Of course, the downside is that each frame has a different amount of sur- rounding context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' however, in our experiments this has little impact on our model’s accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Knowledge distillation Given the presence of multiple decoders in our model, one of which also produces intermediate saliency maps, choosing a distillation approach to supervise the student’s training is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 4, we carry out knowledge distillation by extracting intermediate and final outputs from a hierarchical teacher to supervise intermedi- ate of one student decoder and final student outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Our design of the distillation process is guided by several ob- servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' First and foremost, it is necessary to provide a training signal at the very output of the model, in order to train the final fusion layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Second, carrying out distilla- tion at the representation level, by enforcing similarity be- tween teacher and student features, defeats the purpose of having multiple decoders that are meant to recognize their own distinctive saliency patterns and should therefore be free to independently learn their own features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Also, us- ing saliency maps directly ensures that the output and target have the same size, so that the use of adaptation layers to match feature size of student’s and teacher’s can be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' We therefore choose to use intermediate saliency maps from a hierarchical teacher, HD2S [1], since this makes it possi- ble in a natural way to affect the model at different depths of the encoder, without providing as strong a training signal as internal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' We formalize our knowledge distillation procedure as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Let V be the space of video sequences and S be the space of saliency maps (whether for the entire sequence or for a single frame);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' let M be a family of models such that each element in M is a function M : V → Sn+1, which provides n intermediate and one output saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' We thus define a teacher T ∈ M and student S ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' For simplicity, the notations Si and Ti will indicate the i-th map generated by, respectively, the student and teacher;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' indexes from 1 to n will denote intermediate maps, while index n+1 will refer to the final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Saliency map distance is mea- sured by Kullback-Leibler (KL) divergence: LKL (x, y) = � i yi log yi xi , (1) with i iterating over spatial locations of the saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' At each training iteration, we sample a video sequence v ∈ V and its ground-truth saliency s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' The em- CNN Block 1 CNN Block 2 CNN Block 3 CNN Block 4 Feature Aggregation Decoder 1 Decoder 2 Decoder 3 6 Intermediate Saliency predictions CNN Block 4 Prediction Teacher Prediction Ground Truth Teacher Network Student Network CNN Block 1 CNN Block 2 CNN Block 3 Input Input Blocks of Convolutional layers used for encoders and decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Student network structures are detailed in supplementary material and Teacher network is the same as HD2S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Features from all layers of mobileNet V2 are reorganized and aggregated to features from 4 abstraction level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='4 intermediate maps from student network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='generated from decoder 1 (similar to HD2S ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='1 intermediate map from student network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='generated from decoder 2 (U-Net like) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='1 intermediate map from student network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='generated from decoder 3 (DLA like) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='Final predictions generated by fusing all 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='intermediate maps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='4 intermediate maps from teacher ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='network(HD2S ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='Final predictions generated by teacher ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='network (pseudo labels) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='Ground truth saliency maps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='KL divergence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='losses are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='computed and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='back propagated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='through student ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='Figure 4: Overview of the proposed multi-decoder architecture with hierarchical knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' ployed loss function aims at minimizing the KL divergence between student and teacher maps (both intermediate and final) and between (final) student and ground-truth maps: L = n+1 � i=1 LKL � Si (v) , Ti (v) � + LKL � Sn+1 (v) , s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' (2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='1 Training with auxiliary dataset The usage of unlabeled auxiliary datasets in a knowledge distillation setting has been shown to help boost perfor- mance [21, 32, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Following this approach, we introduce a new video distribution W, and extend the loss function with a term that measures the distance between student’s predicted saliency maps and “pseudo-labels” (which are, in fact, also maps) provided by the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' As a result, given a pair of input videos v ∈ V and w ∈ W, the new loss function becomes: L = n+1 � i=1 LKL � Si (v) , Ti (v) � + n+1 � i=1 LKL � Si (w) , Ti (w) � +LKL � Sn+1 (v) , s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' (3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='2 Channel reduction with teacher assistant Previous works have shown that, with a suitable network de- sign, it is possible to decrease the number of channels in the encoder’s layers, in order to reduce the computational cost, without an excessive loss in accuracy [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Our channel re- duction strategy applies multiple knowledge distillation it- erations: at each of them, a new student is initialized by av- eraging the weights of each pair of consecutive kernels into a new kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Although kernel ordering is essentially ran- dom, this approach has been shown to provide a meaningful initialization to the new student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Additionally, we also ex- plore the “teacher assistant” [26] distillation strategy: rather than using the original teacher to perform knowledge dis- tillation on reduced-channel students, we employ the full- capacity student (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=', before any channel reduction) as a new teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' As a result, by combining the channel reduc- tion and teacher assistant, we encourage the model to distill more information while reducing computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Datasets and Metrics We conduct experiments on DHF1K [36], UCF- Sports [29, 24] and Hollywood2 [23, 24] datasets, com- monly employed to evaluate video saliency prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' DHF1K contains 1000 videos split into 600/100/300 for training, validation, and test (unreleased).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Eye fixations are collected from 17 participants in free-viewing experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' UCF-Sports is a task-driven dataset that includes 150 videos (103 for training, 47 for test) covering 9 sport activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Participants were asked to identify the activity in each video sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Hollywood2 includes 1707 videos ex- tracted from 69 movies and categorized between 12 action classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' At data collection, 3 observers are free-viewing, 12 observers are asked recognize the action, and 4 observers are asked to recognize the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 823 videos are used for training and 884 for test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' We also employ the Kinetic- 400 [17] action recognition benchmark as auxiliary dataset, used by the teacher to generate additional training inputs with pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' For evaluation purposes, we report re- sults in terms of the standard metrics for video saliency prediction [35]: AUC-Judd (AUC-J), AUC-Borji (AUC-B), Linear Correlation Coefficient (CC), Normalized Scanpath Saliency (NSS), and Similarity Metric (SIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Table 1: Comparison with SoA on the DHF1K and Hollywood2 test set in both the MISO and MIMO settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' (a) Prediction accuracy and computational cost on the DHF1K test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' GMACs are estimated for 16 frames, hence the ×16 multiplication for MISO models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Models marked with a ∗ are image saliency models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Models AUC-J SIM sAUC CC NSS GMACs #params Multi-input/single-output (MISO) prediction SalGAN∗ 0.' metadata={'source': 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3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='186 STRA-Net 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='536 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='662 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='478 UNISAL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='934 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='542 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='673 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='901 TinyHD-M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='934 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='686 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='744 Frame GT Output D1 (1) D1 (2) D1 (3) D1 (4) D2 D3 Figure 5: Examples of video saliency maps predicted by the proposed model, as well as intermediate maps by multiple decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Values between parentheses indicate one of the intermediate saliency maps by decoder D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Training procedure Models are trained for 200 epochs using mini-batch stochastic gradient descent, with a mini-batch size of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' The initial learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='01, and it is reduced by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='1 at epochs 100, 150, and 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Input sequence length is 16 frames, spatially resized to 192×256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' We carry out data augmentation by means of random horizontal flips;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' in our experiments, spatial resize and cropping do not lead to significant benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' When the teacher assistant strategy is employed for channel reduction, we perform two additional knowledge distillation, each time training a new student net- work whose encoder contains, respectively, half and a quar- ter of the original number of channel at each encoder layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Performance comparison with state-of-the-art models In these experiments, we report results of our model in both the MISO and MIMO configurations (respectively, TinyHD-S and TinyHD-M), trained with the auxiliary un- labeled dataset but without channel reduction that using the teacher assistant strategy (which introduces trade-offs be- tween accuracy and computational costs that will be dis- cussed later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' We also report the number of multiply- accumulate operations (MAC) carried out by each method2 to generate a 16-frame saliency sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Results on 2Values are computed from official implementations when available and from our own implementations otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 18-PAR4 JASONDUFNER SNDT 484 LIVEDHF1K are shown in Table 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' In the MISO configu- ration, our model is on par with state-of-the-art methods (and even better on NSS), but only employs a fraction of the their computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' In the MIMO configuration, our method sets a new state of the art, outperforming (on four metrics out of five) also UNISAL, which has a sim- ilar number of parameters but is about twice as demand- ing in terms of GMACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 5 presents a few examples of saliency predictions by our model3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' For each example, we also show the intermediate maps provided by each de- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Qualitatively, our model predicts reasonable saliency regions, sometimes identifying additional elements not in- cluded in the ground truth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=', the third example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' In- termediate maps also exhibit a certain variability, although similar patterns can be found in pairs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=', maps 1-2 and maps 3-4 from D1, and maps from D2 and D3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' In general, the highest-level map from D1 (the fourth) mostly affects the output prediction: this is expected, since the correspond- ing architecture matches the teacher’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' However, the fusion layer includes all information from intermediate maps, as shown in the last example, where two salient areas identi- fied by the highest-level map from D1 are discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Table 1b and 2 report results on Hollywood2 and UCF- Sports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' While the model performs very well on the for- mer, especially in the more efficient MIMO setting, ViNet and UNISAL achieve higher accuracy on UCF-Sports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' This may be due to the lower performance of the HD2S teacher on that specific dataset, and to the arguable suitability of UCF-Sports as a video saliency prediction benchmark: the vast majority of its videos has fewer than 100 frames, and user fixations are driven by action classification, rather than free-viewing saliency [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Ablation studies In order to experimentally substantiate our architectural and methodological choices, we carry out a set of abla- tion studies on each component of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' the results of these experiments are reported on the DHF1K validation set, since testing set is not publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' First, we as- sess the effect of our heterogeneous multi-decoder strategy, evaluating the model’s performance under several decoder configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' We carry out this experiment in the MISO configuration, which achieves higher accuracy, as shown in Table 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' In order to demonstrate the importance of com- bining different decoder architectures, Table 3 reports re- sults when using homogeneous decoders in our architec- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Table 3 show that the heterogeneous approach gen- erally performs better than configurations with a single de- coder type, most remarkably in the NSS metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' For the sake of completeness, we also show configurations where a smaller number of homogeneous decoders are employed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 3More examples are provided in the supplementary materials, as well as a visual comparison with state-of-the-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Table 2: Performance comparison on UCF-Sports in both the MISO and MIMO settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Models AUC-J SIM CC NSS Multi-input/single-output prediction ACLNet 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='802 TASED 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='899 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='469 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='582 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='920 ViNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='924 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='522 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='510 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='624 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='280 Multi-input/multi-output prediction SalEMA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='544 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='638 STRA-Net 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='910 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='479 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='593 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='018 UNISAL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='523 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='644 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='381 TinyHD-M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='609 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='234 Table 3: Performance of our architecture with homoge- neous decoders, on the DHF1K validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Number of parameters of models with homogeneous decoders are: 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='57×16 these setups are, of course, more computationally efficient, but exhibit lower performance on average in the accuracy metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' In the second part of our ablation study, we evaluate of the impact of our knowledge distillation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Table 4 re- ports the results obtained by the proposed model, in MISO configuration, when trained on ground-truth maps only, and when gradually adding knowledge distillation terms on DHF1K and on Kinetics-400, using HD2S as teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' The full loss setting achieves better performance on average — as previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' This is most evident in the NSS metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Table 4: Impact of loss terms on our model in the MISO configuration, starting from training on ground-truth (GT) maps only, and gradually adding knowledge distillation terms on DHF1K (target dataset or TD) and on Kinetics- 400 (auxiliary dataset or AD), using HD2S as a teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Loss term AUC-J AUC-B CC NSS SIM GT maps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='9033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='8286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='4864 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='7680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='3765 + K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' on TD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='9058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='8237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='4875 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='8182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='3846 + K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' on AD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='9075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='8244 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='4945 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='8735 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='3887 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Channel reduction with teacher assistant Finally, we investigate further reducing computational costs by means of our channel reduction strategy: mul- tiple distillation steps are carried out, with each student progressively halving its number of encoding and decod- ing features, as described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' We also evalu- ate the performance of this approach when training on the original teacher (HD2S) and when using the “teacher as- sistant” technique, with the full-capacity student used as a teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Table 5 reports results, on both MISO and MIMO settings, after one and two reduction steps steps, respec- tively resulting in models with half (marked as × 1 2) and a quarter (marked as × 1 4) of the original number of convolu- tional features (marked as ×1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Rows with “+TA” denote the use of the full-capacity student as teacher for knowl- edge distillation, rather than HD2S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' As expected, channel reduction introduces a trade-off between retaining the ac- curacy of the original model and reducing computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' As multiply-accumulate operations and model pa- rameters are significantly reduced, accuracy also decreases, most evidently in the NSS and, to a smaller extent, in the SIM metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' It is noteworthy that configurations employing a teacher assistant outperform the counterpart using HD2S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Conclusions In this work, starting from the observation that differ- ent encoder-decoder architectures recognize specific video saliency patterns, we propose a heterogeneous multi- decoder architecture that leverages simpler versions of state-of-the-art decoding strategies to achieve high predic- tion accuracy at a fraction of the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' We train our model in a multi-target knowledge distillation set- ting, where a hierarchical decoder is used as a teacher to supervise a matching internal decoder in our model and the output prediction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' additionally, we employ semi-supervised learning on an unlabeled auxiliary dataset to further im- prove model generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Our model sets new state-of- the-art performance when employed in a multi-input/multi- output setting, while being significantly more efficient in terms of floating-point operations and number of parame- Table 5: Performance of the proposed model when employ- ing channel reduction and teacher assistant distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' (a) Number of parameters of models with reduced channels and GMACs reported on generating 16 output saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' GMACs #params Models ×1 × 1 2 × 1 4 ×1 × 1 2 × 1 4 TinyHD-S 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='12 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='52 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='44 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='94M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='37M 513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='1k TinyHD-M 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='95 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='92 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='92M 1.' metadata={'source': 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model by applying a channel reduction procedure through multiple distillation steps and using the full-capacity student as a teacher, ac- cording to the “teacher assistant” paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' In the resulting model, the number of floating-point operations is approx- imately halved compared to the full-capacity version, and the number of parameters becomes as small as about 500k, taking about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='4 MB storage space without compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Acknowledgments This publication has been financially supported by: Science Foundation Ireland (SFI) under grant number SFI/12/RC/2289 P2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Regione Sicilia, Italy, RehaStart project (grant identifier: PO FESR 2014/2020, Azione 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content='5, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' 08ME6201000222, CUP G79J18000610007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' University of Catania, Piano della Ricerca di Ateneo, 2020/2022, Linea 2D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' MIUR, 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Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Sta3d: Spatiotemporally attentive 3d network for video saliency prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} +page_content=' Pattern Recognition Letters, 147:78–84, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE3T4oBgHgl3EQfmwqR/content/2301.04619v1.pdf'} diff --git a/0NE4T4oBgHgl3EQfyw06/content/tmp_files/2301.05268v1.pdf.txt b/0NE4T4oBgHgl3EQfyw06/content/tmp_files/2301.05268v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..caa11518fd281b20013ed6e4e0fda6c649a56492 --- /dev/null +++ b/0NE4T4oBgHgl3EQfyw06/content/tmp_files/2301.05268v1.pdf.txt @@ -0,0 +1,855 @@ +The non-linear perturbation of a black hole by +gravitational waves. III. Newman-Penrose constants +J Frauendiener1, A Goodenbour2 and C Stevens2 +1Department of Mathematics and Statistics, University of Otago, Dunedin 9016, New +Zealand +2Department of Mathematics and Statistics, University of Canterbury, Christchurch +8041, New Zealand +E-mail: joerg.frauendiener@otago.ac.nz, +alex.goodenbour@pg.canterbury.ac.nz, chris.stevens@canterbury.ac.nz +Abstract. +In this paper we continue our study of the non-linear response of a +Schwarzschild black hole to an ingoing gravitational wave by computing the Newman- +Penrose (NP) constants. The NP constants are five complex, supertranslation-invariant +quantities defined on null infinity I + and although put forward in the 60’s, they +have never been computed in a non-stationary setting. We accomplish this through +a numerical implementation of Friedrich’s generalized conformal field equations whose +semi-global evolution yields direct access to I +. Generalizations of the NP constants’ +integral expressions are made to allow their computation in a more general gauge +that better suits the output of a numerical evolution. Canonical methods of fixing +inherent degrees of freedom in their definitions are discussed. The NP constants are +then computed for a variety of different ingoing wave profiles in axisymmetry, and then +with no symmetry assumptions in 3+1 for which all five are non-zero. +Submitted to: Class. Quantum Grav. +arXiv:2301.05268v1 [gr-qc] 12 Jan 2023 + +The non-linear perturbation of a black hole by gravitational waves +2 +1. Introduction +Gravitational waves are a robust prediction of general relativity. The existence of wave +solutions to the field equations has been known since the early days of the theory, +but there was doubt that wave-like behaviour occurred generically outside of overly- +symmetric exact solutions [25]. The way we characterise radiation today emerged out of +the work of Bondi [4], Sachs [26], Newman and Penrose [18, 19, 21]. Penrose’s procedure +of conformal compactification [22] succinctly encodes the asymptotic fall-off conditions +hard-coded by Bondi and Sachs. The conformal boundary, I , emerges as the natural +place to define gravitational radiation. +This +picture +of +gravitational +radiation +owes +a +great +debt +to +Maxwell’s +electromagnetism. +The isolation of radiative degrees of freedom by Bondi and +Sachs echoes an earlier analysis of electromagnetic radiation via the Liénard-Wiechert +potential and Penrose’s conformal compactification is premised on the conformal- +invariance of zero-rest-mass fields, a fact which was shown for a Maxwell field much +earlier [1, 6]. +This paper is focused on the explicit calculation of another such import from +Maxwell electromagnetism, the Kirchhoff integral formula. Generalised to fields of spin- +s in Minkowski space, it is called the generalised Kirchhoff-d’Adhémar formula and +relates the value of the field at a point to an integral over an arbitrary smooth cut of its +(past) light cone. Formally applied to the spin-2 Weyl spinor on the conformal boundary +I + of an asymptotically flat spacetime, the Kirchhoff-d’Adhémar formula yields a set +of five complex supertranslation invariant quantities on I + which are ostensibly the +components of the Weyl spinor at timelike infinity. These are the NP constants [20], +whose physical interpretation has proved elusive for over half a century. +Nevertheless, much has been said about these constants in the intervening years. +In their original paper, Newman and Penrose make the argument that the existence +of the constants has non-trivial physical significance [20, 24]. It has since been shown +that the vanishing of the NP constants distinguishes between fundamentally different +late-time behaviour of self-gravitating waves [16]. They play an important role in the +early radiative properties of isolated systems close to spatial infinity [17]. Another line +of analysis has found that the NP constants appear as subleading BMS charges [15]. +However, as far as we are aware, the NP constants have never been explicitly computed +in a general space-time. +Explicit numerical computation of quantities at the conformal boundary without +the use of limiting procedures can be done by employing conformal compactification in +a numerical scheme. Friedrich’s conformal field equations regularly extend the Einstein +equations to include the conformal boundary [10–12]. Recently, an initial boundary +value problem (IBVP) framework for the generalised conformal field equations (GCFE) +was presented [3]. This framework puts I + within the computational domain, allowing +for the non-linear perturbation of black hole space-times. +Because the computational domain includes at least a portion of the future null + +The non-linear perturbation of a black hole by gravitational waves +3 +boundary, quantities defined there can be computed with local differential geometrical +methods. Most asymptotic quantities, including the NP constants, are defined on a +2-dimensional cut of I +, therefore one can see how a quantity evolves along the set of +successive cuts. However, in the literature, these quantities are often defined in terms of +a very specific set of coordinates, frame, and conformal factor. These choices are usually +incompatible with the requirements of the numerical scheme. Therefore, to compute a +quantity at null infinity with this scheme, it must be written in a conformally invariant +way. +The aim of this paper is to use the numerical framework provided by the IBVP +formulation of the GCFE to compute for the first time, the NP constants explicitly on +I +. The case considered is the non-linear perturbation of a Schwarzschild space-time +by gravitational waves. The reader is referred to [3] for details of the numerical scheme +and checks of correctness such as constraint convergence tests. +The layout of the paper is as follows: Section 2 summarizes the IBVP framework +for the GCFE. Section 3 presents the NP constants and proves their supertranslation- +invariance in the general form required for their computation. Section 4 presents the +details of aligning the frame of the GCFE with the frame in which the NP constants are +defined and discusses the details of their calculation. Section 5 presents numerical checks +of correctness and results for a range of initial wave profiles and Section 6 concludes +with a brief discussion. We follow conventions of Penrose and Rindler [24] throughout. +2. Overview of the GCFE and its numerical IBVP implementation +We implement Friedrich’s generalized conformal field equations analogously to previous +papers in this series [8] and here just give a brief overview. +The conformal field equations are a regular extension of the Einstein equations +defined on a physical space-time to another conformally related Lorentzian manifold, +related by a conformal factor Θ, where the points at ’infinity’ of the physical space-time +are given by Θ = 0. Imposing the conformal Gauß gauge on the GCFE [12] yields a +system of evolution equations of which most are ordinary differential equations except +those governing the components of the gravitational tensor which form a symmetric +hyperbolic system. These evolution equations are complemented by a set of constraint +equations which are preserved by the evolution. The associated IBVP is completed by +constraint preserving boundary conditions [3] which are used to generate fully non-linear +gravitational dynamics. +The Schwarzschild space-time of mass m written in isotropic coordinates is again +used as the initial space-time. The specific choice of conformal Gauß gauge given by +Friedrich [13] is used by which regular coordinates, frame and conformal factor up to +and beyond null infinity can be defined. +Our numerical implementation is capable of general 3 + 1 dimensional simulations +and we use this capability to generate a complete set of non-trivial NP constants going +beyond the axisymmetric case. + +The non-linear perturbation of a black hole by gravitational waves +4 +For all simulations presented here (excluding convergence tests) we use coordinates +{t, r, θ, φ}‡. +The spatial coordinates are discretized into equidistant points in the +intervals r ∈ [m/2, m/2 + 2m], θ ∈ [0, π) and φ ∈ [0, 2π] with 401, 33 and 64 +points respectively. The temporal discretization is also equidistant in this study with +timestep given by dt = dr/2 giving a Courant-Friedrichs-Lewy number of 0.5. The MPI- +parallelized Python package COFFEE [7] contains all the necessary numerical methods +to evolve this initial boundary value problem. The standard explicit Runge-Kutta fourth +order method is used to march in time, Strand’s fourth order summation-by-parts finite +difference operator (third order on the boundary) [27] is used to approximate radial +derivatives and the simultaneous-approximation-method [5] is used to stably impose +maximally dissipative boundary conditions. +Finally, we use spin-weighted spherical +harmonics to allow for fast and accurate angular derivatives through a pseudo-spectral +implementation of Penrose’s ð-calculus [2]. +Regridding is also performed, whereby +regions outside of future null infinity are chopped away from the computational domain +to maintain a stable evolution. This is also performed inside the black hole to avoid the +singularity. +3. Newman-Penrose constants +The Kirchhoff-d’Adhémar construction in Minkowski space expresses a solution to the +zero-rest-mass field equations at a point P as an integral of an arbitrary smooth cut +of the (past) light cone of P [23]. It is a conformally invariant construction and so +we may apply it specifically to relate a zero-rest-mass field φAB...L at the future (past) +timelike infinity of a conformally rescaled spacetime to a cut of its light cone, future +(past) null infinity. Because the construction is invariant with respect to the cut of the +light cone on which it is evaluated, it gives a set of supertranslation invariant constants +corresponding to the 2s+1 components of the zero-rest-mass field at timelike infinity. In +a spacetime with matter, timelike infinity becomes a singular point of the conformally +rescaled manifold, but the integrand of the Kirchoff-d’Adhémar construction, being +evaluated on null infinity, remains regular and so applied to the spin-2 Weyl spinor, we +are left with a set of five complex supertranslation invariant quantities defined on any +asymptotically flat spacetime. These are absolutely conserved in the sense that they +remain constant even with non-vanishing news. +The physical set-up of the Kirchoff-d’Adhémar construction is as follows. Consider +a point P with a (past or future) light cone I and an arbitrary smooth 3-dimensional +null hypersurface N intersecting I . The intersection has spherical topology and is +labeled C . +The Kirchhoff-d’Adhémar construction will take the form of an integral +evaluated on C whose value is independent of the intersecting null hypersurface N and +thus of the specific intersection C . In fact, any smooth cut of the light cone I can be +‡ The axisymmetric numerical implementation is analogous to the proceeding outline but without the +φ-direction and with optimized spin-weighted spherical harmonic transformations and corresponding +ð-calculus calculations. + +The non-linear perturbation of a black hole by gravitational waves +5 +said to have come about by the intersection of I with some null hypersurface N . +The method of proof consists in showing that the Kirchhoff-d’Adhémar integral +evaluated on two arbitrary cuts of I denoted C and C ′ gives the same result by treating +C − C ′ as the oriented boundary of a 3-dimensional section of the light cone I . The +generalised Stokes’ theorem can be used to relate cut invariance to the vanishing of a +related integral on the region between the cuts. This is explained in detail in [20, 24] +At each point of an intersection C there are two distinguished null directions, +one along the generators of the light cone I and the other along the intersecting null +hypersurface so it is advantageous to use the GHP formalism [14]. We can choose a +spin-frame such that oA points along the intersecting hypersurface, and ιA along I , +normalised so that oAιA = 1. +Formally, the Kirchhoff-d’Adhémar formula reads +φ[U] +�� +P= +� +C +Uþcφ d2C +(3.1) +where φ := φAB...LoAoB . . . oL, U is a weighted scalar satisfying +(i) ¯ðcU = 0, +and +(ii) þ′ +cU = 0. +(3.2) +and the derivative operators with a subscript ’c’ are conformally weighted operators of +the cGHP formalism as introduced in [9]. +In Minkowski spacetime, in a gauge where each cut C is represented as a unit +2-sphere, U will be a component of the spinor ιAιB . . . ιL, i.e., one of the 2s + 1 spin- +weighted spherical harmonics −sYsm with spin-weight −s. +In this gauge, these are +constant, thus trivially propagating along the light cone. +In a curved spacetime, U is a generalisation of these spin-weighted spherical +harmonics to a topological but not necessarily metric sphere. +In this general case, +there are still 2s + 1 independent solutions of (3.2(i)) for U. +4. Calculating the NP constants +In the remainder of this work we focus on the gravitational NP constants, i.e., with +φABCD = ψABCD, the conformally rescaled Weyl spinor. Many system variables of the +GCFE are components of spinors with respect to a certain spin-frame. In general, this +spin-frame, and its associated null tetrad,does not agree with the frame adapted to I +that was used in the definition of the NP constants but we can use null rotations to +transform between the GCFE null frame and the frame adapted to I herein referred +to as a Bondi frame§. +§ This is not strictly correct, since we are referring here only to a single cut, whereas the standard +usage of the term Bondi frame refers to an entire system of cuts parametrised by the retarded Bondi +time. + +The non-linear perturbation of a black hole by gravitational waves +6 +A null rotation mixes one component of a spin-frame into another. For example, a +null rotation of oA around ιA is given by +oA → oA + Y ιA, +ιA → ιA +(4.1) +thus keeping ιA fixed, where Y is a function of the spacetime coordinates. +If we +denote the Bondi spin-frame by OA and IA, the GCFE frame by oA and ιA, and the +corresponding Bondi and GCFE null-tetrad vectors by capital and lowercase letters +respectively, then we may transform between frames by two null successive rotations +(first fixing oA and then the new ιA) which have the combined form +OA = oA + Y (ιA + XoA), +IA = ιA + XoA. +(4.2) +The null rotation functions X and Y are determined by the conditions +∇aΘ = −ANa, +M a∇at = 0, +(4.3) +where the scaling A is fixed given the conformal factor Θ and the above expression of the +adapted spin-frame. These conditions impose that N a points along the null generators of +I and that the complex vector M a lies within the t = const. cuts of I . Appropriately +fixing the freedom in how the frame propagates along the timelike conformal geodesics +of the conformal Gauß gauge allows one to satisfy the second condition automatically, +yielding X = 0. The first condition gives us the value of the null rotation function Y +on I +. +The transformation between the GCFE frame and the Bondi frame is then known +and so we may write components with respect to the Bondi frame in terms of components +with respect to the GCFE frame which are known numerically. As a simple example, +the third component of the gravitational spinor is written in the Bondi frame as +ψABCDOAIBICID = ψABCD(oA + Y ιA)ιBιCιD = ψ3 + Y ψ4. +Both the GCFE and the NP constants are defined with respect to a spin and boost +covariant formalism and so a properly weighted expression with respect to one frame +results in a properly weighted expression with respect to another. +The same process is used to compute the area-form in terms of numerically available +quantities. +4.1. Fixing the behaviour of the frame off I + +With the above two null rotations, the frame on I + is fixed, but we also have some +freedom to choose how our frame changes as we move away from I +. The presentation +of I + in the proof of supertranslation invariance makes use of this and takes κ = 0 +since the intersecting null hypersurface is foliated by a null geodetic congruence [24]. +The Bondi frame so far is only fixed on I + and in order to achieve κ = 0 we need +to enforce that DoA ∝ oA which means that we need to determine the null rotation + +The non-linear perturbation of a black hole by gravitational waves +7 +function Y away from I +. Suppose, we have fixed Y on I + with the above procedure, +then we can get the required result with a third null rotation that becomes the identity +on I + +oA → ˆoA = oA + ZιA, +ιA → ˆιA = ιA, +where Z = O(Θ), +(4.4) +recalling the conformal factor Θ. Under this transformation, +ˆκ = κ − DZ, +(4.5) +where D := La∇a and choosing Z so that κ = DZ on I + we obtain ˆκ = 0 there. +Although the transformation becomes the identity on I +, we must worry about +derivatives of the frame. In the Kirchhoff-d’Adhémar integral, we have a derivative of +the form Dφ where φ = φAB..LoAoB..oL (2s indices). Under this null rotation, +ˆD ˆφ +��� +I + = D ˆφ +(4.6) += D(φA..L(oA + ZιA)..(oL + ZιL)) +(4.7) += Dφ + 2sκφ1 +(4.8) +since the derivative of any term containing powers of Z higher than one will vanish +on I +. +4.2. Computing U +In the NP constant integrand (3.1), the active component would appear to be the term +þcφ since this brings information of the arrival of the field φ at I +, while the quantity +U appears to be somewhat inert, being used to project out certain pieces of information +from the integrand. Before the jump was made to curved spacetime, the Kirchhoff- +d’Adhémar integral could represent the value of the field φ at timelike infinity. In this +case, the quantity U was replaced by components of ιAιB...ιL which are spin-weighted +spherical harmonics in an appropriate frame. When curved spacetime is introduced, +the U takes the role of these components and so different choices of U which satisfy the +underlying equations (3.2), represent what would have been components of φ at timelike +infinity. The job of U is to lend its spin, boost, and conformal weight to the expression, +and so provide alignment between cuts of I + allowing for comparison from cut to cut. +To compute U we must solve the “constraint equation” ¯ðcU = 0 on a cut∥ and evolve +it along the null generators of I + with the evolution equation þ′ +cU = 0. Following +methods from our earlier paper [8], we can expand these operators written in the Bondi +frame in terms of the numerically implemented, standard operators ˜ð, ˜ð′, to obtain +A˜ðU + B˜ð′U + CU = 0. +(4.9) +∥ This term is justified since, by considering the commutator [þ′ +c, ¯ðc] one can show that any U satisfying +the evolution equation þ′ +cU = 0 will satisfy ¯ðcU = 0 on every cut if it satisfies this equation on a single +cut. In this sense, the constraint is propagated by the evolution. + +The non-linear perturbation of a black hole by gravitational waves +8 +Expanding the known coefficients A, B, and C, and the unknown function U in terms of +spin-weighted spherical harmonics, and using the well-known relationship for products +of spin-weighted spherical harmonics in terms of Clebsch-Gordon coefficients (see [23]) +results in a system of homogeneous linear equations for the spectral coefficients of the +function U. There are five linearly independent solutions which span the solution space +to the constraint equation for U. +The evolution equation can similarly be written in terms of numerically available +quantities as, +A∂tU + B˜ðU + C˜ð′U + DU = 0, +(4.10) +and may be evolved along I + by the method of lines given an initial solution to the +constraint equation. An adaptive fourth-order Runge-Kutta method is used since the +numerical output is not linearly spaced in t. +4.3. Fixing a basis in the solution space U +It is clear that solution U of (3.2(ii)) leads to another solution αU provided that α is +a complex constant on the cut C . More generally, any complex linear combination of +solutions will also be a solution. Thus, changing the basis of the solution space U of +(3.2(ii)) will also change the individual values of the five NP constants. Therefore, these +do not, by themselves, carry independent physical information. Only the combination +of the values of the integrals together with the knowledge of the basis of U carries the +full information. +This means that in order to compare NP constants across different spacetimes we +need to make sure that we specify “the same basis” for the solution space for each +spacetime. There are several ways to do this: two complicated ones which are also +physically relevant, and a third easier one which not as physically meaningful but much +more pragmatic. +The first idea that comes to mind is to first conformally rescale the metric on the cut +to make it into a unit-sphere, and, in a second step, to change the coordinate system by +a Möbius transformation so that it becomes a standard polar coordinate system on the +unit-sphere. In this situation, the solutions of (3.2(ii)) are the standard spin-weighted +spherical harmonics Ym := −2Y2m with −2 ≤ m ≤ 2. +While the first step is rather straightforward, the second step leads to a Poisson-type +equation on the sphere with a δ-like source term which is difficult (but not impossible) +to treat numerically. In addition, after the coordinate transformation all quantities must +be transformed which may introduce several numerical errors. +The second way to introduce a basis in U is to make use of the fact that the standard +spin 2 spherical harmonics Ym form an irreducible representation of the group SU(2). +They each are an eigenvector of an infinitesimal generator with different eigenvalue, +and they are obtained one from the other by the action of two ladder operators (very +much akin to the angular momentum algebra of quantum mechanics). Fixing one of +them as being annihilated by one of the ladder operators, one can generate the others + +The non-linear perturbation of a black hole by gravitational waves +9 +by successive application of the other ladder operator. This fixes the complete basis in +terms of the first vector and leaves the freedom of scaling with one complex number. +This can be almost fixed by normalising the vectors with respect to an appropriate +Hermitian product, leaving the remaining freedom of a single phase. +In principle, this program could be carried out but it is very cumbersome. First, +one needs to find the infinitesimal generators of the group action. +This leads to a +series of elliptic equations to be solved on the sphere. Next, one needs to use these +generators to determining the function that is killed by one of the ladder operators, +which is again an elliptic equation on the sphere, and then generate the other functions +by successive application of the other ladder operator. As an alternative, one could solve +the eigenvalue problem for the third operator. Obviously, this procedure is numerically +quite involved and prone to inaccuracies due to successive numerical differentiation. +For this reason, we use a third method to fix a “universal” basis of U , and we +use the “universal structure” that is available to us, namely our numerical setup which +is the same for every spacetime that we compute. Recall that our method is based +on concentric round spheres and that every function we compute can be expanded as +a linear combination of spin-weighted spherical harmonics defined with respect to the +numerical round spheres. Therefore, we proceed as follows: first, on an initial cut, we +compute five linearly independent solutions (uk)k=−2:2 of (3.2(ii)). These have the form +uk = +2 +� +m=−2 +cm +kYm + Zl>2, +k = −2 : 2 +where Zl>2 stands for terms with higher values of l. +Then a straightforward linear +combination of these solutions leads to the universal basis Uk which is defined by +Uk = Yk + Zl>2 +where Zl>2 again stands for higher l terms. We can interpret this basis as being the +deformation of the standard basis provided by the Ym due to the impact of the incoming +gravitational wave. If there was no gravitational wave, then the cut would be spherically +symmetric and the yk would agree with the standard basis. The basis, thus defined, is +then propagated along I + using the evolution equation (3.2(i)). In this process, the +form of the yk will change. This process of fixing a “universal basis” of U leaves no +further freedom (except, of course, for the free phase inherent in the definition of the +spin-weighted spherical harmonics). +4.4. Integrating the Newman-Penrose integrand +At this stage, all elements of the Newman-Penrose integral (3.1) have expressions in +terms of known quantities. Integration can be performed against the basis Uk as defined +in 4.3 by simply computing the s = l = m = 0 spectral coefficient of the complete +integrand and dividing by 2π. The theory shows these five complex numbers, obtained +on a cut, come out the same independently of which cut was chosen for their evaluation. +In the next section we present numerical results that showcase these properties. + +The non-linear perturbation of a black hole by gravitational waves +10 +5. Numerical Results +Using the above procedure, the NP constants were computed with data on I + from a +numerically evolved spacetime modelling the non-linear perturbation of a Schwarzschild +black hole by an incoming gravitational wave pulse. +Because the NP constants are +evaluated on each cut of I + defined as the intersection with a t = const. hypersurface, +we obtain five complex numbers for every t. +The initial mass of the black hole is m = 0.5 for all simulations considered. +5.1. Ingoing wave +The ingoing pulse is defined as the choice of free data q0 of the lightlike, ingoing +characteristic variable on the outer boundary. This is chosen to be a linear combination +of the spin-weighted spherical harmonics 2Y2m for m = 0, 1, 2 and with amplitudes a, b +and c respectively. The choices of these amplitudes will vary in the upcoming sections. +This gives the wave profile on the outer boundary +q0(t, θ, φ) = +� +� +� +[4a +� +2π +15 2Y20 − 2b +� +5 +π 2Y21 + 2c�π +5 2Y22] sin8(8πt) +t ≤ 1 +8 +0 +t > 1 +8 +. +5.2. Checks of correctness +We have demonstrated in several papers [3, 8] that the solutions computed by the GCFE +system converge at the correct order for the 2 + 1 axisymmetric case. Here we show +that this is also true in the general case of 3 + 1 dimensions. We also show that the NP +constants converge to constant values on I +. +(a) θ = π/2 and φ = π. +(b) r = 0.978 and φ = π. +(c) r = 0.978 and θ = π/2. +Figure 1: +The imaginary part of a constraint equation from the Bianchi identity +when +a +single +spatial +coordinate +is +fixed +at +t += +1 +for +r, θ, φ +resolutions +of {101, 9, 16}, {201, 17, 32} and {401, 33, 64} (denoted by Res1, Res2 and Res3 +respectively). The dashed vertical line represents I + in the radial plot. The curves +from top to bottom correspond to increasing resolution. +We use an ingoing wave profile proportional to 2Y22 with a = b = 0, c = i to allow +excitation in the φ-direction. Fig. 1 demonstrates convergence in all spatial directions +at t = 1. + +0 +Resl +Res3 +Res2 +log10 lErrorl +.5 +.10 +-15 +0.4 +0.6 +0.8 +1.0 +r0 +Res1 +Res3 +Res2 +log1o lErrorl +-5 +-10 +-15 +0 +1 +2 +30 +Res1 +Res3 +Res2 +log1o lErrorl +-5 +-10 +-15 +0 +2 +4 +6 +0The non-linear perturbation of a black hole by gravitational waves +11 +Focusing now on the NP constants, we demonstrate how they approach constant +values along I +. Fig. 2 shows a convergence test of the discrepancy from constancy +of the single non-vanishing NP constant in axisymmetry, choosing amplitudes a = 1, +b = c = 0, along I + as the spatial resolution is increased. The resolution in r (number +of intervals) is denoted rres and corresponds to an equivalently scaled resolution θres along +the θ-direction. The coarsest values are rres = 100 and θres = 8, and the resolutions are +doubled in successive simulations. +0.8 +1.0 +1.2 +1.4 +1.6 +Conformal time t +−8 +−6 +−4 +−2 +0 +log10 Error +rres = 100 +rres = 200 +rres = 400 +Figure 2: Convergence of the log10 difference between the magnitude of the NP constant +at time t and at the initial cut with increasing spatial resolution. +5.3. Variable ingoing amplitude +An superficial glance at (3.1) would suggest that due to the linearity of the integrand +and the zero-rest-mass field equations with respect to φ, scaling the amplitude of the +ingoing wave would just scale the NP constants linearly. However, this turns out not +to be the case because φ satisfies the Bianchi equation into which φ enters non-linearly +through the connection coefficients of the covariant derivative [23, §5.7]. Hence, it is +interesting to numerically probe the scaling of the NP constants as the ingoing wave +profile is scaled. +We performed four simulations in axisymmetry with the amplitudes b = c = 0 and +a taking the values 1,2,5, and 10. These are evolved up to t = 1.77 at which point the +system variables start to diverge due to the close ‘conformal’ proximity to i+ at t ≈ 1.79. +Table 1 shows the corresponding single NP constant for each amplitude as well as the +relative error from a linear fit through the origin. Fitting the NP constants to the ansatz + +The non-linear perturbation of a black hole by gravitational waves +12 +αaβ yields α ≈ 0.53865 and β ≈ 0.99803. Fig. 3 shows the log10 deviation of the NP +constant value from the value on the initial cut for each amplitude as a function of t. +The deviation from a linear fit is orders of magnitude greater than the error. This is due +to the amplitude of the initial wave profile entering into the field equations non-linearly +resulting in a non-linear relationship between initial amplitude and Newman-Penrose +constant. +Amplitude +1 +2 +5 +10 +NPC +0.53882 +1.07638 +2.68405 +5.36222 +Rel. Err. +0 +0.00116 +0.00373 +0.00482 +Table 1: The one non-vanishing NP constant for different ingoing wave amplitudes and +the deviation from a linear fit through the origin. This deviation is orders of magnitude +larger than the error for each. This is a result of the amplitude entering non-linearly +into the field equations. +0.8 +1.0 +1.2 +1.4 +1.6 +Conformal time t +−10 +−9 +−8 +−7 +−6 +−5 +−4 +log10 error +a = 1 +a = 2 +a = 5 +a = 10 +Figure 3: The log10 difference of the NP constant from the value on an initial cut as +a function of conformal time t for a variable amplitude of the initial wave profile as +a measure of deviation from constancy due to error. Cumulative error grows as we +integrate along I +. +5.4. Deviation from axisymmetry +In a general asymptotically flat spacetime there are five complex NP constants +corresponding to the five independent solutions to the equations (3.2). In axisymmetry, + +The non-linear perturbation of a black hole by gravitational waves +13 +these collapse to only one independent solution, when the frame and coordinates also +respect the symmetry, because then only the m = 0 modes of a spherical harmonic +expansion remain. +We can investigate the collapse of five NP constants into one by using the initial +wave profile given by (5.1) and using a = i, b = c = ϵ i, where ϵ parametrises a deviation +from axisymmetry. +Six simulations were run for this wave profile for ϵ = 0, 1, 2, 3, 4, 5. Fig. 4 shows +the magnitudes of the corresponding NP constants. Although we do have access to the +full ten real degrees of freedom (five complex degrees of freedom) for each simulation, +the trends can be seen in the behaviour of the magnitudes. Fig. 5 shows the same +quantities but separated by the value of m of the corresponding U so that individual +trends can be seen. +We can see that for ϵ = 0, there is only one non-trivial constant +0 +1 +2 +3 +4 +5 +Amplitude ϵ +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +NPC +m = −2 +m = −1 +m = 0 +m = 1 +m = 2 +Figure 4: The magnitudes of the five complex NP constants as a function of a parameter +ϵ which breaks axisymmetry in the initial wave profile. For ϵ > 0 all constants are non- +zero although most are small. +corresponding to the axisymmetric m = 0 mode, but as non-axisymmetric modes are +introduced for ϵ > 0, all five constants take on non-trivial values and grow with ϵ. +6. Discussion +In this paper, we continue our numerical investigation into the non-linear perturbation +of a Schwarzschild black hole using an initial boundary value problem for the general +conformal field equations. This novel numerical scheme allows us to include I + within +the computational domain and so compute quantities there directly. Thereby, we have +computed the NP constants for the first time in a physically realistic spacetime. +The gauge quantities of the system were fixed by numerical needs which implies +that we were unable to directly use the very specific set of coordinates, frame, and + +The non-linear perturbation of a black hole by gravitational waves +14 +0 +2 +4 +0.0 +0.5 +1.0 +1.5 +m = −2 +0 +2 +4 +0.0 +0.5 +1.0 +1.5 +m = −1 +0 +2 +4 +0.540 +0.541 +m = 0 +0 +2 +4 +Amplitude ϵ +0.00 +0.02 +0.04 +NPC +m = 1 +0 +2 +4 +0.000 +0.005 +0.010 +m = 2 +Figure 5: The magnitudes of the five complex NP constants as a function of a parameter +ϵ which breaks axisymmetry in the initial wave profile split by the value of m of the +corresponding U. Note that m is a label on spherical harmonics, not a mass. +conformal factor typically used when defining quantities at I +. To compute physical +quantities such as the NP constants with data from the numerical simulation, we need +an explicitly conformally invariant expression for the quantity. However, this concept +of conformal invariance is a rather special one, and it might be appropriate to highlight +it again here. +Physical quantities make reference to the physical metric ˜gab of the +spacetime. In our context, the physical metric is represented as ˜gab = Ω−2gab, i.e., in +terms of another metric in the same conformal class and the conformal factor relating +the two. By conformal invariance we do not mean invariance under ˜gab �→ θ2˜gab, but +rather the invariance under (gab, Ω) �→ (θ2gab, θΩ), which corresponds to the free choice +of the splitting of ˜gab into a conformal and a scale part. +For example, the recent analysis of the Bondi-Sachs energy-momentum in this +framework involved generalising the procedure of constructing a basis of translations +with respect to which components of the Bondi-Sachs 4-vector may be taken. +The +standard procedure of choosing the first four spherical harmonics is certainly not +conformally invariant in this sense. +This led to an invariant characterisation of the +Lorentzian metric on the space of BMS translations [9]. Of course, the existence of this +metric still leaves the freedom of Lorentz transformations for the choice of the basis. +We run into the same problem when defining a basis for the quantity U which, +when integrated against the Newman-Penrose integrand, gives the linearly independent +NP constants. Again, it is the solution space which is defined in a conformally invariant +way. But in this case there is no obvious inner product that one could use to select a +basis. Even if there was one, the basis would still be defined only up to the appropriate + +REFERENCES +15 +(pseudo-) orthogonal transformations. We circumvent the non-uniqueness of the basis +in this case by refering to the universal structure that is imposed on the problem by +the numerical setup as explained in Sec. 4.3. This seems to be the best way to ensure +comparability across the different space-times that we investigate. +Acknowledgments +Supported by the Marsden Fund Council from Government funding, managed by Royal +Society Te Ap¯arangi. +The authors would like to thank L. Escobar for sharing the general form of his +SWSH code. +We wish to acknowledge the use of New Zealand eScience Infrastructure (NeSI) high +performance computing facilities, consulting support and/or training services as part of +this research. New Zealand’s national facilities are provided by NeSI and funded jointly +by NeSI’s collaborator institutions and through the Ministry of Business, Innovation & +Employment’s Research Infrastructure programme. URL https://www.nesi.org.nz. +References +[1] +H. Bateman, “The transformation of the electrodynamical equations”, Proc LMS +s2-8, 223–264 (1910). +[2] +F. Beyer, L. Escobar, and J. Frauendiener, “Numerical solutions of Einstein’s +equations for cosmological spacetimes with spatial topology S3 and symmetry +group U(1)”, Phys. Rev. D 93, 043009 (2016). +[3] +F. Beyer, J. Frauendiener, C. Stevens, and B. Whale, “Numerical initial boundary +value problem for the generalized conformal field equations”, Phys. Rev. D 96, +084020 (2017). +[4] +H. Bondi, M. G. van der Burg, and A. W. K. Metzner, “Gravitational waves in +general relativity. VII. Waves from axi-symmetric isolated systems”, Proc. Roy. +Soc. A 269, 21–52 (1962). +[5] +M. H. Carpenter, D. Gottlieb, and S. Abarbanel, “Time-stable boundary +conditions for finite-difference schemes solving hyperbolic systems: methodology +and application to high-order compact schemes”, J. Comp. Phys. 111, 220–236 +(1994). +[6] +E. Cunningham, “The principle of relativity in electrodynamics and an extension +thereof”, Proc. LMS s2-8, 77–98 (1910). +[7] +G. Doulis, J. Frauendiener, C. Stevens, and B. Whale, “COFFEE—An MPI- +parallelized Python package for the numerical evolution of differential equations”, +SoftwareX 10, 100283 (2019). + +REFERENCES +16 +[8] +J. Frauendiener and C. Stevens, “The non-linear perturbation of a black hole by +gravitational waves. I. The Bondi-Sachs mass loss”, Class. Quantum Grav. 38, +194002 (2021). +[9] +J. +Frauendiener +and +C. +Stevens, +“A +new +look +at +the +Bondi–Sachs +en- +ergy–momentum”, Class. Quantum Grav. 39, 025007 (2022). +[10] +H. Friedrich, “On the regular and the asymptotic characteristic initial value +problem for Einstein’s vacuum field equations”, Proc. Roy. Soc. A 375, 169–184 +(1981). +[11] +H. Friedrich, “The asymptotic characteristic initial value problem for Einstein’s +vacuum field equations as an initial value problem for a first-order quasilinear +symmetric hyperbolic system”, Proc. Roy. Soc. A 378, 401–421 (1981). +[12] +H. Friedrich, “Einstein equations and conformal structure: Existence of anti-de +Sitter-type space-times”, J. Geom. Phys. 17, 125–184 (1995). +[13] +H. Friedrich, “Conformal geodesics on vacuum space-times”, Commun. Math. Phys. +235, 513–543 (2003). +[14] +R. P. Geroch, A. Held, and R. Penrose, “A space-time calculus based on pairs of +null directions”, J Math Phys 14, 874–881 (1973). +[15] +H. Godazgar, M. Godazgar, and C. N. Pope, “Subleading BMS charges and fake +news near null infinity”, J. High Energ. Phys. 2019, 143 (2019). +[16] +R. Gómez, J. Winicour, and B. G. Schmidt, “Newman-Penrose constants and the +tails of self-gravitating waves”, Phys. Rev. D 49, 2828–2836 (1994). +[17] +J. A. V. Kroon, “Early radiative properties of the developments of time-symmetric +conformally flat initial data”, Class. Quantum Grav. 20, L53 (2003). +[18] +E. T. Newman and T. W. J. Unti, “Behavior of asymptotically flat empty spaces”, +J Math Phys 3, 891–901 (1962). +[19] +E. T. Newman and R. Penrose, “An approach to gravitational radiation by a +method of spin coefficients”, J Math Phys 3, 566–578 (1962). +[20] +E. T. Newman and R. Penrose, “New conservation laws for zero rest-mass fields +in asymptotically flat space-time”, Proc. Roy. Soc. A 305, 175–204 (1968). +[21] +R. Penrose, “Asymptotic properties of fields and space-times”, Phys. Rev. Lett. +10, 66–68 (1963). +[22] +R. Penrose, “Zero rest-mass fields including gravitation: asymptotic behaviour”, +Proc. Roy. Soc. A 284, 159–203 (1965). +[23] +R. Penrose and W. Rindler, Spinors and Spacetime: Two-spinor calculus and +relativistic fields, Vol. 1 (Cambridge University Press, Cambridge, 1984). +[24] +R. Penrose and W. Rindler, Spinors and Spacetime: Spinor and twistor methods +in space- time geometry, Vol. 2 (Cambridge University Press, 1986). +[25] +N. Rosen, “Plane polarized waves in the general theory of relativity”, Phys. Z. +Sowjetunion 12, 366–372 (1937). + +REFERENCES +17 +[26] +R. +K. +Sachs, +“Gravitational +waves +in +general +relativity. +VIII. +Waves +in +asymptotically flat space-time”, Proc. Roy. Soc. A 270, 103–126 (1962). +[27] +B. Strand, “Summation by parts for finite difference approximations for d/dx”, J. +Comp. Phys. 110, 47–67 (1994). + diff --git a/0NE4T4oBgHgl3EQfyw06/content/tmp_files/load_file.txt b/0NE4T4oBgHgl3EQfyw06/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7f27ca596e70d4ee2c1121980559f19ce0f24ecc --- /dev/null +++ b/0NE4T4oBgHgl3EQfyw06/content/tmp_files/load_file.txt @@ -0,0 +1,532 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf,len=531 +page_content='The non-linear perturbation of a black hole by gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Newman-Penrose constants J Frauendiener1, A Goodenbour2 and C Stevens2 1Department of Mathematics and Statistics, University of Otago, Dunedin 9016, New Zealand 2Department of Mathematics and Statistics, University of Canterbury, Christchurch 8041, New Zealand E-mail: joerg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='frauendiener@otago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='nz, alex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='goodenbour@pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='canterbury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='nz, chris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='stevens@canterbury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='nz Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In this paper we continue our study of the non-linear response of a Schwarzschild black hole to an ingoing gravitational wave by computing the Newman- Penrose (NP) constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The NP constants are five complex, supertranslation-invariant quantities defined on null infinity I + and although put forward in the 60’s, they have never been computed in a non-stationary setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' We accomplish this through a numerical implementation of Friedrich’s generalized conformal field equations whose semi-global evolution yields direct access to I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Generalizations of the NP constants’ integral expressions are made to allow their computation in a more general gauge that better suits the output of a numerical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Canonical methods of fixing inherent degrees of freedom in their definitions are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The NP constants are then computed for a variety of different ingoing wave profiles in axisymmetry, and then with no symmetry assumptions in 3+1 for which all five are non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Submitted to: Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Quantum Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='05268v1 [gr-qc] 12 Jan 2023 The non-linear perturbation of a black hole by gravitational waves 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Introduction Gravitational waves are a robust prediction of general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The existence of wave solutions to the field equations has been known since the early days of the theory, but there was doubt that wave-like behaviour occurred generically outside of overly- symmetric exact solutions [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The way we characterise radiation today emerged out of the work of Bondi [4], Sachs [26], Newman and Penrose [18, 19, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Penrose’s procedure of conformal compactification [22] succinctly encodes the asymptotic fall-off conditions hard-coded by Bondi and Sachs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The conformal boundary, I , emerges as the natural place to define gravitational radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This picture of gravitational radiation owes a great debt to Maxwell’s electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The isolation of radiative degrees of freedom by Bondi and Sachs echoes an earlier analysis of electromagnetic radiation via the Liénard-Wiechert potential and Penrose’s conformal compactification is premised on the conformal- invariance of zero-rest-mass fields, a fact which was shown for a Maxwell field much earlier [1, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This paper is focused on the explicit calculation of another such import from Maxwell electromagnetism, the Kirchhoff integral formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Generalised to fields of spin- s in Minkowski space, it is called the generalised Kirchhoff-d’Adhémar formula and relates the value of the field at a point to an integral over an arbitrary smooth cut of its (past) light cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Formally applied to the spin-2 Weyl spinor on the conformal boundary I + of an asymptotically flat spacetime, the Kirchhoff-d’Adhémar formula yields a set of five complex supertranslation invariant quantities on I + which are ostensibly the components of the Weyl spinor at timelike infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' These are the NP constants [20], whose physical interpretation has proved elusive for over half a century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Nevertheless, much has been said about these constants in the intervening years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In their original paper, Newman and Penrose make the argument that the existence of the constants has non-trivial physical significance [20, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' It has since been shown that the vanishing of the NP constants distinguishes between fundamentally different late-time behaviour of self-gravitating waves [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' They play an important role in the early radiative properties of isolated systems close to spatial infinity [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Another line of analysis has found that the NP constants appear as subleading BMS charges [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' However, as far as we are aware, the NP constants have never been explicitly computed in a general space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Explicit numerical computation of quantities at the conformal boundary without the use of limiting procedures can be done by employing conformal compactification in a numerical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Friedrich’s conformal field equations regularly extend the Einstein equations to include the conformal boundary [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Recently, an initial boundary value problem (IBVP) framework for the generalised conformal field equations (GCFE) was presented [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This framework puts I + within the computational domain, allowing for the non-linear perturbation of black hole space-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Because the computational domain includes at least a portion of the future null The non-linear perturbation of a black hole by gravitational waves 3 boundary, quantities defined there can be computed with local differential geometrical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Most asymptotic quantities, including the NP constants, are defined on a 2-dimensional cut of I +, therefore one can see how a quantity evolves along the set of successive cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' However, in the literature, these quantities are often defined in terms of a very specific set of coordinates, frame, and conformal factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' These choices are usually incompatible with the requirements of the numerical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Therefore, to compute a quantity at null infinity with this scheme, it must be written in a conformally invariant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The aim of this paper is to use the numerical framework provided by the IBVP formulation of the GCFE to compute for the first time, the NP constants explicitly on I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The case considered is the non-linear perturbation of a Schwarzschild space-time by gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The reader is referred to [3] for details of the numerical scheme and checks of correctness such as constraint convergence tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The layout of the paper is as follows: Section 2 summarizes the IBVP framework for the GCFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Section 3 presents the NP constants and proves their supertranslation- invariance in the general form required for their computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Section 4 presents the details of aligning the frame of the GCFE with the frame in which the NP constants are defined and discusses the details of their calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Section 5 presents numerical checks of correctness and results for a range of initial wave profiles and Section 6 concludes with a brief discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' We follow conventions of Penrose and Rindler [24] throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Overview of the GCFE and its numerical IBVP implementation We implement Friedrich’s generalized conformal field equations analogously to previous papers in this series [8] and here just give a brief overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The conformal field equations are a regular extension of the Einstein equations defined on a physical space-time to another conformally related Lorentzian manifold, related by a conformal factor Θ, where the points at ’infinity’ of the physical space-time are given by Θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Imposing the conformal Gauß gauge on the GCFE [12] yields a system of evolution equations of which most are ordinary differential equations except those governing the components of the gravitational tensor which form a symmetric hyperbolic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' These evolution equations are complemented by a set of constraint equations which are preserved by the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The associated IBVP is completed by constraint preserving boundary conditions [3] which are used to generate fully non-linear gravitational dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The Schwarzschild space-time of mass m written in isotropic coordinates is again used as the initial space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The specific choice of conformal Gauß gauge given by Friedrich [13] is used by which regular coordinates, frame and conformal factor up to and beyond null infinity can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Our numerical implementation is capable of general 3 + 1 dimensional simulations and we use this capability to generate a complete set of non-trivial NP constants going beyond the axisymmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The non-linear perturbation of a black hole by gravitational waves 4 For all simulations presented here (excluding convergence tests) we use coordinates {t, r, θ, φ}‡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The spatial coordinates are discretized into equidistant points in the intervals r ∈ [m/2, m/2 + 2m], θ ∈ [0, π) and φ ∈ [0, 2π] with 401, 33 and 64 points respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The temporal discretization is also equidistant in this study with timestep given by dt = dr/2 giving a Courant-Friedrichs-Lewy number of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The MPI- parallelized Python package COFFEE [7] contains all the necessary numerical methods to evolve this initial boundary value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The standard explicit Runge-Kutta fourth order method is used to march in time, Strand’s fourth order summation-by-parts finite difference operator (third order on the boundary) [27] is used to approximate radial derivatives and the simultaneous-approximation-method [5] is used to stably impose maximally dissipative boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Finally, we use spin-weighted spherical harmonics to allow for fast and accurate angular derivatives through a pseudo-spectral implementation of Penrose’s ð-calculus [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Regridding is also performed, whereby regions outside of future null infinity are chopped away from the computational domain to maintain a stable evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This is also performed inside the black hole to avoid the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Newman-Penrose constants The Kirchhoff-d’Adhémar construction in Minkowski space expresses a solution to the zero-rest-mass field equations at a point P as an integral of an arbitrary smooth cut of the (past) light cone of P [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' It is a conformally invariant construction and so we may apply it specifically to relate a zero-rest-mass field φAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='L at the future (past) timelike infinity of a conformally rescaled spacetime to a cut of its light cone, future (past) null infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Because the construction is invariant with respect to the cut of the light cone on which it is evaluated, it gives a set of supertranslation invariant constants corresponding to the 2s+1 components of the zero-rest-mass field at timelike infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In a spacetime with matter, timelike infinity becomes a singular point of the conformally rescaled manifold, but the integrand of the Kirchoff-d’Adhémar construction, being evaluated on null infinity, remains regular and so applied to the spin-2 Weyl spinor, we are left with a set of five complex supertranslation invariant quantities defined on any asymptotically flat spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' These are absolutely conserved in the sense that they remain constant even with non-vanishing news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The physical set-up of the Kirchoff-d’Adhémar construction is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Consider a point P with a (past or future) light cone I and an arbitrary smooth 3-dimensional null hypersurface N intersecting I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The intersection has spherical topology and is labeled C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The Kirchhoff-d’Adhémar construction will take the form of an integral evaluated on C whose value is independent of the intersecting null hypersurface N and thus of the specific intersection C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In fact, any smooth cut of the light cone I can be ‡ The axisymmetric numerical implementation is analogous to the proceeding outline but without the φ-direction and with optimized spin-weighted spherical harmonic transformations and corresponding ð-calculus calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The non-linear perturbation of a black hole by gravitational waves 5 said to have come about by the intersection of I with some null hypersurface N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The method of proof consists in showing that the Kirchhoff-d’Adhémar integral evaluated on two arbitrary cuts of I denoted C and C ′ gives the same result by treating C − C ′ as the oriented boundary of a 3-dimensional section of the light cone I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The generalised Stokes’ theorem can be used to relate cut invariance to the vanishing of a related integral on the region between the cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This is explained in detail in [20, 24] At each point of an intersection C there are two distinguished null directions, one along the generators of the light cone I and the other along the intersecting null hypersurface so it is advantageous to use the GHP formalism [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' We can choose a spin-frame such that oA points along the intersecting hypersurface, and ιA along I , normalised so that oAιA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Formally, the Kirchhoff-d’Adhémar formula reads φ[U] �� P= � C Uþcφ d2C (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='1) where φ := φAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='LoAoB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' oL, U is a weighted scalar satisfying (i) ¯ðcU = 0, and (ii) þ′ cU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='2) and the derivative operators with a subscript ’c’ are conformally weighted operators of the cGHP formalism as introduced in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In Minkowski spacetime, in a gauge where each cut C is represented as a unit 2-sphere, U will be a component of the spinor ιAιB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' ιL, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=', one of the 2s + 1 spin- weighted spherical harmonics −sYsm with spin-weight −s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In this gauge, these are constant, thus trivially propagating along the light cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In a curved spacetime, U is a generalisation of these spin-weighted spherical harmonics to a topological but not necessarily metric sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In this general case, there are still 2s + 1 independent solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='2(i)) for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Calculating the NP constants In the remainder of this work we focus on the gravitational NP constants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=', with φABCD = ψABCD, the conformally rescaled Weyl spinor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Many system variables of the GCFE are components of spinors with respect to a certain spin-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In general, this spin-frame, and its associated null tetrad,does not agree with the frame adapted to I that was used in the definition of the NP constants but we can use null rotations to transform between the GCFE null frame and the frame adapted to I herein referred to as a Bondi frame§.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' § This is not strictly correct, since we are referring here only to a single cut, whereas the standard usage of the term Bondi frame refers to an entire system of cuts parametrised by the retarded Bondi time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The non-linear perturbation of a black hole by gravitational waves 6 A null rotation mixes one component of a spin-frame into another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' For example, a null rotation of oA around ιA is given by oA → oA + Y ιA, ιA → ιA (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='1) thus keeping ιA fixed, where Y is a function of the spacetime coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' If we denote the Bondi spin-frame by OA and IA, the GCFE frame by oA and ιA, and the corresponding Bondi and GCFE null-tetrad vectors by capital and lowercase letters respectively, then we may transform between frames by two null successive rotations (first fixing oA and then the new ιA) which have the combined form OA = oA + Y (ιA + XoA), IA = ιA + XoA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='2) The null rotation functions X and Y are determined by the conditions ∇aΘ = −ANa, M a∇at = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='3) where the scaling A is fixed given the conformal factor Θ and the above expression of the adapted spin-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' These conditions impose that N a points along the null generators of I and that the complex vector M a lies within the t = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' cuts of I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Appropriately fixing the freedom in how the frame propagates along the timelike conformal geodesics of the conformal Gauß gauge allows one to satisfy the second condition automatically, yielding X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The first condition gives us the value of the null rotation function Y on I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The transformation between the GCFE frame and the Bondi frame is then known and so we may write components with respect to the Bondi frame in terms of components with respect to the GCFE frame which are known numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' As a simple example, the third component of the gravitational spinor is written in the Bondi frame as ψABCDOAIBICID = ψABCD(oA + Y ιA)ιBιCιD = ψ3 + Y ψ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Both the GCFE and the NP constants are defined with respect to a spin and boost covariant formalism and so a properly weighted expression with respect to one frame results in a properly weighted expression with respect to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The same process is used to compute the area-form in terms of numerically available quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Fixing the behaviour of the frame off I + With the above two null rotations, the frame on I + is fixed, but we also have some freedom to choose how our frame changes as we move away from I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The presentation of I + in the proof of supertranslation invariance makes use of this and takes κ = 0 since the intersecting null hypersurface is foliated by a null geodetic congruence [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The Bondi frame so far is only fixed on I + and in order to achieve κ = 0 we need to enforce that DoA ∝ oA which means that we need to determine the null rotation The non-linear perturbation of a black hole by gravitational waves 7 function Y away from I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Suppose, we have fixed Y on I + with the above procedure, then we can get the required result with a third null rotation that becomes the identity on I + oA → ˆoA = oA + ZιA, ιA → ˆιA = ιA, where Z = O(Θ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='4) recalling the conformal factor Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Under this transformation, ˆκ = κ − DZ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='5) where D := La∇a and choosing Z so that κ = DZ on I + we obtain ˆκ = 0 there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Although the transformation becomes the identity on I +, we must worry about derivatives of the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In the Kirchhoff-d’Adhémar integral, we have a derivative of the form Dφ where φ = φAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='.LoAoB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='.oL (2s indices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Under this null rotation, ˆD ˆφ ��� I + = D ˆφ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='6) = D(φA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='.L(oA + ZιA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='.(oL + ZιL)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='7) = Dφ + 2sκφ1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='8) since the derivative of any term containing powers of Z higher than one will vanish on I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Computing U In the NP constant integrand (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='1), the active component would appear to be the term þcφ since this brings information of the arrival of the field φ at I +, while the quantity U appears to be somewhat inert, being used to project out certain pieces of information from the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Before the jump was made to curved spacetime, the Kirchhoff- d’Adhémar integral could represent the value of the field φ at timelike infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In this case, the quantity U was replaced by components of ιAιB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='ιL which are spin-weighted spherical harmonics in an appropriate frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' When curved spacetime is introduced, the U takes the role of these components and so different choices of U which satisfy the underlying equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='2), represent what would have been components of φ at timelike infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The job of U is to lend its spin, boost, and conformal weight to the expression, and so provide alignment between cuts of I + allowing for comparison from cut to cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' To compute U we must solve the “constraint equation” ¯ðcU = 0 on a cut∥ and evolve it along the null generators of I + with the evolution equation þ′ cU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Following methods from our earlier paper [8], we can expand these operators written in the Bondi frame in terms of the numerically implemented, standard operators ˜ð, ˜ð′, to obtain A˜ðU + B˜ð′U + CU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='9) ∥ This term is justified since, by considering the commutator [þ′ c, ¯ðc] one can show that any U satisfying the evolution equation þ′ cU = 0 will satisfy ¯ðcU = 0 on every cut if it satisfies this equation on a single cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In this sense, the constraint is propagated by the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The non-linear perturbation of a black hole by gravitational waves 8 Expanding the known coefficients A, B, and C, and the unknown function U in terms of spin-weighted spherical harmonics, and using the well-known relationship for products of spin-weighted spherical harmonics in terms of Clebsch-Gordon coefficients (see [23]) results in a system of homogeneous linear equations for the spectral coefficients of the function U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' There are five linearly independent solutions which span the solution space to the constraint equation for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The evolution equation can similarly be written in terms of numerically available quantities as, A∂tU + B˜ðU + C˜ð′U + DU = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='10) and may be evolved along I + by the method of lines given an initial solution to the constraint equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' An adaptive fourth-order Runge-Kutta method is used since the numerical output is not linearly spaced in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Fixing a basis in the solution space U It is clear that solution U of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='2(ii)) leads to another solution αU provided that α is a complex constant on the cut C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' More generally, any complex linear combination of solutions will also be a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Thus, changing the basis of the solution space U of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='2(ii)) will also change the individual values of the five NP constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Therefore, these do not, by themselves, carry independent physical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Only the combination of the values of the integrals together with the knowledge of the basis of U carries the full information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This means that in order to compare NP constants across different spacetimes we need to make sure that we specify “the same basis” for the solution space for each spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' There are several ways to do this: two complicated ones which are also physically relevant, and a third easier one which not as physically meaningful but much more pragmatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The first idea that comes to mind is to first conformally rescale the metric on the cut to make it into a unit-sphere, and, in a second step, to change the coordinate system by a Möbius transformation so that it becomes a standard polar coordinate system on the unit-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In this situation, the solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='2(ii)) are the standard spin-weighted spherical harmonics Ym := −2Y2m with −2 ≤ m ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' While the first step is rather straightforward, the second step leads to a Poisson-type equation on the sphere with a δ-like source term which is difficult (but not impossible) to treat numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In addition, after the coordinate transformation all quantities must be transformed which may introduce several numerical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The second way to introduce a basis in U is to make use of the fact that the standard spin 2 spherical harmonics Ym form an irreducible representation of the group SU(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' They each are an eigenvector of an infinitesimal generator with different eigenvalue, and they are obtained one from the other by the action of two ladder operators (very much akin to the angular momentum algebra of quantum mechanics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Fixing one of them as being annihilated by one of the ladder operators, one can generate the others The non-linear perturbation of a black hole by gravitational waves 9 by successive application of the other ladder operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This fixes the complete basis in terms of the first vector and leaves the freedom of scaling with one complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This can be almost fixed by normalising the vectors with respect to an appropriate Hermitian product, leaving the remaining freedom of a single phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In principle, this program could be carried out but it is very cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' First, one needs to find the infinitesimal generators of the group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This leads to a series of elliptic equations to be solved on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Next, one needs to use these generators to determining the function that is killed by one of the ladder operators, which is again an elliptic equation on the sphere, and then generate the other functions by successive application of the other ladder operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' As an alternative, one could solve the eigenvalue problem for the third operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Obviously, this procedure is numerically quite involved and prone to inaccuracies due to successive numerical differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' For this reason, we use a third method to fix a “universal” basis of U , and we use the “universal structure” that is available to us, namely our numerical setup which is the same for every spacetime that we compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Recall that our method is based on concentric round spheres and that every function we compute can be expanded as a linear combination of spin-weighted spherical harmonics defined with respect to the numerical round spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Therefore, we proceed as follows: first, on an initial cut, we compute five linearly independent solutions (uk)k=−2:2 of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='2(ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' These have the form uk = 2 � m=−2 cm kYm + Zl>2, k = −2 : 2 where Zl>2 stands for terms with higher values of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Then a straightforward linear combination of these solutions leads to the universal basis Uk which is defined by Uk = Yk + Zl>2 where Zl>2 again stands for higher l terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' We can interpret this basis as being the deformation of the standard basis provided by the Ym due to the impact of the incoming gravitational wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' If there was no gravitational wave, then the cut would be spherically symmetric and the yk would agree with the standard basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The basis, thus defined, is then propagated along I + using the evolution equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='2(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In this process, the form of the yk will change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This process of fixing a “universal basis” of U leaves no further freedom (except, of course, for the free phase inherent in the definition of the spin-weighted spherical harmonics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Integrating the Newman-Penrose integrand At this stage, all elements of the Newman-Penrose integral (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='1) have expressions in terms of known quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Integration can be performed against the basis Uk as defined in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='3 by simply computing the s = l = m = 0 spectral coefficient of the complete integrand and dividing by 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The theory shows these five complex numbers, obtained on a cut, come out the same independently of which cut was chosen for their evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In the next section we present numerical results that showcase these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The non-linear perturbation of a black hole by gravitational waves 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Numerical Results Using the above procedure, the NP constants were computed with data on I + from a numerically evolved spacetime modelling the non-linear perturbation of a Schwarzschild black hole by an incoming gravitational wave pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Because the NP constants are evaluated on each cut of I + defined as the intersection with a t = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' hypersurface, we obtain five complex numbers for every t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The initial mass of the black hole is m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='5 for all simulations considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Ingoing wave The ingoing pulse is defined as the choice of free data q0 of the lightlike, ingoing characteristic variable on the outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This is chosen to be a linear combination of the spin-weighted spherical harmonics 2Y2m for m = 0, 1, 2 and with amplitudes a, b and c respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The choices of these amplitudes will vary in the upcoming sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This gives the wave profile on the outer boundary q0(t, θ, φ) = � � � [4a � 2π 15 2Y20 − 2b � 5 π 2Y21 + 2c�π 5 2Y22] sin8(8πt) t ≤ 1 8 0 t > 1 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Checks of correctness We have demonstrated in several papers [3, 8] that the solutions computed by the GCFE system converge at the correct order for the 2 + 1 axisymmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Here we show that this is also true in the general case of 3 + 1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' We also show that the NP constants converge to constant values on I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' (a) θ = π/2 and φ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' (b) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='978 and φ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' (c) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='978 and θ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Figure 1: The imaginary part of a constraint equation from the Bianchi identity when a single spatial coordinate is fixed at t = 1 for r, θ, φ resolutions of {101, 9, 16}, {201, 17, 32} and {401, 33, 64} (denoted by Res1, Res2 and Res3 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The dashed vertical line represents I + in the radial plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The curves from top to bottom correspond to increasing resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' We use an ingoing wave profile proportional to 2Y22 with a = b = 0, c = i to allow excitation in the φ-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 1 demonstrates convergence in all spatial directions at t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 0 Resl Res3 Res2 log10 lErrorl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='0 r0 Res1 Res3 Res2 log1o lErrorl 5 10 15 0 1 2 30 Res1 Res3 Res2 log1o lErrorl 5 10 15 0 2 4 6 0The non-linear perturbation of a black hole by gravitational waves 11 Focusing now on the NP constants, we demonstrate how they approach constant values along I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 2 shows a convergence test of the discrepancy from constancy of the single non-vanishing NP constant in axisymmetry, choosing amplitudes a = 1, b = c = 0, along I + as the spatial resolution is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The resolution in r (number of intervals) is denoted rres and corresponds to an equivalently scaled resolution θres along the θ-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The coarsest values are rres = 100 and θres = 8, and the resolutions are doubled in successive simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='6 Conformal time t −8 −6 −4 −2 0 log10 Error rres = 100 rres = 200 rres = 400 Figure 2: Convergence of the log10 difference between the magnitude of the NP constant at time t and at the initial cut with increasing spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Variable ingoing amplitude An superficial glance at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='1) would suggest that due to the linearity of the integrand and the zero-rest-mass field equations with respect to φ, scaling the amplitude of the ingoing wave would just scale the NP constants linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' However, this turns out not to be the case because φ satisfies the Bianchi equation into which φ enters non-linearly through the connection coefficients of the covariant derivative [23, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Hence, it is interesting to numerically probe the scaling of the NP constants as the ingoing wave profile is scaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' We performed four simulations in axisymmetry with the amplitudes b = c = 0 and a taking the values 1,2,5, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' These are evolved up to t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='77 at which point the system variables start to diverge due to the close ‘conformal’ proximity to i+ at t ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Table 1 shows the corresponding single NP constant for each amplitude as well as the relative error from a linear fit through the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Fitting the NP constants to the ansatz The non-linear perturbation of a black hole by gravitational waves 12 αaβ yields α ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='53865 and β ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='99803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 3 shows the log10 deviation of the NP constant value from the value on the initial cut for each amplitude as a function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The deviation from a linear fit is orders of magnitude greater than the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This is due to the amplitude of the initial wave profile entering into the field equations non-linearly resulting in a non-linear relationship between initial amplitude and Newman-Penrose constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Amplitude 1 2 5 10 NPC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='53882 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='07638 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='68405 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='36222 Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='00116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='00373 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='00482 Table 1: The one non-vanishing NP constant for different ingoing wave amplitudes and the deviation from a linear fit through the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This deviation is orders of magnitude larger than the error for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This is a result of the amplitude entering non-linearly into the field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='6 Conformal time t −10 −9 −8 −7 −6 −5 −4 log10 error a = 1 a = 2 a = 5 a = 10 Figure 3: The log10 difference of the NP constant from the value on an initial cut as a function of conformal time t for a variable amplitude of the initial wave profile as a measure of deviation from constancy due to error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Cumulative error grows as we integrate along I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Deviation from axisymmetry In a general asymptotically flat spacetime there are five complex NP constants corresponding to the five independent solutions to the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In axisymmetry, The non-linear perturbation of a black hole by gravitational waves 13 these collapse to only one independent solution, when the frame and coordinates also respect the symmetry, because then only the m = 0 modes of a spherical harmonic expansion remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' We can investigate the collapse of five NP constants into one by using the initial wave profile given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='1) and using a = i, b = c = ϵ i, where ϵ parametrises a deviation from axisymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Six simulations were run for this wave profile for ϵ = 0, 1, 2, 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 4 shows the magnitudes of the corresponding NP constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Although we do have access to the full ten real degrees of freedom (five complex degrees of freedom) for each simulation, the trends can be seen in the behaviour of the magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 5 shows the same quantities but separated by the value of m of the corresponding U so that individual trends can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' We can see that for ϵ = 0, there is only one non-trivial constant 0 1 2 3 4 5 Amplitude ϵ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='50 NPC m = −2 m = −1 m = 0 m = 1 m = 2 Figure 4: The magnitudes of the five complex NP constants as a function of a parameter ϵ which breaks axisymmetry in the initial wave profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' For ϵ > 0 all constants are non- zero although most are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' corresponding to the axisymmetric m = 0 mode, but as non-axisymmetric modes are introduced for ϵ > 0, all five constants take on non-trivial values and grow with ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Discussion In this paper, we continue our numerical investigation into the non-linear perturbation of a Schwarzschild black hole using an initial boundary value problem for the general conformal field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This novel numerical scheme allows us to include I + within the computational domain and so compute quantities there directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Thereby, we have computed the NP constants for the first time in a physically realistic spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The gauge quantities of the system were fixed by numerical needs which implies that we were unable to directly use the very specific set of coordinates, frame, and The non-linear perturbation of a black hole by gravitational waves 14 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='5 m = −2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='5 m = −1 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='540 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='541 m = 0 0 2 4 Amplitude ϵ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='04 NPC m = 1 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='010 m = 2 Figure 5: The magnitudes of the five complex NP constants as a function of a parameter ϵ which breaks axisymmetry in the initial wave profile split by the value of m of the corresponding U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Note that m is a label on spherical harmonics, not a mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' conformal factor typically used when defining quantities at I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' To compute physical quantities such as the NP constants with data from the numerical simulation, we need an explicitly conformally invariant expression for the quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' However, this concept of conformal invariance is a rather special one, and it might be appropriate to highlight it again here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Physical quantities make reference to the physical metric ˜gab of the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' In our context, the physical metric is represented as ˜gab = Ω−2gab, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=', in terms of another metric in the same conformal class and the conformal factor relating the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' By conformal invariance we do not mean invariance under ˜gab �→ θ2˜gab, but rather the invariance under (gab, Ω) �→ (θ2gab, θΩ), which corresponds to the free choice of the splitting of ˜gab into a conformal and a scale part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' For example, the recent analysis of the Bondi-Sachs energy-momentum in this framework involved generalising the procedure of constructing a basis of translations with respect to which components of the Bondi-Sachs 4-vector may be taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The standard procedure of choosing the first four spherical harmonics is certainly not conformally invariant in this sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This led to an invariant characterisation of the Lorentzian metric on the space of BMS translations [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Of course, the existence of this metric still leaves the freedom of Lorentz transformations for the choice of the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' We run into the same problem when defining a basis for the quantity U which, when integrated against the Newman-Penrose integrand, gives the linearly independent NP constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Again, it is the solution space which is defined in a conformally invariant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' But in this case there is no obvious inner product that one could use to select a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Even if there was one, the basis would still be defined only up to the appropriate REFERENCES 15 (pseudo-) orthogonal transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' We circumvent the non-uniqueness of the basis in this case by refering to the universal structure that is imposed on the problem by the numerical setup as explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' This seems to be the best way to ensure comparability across the different space-times that we investigate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Acknowledgments Supported by the Marsden Fund Council from Government funding, managed by Royal Society Te Ap¯arangi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' The authors would like to thank L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' Escobar for sharing the general form of his SWSH code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' We wish to acknowledge the use of New Zealand eScience Infrastructure (NeSI) high performance computing facilities, consulting support and/or training services as part of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} +page_content=' New Zealand’s national facilities are provided by NeSI and funded jointly by NeSI’s collaborator institutions and through the Ministry of Business, Innovation & Employment’s Research Infrastructure programme.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE4T4oBgHgl3EQfyw06/content/2301.05268v1.pdf'} diff --git a/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf b/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..70906673acb4e93ae18a4f2d793df5363bdd5a5d --- /dev/null +++ b/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4b8d07b834a9a0f2578b5f0537410e74eaf4e65fbad64946dc1f01f5b5b07e64 +size 178882 diff --git a/0tFRT4oBgHgl3EQfkzf9/vector_store/index.faiss b/0tFRT4oBgHgl3EQfkzf9/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..7b3d51109a02a16a711b2d82a9a68334d4a2ea97 --- /dev/null +++ b/0tFRT4oBgHgl3EQfkzf9/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a7bde347df668af9fa6995ad847071117570c1d9201f6ce45905b87310982bc3 +size 3014701 diff --git 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index 0000000000000000000000000000000000000000..98a42055ca0203aa3ff1aa886ff6c50d8497835e --- /dev/null +++ b/1tFAT4oBgHgl3EQfkB3r/content/tmp_files/2301.08609v1.pdf.txt @@ -0,0 +1,841 @@ +Approximate Quantum Compiling for Quantum Simulation: A +Tensor Network based approach +Niall F. Robertson1, Albert Akhriev1, Jiri Vala2,3 and Sergiy Zhuk1 +1 IBM Quantum, IBM Research Europe - Dublin, IBM Technology Campus, Dublin 15, Ireland +2 Maynooth University, Maynooth, Ireland +3 Tyndall National Institute, Cork, Ireland +Abstract +The simulation of quantum spin chains is a promising candidate for the demonstration of quantum +advantage. One of the main obstacles to achieving this is the noise that arises from implementing +the deep circuits that appear in standard quantum time evolution algorithms. Compiling these deep +circuits into shallower ones is thus a key issue that we address in this work. We use a Tensor Network +based approach to Approximate Quantum Compiling to produce short depth quantum circuits that +simulate the time evolution of the Heisenberg spin chain on up to 100 qubits. Furthermore, we run +these short depth circuits on a ibmq-mumbai - a 27 qubit device - and show that the accuracy of +the measured observables is significantly improved after applying our Tensor Network compilation +scheme. +1 +Introduction +The simulation of quantum many-body systems is a task of immense scientific interest. The study of +quantum dynamics, in particular, allows for the study of thermalisation, many-body localisation, Hub- +bard model physics and the applicability of field theory to out-of-equilibrium phenomena. In all of these +fields there are many open scientific questions whose answers are likely to demand accurate simulation +of quantum dynamics. However, the classical computational requirements of a brute-force approach to +quantum dynamical simulations scales exponentially in the size of the system. Approximate techniques +such as Tensor Networks are thus often called upon. Tensor Networks represent one of the best set of +tools available to simulate time evolution and can also be applied to other problems such as ground state +calculations [1] and machine learning [2, 3]. +Matrix Product States (MPS) are a particular type of Tensor Network that are particularly suited to +describe quantum systems in one dimension. They form a key component of modern implementations of +the well known Density Matrix Renormalisation Group (DMRG) algorithm used to find the ground state +of local Hamiltonians. The DMRG algorithm was designed many years before [4] it was realised that it +could be understood as a variational optimisation algorithm where a Matrix Product State is used as an +Ansatz for the ground state [5]. This insight shed light on the reasons behind the spectacular success of +DMRG; the ground states of local Hamiltonians are only weakly entangled and so too are Matrix Product +States. More precisely, the bipartite entanglement entropy S of the ground state of a local Hamiltonian +satisfies an area law, meaning that the entanglement entropy is proportional to the area of the boundary +of the two subsystems in the bipartition. In 1D, this means that the entanglement entropy is independent +of the system size [6]. This is in contrast to typical states in Hilbert space whose entanglement structures +satisfy a volume law. Matrix Product States are also known to satisfy an area law [5] and thus have the +same entanglement structure as the ground state by design. +Since the weak entanglement of ground states of local Hamiltonians allow for their efficient storage as +Matrix Product States, it is natural to ask if this is also possible for states that are generated by time +evolution as these states are no longer necessarily weakly entangled. It turns out that for many physical +systems of interest, entanglement entropy increases linearly until it saturates, at which point an MPS +1 +arXiv:2301.08609v1 [quant-ph] 20 Jan 2023 + +will no longer be an efficient representation of the state. However, if the initial state is weakly entangled +then the MPS representation can be used to store the state at early times. A paradigmatic example of +this scenario is a quantum quench, whereby a quantum system is initially prepared in the ground state of +some local Hamiltonian, the parameters of the Hamiltonian are subsequently changed very rapidly and +the system then evolves according to Schr¨odinger’s equation. The TEBD algorithm (Time Evolving Block +Decimation) can be used to simulate time evolution after a quantum quench; the state is stored as an MPS +and this MPS is updated as a function of time. Despite the success of DMRG, TEBD and other Tensor +Network algorithms, these approaches are not without limitations. The memory requirements to store +an MPS is characterised by the bond dimension, given by the dimension of the largest matrix used in the +description of the state. For constant approximation error ϵ this bond dimension increases exponentially +with the entanglement entropy and thus with time. Therefore, for a fixed maximum bond dimension, +the error ϵ increases exponentially with time. This limits the applicability of Tensor Network algorithms +to short time simulations. A quantum algorithm however, does not in principle suffer from this issue - +the key difference between a quantum and a classical device being the ability to store highly entangled +states. A quantum computer therefore has the potential to simulate quantum many-body systems for +long times. The accurate simulation of the time evolution of 1D quantum systems is thus a promising +route for the demonstration of quantum advantage in the short term. One such quantum algorithm is +Trotterisation, where a discrete time step dt is used and the time evolution operator is approximated as +a quantum circuit with an error that scales polynomially in dt. The depth of the quantum circuit used +in such an approach increases with decreasing dt, leading to a trade-off between the noise arising from +using deep circuits and the decreasing accuracy of the approximation when dt is increased. A number +of variational quantum algorithms for the simulation of time evolution have therefore been developed +that aim to use shallower circuits [7, 8, 9, 10]. Each of these approaches suffer from a number of issues +such as convergence, runtime and limited device connectivity. As a result, it has been argued that such +variational approaches are not practical for use on near term quantum hardware [11]. +One approach that aims to overcome the issue of deep circuits is Approximate Quantum Compiling +[12, 13, 14], where one defines a parametric circuit of fixed depth and uses techniques from optimisation +to minimise the distance between the parametric circuit and the target circuit of interest - where distance +is defined by some carefully chosen metric. In principle, this approach can lead to short depth circuits +that implement the target circuit of interest within some error tolerance. In practice, a classical imple- +mentation of such an approach [14] is limited to act on a small number of qubits due to the exponential +scaling of the Hilbert space with the number of qubits. +Here we develop a new approach to quantum simulation that combines Matrix Product States, Approx- +imate Quantum Compiling and Trotterisation to produce short depth quantum circuits that implement +the time evolution operator of the Heisenberg spin chain. This approach is scalable thanks to the im- +mense power of Matrix Product States. Figure 1 shows a schematic of our approach: first we apply +Trotterisation classically for the maximum length of time for which we can still store the state as an +MPS. We then apply a Matrix Product State implementation of Approximate Quantum Compiling to +squeeze the circuit (purple box in the figure) to find a much shallower circuit that still reproduces the +same state as Trotterisation, up to some small error in the fidelity. We then use the squeezed circuit as +the input for the Trotter circuit which can now generate a quantum state beyond what can be stored +classically. +2 +Setup +2.1 +The model +We will consider the XXX spin-chain - a paradigmatic model for quantum magnetism - defined by the +Hamiltonian: +HXXX = − +L−1 +� +i=0 +hi,i+1 = − +L−1 +� +i=0 +� +Sx +i Sx +i+1 + Sy +i Sy +i+1 + Sz +i Sz +i+1 +� +, +(1) +where Sx, Sy and Sz are written in terms of Pauli matrices as Sx = σx +2 , Sy = σy +2 and Sz = σz +2 . The +Hamiltonian in (1) is a prototypical example of an integrable 1D model and its dynamical behaviour has +been studied extensively [15], including on a quantum computer [16]. The time evolution of a quantum +2 + +Compress with AQC +... +... +... +... +... +... +... +... +Figure 1: Schematic of our approach: Trotterisation is applied classically (purple box) and then a Matrix +Product State implementation of Approximate Quantum Compiling is applied to compress the first part +of the circuit. Standard Trotterisation is then applied on a quantum device afterwards to simulate longer +times, i.e. times which are beyond what is possible classically. +Rz(θ) +Rz( π +2 ) +Rz(− π +2 ) +Ry(φ) +Ry(λ) +Figure 2: Implementation of two site operator ei(ασx⊗σx+βσy⊗σy+γσz⊗σz) as a quantum circuit. We have +the correspondences θ = π +2 − 2γ, φ = 2α − π +2 and λ = π +2 − 2β. The Hamiltonian in (1) corresponds to +the case α = β = γ = dt +state |ψ(t)⟩ is governed by the Schr¨odinger equation: +|ψ(t)⟩ = e−iHXXXt |ψ(0)⟩ +(2) +where |ψ(0)⟩ is the wavefunction at time t = 0. In this work, we will consider the N´eel state, written as: +|↑↓↑↓ ... ↑↓⟩ where ↑ and ↓ represent up and down spins respectively. The N´eel state for n spins is simply +implemented on n qubits as |1010...10⟩. +The time evolution operator U(t) ≡ e−iHt can be executed as a quantum circuit in a resource efficient +way; we first write the Hamiltonian in (1) as HXXX = H1 + H2 where H1 = − � +i odd +hi,i+1 and H2 = +− � +i even +hi,i+1. Note that all operators in a given sum commute with all other operators in their respective +sums. We then define the Suzuki-Trotter time evolution operator Utrot(dt) in the following way: +U(1) +trot(dt) = +L/2−1 +� +j=0 +U2j,2j+1(dt) +L/2−1 +� +j=1 +U2j−1,2j(dt) = e−iHXXZdt + O(dt2) +(3) +where Ujk(dt) = e−ihjkdt. The exact time evolution operator U(t) is thus approximated by m repeated +applications of Utrot(dt = +t +m), i.e. U(t) ≈ Um +trot(dt = +t +m). As discussed in [16], each Ujk(dt) appearing in +(3) can be implemented by the quantum circuit with just three CNOTs as in Figure 2. We can reduce +the error in the Trotter formula in equation (3) by using higher order expressions [17]. It turns out that +the second order Trotter formula can be implemented on a quantum circuit with only one extra layer in +the circuit [16]. We have: +U(2) +trot(dt) = +L/2−1 +� +j=0 +U2j,2j+1 +�dt +2 +� L/2−1 +� +j=1 +U2j−1,2j (dt) +L/2−1 +� +j=0 +U2j,2j+1 +�dt +2 +� += e−iHXXZdt + O(dt2) +(4) +which can be implemented on a quantum device by the circuit in Figure 4. +3 + +U(dt) +U(dt) +U(dt) +U(dt) +U(dt) +U(dt) +U(dt) +U(dt) +U(dt) +U(dt) +U(dt) +U(dt) +U(dt) +U(dt) +U(dt) +Figure 3: First order Trotter circuit acting on six qubits. +U( dt +2 ) +U(dt) +U(dt) +U( dt +2 ) +U(dt) +U(dt) +U(dt) +U( dt +2 ) +U(dt) +U(dt) +U( dt +2 ) +U(dt) +U(dt) +U(dt) +U( dt +2 ) +U(dt) +U(dt) +U( dt +2 ) +Figure 4: Second order Trotter circuit acting on six qubits. +4 + +A(1) +A(2) +A(3) +A(4) +A(5) +A(6) +Figure 5: Graphical representation of an MPS. There are two matrices A(i) for each qubit at position i. +A(1) +A(2) +A(3) +A(4) +A(5) +A(6) +B(1) +B(2) +B(3) +B(4) +B(5) +B(6) +Figure 6: The inner product ⟨ψ1|ψ2⟩ of two Matrix Product States - see equations (11) and (6). +2.2 +Matrix Product States +An arbitrary quantum state on n qubits can be written in terms of complex variables cj1,...,jn, the number +of which scales as 2n: +|ψ⟩ = +� +{j1,...,jn} +cj1,...,jn |j1, ..., jn⟩ +(5) +where the sum is over all configurations of the binary variables j1, ..., jn. The bipartite entanglement +entropy of an arbitrary quantum state picked at random from Hilbert space satisfies a volume law which, +as was discussed in the introduction, is distinct from area law entanglement in which case the entanglement +entropy of two regions after the bipartition of the system is proportional to the area of the boundary of +the system. A small subset of states in Hilbert space satisfies an area law. The coefficients cj1,...,jn of +such states have a certain structure that we can take advantage of to study classically. Any state |ψ⟩ can +be written in the following way: +cj1,...,jn = A(1) +j1 · A(2) +j2 ... · A(n) +jn +(6) +where the Aj are χj × χj+1 dimensional matrices. Quantum states of the form (6) are known as Matrix +Product States (MPS). The maximum value of χj is referred to as the bond dimension of the MPS. We +can represent an MPS graphically as in Figure 5. We associate one matrix A(i) to each qubit. Note +that for each qubit i we have two matrices. We thus have a total of 2n matrices to keep track of. The +bond dimension χj can be seen as a measure of the entanglement between the two subsystems when a +bipartition is made at qubit j. Therefore, states in Hilbert space that satisfy an area law - and therefore +have a low bond dimension in their MPS representation - can be efficiently stored as Matrix Product +States. States that satisfy a volume law will have a bond dimension that is exponential in the number of +qubits. We will consider in this work the non-trivial dynamics governed by equation (2). As discussed in +the introduction, the bipartite entanglement entropy of a ground state of a one-dimensional Hamiltonian +that has a gap between its ground state and its excited state is independent of the size of the subsystems. +The ground state of such a system - and hence the initial state in our setup - can be efficiently stored +as an MPS. One can then use an algorithm such as TEBD (Time Evolving Block Decimation) [18] to +update the MPS as a function of time to study the dynamics of the system. However, the entanglement +entropy of the state increases linearly with time, hence the bond dimension χ that is required to keep +the error constant diverges exponentially with time. To simulate for longer times, a quantum computer +would be needed. In section 2.3, we will discuss how Matrix Product States can be leveraged to reduce +the resource requirements for this simulation problem when implemented on a quantum device. +2.3 +Matrix Product States applied to Approximate Quantum Compiling +j +k +Ry(θ1) +Rz(θ2) +Ry(θ3) +Rx(θ4) +Figure 7: CNOT block forms the basic building block of our circuit ansatz. +5 + +Rz(θ1) +Ry(θ2) +Rz(θ3) +Rz(θ4) +Ry(θ5) +Rz(θ6) +Rz(θ7) +Ry(θ8) +Rz(θ9) +Rz(θ10) +Ry(θ11) +Rz(θ12) +Figure 8: Parameterised circuit inspired by the structure of the first order Trotter circuit in Figure 3. +Rz(θ1) +Ry(θ2) +Rz(θ3) +Rz(θ4) +Ry(θ5) +Rz(θ6) +Rz(θ7) +Ry(θ8) +Rz(θ9) +Rz(θ10) +Ry(θ11) +Rz(θ12) +Figure 9: Parameterised circuit inspired by the structure of the second order Trotter circuit in Figure 4. +Approximate quantum compiling (AQC) involves the design of a parametric quantum circuit with +fixed depth - the parameters are then adjusted to bring it as close as possible to the target, where “close” +is defined via some carefully chosen metric, see below. As discussed in [12], one can use so-called CNOT +blocks to construct a natural circuit Ansatz. A CNOT block is a CNOT gate followed by single qubit +rotations (see Figure 7). A block with a CNOT gate acting on a “control” qubit j and “target” qubit +k is written as CUjk(θ1, θ2, θ3, θ4). For a given hardware connectivity, one can then write down a fully +parameterised circuit as: +Vct(θ) =CUct(L) (θ3n+4L−3, . . . , θ3n+4L) · · · CUct(1) (θ3n+1, . . . , θ3n+4) +[Rz (θ1) Ry (θ2) Rz (θ3)] ⊗ · · · ⊗ [Rz (θ3n−2) Ry (θ3n−1) Rz (θ3n)] +(7) +The position of the CNOT blocks in the parameterised circuit can be customised to suit the particular +target circuit that one is interested in. Here we are interested in finding a circuit that implements the +unitary time evolution operator as in equation (2). We thus consider a structure inspired by the first and +second-order Trotter circuits in Figures 3 and 4 respectively. Recall that each block U(dt) in Figures 3 +and 4 represents the 2-qubit sub-circuit with three CNOTs in Figure 2; it is therefore natural to consider +a circuit Ansatz with sub-circuits each with three CNOT blocks as in Figures 8 and 9, such that the +circuit Ansatz mimics the structure of the first and second order Trotter circuits. In the notation of [14], +the parameterised circuits in Figures 8 and 9 correspond to n = 4 qubits, l = 2 layers and b = 3 CNOT +blocks in each layer. In both Figure 8 and Figure 9 there are three rotation gates acting on each qubit at +the beginning of the circuit. In the examples considered in this work we will take the initial state to be |0⟩ +- the initial rotation gate Rz(θ) is redundant for these cases but is necessary for more general initial states. +One can define the distance between the target and parameterised circuit via a number of different +metrics. Here we use a cost function based on the Hilbert-Schmidt test: +Cstate +hs += 1 − | ⟨0| V †(θ) |ψ0⟩ |2 +(8) +The goal of AQC is to tune the parameters θ to minimise the cost function under consideration. Note +that here we are considering the application of AQC to state preparation as opposed to full circuit com- +pilation. More precisely, this means that our cost function is designed such that it is minimised when the +action of V (θ) on the initial state |0⟩ produces a state that is as close as possible to a target state |ψ0⟩ +(up to some global phase). This is in contrast to the situation where one starts with some target circuit +U and the cost function is designed to bring the full matrix V (θ) as close as possible to U. +6 + +As pointed out in [19], the gradient of the cost function in (8) vanishes exponentially. This observation +lead to the distinction between global and local cost functions; local cost functions have only polynomially +vanishing gradients in some cases of interest - see [19, 20, 14] for details. As was shown in [14], the Hilbert- +Schmidt test - which is a global cost function - can be turned into a local one by adding several “bit-flip” +terms which increases the magnitude of the gradient: +Cstate +lhs += 1−| ⟨0| V †(θ) |ψ0⟩ |2 − +�n − 1 +n +� +n +� +j=1 +| ⟨0| XjV †(θ) |ψ0⟩ |2 +− +�n − 2 +n +� � +j vs Itl_gt> and Itl> vs Itl gt>, num.qubits: 50 +1.00 +0.99 +0.98 +0.97 +0.96 +fidelity +0.95 +0.94 +0.93 +0.92 +Ansatz +0.91 +Trotter +0.90 +1.2 +2.4 +3.6 +4.8 +6.0 +7.2 +evolution time +13 +６ +9 +12 +15 +18 +number of layers in ansatz +13 +６ +９ +12 +15 +18 +number of Trotter stepsFidelity: lal> vs Itl_gt> and Itl> vs Itl gt>, num.qubits: 50 +1.00 +0.99 +0.98 +0.97 +0.96 +fidel +0.95 +0.94 +0.93 +0.92 +0.91 +Ansatz +Trotter +0.90 +1.2 +2.4 +3.6 +4.8 +6.0 +7.2 +evolution time +2 +12 +4 +6 +8 +10 +number of layers in ansatz +13 +６ +9 +12 +15 +18 +number of Trotter stepsFigure 12: 50 qubits: the maximum number of layers in the parametric circuit is 9 while it is 18 for the +Trotter circuit. Both circuits achieve very similar fidelities despite the parametric circuit being half the +depth of the Trotter circuit. +Figure 13: 100 qubits: The maximum number of layers in the parametric circuit and the Trotter circuit +is 18. It can be seen that the fidelity of the Trotter circuit decays rapidly but that of the parametric +circuit remains high. +same data for 100 qubits in Figures 13 and 14. +Now we would like to consider how these results affect the implementation on a real quantum device. +We consider a 20 qubit spin-chain on the 27 qubit device ibmq-mumbai. First we plot the fidelity results +for 20 qubits in Figure 15. We ran the resulting parametric circuit on ibmq-mumbai and in Figure 16 +we plot the expectation values ⟨ψ(t)|Sz +0|ψ(t)⟩ as obtained from the quantum device using the parametric +AQC circuit, the Trotter circuit and from a classical Tensor Network simulation. We observe that this +observable is more accurate when obtained with the AQC circuit due to its reduced depth. Note that +the difference between the results from the simulation and the results from the quantum device would be +greatly reduced after applying error mitigation [21, 22]. We have not attempted to apply error mitigation +to either circuit as this would be outside the scope of this work. +We expect that, since our Tensor +Network compilation scheme greatly reduces the noise of the circuit, any error mitigation scheme would +be enhanced by our approach. +9 + +Fidelity: lal> vs |tl_gt> and Itl> vs Itl_gt>, num.qubits: 50 +1.00 +Ansatz +0.99 +Trotter +0.98 +0.97 +0.96 +fideli +0.95 +0.94 +0.93 +0.92 +0.91 +0.90 +1.2 +2.4 +3.6 +4.8 +6.0 +7.2 +evolution time +2 +4 +8 +９ +6 +number of layers in ansatz +13 +６ +9 +12 +15 +18 +number of Trotter stepsFidelity: lai) and Iti> vs. ground-truth state, Ngqubits: 1o0 +1.00 +0.99 +0.98 +0.97 +0.96 +0.95 +p! +0.94 +0.93 +0.92 +0.91 +Ansatz +Trotter +0.90 +1.2 +2.4 +3.6 +4.8 +6.0 +7.2 +evolution time +13 +６ +9 +12 +15 +18 +number of layers in ansatz +13 +6 +９ +12 +15 +18 +number of Trotter stepsFigure 14: 100 qubits: The maximum number of layers in the parametric circuit is 12 while it is 18 for +the Trotter circuit. +Figure 15: The maximum depth of the parametric circuit is half that of the Trotter circuit - there are 9 +and 18 layers respectively. These 20 qubit circuits were implemented on ibmq-mumbai - see Figure 16. +4 +Discussion +In this paper we applied Tensor Network methods to Quantum Compiling and demonstrated their efficacy +on the 27 qubit device ibmq-mumbai. Our method is similar in spirit to [23] where Matrix Product States +were used to prepare the initial state for VQE to find the ground state of some Hamiltonian - here we +use Matrix Product States to prepare a short depth quantum circuit that simulates the time evolution of +a 1D Hamiltonian. We chose the XXX Hamiltonian in equation (1) because it has been well studied, but +we would be particularly interested to apply the compilation methods developed here to non-integrable +systems by e.g. adding a random field to the Hamiltonian in (1) and studying phenomena of scientific +interest such as many-body localisation. +We have shown results of our simulations on up to 100 qubits. In principle we can significantly increase +the number of qubits and the length of time to which we apply our MPS compilation scheme; the limiting +factor at present seems to be the particular implementation that we apply to SVD and to calculate the +gradient. We believe that both of these can be improved significantly, in particular by using an efficient +parallel implementation - this is the subject of ongoing work. In our current framework we use the Qiskit +10 + +Fidelity: lai) and Iti> vs. ground-truth state, Ngqubits: 1o0 +1.00 +Ansatz +0.99 +Trotter +0.98 +0.97 +0.96 +0.95 +p! +0.94 +0.93 +0.92 +0.91 +0.90 +1.2 +2.4 +3.6 +4.8 +6.0 +7.2 +evolution time +2 +12 +4 +6 +8 +10 +number of layers in ansatz +13 +６ +9 +12 +15 +18 +number of Trotter stepsFidelity: lai> and Iti> vs. ground-truth state, Nqubits: 20 +1.000 +0.995 +0.990 +0.985 +0.980 +0.975 +fide +0.970 +0.965 +0.960 +Ansatz +0.955 +Trotter +0.950 +1.2 +2.4 +3.6 +4.8 +6.0 +7.2 +evolution time +2 +4 +９ +6 +7 +8 +number of layers in ansatz +13 +６ +９ +12 +15 +18 +number of Trotter stepsFigure 16: The expectation value of Sz +0 vs time for a chain of 20 qubits as measured on the 27 qubit +quantum device ibmq-mumbai. +The circuit produced from our MPS implementation of AQC uses is +shallower than the Trotter circuit, and thus produces an expectation value that is much closer to the true +value plotted in the blue curve, obtained by classical Tensor Network simulations. +MPS package which is designed for generic situations in which long range connectivity may be required +and thus does not take advantage of the short range structure of the circuits in Figures 8 and 9. +5 +Acknowledgements +This work was funded by the Disruptive Technologies Innovation Fund (DTIF), by Enterprise Ireland, +under project number DTIF2019-090 (project QCoIR) and also supported by IBM Quantum. +11 + +0.0 - ++ +-0.1 ++ +0.2 +0.3 +-0.4 +Simulation +-0.5 +ibm mumbai: Trotter circuit ++ +ibm mumbai: AQC circuit +-0.6. +0.D +0.5 +15 +2D +25 +3.D +3.5 +4.D +tReferences +[1] Frank Verstraete and J Ignacio Cirac. +Matrix product states represent ground states faithfully. +Physical review b, 73(9):094423, 2006. +[2] Edwin Stoudenmire and David J Schwab. 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Syn- +ergy between quantum circuits and tensor networks: Short-cutting the race to practical quantum +advantage. arXiv preprint arXiv:2208.13673, 2022. +13 + diff --git a/1tFAT4oBgHgl3EQfkB3r/content/tmp_files/load_file.txt b/1tFAT4oBgHgl3EQfkB3r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e11f3b89ac0ad41c25c6c3aedade845bb98a058c --- /dev/null +++ b/1tFAT4oBgHgl3EQfkB3r/content/tmp_files/load_file.txt @@ -0,0 +1,460 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf,len=459 +page_content='Approximate Quantum Compiling for Quantum Simulation: A Tensor Network based approach Niall F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Robertson1, Albert Akhriev1, Jiri Vala2,3 and Sergiy Zhuk1 1 IBM Quantum, IBM Research Europe - Dublin, IBM Technology Campus, Dublin 15, Ireland 2 Maynooth University, Maynooth, Ireland 3 Tyndall National Institute, Cork, Ireland Abstract The simulation of quantum spin chains is a promising candidate for the demonstration of quantum advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' One of the main obstacles to achieving this is the noise that arises from implementing the deep circuits that appear in standard quantum time evolution algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Compiling these deep circuits into shallower ones is thus a key issue that we address in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' We use a Tensor Network based approach to Approximate Quantum Compiling to produce short depth quantum circuits that simulate the time evolution of the Heisenberg spin chain on up to 100 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Furthermore, we run these short depth circuits on a ibmq-mumbai - a 27 qubit device - and show that the accuracy of the measured observables is significantly improved after applying our Tensor Network compilation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' 1 Introduction The simulation of quantum many-body systems is a task of immense scientific interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The study of quantum dynamics, in particular, allows for the study of thermalisation, many-body localisation, Hub- bard model physics and the applicability of field theory to out-of-equilibrium phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' In all of these fields there are many open scientific questions whose answers are likely to demand accurate simulation of quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' However, the classical computational requirements of a brute-force approach to quantum dynamical simulations scales exponentially in the size of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Approximate techniques such as Tensor Networks are thus often called upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Tensor Networks represent one of the best set of tools available to simulate time evolution and can also be applied to other problems such as ground state calculations [1] and machine learning [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Matrix Product States (MPS) are a particular type of Tensor Network that are particularly suited to describe quantum systems in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' They form a key component of modern implementations of the well known Density Matrix Renormalisation Group (DMRG) algorithm used to find the ground state of local Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The DMRG algorithm was designed many years before [4] it was realised that it could be understood as a variational optimisation algorithm where a Matrix Product State is used as an Ansatz for the ground state [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' This insight shed light on the reasons behind the spectacular success of DMRG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' the ground states of local Hamiltonians are only weakly entangled and so too are Matrix Product States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' More precisely, the bipartite entanglement entropy S of the ground state of a local Hamiltonian satisfies an area law, meaning that the entanglement entropy is proportional to the area of the boundary of the two subsystems in the bipartition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' In 1D, this means that the entanglement entropy is independent of the system size [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' This is in contrast to typical states in Hilbert space whose entanglement structures satisfy a volume law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Matrix Product States are also known to satisfy an area law [5] and thus have the same entanglement structure as the ground state by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Since the weak entanglement of ground states of local Hamiltonians allow for their efficient storage as Matrix Product States, it is natural to ask if this is also possible for states that are generated by time evolution as these states are no longer necessarily weakly entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' It turns out that for many physical systems of interest, entanglement entropy increases linearly until it saturates, at which point an MPS 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='08609v1 [quant-ph] 20 Jan 2023 will no longer be an efficient representation of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' However, if the initial state is weakly entangled then the MPS representation can be used to store the state at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' A paradigmatic example of this scenario is a quantum quench, whereby a quantum system is initially prepared in the ground state of some local Hamiltonian, the parameters of the Hamiltonian are subsequently changed very rapidly and the system then evolves according to Schr¨odinger’s equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The TEBD algorithm (Time Evolving Block Decimation) can be used to simulate time evolution after a quantum quench;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' the state is stored as an MPS and this MPS is updated as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Despite the success of DMRG, TEBD and other Tensor Network algorithms, these approaches are not without limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The memory requirements to store an MPS is characterised by the bond dimension, given by the dimension of the largest matrix used in the description of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' For constant approximation error ϵ this bond dimension increases exponentially with the entanglement entropy and thus with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Therefore, for a fixed maximum bond dimension, the error ϵ increases exponentially with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' This limits the applicability of Tensor Network algorithms to short time simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' A quantum algorithm however, does not in principle suffer from this issue - the key difference between a quantum and a classical device being the ability to store highly entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' A quantum computer therefore has the potential to simulate quantum many-body systems for long times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The accurate simulation of the time evolution of 1D quantum systems is thus a promising route for the demonstration of quantum advantage in the short term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' One such quantum algorithm is Trotterisation, where a discrete time step dt is used and the time evolution operator is approximated as a quantum circuit with an error that scales polynomially in dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The depth of the quantum circuit used in such an approach increases with decreasing dt, leading to a trade-off between the noise arising from using deep circuits and the decreasing accuracy of the approximation when dt is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' A number of variational quantum algorithms for the simulation of time evolution have therefore been developed that aim to use shallower circuits [7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Each of these approaches suffer from a number of issues such as convergence, runtime and limited device connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' As a result, it has been argued that such variational approaches are not practical for use on near term quantum hardware [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' One approach that aims to overcome the issue of deep circuits is Approximate Quantum Compiling [12, 13, 14], where one defines a parametric circuit of fixed depth and uses techniques from optimisation to minimise the distance between the parametric circuit and the target circuit of interest - where distance is defined by some carefully chosen metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' In principle, this approach can lead to short depth circuits that implement the target circuit of interest within some error tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' In practice, a classical imple- mentation of such an approach [14] is limited to act on a small number of qubits due to the exponential scaling of the Hilbert space with the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Here we develop a new approach to quantum simulation that combines Matrix Product States, Approx- imate Quantum Compiling and Trotterisation to produce short depth quantum circuits that implement the time evolution operator of the Heisenberg spin chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' This approach is scalable thanks to the im- mense power of Matrix Product States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Figure 1 shows a schematic of our approach: first we apply Trotterisation classically for the maximum length of time for which we can still store the state as an MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' We then apply a Matrix Product State implementation of Approximate Quantum Compiling to squeeze the circuit (purple box in the figure) to find a much shallower circuit that still reproduces the same state as Trotterisation, up to some small error in the fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' We then use the squeezed circuit as the input for the Trotter circuit which can now generate a quantum state beyond what can be stored classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' 2 Setup 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='1 The model We will consider the XXX spin-chain - a paradigmatic model for quantum magnetism - defined by the Hamiltonian: HXXX = − L−1 � i=0 hi,i+1 = − L−1 � i=0 � Sx i Sx i+1 + Sy i Sy i+1 + Sz i Sz i+1 � , (1) where Sx, Sy and Sz are written in terms of Pauli matrices as Sx = σx 2 , Sy = σy 2 and Sz = σz 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The Hamiltonian in (1) is a prototypical example of an integrable 1D model and its dynamical behaviour has been studied extensively [15], including on a quantum computer [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The time evolution of a quantum 2 Compress with AQC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Figure 1: Schematic of our approach: Trotterisation is applied classically (purple box) and then a Matrix Product State implementation of Approximate Quantum Compiling is applied to compress the first part of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Standard Trotterisation is then applied on a quantum device afterwards to simulate longer times, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' times which are beyond what is possible classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Rz(θ) Rz( π 2 ) Rz(− π 2 ) Ry(φ) Ry(λ) Figure 2: Implementation of two site operator ei(ασx⊗σx+βσy⊗σy+γσz⊗σz) as a quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' We have the correspondences θ = π 2 − 2γ, φ = 2α − π 2 and λ = π 2 − 2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The Hamiltonian in (1) corresponds to the case α = β = γ = dt state |ψ(t)⟩ is governed by the Schr¨odinger equation: |ψ(t)⟩ = e−iHXXXt |ψ(0)⟩ (2) where |ψ(0)⟩ is the wavefunction at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' In this work, we will consider the N´eel state, written as: |↑↓↑↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' ↑↓⟩ where ↑ and ↓ represent up and down spins respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The N´eel state for n spins is simply implemented on n qubits as |1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='10⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The time evolution operator U(t) ≡ e−iHt can be executed as a quantum circuit in a resource efficient way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' we first write the Hamiltonian in (1) as HXXX = H1 + H2 where H1 = − � i odd hi,i+1 and H2 = − � i even hi,i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Note that all operators in a given sum commute with all other operators in their respective sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' We then define the Suzuki-Trotter time evolution operator Utrot(dt) in the following way: U(1) trot(dt) = L/2−1 � j=0 U2j,2j+1(dt) L/2−1 � j=1 U2j−1,2j(dt) = e−iHXXZdt + O(dt2) (3) where Ujk(dt) = e−ihjkdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The exact time evolution operator U(t) is thus approximated by m repeated applications of Utrot(dt = t m), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' U(t) ≈ Um trot(dt = t m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' As discussed in [16], each Ujk(dt) appearing in (3) can be implemented by the quantum circuit with just three CNOTs as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' We can reduce the error in the Trotter formula in equation (3) by using higher order expressions [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' It turns out that the second order Trotter formula can be implemented on a quantum circuit with only one extra layer in the circuit [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' We have: U(2) trot(dt) = L/2−1 � j=0 U2j,2j+1 �dt 2 � L/2−1 � j=1 U2j−1,2j (dt) L/2−1 � j=0 U2j,2j+1 �dt 2 � = e−iHXXZdt + O(dt2) (4) which can be implemented on a quantum device by the circuit in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' 3 U(dt) U(dt) U(dt) U(dt) U(dt) U(dt) U(dt) U(dt) U(dt) U(dt) U(dt) U(dt) U(dt) U(dt) U(dt) Figure 3: First order Trotter circuit acting on six qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' U( dt 2 ) U(dt) U(dt) U( dt 2 ) U(dt) U(dt) U(dt) U( dt 2 ) U(dt) U(dt) U( dt 2 ) U(dt) U(dt) U(dt) U( dt 2 ) U(dt) U(dt) U( dt 2 ) Figure 4: Second order Trotter circuit acting on six qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' 4 A(1) A(2) A(3) A(4) A(5) A(6) Figure 5: Graphical representation of an MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' There are two matrices A(i) for each qubit at position i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' A(1) A(2) A(3) A(4) A(5) A(6) B(1) B(2) B(3) B(4) B(5) B(6) Figure 6: The inner product ⟨ψ1|ψ2⟩ of two Matrix Product States - see equations (11) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='2 Matrix Product States An arbitrary quantum state on n qubits can be written in terms of complex variables cj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=',jn, the number of which scales as 2n: |ψ⟩ = � {j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=',jn} cj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=',jn |j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=', jn⟩ (5) where the sum is over all configurations of the binary variables j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=', jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The bipartite entanglement entropy of an arbitrary quantum state picked at random from Hilbert space satisfies a volume law which, as was discussed in the introduction, is distinct from area law entanglement in which case the entanglement entropy of two regions after the bipartition of the system is proportional to the area of the boundary of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' A small subset of states in Hilbert space satisfies an area law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The coefficients cj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=',jn of such states have a certain structure that we can take advantage of to study classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Any state |ψ⟩ can be written in the following way: cj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=',jn = A(1) j1 · A(2) j2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' · A(n) jn (6) where the Aj are χj × χj+1 dimensional matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Quantum states of the form (6) are known as Matrix Product States (MPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The maximum value of χj is referred to as the bond dimension of the MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' We can represent an MPS graphically as in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' We associate one matrix A(i) to each qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Note that for each qubit i we have two matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' We thus have a total of 2n matrices to keep track of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The bond dimension χj can be seen as a measure of the entanglement between the two subsystems when a bipartition is made at qubit j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Therefore, states in Hilbert space that satisfy an area law - and therefore have a low bond dimension in their MPS representation - can be efficiently stored as Matrix Product States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' States that satisfy a volume law will have a bond dimension that is exponential in the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' We will consider in this work the non-trivial dynamics governed by equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' As discussed in the introduction, the bipartite entanglement entropy of a ground state of a one-dimensional Hamiltonian that has a gap between its ground state and its excited state is independent of the size of the subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' The ground state of such a system - and hence the initial state in our setup - can be efficiently stored as an MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' One can then use an algorithm such as TEBD (Time Evolving Block Decimation) [18] to update the MPS as a function of time to study the dynamics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' However, the entanglement entropy of the state increases linearly with time, hence the bond dimension χ that is required to keep the error constant diverges exponentially with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' To simulate for longer times, a quantum computer would be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' In section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='3, we will discuss how Matrix Product States can be leveraged to reduce the resource requirements for this simulation problem when implemented on a quantum device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content='3 Matrix Product States applied to Approximate Quantum Compiling j k Ry(θ1) Rz(θ2) Ry(θ3) Rx(θ4) Figure 7: CNOT block forms the basic building block of our circuit ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' 5 Rz(θ1) Ry(θ2) Rz(θ3) Rz(θ4) Ry(θ5) Rz(θ6) Rz(θ7) Ry(θ8) Rz(θ9) Rz(θ10) Ry(θ11) Rz(θ12) Figure 8: Parameterised circuit inspired by the structure of the first order Trotter circuit in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Rz(θ1) Ry(θ2) Rz(θ3) Rz(θ4) Ry(θ5) Rz(θ6) Rz(θ7) Ry(θ8) Rz(θ9) Rz(θ10) Ry(θ11) Rz(θ12) Figure 9: Parameterised circuit inspired by the structure of the second order Trotter circuit in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Approximate quantum compiling (AQC) involves the design of a parametric quantum circuit with fixed depth - the parameters are then adjusted to bring it as close as possible to the target, where “close” is defined via some carefully chosen metric, see below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' As discussed in [12], one can use so-called CNOT blocks to construct a natural circuit Ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' A CNOT block is a CNOT gate followed by single qubit rotations (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' A block with a CNOT gate acting on a “control” qubit j and “target” qubit k is written as CUjk(θ1, θ2, θ3, θ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' For a given hardware connectivity, one can then write down a fully parameterised circuit as: Vct(θ) =CUct(L) (θ3n+4L−3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' , θ3n+4L) · · · CUct(1) (θ3n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' , θ3n+4) [Rz (θ1) Ry (θ2) Rz (θ3)] ⊗ · · · ⊗ [Rz (θ3n−2) Ry (θ3n−1) Rz (θ3n)] (7) The position of the CNOT blocks in the parameterised circuit can be customised to suit the particular target circuit that one is interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Here we are interested in finding a circuit that implements the unitary time evolution operator as in equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' We thus consider a structure inspired by the first and second-order Trotter circuits in Figures 3 and 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Recall that each block U(dt) in Figures 3 and 4 represents the 2-qubit sub-circuit with three CNOTs in Figure 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' it is therefore natural to consider a circuit Ansatz with sub-circuits each with three CNOT blocks as in Figures 8 and 9, such that the circuit Ansatz mimics the structure of the first and second order Trotter circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' In the notation of [14], the parameterised circuits in Figures 8 and 9 correspond to n = 4 qubits, l = 2 layers and b = 3 CNOT blocks in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' In both Figure 8 and Figure 9 there are three rotation gates acting on each qubit at the beginning of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' In the examples considered in this work we will take the initial state to be |0⟩ the initial rotation gate Rz(θ) is redundant for these cases but is necessary for more general initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' One can define the distance between the target and parameterised circuit via a number of different metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Here we use a cost function based on the Hilbert-Schmidt test: Cstate hs = 1 − | ⟨0| V †(θ) |ψ0⟩ |2 (8) The goal of AQC is to tune the parameters θ to minimise the cost function under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' Note that here we are considering the application of AQC to state preparation as opposed to full circuit com- pilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' More precisely, this means that our cost function is designed such that it is minimised when the action of V (θ) on the initial state |0⟩ produces a state that is as close as possible to a target state |ψ0⟩ (up to some global phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' This is in contrast to the situation where one starts with some target circuit U and the cost function is designed to bring the full matrix V (θ) as close as possible to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' 6 As pointed out in [19], the gradient of the cost function in (8) vanishes exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' This observation lead to the distinction between global and local cost functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' local cost functions have only polynomially vanishing gradients in some cases of interest - see [19, 20, 14] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFAT4oBgHgl3EQfkB3r/content/2301.08609v1.pdf'} +page_content=' As was shown in [14], the Hilbert- Schmidt test - which is a global cost function - can be turned into a local one by adding several “bit-flip” terms which increases the magnitude of the gradient: Cstate lhs = 1−| ⟨0| V †(θ) |ψ0⟩ |2 − �n − 1 n � n � j=1 | ⟨0| XjV †(θ) |ψ0⟩ |2 − �n − 2 n � � j