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+{"text":"\\section{Introduction}\n\\label{SEC:introduction}\n\n\nHelicity scattering amplitudes in Quantum Field Theory (QFT) encode the full dependence\non the spin degrees of freedom of the particles involved in the scattering, and are the building blocks \nfor computing various kinds of physical observables through which we try to understand \nthe interactions among particles observed in nature.\nThe incorporation of spin degrees of freedom, or polarization effects, in terms of  \nspin- respectively polarization-dependent physical observables, leads to a richer phenomenology.\nSuch observables offer valuable means to discriminate different dynamical models, \nin particular for discovering potential Beyond-Standard-Model effects.\nFor a review of the role of particle polarizations in testing the Standard Model and searching for new physics, \nwe refer to refs.~\\cite{Lampe:1998eu,Leader:2001gr,Accomando:1997wt,MoortgatPick:2005cw} \nand references therein.\n\n\nUnlike physical observables, individual scattering amplitudes in QFT generally possess \ninfrared\\footnote{We use the term ``infrared'' (IR) to denote both soft and collinear divergences.} \n(IR) and ultraviolet (UV) divergences, and thus a regularization scheme (RS) for handling these\nintermediate divergences needs to be introduced. \nDimensional regularization~\\cite{tHooft:1972tcz,Bollini:1972ui} is by far the most convenient one \nto use in gauge theories as it respects gauge and Lorentz invariance\\footnote{We ignore the treatment of\n$\\gamma_5$ in dimensional regularization for the moment.}\nand allows one to handle both UV and IR divergences in the same manner.\nThe key ingredient of dimensional regularization is the analytic continuation of \nloop momenta to $D=4-2\\epsilon$ space-time dimensions with indefinite $\\epsilon$. \nHaving done this, one is still left with some freedom regarding the dimensionality of the momenta\nof the external particles, of algebraic objects like the space-time metric tensor and Dirac matrices, \nas well as the number of polarizations of both external and internal particles. \nThis gives rise to different dimensional regularization variants (for a review see e.g. \nref.~\\cite{Gnendiger:2017pys} and references therein), which in general leads to different \nexpressions for singular amplitudes. \nApparently the RS dependence is intimately connected to the singularity structures of amplitudes, \nwhich fortunately obey a nice factorization form at the amplitude level~\\cite{Sen:1982bt,Collins:1989bt,Catani:1998bh,Sterman:2002qn,Dixon:2008gr,Gardi:2009qi,Gardi:2009zv,Becher:2009cu,Becher:2009kw,Becher:2009qa,Feige:2014wja}.\nThe result for a physical quantity, such as a physical cross section which is free of any such divergence, \nmust not depend on the RS that has been used. \nHowever, in practice, such a result is obtained as a sum of several partial contributions, \nwhich usually are individually divergent and computed separately before being combined. \nTherefore, these intermediate results can depend on the RS, and have to be computed consistently \nto ensure the cancellation of the spurious RS-dependence.\n\n\nThe conventional dimensional regularization (CDR)\\footnote{By the acronym ``CDR'' \nwe refer in this article to the usual CDR~\\cite{Collins:1984xc} where, \nin addition, $\\gamma_5$ is treated by Larin's prescription~\\cite{Larin:1991tj,Larin:1993tq}.} \nscheme~\\cite{Collins:1984xc} is a very popular RS, where all vector bosons are treated as D-dimensional objects. \nIt is conceptually the simplest one and does guarantee a consistent treatment.\nIt is typically employed in calculating (unpolarized) amplitude interferences \nwhere the sum over the polarizations of an external particle is conveniently made\nby using the respective unpolarized Landau density matrix. \nFor computing helicity amplitudes at the loop level, the two commonly used RS \nare the \\textquotesingle t Hooft-Veltman (HV) scheme~\\cite{tHooft:1972tcz} \nand the Four-Dimensional-Helicity (FDH) scheme~\\cite{Bern:1991aq,Bern:2002zk}. \nIn the FDH, the usage of spinor-helicity representations~\\cite{DeCausmaecker:1981wzb,DeCausmaecker:1981jtq,Gunion:1985vca,Kleiss:1985yh,Xu:1986xb,Kleiss:1986qc,Dittmaier:1998nn,Schwinn:2005pi,Arkani-Hamed:2017jhn}\nand unitarity-cut based methods~\\cite{Bern:1994zx,Bern:1994cg,Bern:1997sc,Britto:2004nc,Bern:2007dw} lead to compact expressions \nfor helicity amplitudes, which are computationally very advantageous, \nwhile the proper renormalization procedure for non-supersymmetric theories beyond one loop order \nrequires some expertise~\\cite{Kilgore:2011ta,Kilgore:2012tb,Gnendiger:2016cpg,Gnendiger:2017rfh}.\nAnother widely used dimensional regularization variant, the Dimensional-Reduction (DRED) scheme~\\cite{Siegel:1979wq}, \nwas initially devised for application to supersymmetric theories and was later shown to be applicable also  \nto non-supersymmetric theories~\\cite{Capper:1979ns,Jack:1993ws}. \nThe DRED and FDH have much in common, while there are also subtle differences \nbetween the two~\\cite{Bern:2002zk,Kilgore:2012tb,Broggio:2015dga,Gnendiger:2017pys}.\n\n\nFor computing D-dimensional helicity amplitudes, especially for amplitudes at the loop level, \none typically uses the projection method~\\cite{Karplus:1950zza,Kniehl:1990iva,Binoth:2002xg} \nwhich is based on Lorentz covariant tensor decomposition of scattering amplitudes (with external state vectors being stripped off).\nThe entire dependence of loop amplitudes on loop integrals is encoded in the Lorentz invariant \ndecomposition coefficients which multiply the relevant Lorentz tensor structures. \nLorentz tensor decomposition is employed, for instance, at one loop in the \nPassarino-Veltman reduction procedure~\\cite{Passarino:1978jh}, and also in  \nthe systematic constructions of dimensionally regularized QCD helicity amplitudes~\\cite{Abreu:2018jgq,Boels:2018nrr}. \nDuring the last decades, there have been many computations done for high order QCD corrections to scattering amplitudes using the projection method, for example,~\\cite{Glover:2003cm,Glover:2004si,Binoth:2006hk,Gehrmann:2011aa,Gehrmann:2013vga,Gehrmann:2015ora,vonManteuffel:2015msa,Bernreuther:2004ih,Bernreuther:2004th,Bernreuther:2005gw,Bernreuther:2005rw,Aglietti:2004nj,Moch:2005id,Gehrmann:2005pd,Bonciani:2008wf,Bonciani:2008az,Asatrian:2008uk,Beneke:2008ei,Bell:2008ws,Assadsolimani:2014oga,Ablinger:2017hst,Liu:2017axv,Henn:2016tyf,Lee:2018nxa,Lee:2018rgs,Ablinger:2018yae,Ahmed:2015qpa,Borowka:2016ehy,Heinrich:2017bvg,Chen:2017jvi,Jones:2018hbb,Anastasiou:2018fjr,Maltoni:2018zvp,vonManteuffel:2019wbj}.\n\n\nDespite being very generic, versatile, and widely used in many high-order applications, \nthere are a few aspects of the Lorentz tensor decomposition approach that makes the traditional projection method \nnot so easy to be carried out in certain cases, as will be discussed in the next section.\nFor example, besides facing complexities in deriving D-dimensional projectors for tensor \ndecomposition coefficients in some multiple-parton, multiple-scale scattering processes, \nevanescent Lorentz structures\\footnote{The evanescent Lorentz structures appearing in a Lorentz tensor decomposition should not be confused with \\textit{operator mixings} in the renormalization of composite operators in effective field theories \\cite{Collins:1984xc,Buras:1989xd}, nor with evanescent terms in the DRED or FDH regularized Lagrangian~\\cite{vanDamme:1984ig,Jack:1993ws,Jack:1994bn,Stockinger:2005gx,Harlander:2006rj,Bern:2002zk,Kilgore:2011ta,Kilgore:2012tb,Gnendiger:2016cpg,Gnendiger:2017rfh}.} can appear in the D-dimensional basis for the loop amplitudes in question. \nTheir presence can lead to intermediate spurious poles in the resulting D-dimensional projectors~\\cite{Binoth:2002xg,Glover:2004si,Gehrmann:2015ora}. \nFurthermore, when there are several external fermions involved in the scattering~\\cite{Glover:2004si,Abreu:2018jgq}, \nthe complete and linearly independent set of basis \nstructures in D dimensions will generally increase with the perturbative order at which the \nvirtual amplitude is computed (as the Dirac algebra is formally infinite-dimensional in non-integer D dimensions).  \n\n\nGiven the impressive long list of high-order QCD calculations of important phenomenological consequences \ndone in CDR and, moreover, having in mind the aforementioned critical features of D-dimensional Lorentz tensor decomposition, it should be justified to think of possible add-ons in order to facilitate the computations of polarized amplitudes in a way fully compatible with CDR. \nIn this article we propose an alternative regularization prescription of external states \n(for both bosons and fermions) in order to avoid Lorentz tensor decomposition in the conventional projection method \nfor extracting helicity amplitudes. The prescription outlined below is devised to be fully compatible with CDR \nso that certain results known in CDR can be directly recycled.\n\n\nAs will become clear in following sections, the idea is based on the following simple observation. \nIn 4 dimensions, there are only four linearly independent Lorentz 4-vectors, \nand hence any Lorentz 4-vector can be expressed linearly using just three linearly independent \nLorentz 4-vectors with the aid of the Levi-Civita tensor.\nTherefore all polarization vectors can be built up by just using three linearly \nindependent external momenta in a Lorentz covariant way, provided that there are \nenough linearly independent momenta involved in the process.  \nThis basic mathematical fact is of course well known, and without surprise it was already \nexploited about forty years ago in calculating (tree-level) multiple photon bremsstrahlung processes \nin massless QED~\\cite{DeCausmaecker:1981wzb,DeCausmaecker:1981jtq}.\nIt was initially used for simplifying the massless QED vertex by rewriting the \nslashed photon polarization vector in terms of the slashed momenta of external charged fermions\n(from which the photon was radiated), a trick that preluded the introduction of the   \n4-dimensional massless spinor-helicity formalism~\\cite{Gunion:1985vca,Kleiss:1985yh,Xu:1986xb,Kleiss:1986qc}.\nIn this article, instead of seeking simplifications of the gauge interaction vertices of fermions in \n4-dimensional massless theories, this mathematical fact is employed for finding a CDR-compatible \nway to directly project out polarized loop amplitudes, circumventing Lorentz tensor decomposition. \nFurthermore, despite being different from CDR, we would like to argue that thanks to the \namplitude-level factorization of UV and IR singularities, such a prescription can be used\nin a hybrid way together with results known in CDR to obtain RS-independent finite remainders \nof loop amplitudes, without the need to recalculate the integrated subtraction coefficients \ninvolved in an IR subtraction framework. \nIn other words, we will show that such a hybrid CDR-compatible prescription is unitary\nin the sense defined in refs.~\\cite{vanDamme:1984ig,Catani:1996pk}.~\\\\\n\n\nThe article is organized as follows. \nIn the next section, the conventional projection method for computing polarized\namplitudes is reviewed with comments on a few aspects which motivated the work\npresented in this article.\nIn section~\\ref{SEC:Prescription} the proposed prescription to obtain polarized \ndimensionally regularized scattering amplitudes is outlined in detail.\nSection~\\ref{SEC:unitarity} is devoted to the discussion of the unitarity of the \nhybrid regularization prescription of section~\\ref{SEC:Prescription}.\nIn particular we show that pole-subtracted RS-independent finite remainders\nare always obtained and furthermore demonstrate this feature in the context of an IR subtraction method.\nIn section~\\ref{SEC:examples}, we provide two simple examples of calculating finite remainders \nof one-loop virtual amplitudes in order to illustrate the usage of the prescription and to comment \non a few practical points worthy of attention. \nWe conclude in section~\\ref{SEC:conclusion}.\n\n\n\\section{A Recap of the Projection Method}\n\\label{SEC:projectionmethod}\n\nIn this section, we review the projection method for computing polarized amplitudes, \nand discuss a few aspects that motivated the work in this article.\n\n\nThe projection method~\\cite{Karplus:1950zza,Kniehl:1990iva,Binoth:2002xg}, \nbased on Lorentz covariant tensor decomposition, can be used to obtain helicity amplitudes \nfor a generic scattering process at any loop order. \nThe entire dependence of scattering amplitudes on loop integrals \nis encoded in their Lorentz-invariant decomposition coefficients that multiply \nthe corresponding Lorentz tensor structures and are independent of the external \nparticles' polarization vectors.\nThese Lorentz-invariant decomposition coefficients are sometimes called \\textit{form factors} \nof the amplitudes, a relativistic generalization of the concept of charge distributions. \nIn order to extract these form factors containing dimensionally regularized loop integrals, \nprojectors defined in D dimensions should be constructed and subsequently applied directly \nto the Feynman-diagrammatic expression of the amplitude, which can proceed diagram by diagram.\n\n\n\\subsection{Gram matrix and projectors}\n\\label{SEC:projectionmethod:recap}\n\nScattering amplitudes in QFT with Poincar\\'e symmetry are multi-linear in the \nstate vectors of the external particles, i.e., proportional to the tensor product \nof all external polarization vectors, to all loop orders in perturbative calculations, \nas manifestly shown by the Feynman diagram representations. \nThe color structure of QCD amplitudes can be conveniently described using\nthe color-decomposition~\\cite{Berends:1987cv,Mangano:1987xk,Mangano:1987kp,Mangano:1988kk,Bern:1990ux} \nor the color-space formalism of ref.~\\cite{Catani:1996vz}. \nQCD amplitudes are thus viewed as abstract vectors in the color space of external colored particles.\nSince projecting QCD amplitudes onto the factorized color space and spin (Lorentz) structures \ncan be done independently of each other, we suppress for ease of notation possible color \nindices of scattering amplitudes in the following discussions.\n\n  \nAs nicely summarized and exploited in~\\cite{Boels:2017gyc,Boels:2018nrr},   \nevery scattering amplitude is a vector in a linear space spanned by a finite set of Lorentz \ncovariant structures, in dimensional regularization at any given perturbative order.  \nThese structures are constrained by physical requirements such as on-shell kinematics \nand symmetries of the dynamics. Scattering amplitudes can thus be written as a linear combination of \na set of chosen Lorentz basis structures, where the decomposition coefficients are functions \nof Lorentz invariants of external kinematics. All non-rational dependence\nof the decomposition coefficients on external kinematics appear via loop integrals.   \nThis implies the following linear ansatz for a scattering amplitude \n$\\hat{\\mathcal{M}}$ at a fixed perturbative order, \n\\begin{eqnarray} \\label{EQ:ampFFsprimitive}\n\\hat{\\mathcal{M}} = \\sum_{n=1}^{N_P} c_n ~\\hat{T}_n \\, , \n\\end{eqnarray}\nwhere each form factor $c_n$ is a function of Lorentz invariants of external momenta, \nand each Lorentz structure $\\hat{T}_n$ is multi-linear in the external polarization \nstate vectors. In general, $\\hat{T}_n$ contains contractions of external gauge bosons' \npolarization state vectors with either the space-time metric tensor connecting two different \npolarizations or with external momenta, and contains also products of Dirac matrices sandwiched \nbetween external on-shell spinors. \nThe Levi-Civita tensor can also occur if the scattering process involves \nparity-violating vertices. The complete and linearly independent set of Lorentz structures \nfor $\\hat{\\mathcal{M}}$ at any given perturbative order depends on its \nsymmetry properties as well as the Lorentz and Dirac algebra in use.\n\nNote that, as discussed in detail for the four-quark scattering amplitude \n$q \\bar{q} \\rightarrow Q\\bar{Q}$ in~\\cite{Glover:2004si,Abreu:2018jgq}, \nthe complete and linearly independent set of D-dimensional basis structures must in general \nbe enlarged according to the perturbative order at which $q \\bar{q} \\rightarrow Q\\bar{Q}$ \nis computed, because the Dirac algebra is infinite-dimensional for non-integer dimensions. \nAt each perturbative order only a finite number of linearly independent Lorentz structures \ncan appear in an amplitude, as is evident from inspecting the corresponding Feynman diagrams\nwhich is a set of finite elements.~\\\\\n\n\nTo be specific, we consider in the following the Lorentz tensor decomposition of scattering amplitudes \nin CDR at fixed order in perturbation theory.\nIn the following discussion of the projection method, we investigate also how to uncover linear dependent relations \namong a set of (preliminarily chosen) Lorentz tensor structures arising from on-shell constraints, \nwithout making explicit reference to the origin of these linear dependencies.\n\n\nLet us assume that by construction the set of the $N_P$  Lorentz structures \n$\\hat{T}_n$ in eq.~(\\ref{EQ:ampFFsprimitive}), denoted by $\\mathbf{T}_{P} \\equiv \\{\n\\hat{T}_1, \\cdots, \\hat{T}_{N_P}\\}$, is linearly complete for the $\\hat{\\mathcal{M}}$ in question,\nbut the  $\\hat{T}_n$ may not be linearly independent of each other. \nFor an analogy we recall the  representation of QCD amplitudes in terms of a set of \ncolor structures in color space without demanding linear independence of these color structures.\nLet us thus call eq.~(\\ref{EQ:ampFFsprimitive}) a \\textit{primitive} Lorentz covariant decomposition \nof $\\hat{\\mathcal{M}}$. \nPossible linear relations among the $N_P$ Lorentz structures $\\hat{T}_n$ due to Lorentz and\/or Dirac algebra \nand also on-shell constraints, such as equations of motion as well as transversality satisfied \nby external state vectors, can be uncovered by computing their  $N_P \\times N_P$ Gram matrix \n$\\hat{\\mathrm{\\mathbf{G}}}$, whose matrix elements are defined as \n\\begin{eqnarray} \\label{EQ:grammatrix}\n\\hat{\\mathrm{\\mathbf{G}}}_{ij} = \\langle \\hat{T}^{\\dagger}_i , \\hat{T}_j \\rangle.\n\\end{eqnarray} \nThe symbol $\\langle \\hat{T}^{\\dagger}_i, \\hat{T}_j \\rangle$ denotes the Lorentz invariant inner product \nbetween these two linear Lorentz structures. It is typically defined as the trace of the matrix product \nof $\\hat{T}_i$'s Hermitian conjugate, i.e. $\\hat{T}^{\\dagger}_i$, and $\\hat{T}_j$ with tensor products \nof external state vectors (spinors) being substituted by the corresponding unpolarized Landau density matrices.  \nIn other words, this Lorentz invariant quantity can be viewed as the unpolarized interference\nbetween two linear Lorentz structures $\\hat{T}_i$ and $\\hat{T}_j$ over all helicity states of \nexternal particles in accordance with certain polarization sum rules (encoded in the unpolarized Landau \ndensity matrices).  \n\n\nThis $N_P \\times N_P$ Gram matrix $\\hat{\\mathrm{\\mathbf{G}}}$ in eq.~(\\ref{EQ:grammatrix}) can then be \nused to determine the linearly independent subset of $\\mathbf{T}_{P}$ spanning the \nvector space where the considered amplitude $\\hat{\\mathcal{M}}$ lives. \nIf the determinant of $\\hat{\\mathrm{\\mathbf{G}}}$ is not identically zero, then the set $\\mathbf{T}_{P}$\nis both  complete and linearly independent, and thus qualifies as a basis of the vector space \nwhere $\\hat{\\mathcal{M}}$ lives.   \nOtherwise, $\\hat{\\mathrm{\\mathbf{G}}}$ is not a full-rank matrix, and its matrix rank \n$N_R \\equiv \\mathrm{R}[\\hat{\\mathrm{\\mathbf{G}}}]$ tells us the number of \nlinearly independent members of $\\mathbf{T}_{P}$. \nSince $\\mathbf{T}_{P}$ is assumed to be linearly complete w.r.t. $\\hat{\\mathcal{M}}$ by \nconstruction, $N_R$ is thus the number of basis elements of a linear basis of the vector space that \ncontains $\\hat{\\mathcal{M}}$.\n\n\nThe number $N_P-N_R$ of linear dependent relations in $\\mathbf{T}_{P}$ can be extracted from \nthe \\textit{null-space} of this Gram matrix $\\hat{\\mathrm{\\mathbf{G}}}$. \nTechnically, the null-space of a matrix (not necessarily a square matrix) is \nthe solution space of the corresponding homogeneous system of linear algebraic equations\ndefined by taking this matrix as the system's coefficient matrix. \nThe null-space of $\\hat{\\mathrm{\\mathbf{G}}}$ can be conveniently represented as \na list of linearly independent $N_P$-dimensional basis vectors of the \nsolution space of the homogeneous linear algebraic system defined by \n$\\hat{\\mathrm{\\mathbf{G}}}$. The number of members of this list is equal to the dimension of \n$\\hat{\\mathrm{\\mathbf{G}}}$ minus its matrix rank, i.e., $N_P-N_R$. \nFor the information we would like to extract\\footnote{To just\nidentify the linearly dependent columns and\/or rows of the multivariate Gram matrix, numerical samples of this matrix \nat a few test points are usually enough.}, this null-space provides \nthe complete set of linear combination coefficients (being rational in the external kinematics) \nof the column vectors of $\\hat{\\mathrm{\\mathbf{G}}}$ that lead to vanishing $N_P$-dimensional vectors. \nAfter having removed those linearly dependent columns (and their corresponding transposed rows),\n we end up with a reduced full-rank Gram matrix \namong the thus-selected linearly independent set of Lorentz structures, denoted by $\\mathbf{T}_{R}$. \nThe set $\\mathbf{T}_{R}$ can then be directly taken as the basis of the vector space of $\\hat{\\mathcal{M}}$. \n\n\nElimination of redundancies in the set $\\mathbf{T}_{P}$ for $\\hat{\\mathcal{M}}$ involving external gauge bosons, \ndue to Ward identities of local gauge interactions, can be effectively accounted for by choosing physical \npolarization sum rules for those external gauge bosons (with their reference vectors chosen as momenta of other external particles).\nThis point can be easily seen once we realize that any unphysical structure,  \nwhich may happen to be just one specific $\\hat{T}_n$ or a linear combination of some of them\n(with rational coefficients in external kinematics), \ngets nullified by the physical polarization sum rules of external gauge bosons.\nNotice, however, reduction in the number of linearly independent basis structures \nof $\\hat{\\mathcal{M}}$ due to additional process-specific symmetries such as \ncharge, parity, and\/or Bose symmetry is not achieved by analyzing $\\hat{\\mathrm{\\mathbf{G}}}$ in this way. \nInstead they have to be accounted for from the outset when determining \nthe primitive set $\\mathbf{T}_P$ in eq.~(\\ref{EQ:ampFFsprimitive}). ~\\\\\n\n\nIn terms of the thus-determined basis $\\mathbf{T}_{R}$, the linear decomposition of $\\hat{\\mathcal{M}}$ \ncan be recast into \n\\begin{eqnarray} \\label{EQ:ampFFs}\n\\hat{\\mathcal{M}} = \\sum_{n=1}^{N_R} \\tilde{c}_n ~\\hat{T}_n ~,~\n\\end{eqnarray}\nand the Gram matrix $\\hat{\\mathrm{\\mathbf{G}}}_R$ of $\\mathbf{T}_{R}$ with matrix elements defined \nsimilarly as eq.~(\\ref{EQ:grammatrix}) is now an invertible $N_R \\times N_R$ matrix.~\\\\\n\n\nNow we are ready to discuss projectors $\\hat{P}_n$ for the \nLorentz decomposition coefficients (or form factors) $\\tilde{c}_n$ of $\\hat{T}_n$ \nin eq.~(\\ref{EQ:ampFFs}).\nThey are defined by\n\\begin{eqnarray} \\label{EQ:projectordef}\n\\tilde{c}_n = \\langle \\hat{P}^{\\dagger}_n , \\hat{\\mathcal{M}} \\rangle~ ~~ \n\\text{for any $n ~\\in~ \\{1, \\cdots, N_R\\}$} \\, ,\n\\end{eqnarray} \nwhere the same Lorentz-invariant inner product operation as in eq.~(\\ref{EQ:grammatrix}) is \nused in the above projection. \nThe defining equation (\\ref{EQ:projectordef}) of $\\hat{P}_n$ holds for any linear object \n from the vector space spanned by the basis $\\mathbf{T}_{R}$, rather than just for a particular \nscattering amplitude $\\hat{\\mathcal{M}}$.\nInserting eq.~(\\ref{EQ:ampFFs}) into \neq.~(\\ref{EQ:projectordef}) then, taking the aforementioned property into account,  the defining equation  \n for the  projectors translates into \n\\begin{eqnarray} \\label{EQ:projectordefequvi}\n\\langle \\hat{P}^{\\dagger}_n , \\hat{T}_m \\rangle = \\delta_{nm} \\quad \\text{for any $n,m ~\\in~ \\{1, \\cdots, N_R\\}$} \\,. \n\\end{eqnarray}\n\n\nEach projector $\\hat{P}^{\\dagger}_n$ can be expressed in terms of a linear combination \nof Hermitian conjugate members of $\\mathbf{T}_{R}$ that span also a  vector space.\nWe thus write \n\\begin{eqnarray} \\label{EQ:projectoransatz}\n\\hat{P}^{\\dagger}_n = \\sum_{k=1}^{N_R} \\left(\\hat{\\mathrm{\\mathbf{H}}}\\right)_{nk} ~ \\hat{T}^{\\dagger}_k \\, ,\n\\end{eqnarray}\nwhere the elements $\\left(\\hat{\\mathrm{\\mathbf{H}}}\\right)_{nk}$ are to be determined. \nInserting eq.~(\\ref{EQ:projectoransatz}) into eq.~(\\ref{EQ:projectordefequvi}) and\n using the definition of Gram matrix elements we get\n\\begin{eqnarray}\n\\sum_{k=1}^{N_R} \\left(\\hat{\\mathrm{\\mathbf{H}}}\\right)_{nk} ~ \\left(\\hat{\\mathrm{\\mathbf{G}}}_R\\right)_{km} = \\delta_{nm}~, \n \\quad \\text{i.e.,} \\; \\hat{\\mathrm{\\mathbf{H}}} ~ \\hat{\\mathrm{\\mathbf{G}}}_R = \\hat{1}. \n\\end{eqnarray}\nRecall that $\\hat{\\mathrm{\\mathbf{G}}}_R$ is invertible by the aforementioned trimming procedure.  \nThis then answers the question of how to construct the projectors $\\hat{P}^{\\dagger}_n$ \n from linear combinations of the Hermitian conjugates of $\\mathbf{T}_{R}$.\nIn the special and ideal case of a norm-orthogonal basis $\\mathbf{T}_{R}$, \nits Gram matrix $\\hat{\\mathrm{\\mathbf{G}}}_R$ is equal to the identity matrix of dimension $N_R$ \nand hence $\\hat{\\mathrm{\\mathbf{H}}}  = \\hat{\\mathrm{\\mathbf{G}}}_R^{-1} = \\hat{1} $.\nSubsequently, we have $\\hat{P}^{\\dagger}_n = \\hat{T}^{\\dagger}_n$, as is well known\nfor a norm-orthogonal basis.~\\\\\n\n\nBy taking the Dirac traces and keeping all Lorentz indices in D dimensions in the projection, \nthese Lorentz-invariant tensor decomposition coefficients, or form factors, \nare evaluated in D dimensions. These form factors are independent of the external polarization vectors, \nand all their non-rational dependence on external momenta is confined to loop integrals. \nScalar loop integrals appearing in these form factors can be reduced to a finite \nset of master integrals with the aid of linear integration-by-parts identities~\\cite{Tkachov:1981wb,Chetyrkin:1981qh}. \nOnce these dimensionally regularized form factors have been determined, external particles' state vectors \ncan be conveniently chosen in 4 dimensions, leading to helicity amplitudes in accordance \nwith the HV scheme. \nIn fact, once the (renormalized) virtual amplitudes are available at hand in such a \nD-dimensional tensor-decomposed form (with all (singular) Lorentz-invariant form factors computed \nin D dimensions), then changing the regularization convention for the external particles' states \nconsistently in both the virtual amplitude and the corresponding IR-subtraction terms, \nshould not alter the finite remainder that is left after subtracting all poles, \nalthough the individual singular pieces do change accordingly.\n\n\n\\subsection{Comments on the D-dimensional projection}\n\\label{SEC:projectionmethod:comments}\n\nWe now discuss a few delicate aspects of the Lorentz tensor decomposition in D dimensions \nthat motivated the work presented in this article. \n\n\nIn general, the Gram matrix $\\hat{\\mathrm{\\mathbf{G}}}$ or $\\hat{\\mathrm{\\mathbf{G}}}_R$ \ncomputed using Lorentz and Dirac algebra in CDR depends on the space-time dimension D. \n We can examine its 4-dimensional limit by inserting \n$D=4-2\\epsilon$ and check whether its determinant power-expanded in $\\epsilon$ is zero or not\nin the limit $\\epsilon = 0$. A  determinant vanishing at $\\epsilon = 0$ implies\nthe presence of Lorentz structures in the D-dimensional \nlinearly independent basis set $\\mathbf{T}_{R}$ that are redundant in  4 dimensions. \n\n\nTo be more specific, we can compute the matrix rank of $\\hat{\\mathrm{\\mathbf{G}}}_R$ \nat $D =4$, denoted by $\\mathrm{R}[\\hat{\\mathrm{\\mathbf{G}}}_R^{D=4}]$, and  \nthe difference $N_R -\\mathrm{R}[\\hat{\\mathrm{\\mathbf{G}}}_R^{D=4}]$\ntells us the number of Lorentz structures  \nappearing in $\\mathbf{T}_R$ that are redundant in D=4. \nFurthermore, if we compute the null-space of the 4-dimensional limit of \n$\\hat{\\mathrm{\\mathbf{G}}}$, then we can explicitly uncover all these special linear \nrelations among $\\hat{T}_n$ due to the constraint of integer dimensionality\\footnote{Any potential non-linear \nrelation among the $\\hat{T}_n$ is irrelevant here as we use a linear basis.} \nin a similar way as one identifies  $\\mathbf{T}_{R}$ out of $\\mathbf{T}_{P}$.\nThese special linear relations can be used to construct exactly the number \n$N_R -\\mathrm{R}[\\hat{\\mathrm{\\mathbf{G}}}_R^{D=4}]$ of \nevanescent Lorentz structures out of $\\mathbf{T}_{R}$ \n that are non-vanishing in D dimensions but vanishing in 4 dimensions. \n In this way, the original basis set $\\mathbf{T}_R$  \ncan be re-cast into a union of two subsets: one is linearly independent and complete \nin 4 dimensions, and the other one only consists of $N_R -\\mathrm{R}[\\hat{\\mathrm{\\mathbf{G}}}_R^{D=4}]$ evanescent Lorentz structures. \nSuch a reformulation of the Lorentz tensor decomposition basis in D dimensions \ncan be very useful in exhibiting the additional non-four-dimensional structures\ninvolved in the virtual amplitude.~\\\\\n\n\nIn case the number of structures in $\\mathbf{T}_{R}$ is not very small \n(say, not less than 10) and if there are several kinematic variables involved, \nalgebraically inverting $\\hat{\\mathrm{\\mathbf{G}}}_R$ can be computationally \nquite cumbersome~\\cite{Boels:2018nrr}. \nMoreover, the resulting projectors constructed in the above fashion may be hardly \nusable if the amplitudes themselves are already quite complicated. \nThis situation occurs naturally in multiple-parton multiple-scale scattering processes.\nPossible simplifications may be obtained by suitably recombining the linear basis structures \nin $\\mathbf{T}_{R}$ classified into several groups, such that they are mutually orthogonal \nor decoupled from each other~\\cite{Boels:2018nrr}. \nFor example, we could divide the set of tensor structures into symmetric and anti-symmetric\nsectors, and also choose the anti-symmetrized product basis for strings of Dirac matrices~\\cite{Dugan:1990df,Gehrmann:2011aa}. \nThis amounts to choosing the basis structures in $\\mathbf{T}_{R}$ such that a partial triangularization \nof the corresponding Gram matrix $\\hat{\\mathrm{\\mathbf{G}}}_R$ is achieved already by construction. \nThis will facilitate the subsequent inversion operation, and also make the results simpler. \nIn addition, in case the set of tensor structures all observe factorized forms in terms of products \nof a smaller set of lower rank tensor structures, then this factorization can also be exploited \nto greatly facilitate the construction of projectors~\\cite{Binoth:2002xg}. \nAlternatively, it is also a good practice to ``compactify'' the vector space as much as possible, \nbefore the aforementioned construction procedure is applied, by employing all possible physical constraints \nand symmetries, such as parity and\/or charge symmetry of the amplitudes in question, \nand also by fixing the gauge of the external gauge bosons~\\cite{Gehrmann:2013vga,vonManteuffel:2015msa,Boels:2017gyc,Boels:2018nrr}. \n\n\nOther than the aforementioned technical complexity in inverting the Gram matrix,\nthere is another delicate point about the Lorentz tensor decomposition approach in D dimensions,\nas already briefly mentioned above. \nIn cases where the external state consists only of bosons, a list of fixed number of Lorentz tensor structures \nis indeed linearly complete in D dimensions to all orders in perturbation theory~\\cite{Binoth:2002xg,Gehrmann:2013vga}. \nHowever, if there are external fermions involved in the scattering, the complete and linearly independent set of basis \nstructures will generally increase with the perturbative order at which the scattering amplitude is computed, \nbecause the Dirac algebra is formally infinite dimensional in non-integer D dimensions, as discussed \nfor the four-quark scattering amplitude $q \\bar{q} \\rightarrow Q\\bar{Q}$ in~\\cite{Glover:2004si,Abreu:2018jgq}.\nOf course, at each given perturbative order only a finite number of linearly \nindependent Lorentz structures can appear in an amplitude, because the corresponding Feynman diagrams are \njust a set of finite elements. These additional D-dimensional Lorentz structures \nare either evanescent by themselves or will lead to additional evanescent structures of \nthe same number computed by the procedure discussed above. \n\n\nThe last comment we would like to make about the projection method in D dimensions \nis the possible appearance of intermediate spurious poles in these projectors~\\cite{Binoth:2002xg,Glover:2004si,Gehrmann:2015ora}, \nwhich are closely related to the presence of the aforementioned evanescent Lorentz structures \nin the D-dimensional linearly independent basis. \nSince the presence of evanescent Lorentz structures in the D-dimensional basis\nimplies a Gram matrix that vanishes in 4 dimensions, \none expects that projectors resulting from its inverse can contain poles in $D-4=-2\\epsilon$,\nfor instance in~\\cite{Binoth:2002xg} for four-photon scattering. \nOf course, all intermediate spurious poles generated this way in individual \nform factors projected out should cancel in the physical amplitudes composed out of them, \nsuch as helicity amplitudes or linearly polarized amplitudes.~\\\\\n\n\nAll these sometimes cumbersome issues discussed above motivated the work that will be presented in the following: \nthe construction of  simple and general polarized amplitude projectors in D dimensions \nthat avoids conventional Lorentz tensor decomposition, yet is still fully compatible with CDR.\n\n\n\\section{The Prescription}\n\\label{SEC:Prescription}\n\nThe idea behind the proposed prescription to obtain polarized dimensionally\nregularized scattering amplitudes can be briefly outlined as follows, \nwith details to be exposed in the subsequent subsections. \n\n\n\nFor external gauge bosons of a scattering amplitude, massless and\/or massive, \nwe decompose each external polarization  vector \nin terms of external momenta. \nWe then keep the form of Lorentz covariant decomposition fixed while formally promote \nall its open Lorentz indices, which are now all carried by external momenta, \nfrom 4 dimensions to D dimensions, like every Lorentz vector in CDR. \nIf external fermions are present in the scattering amplitude,  \nstrings of Dirac matrices sandwiched between external on-shell spinors will show up. \nFor each open fermion line, we first rewrite this quantity as a trace of products of \nDirac matrices with the aid of external spinors' Landau density matrices, \nup to an overall Lorentz-invariant normalization factor.\nThe space-like polarization vectors of a massive spinor can also be represented \nin terms of external momenta.  Again, once such a momentum basis representation \nis established in 4 dimensions, the Lorentz covariant form will be kept fixed \nwhile all open Lorentz and Dirac indices, carried by either external momenta or \nDirac matrices, will be respectively promoted in accordance with CDR.\n\n\nAs scattering amplitudes are multi-linear in the state vectors of the external particles \nto all loop orders in perturbation theory, the tensor products of momentum basis representations \nof all external gauge bosons and all properly re-written external spinor products, \nwith their open indices promoted accordingly as in CDR, \nwill be taken as the \\textit{external} projectors for polarized amplitudes. \nHelicity amplitude projectors of a generic scattering process defined in this way\nnaturally obey a simple factorized pattern as the tensor product of \nthe respective polarization projector of each external gauge boson and open fermion line. \nFeatures and subtleties worthy of attention during these rewriting procedures  \nwill be discussed and explained below.\n\n\n\\subsection{Momentum basis representations of polarization vectors}\n\\label{SEC:prescription:MBR}\n\nLet us start with the cases with only bosons in the external states. \nWe recall that the polarization vector $\\varepsilon^{\\mu}_{\\lambda}(p)$ of a physical vector-boson state \nof momentum $p^\\mu$ has to satisfy $\\varepsilon^{\\mu}_{\\lambda}(p)~ p_{\\mu} = 0$. \nHere the subscript $\\lambda$ labels the number of physical spin degrees of freedom, \ni.e., $\\lambda=1,2,3$ in D = 4.\nBy convention the physical polarization state vectors are orthogonal and normalized by \n$\\varepsilon^{*}_{\\lambda}(p) \\cdot \\varepsilon_{\\lambda'}(p) = -\\delta_{\\lambda \\lambda'}$.\nThe polarization vectors of a massless gauge boson obey an additional condition \nin order to encode the correct number of physical spin degrees of freedom.\nIn practice, this additional condition is usually implemented by introducing an auxiliary \nreference vector $\\hat{r}_{\\mu}$ that is not aligned with the boson's momentum \nbut otherwise arbitrary, to which the physical polarization vectors have to be orthogonal, \n$\\varepsilon^{\\mu}_{\\lambda}(p)~ \\hat{r}_{\\mu} = 0$.\nThus, the reference vector $\\hat{r}_{\\mu}$ and the bosons' momentum $p_{\\mu}$ define \na plane to which the massless gauge boson's physical polarization vectors are orthogonal.\nWe also recall that in CDR the number of physical polarizations of a massless gauge\nboson in D dimension is taken to be D-2. This is in contrast to our prescription, \nwhere the number of physical polarizations remains two in D dimensions, see below.\n\n\\subsubsection{The $2\\to 2$ scatterings among massless gauge bosons}\n\\label{SEC:prescription:MBR:2to2}\n\nLet us first consider a prototype  $2\\to 2$ scattering among 4 external \nmassless gauge bosons:\n\\begin{align} \\label{momentaassignment}\ng_1(p_1) + g_2(p_2) \\rightarrow g_3(p_3) + g_4(p_4), \n\\end{align}\nwith on-shell conditions $p_j^2 = 0,~ j=1,...,4.$\nThe  Mandelstam variables  associated with \\eqref{momentaassignment}\n\\begin{eqnarray} \\label{EQ:kinematicinvariants}\n s \\equiv \\left(p_1 + p_2 \\right)^2 = \\left(p_3 + p_4 \\right)^2 ~, \\qquad\n t \\equiv \\left(p_2 - p_3 \\right)^2 = \\left(p_1 - p_4 \\right)^2\n\\end{eqnarray} \nencode the independent external kinematic invariants.\n\nThe representation of the gauge bosons' polarization state vectors \nin terms of three linearly independent external momenta, $p_1, p_2, p_3$ \n can be determined in the following way. \nWe first write down a Lorentz covariant parameterization ansatz for the linear representation \nand then solve the aforementioned orthogonality and normalization conditions \nfor the linear decomposition coefficients. Once we have established a definite Lorentz \ncovariant decomposition form in 4 dimensions solely in terms of external momenta and \nkinematic invariants, this form will be used as the definition of \nthe corresponding polarization state vector in D dimensions. \n\n\nWhile the decomposition of polarization state vectors in terms of external momenta \nis Lorentz covariant, it is always helpful to have in mind \na particular reference frame so that a clear geometric picture can be used to\nillustrate the  choices of and constraints on polarization state vectors.  \nTo this end, we consider in the following discussion the center-of-mass frame of \nthe two incoming particles, as illustrated in figure~\\ref{FIG:coordinatesystem}, \nwhere the beam axis is taken as the Z-axis with its positive direction set along $p_1$.\nFurthermore, the scattering plane determined by $p_1$ and $p_3$ is set as the X-O-Z plane \nwith $p_3$ having a non-negative X-component by definition. \nThe positive direction of the Y-axis of the coordinate system will be determined \naccording to the \\textit{right-hand} rule. The reference frame is shown \nin figure~\\ref{FIG:coordinatesystem}.~\\\\\n\n\\begin{figure}[tbh!]\n\\centering\n\\includegraphics[width=12cm,height=6cm]{CMSframecoordinates.png}\n\\caption{The chosen coordinate system in the center-of-mass reference\nframe of the two incoming particles.}\n\\label{FIG:coordinatesystem}\n\\end{figure}\n\nLet's now come to the momentum basis representations of the polarization vectors \nin this reference frame. \nThere are two common basis choices regarding the transverse polarization states, \nthe linear  and the circular polarization basis, \nthe latter represents  helicity eigenstates of gauge bosons.\nThese two bases can be related via a ${\\pi}\/{2}$ rotation in the complex plane. \nIn the following, we will first establish a Lorentz covariant decomposition of \na set of elementary linear polarization state vectors in terms of external 4-momenta \nand then compose circular polarization states of all external gauge bosons, \ni.e., their helicity eigenstates, out of these elementary ones.\nIt is beneficial to postpone such an explicit basis transformation \nuntil the very last step of the calculation, as will become clear from \nthe later discussions.\n\n\nFor the two initial-state massless gauge bosons $g_1(p_1)$ and $g_2(p_2)$, \nwhose momenta are taken as the reference momenta for each other, \nwe first introduce a common linear polarization state vector $\\varepsilon_{X}^{\\mu}$ \nalong the X-axis direction, i.e., transverse to the beam axis but within the X-O-Z plane. \nThe  set of equations that determines $\\varepsilon_{X}^{\\mu}$  reads: \n\\begin{eqnarray} \\label{EQ:XpolEQs}\n&&\\varepsilon^{\\mu}_{X}  = c^{X}_1~ p^{\\mu}_1 + c^{X}_2~ p^{\\mu}_2 + c^{X}_3~ p^{\\mu}_3  \\, ,\\nonumber\\\\ \n&&\\varepsilon_{X} \\cdot p_1 = 0 \\, ,\\nonumber\\\\\n&&\\varepsilon_{X} \\cdot p_2 = 0 \\, ,\\nonumber\\\\\n&&\\varepsilon_{X} \\cdot \\varepsilon_{X} = -1 \\, .\n\\end{eqnarray}\nSolving eq.~(\\ref{EQ:XpolEQs}) for the coefficients $c^{X}_1,~ c^{X}_2,~ c^{X}_3$,\nand subsequently inserting the solution back to the first line of eq.~(\\ref{EQ:XpolEQs}),  \nwe obtain the following momentum basis representation for $\\varepsilon_{X}$: \n\\begin{eqnarray} \\label{EQ:XpolMBR}\n\\varepsilon^{\\mu}_{X}  = \\mathcal{N}_{X} \n\\Big(t~ p^{\\mu}_1 + (-s-t)~ p^{\\mu}_2 + s~ p^{\\mu}_3 \\Big) \\, ,\n\\end{eqnarray}\nwhere $\\mathcal{N}^{-2}_{X} = -t s (s + t)$.\nNotice that, as will be made clear later, the overall Lorentz-invariant normalization \nfactor $\\mathcal{N}_{X}$ needs to be included  only in the very last step \nof the computation of polarized loop amplitudes, for instance after UV renormalization \nand IR subtraction if an IR subtraction method is employed. \nTherefore, we never have to deal with $\\mathcal{N}_{X}$, i.e., with a square root explicitly \nin the intermediate stages. \n If we choose to incorporate the overall normalization factors only at the level of \nsquared amplitudes (or interferences), then  square roots of kinematic invariants never appear.\nFurthermore, we can always by convenience define this overall normalization factor \nsuch that the coefficients exhibited in eq.~(\\ref{EQ:XpolMBR}) are polynomials\nin the external kinematic invariants (rather than rational functions). This can be helpful \nas computer algebra systems are typically more efficient when dealing with polynomials only.\n\n\n\nConcerning the two final-state massless gauge bosons $g_3(p_3)$ and $g_4(p_4)$, \nwhose momenta are also taken as reference momenta for each other, \nwe can introduce a common linear polarization state vector $\\varepsilon_{T}^{\\mu}$\ndefined to be transverse to $p_3$ and $p_4$ but still lying within the X-O-Z plane,\n in analogy to  $\\varepsilon_{X}^{\\mu}$. \nThe definition of $\\varepsilon_{T}^{\\mu}$ then translates into the following set of equations:\n\\begin{eqnarray} \\label{EQ:TpolEQs}\n&&\\varepsilon^{\\mu}_{T}  = c^{T}_1~ p^{\\mu}_1 + c^{T}_2~ p^{\\mu}_2 + c^{T}_3~ p^{\\mu}_3  \\, , \\nonumber\\\\ \n&&\\varepsilon_{T} \\cdot p_3 = 0 \\, , \\nonumber\\\\\n&&\\varepsilon_{T} \\cdot p_4 = 0 \\, , \\nonumber\\\\\n&&\\varepsilon_{T} \\cdot \\varepsilon_{T} = -1 \\, .\n\\end{eqnarray}\nSolving eq.~(\\ref{EQ:TpolEQs}) for the coefficients $c^{T}_1,~ c^{T}_2,~ c^{T}_3$, \n one  obtains\n\\begin{eqnarray} \\label{EQ:TpolMBR}\n\\varepsilon^{\\mu}_{T}  = \\mathcal{N}_{T} \n\\Big(t~ p^{\\mu}_1 + (s+t)~ p^{\\mu}_2 + (-s-2 t)~ p^{\\mu}_3 \\Big) \\, ,\n\\end{eqnarray}\nwhere $\\mathcal{N}_{T}^{~-2} = -t s (s + t)$.\nThe comments given above on $\\varepsilon_{X}^{\\mu}$ apply here as well.\n\n\nThe last elementary polarization state vector needed for constructing\n helicity eigenstates of all four external massless gauge bosons is the one \northogonal to $p_1$, $p_2$, and $p_3$, denoted by $\\varepsilon_{Y}$,\nwhich is thus perpendicular to the X-O-Z plane. \nIn 4 dimensions, we obtain it using the Levi-Civita tensor:\n\\begin{eqnarray} \\label{EQ:YpolMBR}\n\\varepsilon^{\\mu}_{Y}  = \\mathcal{N}_{Y}~ \\epsilon^{\\nu\\rho\\sigma\\mu} p_{1\\nu} p_{2 \\rho} p_{3 \\sigma}  \n= \\mathcal{N}_{Y}~ \\epsilon^{\\mu}_{p_1 p_2 p_3}  \\, ,\n\\end{eqnarray}\nwhere $\\mathcal{N}_{Y}^{~-2} = -s t (s + t)\/4$, \nand in the last line we introduced the notation \n$\\epsilon^{\\mu}_{p_1 p_2 p_3} \\equiv \\epsilon^{\\nu\\rho\\sigma\\mu} p_{1\\nu} p_{2 \\rho} p_{3 \\sigma}$. \n We use the convention $\\epsilon^{0123} = +1$ and  \n$\\epsilon_{\\mu\\nu\\rho\\sigma} = -\\epsilon^{\\mu\\nu\\rho\\sigma}$.\n\n\nA comment concerning $\\epsilon^{\\mu\\nu\\rho\\sigma}$ is appropriate here.\nThe above polarization state vectors will be eventually used in D-dimensional calculations. \nTo this end, following~\\cite{Larin:1991tj,Larin:1993tq,Moch:2015usa}, \nwe will treat $\\epsilon^{\\mu\\nu\\rho\\sigma}$ merely as a symbol denoting an object whose \nalgebraic manipulation rules consist of the following two statements.\n\\begin{itemize}\n\\item \nAntisymmetry: it is completely anti-symmetric regarding any odd permutation of its arguments.\n\\item \nContraction Rule\\footnote{There is a subtle point concerning this when there are \nmultiple Levi-Civita tensors in the contraction, related to the choice of pairing, \nas will be briefly commented on in section \\ref{SEC:examples:eeQQ}.}: \nthe product of two $\\epsilon^{\\mu\\nu\\rho\\sigma}$ is replaced by a combination \nof products of space-time metric tensors $g^{\\mu\\nu}$ of the same tensor rank according to \nthe following fixed pattern:\n\\begin{eqnarray} \\label{EQ:LeviCivitaContRule}\n\\epsilon^{\\mu\\nu\\rho\\sigma} \\epsilon^{\\mu'\\nu'\\rho'\\sigma'} \n= \\mathrm{Det}\\Big[g^{\\alpha \\alpha'} \\Big]~, \n\\text{~  with $\\alpha = \\mu,~\\nu,~\\rho,~\\sigma$ and $\\alpha' = \\mu',~\\nu',~\\rho',~\\sigma'$,} \n\\end{eqnarray}\nwhich agrees with the well-known mathematical identity for Levi-Civita tensors in 4 dimensions.\n\\end{itemize}\n\n\nUsing eq.~(\\ref{EQ:LeviCivitaContRule}) with the D-dimensional space-time metric-tensor \nin determining $\\mathcal{N}_{Y}$ in eq.~(\\ref{EQ:YpolMBR}), \none gets ${\\mathcal{N}_{Y}^{-2}}=(3-D) s t (s + t)\/4$ with $D = 4 - 2 \\epsilon$. \n Because $\\mathcal{N}_{Y}$ is an overall normalization factor \nwhich must be used consistently in computing both the (singular) \nvirtual loop amplitudes, the UV-renormalization counter-terms, \nas well as potential IR subtraction terms, \n it is merely a normalization convention \nwhether the explicit $D$ appearing in $\\mathcal{N}_{Y}$ is set to 4 or to $4 - 2 \\epsilon$, \non which the final 4-dimensional finite remainder should not depend\n(albeit the individual singular objects do of course differ).\nThis point can be made even more transparent if one chooses to incorporate this overall normalization \nfactor only in the very last stage of the consistent computation of finite remainders \nwhere the 4-dimensional limit has already been explicitly taken. \n~\\\\\n\n\nThe circular polarization state vectors of all four external massless gauge bosons,\nnamely their helicity eigenstates, can be easily constructed from the three\n linear polarization states given above by a suitable ${\\pi}\/{2}$ rotation in the complex plane. \n The two helicity eigenstates of each gauge boson are given by  \n\\begin{eqnarray} \\label{EQ:LP2HLmassless}\n\\varepsilon_{\\pm}(p_1;p_2) &=& \\frac{1}{\\sqrt{2}} \\left( \\varepsilon_{X} \\pm \ni * \\varepsilon_{Y} \\right) \\, , \\nonumber\\\\\n\\varepsilon_{\\pm}(p_2;p_1) &=& \\frac{1}{\\sqrt{2}} \\left( \\varepsilon_{X} \\mp \ni * \\varepsilon_{Y} \\right)  \\, ,\\nonumber\\\\\n\\varepsilon_{\\pm}(p_3;p_4) &=& \\frac{1}{\\sqrt{2}} \\left( \\varepsilon_{T} \\pm \ni * \\varepsilon_{Y} \\right)  \\, ,\\nonumber\\\\\n\\varepsilon_{\\pm}(p_4;p_3) &=& \\frac{1}{\\sqrt{2}} \\left( \\varepsilon_{T} \\mp \ni * \\varepsilon_{Y} \\right) \\, ,\n\\end{eqnarray}\nwhere the first argument of $\\varepsilon_{\\pm}(p;r)$ is the particle's momentum while the second \nshows the reference momentum. Eq.~(\\ref{EQ:LP2HLmassless}) shows that the helicity flips \nonce the particle's 3-momentum gets reversed or if the polarization vector is subject to complex conjugation. \nFurthermore, owing to the Ward identities fulfilled by the gauge amplitudes, the representations\nof helicity state vectors in eq.~(\\ref{EQ:LP2HLmassless}) can be further reduced respectively \nfor each gauge boson by removing the component proportional to the gauge boson's own 4-momentum. \nFor instance, for the gauge boson $g_1$ with 4-momentum $p_1$, the component of $\\varepsilon_{X}$ \nproportional to $p_1^{\\mu}$ in eq.~(\\ref{EQ:XpolMBR}) can be safely dropped when \nconstructing $\\varepsilon_{\\pm}(p_1;p_2)$, and similar reductions hold also for the \nother gauge bosons. \nHowever, as will become clear in the following discussions, it is beneficial to project  \npolarized amplitudes first in the linear polarization basis and then have the helicity amplitudes\ncomposed at the last stage of the computation.\nSince these elementary linear polarization state vectors will be used to construct helicity states \nof several scattered particles, we should keep their complete momentum basis representation forms \nas given by eqs. \\eqref{EQ:XpolMBR}, \\eqref{EQ:TpolMBR}, and \\eqref{EQ:YpolMBR}.~\\\\\n\n \n\n\nWe emphasize again that in our prescription the number of physical polarizations in D dimensions \nof a massless gauge boson remains two, see eq.~\\eqref{EQ:LP2HLmassless}. \nIn order to illustrate resulting differences to CDR \nlet us do a simple exercise about polarization sums. \nIn CDR the sum over the physical polarizations of a massless gauge boson $g_1$ with 4-momentum \n$p_1^{\\mu}$ and gauge reference vector $r^{\\mu}=p^{\\mu}_2$ (cf. eq.~\\eqref{momentaassignment}) is \n\\begin{eqnarray}\\label{EQ:polsumCDRPhys}\n\\sum_{\\bar{\\lambda} = \\pm,~ D-4} \\bar{\\varepsilon}_{\\bar{\\lambda}}^{~\\mu}(p_1;p_2)\\bar{\\varepsilon}_{\\bar{\\lambda}}^{~*\\nu}(p_1;p_2)\n&=&  -g^{\\mu\\nu} + \\frac{p_1^{\\mu} p_2^{\\nu} + p_2^{\\mu} p_1^{\\nu}}{p_1 \\cdot p_2} \\nonumber\\\\\n&=&  -g^{\\mu\\nu} + \\frac{2}{s} \\left(p_1^{\\mu} p_2^{\\nu} + p_2^{\\mu} p_1^{\\nu} \\right)\n\\end{eqnarray}\nwhich is also the unpolarized Landau density matrix of the polarization states of $g_1$.\nAll Lorentz indices in \\eqref{EQ:polsumCDRPhys} are D dimensional and the $\\bar{\\lambda}$ labels\nthe polarization states in D dimensions.\nOn the other hand, in our prescription we sum over just the two transverse polarization states of \n$g_1$ that are defined by their respective momentum basis representations in \neqs.~(\\ref{EQ:XpolMBR}), \\eqref{EQ:TpolMBR}.  \nWe get \n\\begin{eqnarray}\\label{EQ:polsumMBR}\n\\sum_{\\lambda = X,Y} \\varepsilon^{~\\mu}_{\\lambda}(p_1;p_2)\\varepsilon^{~*\\nu}_{\\lambda}(p_1;p_2) \n&=& \\frac{1}{D-3} \\left(-g^{\\mu\\nu} + \\frac{D-2}{s} \\left(p_1^{\\mu} p_2^{\\nu} + p_2^{\\mu} p_1^{\\nu} \\right) \\right) \\nonumber\\\\\n&+&\\frac{4-D}{D-3} \n\\Bigg(\\frac{t}{s (s+t)} p_1^{\\mu} p_1^{\\nu} \n+\\frac{s+t}{s t} p_2^{\\mu} p_2^{\\nu}\n+\\frac{s}{t (s+t)} p_3^{\\mu} p_3^{\\nu} \n\\nonumber\\\\&&~~~\n+\\frac{1}{s+t} \\left(p_1^{\\mu} p_3^{\\nu} + p_3^{\\mu} p_1^{\\nu} \\right)\n-\\frac{1}{t} \\left(p_2^{\\mu} p_3^{\\nu} + p_3^{\\mu} p_2^{\\nu} \\right)\n\\Bigg)\n\\end{eqnarray}\nwhere, as part of the definition of this expression, we have rewritten the \nproduct of two Levi-Civita tensors in $\\varepsilon^{\\mu}_{Y}(p_1;p_2)\\varepsilon^{*\\nu}_{Y}(p_1;p_2)$ \nin terms of space-time metric tensors.  \nApparently eq.~(\\ref{EQ:polsumMBR}) is not identical to eq.~(\\ref{EQ:polsumCDRPhys}),\\footnote{Note that \nwith our prescription unpolarized squared amplitudes are supposed to be computed by incoherently summing over\nhelicity amplitudes, and not by using polarization sums like \\eqref{EQ:polsumMBR}.} but the two \nexpressions agree of course in D = 4 dimensions.~\\\\ \n\n\n\nBefore we move on to establish explicit momentum basis representations of longitudinal polarization\nvectors for massive vector bosons and also for massive fermions, let us emphasize  that \nby construction these momentum basis representations of polarization state vectors \nfulfill all the defining physical constraints, i.e., orthogonality to momenta and \nreference vectors, which are assured even if the open Lorentz indices (carried\nby either the external momenta or the Levi-Civita symbol) are taken to be  D-dimensional. \n\n\nMathematically, the procedure of determining norm-orthogonal polarization vectors\neqs.~\\eqref{EQ:XpolEQs}, \\eqref{EQ:TpolEQs} from a given set of linearly independent momenta \nin 4 dimensions resembles the Gram-Schmidt orthogonalization procedure. \nOur key insight here is that we establish these Lorentz covariant decomposition \nrepresentations in 4 dimensions in a form that facilitates the subsequent promotion \nof their open Lorentz indices from 4 to D, resulting in expressions which will be \ntaken as their definitions in D dimensions.\n\n\n\\subsubsection{Massive particles in the final state}\n\nNext we consider the scattering process eq.~(\\ref{momentaassignment}) but \nwith massive final-state vector bosons, for instance W or Z bosons, \nwith on-shell conditions \n\\begin{eqnarray} \\label{EQ:onshellmassive}\np_1^2 = p_2^2 = 0~,~~ p_3^2 = p_4^2 = m^2 \\, . \n\\end{eqnarray}\n\n\nConcerning the three elementary transverse polarization state vectors, \n$\\varepsilon^{\\mu}_{X}~,~ \\varepsilon^{\\mu}_{T}~,~ \\varepsilon^{\\mu}_{Y}$, \nthe above constructions can be repeated but with slightly different \nkinematics.\nIt is straightforward to arrive at the following explicit representations:\n\\begin{eqnarray} \\label{EQ:TranspolMBRmassive}\n\\varepsilon^{\\mu}_{X} &=& \\mathcal{N}_{X} \\, \n\\Big((t-m^2)~ p^{\\mu}_1 + (-s-t+m^2)~ p^{\\mu}_2 + s~ p^{\\mu}_3 \\Big) \\, ,\\nonumber\\\\\n\\varepsilon^{\\mu}_{T} &=&  \\mathcal{N}_{T} \n\\Big((t+m^2)~ p^{\\mu}_1 + (s+t-3m^2)~ p^{\\mu}_2 + (-s-2 t+2m^2)~ p^{\\mu}_3 \\Big) \\, ,\\nonumber\\\\\n\\varepsilon^{\\mu}_{Y} &=& \\mathcal{N}_{Y}~ \\epsilon^{\\mu}_{p_1 p_2 p_3} \\, ,\n\\end{eqnarray}\nwith the normalization factors \n\\begin{eqnarray}\n\\mathcal{N}_{X}^{~-2} &=& s \\left(-t (s + t) + 2 m^2 t - m^4 \\right) \\, ,\\nonumber\\\\\n\\mathcal{N}_{T}^{~-2} &=& - s t (s + t) + 2 m^2 t (3 s + 2 t) - m^4 (s + 8 t) + 4 m^6 \\, , \\nonumber\\\\\n\\mathcal{N}_{Y}^{~-2} &=& \\frac{1}{4} s \\left(-t (s + t) + 2 m^2 t - m^4\\right) \\, ,\n\\end{eqnarray}\nwhich, as already emphasized above, can always be conveniently chosen to be \nincorporated only at the very last stage of the computation.~\\\\\n\n\nCompared to the massless case, the helicity eigenstates of massive gauge bosons are \nreference-frame dependent and their helicities are not Lorentz-invariant.  \nHelicity eigenstates constructed from the above elementary linear polarization state \nvectors are defined in the center-of-mass reference frame of the two colliding particles.\nThe third physical polarization state of a massive gauge boson is described by \nthe longitudinal polarization vector (defined in the same reference frame), \nwhich has its spatial part aligned with the momentum of the boson.\nFor the massive particle $g_3(p_3)$ these conditions translate into \nthe following set of equations for its longitudinal polarization vector $\\varepsilon_{L3}^{\\mu}$:\n\\begin{eqnarray} \\label{EQ:L3polEQs}\n&&\\varepsilon_{L3}^{\\mu}  = c^{L3}_1 \\left( p^{\\mu}_1 + p^{\\mu}_2 - p^{\\mu}_3 \\right) \n+ c^{L3}_2 p^{\\mu}_3  \\, , \\nonumber\\\\ \n&&\\varepsilon_{L3}^{\\mu} \\cdot p_3 = 0 \\, ,\\nonumber\\\\\n&&\\varepsilon_{L3} \\cdot \\varepsilon_{L3} = -1 \\, .\n\\end{eqnarray}\nSolving eq.~(\\ref{EQ:L3polEQs}) for $c^{L3}_1,~ c^{L3}_2$, one obtains \n\\begin{eqnarray} \\label{EQ:L3polMBR}\n\\varepsilon^{\\mu}_{L3}  = \\mathcal{N}_{L3} \n\\Big(-2 m^2~ \\left( p^{\\mu}_1 + p^{\\mu}_2 \\right) + s~ p^{\\mu}_3 \\Big) \\, ,\n\\end{eqnarray}\nwhere $\\mathcal{N}_{L3}^{~-2} = s m^2 (s-4 m^2)$. \nFor the massive vector boson $g_4(p_4)$ one gets for its longitudinal polarization vector $\\varepsilon_{L4}^{\\mu}$: \n\\begin{eqnarray} \\label{EQ:L4polMBR}\n\\varepsilon^{\\mu}_{L4}  = \\mathcal{N}_{L4} \n\\Big((s -2 m^2)~ \\left( p^{\\mu}_1 + p^{\\mu}_2 \\right) - s~ p^{\\mu}_3 \\Big) \\, ,\n\\end{eqnarray}\nwhere $\\mathcal{N}_{L4} = \\mathcal{N}_{L3}$.\n By construction the defining physical properties, such as orthogonality to the momenta, \nare fulfilled by these momentum basis representations, even if \ntheir open Lorentz indices are taken to be  D-dimensional. \nWe emaphasize that in our prescription the number of physical polarizations of a massive vector boson \nremains three in D dimensions.~\\\\\n\n\nThere are also polarization vectors associated with massive fermions. \nThe helicity eigenstate of a massive fermion with 4-momentum $k$ can be described by \na Dirac spinor, e.g.~$u(k,S_k)$, characterized by the normalized space-like polarization \nvector $S_k^{\\mu}$. And component-wisely  \n\\begin{eqnarray} \nS_k^{\\mu} = \\left(\\frac{|\\vec{k}|}{m}, \\frac{k^0}{m} \\frac{\\vec{k}}{|\\vec{k}|}\\right) \\, ,\n\\end{eqnarray} \nwhere $k^0$ and $m$ are, respectively, the energy and mass of the massive fermion,\nwhile $\\vec{k}$ represents its 3-momentum. \nInterestingly, this polarization vector has the same momentum basis decomposition form \nas the longitudinal polarization vector of a massive vector boson \n(of the same momentum), provided the same external kinematic configuration applies. \nBy identifying $p_3^{\\mu} = k^{\\mu}$, eq.~(\\ref{EQ:L3polMBR}) can be viewed as the momentum \nbasis representation of $S_k^{\\mu}$ for the same external kinematic configuration as above. \nNamely, \n\\begin{eqnarray} \\label{EQ:PLpolMBR}\nS_k^{\\mu} = \\mathcal{N}_{S_k} \n\\Big(-2 m^2~\\left( p^{\\mu}_1 + p^{\\mu}_2 \\right) + s~ k^{\\mu} \\Big)\n\\end{eqnarray}  \nwith $\\mathcal{N}_{S_k}^{~-2} = s m^2 (s-4 m^2)$.\nThis is because the set of norm-orthogonal conditions that $S^{\\mu}$ has to fulfill, \nnamely $k \\cdot S = 0~,~ S \\cdot S = -1~,~ \\vec{S} \\parallel \\vec{k}$, \nwhich are sufficient to determine it up to an overall phase, are exactly \nthe same as those that the longitudinal polarization  vector \nin eq.~(\\ref{EQ:L3polMBR}) has to fulfill. \n\n\n\\subsection{Normalized tensor products of external spinors}\n\\label{SEC:prescription:NTS}\n\n\nIn cases where external fermions are involved in scattering amplitudes, \nstrings of Dirac matrices sandwiched between external on-shell spinors \nwill show up. In order to evaluate each  open fermion line using \ntrace techniques, we employ the standard trick of multiplying and dividing \nthis quantity by appropriate auxiliary Lorentz-invariant spinor inner products,\nwhich can be traced back to~ref.~\\cite{Bjorken:1966kh}.\nPulling out the chosen overall Lorentz-invariant normalization factor, \nthe rest can be cast into a trace of products of Dirac matrices with the aid \nof Landau density matrices of external spinors.\nThe momentum basis representations of (massive) fermion's space-like \npolarization vectors, such as eq.~(\\ref{EQ:PLpolMBR}), can be used \n in these density matrices.\nFor massless fermions, the spin density matrices are reduced to \nleft- respectively right-chirality projectors, which thus spares us from introducing \nany explicit polarization vector in this case. \nThis is because helicity states of massless fermions coincide with chiral spinors. ~\\\\\n\n\nFrom a single open fermion line in a Feynman diagram, we get a contribution \nwhich can be generically written as $\\langle \\psi_A|\\hat{\\mathrm{M}}|\\psi_B \\rangle$.\nThe symbol $\\hat{\\mathrm{M}}$ denotes a product of Dirac matrices with their Lorentz indices \neither contracted or left open, and $|\\psi_A\\rangle,~|\\psi_B \\rangle$ stand for \nthe two external on-shell Dirac spinors, either of $u$-type or $v$-type. \nViewed as a spinor inner product, $\\langle \\psi_A|\\hat{\\mathrm{M}}|\\psi_B \\rangle$  \ncan always be rewritten as a trace of a product of Dirac-matrices \nin the Dirac-spinor space:\n\\begin{eqnarray} \\label{EQ:SFLtrace1}\n\\langle \\psi_A|\\hat{\\mathrm{M}}|\\psi_B \\rangle = \\mathrm{Tr} \\Big{[} \n|\\psi_B \\rangle  \\langle \\psi_A|\\hat{\\mathrm{M}} \\Big{]}.\n\\end{eqnarray}\nThis formal rewriting is not really useful unless we can further \nexploit the matrix structure of the external spinors' tensor product $|\\psi_B \\rangle \\langle \\psi_A|$ \nin the spinor space (explicitly in terms of elementary Dirac matrices). \nTo this end, we rewrite $|\\psi_B \\rangle \\langle \\psi_A|$ \nby introducing an auxiliary spinor inner product along the following line:\n\\begin{eqnarray} \\label{EQ:TPextsps1}\n|\\psi_B \\rangle \\langle \\psi_A| \n&=& \n\\frac{\\langle \\psi_B|\\hat{\\mathrm{N}}|\\psi_A \\rangle }{\\langle \\psi_B|\\hat{\\mathrm{N}}|\\psi_A \\rangle }\n|\\psi_B \\rangle  \\langle \\psi_A| \\nonumber\\\\\n&=& \\frac{1}{\\langle \\psi_B|\\hat{\\mathrm{N}}|\\psi_A \\rangle}\n|\\psi_B \\rangle \\langle \\psi_B|\\hat{\\mathrm{N}}|\\psi_A \\rangle \\langle \\psi_A| \\nonumber\\\\\n&=& \\mathcal{N}_{AB}~\n|\\psi_B \\rangle  \\langle \\psi_B| \\hat{\\mathrm{N}} |\\psi_A \\rangle \\langle \\psi_A| \\, , \n\\end{eqnarray}\nwhere $\\mathcal{N}_{AB} \\equiv ({\\langle \\psi_B|\\hat{\\mathrm{N}}|\\psi_A \\rangle})^{-1}$.\n The auxiliary matrix $\\hat{\\mathrm{N}}$ is only required to have a non-vanishing\nmatrix element $\\langle \\psi_B|\\hat{\\mathrm{N}}|\\psi_A \\rangle$ and otherwise can be chosen \nto be as simple as desired.\nFor instance, for massive external spinors of some particular helicity configurations,\n $\\hat{\\mathrm{N}}$ may be chosen to be the identity matrix in spinor space, provided that \nthe spinor inner products between those helicity spinors are not vanishing. \nA generally valid and simple choice is $\\hat{\\mathrm{N}} = \\gamma_{\\mu} p^{\\mu}$ \nwith a 4-momentum $p^{\\mu}$ that is not linearly dependent on the on-shell momenta  $p_A$ and $p_B$\nof $\\langle \\psi_A|$ and $|\\psi_B \\rangle$, respectively.  \n\n\nWe manipulate eq.~(\\ref{EQ:TPextsps1}) further by first substituting  Landau density matrices for \n$|\\psi_A \\rangle \\langle \\psi_A|$ and $|\\psi_B \\rangle \\langle \\psi_B|$,  conventionally given by \n\\begin{eqnarray} \\label{EQ:LDMofDSP}\nu(p,S_p) \\otimes \\bar{u}(p,S_p) &=& \\left(\\slashed{p} + m \\right) \\frac{1+\\gamma_5 \\slashed{S}_p}{2} \\, , \\nonumber\\\\\nv(p,S_p) \\otimes \\bar{v}(p,S_p) &=& \\left(\\slashed{p} - m \\right) \\frac{1+\\gamma_5 \\slashed{S}_p}{2} \\, .\n\\end{eqnarray} \nThen we simplify the resulting composite Dirac matrix object \nbefore finally obtaining a form that is suitable for being unambiguously used \nin eq.~(\\ref{EQ:SFLtrace1}) with the trace to be done in D dimensions.  \n\n\nThere are several equivalent forms of these on-shell Dirac-spinors' projectors \nin 4 dimensions. In particular, one may commute the on-shell projection \noperator $\\slashed{p} \\pm m$ and the polarization projection operator \n$({1+\\gamma_5 \\slashed{S}_p})\/{2}$ using  $p \\cdot S_p = 0$.\nHowever, it is well known that a fully anticommuting $\\gamma_5$ can not be\nimplemented in dimensional regularization in an algebraically consistent way~\\cite{Collins:1984xc}, \nif we still want this object to coincide with the usual $\\gamma_5$ in 4 dimensions. \nIn this article, we adopt the  $\\gamma_5$ prescription of ref.~\\cite{Larin:1991tj,Larin:1993tq}, \nconventionally known as Larin's scheme, whose equivalent but more efficient implementations in high-order \nperturbative calculations are discussed in ref.~\\cite{Moch:2015usa}.  \nIn our work, all appearances of $\\gamma_5$ matrices, originating either from \ninteraction vertices or external polarization projectors, should be \nregarded just for bookkeeping purposes and their interpretations shall be based \non~\\cite{Larin:1991tj,Larin:1993tq,Moch:2015usa}. \nAs a consequence of this prescription, $\\gamma_5$ no longer anticommutes with all \nDirac $\\gamma$ matrices, and 4-dimensional equivalent forms of eq.~(\\ref{EQ:LDMofDSP}) \nare no longer necessarily algebraically equivalent in D dimensions. \n\n\nIn order to eliminate potential ambiguities --- after having  simplified \\eqref{EQ:SFLtrace1}, \\eqref{EQ:LDMofDSP}\nusing 4-dimensional Lorentz and Dirac algebra as much as possible ---, we should agree on one definite fixed form of \n\\eqref{EQ:SFLtrace1}, \\eqref{EQ:LDMofDSP}, solely in terms of a string of Dirac $\\gamma$ matrices with \nfixed product ordering, the Levi-Civita tensor, and external momenta. \nWe may call these their canonical forms in 4 dimensions. \nThis allows an unambiguous interpretation\\footnote{This is up to a potential subtlety \nrelated to the contraction order among multiple Levi-Civita tensors~\\cite{Moch:2015usa}, \nas will be commented on in section~\\ref{SEC:examples:eeQQ}.} \nof the expression in D dimensions where it will be manipulated according to the $D$-dimensional algebra \nafter being inserted back into eq.~(\\ref{EQ:SFLtrace1}).\n\n\nLet us now be more specific about this by working out a representative case, \na single open fermion line with two massive external $u$-type spinors, $u(p_A,S_A)$ and $u(p_B,S_B)$.\nWe choose $\\hat{\\mathrm{N}} = \\slashed{q}$ where $q^{\\mu}$ is a 4-momentum \nthat is linearly independent of $p_A$ and $p_B$. \nPulling out the normalization factor $\\mathcal{N}_{AB} = \\left({\\bar{u}(p_A,S_A) \\slashed{q} u(p_B,S_B)}\\right)^{-1}$, \neq.~(\\ref{EQ:TPextsps1}) reads in this case:\n\\begin{eqnarray}\\label{EQ:TPextspsaux}\n\\frac{1}{\\mathcal{N}_{AB}}~ u(p_B,S_B) \\otimes \\bar{u}(p_A,S_A) \n&=& \n\\left(\\slashed{p}_B + m \\right) \n\\frac{1+\\gamma_5 \\slashed{S}_B}{2}\n\\slashed{q}\n\\frac{1+\\gamma_5 \\slashed{S}_A}{2} \n\\left(\\slashed{p}_A + m \\right) \\, ,\n\\end{eqnarray}\n which can be brought into the form\n\\begin{eqnarray} \\label{EQ:TPextsps2}\n\\frac{1}{\\mathcal{N}_{AB}}~ u(p_B,S_B) \\otimes \\bar{u}(p_A,S_A) \n&=& \\left(\\slashed{p}_B + m \\right) \n\\frac{1}{4}\\slashed{q}\n\\left(\\slashed{p}_A + m \\right) \\nonumber\\\\\n&+& \\left(\\slashed{p}_B + m \\right) \n\\frac{1}{4} \n\\left(\\frac{-i}{3!} \\epsilon_{\\gamma \\gamma \\gamma S_B} \\right) \\slashed{q}\n\\left(\\slashed{p}_A + m \\right) \\nonumber\\\\\n&+& \\left(\\slashed{p}_B + m \\right) \n\\frac{1}{4}\\slashed{q} \n\\left(\\frac{-i}{3!} \\epsilon_{\\gamma \\gamma \\gamma S_A} \\right)\n\\left(\\slashed{p}_A + m \\right) \\nonumber\\\\\n&+& \\left(\\slashed{p}_B + m \\right) \n\\frac{1}{4} \\slashed{S}_B \\slashed{q} \\slashed{S}_A \n\\left(\\slashed{p}_A + m \\right) \\, .\n\\end{eqnarray} \nStrictly speaking, eq.~(\\ref{EQ:TPextsps2}) is identical to \\eqref{EQ:TPextspsaux}\nonly in 4 dimensions. The unambiguous eq.~(\\ref{EQ:TPextsps2}), which no longer contains \nany explicit $\\gamma_5$, will be taken as the definition of  $u(p_B,S_B) \\otimes \\bar{u}(p_A,S_A)$ \nwhen it is inserted into eq.~(\\ref{EQ:SFLtrace1}) and manipulated in accordance with \nthe D-dimensional algebra.\n\n\nNotice that in eq.~\\eqref{EQ:TPextspsaux} the auxiliary matrix \n$\\slashed{q}$ and the polarization projection operators were \nplaced inside the on-shell projection operators $\\slashed{p}_{\\tiny{A\/B}} +  m$,\na point which will be explained and become clear in section~\\ref{SEC:unitarity:PSA}.  \nWe emphasize again that the momentum basis representations of the\nmassive fermions' helicity polarization vectors $S^{\\mu}_A$ and $S^{\\mu}_B$\nwill be eventually inserted, whose open Lorentz indices are carried \nby external momenta that are assumed to be D dimensional.\nSimilar rewritings and definitions like eq.~(\\ref{EQ:TPextsps2}) can be made \nalso for fermion lines with external $v$-type  spinors, whose Landau density \nmatrices are given in eq.~(\\ref{EQ:LDMofDSP}). \n\n\n\nIn practice, it is very convenient to keep projections associated with each of \nthe four terms in eq.~(\\ref{EQ:TPextsps2}) separate from each other, for two reasons.\nFirst, this organization is in accordance with the power of the Levi-Civita tensor \nappearing in the terms, which is advantageous especially when Feynman diagram expressions are \nalso split into terms with even and odd products of $\\gamma_5$ (arising from axial vertices).  \nSecond, for a fermion with fixed momentum, its polarization vector, e.g. $S_A$ or $S_B$ \nin eq.(\\ref{EQ:TPextsps2}), changes just by an overall minus sign when its helicity is flipped.\nTherefore, the expressions of eq.~(\\ref{EQ:SFLtrace1}) for the four different helicity configurations\ncan all be obtained by suitably combining the traces in eq.~(\\ref{EQ:SFLtrace1}) of the product \nof $\\hat{\\mathrm{M}}$ and each of the four terms in eq.~(\\ref{EQ:TPextsps2}).\n\nNotice that in general the normalization factor $\\mathcal{N}_{AB}$ in eq.~(\\ref{EQ:SFLtrace1}) \ndepends on the helicities of the external fermions A and B, \nas will be explicitly shown in the example given in section~\\ref{SEC:examples:eeQQ}.\nUsing \\eqref{EQ:TPextsps2} we need to compute these four individual projections separately just once, \nout of which all four different helicity configurations can be obtained. \n\n\nOnce a definite unambiguous form of the right-hand side of eq.~(\\ref{EQ:TPextsps2}) has been established \nin 4 dimensions, it will be kept fixed while all open Lorentz and Dirac indices will be promoted \nin accordance with computations in CDR. \nAdditionally, just like the aforementioned normalization factors associated with the gauge boson's polarization vectors, \nthe factor $\\mathcal{N}_{AB}$ in eq.~(\\ref{EQ:SFLtrace1}) is an overall normalization factor \nwhich must be adopted consistently in computing all amplitudes involved in the calculations of finite remainders. \nIf one chooses to incorporate this overall normalization factor only in the very last stage of calculating \nfinite remainders where the 4 dimensional limit can already be taken, \nit is then evident that we can evaluate these Lorentz invariant factors in 4 dimensions.~\\\\\n\n\nAs already mentioned above, in the massless limit the spin density matrices in eq.~(\\ref{EQ:LDMofDSP}) \nare reduced to left- or right-chiral projectors. Thus no polarization vectors are needed.\nFor instance, the massless limit of eq.~(\\ref{EQ:TPextsps2}) with $++$ helicity configuration reads: \n\\begin{eqnarray} \\label{EQ:TPextsps2massless}\n\\frac{1}{\\mathcal{N}_{AB}}~ u(p_B,+) \\otimes \\bar{u}(p_A,+) \n&=& \n\\slashed{p}_B  \n\\frac{1-\\gamma_5 }{2}\n\\slashed{q}\n\\frac{1+\\gamma_5}{2} \n\\slashed{p}_A \\nonumber\\\\\n&=& \\frac{1}{2}  \n\\Bigg(\n\\slashed{p}_B  \n\\slashed{q} \n\\slashed{p}_A \n- \\slashed{p}_B  \n\\left(\\frac{-i}{3!} \\epsilon_{\\gamma \\gamma \\gamma q} \\right)\n\\slashed{p}_A \\Bigg). \n\\end{eqnarray}\nThe remarks below eq.~\\eqref{EQ:TPextsps2} concerning the use of this equation in D-dimensional calculations\napply also to \\eqref{EQ:TPextsps2massless}.\nThe above reformulations of tensor products of two external helicity spinors, \nsuch as eq.~(\\ref{EQ:TPextsps2}) and eq.~(\\ref{EQ:TPextsps2massless}), \ncan be applied to each single open fermion line, besides using for each external boson\na momentum basis representation of its polarization vector. ~\\\\\n\n\n\nTo summarize, the tensor product of momentum basis representations of all external gauge bosons' polarization vectors \nand all properly re-written external spinor products, such as those given by \neqs.~\\eqref{EQ:XpolMBR} - \\eqref{EQ:YpolMBR}, and eq.~(\\ref{EQ:TPextsps2}), (\\ref{EQ:TPextsps2massless}), \nwith their open indices promoted in accordance with CDR, will be taken as the external projectors for polarized amplitudes. \nPolarized amplitudes are thus first projected in the linear polarization basis for external gauge bosons \nand the basis indicated by eq.~(\\ref{EQ:TPextsps2}), (\\ref{EQ:TPextsps2massless}) for each open fermion line.\nIt is a good practice to first combine Levi-Civita tensors that appear in external projectors in order to \nreach an unambiguous canonical form that is homogeneous in the Levi-Civita tensor whose power is at most one\\footnote{See the end of \nsection~\\ref{SEC:examples:eeQQ} for more discussions about this.}. \nHelicity amplitudes can be subsequently obtained from these these polarized amplitudes by\nlinear combinations, such as those implied in eq.~(\\ref{EQ:LP2HLmassless}). \nFor instance, the transformation matrix of polarized scattering amplitudes \namong four massless gauge bosons from the linear  to the circular polarization basis \nis a $16\\times16$ constant matrix that can be extracted from eq.~(\\ref{EQ:LP2HLmassless}). \nLikewise, constant transformation matrices can also be extracted from eq.~(\\ref{EQ:TPextsps2}) \nfor massive fermion lines and eq.~(\\ref{EQ:TPextsps2massless}) for massless fermion lines.~\\\\\n\n\nEventually every helicity amplitude thus composed is manifestly given as a function of Lorentz invariant \nvariables solely made out external momenta. \nThis is owing to the fact that the momentum basis representations of polarization vectors \nallow us to find a Lorentz covariant representation of the tensor product of external particle states \nsolely in terms of external momenta and algebraic constants (such as the metric tensor, the\nLevi-Civita tensor and Dirac matrices). \nSubsequently this makes it feasible to directly take these objects as the external polarization projectors.\nFrom the point of view of the projection method as outlined in section~\\ref{SEC:projectionmethod:recap},\nthe set of external polarization projectors described above might be loosely viewed as \na special choice of Lorentz decomposition basis which by construction are orthogonal among each other.\nConsequently, the corresponding Gram matrix is diagonal and its inversion is trivial. \nFurthermore, each structure that arises from such a decomposition is directly related to a physical quantity, \nand therefore its singularity pattern is protected by physical constraints obeyed by these physical quantities. \nIn this way the issues related to the conventional form factor decomposition as discussed \nin section~\\ref{SEC:projectionmethod:comments} are circumvented. \n\n\n\\subsection{Comments on other processes} \n\\label{SEC:prescription:comments}\n\n\nIn the preceding subsections we have discussed a prototype $2 \\to 2$ scattering process  \nwhere there are only three linearly independent external momenta and consequently \nthe Lorentz invariant scattering amplitude cannot contain a term composed of one Levi-Civita tensor \nfully contracted with external momenta.\nThis fact can lead to a reduction of terms that are to be included in external projectors. \nFor instance, if the $2\\to 2$ scattering process is parity-invariant, then all terms in the external projectors \nthat are linear in Levi-Civita tensor can be dropped from the outset. \nThis simplification does no longer occur if there are more than four particles involved in the scattering, \ne.g. in a  $2\\to 3$ process.\nWe comment on a few technical aspects in handling these cases.~\\\\ \n\n \n\\subsubsection*{$2\\to 3$ scattering.} \n\nWe have seen in section~\\ref{SEC:prescription:MBR} and \\ref{SEC:prescription:NTS}  \nthat three linearly independent external momenta are sufficient to build momentum basis representations \nof external polarization vectors, and the concrete decomposition coefficients depend on the particular \nkinematics. For constructing momentum basis representations of polarization vectors for final-state particles, \nit is convenient to take a group of three linearly independent external momenta of which two are always \nchosen to be the momenta of the initial-state (massless) particles and the third one is the one of the \nfinal state particle in question. \n\n\nFor convenience, let us document below the momentum basis representations of linear polarization \nvectors introduced in section~\\ref{SEC:prescription:MBR}, but without specializing the external kinematic \nconfiguration.\nWe consider a generic configuration with two massless initial state particles with momenta $p_1$ and $p_2$ \n(which is applicable to most of the phenomenologically interesting high-energy scattering processes), \nwhile the mass of the particular final state particle, with  momentum labeled as $p_3$, in question is \nleft unspecified. These three external momenta are assumed to be linearly independent. \nNo specification is made of the kinematics of the other particles in the final state.  \n\n\nWe introduce the following symbols for kinematic invariants:\n\\begin{eqnarray}\ns_{12} = 2~ p_1 \\cdot p_2 ~,~~  s_{13} = 2~ p_1 \\cdot p_3 ~,~~  \ns_{23} = 2~ p_2 \\cdot p_3 ~,~~ m^2 = p_3 \\cdot p_3 ~,\n\\end{eqnarray} \nwhich are assumed to be independent of each other. Repeating the construction made \nin section~\\ref{SEC:prescription:MBR}, we obtain for this generic kinematic setting:\n\\begin{eqnarray} \\label{EQ:TranspolMBRmassiveGeneric}\n\\varepsilon^{\\mu}_{X} &=& \\mathcal{N}_{X} \n\\Big((-s_{23})~ p^{\\mu}_1 + (-s_{13})~ p^{\\mu}_2 + s_{12}~ p^{\\mu}_3 \\Big) , \\nonumber\\\\\n\\varepsilon^{\\mu}_{T} &=&  \\mathcal{N}_{T} \n\\Big((s_{23} (s_{13} + s_{23}) - 2 m^2 s_{12} )~ p^{\\mu}_1 + (-s_{13} (s_{13} + s_{23}) + 2 m^2 s_{12})~ p^{\\mu}_2 + (s_{12} (s_{13} - s_{23}))~ p^{\\mu}_3 \\Big), \\nonumber\\\\\n\\varepsilon^{\\mu}_{Y} &=& \\mathcal{N}_{Y}~ \\epsilon^{\\mu}_{p_1 p_2 p_3}, \\nonumber\\\\\n\\varepsilon^{\\mu}_{L3}&=& \\mathcal{N}_{L3} \n\\Big(-2 m^2~ \\left( p^{\\mu}_1 + p^{\\mu}_2 \\right) + (s_{13} + s_{23}) ~ p^{\\mu}_3 \\Big) \\, ,\n\\end{eqnarray}\nwith normalization factors \n\\begin{eqnarray}\n\\mathcal{N}_{X}^{~-2} &=& s_{12} \\left(s_{13} s_{23} - m^2 s_{12}\\right) \\, ,\\nonumber\\\\\n\\mathcal{N}_{T}^{~-2} &=& s_{12} \\left( s_{13} s_{23} (s_{13} + s_{23})^2 - m^2 s_{12} (s_{13}^2 + 6 s_{13} s_{23} + s_{23}^2) + 4 m^4 s_{12}^2 \\right) \\, , \\nonumber\\\\\n\\mathcal{N}_{Y}^{~-2} &=& \\frac{1}{4} s_{12} \\left(s_{13} s_{23} - m^2 s_{12}\\right) \\, , \\nonumber\\\\\n\\mathcal{N}_{L3}^{~-2} &=& m^2 \\left((s_{13} + s_{23})^2 - 4 m^2 s_{12}\\right) \\, .\n\\end{eqnarray}\nAll previous comments on polarization vectors and normalization factors apply here as well.\n~\\\\\n\n\nDue to the presence of four linearly independent external momenta in a $2\\to 3$ process, \nthe aforementioned reduction of terms in the external projectors for $2\\to 2$  processes \nno longer applies in general. \nHowever, this fact also offers an opportunity to eliminate the explicit appearance of \nthe Levi-Civita tensor $\\epsilon^{\\mu\\nu\\rho\\sigma}$ from external projectors \n(which may already be pre-processed to be at most linear in $\\epsilon^{\\mu\\nu\\rho\\sigma}$),\nby applying the trick used in defining the van Neerven-Vermaseren basis~\\cite{vanNeerven:1983vr}.\nTo be more specific, let us consider a $2\\to 3$ scattering process where the four linearly independent \n4-momenta are denoted by $p_1, p_2,p_3,p_4$.\n A single power $\\epsilon^{\\mu\\nu\\rho\\sigma}$ in external polarization projectors can be rewritten as \n\\begin{eqnarray} \\label{EQ:NVtrick}\n\\epsilon^{\\mu\\nu\\rho\\sigma} &=& \n\\frac{\\epsilon_{p_1 p_2 p_3 p_4}}{\\epsilon_{p_1 p_2 p_3 p_4}} \n~\\epsilon^{\\mu\\nu\\rho\\sigma} \\nonumber\\\\\n&=& \\Delta ~  \n\\Big(\np_1^{\\rho } p_2^{\\nu } p_3^{\\mu } p_4^{\\sigma }-p_1^{\\nu } p_2^{\\rho } p_3^{\\mu } p_4^{\\sigma }-p_1^{\\rho } p_2^{\\mu } p_3^{\\nu } p_4^{\\sigma }+p_1^{\\mu } p_2^{\\rho } p_3^{\\nu } p_4^{\\sigma }+p_1^{\\nu } p_2^{\\mu } p_3^{\\rho } p_4^{\\sigma }-p_1^{\\mu } p_2^{\\nu} p_3^{\\rho } p_4^{\\sigma }\n\\nonumber\\\\&&~~~~~\n-p_1^{\\rho } p_2^{\\nu } p_3^{\\sigma } p_4^{\\mu }+p_1^{\\nu } p_2^{\\rho } p_3^{\\sigma } p_4^{\\mu}+p_1^{\\rho } p_2^{\\sigma } p_3^{\\nu } p_4^{\\mu }-p_1^{\\sigma } p_2^{\\rho } p_3^{\\nu } p_4^{\\mu }-p_1^{\\nu } p_2^{\\sigma }\np_3^{\\rho } p_4^{\\mu }+p_1^{\\sigma } p_2^{\\nu } p_3^{\\rho } p_4^{\\mu }\n\\nonumber\\\\&&~~~~~\n+p_1^{\\rho } p_2^{\\mu } p_3^{\\sigma } p_4^{\\nu }-p_1^{\\mu }p_2^{\\rho } p_3^{\\sigma } p_4^{\\nu }-p_1^{\\rho } p_2^{\\sigma } p_3^{\\mu } p_4^{\\nu }+p_1^{\\sigma } p_2^{\\rho } p_3^{\\mu } p_4^{\\nu\n   }+p_1^{\\mu } p_2^{\\sigma } p_3^{\\rho } p_4^{\\nu }-p_1^{\\sigma } p_2^{\\mu } p_3^{\\rho } p_4^{\\nu }\n\\nonumber\\\\&&~~~~~\n-p_1^{\\nu } p_2^{\\mu } p_3^{\\sigma } p_4^{\\rho }+p_1^{\\mu } p_2^{\\nu } p_3^{\\sigma } p_4^{\\rho }+p_1^{\\nu } p_2^{\\sigma } p_3^{\\mu } p_4^{\\rho}-p_1^{\\sigma } p_2^{\\nu } p_3^{\\mu } p_4^{\\rho }-p_1^{\\mu } p_2^{\\sigma } \np_3^{\\nu } p_4^{\\rho }+p_1^{\\sigma } p_2^{\\mu } p_3^{\\nu } p_4^{\\rho } \\Big),\n\\nonumber\\\\\n\\end{eqnarray}\nwhere the normalization factor $\\Delta \\equiv \\frac{1}{\\epsilon_{p_1 p_2 p_3 p_4}}$ \ncan be conveniently pulled out and grouped together with other normalization factors of external projectors\n(and  used consistently through out the whole calculation). \nThe treatment of the Levi-Civita tensor in eq.~(\\ref{EQ:NVtrick}) complies with the two rules listed\nin section~\\ref{SEC:prescription:MBR}. \nIn this way,  no Levi-Civita tensor appears in external polarization projectors  \nfor $2 \\to 3$ scattering amplitudes any more, up to a global normalization factor, \nand hence it is manifest that the form of external projectors can be unambiguously constructed.\n\n\n\\subsubsection*{$1\\to 2$ decay.} \n\nFor a $1\\to 2$ decay amplitude, the conventional Lorentz tensor decomposition and projection method \ncan be carried out quite simply (due to the limited number of basis structures and scales). \nFor instance, for the fermion's gauge interaction vertex a general form factor decomposition\ncan be found in~\\cite{Hollik:1998vz}. \nHere we briefly comment on how one can compute polarized $1\\to 2$ decay amplitudes if one wants \nto use the above prescription. \n\n\nThe computation requires the introduction of an intermediate auxiliary reference-vector, \ndenoted by $\\hat{r}^{\\mu}$, which will be formally treated on the same footing \nas an external 4-momentum.\nThe reference-vector $\\hat{r}^{\\mu}$ may be associated with the polarization vector of \nthe decaying particle (in which case it has a physical meaning), or chosen to be \nan auxiliary coordinate-system-specific vector merely for intermediate usage. \nThe important point we would like to emphasize here is that \nthe definition of $\\hat{r}^{\\mu}$ can be achieved by simply specifying the values of \na complete set of quadratic Lorentz invariant products between $\\hat{r}^{\\mu}$ \nand two linearly independent external momenta, which we denote by $p_1$ and $p_2$.  \nFor instance, the normalized space-like $\\hat{r}^{\\mu}$ can be implicitly specified by \n\\begin{eqnarray} \\label{EQ:defRV}\n\\hat{r} \\cdot p_1 = 0~, ~~ \\hat{r} \\cdot p_2 = 0~,~~ \\hat{r} \\cdot \\hat{r} = -1, \n\\end{eqnarray}\nwhich guarantees that it lies in the plane transverse to $p_1$ and $p_2$. \nThis set of assignments \\eqref{EQ:defRV} is sufficient to algebraically manipulate\n$\\hat{r}$ in the computation of polarized $1\\to 2$ decay amplitudes. \nThere is no need for its explicit component-wise specification in a definite coordinate-reference system. \nWith the aid of the thus-defined $\\hat{r}$, all procedures outlined above for the \n$2 \\to 2$ scattering processes, discussed in section~\\ref{SEC:prescription:MBR}, can be repeated here. \nTo be a bit more specific, in this case the set of three linearly independent 4-vectors \n$\\{ p_1, p_2, \\hat{r}\\}$ will take over the roles that were played by the three linearly independent \nexternal momenta $\\{ p_1, p_2, p_3\\}$ in the $2 \\to 2$ scattering processes.\nIn fact the $\\hat{r}$ defined in eq.~(\\ref{EQ:defRV}) fulfills the same set of conditions\nthat $\\varepsilon_{X}$ in eq.~(\\ref{EQ:XpolEQs}) satisfies. \nMoreover, it never appears in Feynman propagators\\footnote{This means that the sectors of loop \nintegrals appearing in the projected amplitudes will not be enlarged by the introduction \nof this external reference-vector $\\hat{r}$.}, and the Lorentz invariants appearing in \nthe resulting projections are still just those made out of $p_1$ and $p_2$ (as the right-hand side \nof eq.~(\\ref{EQ:defRV}) are all constants).\nIn the end the physical decay rates are independent of the choice of this auxiliary $\\hat{r}$. \nIn the case of a scalar decaying into a pair of fermions, the introduction of such an auxiliary vector \ncould be avoided because the helicity polarization vector of a massive fermion, eq.~(\\ref{EQ:PLpolMBR}), \nmakes no reference at all to any transverse direction w.r.t. its momentum. \n\n\n\\section{Unitarity of the Prescription}\n\\label{SEC:unitarity}\n\nThe potential RS dependence of amplitudes is intimately connected to the\nstructure of their UV and IR singularities. Fortunately, in QCD they obey a \nfactorized form at the amplitude level~\\cite{Sen:1982bt,Collins:1989bt,Catani:1998bh,Sterman:2002qn,Dixon:2008gr,Gardi:2009qi,Gardi:2009zv,Becher:2009cu,Becher:2009kw,Becher:2009qa,Feige:2014wja}. \nThe final result for a physical quantity, for instance a cross section, \nis of course finite and must not depend on the RS used. \n\n\nThe usage of the polarization projectors defined in the previous sections \nyields helicity amplitudes that differ in general from those defined in many existing dimensional regularization \nvariants, in particular the CDR. In this section, we argue that our prescription \nof external state vectors will however lead to the same RS-independent finite remainders as for instance\nin CDR, and can therefore be used in a hybrid way with CDR to achieve a maximal convenience \nowing to the amplitude-level factorization of UV and IR singularities in QCD amplitudes. \n\n\\subsection{Pole subtracted amplitudes}\n\\label{SEC:unitarity:PSA}\n\n\nWe recall that in the D-dimensional Lorentz decomposition representation \nof a scattering amplitude, the Lorentz-invariant form factors encode all dependence \non dimensionally regularized loop integrals and are independent of the external polarization vectors.\nOnce the (renormalized) loop amplitudes are available in such a tensor decomposed form, \nwith all (singular) Lorentz-invariant form factors computed in D dimensions, \nthen merely changing the RS for the external particles' state vectors, consistently both for the loop amplitudes \nand the corresponding IR subtraction terms, should not alter the finite remainders resulting from \nsubtracting all poles and subsequently taking the 4-dimensional limit\\footnote{The equivalence between CDR and HV \nin leading to the same RS-independent finite remainders with the identical set of renormalization constants \nand anomalous dimensions~\\cite{Kilgore:2012tb,Broggio:2015dga} can be appreciated this way, \nand the same arguments apply here as well.}.  \nBecause in the form-factor representation of an amplitude the loop-integral dependent part \nis separated from the part depending on the external states, it is thus unambiguous to implement \nwhatever non-CDR convention for external state vectors in the computation of singular amplitudes.\nThe crucial question for our purpose is \nwhether our non-CDR prescription for external state vectors can still be unambiguously and \ndirectly applied in the computation of amplitudes without performing the form factor decomposition first. \n\n\nIn our prescription all open Lorentz indices of the polarization projectors \ndefined in section~\\ref{SEC:Prescription} are set to be D-dimensional and no dimensional \nsplitting is ever introduced, just like in CDR. Thus, commutation between Lorentz \nindex contraction and loop integration is preserved within our prescription. \nThis means that applying our polarization projectors directly to the original \nFeynman-diagrammatic representation of a loop amplitude should lead to the same \npolarized amplitudes as would be obtained by applying these projectors \nto the D-dimensional form-factor decomposition representation of that amplitude.\nNo matter whether or not  evanescent Lorentz structures appear explicitly \nor implicitly in the form-factor decomposition of the  loop amplitude, \nthey are taken into account exactly as they are in the original Feynman-diagrammatic \nrepresentation of this amplitude.\nFrom this perspective we could already expect to end up with the same \n(4-dimensional) finite remainder as one would obtain from a computation purely within CDR.~\\\\\n\n\nBelow we demonstrate this crucial point more clearly via providing an alternative formulation of \nfinite remainders introduced in the proposed prescription, which also helps to clarify a few points \nalluded in the preceding section.\nLet us consider the finite remainders of amplitudes in CDR as defined by the celebrated amplitude-level \nfactorization formula. \nSingularities in the dimensionally regularized QCD amplitudes are known to \nfactorize~\\cite{Sen:1982bt,Collins:1989bt,Catani:1998bh,Sterman:2002qn,Dixon:2008gr,Gardi:2009qi,Gardi:2009zv,Becher:2009cu,Becher:2009kw,Becher:2009qa,Feige:2014wja}.\nFor our purpose, we can sketch this factorization property \nof a bare QCD scattering amplitude $\\hat{\\mathcal{A}}(\\epsilon)$ among several resolved \nexternal particles (with fixed external kinematics) schematically as follows:\n\\begin{eqnarray}\\label{EQ:AmpPoleFactorization}\n\\hat{\\mathcal{A}}(\\epsilon) = \\hat{\\mathcal{Z}}_{\\mathrm{IR}}(\\epsilon)~ \n\\mathcal{Z}_{\\mathrm{UV}}(\\epsilon)~ \\hat{\\mathcal{F}}(\\epsilon) \\, ,\n\\end{eqnarray}\nwhere\\footnote{The need of mass renormalizations in the case of massive quarks is understood.}\nwe have suppressed the dependence of the quantities on external kinematics and masses\nas well as on auxiliary dimensional scales except the dimensional regulator $\\epsilon$, \n(for a detailed exposition, see e.g.~\\cite{Catani:1998bh,Feige:2014wja,Broggio:2015dga,Magnea:2018ebr} \nand references therein).\nThe bare amplitude $\\hat{\\mathcal{A}}(\\epsilon)$ and the finite pole-subtracted amplitude \n$\\hat{\\mathcal{F}}(\\epsilon)$ should be viewed as vectors in the color space of the external particles, \nand the multiplicative singular IR-factor $\\hat{\\mathcal{Z}}_{\\mathrm{IR}}(\\epsilon)$ as a matrix. \nThe RS-dependent singular factors $\\mathcal{Z}_{\\mathrm{UV}}(\\epsilon)$ and $\\hat{\\mathcal{Z}}_{\\mathrm{IR}}(\\epsilon)$ \nencode all UV and IR pole-singularities of $\\hat{\\mathcal{A}}(\\epsilon)$, and are independent \nof the detailed kinematic configuration, such as polarization states, of the external resolved particles. \n(This is the meaning of ``factorization''.)\nBy the very meaning of pole factorization in eq.~(\\ref{EQ:AmpPoleFactorization}), \n$\\hat{\\mathcal{F}}(\\epsilon)$ is regular in $\\epsilon$ and has a finite 4-dimensional limit, \n$\\hat{\\mathcal{F}}(\\epsilon = 0)$. \nWe call this quantity the (4-dimensional) \\textit{finite remainder} of $\\hat{\\mathcal{A}}(\\epsilon)$ \ndefined by subtracting all poles minimally by the multiplicative factors \nas sketched in eq.~(\\ref{EQ:AmpPoleFactorization}).  \n\n\nWe may summarize this by the following expression for the finite remainder \n$\\hat{\\mathcal{F}}_{4} \\equiv \\hat{\\mathcal{F}}(\\epsilon = 0)$, namely \n\\begin{eqnarray}\\label{EQ:AmpsFiniteRemainder}\n\\hat{\\mathcal{F}}_{4} = \n\\Bigg(\n\\hat{\\mathcal{Z}}^{-1}_{\\mathrm{IR};\\mathrm{CDR}}(\\epsilon)~ \n\\mathcal{Z}^{-1}_{\\mathrm{UV};\\mathrm{CDR}}(\\epsilon)~\\hat{\\mathcal{A}}_{\\mathrm{CDR}}(\\epsilon) \n\\Bigg)_{\\epsilon = 0}\n\\end{eqnarray}\nwhere we added the subscript ``CDR'' to all singular RS-dependent quantities given in CDR.  \nFor the point to be demonstrated here, the concrete expressions of these singular multiplicative factors \ntaken from CDR are irrelevant. \nThe claim is that replacing all CDR-regularized external states of the fixed-angle bare scattering\namplitude $\\hat{\\mathcal{A}}_{\\mathrm{CDR}}(\\epsilon)$  by their respective counterparts \ngiven in terms of momentum basis representations defined in section~\\ref{SEC:Prescription} \nwill still result in the same finite remainder $\\hat{\\mathcal{F}}_{4}$, \nwhere all poles have been subtracted in a minimal way by the same untouched \n$\\hat{\\mathcal{Z}}^{-1}_{\\mathrm{IR};\\mathrm{CDR}}(\\epsilon) ~ \\mathcal{Z}^{-1}_{\\mathrm{UV};\\mathrm{CDR}}(\\epsilon)$, \nwithout appealing to the Lorentz tensor decomposition representation of $\\hat{\\mathcal{A}}_{\\mathrm{CDR}}(\\epsilon)$. \n\n\n\nIn order to facilitate the discussion, let us exhibit the dependence of \n$\\hat{\\mathcal{A}}_{\\mathrm{CDR}}(\\epsilon)$ on the CDR-regularized polarization state \n$\\bar{\\varepsilon}_{\\bar{\\lambda}}(p_i,r_i)$ of a \\textit{representative} external \nmassless gauge boson with momentum $p_i$ and reference vector $r_i$. \nBecause the bare scattering amplitude $\\hat{\\mathcal{A}}_{\\mathrm{CDR}}$ is linear in \n$\\bar{\\varepsilon}_{\\bar{\\lambda}}(p_i,r_i)$,  \nwe write \n\\begin{equation} \\label{eq:ACDReps}\n\\hat{\\mathcal{A}}_{\\mathrm{CDR}} \n\\Big(\\epsilon; \\bar{\\varepsilon}_{\\bar{\\lambda}}(p_i,r_i) \\Big)\n= g_{\\mu \\nu}~\n\\Big(\\hat{\\mathcal{A}}_{\\mathrm{CDR}}\\Big)^{\\mu} ~  \n\\bar{\\varepsilon}^{~\\nu}_{\\bar{\\lambda}}(p_i,r_i) \\, .\n\\end{equation}\nwhere we have introduced a compact notation $\\Big(\\hat{\\mathcal{A}}_{\\mathrm{CDR}}\\Big)^{\\mu}$.\nFor the pole-subracted amplitude we have \n\\begin{eqnarray}\\label{EQ:AmpsFiniteRemainderCDR}\n\\hat{\\mathcal{F}}_{\\mathrm{CDR}}\\Big( \\epsilon; \\bar{\\varepsilon}_{\\bar{\\lambda}}(p_i,r_i) \\Big) &\\equiv& \n\\hat{\\mathcal{Z}}^{-1}_{\\mathrm{IR};\\mathrm{CDR}}\\left(\\epsilon\\right)~ \n\\mathcal{Z}^{-1}_{\\mathrm{UV};\\mathrm{CDR}}\\left(\\epsilon\\right)~\n\\hat{\\mathcal{A}}_{\\mathrm{CDR}}\n\\Big(\\epsilon; \\bar{\\varepsilon}_{\\bar{\\lambda}}(p_i,r_i) \\Big) \\nonumber\\\\\n&=& \\hat{\\mathcal{F}}_{4}\\Big(\\varepsilon_{\\lambda}(p_i,r_i) \\Big) + \\mathcal{O}(\\epsilon) \\, ,\n\\end{eqnarray} \nwhose limit at $\\epsilon = 0$ is precisely the finite remainder $\\hat{\\mathcal{F}}_{4}$ \nin eq.~(\\ref{EQ:AmpsFiniteRemainder}) with 4-dimensional external polarization  vector\n$\\varepsilon_{\\lambda}(p_i,r_i)$. \nNow we multiply this regular finite quantity by a generalized D-dependent \nLorentz-invariant norm-orthogonal factor $\\Delta_{\\bar{\\lambda} \\lambda}$ defined by\n\\begin{eqnarray}\\label{EQ:generalizedDelta}\n\\Delta_{\\bar{\\lambda} \\lambda} \n&\\equiv& - \\bar{\\varepsilon}^{~*}_{\\bar{\\lambda}}(p_i,r_i) \\cdot \n\\varepsilon^{\\text{MBR}}_{\\lambda}(p_i,r_i) \\nonumber\\\\\n&=& \\delta_{\\bar{\\lambda} \\lambda} + \\mathcal{O}(\\epsilon). \n\\end{eqnarray}\nHere $\\varepsilon^{\\text{MBR}}_{\\lambda}$ refers to a polarization vector for a massless\ngauge boson of our prescription\\footnote{The acronym ``MBR'' denotes\nmometum basis representation.} of section~\\ref{SEC:Prescription}, and the dot product in  \n\\eqref{EQ:generalizedDelta} refers to the D-dimensional Minkowski scalar product. \nWe recall that  the polarization index $\\bar{\\lambda}$ labels the $D-2$ polarization\nstates of CDR while in our prescription the index $\\lambda$ of $\\varepsilon^{\\text{MBR}}_{\\lambda}$\ntakes only two values for massless gauge bosons (and three for massive ones), \nwhich are $\\pm$ in helicity basis for polarizations.   \nThe 4-dimensional limits of these simple Lorentz-invariant contractions \n$\\Delta_{\\bar{\\lambda} \\lambda}$ are the norm-orthogonal factors \n(i.e.~the Kronecker deltas) among different 4-dimensional physical \npolarization\/helicity states. \n\n\nNext we consider the sum of products \n\\begin{equation} \\label{EQ:AmpsFiniteRemainderMBRstartpoint}\n\\sum_{\\bar{\\lambda} = \\pm,~ D-4}\n\\hat{\\mathcal{F}}_{\\mathrm{CDR}}\\Big( \\epsilon; \\bar{\\varepsilon}_{\\bar{\\lambda}}(p_i,r_i) \\Big)\n~\\Delta_{\\bar{\\lambda} \\lambda}.\n\\end{equation} \nAs exhibited in eqs.~(\\ref{EQ:AmpsFiniteRemainderCDR}) and \\eqref{EQ:generalizedDelta}, both \n$\\hat{\\mathcal{F}}_{\\mathrm{CDR}}\\Big( \\epsilon; \\bar{\\varepsilon}_{\\bar{\\lambda}}(p_i,r_i) \\Big)$ \nand $\\Delta_{\\bar{\\lambda} \\lambda}$ are regular in $\\epsilon$. \nThus they can be power expanded in $\\epsilon$, and their 4-dimensional limits can be taken separately before being \nmultiplied together and subsequently summed over polarizations. \nProceeding in this way, we first insert the $\\epsilon$-expanded expressions of these two factors given\nabove, and the resulting quantity is precisely the finite remainder  \n$\\hat{\\mathcal{F}}_{4}\\Big(\\varepsilon_{\\lambda}(p_i,r_i) \\Big)$ of eq.~(\\ref{EQ:AmpsFiniteRemainder}):\n\\begin{equation} \\label{EQ:AmpsFiniteRemainderMBRleft}\n\\sum_{\\bar{\\lambda} = \\pm,~ D-4} \n\\hat{\\mathcal{F}}_{\\mathrm{CDR}}\\Big( \\epsilon; \\bar{\\varepsilon}_{\\bar{\\lambda}}(p_i,r_i) \\Big)\n~\\Delta_{\\bar{\\lambda} \\lambda}\n= \\hat{\\mathcal{F}}_{4}\\Big(\\varepsilon_{\\lambda}(p_i,r_i) \\Big) + \\mathcal{O}(\\epsilon). \n\\end{equation}\n\n\nOn the other hand, we can first perform the polarization sum in \\eqref{EQ:AmpsFiniteRemainderMBRstartpoint}\nin D dimensions and take the 4-dimensional limit afterwards. \nProceeding this way, we have \n\\begin{eqnarray}\\label{EQ:AmpsFiniteRemainderMBRright1}\n&&\\sum_{\\bar{\\lambda} = \\pm,~ D-4}\n\\hat{\\mathcal{F}}_{\\mathrm{CDR}}\\Big( \\epsilon; \\bar{\\varepsilon}_{\\bar{\\lambda}}(p_i,r_i) \\Big)\n~\\Delta_{\\bar{\\lambda} \\lambda}  \\nonumber\\\\\n&=& \n - \\sum_{\\bar{\\lambda} = \\pm,~ D-4} \n\\hat{\\mathcal{Z}}^{-1}_{\\mathrm{IR};\\mathrm{CDR}}\\left(\\epsilon\\right)~ \n\\mathcal{Z}^{-1}_{\\mathrm{UV};\\mathrm{CDR}}\\left(\\epsilon\\right)~\n\\hat{\\mathcal{A}}_{\\mathrm{CDR}} \n\\Big(\\epsilon;~ \\bar{\\varepsilon}_{\\bar{\\lambda}}(p_i,r_i) \\Big)~ \n\\bar{\\varepsilon}^{~*}_{\\bar{\\lambda}}(p_i,r_i) \\cdot \n\\varepsilon^{\\text{MBR}}_{\\lambda}(p_i,r_i) \\nonumber\\\\\n&=& \n- \\hat{\\mathcal{Z}}^{-1}_{\\mathrm{IR};\\mathrm{CDR}}\\left(\\epsilon\\right)~ \n\\mathcal{Z}^{-1}_{\\mathrm{UV};\\mathrm{CDR}}\\left(\\epsilon\\right)~\n\\sum_{\\bar{\\lambda} = \\pm,~ D-4} \n\\Big(\\hat{\\mathcal{A}}_{\\mathrm{CDR}}\\Big)_{\\mu} ~ \n\\bar{\\varepsilon}_{\\bar{\\lambda}}^{~\\mu}(p_i,r_i) ~\n\\bar{\\varepsilon}^{~*\\nu}_{\\bar{\\lambda}}(p_i,r_i) ~\n\\varepsilon^{\\text{MBR}}_{\\lambda,~\\nu} (p_i,r_i) \\, ,\\nonumber\\\\\n\\end{eqnarray} \nwhere we have used the fact that $\\hat{\\mathcal{A}}_{\\mathrm{CDR}} \n\\Big(\\epsilon;~ \\bar{\\varepsilon}_{\\bar{\\lambda}}(p_i,r_i) \\Big)$ is linear \nin the external polarization vector $\\bar{\\varepsilon}_{\\bar{\\lambda}}(p_i,r_i)$.\nNow we employ  eq.~(\\ref{EQ:polsumCDRPhys}) for summing over the physical polarizations\nof the CDR-regularized external gauge boson\\footnote{Note that here we should sum over \nphysical polarizations only, especially in the case of gluons, which ensures that unphysical components \nsuch as scalar and longitudinal polarizations are absent from the outset. \nWith this choice there is no need to incorporate diagrams involving ghost fields \nin the external states (when there are multiple external non-Abelian gauge bosons).}\nand obtain\n\\begin{eqnarray}\\label{EQ:AmpsFiniteRemainderMBRright2}\n&& -\\sum_{\\bar{\\lambda} = \\pm,~ D-4} \n\\Big(\\hat{\\mathcal{A}}_{\\mathrm{CDR}}\\Big)_{\\mu} ~ \n\\bar{\\varepsilon}_{\\bar{\\lambda}}^{~\\mu}(p_i,r_i) ~\n\\bar{\\varepsilon}^{~*\\nu}_{\\bar{\\lambda}}(p_i,r_i) ~\n\\varepsilon^{\\text{MBR}}_{\\lambda,~\\nu} (p_i,r_i) \\nonumber\\\\\n&=& \n\\Big(\\hat{\\mathcal{A}}_{\\mathrm{CDR}}\\Big)_{\\mu} \n~\\Bigg(\ng^{\\mu\\nu} - \\frac{p_i^{\\mu} r_i^{\\nu} + r_i^{\\mu} p_i^{\\nu}}{p_i \\cdot r_i}\n\\Bigg)~ \n\\varepsilon^{\\text{MBR}}_{\\lambda,~\\nu} (p_i,r_i) \\nonumber\\\\\n&=& \n\\Big(\\hat{\\mathcal{A}}_{\\mathrm{CDR}}\\Big)_{\\mu} \n~\\Bigg(\ng^{\\mu\\nu} \n\\Bigg)~ \n\\varepsilon^{\\text{MBR}}_{\\lambda,~\\nu} (p_i,r_i) \\nonumber\\\\\n&=&  \\hat{\\mathcal{A}}_{\\mathrm{CDR}} \n\\Big(\\epsilon;~ \\varepsilon^{\\text{MBR}}_{\\lambda}(p_i,r_i)\\Big)\n\\end{eqnarray} \nwhere we have used the orthogonality of $\\varepsilon^{\\text{MBR}}_{\\lambda}(p_i,r_i)$ \nw.r.t. the particle's momentum $p_i$ and its reference vector $r_i$ in D dimensions, \nwhich $\\varepsilon^{\\text{MBR}}_{\\lambda}(p_i,r_i)$  has to satisfy by construction.\nInserting  eq.~(\\ref{EQ:AmpsFiniteRemainderMBRright2}) back into \neq.~(\\ref{EQ:AmpsFiniteRemainderMBRright1}) we end up with \n\\begin{eqnarray}\\label{EQ:AmpsFiniteRemainderMBRright3}\n\\sum_{\\bar{\\lambda} = \\pm,~ D-4}\n\\hat{\\mathcal{F}}_{\\mathrm{CDR}}\\Big( \\epsilon;~ \\bar{\\varepsilon}_{\\bar{\\lambda}}(p_i,r_i) \\Big)\n~\\Delta_{\\bar{\\lambda} \\lambda} \n&=& \n\\hat{\\mathcal{F}}_{\\mathrm{CDR}}\\Big( \\epsilon;~ \\varepsilon^{\\text{MBR}}_{\\lambda}(p_i,r_i) \\Big) \\, ,\n\\end{eqnarray}  \nwhose left-hand side has, according to  eq.~(\\ref{EQ:AmpsFiniteRemainderMBRleft}), \na 4-dimensional limit that is equal to the  finite remainder \n$\\hat{\\mathcal{F}}_{4}\\Big(\\varepsilon_{\\lambda}(p_i,r_i) \\Big)$ given in eq.~(\\ref{EQ:AmpsFiniteRemainder}).\nNotice that eq.~(\\ref{EQ:AmpsFiniteRemainderMBRright3}) is an identity holding to all orders in $\\epsilon$.\nThe right-hand side of \\eqref{EQ:AmpsFiniteRemainderMBRright3}, more explicitly, \n\\begin{equation} \\label{eq:rhsFcdr}\n\\hat{\\mathcal{F}}_{\\mathrm{CDR}}\\Big( \\epsilon;~ \\varepsilon^{\\text{MBR}}_{\\lambda}(p_i,r_i) \\Big) \n= \\hat{\\mathcal{Z}}^{-1}_{\\mathrm{IR};\\mathrm{CDR}}\\left(\\epsilon\\right)~ \n\\mathcal{Z}^{-1}_{\\mathrm{UV};\\mathrm{CDR}}\\left(\\epsilon\\right)~\n\\hat{\\mathcal{A}}_{\\mathrm{CDR}} \n\\Big(\\epsilon;~ \\varepsilon^{\\text{MBR}}_{\\lambda}(p_i,r_i)\\Big)\n\\end{equation}\nis exactly the quantity suggested by our prescription.\nIn order to avoid confusion we emphasize that the subscript ``CDR'' on $\\hat{\\mathcal{F}}_{\\mathrm{CDR}}$\nat the right-hand side of \\eqref{EQ:AmpsFiniteRemainderMBRright3}, and on $\\hat{\\mathcal{F}}_{\\mathrm{CDR}}$\nand $\\hat{\\mathcal{A}}_{\\mathrm{CDR}}$ in eq.~\\eqref{eq:rhsFcdr} means that these are the respective \nCDR expressions with the exception that the CDR polarization vector of the external gluon with momentum $p_i$\nis replaced by the polarization vector of our MBR prescription. If there are more gluons in the external\nstate then the procedure outlined by eqs.~\\eqref{eq:ACDReps} - \\eqref{eq:rhsFcdr} can be iterated.\n\n\nWhat the above reformulations show is that, to all orders in $\\epsilon$, the \n$\\hat{\\mathcal{F}}_{\\mathrm{CDR}}\\Big(\\epsilon;~\\varepsilon^{\\text{MBR}}_{\\lambda}(p_i,r_i) \\Big)$ \ncan be formally viewed as an unpolarized interference between $\\hat{\\mathcal{F}}_{\\mathrm{CDR}}\\Big( \\epsilon;~ \\bar{\\varepsilon}_{\\bar{\\lambda}}(p_i,r_i) \\Big)$ \nand the Lorentz-invariant generalized norm-orthogonal factor defined in eq.~(\\ref{EQ:generalizedDelta}), \nusing physical polarization sum rules for all CDR external states. \nThe unpolarized Landau density matrices of external gauge bosons reduce into the unique spacetime metric tensor \nby the virtue of the built-in orthogonality between $\\varepsilon^{\\text{MBR}}_{\\lambda}(p_i,r_i)$ and $p_i,r_i$.~\\\\\n\n\n\nAn analogous reformulation can be made for external fermions in the scattering amplitude.\nIn fact, for each open fermion line such a reformulation is more straightforward than \nin the above gauge boson case, because there is no redundancy in the spinor representation of the Lorentz algebra \nand the number of the polarization\/helicity states of a fermion is two both in CDR and in our prescription.\nThe unpolarized Landau density matrix of an external fermion is the well-known projection operator  \nonto the space of on-shell Dirac-spinors. \nAfter performing a similar reformulation of an open fermion line in the scattering amplitude, \ndenoted by $\\langle \\psi_A|\\hat{\\mathrm{M}}|\\psi_B \\rangle$ as in eq.~(\\ref{EQ:SFLtrace1}),\nwe end up with the following replacement:\n\\begin{eqnarray} \\label{EQ:fermionlinerewritten}\n\\langle \\psi^{\\mathrm{CDR}}_A|\\hat{\\mathrm{M}}|\\psi^{\\mathrm{CDR}}_B \\rangle \n~\\longrightarrow~ \n\\mathrm{Tr} \\Big{[} \n\\frac{\\hat{P}_{on}\\left(p_B,m_B\\right)}{2 f_B~ m_B}\n\\Big(\n~|\\psi^{\\mathrm{MBR}}_B \\rangle \\langle \\psi^{\\mathrm{MBR}}_A|~\n\\Big)\n\\frac{\\hat{P}_{on}\\left(p_A,m_A\\right)}{2 f_A~ m_A}\n~\\hat{\\mathrm{M}}\n\\Big{]} \\nonumber\\\\\n\\end{eqnarray}\nwhere $\\hat{P}_{on}(p,m)=(\\slashed{p} \\pm m)$ denotes the aforementioned on-shell projection operator \nfor a $u$- respectively $v$-type Dirac spinor with momentum $p$ and mass $m$,\nand $|\\psi^{\\mathrm{MBR}}_B \\rangle \\langle \\psi^{\\mathrm{MBR}}_A|$\nis exactly the matrix  \\eqref{EQ:TPextsps1} that was further discussed in \neqs.~\\eqref{EQ:LDMofDSP} - (\\ref{EQ:TPextsps2}).\nThe appearance of ${1}\/{(2 f_A~ m_A)}$ and ${1}\/{(2 f_B~ m_B)}$ in eq.~(\\ref{EQ:fermionlinerewritten})\nis due to the conventional choice of normalization factors of on-shell Dirac spinors.\nHere the factors $f_A, f_B = 1 ~(-1)$ when the fermion $A$ respectively\n$B$ is associated with a $u$-type ($v$-type) spinor.\n\n\nQuantities that are sandwiched between the pair of on-shell projection operators, \n$\\hat{P}_{on}(p_A,m_A)$ and $\\hat{P}_{on}(p_B,m_B)$, associated with the two external spinors of \nthe open fermion line, can be manipulated and simplified according to the 4-dimensional \nLorentz\/Dirac-algebra. We just have to agree on one definite form that will be\ntaken as its canonical form (out of all its forms that are  equivalent in 4 dimensions)\nand used unambiguously in D-dimensional algebraic computations. \nThis pair of on-shell projection operators sets the domain where matrices related to \nexternal fermions' states, namely $|\\psi^{\\mathrm{MBR}}_B \\rangle \\langle \\psi^{\\mathrm{MBR}}_A|$,\ncan be manipulated and moved around using just 4 dimensional Lorentz\/Dirac-algebra.\nWhile, in general, moving any of these matrices beyond this range must be done in accordance with the \nD-dimensional Lorentz\/Dirac-algebra in order to not introduce artificial terms by mistake. \nFor instance, the object $\\gamma_{\\mu} \\gamma_{\\nu} \\gamma_{\\rho} S_{\\sigma} \\epsilon^{\\mu\\nu\\rho\\sigma}$ \ncommutes with $\\slashed{P}$ in 4 dimensions because of the orthogonality condition\n$S \\cdot P = 0$. However, this is no longer true w.r.t. the D-dimensional algebra, \nand there is thus a non-vanishing evanescent commutator resulting from \ninterchanging the product order between the two.\nIn section~\\ref{SEC:examples:eeQQ} we will briefly comment on this subtle point again. \n\n\nFinally, in order to bring the external projector in eq.~(\\ref{EQ:fermionlinerewritten}) into a form analogous to \neq.~(\\ref{EQ:SFLtrace1}) with the tensor product of external spinors given by eq.~(\\ref{EQ:TPextsps2}),\nthe following defining property of the on-shell projection operators, valid for $p^2=m^2$ in D dimensions, \ncan be used:\n\\begin{eqnarray}\\label{EQ:onshellprojectorID}\n\\hat{P}_{on}(p,m) \\frac{\\hat{P}_{on}(p,m)}{2 f~ m} = \\hat{P}_{on}(p,m) \\, ,\n\\end{eqnarray}\nwhere $f =\\pm 1$ depending on whether  ${\\hat P}_{on}$ is associated with a $u$-type or $v$-type spinor.\nNotice also that such an identity has a continuous limit at $m \\rightarrow 0$, \ndespite the superficial appearance of the singular $\\frac{1}{m}$ factor \nwhich does prevent setting $m=0$ directly in eq.(\\ref{EQ:onshellprojectorID}).\nSuch an alternative perspective thus helps to explain the choice made  \nin eq.~\\eqref{EQ:TPextsps2} where the polarization projection operators \nwere placed inside the on-shell projection operators.~\\\\\n\n\n\nWe thus achieved what we aimed at in this subsection.\nWe found an alternative formulation of pole-subtracted finite amplitudes  \nwhich helps to prove the following claim:\ndespite the fact that usage of the polarization projectors defined \nin section~\\ref{SEC:Prescription} results in helicity amplitudes different from those in CDR, \nreplacing all CDR-regularized external polarization states of \n$\\hat{\\mathcal{A}}_{\\mathrm{CDR}}(\\epsilon)$ in eq.~(\\ref{EQ:AmpPoleFactorization})\nby their counterparts given in terms of momentum basis representations \nconstructed in section~\\ref{SEC:Prescription} still results in the  same \nRS-independent finite remainder, where all poles are chosen to be subtracted \nby the same factorized (singular) coefficients given in CDR, without appealing \nto Lorentz tensor decomposition representations of $\\hat{\\mathcal{A}}_{\\mathrm{CDR}}(\\epsilon)$.\nThe validity of this statement is not confined \nto one-loop or next-to-leading order (NLO) corrections to a Born-level scattering \namplitude, but holds as long as the amplitude-level factorization \nformula sketched in eq.~(\\ref{EQ:AmpPoleFactorization}) holds in CDR.\n\n\n\\subsection{Finite remainders in an IR subtraction framework}\n\\label{SEC:unitarity:FRIR}\n\n\nIn this subsection, we move on and analyze finite remainders defined in an IR-subtraction method \nthat are obtained with our MBR prescription for external polarization vectors.\nWe will then show that this hybrid CDR-compatible prescription is unitary as defined in the sense of refs.~\\cite{vanDamme:1984ig,Catani:1996pk}.~\\\\ \n\n\nIn practice the finite RS-independent physical observables at NLO and beyond are usually computed \nas combinations of separate, in general UV and\/or IR divergent contributions living in different \npartonic phase spaces. (UV renormalization is understood in what follows.)\nTo render individual contributions from each partonic phase space IR-finite and RS-independent respectively, \none can add and subtract properly defined auxiliary IR-subtraction terms. \nThe introduction of these auxiliary terms are designed to ensure the cancellation of all intermediate \nIR-divergences of amplitudes in each partonic phase space, while on the other hand  \nthey leave no trace in the final properly combined physical observables. \nThis is the idea of IR-subtraction methods~\\cite{Ellis:1980wv,Kunszt:1992tn}, \nwhich are nowadays available in many different versions (e.g.,~\\cite{Frixione:1995ms,Catani:1996vz,Nagy:2003qn,Kosower:1997zr,GehrmannDeRidder:2005cm,Czakon:2010td,Czakon:2014oma,Caola:2017dug,Magnea:2018hab,Herzog:2018ily,Catani:2007vq,Somogyi:2006da}). \n\n\nLet us now sketch an IR-subtraction method by only being explicit about aspects that are relevant for \nshowing that our MBR prescription of external states is unitary. \n\n\nAssume that the Born-level scattering amplitude $\\mathcal{A}_n$ lives in a n-particle \nphase space, and we consider an IR-safe observable defined by the measurement function $F_J$. \nThe leading-order (LO) observable $\\sigma_{\\rm LO}$ is given by  \n\\begin{eqnarray}\\label{EQ:LOamp}\n\\sigma_{\\rm LO} & = & \\int_{d\\Phi_n} |\\mathcal{A}_n|^2 ~ \\mathit{F}_{\\mathit{J}}^{~(n)},\n\\end{eqnarray}\nwhere we suppressed all pre-factors related to spin averaging for the initial state \nand the incident flux.\nThe NLO QCD correction $\\sigma_{\\rm NLO}$ consists of real radiations $\\int_{d\\Phi_{n+1}} d\\sigma_{\\rm NLO}^{\\mathcal{R}}$\nin the (n+1)-particle phase space and the (renormalized) virtual corrections \n$\\int_{d\\Phi_{n}} d\\sigma_{\\rm NLO}^{\\mathcal{V}}$ in the n-particle phase space. \nTo render individual contributions in each of these two phase spaces finite, \none adds and subtracts an appropriate IR-subtraction term $d\\sigma^{\\mathcal{S}}$. \nSubsequently $\\sigma_{\\rm NLO}$ can then be rewritten in an IR subtraction method as follows\\footnote{\nFor the sake of simplicity, we suppressed here an initial-state collinear subtraction \nterm related to the (re)definition of parton-distribution functions, \nwhich does not add any additional conceptual complexity to what we want to show.}    \n\\begin{eqnarray}\\label{EQ:NLOamp}\n\\sigma_{\\rm NLO} & = & \\int_{d\\Phi_{n+1}} d\\sigma_{\\rm NLO}^{\\mathcal{R}} + \n                      \\int_{d\\Phi_{n}} d\\sigma_{\\rm NLO}^{\\mathcal{V}} \\nonumber\\\\ \n                  & = & \\int_{d\\Phi_{n+1}} |\\mathcal{A}_{n+1}^{\\mathcal{R}}|^2 ~ \n                        \\mathit{F}_{\\mathit{J}}^{~(n+1)} \n                      +  \\left(   \n                         \\int_{d\\Phi_{n+1}} d\\sigma^{\\mathcal{S}} ~\\mathit{F}_{\\mathit{J}}^{~(n)}\n                         - \n                         \\int_{d\\Phi_{n+1}} d\\sigma^{\\mathcal{S}} ~\\mathit{F}_{\\mathit{J}}^{~(n)}\n                           \\right) \n                       + \\int_{d\\Phi_{n}} 2\\mathrm{Re}\\left[\\mathcal{A}_{n}^{*} \\mathcal{A}_{n}^{\\mathcal{V}}\\right] ~ \n                         \\mathit{F}_{\\mathit{J}}^{~(n)} \\nonumber\\\\\n                   & = & \\int_{d\\Phi_{n+1}} \\Bigg[ \\left( \n                        |\\mathcal{A}_{n+1}^{\\mathcal{R}}|^2 ~ \n                        \\mathit{F}_{\\mathit{J}}^{~(n+1)}\\right)_{\\epsilon = 0} \n                         -  \n                       \\left(  d\\sigma^{\\mathcal{S}} ~ \\mathit{F}_{\\mathit{J}}^{~(n)}\n                       \\right)_{ \\epsilon = 0}  \\Bigg] \n                      + \\int_{d\\Phi_{n}}  \\left[ 2~ \\mathrm{Re}\\left[\\mathcal{A}_{n}^{*} \\mathcal{A}_{n}^{\\mathcal{V}}\\right]  \n                       + \\int_{1} d\\sigma^{\\mathcal{S}} \n                        \\right]_{ \\epsilon = 0} \\mathit{F}_{\\mathit{J}}^{~(n)}. \\nonumber\\\\\n\\end{eqnarray}\nBy construction, the subtraction term $d\\sigma^{\\mathcal{S}}$ should have the same local \nIR-singular behavior as the squared real-radiation matrix $|\\mathcal{A}_{n+1}^{\\mathcal{R}}|^2$ \neverywhere in the (n+1)-particle phase space (subject to the constraint implied by $\\mathit{F}_{\\mathit{J}}$). \nConsequently, the resulting subtracted phase-space integrand \n$\\left[ \\left(|\\mathcal{A}_{n+1}^{\\mathcal{R}}|^2 ~ \\mathit{F}_{\\mathit{J}}^{~(n+1)}\\right)_{\\epsilon = 0}  -  \n\\left(  d\\sigma^{\\mathcal{S}} ~\\mathit{F}_{\\mathit{J}}^{~(n)} \\right)_{ \\epsilon = 0}  \\right]$ \ncan be numerically evaluated and integrated over the phase space in 4 dimensions, \nas indicated by $\\epsilon = 0$.\nNotice that it is $\\mathit{F}_{\\mathit{J}}^{~(n)}$ that is associated with \n$d\\sigma^{\\mathcal{S}}$, the same as for virtual corrections living in n-particle phase space.\nThe integration of $d\\sigma^{\\mathcal{S}}$ over the unresolved phase space \nhas to be done in D dimensions with the IR unresolved partonic d.o.f. regularized in the same way as \nthose in the virtual correction $2~\\mathrm{Re}\\left[\\mathcal{A}_{n}^{*} \\mathcal{A}_{n}^{\\mathcal{V}}\\right]$,\nfollowing from the unitarity constraint. \nThe resulting IR singularities that appear as poles in $\\epsilon$ must cancel  \nthose appearing in $2~\\mathrm{Re}\\left[\\mathcal{A}_{n}^{*} \\mathcal{A}_{n}^{\\mathcal{V}}\\right]$, \nwhich renders the  quantity in the second square bracket of \nthe last line of eq.~(\\ref{EQ:NLOamp}) finite in 4 dimensions as well.\n\n\nIn order that eq.~(\\ref{EQ:NLOamp}) is useful in practice, one must be able to perform  \nthe D-dimensional integration $\\int_{1} d\\sigma^{\\mathcal{S}}$,\neither analytically or numerically. \nThanks to the IR factorization, $d\\sigma^{\\mathcal{S}}$ and likewise its integrated counterpart \n$\\int_{1} d\\sigma^{\\mathcal{S}}$ can be constructed, schematically,\nas a convoluted product of certain universal (process-independent) multiplicative coefficient \nand the (process-specific) squared Born amplitude  \n$|\\mathcal{A}_n|^2$:\n\\begin{eqnarray}\\label{EQ:IRsubtractionterm}\nd\\sigma^{\\mathcal{S}} &=& \\left( d \\hat{I}_{RS} \\right) \\otimes |\\mathcal{A}_n|^2~,\\nonumber\\\\\n\\int_{1} d\\sigma^{\\mathcal{S}} &=& \\hat{I}_{RS} \\otimes |\\mathcal{A}_n|^2.\n\\end{eqnarray}\nThe factor $\\hat{I}_{RS}$ plays a similar role as the multiplicative factors \n$\\hat{\\mathcal{Z}}_{\\mathrm{IR}}(\\epsilon)$ in eq.~(\\ref{EQ:AmpPoleFactorization}).\nAt NLO it encodes all IR pole-singularities and is to be viewed \nas an operator in the color space of the external particles.\n\n\nIn fact each variant of an IR-subtraction method can be seen as providing a concrete constructive\nprescription for the integral representations of factorized IR-subtraction coefficients, \nlike the factor  $\\hat{I}_{RS}$, that contain all the explicit pole-singularities of the \nloop amplitudes (after multiplication with certain relevant process-dependent hard-scattering amplitudes). \nThe crucial point relevant for the following discussion is that \nthese integral representations are based on the amplitude-level IR factorization, \nand are manifestly independent of the polarization states of external particles which \nappear in the (remaining) hard-scattering matrix elements.\\footnote{The dependence of factorized collinear \n$\\epsilon$-pole singularities on the polarization of a parent parton drops once one\nsums over the polarizations of all other particles and also integrates over all \nunresolved degrees of freedom in the collinear limit, notably the transverse plane of the radiated partons \n(which essentially eliminates any preference in the transverse direction).}\n\n\nAll quantities in eq.~(\\ref{EQ:NLOamp}) that contain explicit IR-divergences, i.e.\npoles in $\\epsilon$, contain RS-dependent pieces in their truncated \nLaurent series to order $\\epsilon^0$, especially the integrated $\\hat{I}_{RS}$. \nAt NLO, this concerns only $\\int_{d\\Phi_{n}} d\\sigma_{\\rm NLO}^{\\mathcal{V}}$ and \n$\\int_{1} d\\sigma^{\\mathcal{S}} = \\hat{I}_{RS} \\otimes |\\mathcal{A}_n|^2$\nthat live in the same n-particle phase space. \nBy appealing to an IR-subtraction method the unitarity constraint, originally imposed \nbetween the calculations of $\\int_{d\\Phi_{n+1}} d\\sigma_{\\rm NLO}^{\\mathcal{R}}$ and \n$\\int_{d\\Phi_{n}} d\\sigma_{\\rm NLO}^{\\mathcal{V}}$ is translated into the following ``locally distributed'' \nversion: \nwe just need to make sure that contributions associated with the same partonic phase space \nare computed consistently with a unitarity-respecting prescription, while \npole-subtracted 4-dimensional remainders living in different partonic phase spaces \ncan be computed independently of each other (using different methods).\nThus, as argued in ref.~\\cite{Catani:1996pk}, IR subtraction methods offer a \nconvenient way to isolate and investigate the RS-dependence of individual \nsingular pieces and subsequently ensure the unitarity of regularization prescriptions \nused in the calculation.~\\\\ \n\n\nWith the skeleton of an IR-subtraction framework ready, we can discuss \nhow each of the two square brackets in the last line of eq.~(\\ref{EQ:NLOamp}) should be evaluated \nwith our proposed prescription in order to ensure a correct NLO observable $\\sigma_{\\rm NLO}$.\n \n\nFirst, the subtraction of implicit IR-singularities in $d\\sigma_{\\rm NLO}^{\\mathcal{R}}$, \ni.e. terms in the first square bracket of the last line of eq.~(\\ref{EQ:NLOamp}), \nis to be done at the integrand level of phase-space integrals. This results in a subtracted \nreal-radiation contribution that is numerically integrable in 4 dimensions. \nIn the 4-dimensional limit ($\\epsilon = 0$) the external polarization states \ndefined by the momentum basis representations given in section~\\ref{SEC:Prescription}, \nall coincide with their respective standard 4-dimensional expressions. \nTherefore the RS-independence of the finite remainders of real-radiation contributions \nassociated with the 4-dimensional (n+1)-particle phase space is manifest \nas dimensional regularization can be avoided from the outset. \nThus we just have to make sure that in this hybrid prescription, \nthe integral-level subtraction of explicit $\\epsilon$-pole singularities \nin  $2~\\mathrm{Re}\\left[\\mathcal{A}_{n}^{*} \\mathcal{A}_{n}^{\\mathcal{V}}\\right]$, \ni.e. the second square bracket of the last line of eq.~(\\ref{EQ:NLOamp}),  \nis also done in a unitarity-respecting way so as to lead to the \ncorrect RS-independent finite remainder in the n-particle phase space.\n\n\nTo this end, we can proceed in two ways. \nWe could devise a proof analogous to the previous subsection, but now applied to the finite remainder \n$\\left[ 2\\mathrm{Re}\\left[\\mathcal{A}_{n}^{*} \\mathcal{A}_{n}^{\\mathcal{V}}\\right] + \\hat{I}_{RS} \\otimes |\\mathcal{A}_n|^2 \\right]_{\\epsilon = 0}$, \nwhere the integrated factor $\\hat{I}_{RS}$ plays a similar role\nas the perturbatively-expanded multiplicative factor $\\hat{\\mathcal{Z}}_{\\mathrm{IR}}(\\epsilon)$ in eq.~(\\ref{EQ:AmpPoleFactorization}). \nAlternatively, we argue in this subsection that the unitarization recipe of ref.~\\cite{Catani:1996pk} \nis indeed respected by our hybrid prescription. \nWe examine this now one by one. \n\\begin{enumerate}\n\\item \nThe external partons in the Born-level hard-scattering matrix element $\\mathcal{A}_n$ \nof the factorized IR-subtraction term $\\hat{I}_{RS} \\otimes |\\mathcal{A}_n|^2$ have to be \ntreated like the external partons in the virtual loop amplitude $\\mathcal{A}_{n}^{\\mathcal{V}}$ \n(of the same external kinematic configuration).\n\n\nThis is guaranteed by applying the same set of polarization projectors defined in \nsection~\\ref{SEC:Prescription} consistently to $\\mathcal{A}_n$ at LO and $\\mathcal{A}^{\\mathcal{V}}_n$ \nat NLO, computed respectively to the required powers in $\\epsilon$.\n\n\\item \nThe parent parton and its (soft and collinear) daughter partons involved in the \nintegral representation of the factorized process-independent (singular) coefficient function \n$\\hat{I}_{RS}$ have to be treated like the corresponding partons inside the\nloop integrals of $\\mathcal{A}_{n}^{\\mathcal{V}}$.\n\n\nThis is guaranteed by performing integrals involving IR-unresolved d.o.f.\nconsistently regularized with CDR. In particular, the phase-space integrals \nin $\\hat{I}_{RS}$ are done in D dimensions like D-dimensional loop integrals \nsubject to Cutkosky cuts. \n\\end{enumerate}\n\nConcerning the first point, as long as there is an unambiguous and consistent way of \ndirectly applying such a non-CDR regularization convention of external states \nin the computation of the virtual loop amplitude $\\mathcal{A}_{n}^{\\mathcal{V}}$ \n(without appealing to its Lorentz tensor decomposition representation), \nthen the demonstration is completed. \nSimilar as in section~\\ref{SEC:unitarity:PSA}, this point is guaranteed in our\nprojection prescription by the fact that all open Lorentz indices \nof the polarization projectors defined in section~\\ref{SEC:Prescription} are taken to be\na D-dimensional and no dimensional splitting is ever introduced, just like in CDR. \n\n\n\nThus we have argued that our hybrid prescription can be conveniently \nused in a NLO IR subtraction framework to correctly obtain all RS-independent \nfinite remainders needed for computing physical observables, \nwith the (process-independent) integrated IR-subtraction coefficients directly \ntaken from CDR. \nIn other words, we have argued that our hybrid CDR-compatible prescription is unitary.~\\\\\n\n\nAlthough beyond the scope of this article, it is possible, by analogy to the NLO case, \nto ensure unitarity of the prescription at NNLO and beyond, owing to the following generic features \nof an IR subtraction method (which the above NLO discussions essentially rely on).\n\\begin{itemize}\n\\item \nIn a typical IR subtraction framework, all \\textit{explicit} IR-singularities \nin loop amplitudes, manifested as poles in $\\epsilon$, are always subtracted \nby IR subtraction terms whose constructions are based on amplitude-level singularity factorization \nformulae, and the factorized IR-subtraction coefficients are independent of all \nexternal polarization states;\n\\item  \nAny potential \\textit{implicit} IR singularity of the ($\\epsilon$-pole-free) finite remainders \nwill always be further subtracted at the integrand level of phase-space integrals \nover the external kinematics, and will be directly evaluated in 4 dimensions without employing \ndimensional regularization.\n\\end{itemize}\nThus concerning the 4-dimensional integrand level subtractions of implicit IR-singularities\nin those finite remainders, their $\\epsilon$-suppressed terms are never needed \nbecause the phase-space integration over the external kinematics is done (numerically) in 4 dimensions.\nWe leave a detailed exposition of this at NNLO for a future publication.\n\n\n\n\\section{Examples at NLO QCD}\n\\label{SEC:examples}\n\nThe polarization projectors constructed in section~\\ref{SEC:Prescription} are independent of \nthe loop order of virtual amplitudes, regardless of possible evanescent Lorentz structures \nthat may be generated in D dimensions. To illustrate its usage without being overwhelmed by \nirrelevant complications, we consider two prototype examples of a NLO QCD virtual amplitude,\nthe 1-loop QCD corrections to $gg \\rightarrow gg$ and to $e^+ e^- \\rightarrow Q \\bar{Q}$,  \nin order to show that RS-independent finite remainders are indeed obtained as\n was discussed in the preceding sections. We will comment along the way points worthy of attention. \n\n\n\\subsection{$gg \\rightarrow gg$}\n\\label{SEC:examples:gggg}\n\nBecause singular amplitudes are RS-dependent, it is only meaningful for our purpose\nto compare properly defined finite remainders between computations done \nusing different regularization schemes. \nAs discussed in section~\\ref{SEC:unitarity:FRIR}, the IR-subtracted \nreal-radiation contribution at NLO is obviously RS-independent. \nThus we just need to show  how the same finite remainders \nof the virtual corrections are obtained in computations using our\n hybrid prescription and in CDR.~\\\\\n\nWe consider the scattering process among 4 massless gluons:\n\\begin{equation} \\label{EQ:gggg}\ng_1(p_1)~+~ g_2(p_2) \\to g_3(p_3)~+~g_4(p_4),\n\\end{equation}\nin  QCD without fermions for simplicity. The Mandelstam variables are given in\n eq.~(\\ref{EQ:kinematicinvariants}).\n\n\nThe corresponding scattering amplitude perturbatively expanded up to NLO reads \n\\begin{eqnarray} \\label{EQ:ggggAMP}\n\\Big|\\mathcal{A}_{gggg} \\Big\\rangle = \\Big|\\mathcal{A}^{[\\text{tree}]}_{gggg} \\Big\\rangle + \\Big|\\mathcal{A}^{[\\text{1-loop}]}_{gggg} \\Big\\rangle  \n+ \\mathcal{O}(\\alpha^{3}_s) \\, , \n\\end{eqnarray}\nwhich is a vector in the color space of the external gluons. \nThe 1-loop virtual amplitudes were computed in refs.~\\cite{Bern:1991aq,Schubert:2001he,Binoth:2006hk}.\n For representing color structures of multi-gluon scattering amplitudes, \nlike eq.~(\\ref{EQ:ggggAMP}), it is most convenient to perform a color decomposition \nusing the choice of basis of refs.~\\cite{Berends:1987cv,Mangano:1987xk,Mangano:1987kp,Mangano:1988kk,Bern:1990ux}. \nIt is well known that tree-level QCD amplitudes with n external gluons can be decomposed\ninto color-ordered partial amplitudes, multiplied by associated single color traces \n(over all noncylic permutations of fundamental color generators).\nDecomposition of color structures of one-loop QCD amplitudes can be done in a similar\nway but with an extended color basis including products of two \ncolor traces\\footnote{This can be easily understood by combing the statement about tree-level color decomposition\nand the Fierz identities of SU(N) color algebra.}. \n\n\nFor the amplitude eq.~(\\ref{EQ:ggggAMP}) we generate symbolic expressions \nof all contributing Feynman diagrams using QGRAF~\\cite{Nogueira:1991ex}, \nand subsequently decompose them as  follows:\n\\begin{eqnarray}\\label{EQ:colordecomposition}\n\\Big|\\mathcal{A}^{[\\text{tree}]}_{gggg} \\Big\\rangle  &=& \\sum_{i=1}^{6} \\mathcal{A}^{[\\text{tree}]}_{gggg}\n(i) ~ |c_i \\rangle~,~ \\nonumber\\\\\n\\Big|\\mathcal{A}^{[\\text{1-loop}]}_{gggg} \\Big\\rangle &=& \\sum_{i=1}^{9} \\mathcal{A}^{[\\text{1-loop}]}_{gggg} (i) ~ |c_i \\rangle~,~\n\\end{eqnarray}\nusing the following basis of 9 color structures: \n\\begin{eqnarray}\\label{EQ:colorbasis}\n|c_1\\rangle &=& \\mathrm{Tr}\\Big[T_1~ T_2~ T_3~ T_4 \\Big]~,~~\n|c_2\\rangle = \\mathrm{Tr}\\Big[T_1~ T_2~ T_4~ T_3 \\Big] ~,~~\n|c_3\\rangle = \\mathrm{Tr}\\Big[T_1~ T_3~ T_4~ T_2 \\Big] \\nonumber\\\\\n|c_4\\rangle &=& \\mathrm{Tr}\\Big[T_1~ T_3~ T_2~ T_4 \\Big] ~,~~\n|c_5\\rangle = \\mathrm{Tr}\\Big[T_1~ T_4~ T_3~ T_2 \\Big] ~,~~\n|c_6\\rangle = \\mathrm{Tr}\\Big[T_1~ T_4~ T_2~ T_3 \\Big] \\nonumber\\\\\n|c_7\\rangle &=& \\mathrm{Tr}\\Big[T_1~ T_2\\Big] \\mathrm{Tr}\\Big[T_3~ T_4 \\Big]~,~\n|c_8\\rangle = \\mathrm{Tr}\\Big[T_1~ T_3\\Big] \\mathrm{Tr}\\Big[T_2~ T_4 \\Big] ~,~\n|c_9\\rangle = \\mathrm{Tr}\\Big[T_1~ T_4\\Big] \\mathrm{Tr}\\Big[T_2~ T_3 \\Big] ~,\\nonumber\\\\\n\\end{eqnarray}\nwhere the subscripts of the color generators label the associated gluons while  \ntheir explicit color indices are suppressed.\nThese 9 color structures are linearly independent, \nas can be checked by computing its Gram matrix. The amplitude\n$\\Big|\\mathcal{A}^{[\\text{tree}]}_{gggg} \\Big\\rangle$ involves \nonly the first 6 non-cylic single color traces given in eq.~\\eqref{EQ:colorbasis}, \nwhich can be further reduced to 4 structures by reflection symmetries. The color structures\n$|c_7\\rangle,~ |c_8\\rangle,~|c_9\\rangle$ are needed in addition to represent \n$\\Big|\\mathcal{A}^{[\\text{1-loop}]}_{gggg} \\Big\\rangle$. \n\n\n\nEach of the color decomposition coefficients $\\mathcal{A}^{[\\text{tree}]}_{gggg}\n(i),~\\mathcal{A}^{[\\text{1-loop}]}_{gggg} (i)$ is a function of external kinematics \nand polarization state vectors, to which we now apply the polarization \nprojectors prescribed in section~\\ref{SEC:Prescription}.\nWe extract polarized amplitudes in the linear polarization basis for all four external gluons (cf. section~\\ref{SEC:prescription:MBR:2to2}), \nfrom which helicity amplitudes can be easily obtained. \nBecause the reaction (\\ref{EQ:gggg}) is parity-invariant the scattering amplitude does not contain terms involving \n$\\gamma_5$ or an odd number of Levi-Civita tensors. \nWe thus need to consider only the following 8 linear polarization projectors, \nwhich are even in $\\varepsilon_Y$, respectively in the number of Levi-Civita tensors:\n\\begin{eqnarray}\\label{EQ:LPgggg8}\n&&\\varepsilon^{\\mu_1}_X \\varepsilon^{\\mu_2}_X \\varepsilon^{\\mu_3}_T \\varepsilon^{\\mu_4}_T ~,~ \n\\varepsilon^{\\mu_1}_X \\varepsilon^{\\mu_2}_X \\varepsilon^{\\mu_3}_Y \\varepsilon^{\\mu_4}_Y ~,~ \n\\varepsilon^{\\mu_1}_X \\varepsilon^{\\mu_2}_Y \\varepsilon^{\\mu_3}_T \\varepsilon^{\\mu_4}_Y ~,~ \n\\varepsilon^{\\mu_1}_X \\varepsilon^{\\mu_2}_Y \\varepsilon^{\\mu_3}_Y \\varepsilon^{\\mu_4}_T ~,~ \\nonumber\\\\\n&&\\varepsilon^{\\mu_1}_Y \\varepsilon^{\\mu_2}_X \\varepsilon^{\\mu_3}_T \\varepsilon^{\\mu_4}_Y ~,~\n\\varepsilon^{\\mu_1}_Y \\varepsilon^{\\mu_2}_X \\varepsilon^{\\mu_3}_Y \\varepsilon^{\\mu_4}_T ~,~\n\\varepsilon^{\\mu_1}_Y \\varepsilon^{\\mu_2}_Y \\varepsilon^{\\mu_3}_T \\varepsilon^{\\mu_4}_T ~,~\n\\varepsilon^{\\mu_1}_Y \\varepsilon^{\\mu_2}_Y \\varepsilon^{\\mu_3}_Y \\varepsilon^{\\mu_4}_Y.\n\\end{eqnarray}\nFor the sake of simplicity of notation, the arguments of these polarization vectors are suppressed \nwhile their subscripts at the open Lorentz indices indicate the associated gluons.\n\n\nThe number of linear polarization projectors in eq.~(\\ref{EQ:LPgggg8}) equals\nthe number of independent helicity amplitudes, taking into account the parity symmetry of \nthe scattering amplitude. We do not consider additional relations among the \nlinear polarized amplitudes arising from Bose symmetry, which involve kinematic crossings.\nThe set of 8 linear polarization projectors in eq.~(\\ref{EQ:LPgggg8}) are sufficient for \nany parity-even scattering amplitude among four external massless bosons to any loop order, \nirrespective of any possible (evanescent) Lorentz structures therein.\\footnote{In case a $2\\to 2$ amplitude \ninvolves parity-violating couplings, 8 linear polarization projectors containing an odd number of \n$\\varepsilon_Y$ (or Levi-Civita tensors) can be used in addition.} \n\n\nWe insert the expressions (\\ref{EQ:XpolMBR}), \\eqref{EQ:TpolMBR}, and \\eqref{EQ:YpolMBR} for the \npolarization vectors in (\\ref{EQ:LPgggg8}). \nAll the Lorentz algebra is carried out using FORM~\\cite{Vermaseren:2000nd}. \nLet us emphasize again that, in order to avoid possible ambiguities in the definition and application \nof these external projectors, all pairs of Levi-Civita tensors in eq.~(\\ref{EQ:LPgggg8})\nare replaced according to the contraction rule eq.~(\\ref{EQ:LeviCivitaContRule}) \nbefore being used in the projection. Then the projectors (\\ref{EQ:LPgggg8}) are expressed \nsolely in terms of external momenta and space-time metric tensors.\nAfter pulling out the normalization factors as prescribed in section~(\\ref{SEC:Prescription}), \nthe resulting tensor projectors (which have only a polynomial dependence on external momenta and kinematics) \nwill be applied to the color stripped amplitudes \n$\\mathcal{A}^{[\\text{tree}]}_{gggg}(i), ~\\mathcal{A}^{[\\text{1-loop}]}_{gggg} (i)$.  \nWe use the convention to set the variable  $D=4$ in the projectors (\\ref{EQ:LPgggg8}), \nin particular in the normalization factors that are pulled out. Of course this convention is used both \nfor the amplitudes and the associated UV and\/or IR subtraction terms.\nThen the normalization factors pulled out from the respective projectors \\eqref{EQ:LPgggg8} are \n\\begin{eqnarray}\\label{EQ:LPnormfactors}\n&&\\mathcal{N}_{XXTT} = \\frac{1}{s^2 t^2 (s + t)^2} ~,~\n\\mathcal{N}_{XXYY} = \\frac{4}{s^2 t^2 (s + t)^2} ~,~\n\\nonumber\\\\\n&&\n\\mathcal{N}_{XYTY} = \\frac{4}{s^2 t^2 (s + t)^2} ~,~\n\\mathcal{N}_{XYYT} = \\frac{4}{s^2 t^2 (s + t)^2} ~,~\n\\nonumber\\\\\n&&\n\\mathcal{N}_{YXTY} = \\frac{4}{s^2 t^2 (s + t)^2} ~,~\n\\mathcal{N}_{YXTX} = \\frac{4}{s^2 t^2 (s + t)^2} ~,~\n\\nonumber\\\\\n&&\n\\mathcal{N}_{YYTT} = \\frac{4}{s^2 t^2 (s + t)^2} ~,~\n\\mathcal{N}_{YYYY} = \\frac{16}{s^2 t^2 (s + t)^2} ~.\n\\end{eqnarray}\n\n\nThe linear polarized amplitudes projected out by applying eq.~(\\ref{EQ:LPgggg8}) \nto $\\mathcal{A}^{[\\text{1-loop}]}_{gggg}(i)$ contain both UV and IR singularities,\nmanifested as poles in $\\epsilon$. We are only interested in the finite remainders\ndefined by subtracting all these singularities in accordance with a certain convention.\nFor our purpose, there is no need to stick to \na specific IR-subtraction scheme. All we need to know is a factorization formula \nproviding us  with a set of terms that capture all singularities in \n$\\mathcal{A}^{[\\text{1-loop}]}_{gggg}(i)$ \n(with the process-independent singular coefficients obtained in CDR). \nTo be specific, we choose to define the finite remainders of the virtual amplitude\n$\\Big|\\mathcal{A}^{[\\text{1-loop}]}_{gggg} \\Big\\rangle$ by the following explicit  \non-shell UV and IR subtraction terms:\n\\begin{eqnarray}\\label{EQ:ggggUVIRcounterterms}\n\\Big| \\mathcal{A}^{[\\text{UV}]}_{gggg} \\Big\\rangle  \n&=& \\Big(-\\frac{11}{3} \\mathrm{N}_c \\frac{1}{\\epsilon}  + \\mathcal{O}(\\epsilon^0) \\Big)~\n\\Big|\\mathcal{A}^{[\\text{tree}]}_{gggg} \\Big\\rangle \\, , \\nonumber\\\\ \n\\Big|\\mathcal{A}^{[\\text{IR}]}_{gggg} \\Big\\rangle &=& 2\\mathrm{N}_c\n\\left(\\frac{-2}{\\epsilon^2} + \\frac{1}{\\epsilon}\\left(\\frac{-11}{3} + \\mathrm{log}\\left(\\frac{-s_{12}}{\\mu^2_{DR}}\\right) + \\mathrm{log}\\left(\\frac{+s_{23}}{\\mu^2_{DR}}\\right)\\right) + \\mathcal{O}(\\epsilon^0) \\right)\n2 \\mathcal{A}^{[\\text{tree}]}_{gggg}(1;\\epsilon) ~ \\Big|c_1 \\Big\\rangle \\, \\nonumber\\\\\n&+& 2\\mathrm{N}_c\n\\left(\\frac{-2}{\\epsilon^2} + \\frac{1}{\\epsilon}\\left(\\frac{-11}{3} + \\mathrm{log}\\left(\\frac{-s_{12}}{\\mu^2_{DR}}\\right) + \\mathrm{log}\\left(\\frac{+s_{24}}{\\mu^2_{DR}}\\right)\\right) + \\mathcal{O}(\\epsilon^0) \\right)\n2 \\mathcal{A}^{[\\text{tree}]}_{gggg}(2;\\epsilon) ~ \\Big|c_2 \\Big\\rangle \\, \\nonumber\\\\\n&+& 2\\mathrm{N}_c\n\\left(\\frac{-2}{\\epsilon^2} + \\frac{1}{\\epsilon}\\left(\\frac{-11}{3} + \\mathrm{log}\\left(\\frac{+s_{13}}{\\mu^2_{DR}}\\right) + \\mathrm{log}\\left(\\frac{+s_{23}}{\\mu^2_{DR}}\\right)\\right) + \\mathcal{O}(\\epsilon^0) \\right)\n \\mathcal{A}^{[\\text{tree}]}_{gggg}(4;\\epsilon) ~ \\Big|c_4 \\Big\\rangle \\, \\nonumber\\\\\n&+& 2\\mathrm{N}_c\n\\left(\\frac{-2}{\\epsilon^2} + \\frac{1}{\\epsilon}\\left(\\frac{-11}{3} + \\mathrm{log}\\left(\\frac{+s_{14}}{\\mu^2_{DR}}\\right) + \\mathrm{log}\\left(\\frac{+s_{24}}{\\mu^2_{DR}}\\right)\\right) + \\mathcal{O}(\\epsilon^0) \\right)\n \\mathcal{A}^{[\\text{tree}]}_{gggg}(6;\\epsilon) ~ \\Big|c_6 \\Big\\rangle \\, ,\\nonumber\\\\\n\\end{eqnarray}\nwhere $s_{ij} \\equiv 2 p_i \\cdot p_j$, and $\\mu_{DR}$ denotes the auxiliary mass parameter of dimensional  regularization. \nThe IR factorization coefficients listed in eq.~(\\ref{EQ:ggggUVIRcounterterms}) are\nextracted from the known singular part of $\\mathcal{A}^{[\\text{1-loop}]}_{gggg}(i)$ \ngiven in ref.~\\cite{Bern:1991aq}. The IR singular pieces of eq.~(\\ref{EQ:ggggUVIRcounterterms}) \nare the same for all IR subtraction methods (for the same renormalized virtual amplitude).\nIn our consideration the finite remainders of virtual amplitudes are defined by \nsubtracting all pole singularities of the loop amplitude~(\\ref{EQ:colordecomposition}) by means of \neq.~(\\ref{EQ:ggggUVIRcounterterms}).~\\\\\n\n\nWith these ingredients and prescriptions it is straightforward to get the analytic results for all \n8 non-vanishing finite remainders of the interferences between \n$\\Big|\\mathcal{A}^{[\\text{1-loop}]}_{gggg} \\Big\\rangle$ and $\\Big|\\mathcal{A}^{[\\text{tree}]}_{gggg} \\Big\\rangle$ \nin linear polarization basis.\\footnote{We used the library from the Package-X~\\cite{Patel:2015tea} for one-loop integrals.}\nThe finite remainder of unpolarized interferences in 4 dimensions is obtained by summing over these 8 quantities.\nOn the other hand, the finite remainder of the unpolarized interferences of the same scattering process \ncan  be computed within CDR using a polarization sum formula like \\eqref {EQ:polsumCDRPhys} for each of the \n4 external gluons. We have checked analytically that both ways lead to the same finite expression.~\\\\\n\n\nThe constant transformation matrix from the linearly polarized amplitudes projected out \nusing eq.~(\\ref{EQ:LPgggg8}) to helicity amplitudes can be read off from the defining relations \neq.~(\\ref{EQ:LP2HLmassless}). \nIn order to obtain the finite remainders of helicity amplitudes it is advantageous to perform such a \ntransformation only at the very last stage of the computation, e.g.~,~at the level of finite remainders \nof linear polarized amplitudes.\n\n\nFinally we remark that we also computed the helicity amplitudes \nby first obtaining the Lorentz tensor decomposition representation \nof $\\Big|\\mathcal{A}^{[\\text{1-loop}]}_{gggg} \\Big\\rangle$, \nusing the form factor projectors\\footnote{We did not \nconsider reductions owing to Bose symmetry involving kinematic crossings, and hence we\nextracted an over-complete set of 20 form factors that are left after\nimposing the transversality constraint and gauge-fixings indicated by choices of reference vectors in \neq.~(\\ref{EQ:LP2HLmassless}).} given in ref.~\\cite{Binoth:2002xg},\nand then evaluating contractions between Lorentz structures and external polarization vectors in 4 dimensions. \nThis amounts to obtaining helicity amplitudes defined in the HV scheme. \nWe confirm numerically that for all helicity amplitudes defined by the \nsingularity subtraction terms listed in eq.~(\\ref{EQ:ggggUVIRcounterterms}) the same finite remainders \nare obtained at a few chosen test points (while the unsubtracted helicities amplitudes \ndiffer starting from the subleading power in $\\epsilon$). \n\n\n\\subsection{$e^+ e^- \\rightarrow Q \\bar{Q}$}\n\\label{SEC:examples:eeQQ}\n\nNext we consider quark-pair production in $e^+e^-$ collisions:\n\\begin{equation} \\label{EQ:eeQQ}\ne^-(p_1) ~+~ e^+(p_2) \\to Z^* \\to Q(p_3)~+~{\\bar Q}(p_4)~,\n\\end{equation}\nmediated by a Z-boson where $Q$ denotes a massive quark with mass \n$m$, i.e., $p_3^2 = p_4^2 = m^2$, and the electron (positron) is taken to be massless. \nThe corresponding bare scattering amplitude perturbatively expanded to NLO in QCD reads\n\\begin{eqnarray}\\label{EQ:eeQQAMP} \n\\Big|\\mathcal{A}_{eeQQ} \\Big\\rangle &=&  \n\\mathcal{A}^{[\\text{tree}]}_{eeQQ}(1_{e^-}, 2_{e^+}, 3_{Q}, 4_{\\bar{Q}}) ~\\delta_{i_3 i_4} \\nonumber\\\\\n&+&\\frac{\\alpha_s}{4 \\pi} \\bar{C}( \\epsilon ) \n\\mathcal{A}^{[\\text{1-loop}]}_{eeQQ}(1_{e^-}, 2_{e^+}, 3_{Q}, 4_{\\bar{Q}}) ~2\\mathrm{C}_F \\delta_{i_3 i_4}\n+ \\mathcal{O}(\\alpha_s^2) \\, ,\n\\end{eqnarray} \nwhere $i_3$ ($i_4$) denotes the color index of the heavy quark (antiquark), \n$\\mathrm{C}_F=({N_c^2-1})\/({2N_c})$, and $\\bar{C}(\\epsilon) \\equiv \n( 4 \\pi )^\\epsilon e^{-\\epsilon \\gamma_E }$ with $\\gamma_E = 0.57721 \\ldots $ \ndenoting the Euler--Mascheroni constant.\nIn eq.~(\\ref{EQ:eeQQAMP}) we introduced symbolic labels $i_X$ in order to encode the dependence on  \nthe momentum $p_i$ and helicity $\\lambda_i$ of an external particle $i$ of type $X$.\nThese 1-loop QCD corrections were first computed in ref.~\\cite{Jersak:1981sp}.\n\n\nBecause we work to the lowest order in electroweak couplings, the UV renormalization counterterms \ncan be introduced by the following replacement of the bare coupling vertex of the Z boson and the heavy quark:\n\\begin{eqnarray} \\label{EQ:eeQQUVcounterterms}\n\\Big( v_Q \\gamma^{\\mu} + a_Q \\gamma^{\\mu} \\gamma_5 \\Big) \n\\rightarrow \nZ^{[1]}_{\\psi,OS}(\\epsilon,\\alpha_s)\n\\Big( v_Q \\gamma^{\\mu} + Z^{ns}_5(\\alpha_s) ~a_Q  \n\\frac{-i}{3!} \\epsilon^{\\mu \\nu \\rho \\sigma} \n\\gamma_{\\nu} \\gamma_{\\rho} \\gamma_{\\sigma} \\Big) \\, .\n\\end{eqnarray}\nHere $v_Q$ and $a_Q$ denote the vector and axial vector couplings of $Q$,\n\\begin{equation*}\n Z^{[1]}_{\\psi,OS}(\\epsilon,\\alpha_s) = \n-\\frac{\\alpha_s}{4\\pi} (4\\pi)^\\epsilon ~ \\Gamma(1+\\epsilon)\\frac{1}{\\epsilon} \n\\left(\\frac{\\mu_{DR}^2}{m^2}\\right)^\\epsilon \\mathrm{C}_F  \\frac{(3-2\\epsilon)}{(1-2\\epsilon)} + \\mathcal{O}(\\alpha_s^2) \\, ,\n\\end{equation*}\nand we use Larin's prescription~\\cite{Larin:1991tj,Larin:1993tq} for the non-singlet axial vector current \nwhich involves $Z^{ns}_5(\\alpha_s) = 1 +  \\frac{\\alpha_s}{4\\pi} \\left( -4 \\mathrm{C}_F \\right)+ \\mathcal{O}(\\alpha_s^2)$.\n\n\nFor subtracting the IR singularities of the renormalized 1-loop amplitude $\\mathcal{A}^{[\\text{1-loop,R}]}_{eeQQ}$, \nwe use the antenna subtraction method \\cite{Kosower:1997zr,GehrmannDeRidder:2005cm}.\nThe antenna subtraction term needed here reads \\cite{GehrmannDeRidder:2009fz}: \n\\begin{eqnarray}\\label{EQ:eeQQIRcounterterms} \n\\Big|\\mathcal{A}^{[\\text{IR}]}_{eeQQ} \\Big\\rangle &=&  \n\\frac{\\alpha_s}{4 \\pi} \\bar{C}( \\epsilon ) \n\\mathcal{A}^0_3\\left(\\epsilon, \\frac{\\mu_{DR}^2}{s}; y \\right)\n\\mathcal{A}^{[\\text{tree}]}_{eeQQ}(1_{e^-}, 2_{e^+}, 3_{Q}, 4_{\\bar{Q}}) ~2\\mathrm{C}_F \\delta_{i_3 i_4}\n+ \\mathcal{O}(\\alpha_s^2), \n\\end{eqnarray} \nwhere $y=\\frac{1-\\beta}{1+\\beta}~,~ \\beta=\\sqrt{1-4m^2\/s}$, and $\\mathcal{A}^0_3\\left(\\epsilon, \\frac{\\mu_{DR}^2}{s}; y \\right)$ denotes the \nintegrated three-parton tree-level massive quark-antiquark antenna function \ngiven in \\cite{GehrmannDeRidder:2009fz,Abelof:2011jv}. \n\n\nBecause we take the leptons to be  massless, there are only 8 non-vanishing helicity amplitudes\nwhich, in the absence of parity symmetry\\footnote{In the Standard Model  the 1-loop scattering\namplitude of (\\ref{EQ:eeQQ}) still respects the combined symmetry of parity and charge conjugation,\nwhich relates the helicity amplitude with helicity configuration $+- ++$ to $+- --$, and similarly $-+ ++$ to $-+ --$.},\ndiffer from each other. \nWe now consider the extraction of polarized amplitudes in the helicity basis both at the tree level and the 1-loop level.\nFollowing the discussion of section~\\ref{SEC:prescription:NTS}, \nwe choose to attach an auxiliary spinor inner product \n\\begin{equation}\\label{EQ:flHLnormfactors}\n\\mathcal{N}_{\\lambda_{e} \\lambda_{Q} \\lambda_{\\bar{Q}}} = \n\\bar{u}(p_1, \\lambda_{e}) \\slashed{p}_3 v(p_2, -\\lambda_{e}) \\otimes  \n\\bar{v}(p_4, \\lambda_{\\bar{Q}}) \\slashed{p}_1 u(p_3, \\lambda_{Q}) \n\\end{equation}\nto each helicity amplitude characterized by $\\lambda_{e},~ \\lambda_{Q},~\\lambda_{\\bar{Q}}$. \nThis factor is to be removed by numerical division  at the end of the computation in 4 dimensions. \nPulling off $\\mathcal{N}^{~ -1}_{\\lambda_{e} \\lambda_{Q} \\lambda_{\\bar{Q}}}$  \nfrom each helicity amplitude, the polarization projections can be most conveniently performed, in analogy to eq.~\\eqref{EQ:SFLtrace1}, using \nthe following 8 regrouped projectors according to eqs.~(\\ref{EQ:TPextsps2}), \\eqref{EQ:TPextsps2massless}:\n\\begin{eqnarray} \\label{EQ:LPeeQQ8}\n\\hat{\\mathrm{P}}_1 &=&  \\Big(\\slashed{p}_1  \\slashed{p}_3 \\slashed{p}_2 \\Big) \\otimes\n\\Big( \\left(\\slashed{p}_4 - m \\right) \\slashed{p}_1 \\left(\\slashed{p}_3 + m \\right) \\Big)~,~\\nonumber\\\\\n \\hat{\\mathrm{P}}_2 &=&  \\Big(\\slashed{p}_1  \\slashed{p}_3 \\slashed{p}_2 \\Big) \\otimes\n\\Big( \\left(\\slashed{p}_4 - m \\right) \\left(\\frac{-i}{3!} \\epsilon_{\\gamma \\gamma \\gamma S_{\\bar{Q}}} \\right) \\slashed{p}_1 \\left(\\slashed{p}_3 + m \\right) \\Big)~,~\\nonumber\\\\\n\\hat{\\mathrm{P}}_3 &=&  \\Big(\\slashed{p}_1  \\slashed{p}_3 \\slashed{p}_2 \\Big) \\otimes\n\\Big( \\left(\\slashed{p}_4 - m \\right) \\slashed{p}_1 \\left(\\frac{-i}{3!} \\epsilon_{\\gamma \\gamma \\gamma S_{Q}} \\right) \\left(\\slashed{p}_3 + m \\right) \\Big)~,~\\nonumber\\\\\n \\hat{\\mathrm{P}}_4 &=&  \\Big(\\slashed{p}_1  \\slashed{p}_3 \\slashed{p}_2 \\Big) \\otimes\n\\Big( \\left(\\slashed{p}_4 - m \\right) \\slashed{S}_{\\bar{Q}} \\slashed{p}_1 \\slashed{S}_{Q}  \\left(\\slashed{p}_3 + m \\right) \\Big)~,~\\nonumber\\\\\n\\hat{\\mathrm{P}}_5 &=&  \\Big(\\slashed{p}_1 \\frac{i}{3!}\\epsilon_{\\gamma \\gamma \\gamma p_3}  \\slashed{p}_2 \\Big) \\otimes\n\\Big( \\left(\\slashed{p}_4 - m \\right) \\slashed{p}_1 \\left(\\slashed{p}_3 + m \\right) \\Big)~,~\\nonumber\\\\\n \\hat{\\mathrm{P}}_6 &=&  \\Big(\\slashed{p}_1 \\frac{i}{3!}\\epsilon_{\\gamma \\gamma \\gamma p_3}  \\slashed{p}_2 \\Big) \\otimes\n\\Big( \\left(\\slashed{p}_4 - m \\right) \\left(\\frac{-i}{3!} \\epsilon_{\\gamma \\gamma \\gamma S_{\\bar{Q}}} \\right) \\slashed{p}_1 \\left(\\slashed{p}_3 + m \\right) \\Big)~,~\\nonumber\\\\\n\\hat{\\mathrm{P}}_7 &=&  \\Big(\\slashed{p}_1 \\frac{i}{3!}\\epsilon_{\\gamma \\gamma \\gamma p_3}  \\slashed{p}_2 \\Big) \\otimes\n\\Big( \\left(\\slashed{p}_4 - m \\right) \\slashed{p}_1 \\left(\\frac{-i}{3!} \\epsilon_{\\gamma \\gamma \\gamma S_{Q}} \\right) \\left(\\slashed{p}_3 + m \\right) \\Big)~,~\\nonumber\\\\\n \\hat{\\mathrm{P}}_8 &=&  \\Big(\\slashed{p}_1 \\frac{i}{3!}\\epsilon_{\\gamma \\gamma \\gamma p_3} \\slashed{p}_2 \\Big) \\otimes\n\\Big( \\left(\\slashed{p}_4 - m \\right) \\slashed{S}_{\\bar{Q}} \\slashed{p}_1 \\slashed{S}_{Q}  \\left(\\slashed{p}_3 + m \\right) \\Big)~,\n\\end{eqnarray}\nwhere the momentum basis representations of the two helicity polarization vectors  \n$S_{Q}^{\\mu}$ and $S_{\\bar{Q}}^{\\mu}$, in analogy to  eq.~(\\ref{EQ:PLpolMBR}), \nwill be inserted during the computation\\footnote{This insertion can \nconveniently be done after having performed the Dirac traces and having used \n$p_3 \\cdot S_{Q}=p_4 \\cdot S_{\\bar{Q}}=0$ and $S_{Q} \\cdot S_{Q} = S_{\\bar{Q}} \\cdot S_{\\bar{Q}} = -1$.}\nso that eventually the resulting projections are functions of the external momenta only.\nOf course, the manipulation of Dirac matrices associated with two disconnected fermion lines \n(separated by $\\otimes$ in eq.~(\\ref{EQ:LPeeQQ8})) can be performed independently and should not be confused. \nNotice that the set of polarization projectors in eq.~(\\ref{EQ:LPeeQQ8}) is also sufficient for computing virtual \namplitudes that involve box contributions, for instance $q(p_1)~\\bar{q}(p_2) \\to Q(p_3)~{\\bar Q}(p_4)$ in QCD, \nirrespective of any possible evanescent \nLorentz structure that can be generated at high loop orders in D dimensions. \nIn case $q(p_1)~\\bar{q}(p_2) \\to Q(p_3)~{\\bar Q}(p_4)$ is parity invariant, \nwhich is the case if one considers only QCD interactions, \nthen $\\hat{\\mathrm{P}}_2~,\\hat{\\mathrm{P}}_3~,\\hat{\\mathrm{P}}_5~,\\hat{\\mathrm{P}}_8$ \ncan be safely discarded and only 4 projectors are needed. \n\n\nIn the simple example considered here, where the amplitude \\eqref{EQ:eeQQAMP} involves only 3-point\nvertex functions, there is not much technical advantage in using eq.~(\\ref{EQ:LPeeQQ8}) instead of  \nthe conventional form factor decomposition. If one nevertheless chooses to use the projectors \n(\\ref{EQ:LPeeQQ8}) for computing helicity amplitudes including QCD corrections, \none can compute the trace \\eqref{EQ:SFLtrace1} of the string of Dirac matrices along the lepton line,\nboth for the renormalized amplitude and the IR subtraction term \\eqref{EQ:eeQQIRcounterterms},\nin 4 dimensions, because the lepton line receives no QCD correction and remains purely tree level. \nIn this case we can replace $\\frac{i}{3!}\\epsilon_{\\gamma \\gamma \\gamma p_3}$ \nin eq.~(\\ref{EQ:LPeeQQ8}) by $\\slashed{p}_3 \\gamma_5$. \n\n\nHelicity amplitudes can be assembled by linear combinations of the \nprojections made with (\\ref{EQ:LPeeQQ8}), and the linear combination coefficients can be \nread off from eqs.~(\\ref{EQ:TPextsps2}), \\eqref{EQ:TPextsps2massless}. \nIt is convenient to perform such a transformation only in the \nlast stage of the computation at the level of 4-dimensional finite remainders. \nThe explicit form of the overall normalization factor given in \neq.~(\\ref{EQ:flHLnormfactors}) is usually needed only at the level  \nof squared amplitudes (or interferences).\nThe squared modulus of  $\\mathcal{N}_{\\lambda_{e} \\lambda_{Q} \\lambda_{\\bar{Q}}}$ is\n\\begin{eqnarray}\n\\Big| \\mathcal{N}_{\\lambda_{e} \\lambda_{Q} \\lambda_{\\bar{Q}}} \\Big|^2 &=& \n-\\frac{m^4 - 2 m^2 t + t (s + t)}{2} \\Bigg(\n\\lambda_{Q} \\lambda_{\\bar{Q}} ~\\Big(\n2(m^2 - t)  \n(p_1 \\cdot S_{Q} ~ p_3 \\cdot S_{\\bar{Q}} - p_1 \\cdot S_{\\bar{Q}} ~ p_4 \\cdot S_{Q}) \\nonumber\\\\\n&&~~~~ + \n2 s ~(p_1 \\cdot S_{\\bar{Q}} ~ p_4 \\cdot S_{Q} - p_1 \\cdot S_{Q} ~ p_1 \\cdot S_{\\bar{Q}} ) \n\\Big) \\nonumber\\\\\n&&~~ + (m^2 - t) (m^2 - s - t) \\left(-1 + \\lambda_{Q} \\lambda_{\\bar{Q}} ~ S_{Q} \\cdot S_{\\bar{Q}} \\right)\n\\Bigg)\\nonumber\\\\\n&=&\n\\frac{1}{2} (m^2 - t) (m^2 - s - t) (m^4 - 2 m^2 t + t (s + t)) \\nonumber\\\\\n&-& \\lambda_{Q} \\lambda_{\\bar{Q}}~\n\\frac{ (m^4 - 2 m^2 t + t (s + t)) (4 m^6 + s t (s + t) - m^4 (3 s + 8 t) + m^2 (s^2 + 2 s t + 4 t^2))} {2 (s - 4 m^2)}, \\nonumber\\\\\n\\end{eqnarray}\nwhere we have inserted momentum basis representations of \n$S_{Q}^{\\mu}$ and $S_{\\bar{Q}}^{\\mu}$ that are given in analogy to eq.~(\\ref{EQ:PLpolMBR}). \nIn case the normalization factors are to be included at the amplitude level, \nwe can use for their computation either the concrete 4-dimensional representations\nof spinors and Dirac matrices, as listed for instance in~\\cite{Murayama:1992gi}, \nor employ the 4-dimensional spinor-helicity representation of \nthese objects~\\cite{Gunion:1985vca,Kleiss:1985yh,Xu:1986xb,Kleiss:1986qc,Dittmaier:1998nn,Schwinn:2005pi,Arkani-Hamed:2017jhn}.\n\n\n\nWith the ingredients just outlined we computed the finite remainders of the interferences between the  \ntree-level and 1-loop helicity amplitudes, multiplied, for convenience, with the inverse square of the Z-boson propagator: \n\\begin{equation}\\label{EQ:tr1lohel}\n\\left(s-m^2_Z\\right)^2 \\times 2~ {\\rm Re}\\Big[\\mathcal{A}^{[\\text{tree}]*}_{eeQQ}(1_{e^-}, 2_{e^+}, 3_{Q}, 4_{\\bar{Q}})~\n \\mathcal{A}^{[\\text{1-loop}]}_{eeQQ}(1_{e^-}, 2_{e^+}, 3_{Q}, 4_{\\bar{Q}}) \\Big] \\, .\n\\end{equation}\nWe calculated \\eqref{EQ:tr1lohel} analytically using FORM~\\cite{Vermaseren:2000nd} and the involved loop integrals \nwith Package-X~\\cite{Patel:2015tea}. Table~\\ref{TAB:numbersFinInf}  contains the finite remainders of \\eqref{EQ:tr1lohel}\nfor all helicity configurations evaluated at the test point $m=17.3$ GeV, $s= 10^6$ $({\\rm GeV})^2$, $t = -90$ $({\\rm GeV})^2$.\n($v_e$ and $a_e$ denote the vector and axial vector couplings of electron.)\n\n\\vspace{2mm}\n\\begin{table}[tbh!]\n\\begin{center}\n\\begin{tabular}{|c|l|}\n\\hline\nHelicities  & $~~~~~~~~~~~~~~~~~~~$ Finite remainders of the interferences \\eqref{EQ:tr1lohel}  in units of $({\\rm GeV})^2$ \\\\\n\\hline \n$+-,++$ & $-1.4211829*10^6 ~a_e^2 v_Q^2 - 2.8423658*10^6 ~a_e v_e v_Q^2-1.4211829*10^6 ~v_e^2 v_Q^2$ \\\\\n\\hline\n\\multirow{3}{*}{$+-,+-$} & $~~ 2.4731876*10^4 ~a_e^2 a_Q^2 + 4.9463752*10^4 ~a_e a_Q^2 v_e + 2.4731876*10^4 ~a_Q^2 v_e^2 $ \\\\\n                         & $~+ 4.9178930*10^4 ~a_e^2 a_Q v_Q + 9.8357861*10^4 ~a_e a_Q v_e v_Q + 4.9178930*10^4 ~a_Q v_e^2 v_Q $ \\\\ \n                         & $~+ 2.4446875*10^4 ~a_e^2 v_Q^2 + 4.8893750*10^4 ~a_e v_e v_Q^2 + 2.4446875*10^4 ~v_e^2 v_Q^2$ \\\\\n\\hline\n\\multirow{3}{*}{$+-,-+$} & $~~ 3.0551961 *10^{12}~a_e^2 a_Q^2 + 6.1103923 *10^{12}~a_e a_Q^2 v_e + 3.0551961 *10^{12}~a_Q^2 v_e^2 $ \\\\\n                         & $~+ 6.0752075 *10^{12}~a_e^2 a_Q v_Q -1.2150415 * 10^{13}~a_e a_Q v_e v_Q -6.0752075 *10^{12}~a_Q v_e^2 v_Q $ \\\\ \n                         & $~+ 3.0199891 *10^{12}~a_e^2 v_Q^2 + 6.0399783 *10^{12}~a_e v_e  v_Q^2 + 3.0199891 *10^{12}~v_e^2 v_Q^2$ \\\\\n\\hline\n$+-,--$ & $-1.4211829*10^6 ~a_e^2 v_Q^2 - 2.8423658*10^6 ~a_e v_e v_Q^2-1.4211829*10^6 ~v_e^2 v_Q^2$ \\\\\n\\hline\n$-+,++$ & $-1.4211829*10^6 ~a_e^2 v_Q^2 + 2.8423658*10^6 ~a_e v_e v_Q^2-1.4211829*10^6 ~v_e^2 v_Q^2$ \\\\\n\\hline\n\\multirow{3}{*}{$-+,+-$} & $~~ 3.0551961 *10^{12}~a_e^2 a_Q^2 - 6.1103923 *10^{12}~a_e a_Q^2 v_e + 3.0551961 *10^{12}~a_Q^2 v_e^2 $ \\\\\n                         & $~+ 6.0752075 *10^{12}~a_e^2 a_Q v_Q -1.2150415 * 10^{13}~a_e a_Q v_e v_Q +6.0752075 *10^{12}~a_Q v_e^2 v_Q $ \\\\ \n                         & $~+ 3.0199891 *10^{12}~a_e^2 v_Q^2 - 6.0399783 *10^{12}~a_e v_e  v_Q^2 + 3.0199891 *10^{12}~v_e^2 v_Q^2$ \\\\\n\\hline\n\\multirow{3}{*}{$-+,-+$} & $~~ 2.4731876*10^4 ~a_e^2 a_Q^2 - 4.9463752*10^4 ~a_e a_Q^2 v_e + 2.4731876*10^4 ~a_Q^2 v_e^2 $ \\\\\n                         & $~+ 4.9178930*10^4 ~a_e^2 a_Q v_Q + 9.8357861*10^4 ~a_e a_Q v_e v_Q - 4.9178930*10^4 ~a_Q v_e^2 v_Q $ \\\\ \n                         & $~+ 2.4446875*10^4 ~a_e^2 v_Q^2 - 4.8893750*10^4 ~a_e v_e v_Q^2 + 2.4446875*10^4 ~v_e^2 v_Q^2$ \\\\\n\\hline\n$-+,--$ & $-1.4211829*10^6 ~a_e^2 v_Q^2 + 2.8423658*10^6 ~a_e v_e v_Q^2-1.4211829*10^6 ~v_e^2 v_Q^2$ \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{TAB:numbersFinInf} \nNumerical values of the finite remainders of the interferences \\eqref{EQ:tr1lohel}\nat the test point $m=17.3$ GeV, $s= 10^6$ $({\\rm GeV})^2$, $t = -90$ $({\\rm GeV})^2$.}\n\\end{center}\n\\end{table} \nThe interferences were computed to about 30 significant digits while only the\nfirst 8 significant digits are shown in table~\\ref{TAB:numbersFinInf}\nfor simplicity.\\footnote{For this reason there is no rounding in the shown digits.} \n CP invariance dictates that the helicity configurations \n$+- ++$ and $+- --$ yield identical expressions, and likewise $-+ ++$ and $-+ --$.\nThe large differences between the values of these helicity amplitudes are  \ndue to the particular kinematic point considered: it corresponds to \na high-energy (small mass) limit of the scattering amplitude \nin the near-forward scattering region.\n\n\nWe computed also the finite remainder of the \nunpolarized interferences \\eqref{EQ:tr1lohel} within CDR at the same  kinematic point\nwith the renormalized virtual amplitudes from refs.~\\cite{Bernreuther:2004ih,Bernreuther:2004th} \navailable in a form factor decomposed form.\nFor this unpolarized interference we obtain\n\\begin{equation*}\n6.1103923*10^{12} ~\\Big(a_e^2 a_Q^2 + v_e^2 a_Q^2 \\Big)\n~-~ 2.4300829*10^{13} ~a_e v_e a_Q v_Q ~+~ \n6.0399727*10^{12} ~\\Big(a_e^2 v_Q^2 + v_e^2 v_Q^2 \\Big),\n\\end{equation*}\nwhich precisely reproduces the sum of all helicity configurations listed in table~\\ref{TAB:numbersFinInf}. \n(The expressions coincide by the first 26 digits out of total 30 significant digits).~\\\\\n\n\nLet us comment on a point that was already alluded to in section~\\ref{SEC:prescription:NTS} \nand discussed in section~\\ref{SEC:unitarity:PSA}. It concerns the  placing of Dirac matrices \nbetween pairs of on-shell projection operators.\nMoving the matrix $\\left(\\frac{-i}{3!} \\epsilon_{\\gamma \\gamma \\gamma S_{Q}} \\right)$ around \nin the external projectors in eq.~\\eqref{EQ:LPeeQQ8} according to the 4-dimensional algebra \nbetween the pair of on-shell projection operators, \n$\\left(\\slashed{p}_4 - m \\right)$ and $\\left(\\slashed{p}_3 + m \\right)$,\nalways leads to the same finite remainders documented in table~\\ref{TAB:numbersFinInf} \nusing the fixed singularity subtraction terms defined by \n\\eqref{EQ:eeQQUVcounterterms} and \\eqref{EQ:eeQQIRcounterterms}. \nYet, as expected, these different choices result in different (unsubtracted) singular virtual amplitudes.\nOnce we decide to move $\\left(\\frac{-i}{3!} \\epsilon_{\\gamma \\gamma \\gamma S_{Q}} \\right)$ \nbeyond $\\left(\\slashed{p}_4 - m \\right)$ or $\\left(\\slashed{p}_3 + m \\right)$,\nthis operation has to be made in accordance with the D-dimensional algebra in order to end up \nwith the same finite remainders. \nFor instance, the commutator between \n$\\left(\\frac{-i}{3!} \\epsilon_{\\gamma \\gamma \\gamma S_{Q}} \\right)$\nand $\\slashed{p}_3$, which vanishes in 4 dimensions because of\n$p_3 \\cdot S_{Q} = 0$, must not be omitted.~\\\\\n\n\nWe conclude this subsection with a remark on a subtle point concerning the specification \nof a definite contraction order among multiple Levi-Civita tensors, in order to reach \nan unambiguous canonical form for a projector as well as for the resulting projection in D dimensions. \nAs discussed in ref.~\\cite{Moch:2015usa}, the contraction of four Levi-Civita tensors \ncan lead to different expressions in D dimensions depending on the choice of pairings,  \nwhich are not algebraically identical due to the lack of a Schouten identity.\nThis issue is of no concern for the amplitude of (\\ref{EQ:eeQQAMP}), \nespecially if we do the trace over the lepton line using 4-dimensional Dirac algebra \nbefore dealing with the heavy quark line. \nNevertheless, in more general situations to which our projector prescriptions also apply,\none should pair Levi-Civita tensors from inner vertices (of the same fermion line) \nin the contraction~\\cite{Moch:2015usa}, leaving all other Levi-Civita tensors appearing \nin the external projectors in a different category that are to be manipulated among themselves. \nOnce a definite choice of pairing and ordering of Levi-Civita tensors in the contraction \nis made, it should be consistently applied in the computations of all terms that contribute to\na (renormalized and subtracted) helicity amplitude.\nAlternatively, if several $\\gamma_5$ matrices from axial non-singlet current vertices\nand\/or pseudoscalar vertices are present along a fermion line, one can resort to \na fully anti-commuting $\\gamma_5$ and use the rule $\\gamma_5^2 = 1$ in $D$ dimensions~\\cite{Chanowitz:1979zu}, \nin order to bring the power of $\\gamma_5$ down to 0 or 1 before further manipulations. \nThis shall lead to the same final result one would get with a thorough implementation of Larin's \nprescription of non-singlet axial vector vertices and pseudoscalar vertices~\\cite{Larin:1993tq,Moch:2015usa}, \nalbeit it is computationally more convenient.\n\n\n\\section{Conclusions}\n\\label{SEC:conclusion}\n\nHelicity amplitudes encode the full dependence of scattering processes \non the polarizations of the external particles with spin and these amplitudes \nenrich the phenomenology of high energy physics. \nDifferent methods for computing dimensionally regularized helicity amplitudes exist, \ndepending on the particular regularization scheme used. \nFor the computation of helicity loop amplitudes with a scheme such as HV, \nthe projection method based on Lorentz covariant tensor decomposition is a widely used choice.\nDespite being very generic and versatile, there are a few delicate aspects of \nthe Lorentz tensor decomposition, as discussed in section~\\ref{SEC:projectionmethod:comments}, \nwhich makes the D-dimensional projection sometimes cumbersome to be carried out \nfor certain multiple-parton multiple-scale scatterings.~\\\\\n\n\nThe aim of this article was to formulate an alternative prescription to obtain polarized \ndimensionally regularized amplitudes, providing a recipe for constructing simple \nand general polarized amplitude projectors in D dimensions, which circumvents  \nthe conventional Lorentz tensor decomposition and difficulties associated with it.\nThe polarization projectors devised in section~\\ref{SEC:Prescription} are based on \nthe momentum basis representations of external state vectors, and all their open \nLorentz indices are taken to be D-dimensional. This avoids dimensional splitting \nwhen applied to loop amplitudes. \nThe momentum basis representations of external gauge boson's polarization vectors \nas well as polarization vectors of massive fermions were discussed in detail\nin section~\\ref{SEC:prescription:MBR}. \n\n\nAs shown in section~\\ref{SEC:Prescription}, it is quite straightforward to construct these projectors,\nand their structures depend only on the masses and spins of the external particles. \nThe construction procedure requires almost no knowledge of the Lorentz structures \npresent in the loop amplitude, nor whether or not they are linearly independent of each other (in D dimensions).\nIn particular, there is no need to trim any unphysical Lorentz structure off the original Feynman-diagrammatic \nrepresentation of the amplitude before applying these external projectors. \nThe number and forms of these projectors are truly independent of the loop order of the virtual amplitude \nas well as of possible evanescent Lorentz structures that could be generated in D dimensions. \nConstraints from symmetry properties such as parity symmetry can be accounted for in a simple way \nin terms of this set of projectors. \n\n\nFrom the point of view of the projection method as recapped in section~\\ref{SEC:projectionmethod:recap},\nthe set of projectors prescribed in this article may be loosely viewed as a special choice of \nLorentz decomposition basis structures which by construction are orthogonal to each other. \nFurthermore, each of these decomposition structures is directly related to a physical quantity, \nand thus patterns of (explicit and\/or implicit) singularities therein are protected by \nphysical conditions observed by these physical quantities. \nIn this way the issues related to the conventional form factor decomposition as discussed \nin section~\\ref{SEC:projectionmethod:comments} are avoided.~\\\\\n\n\n\nThe usage of these D-dimensional polarized amplitude projectors results in helicity amplitudes \nwhich are eventually expressed solely in terms of Lorentz invariants made out of external momenta. \nThe resulting helicity amplitudes (and the incoherent sum of their squared moduli) \nare, however, different from those defined in many existing dimensional regularization schemes, \nin particular CDR.\nDespite being different from CDR, owing to the amplitude-level factorization of UV and IR singularities \ncombined with the crucial commutation between D-dimensional Lorentz index contraction and loop integration, \nour prescription for external states can be used in a hybrid way with CDR to obtain the same \nfinite remainders of loop amplitudes as in CDR, without  having to re-calculate the (process-independent) \npole-subtraction coefficients. \nThis was demonstrated in section~\\ref{SEC:unitarity:PSA} in a formal way for minimally pole-subtracted amplitudes. \nThe validity of our argumentation is not confined to one-loop corrections to the Born amplitudes, but  \npersists as long as the amplitude-level factorization formulae as sketched in eq.~(\\ref{EQ:AmpPoleFactorization}) \nhold in CDR. \nSubsequently, the same issue was discussed in section~\\ref{SEC:unitarity:FRIR} for finite remainders \ndefined in an IR subtraction method, where we argued that the unitarization recipe in ref.~\\cite{Catani:1996pk} \nis properly respected by our method. Thus we have shown that our hybrid CDR-compatible prescription is unitary.\nWe emphasize again that in order to unambiguously and consistently apply our prescription for\nexternal states to the calculation of loop amplitudes in D dimensions, there is no need to appeal to \ntheir Lorentz tensor decomposition representations.\n\n\n\nIn order to illustrate the usage of our hybrid prescription in practical applications, \nwe discussed in section~\\ref{SEC:examples} the construction of polarization projectors \nfor $gg \\rightarrow gg$ and $e^+ e^- \\rightarrow Q \\bar{Q}$, \nand computed their RS-independent finite remainders at one-loop order. \nWhile the arguments presented in section~\\ref{SEC:unitarity:FRIR}, as well as \nthe examples of section~\\ref{SEC:examples}, refer to NLO computations,\nit is possible to ensure unitarity of the prescription at NNLO in QCD and beyond,\nwith the aid of an IR-subtraction method as briefly commented on at the end of the section~\\ref{SEC:unitarity:FRIR}. \nThis is, however, beyond the scope of the current article, and we leave a detailed exposition \nof this in a future publication of polarized calculations where a NNLO subtraction method \nwill be employed.~\\\\\n\n\nGiven the impressive list of calculations of unpolarized observables done using the CDR \n(with Larin's prescription of $\\gamma_5$), we hope that, with this add-on, the resulting hybrid \nCDR-compatible prescription offers a convenient and efficient set-up for computing \nphysical observables associated with polarization effects for phenomenologically interesting processes \nin perturbative QCD.\n\n\n\\section*{Acknowledgments}\n\nThe author is grateful to W.~Bernreuther, M.~Czakon, and G.~Heinrich for discussions and comments on the manuscript.\nThe author also wishes to thank S.~Jahn, S.~Jones and M.~Kerner for helpful discussions and feedback on the draft, \nand T.~Ahmed, M.~Capozi, H.~Luo, J.~Schlenk, Z.G.~Si, Y.~Zhang for reading the manuscript. \n\n\n\n\\bibliographystyle{JHEP}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAfter its hype finally receded about half a decade ago, rather few advances in Semantic Desktop (SemDesk) research have been reported.\nAn overview of (modern) SemDesks can be found in \\cite{DraganD12}:\nExisting implementations are, for example, reproached for being rather complicated to use, not scaling well (thus draining lots of system resources), and there is still no real \"killer app\" available.\nConcerning SemDesk applications, two categories could be observed:\nnewly created semantic applications and plug-ins to enhance traditional, non-semantic ones \\cite{DraganD12}.\n\nAs a successor to the \\textit{NEPOMUK Semantic Desktop}\\footnote{\\url{www.semanticdesktop.org}}, DFKI's Smart Data \\& Knowledge Services department developed its own prototype\\footnote{\nmeanwhile spanning over six years of permanent usage in the department and a group knowledge graph having approx. 2.6 million triple statements\n} \\cite{maus2013weaving} making SemDesk technology ready for 24\/7 usage in practice, covering both, private and corporate scenarios.\nAfter lessons learned in past\\footnote{\ne.g. \\textit{ForgetIT} (\\url{www.forgetit-project.eu}) and \\textit{supSpaces} (\\url{www.supspaces.de})\n} and still ongoing projects\\footnote{\ne.g. \\textit{Managed Forgetting} (\\url{www.spp1921.de\/projekte\/p4.html.de})\n}, we now propose \\textit{Context Spaces} as an extension of this prototype addressing the issues mentioned before.\n\n\\section{Approach}\n\\noindent\n\\textbf{Context Spaces.}\nOne of SemDesk's cornerstones is the Personal Information Model (PIMO) \\cite{sauermann2007pimo}, which tries to represent a user's mental model as good as possible.\nInformation items (files, mails, bookmarks, ...) that are related to each other in a person's mind, but are separated on their computer (file system, mail client, web browser, ...), can thus be interlinked.\nWith \\textit{Context Spaces} (or \\textit{cSpaces} for short) we extend this idea by explicitly (and additionally) associating items with contexts of the user (see lower left of Figure \\ref{fig_cspaces}).\n\\vspace{-0.4cm}\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\textwidth]{img\/cspaces.png}\n\\caption{Conceptual overview of the cSpaces Semantic Desktop}\n\\label{fig_cspaces}\n\\end{figure}\n\\vspace{-0.4cm}\nThis is based on the intuition that every activity is performed in a certain context.\nHence, each information item stored on a person's computing device can be associated with one or more contexts (association strength may vary depending on the user's current context awareness).\nWe therefore assume that users are explicitly aware of the concept of context \\cite{GomezPerez2009} and that they are also aware of their current context (at least most of the time).\nExamples of such contexts are: \\textit{Spain holiday 2017}, \\textit{prepare ESWC18 paper}, or \\textit{my childhood memories}.\nWe do not enforce a certain definition of context: users should be able to stick to their own conceptualization as much as possible.\nHowever, we do assume that contexts express a certain relatedness of its elements.\nBesides being a kind of container for things, they may also be strongly related to (calendar) events, tasks or cases.\nContext hierarchies are also possible.\nMore details about our context model, which is an extension of \\cite{SchwarzContextModel}, will be presented in another paper.\nInstead of just having context as passive metadata, we in addition see it as an accessible element users can interact with (create new (sub)contexts, split or merge them, add\/remove elements, etc.).\n\nSemDesk user studies \\cite{SauermannEval} revealed that people omitted rather specific relations in favor of basic ones (like \\textit{isRelatedTo} or \\textit{isPartOf}), whereas the system is formal where possible, e.g. representing calendar events or address book contacts.\nThis matches our idea of providing a low effort opportunity to already keep things a bit more tidied up when simply associating them with a certain context (or multiple).\nAdditionally, some of these associations may also be inferred by the system reducing manual effort even more, e.g. a received email reply can automatically be associated with the original mail's context.\nMore advanced features supporting the user will be discussed in the section after next\n\n\\noindent\n\\textbf{Transparent Integration.}\nUsing contexts as an explicit interaction element only makes sense if applications also respect them.\nLike illustrated in Figure \\ref{fig_cspaces}, we therefore integrate cSpaces into the rest of the system using standard protocols like \\textit{Server Message Block (SMB)} for files, \\textit{IMAP} for mails, \\textit{CalDAV} for calendar entries, and \\textit{CardDAV} for contacts.\nFor web browsers, we use \\textit{Web\\-Extensions}\\footnote{\\url{https:\/\/wiki.mozilla.org\/WebExtensions}}, which provide cross-browser functionality and an integration level similar to having an underlying protocol.\nApplications are thus able to transparently operate on the knowledge graph (PIMO) managed by our app.\nEspecially in corporate scenarios, it is very convenient if users may just work with the resources in their contexts without caring whether they are actually spread across various sources like intranet shares, for example.\n\nUtilizing only standard protocols has certain limitations due to their rather basic, low-level character.\nSome activities, like writing a note or comment about a resource, can become inconvenient or non-intuitive.\nTo avoid this, we provide an additional sidebar as a single interaction point for using advanced features.\nUsers therefore do not need to learn a new (plugged-in) interface for each of their applications.\nThey can just keep using them the usual way having only the sidebar as a new UI to familiarize with.\nFrom the development point of view, the effort of creating and maintaining plug-ins needed for higher level functionality is comparatively low to that of earlier SemDesks.\nThey can be realized as \\textit{headless plug-ins} having very little functionality, often just the capability of \\textit{sending out} in-app-events to the sidebar (that is why we also shortly call them \"plug-outs\").\nIn addition, their corresponding UI elements and logic are located in the sidebar, where they can be easily reused.\nPlug-outs for different mail clients could, for example, share the same tagging UI\n\n\\noindent\n\\textbf{Self-Reorganization.}\nFeatures discussed so far primarily aim at our system's ease of use.\nThe other aspects mentioned in the beginning (scalability, missing \"killer app\") will be addressed using \\textit{Managed Forgetting}, by which we understand an escalating set of measures: temporal hiding, condensation, adaptive synchronization, archiving and deletion of resources and parts of the knowledge graph \\cite{forgetitbook}.\nBy having users work on cSpaces, we gather rich contextual information about all of their resources, which allows the system to semi-automatically help them in organizing their stuff.\nThus, cSpaces are continuously spawned, retracted, merged, condensed, or forgotten.\nAs an example, let us assume we do a consulting job for company XY.\nThe contract involves five meetings about different topics.\nOur system could represent this by having an overall cSpace containing general information about XY, e.g. contact and contract information.\nFor each meeting, there could be an individual sub-cSpace about its respective topic.\nSeveral months after the job has been completed, the system starts to remove details, e.g. train schedule to get to the meeting or auxiliary material for doing the presentation.\nAfter some years have passed, the sub-cSpaces could be merged with their parent, since the separation into different meetings is not relevant anymore.\nOnly the most important items, e.g. individual reports or an overall final report, are kept.\nAll other items are either condensed, moved to an archive or deleted completely (which can be adjusted by the user on a general level).\nAn item's current and estimated future value for the user are therefore continuously assessed resulting in different forgetting measures like temporal hiding (e.g. some items during one of the meetings), deletion, etc.\nThis especially means that the system is able to reorganize itself to a certain extent, which especially includes a kind of tidying-up-itself functionality.\nSome of the described features have already been implemented and successfully used in our research and industry prototypes \\cite{manaforge, pimodiary}, however most of them are still under heavy development.\\\\[-0.3cm]\n\n\\noindent\n\\textbf{Demo.}\nIn an early proof-of-concept implementation based on \\cite{maus2013weaving}, we already realized some of the file system, browser and calendar parts.\nThe screenshots in Figure \\ref{fig_scr} show a typical feature of our system:\n\\vspace{-0.45cm}\n\\begin{figure}\n\\centering\\includegraphics[width=1\\textwidth]{img\/screenshot.png}\n\n\\caption{\nScreenshot showing sidebar, file explorer and browser before (left half) and after a context switch (right half), illustrating the effects of a dynamic reorganization of the system.\n(Note: windows were rearranged for easier comparison.)\n}\n\\label{fig_scr}\n\\end{figure}\n\\vspace{-0.45cm}\nthe user selects a different context using the sidebar.\nAs a consequence, the \\textit{current context}, available as a folder in the file system as well as the browser, is dynamically reorganized by our app.\nNote that the system tries to present meaningful views on the current context in each app: e.g., the view in the browser only contains web links.\nTo really get an impression of how the interaction with the system looks like, we kindly refer the reader to our online demo video\\footnote{\\url{https:\/\/pimo.opendfki.de\/cSpaces\/}}, which also shows some additional features.\n\n\\section{Conclusion \\& Outlook}\nIn this paper, we presented a new SemDesk prototype based context spaces that users directly interact with and work on.\nThe system is transparently integrated using mostly standard protocols complemented by a sidebar for advanced features.\nUsers may thus stick to their favorite applications which should strongly contribute to the overall ease of use.\nLearning efforts are presumably low due to the sidebar being the only new UI that is introduced.\nBy exploiting its collected context information and applying features of Managed Forgetting, the system is able to dynamically reorganize itself which also includes a kind of tidying-up-itself functionality.\nWe therefore expect it to be more scalable than its predecessors while providing new levels of user support.\n\nNevertheless, a lot of functionality still needs to be fully implemented and evaluated.\nWe plan to do extensive user studies once the system matures.\\\\\n\n\\noindent\n\\textbf{Acknowledgements.}\nThis work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -- DE 420\/19-1.\n\n\\bibliographystyle{splncs03}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}The study of algorithmic randomness has been an\nactive area of research in recent years.  The basic problem is to\nquantify the randomness of a single real number. Here we think of a\nreal $r \\in [0,1]$ as an infinite sequence of 0's and 1's, i.e as an\nelement in $\\TN$. There are three basic approaches to algorithmic\nrandomness: the measure-theoretic approach of \\ml tests,\nthe incompressibility approach of Kolmogorov complexity, and the\nbetting approach in terms of martingales.  All three approaches have\nbeen shown to yield the same notion of (algorithmic) randomness. The\npresent paper will consider only the measure-theoretic approach.  A\nreal $x$ is \\ml random if for any effective sequence $S_1,\nS_2, \\dots$ of c. e. open sets with $\\mu(S_n) \\leq 2^{-n}$, $x \\notin\n\\cap_n S_n$.  For background and history of algorithmic randomness we\nrefer to \\cite{DH:book,Nies:book}.\n\nIn a series of recent papers \\cite{BCRW08,BCD06,BCDW08,BCR06},\nG. Barmpalias, S. Dashti, R. Weber and the authors have defined a\nnotion of (algorithmic) randomness for closed sets and continuous\nfunctions on $2^{\\N}$.  Some definitions are needed.  For a finite\nstring $\\sigma \\in\\{0,1\\}^n$, let $|\\sigma| = n$.  For two strings\n$\\sigma,\\tau$, say that $\\tau$ \\emph{extends} $\\sigma$ and write\n$\\sigma \\sqsubseteq \\tau$ if $|\\sigma| \\leq |\\tau|$ and $\\sigma(i) =\n\\tau(i)$ for $i < |\\sigma|$. For $x \\in \\TN$, $\\sigma \\sqsubset x$\nmeans that $\\sigma(i) = x(i)$ for $i < |\\sigma|$. Let $\\sigma^{\\frown}\n\\tau$ denote the concatenation of $\\sigma$ and $\\tau$ and let\n$\\sigma^{\\frown} i$ denote $\\sigma^{\\frown}(i)$ for $i=0,1$. Let\n$x\\lceil n =(x(0),\\dots,x(n-1))$. Two reals $x$ and $y$ may be coded\ntogether into $z = x \\oplus y$, where $z(2n) = x(n)$ and $z(2n+1) =\ny(n)$ for all $n$.\nFor a finite string $\\sigma$, let $I(\\sigma)$ denote $\\{x \\in\n\\TN:\\sigma \\sqsubset x\\}$. We shall call $I(\\sigma)$ the\n\\emph{interval} determined by $\\sigma$. Each such interval is a clopen\nset and the clopen sets are just finite unions of intervals. We let\n$\\B$ denote the Boolean algebra of clopen sets.\n\nNow a closed set $P$ may be identified with a tree\n$T_P\\subseteq \\{0,1\\}^*$ where $T_P = \\{\\sigma: P \\cap I(\\sigma)\n\\neq\\emptyset\\}$. Note that $T_P$ has no dead ends. That is, if\n$\\sigma\\in T_P$, then either $\\sigma^{\\frown}0 \\in T_P$ or\n$\\sigma^{\\frown}1\\in T_P$.\nset of infinite paths through $T$. It is well-known that $P \\subseteq\n\\TN$ is a closed set if and only if $P = [T]$ for some tree $T$.  $P$\nis a $\\Pi^0_1$ class, or an effectively closed set, if $P = [T]$ for\nsome computable tree $T$; equivalently $T_P$ is a \\pz set. The\ncomplexity of the closed set $P$ is generally identified with that of\n$T_P$.  Thus $P$ is said to be a $\\Pi^0_2$ closed set if $T_P$ is\n$\\Pi^0_2$; in this case $P = [T]$ for some $\\Delta^0_2$ tree $T$.  The\ncomplement of a $\\Pi^0_1$ class is sometimes called a c.e. open\nset. We remark that if $P$ is a $\\Pi^0_1$ class, then $T_P$ is a\n$\\Pi^0_1$ set, but it is not, in general, computable.\nThere is a natural effective enumeration $P_0, P_1, \\dots$ of the\n$\\Pi^0_1$ classes and thus an enumeration of the c.e. open sets. Thus\nwe can say that a sequence $S_0,S_1,\\dots$ of c.e. open sets is\n\\emph{effective} if there is a computable function, $f$, such that\n$S_n = \\TN - P_{f(n)}$ for all $n$. For a detailed development of\n$\\Pi^0_1$ classes, see~\\cite{Ceta}.\n\nIt was observed in \\cite{BCD06} that there is a natural isomorphism\nbetween the space $\\C$ of nonempty closed subsets of $\\{0,1\\}^{\\N}$\nand the space $\\{0,1,2\\}^{\\N}$ (with the product topology)\ndefined as follows. Given a nonempty closed\n$Q\\subseteq \\TN$, let $T = T_Q$ be the tree without dead ends such\nthat $Q =[T]$. Let $\\sigma_0, \\sigma_1, \\ldots$ enumerate the elements\nof $T$ in order, first by length and then lexicographically. We then\ndefine the code $x = x_Q = x_T$ by recursion such that for each $n$,\n$x(n) =2$ if both $\\sigma_n\\fr 0$ and $\\sigma_n\\fr 1$ are in $T$,\n$x(n) =1$ if $\\sigma_n\\fr 0 \\notin T$ and $\\sigma_n\\fr 1 \\in T$, and\n$x(n) =0$ if $\\sigma_n\\fr 0 \\in T$ and $\\sigma_n\\fr 1 \\notin T$. For a\nfinite tree $T \\subseteq \\{0,1\\}^{\\leq n}$, the finite code $\\rho_T$ is\nsimilarly defined, ending with $\\rho_T(k)$ where $\\sigma_k$ is the\nlexicographically last element of $T \\cap \\{0,1\\}^n$.\n\nWe defined in \\cite{BCD06} a measure $\\mu^*$ on the space ${\\mathcal\nC}$ of closed subsets of $\\TN$ as follows.\n\\begin{equation}\n\\mu^*({\\mathcal X}) = \\mu(\\{x_Q:Q \\in {\\mathcal X}\\})\n\\end{equation}\nfor any ${\\mathcal X} \\subseteq {\\mathcal C}$ and $\\mu$ is the\nstandard measure on $\\{0,1,2\\}^{\\N}$.  Informally this means that\ngiven $\\sigma \\in T_Q$, there is probability $\\frac13$ that both\n$\\sigma^{\\frown}0 \\in T_Q$ and $\\sigma^{\\frown}1 \\in T_Q$ and, for\n$i=0,1$, there is probability $\\frac13$ that only $\\sigma^{\\frown}i\n\\in T_Q$. In particular, this means that $Q \\cap I(\\sigma) \\neq\n\\emptyset$ implies that for $i=0,1$, $Q \\cap I(\\sigma^{\\frown}i) \\neq\n\\emptyset$ with probability $\\frac23$.\n\nBrodhead, Cenzer, and Dashti \\cite{BCD06} defined a closed set\n$Q\\subseteq 2^{\\N}$ to be (Martin-L\\\"{o}f) random if $x_Q$ is\n(Martin-L\\\"{o}f) random.  Note that the equal probability of $\\frac13$\nfor the three cases of branching allows the application of Schnorr's\ntheorem that \\ml randomness is equivalent to prefix-free Kolmogorov\nrandomness.  Then in \\cite{BCD06,BCDW08}, the following results are\nproved. No $\\Pi^0_1$ class is random but there is a random\n$\\Delta^0_2$ closed set. Every random closed set contains a random\nmember but not every member is random. Every random real belongs to\nsome random closed set. Every random $\\Delta^0_2$ closed set contains\na random $\\Delta^0_2$ member.  Every random closed set is perfect and\ncontains no computable elements (in fact, it contains no\n$n$-c.e.\\ elements).  Every random closed set has measure 0. A random\nclosed set is a specific type of random recursive construction, as\nstudied by Graf, Mauldin and Williams \\cite{GMW88}. McLinden and\nMauldin \\cite{MM09} showed that the Hausdorff dimension of a random\nclosed set is $log_2(4\/3)$.\n\nJust as an effectively closed set in $\\TN$ may be viewed as the set of\ninfinite paths through a computable tree $T \\subseteq \\{0,1\\}^*$, an\nalgorithmically random closed set in $\\TN$ may be viewed as the set of\ninfinite paths through an algorithmically random tree $T$. Diamondstone and\nKjos-Hanssen \\cite{DK09,KH09} give an alternative definition of random closed\nsets according to the Galton-Watson distribution and show that this definition\nproduces the same family of algorithmically random closed sets. The effective\nHausdorff dimension of members of random closed sets is studied in \\cite{DK09}.\n\nIn the present paper we will examine the notion of computable capacity\nand its relation to computable measures on the space $\\C$ of nonempty\nclosed sets.  In section two, we present a family of computable\nmeasures on $\\C$ and show how they induce capacities. We define the\nnotion of computable capacity and present an effective version of\nChoquet's theorem that every capacity can be obtained from a measure\n$\\mu^*$ on the space of closed sets. The main theorem of section three\ngives conditions under which the capacity $\\T(Q)$ of a $\\mu^*$-random\nclosed set $Q$ is either equal to $0$ or $> 0$.  We also construct a\n$\\Pi^0_1$ class with Lebesgue measure zero but with positive capacity,\nfor each capacity of a certain type.\n\n\\section{Computable Measure and Capacity on the Space of Closed Sets}\n\nIn this section, we describe the hit-or-miss topology on the space\n$\\C$ of closed sets, we define certain probability measures $\\mu_d$ on\nthe space $\\{0,1,2\\}^{\\N}$ and the corresponding measures $\\mu^*_d$ on\nthe homeomorphic space $\\C$. We present an effective version of\nChoquet's theorem connecting measure and capacity.\n\nThe standard (\\emph{hit-or-miss}) topology \\cite{D77} (p. 45)\non the space $\\C$ of closed sets is given by a sub-basis of sets of two types,\nwhere $U$ is any open set in $2^{\\N}$.\n\\[\nV(U) = \\{K: K \\cap U \\neq \\emptyset\\}; \\qquad \\qquad W(U) = \\{K: K \\subseteq U\\}\n\\]\n\nNote that $W(\\emptyset) = \\{\\emptyset\\}$ and that $V(\\TN) = \\C\n\\setminus \\{\\emptyset\\}$, so that $\\emptyset$ is an isolated element\nof $\\C$ under this topology.  Thus we may omit $\\emptyset$ from $\\C$\nwithout complications.\n\nA basis for the hit-or-miss topology may be formed by taking finite\nintersections of the basic open sets. We want to work with the\nfollowing simpler basis.  For each $n$ and each finite tree $A\n\\subseteq\\{0,1\\}^{\\leq n}$, let\n\n\\[\nU_A = \\{K\\in \\C: (\\forall \\sigma \\in A) (K \\cap I(\\sigma) \\neq \\emptyset)\\\n\\&\\ (\\forall \\sigma \\notin A) (K \\cap I(\\sigma) = \\emptyset) \\}.\n\\]\nThat is,\n\\[\nU_A = \\{K \\in \\C: T_K \\cap \\{0,1\\}^{\\leq n} = A\\}.\n\\]\nNote that the sets $U_A$ are in fact clopen. That is, for any tree $A\n\\subseteq \\{0,1\\}^{\\leq n}$, define the tree $A' = \\{\\sigma \\in\n\\{0,1\\}^{\\leq n}: (\\exists \\tau \\in \\{0,1\\}^n \\setminus A) \\sigma\n\\sqsubseteq \\tau\\}$. Then $U_{A'}$ is the complement of $U_A$.\n\nFor any finite $n$ and any tree $T \\subseteq \\{0,1\\}^{\\leq n}$, define the clopen set\n$[T] = \\cup_{\\sigma \\in T} I(\\sigma)$. Then $K \\cap [T] \\neq \\emptyset$\nif and only if there exists some $A \\subseteq \\{0,1\\}^{\\leq n}$ such that $K\n\\in U_A$ and $A \\cap T \\neq \\emptyset$. That is,\n\\[\nV([T]) = \\bigcup\\{U_A: A \\cap T \\neq \\emptyset\\}.\n\\]\nSimilarly, $K \\subseteq [T]$\nif and only if there exists some $A\\subseteq \\{0,1\\}^n$ such that $K\n\\in U_A$ and $A \\subseteq T$. That is,\n\\[\nW([T]) = \\bigcup \\{U_A: A \\subseteq T\\}.\n\\]\nThe following lemma can now be easily verified.\n\n\\begin{lemma} The family of sets $\\{U_A: A \\subseteq \\{0,1\\}^{\\leq n}\\ \\text{a tree}\\}$\nis a basis of clopen sets for the hit-or-miss topology on $\\C$.\n\\end{lemma}\n\nRecall the mapping from $\\C$ to $\\{0,1,2\\}^{\\N}$ taking $Q$ to $x_Q$.\nIt can be shown that this is in fact a homeomorphism.\n(See Axon \\cite{Ax10} for details.) Let $\\B^*$ be the family of clopen sets\nin $\\C$; each set is a finite union of basic sets of the form $U_A$ and thus $\\B^*$\nis a computable atomless Boolean algebra.\n\n\\begin{proposition} The space $\\C$ of nonempty closed subsets of $\\TN$ is homeomorphic\nto the space $\\{0,1,2\\}^{\\N}$. Furthermore, the corresponding map from $\\B$ to $\\B^*$\nis a computable isomorphism.\n\\end{proposition}\n\nNext we consider probability measures $\\mu$ on the space $\\{0,1,2\\}^{\\N}$ and the\ncorresponding measures $\\mu^*$ on $\\C$ induced by $\\mu$.\n\nA probability measure on $\\{0,1,2\\}^{\\N}$ may be defined as in \\cite{RSta}\nfrom a function $d: \\{0,1,2\\}^* \\to [0,1]$ such that $d(\\lambda) = 1$ and,\nfor any $\\sigma \\in\\{0,1,2\\}^*$,\n\\[\nd(\\sigma) = \\sum_{i=0}^2 d(\\sigma \\fr i).\n\\]\nThe corresponding measure $\\mu_d$ on $\\{0,1,2\\}^{\\N}$ is then defined\nby letting $\\mu_d(I(\\sigma)) = d(\\sigma)$. Since the intervals\n$I(\\sigma)$ form a basis for the standard product topology on\n$\\{0,1,2\\}^{\\N}$, this will extend to a measure on all Borel sets.  If\n$d$ is computable, then $\\mu_d$ is said to be computable. The measure\n$\\mu_d$ is said to be \\emph{nonatomic} or \\emph{continuous} if\n$\\mu_d(\\{x\\}) = 0$ for all $x \\in \\{0,1,2\\}^{\\N}$. We will say that\n$\\mu_d$ is \\emph{bounded} if there exist bounds $b,c \\in (0,1)$ such\nthat, for any $\\sigma \\in \\{0,1,2\\}^*$ and $i \\in \\{0,1,2\\}$,\n\\[\nb \\cdot d(\\sigma) < d(\\sigma \\fr i) < c \\cdot d(\\sigma).\n\\]\nIt is easy to see that any bounded measure must be continuous. We will\nsay that the measure $\\mu_d$ is \\emph{regular} if there exist constants\n$b_0,b_1,b_2$ with $b_0+b_1+b_2 = 1$ such that for all $\\sigma$ and\nfor $i \\leq 2$, $d(\\sigma \\fr i) = b_i d(\\sigma)$.\n\nNow let $\\mu_d^*$ be defined by \\[\n\\mu_d^*({\\mathcal X}) = \\mu_d(\\{x_Q: Q \\in {\\mathcal X}\\}).\n\\]\nLet us say that a measure $\\mu^*$ on $\\C$ is computable if the\nrestriction of $\\mu^*$ to $\\B^*$ is computable.\n\n\\begin{proposition}\nFor any computable $d$, the measure $\\mu^*_d$ is a computable measure on $\\C$.\n\\end{proposition}\n\n\\begin{proof}\nFor any tree $A \\subseteq \\{0,1\\}^{\\leq n}$, it is easy to see that\n\n\\[\nK \\in U_A \\iff \\rho_A \\sqsubset x_K,\n\\]\nso that $\\mu_d^*(U_A) = \\mu_d(I(\\rho_A))$.\n\\end{proof}\n\nWe are now ready to define capacity.  For details on capacity and random set\nvariables, see \\cite{Ng06}.\n\n\\begin{definition}\nA \\emph{capacity} on $\\C$ is a function $\\T: \\C \\to [0,1]$ with\n$\\T(\\emptyset) =0$ such that\n\\begin{itemize}\n\\item[(i)] $\\T$ is monotone increasing, that is,\n\\[\nQ_1 \\subseteq Q_2\n  \\longrightarrow \\T (Q_1) \\leq \\T(Q_2).\n\\]\n\\item[(ii)] $\\T$ has the \\emph{alternating of infinite order} property, that is,\nfor $n \\geq 2$ and any $Q_1, \\dots, Q_n \\in \\C$\n\\[\n\\T(\\bigcap_{i=1}^n Q_i) \\leq \\sum \\{(-1)^{|I|+1} \\T(\\bigcup_{i \\in\n  I}Q_i): \\emptyset \\neq I \\subseteq \\{1,2,\\dots,n\\} \\}.\n\\]\n\\item[(iii)] If $Q = \\cap_n Q_n$ and $Q_{n+1} \\subseteq Q_n$ for all\n$n$, then $\\T(Q) = lim_{n \\to \\infty} \\T(Q_n)$.\n\\end{itemize}\n\\end{definition}\n\nWe will also assume, unless otherwise specified, that the capacity\n$\\T(2^N) = 1$.\n\nWe will say that a capacity $\\T$ is computable if it is computable on\nthe family of clopen sets, that is, if there is a computable function $F$ from\nthe Boolean algebra $\\B$ of clopen sets into $[0,1]$ such that\n$F(B) = \\T(B)$ for any $B \\in \\B$.\n\nDefine $\\T_{d}(Q)=\\mu_d^*(V(Q))$.  That is, $\\T_{d}(Q)$ is the\nprobability that a randomly chosen closed set meets $Q$. Here is\nthe first result connecting measure and capacity.\n\n\\begin{theorem} \\label{th1}\nIf $\\mu^{*}_{d}$ is a (computable) probability measure on $\\C$, then\n$\\T_{d}$ is a (computable) capacity.\n\\end{theorem}\n\n\\begin{proof}\nCertainly $\\T_d(\\emptyset) = 0$. The\nalternating property follows by basic probability. For (iii), suppose\nthat $Q = \\cap_n Q_n$ is a decreasing intersection. Then by\ncompactness, $Q \\cap K \\neq \\emptyset$ if and only if $Q_n \\cap K \\neq\n\\emptyset$ for all $n$. Furthermore, $V(Q_{n+1}) \\subseteq V(Q_n)$ for\nall $n$. Thus\n\n\\[\n\\T_d(Q) = \\mu^*_d(V(Q)) = \\mu^*_d(\\cap_n V(Q_n)) = lim_n \\mu^*_d(V(Q_n)) =\nlim_n\\T_d(Q_n).\n\\]\nIf $d$ is computable, then $\\T_d$ may be computed as\nfollows.  For any clopen set $I(\\sigma_1) \\cup \\dots \\cup\nI(\\sigma_k)$ where each $\\sigma_i \\in \\{0,1\\}^n$, we compute the\nprobability distribution for all trees of height $n$ and add the\nprobabilities of those trees which contain one of the\n$\\sigma_i$.\n\\end{proof}\n\nChoquet's Capacity Theorem states that any capacity $\\T$ is determined by a measure,\nthat is $\\T = \\T_d$ for some $d$. See \\cite{Ng06} for details. We now give an\neffective version of Choquet's theorem.\n\n\\begin{theorem} [Effective Choquet Capacity Theorem] \\label{th2}\nIf $\\T$ is a computable capacity, then there is a computable measure\n$\\mu_d^*$ on the space of closed sets such that $\\T =\n\\T_d$. \\end{theorem}\n\n\\begin{proof}\nGiven the values $\\T(U)$ for all clopen sets $I(\\sigma_1)\\cup \\dots\n\\cup I(\\sigma_k)$ where each $\\sigma_i \\in \\{0,1\\}^n$, there is in\nfact a unique probability measure $\\mu_d$ on these clopen sets such\nthat $\\T = \\T_d$ and this can be computed as follows.\n\nSuppose first that $\\T(I(i)) = a_i$ for $i < 2$ and note that each\n$a_i \\leq 1$ and $a_0 + a_1 \\geq 1$ by the alternating property.  If\n$\\T = \\T_d$, then we must have $d((0)) + d((2)) = a_0$ and $d((1)) +\nd((2)) = a_1$ and also $d((0)) + d((1)) + d((2)) = 1$, so that $d((2))\n= a_0 + a_1 - 1$, $d((0)) = 1 - a_1$ and $d((1)) = 1 - a_0$. This will\nimply that $\\T(I\\tau)) = \\T_d(I(\\tau))$ when $|\\tau| = 1$. Now suppose that\nwe have defined $d(\\tau)$ and that $\\tau$ is the code for a finite\ntree with elements $\\sigma_0,\\dots,\\sigma_n =\\sigma$ and thus $d(\\tau\n\\fr i)$ is giving the probability that $\\sigma$ will have one or both\nimmediate successors. We proceed as above. Let $\\T(I(\\sigma \\fr i)) =\na_i \\cdot \\T(I(\\sigma))$ for $i<2$. Then as above $d(\\tau \\fr 2) =\nd(\\tau) \\cdot (a_0 + a_1 - 1)$ and $d(\\tau \\fr i) = d(\\tau) \\cdot (1 -\na_i)$ for each $i$.\n\\end{proof}\n\n\n\\section{When is $\\T(Q)=0$?}\n\nIn this section, we compute the capacity of a random closed set under certain\nprobability measures. We construct a \\pz class with measure zero but\nwith positive capacity.\n\nWe say that $K \\in \\C$ is \\emph{$\\mu_d^*$-random} if $x_K$ is \\ml random\nwith respect to the measure $\\mu_d$. (See \\cite{RSta} for details.)\n\nOur next  result shows that the $\\T_d$ capacity of a $\\mu^*_d$-random closed set\ndepends on the particular measure.\n\n\\begin{theorem}\\label{th4}\nLet $d$ be the uniform measure with $b_0 = b_1 = b > 0$ and $b_2 =\n1-2b > 0$ and let $\\ob = 1 - \\frac{\\sqrt 2}2$.  Then\n\\begin{itemize}\n\\item[(a)] If $b \\geq \\ob$, then\nfor any $\\mu_d^*$-random closed set $R$, $\\T_d(R) = 0$.\n\\item[(b)] If $b < \\ob$, then there is a\n$\\mu_d^*$-random closed set $R$ with $\\T_d(R) > 0$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\nFix $d$ as described above so that $d(\\sigma \\fr i) = d(\\sigma) \\cdot\nb$ and let $\\mu^* = \\mu_d^*$. We will compute the probability,\ngiven two closed sets $Q$ and $K$, that $Q \\cap K$ is nonempty.\nHere we define the usual product measure\non the product space $\\C \\times \\C$ of pairs $(Q,K)$ of nonempty closed sets\nby letting $\\mu^2(U_A \\times U_B) = \\mu^*(U_A) \\cdot \\mu^*(U_B)$\nfor arbitrary  subsets $A,B$ of $\\{0,1\\}^n$.\n\nLet\n\\[\nQ_n = \\bigcup \\{I(\\sigma): \\sigma \\in \\{0,1\\}^n\\ \\&\\ Q \\cap I(\\sigma)\\neq \\emptyset\\}\n\\]\nand similarly for $K_n$. Then $Q \\cap K \\neq \\emptyset$ if and only if\n$Q_n \\cap K_n \\neq \\emptyset$ for all $n$. Let $p_n$ be the\nprobability that $Q_n \\cap K_n \\neq \\emptyset$ for two arbitrary\nclosed sets $K$ and $Q$, relative to our measure $\\mu^*$. It is\nimmediate that $p_1 = 1 - 2b^2$, since $Q_1 \\cap K_1 = \\emptyset$ only\nwhen $Q_1 = I(i)$ and $K_1 = I(1-i)$. Next we will determine the\nquadratic function $f$ such that $p_{n+1} = f(p_n)$. There are 9\npossible cases for $Q_1$ and $K_1$, which break down into 4 distinct\ncases in the computation of $p_{n+1}$.\n\n\\medskip\n\n{\\bf Case (i)}: As we have seen, $Q_1 \\cap K_1 = \\emptyset$ with\nprobability $1 - 2b^2$.\n\n\\medskip\n\n{\\bf Case (ii)}: There are two chances that $Q_1 = K_1 = I(i)$, each\nwith probability $b^2$ so that $Q_{n+1} \\cap K_{n+1} \\neq \\emptyset$\nwith (relative) probability $p_n$.\n\n\\medskip\n\n{\\bf Case (iii)}: There are four chances where $Q_1 = \\TN$ and $K_1\n=I(i)$ or vice versa, each with probability $b \\cdot (1-2b)$, so that\nonce again $Q_{n+1} \\cap K_{n+1} \\neq\\emptyset$ with relative probability\n$p_n$.\n\n\\medskip\n\n{\\bf Case (iv)}: There is one chance that $Q_1 = K_1 = \\TN$, with\nprobability $(1 - 2b)^2$, in which case $Q_{n+1} \\cap K_{n+1} \\neq\n\\emptyset$ with relative probability $1 - (1 -p_n)^2 = 2p_n -\np_n^2$. This is because $Q_{n+1} \\cap K_{n+1} = \\emptyset$ if and only\nif both $Q_{n+1} \\cap I(i) \\cap K_{n+1} = \\emptyset$ for both $i=0$\nand $i=1$.\n\n\\medskip\n\nAdding these cases together, we see that\n\\[\np_{n+1} = [2b^2 + 4b (1-2b)] p_n + (1 - 2b)^2 (2p_n - p_n^2) = (2b^2 -\n4b + 2) p_n  - (1 - 4b + 4b^2) p_n^2.\n\\]\n\nNext we investigate the limit of the computable sequence $<p_n>_{n \\in\n \\omega}$. Let $f(p) =  (2b^2 - 4b + 2) p  - (1 - 4b + 4b^2) p^2$.\nNote that $f(0) = 0$ and $f(1) = 1 - 4b^2 < 1$.\nIt is easy to see that the fixed points of $f$ are $p=0$ and $p =\n \\frac{2b^2 -4b+1}{(1-2b)^2}$. Note that since $b < \\frac 12$, the\n denominator is not zero and hence is always positive.\n\nNow consider the function $g(b) = 2b^2 - 4b +1 = 2 (b-1)^2 - 1$, which has\npositive root $\\ob$ and is decreasing for $0 \\leq b \\leq 1$. There are\nthree cases to consider when comparing $b$ with $\\ob$.\n\n\\medskip\n\n{\\bf Case 1}: If $b > \\ob$, then $g(b) < 0$ and hence the other fixed\npoint of $f$ is negative. Furthermore, $2b^2 - 4b +2 < 1$ so that\n$f(p) < p$ for all $p > 0$. It follows that the sequence $\\{p_n: n \\in\n\\N\\}$ is decreasing with lower bound zero and hence must converge to a\nfixed point of $f$ (since $p_{n+1} = f(p_n)$). Thus $lim_n p_n = 0$.\n\n{\\bf Case 2}: If $b = \\ob$, then $g(b) = 0$ and $f(p) = p - (4b-1)\np^2$, so that $p=0$ is the unique fixed point of $f$. Furthermore,\n$4b-1 = 3 - 2 \\sqrt2 > 0$, so again $f(p) < p$ for all $p$. It follows\nagain that $lim_n p_n = 0$.\n\nIn these two cases, we can define a \\ml test to prove that $T_d(R) =\n0$\nfor any $\\mu$-random closed set $R$.\n\nFor each $m, n \\in \\omega$, let\n\\[\nB_m = \\{(K,Q): K_m \\cap Q_m \\neq \\emptyset\\},\n\\]\nso that $\\mu^*(B_m) = p_m$ and let\n\\[\nA_{m,n} = \\{Q: \\mu^*(\\{K: K_m \\cap Q_m \\neq \\emptyset\\}) \\geq\n2^{-n}\\}.\n\\]\n\n\\begin{claim} \\label{c1} For each $m$ and $n$, $\\mu^*(A_{m,n}) \\leq 2^n \\cdot p_m$.\n\\end{claim}\n\n\\emph{Proof of Claim \\ref{c1}}. Define the Borel measurable function $F_m: \\C\n\\times \\C: \\to \\{0,1\\}$ to be the characteristic function of\n$B_m$. Then\n\\[\np_m = \\mu^2(B_m) = \\int_{Q \\in \\C} \\int_{K \\in \\C} F(Q,K) dK dQ.\n\\]\nNow for fixed $Q$,\n\\[\n\\mu^*(\\{K: K_m \\cap Q_m \\neq \\emptyset\\}) = \\int_{K \\in \\C} F(Q,K) dK,\n\\]\nso that for $Q \\in A_{m,n}$, we have $\\int_{K \\in \\C} F(Q,K) dK \\geq 2^{-n}$.\nIt follows that\n\\begin{align*}\np_m = \\int_{Q \\in \\C} \\int_{K \\in \\C} F(Q,K) dK dQ &\\geq \\int_{Q \\in\n  A_{m,n}} \\int_{K \\in \\C} F(Q,K) dK dQ \\\\\n&\\geq \\int_{Q \\in A_{m,n}}\n2^{-n} dQ = 2^{-n} \\mu^*(A_{m,n}).\n\\end{align*}\nMultiplying both sides by $2^n$ completes the proof of Claim \\ref{c1}. $\\qed$\n\n\\medskip\n\nSince the computable sequence $<p_n>_{n \\in \\omega}$ converges to 0,\nthere must be a computable subsequence $m_0,m_1,\\dots$ such that\n$p_{m_n} < 2^{-2n-1}$ for all $n$. We can now define our \\ml test. Let\n\n\\[\nS_r = A_{m_r,r}\n\\]\nand let\n\\[\nV_n = \\cup_{r>n} S_r.\n\\]\nIt follows that\n\\[\n\\mu^*(A_n) \\leq 2^{n+1}\\mu^*(B_{m_n}) < 2^{n+1} 2^{-2n-1} = 2^{-n}\n\\]\nand therefore\n\\[\n\\mu^*(V_n) \\leq \\sum_{r>n} 2^{-r} = 2^{-n}\n\\]\nNow suppose that $R$ is a random closed set. The sequence $\\la V_n\n\\ra_{n \\in \\omega}$ is a computable sequence of c.e. open sets with\nmeasure $\\leq 2^{-n}$, so that there is some $n$ such that $R \\notin\nS_n$. Thus for all $r > n$, $\\mu^*(\\{K: K_{m_r} \\cap R_{m_r} \\neq\n\\emptyset\\}) < 2^{-r}$ and it follows that\n\\[\n\\mu^*(\\{K: K \\cap R \\neq \\emptyset\\}) = lim_n \\mu^*(\\{K: K_{m_n} \\cap\nR_{m_n} \\neq \\emptyset\\}) = 0.\n\\]\nThus $\\T_d(R) = 0$, as desired.\n\n\\medskip\n\n{\\bf Case 3}:\nFinally, suppose that $b < \\ob$. Then $0 < 2b^2 - 4b +1 < 1$, so that\n$f$ has a positive fixed point $m_b = \\frac{2b^2 -4b+1}{(1-2b)^2}$.\nIt is clear that $f(p) > p$ for $0 < p < m_b$ and $f(p) < p$ for $m_b\n< p$. Furthermore, the function $f$ has its maximum at $p =\n[\\frac{1-b}{1-2b}]^2 > 1$, so that $f$ is monotone increasing on\n$[0,1]$ and hence $f(p) > f(m_b) = m_b$ whenever $p > m_b$.  Observe that $p_0\n= 1 > m_b$ and hence the sequence $\\{p_n: n \\in \\N\\}$ is decreasing\nwith lower bound $m_b$. It follows that $lim_n p_n = m_b > 0$.\n\nNow $B = \\{(Q,K): Q \\cap K \\neq \\emptyset\\} = \\cap_n B_n$ is the\nintersection of a decreasing sequence of sets and hence $\\mu^2(B) =\nlim_n p_ = m_b >0$.\n\n\\begin{claim} \\label{c2} $\\mu^*(\\{Q: \\mu^*(\\{K: K \\cap Q \\neq\n\\emptyset\\}) > 0\\}) \\geq m_b$.\n\\end{claim}\n\n\\emph{Proof of Claim \\ref{c2}}. Let $B = \\{(K,Q): K \\cap Q \\neq \\emptyset$,\nlet $A = \\{Q: \\mu^*(\\{K: K \\cap Q \\neq \\emptyset\\}) > 0\\}$ and suppose\nthat $\\mu^*(A) < m_b$.  As in the proof of Claim \\ref{c1}, we have\n\\[\nm_b = \\mu^2(B) = \\int_{Q \\in \\C} \\int_{K \\in \\C} F(Q,K) dK dQ.\n\\]\nFor $Q \\notin A$, we have $\\int_{K \\in Q} F(Q,K) dK = \\mu^*(\\{K: K\n\\cap Q \\neq \\emptyset\\}) = 0$, so that\n\\[\nm_b = \\int_{Q \\in A} \\int_{K \\in Q} F(Q,K) dK dQ \\leq \\int_{Q \\in A} dQ = \\mu^*(A),\n\\]\nwhich completes the proof of Claim \\ref{c2}. $\\qed$\n\n\\begin{claim} \\label{c3} $\\{Q: \\T_d(Q) \\geq m_b\\}$ has positive measure.\n\\end{claim}\n\n\\emph{Proof of Claim \\ref{c3}}. Recall that $T_d(Q) = \\mu^*(\\{K: Q \\cap K \\neq \\emptyset\\})$.\nLet $B = \\{(K,Q): K \\cap Q \\neq \\emptyset$, let $A = \\{Q: T_d(Q) \\geq m_b\\}$\nand suppose that $\\mu^*(A) = 0$. As\nin the proof of Claim \\ref{c1}, we have\n\\[\nm_b = \\mu^2(B) = \\int_{Q \\in \\C} T_d(Q) dQ.\n\\]\nSince $\\mu^*(A) = 0$, it follows that for any $B \\subseteq \\C$, we have\n\\[\n\\int_{Q \\in B} T_d(Q) dQ \\leq m_b \\mu^*(B).\n\\]\nFurthermore, $T_d(Q) < m_b$ for almost all $Q$, so there exists some $P$ with\n$T_d(P) < m_b - \\epsilon$ for some positive $\\epsilon$. This means that for some\n$n$, $\\mu^*(\\{K: P_n \\cap K_n \\neq \\emptyset\\}) < m_b - \\epsilon$. Then for \\emph{any}\nclosed set $Q$ with $Q_n = P_n$, we have $T_d(Q) < m_b - \\epsilon$. But\n$E = \\{Q: Q_n = P_n\\}$ has positive measure, say $\\delta > 0$. Then we have\n\n\\begin{align*}\nm_b = \\int_{Q \\in \\C} T_d(Q) dQ &= \\int_{Q \\in E} T_d(Q) dQ\\ +\\ \\int_{Q \\notin E} T_d(Q) dQ \\\\\n&\\leq \\ \\delta (m_b - \\epsilon) + (1- \\delta) m_b = m_b - \\epsilon \\delta < m_b.\n\\end{align*}\n\nThis contradiction demonstrates Claim \\ref{c3}. $\\qed$\n\nSince the set of $\\mu*$-random closed sets has measure one, there must\nbe a random closed set $R$ such that $\\T_d(R) \\geq m_b$ and in\nparticular, there is a $\\mu^*$-random closed set with positive capacity.\n\\end{proof}\n\nThus for certain measures, there exists a random closed set with\nmeasure zero but with positive capacity. For the standard measure, a random closed set\nhas capacity zero.\n\n\\begin{corollary}\n\\label{th3}\nLet $d$ be the uniform measure with $b_0 = b_1 = b_2 = \\frac13$. Then\nfor any $\\mu_d^*$-random closed set $R$, $\\T_d(R) = 0$.\n\\end{corollary}\n\nA random closed set may not be effectively closed. But we can also\nconstruct an effectively closed set with measure zero and with\npositive capacity.\n\n\\begin{theorem}  \\label{th5}\nFor the regular measure $\\mu_d$ with $b = b_1 = b_2$, there is a\n$\\Pi^0_1$ class $Q$ with Lebesgue measure zero and positive capacity\n$\\T_d(Q).$\n\\end{theorem}\n\n\\begin{proof}\nFirst let us compute the capacity of $X_n = \\{x: x(n) =0\\}$. For\n$n=0$, we have $\\T_d(X_0) = 1 - b$. That is, $Q$ meets $X_0$ if and\nonly if $Q_0 = I(0)$ (which occurs with probability $b$), or $Q_0 =\n\\TN$ (which occurs with probability $1 - 2b$. Now the probability\n$\\T_d(X_{n+1})$ that an arbitrary closed set $K$ meets $X_{n+1}$ may\nbe calculated in two distinct cases.  As in the proof of Theorem\n\\ref{th3}, let\n\\[\nK_n = \\bigcup \\{I(\\sigma): \\sigma \\in \\{0,1\\}^n\\ \\&\\ K \\cap I(\\sigma)\\neq \\emptyset\\}\n\\]\n\n{\\bf Case I} If $K_0 = \\TN$, then $\\T_d(X_{n+1}) = 1 - (1- \\T_d(X_n))^2$.\n\n\\medskip\n\n{\\bf Case II} If $K_0 = I((i))$ for some $i<2$, then $\\T_d(X_{n+1}) = \\T_d(X_n)$.\n\n\\medskip\n\nIt follows that\n\\begin{align*}\n\\T_d(X_{n+1}) &= 2b \\T_d(X_n) + (1-2b) (2 \\T_d(X_n)\n- (\\T_d(X_n))^2) \\\\\n&= (2-2b) \\T_d(X_n) - (1-2b) (\\T_d(X_n))^2\n\\end{align*}\n\nNow consider the function $f(p) = (2-2b) p - (1-2b) p^2$, where $0 < b\n < \\frac 12$. This function has the properties that $f(0) = 0$, $f(1) =\n 1$ and $f(p) > p$ for $0 < p < 1$. Since $\\T_d(X_{n+1})\n= f(\\T_d(X_n))$, it follows that $lim_n \\T_d(X_n) = 1$ and is the limit\nof a computable sequence.\n\nFor any $\\sigma = (n_0, n_1, \\dots, n_k)$, with $n_0 < n_1 < \\cdots\n<n_k$, similarly define $X_{\\sigma} = \\{x: (\\forall i<k) x(n_i) =\n0\\}$.  A similar argument to that above shows that $lim_n\n\\T_d(X_{\\sigma \\fr n}) \/ \\T_d(X_{\\sigma}) = 1$.\n\nNow consider the decreasing sequence $c_k = \\frac{2^{k+1} +\n1}{2^{k+2}}$ with limit $\\frac12$.  Choose $n = n_0$ such that\n$\\T_d(X_n) \\geq \\frac 34 = c_0$ and for each $k$, choose $n = n_{k+1}$\nsuch that $\\T_d(X_{(n_0,\\dots,n_k,n)}) \\geq c_{k+1}$. This can be done\nsince $c_{k+1} < c_k$. Finally, let $Q = \\bigcap_k\nX_{(n_0,\\dots,n_k)}$.  Then $\\T_d(Q) = lim_k \\T_d(X_{(n_0,\\dots,n_k)})\n\\geq lim_k c_k=\\frac12$.\n\\end{proof}\n\n\\section{Conclusions and Future Research}\n\nIn this paper, we have established a connection between measure and\ncapacity for the space $\\C$ of closed subsets of $\\TN$. We showed that\nfor a computable\nmeasure $\\mu^*$, a computable capacity may be defined by letting\n$\\T(Q)$ be the measure of the family of closed sets $K$ which have\nnonempty intersection with $Q$. We have proved an effective\nversion of the Choquet's theorem by showing that every computable\ncapacity may be obtained from a computable measure in this way.\n\nFor the uniform measure $\\mu$ under which a node $\\sigma$ in $T$ has\nexactly one immediate extension $\\sigma \\fr i$ with probability $b$\nfor $i=0,1$ (and hence $\\sigma$ has both extensions with probability\n$1-2b$), we have established conditions on $b$ that characterize when\nthe capacity of a random closed set equals zero or is $>0$. We have\nalso constructed for each such measure an effectively closed set with\npositive capacity and with Lebesgue measure zero.\n\n\nIn future work, we plan to extend our results to more general measures\nwhere for each string $\\sigma \\in T_Q$, the probability that $\\sigma\n\\fr i \\in T_Q$ depends on $\\sigma$. For example, such a measure on the\nspace of closed sets may be defined  by making the probability that\nboth extensions $\\sigma \\fr i$ of a node $\\sigma \\in T$ belong to $T$\nequal to $1 - \\frac 2n$ and the probability that just one extension\nbelongs to $T$ equal to $\\frac 1n$, where $n = |\\sigma|$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgements}\nThis work was supported in parts by the Siberian Branch of the Russian Academy of Science (project No. 0305-2017-0021), a Leverhulme Trust Research Project Grant RPG-2017-143 and by STFC (AWAKE-UK, Cockroft Institute core and UCL consolidated grants), United Kingdom; a Deutsche Forschungsgemeinschaft project grant PU 213-6\/1 ``Three-dimensional quasi-static simulations of beam self-modulation for plasma wakefield acceleration''; the National Research Foundation of Korea (Nos.\\ NRF-2015R1D1A1A01061074 and NRF-2016R1A5A1013277); the Portuguese FCT---Foundation for Science and Technology, through grants CERN\/FIS-TEC\/0032\/2017, PTDC-FIS-PLA-2940-2014, UID\/FIS\/50010\/2013 and SFRH\/IF\/01635\/2015; NSERC and CNRC for TRIUMF's contribution; and the Research Council of Norway. M. Wing acknowledges the support of the Alexander von Humboldt Stiftung and DESY, Hamburg. The AWAKE collaboration acknowledge the SPS team for their excellent proton delivery.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $W=\\left\\{ W_{t}:t\\geq 0\\right\\} $ be a standard Brownian motion\ninitialized at zero, set $W_{t}^{\\ast }=\\max_{s\\leq t}\\left\\vert\nW_{s}\\right\\vert $ and write $\\mathcal{F}_{t}^{W}=\\sigma \\left\\{ W_{u}:u\\leq\nt\\right\\} $, $t\\geq 0$. In \\cite{DS}, Davis and Suh proved the following\nresult.\n\n\\begin{theorem}[{\\protect\\cite[Th. 1.1]{DS}}]\n\\label{Th : DavisSuh}For every $p>0$ and every $c\\in \\mathbb{R}$, set\n\\begin{eqnarray}\nY_{t} &=&Y_{t}\\left( c,p\\right) =\\left( W_{t}^{\\ast }\\right) ^{p-2}\\left[\nW_{t}^{2}-t\\right] +c\\left( W_{t}^{\\ast }\\right) ^{p}\\text{, \\ \\ }t>0\\text{,}\n\\label{Y} \\\\\nY_{0}\\left( c,p\\right) &=&Y_{0}=0.  \\notag\n\\end{eqnarray}\n\n\\begin{enumerate}\n\\item For every $p\\in (0,2]$, the process $Y_{t}$ is a $\\mathcal{F}_{t}^{W}$%\n-submartingale if, and only if, $c\\geq \\frac{2-p}{p}.$\n\n\\item For every $p\\in \\lbrack 2,+\\infty )$, the process $Y_{t}$ is a $%\n\\mathcal{F}_{t}^{W}$-supermartingale if, and only if, $c\\leq \\frac{2-p}{p}.$\n\\end{enumerate}\n\\end{theorem}\n\n\\bigskip\n\nAs pointed out in \\cite[p. 314]{DS} and in Section \\ref{S : BDG} below, part\n1 of Theorem \\ref{Th : DavisSuh} can be used to derive explicit expressions\nof the constants appearing in the Burkholder-Davis-Gundy (BDG) inequalities\n(see \\cite{Burk73}, or \\cite[Ch. IV, \\S 4]{RY}). The proof of Theorem \\ref%\n{Th : DavisSuh} given in \\cite{DS} uses several delicate estimates related\nto a class of Brownian hitting times: such an approach can be seen as a\nramification of the discrete-time techniques developed in \\cite{Burkhj}. In\nparticular, in \\cite{DS} it is observed that the submartingale (or\nsupermartingale) characterization of $Y_{t}\\left( c,p\\right) $ basically\nrelies on the properties of the random subset of $[0,+\\infty )$ composed of\nthe instants $t$ where $\\left\\vert W_{t}\\right\\vert =W_{t}^{\\ast }$. The aim\nof this note is to bring this last connection into further light, by\nproviding an elementary proof of Theorem \\ref{Th : DavisSuh}, based on a\ndirect application of It\\^{o} formula and on an appropriate version of the\nDoob-Meyer decomposition of submartingales. We will see that our techniques\nlead naturally to some substantial generalizations (see Theorem \\ref{T : DS}\nbelow).\n\n\\bigskip\n\nThe rest of the paper is organized as follows. In Section \\ref{S : Gen} we\nstate and prove a general result involving a class of stochastic processes\nthat are functions of a positive submartingale and of a monotone\ntransformation of its maximum. In Section \\ref{S : Proof} we focus once\nagain on the Brownian setting, and establish a generalization of Theorem \\ref%\n{Th : DavisSuh}. Section \\ref{S : BDG} deals with an application of the\nprevious results to (strong) BDG inequalities. Finally, in Section \\ref{S :\nBal} we provide an explicit connection with some classic \\textsl{balayage\nformulae} for continuous-time semimartingales (see e.g. \\cite{MY}).\n\nAll the objects appearing in the subsequent sections are defined on a common\nprobability space $\\left( \\Omega ,\\mathfrak{A},\\mathbb{P}\\right) $.\n\n\\section{A general result \\label{S : Gen}}\n\nThroughout this section, $\\mathcal{F}=\\left\\{ \\mathcal{F}_{t}:t\\geq\n0\\right\\} $ stands for a filtration satisfying the usual conditions. We will\nwrite $X=\\left\\{ X_{t}:t\\geq 0\\right\\} $ to indicate a \\textsl{continuous\\ }$%\n\\mathcal{F}_{t}$-\\textsl{submartingale }issued from zero and such that $%\n\\mathbb{P}\\left\\{ X_{t}\\geq 0\\text{, \\ }\\forall t\\right\\} =1$. We will\nsuppose that the Doob-Meyer decomposition of $X$ (see for instance \\cite[Th.\n1.4.14]{KS}) is of the type $X_{t}=M_{t}+A_{t}$, $t\\geq 0$, where $M$ is a\n\\textsl{square-integrable} continuous $\\mathcal{F}_{t}$-martingale issued\nfrom zero, and $A$ is an increasing (integrable) natural process. We assume\nthat $A_{0}=M_{0}=0$; the symbol $\\left\\langle M\\right\\rangle =\\left\\{\n\\left\\langle M\\right\\rangle _{t}:t\\geq 0\\right\\} $ stands for the quadratic\nvariation of $M$. We note $X_{t}^{\\ast }=\\max_{s\\leq t}X_{s}$, and we also\nsuppose that $\\mathbb{P}\\left\\{ X_{t}^{\\ast }>0\\right\\} =1$ for every $t>0$.\nThe following result is a an extension of Theorem \\ref{Th : DavisSuh}.\n\n\\begin{theorem}\n\\label{Th : general}Fix $\\varepsilon >0$.\n\n\\begin{enumerate}\n\\item Suppose that the function $\\phi :(0,+\\infty )\\mapsto \\mathbb{R}$ is of\nclass $C^{1}$, non-increasing, and such that\n\\begin{equation}\n\\mathbb{E[}\\int_{\\varepsilon }^{T}\\phi \\left( X_{s}^{\\ast }\\right)\n^{2}d\\left\\langle M\\right\\rangle _{s}]<+\\infty \\text{,}  \\label{int}\n\\end{equation}%\nfor every $T>\\varepsilon $. For every $x\\geq z>0$, we set\n\\begin{equation}\n\\Phi \\left( x,z\\right) =-\\int_{z}^{x}y\\phi ^{\\prime }\\left( y\\right) dy;\n\\label{PHI}\n\\end{equation}%\nthen, for every $\\alpha \\geq 1$ the process\n\\begin{equation}\nZ_{\\varepsilon }\\left( \\phi ,\\alpha ;t\\right) =\\phi \\left( X_{t}^{\\ast\n}\\right) \\left( X_{t}-A_{t}\\right) +\\alpha \\Phi \\left( X_{t}^{\\ast\n},X_{\\varepsilon }^{\\ast }\\right) \\text{, \\ }t\\geq \\varepsilon \\text{,}\n\\label{subsup}\n\\end{equation}%\nis a $\\mathcal{F}_{t}$-submartingale on $[\\varepsilon ,+\\infty )$.\n\n\\item Suppose that the function $\\phi :(0,+\\infty )\\mapsto \\mathbb{R}$ is of\nclass $C^{1}$, non-decreasing and such that (\\ref{int}) holds for every $%\nT>\\varepsilon .$ Define $\\Phi \\left( \\cdot ,\\cdot \\right) $ according to (%\n\\ref{PHI}), and $Z_{\\varepsilon }\\left( \\phi ,\\alpha ;t\\right) $ according\nto (\\ref{subsup}). Then, for every $\\alpha \\geq 1$ the process $%\nZ_{\\varepsilon }\\left( \\phi ,\\alpha ;t\\right) $ is a $\\mathcal{F}_{t}$%\n-supermartingale on $[\\varepsilon ,+\\infty )$.\n\\end{enumerate}\n\\end{theorem}\n\n\\bigskip\n\n\\textbf{Remarks. }(i) Note that the function $\\phi \\left( y\\right) $ (and $%\n\\phi ^{\\prime }\\left( y\\right) $) need not be defined at $y=0$.\n\n(ii) In Section \\ref{S : Proof}, where we will focus on the Brownian\nsetting, we will exhibit specific examples where the condition $\\alpha \\geq\n1 $ is necessary and sufficient to have that the process $Z_{\\varepsilon\n}\\left( \\alpha ,\\phi ;t\\right) $ is a submartingale (when $\\phi $ is\nnon-increasing) or a supermartingale (when $\\phi $ is non-decreasing).\n\n\\bigskip\n\n\\textbf{Proof of Theorem \\ref{Th : general}. }(\\textsl{Proof of Point 1.})\nObserve first that, since $M_{t}=X_{t}-A_{t}$ is a continuous martingale, $%\nX^{\\ast }$ is non-decreasing and $\\phi $ is differentiable, then a standard\napplication of It\\^{o} formula gives that\n\\begin{eqnarray}\n\\phi \\left( X_{t}^{\\ast }\\right) \\left( X_{t}-A_{t}\\right) -\\phi \\left(\nX_{\\varepsilon }^{\\ast }\\right) \\left( X_{\\varepsilon }-A_{\\varepsilon\n}\\right) &=&\\phi \\left( X_{t}^{\\ast }\\right) M_{t}-\\phi \\left(\nX_{\\varepsilon }^{\\ast }\\right) M_{\\varepsilon }  \\notag \\\\\n&=&\\int_{\\varepsilon }^{t}\\phi (X_{s}^{\\ast })dM_{s}+\\int_{\\varepsilon\n}^{t}\\left( X_{s}-A_{s}\\right) \\phi ^{\\prime }\\left( X_{s}^{\\ast }\\right)\ndX_{s}^{\\ast }.  \\label{uyu}\n\\end{eqnarray}%\nThe assumptions in the statement imply that the application $\\widetilde{M}%\n_{\\varepsilon ,t}:=\\int_{\\varepsilon }^{t}\\phi (X_{s}^{\\ast })dM_{s}$ is a\ncontinuous square integrable $\\mathcal{F}_{t}$-martingale on $[\\varepsilon\n,+\\infty )$. Moreover, the continuity of $X$ implies that the support of the\nrandom measure $dX_{t}^{\\ast }$ (on $[0,+\\infty )$) is contained in the\n(random) set $\\left\\{ t\\geq 0:X_{t}=X_{t}^{\\ast }\\right\\} $, thus yielding\nthat\n\\begin{eqnarray*}\n\\int_{\\varepsilon }^{t}\\left( X_{s}-A_{s}\\right) \\phi ^{\\prime }\\left(\nX_{s}^{\\ast }\\right) dX_{s}^{\\ast } &=&\\int_{\\varepsilon }^{t}\\left(\nX_{s}^{\\ast }-A_{s}\\right) \\phi ^{\\prime }\\left( X_{s}^{\\ast }\\right)\ndX_{s}^{\\ast } \\\\\n&=&-\\int_{\\varepsilon }^{t}A_{s}\\phi ^{\\prime }\\left( X_{s}^{\\ast }\\right)\ndX_{s}^{\\ast }-\\Phi \\left( X_{t}^{\\ast },X_{\\varepsilon }^{\\ast }\\right) ,\n\\end{eqnarray*}%\nwhere $\\Phi $ is defined in (\\ref{PHI}). As a consequence,\n\\begin{equation}\nZ_{\\varepsilon }\\left( \\phi ,\\alpha ;t\\right) =\\widetilde{M}_{\\varepsilon\n,t}+\\int_{\\varepsilon }^{t}(-A_{s}\\phi ^{\\prime }\\left( X_{s}^{\\ast }\\right)\n)dX_{s}^{\\ast }+\\left( \\alpha -1\\right) \\Phi \\left( X_{t}^{\\ast\n},X_{\\varepsilon }^{\\ast }\\right) \\text{.}  \\label{a}\n\\end{equation}%\nNow observe that the application $t\\mapsto \\Phi \\left( X_{t}^{\\ast\n},X_{\\varepsilon }^{\\ast }\\right) $ is non-decreasing (a.s.-$\\mathbb{P)}$,\nand also that, by assumption, $-A_{s}\\phi ^{\\prime }\\left( X_{s}^{\\ast\n}\\right) \\geq 0$ for every $s>0$. This entails immediately that $%\nZ_{\\varepsilon }\\left( \\phi ,\\alpha ;t\\right) $ is a $\\mathcal{F}_{t}$%\n-submartingale for every $\\alpha \\geq 1$.\n\n(\\textsl{Proof of Point 2.}) By using exactly the same line of reasoning as\nin the proof of Point 1., we obtain that\n\\begin{equation}\nZ_{\\varepsilon }\\left( \\phi ,\\alpha ;t\\right) =\\int_{\\varepsilon }^{t}\\phi\n(X_{s}^{\\ast })dM_{s}+\\int_{\\varepsilon }^{t}(-A_{s}\\phi ^{\\prime }\\left(\nX_{s}^{\\ast }\\right) )dX_{s}^{\\ast }+\\left( \\alpha -1\\right) \\Phi \\left(\nX_{t}^{\\ast },X_{\\varepsilon }^{\\ast }\\right) .  \\label{aa}\n\\end{equation}%\nSince (\\ref{int}) is in order, we deduce that $t\\mapsto \\int_{\\varepsilon\n}^{t}\\phi (X_{s}^{\\ast })dM_{s}$ is a continuous (square-integrable) $%\n\\mathcal{F}_{t}$-martingale on $[\\varepsilon ,+\\infty )$. Moreover, $%\n-A_{s}\\phi ^{\\prime }\\left( X_{s}^{\\ast }\\right) \\leq 0$ for every $s>0$,\nand we also have that $t\\mapsto \\Phi \\left( X_{t}^{\\ast },X_{\\varepsilon\n}^{\\ast }\\right) $ is a.s. decreasing. This implies that $Z_{\\varepsilon\n}\\left( \\phi ,\\alpha ;t\\right) $ is a $\\mathcal{F}_{t}$-supermartingale for\nevery $\\alpha \\geq 1$. \\ \\ $\\blacksquare $\n\n\\bigskip\n\nThe next result allows to characterize the nature of the process $Z$\nappearing in (\\ref{subsup}) on the whole positive axis. Its proof can be\nimmediately deduced from formulae (\\ref{a}) (for Part 1) and (\\ref{aa}) (for\nPart 2).\n\n\\begin{proposition}\n\\label{P : epsilon}Let the assumptions and notation of this section prevail.\n\n\\begin{enumerate}\n\\item Consider a decreasing function $\\phi :(0,+\\infty )\\mapsto \\mathbb{R}$\nverifying the assumptions of Part 1 of Theorem \\ref{Th : general} and such\nthat\n\\begin{equation}\n\\Phi \\left( x,0\\right) :=-\\int_{0}^{x}y\\phi ^{\\prime }\\left( y\\right) dy%\n\\text{ is finite }\\forall x>0\\text{.}  \\label{f1}\n\\end{equation}%\nAssume moreover that\n\\begin{equation}\n\\mathbb{E[}\\int_{0}^{T}\\phi \\left( X_{s}^{\\ast }\\right) ^{2}d\\left\\langle\nM\\right\\rangle _{s}]<+\\infty \\text{,}  \\label{f2}\n\\end{equation}%\nand also\n\\begin{eqnarray}\n&&\\phi \\left( X_{\\varepsilon }^{\\ast }\\right) M_{\\varepsilon }=\\phi \\left(\nX_{\\varepsilon }^{\\ast }\\right) \\left( X_{\\varepsilon }-A_{\\varepsilon\n}\\right) \\text{ converges to zero in }L^{1}\\left( \\mathbb{P}\\right) ,\\text{\nas }\\varepsilon \\downarrow 0,  \\label{f3} \\\\\n&&\\Phi \\left( X_{t}^{\\ast },0\\right) \\in L^{1}\\left( \\mathbb{P}\\right) \\text{%\n.}  \\label{f4}\n\\end{eqnarray}%\nThen, for every $\\alpha \\geq 1$ the process\n\\begin{equation}\nZ\\left( \\phi ,\\alpha ;t\\right) =\\left\\{\n\\begin{array}{ll}\n0 & \\text{for }t=0 \\\\\n\\phi \\left( X_{t}^{\\ast }\\right) \\left( X_{t}-A_{t}\\right) +\\alpha \\Phi\n\\left( X_{t}^{\\ast },0\\right) & \\text{for }t>0%\n\\end{array}%\n\\right. ,  \\label{zz}\n\\end{equation}%\nis a $\\mathcal{F}_{t}$-submartingale.\n\n\\item Consider an increasing function $\\phi :(0,+\\infty )\\mapsto \\mathbb{R}$\nas in Part 2 of Theorem \\ref{Th : general} and such that assumptions (\\ref%\n{f1})--(\\ref{f4}) are satisfied. Then, for every $\\alpha \\geq 1$ the process\n$Z\\left( \\phi ,\\alpha ;t\\right) $ appearing in (\\ref{zz}) is a $\\mathcal{F}%\n_{t}$-supermartingale.\n\\end{enumerate}\n\\end{proposition}\n\n\\bigskip\n\n\\textbf{Remarks. }(i) A direct application of the Cauchy-Schwarz inequality\nshows that a sufficient condition to have (\\ref{f3}) is the following:%\n\\begin{equation}\n\\lim_{\\varepsilon \\downarrow 0}\\mathbb{E}\\left[ \\phi \\left( X_{\\varepsilon\n}^{\\ast }\\right) ^{2}\\right] \\times \\mathbb{E}\\left[ M_{\\varepsilon }^{2}%\n\\right] =\\lim_{\\varepsilon \\downarrow 0}\\mathbb{E}\\left[ \\phi \\left(\nX_{\\varepsilon }^{\\ast }\\right) ^{2}\\right] \\times \\mathbb{E}\\left[\n\\left\\langle M\\right\\rangle _{\\varepsilon }\\right] =0  \\label{cv0}\n\\end{equation}%\n(observe that $\\lim_{\\varepsilon \\downarrow 0}\\mathbb{E}\\left[\nM_{\\varepsilon }^{2}\\right] =0$, since $M_{0}=0$ by assumption). In other\nwords, when (\\ref{cv0}) is verified the quantity $\\mathbb{E}\\left[\nM_{\\varepsilon }^{2}\\right] $ `takes care' of the possible explosion of $%\n\\varepsilon \\mapsto \\mathbb{E}\\left[ \\phi \\left( X_{\\varepsilon }^{\\ast\n}\\right) ^{2}\\right] $ near zero.\n\n(ii) Let $\\phi $ be non-increasing or non-decreasing on $\\left( 0,+\\infty\n\\right) $, and suppose that $\\phi $ satisfies the assumptions of Theorem \\ref%\n{Th : general} and Proposition \\ref{P : epsilon}. Then, the process $%\nt\\mapsto \\int_{0}^{t}\\phi (X_{s}^{\\ast })dM_{s}$ is a continuous\nsquare-integrable $\\mathcal{F}_{t}^{W}$-martingale. Moreover, for any choice\nof $\\alpha \\in \\mathbb{R}$, the process $Z\\left( \\phi ,\\alpha ;t\\right) $, $%\nt\\geq 0$, defined in (\\ref{zz}) is a semimartingale, with canonical\ndecomposition given by\n\\begin{equation*}\nZ\\left( \\phi ,\\alpha ;t\\right) =\\int_{0}^{t}\\phi (X_{s}^{\\ast\n})dM_{s}+\\int_{0}^{t}\\left( (\\alpha -1)X_{s}^{\\ast }-A_{s}\\right) \\phi\n^{\\prime }\\left( X_{s}^{\\ast }\\right) dX_{s}^{\\ast }.\n\\end{equation*}\n\n\\section{A generalization of Theorem \\protect\\ref{Th : DavisSuh} \\label{S :\nProof}}\n\nThe forthcoming Theorem \\ref{T : DS} is a generalization of Theorem \\ref{Th\n: DavisSuh}. Recall the notation: $W$ is a standard Brownian motion issued\nfrom zero, $W_{t}^{\\ast }=\\max_{s\\leq t}\\left\\vert W_{s}\\right\\vert $ and $%\n\\mathcal{F}_{t}^{W}=\\sigma \\left\\{ W_{u}:u\\leq t\\right\\} $. We also set for\nevery $m\\geq 1$, every $p>0$ and every $c\\in \\mathbb{R}$:\n\\begin{eqnarray}\nJ_{t} &=&J_{t}\\left( m,c,p\\right) =\\left( W_{t}^{\\ast }\\right) ^{p-m}\\left[\n\\left\\vert W_{t}\\right\\vert ^{m}-A_{m,t}\\right] +c\\left( W_{t}^{\\ast\n}\\right) ^{p}\\text{, \\ \\ }t>0\\text{,}  \\label{ggei} \\\\\nJ_{0}\\left( m,c,p\\right) &=&J_{0}=0,  \\notag\n\\end{eqnarray}%\nwhere $t\\mapsto A_{m,t}$ is the increasing natural process in the Doob-Meyer\ndecomposition of the $\\mathcal{F}_{t}^{W}$-submartingale $t\\mapsto\n\\left\\vert W_{t}\\right\\vert ^{m}$. Of course, $J_{t}\\left( 2,c,p\\right)\n=Y_{t}\\left( c,p\\right) $, as defined in (\\ref{Y}).\n\n\\begin{theorem}\n\\label{T : DS}Under the above notation:\n\n\\begin{enumerate}\n\\item For every $p\\in (0,m]$, the process $J_{t}$ is a $\\mathcal{F}_{t}^{W}$%\n-submartingale if, and only if, $c\\geq \\frac{m-p}{p}.$\n\n\\item For every $p\\in \\lbrack m,+\\infty )$, the process $J_{t}$ is a $%\n\\mathcal{F}_{t}^{W}$-supermartingale if, and only if, $c\\leq \\frac{m-p}{p}.$\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nRecall first the following two facts: (i) $W_{t}^{\\ast }\\overset{law}{=}%\n\\sqrt{t}W_{1}^{\\ast }$ (by scaling), and (ii) there exists $\\eta >0$ such\nthat $\\mathbb{E}\\left[ \\exp (\\eta \\left( W_{1}^{\\ast }\\right) ^{-2})\\right]\n<+\\infty $ (this can be deduced e.g. from \\cite[Ch. II, Exercice 3.10]{RY}),\nso that the random variable $\\left( W_{1}^{\\ast }\\right) ^{-1}$ has finite\nmoments of all orders. Note also that the conclusions of both Point 1 and\nPoint 2 are trivial in the case where $p=m$. In the rest of the proof we\nwill therefore assume that $p\\neq m$.\n\nTo prove Point 1, we shall apply Theorem \\ref{Th : general} and Proposition %\n\\ref{P : epsilon} in the following framework: $X_{t}=\\left\\vert\nW_{t}\\right\\vert ^{m}$ and $\\phi \\left( x\\right) =x^{\\frac{p-m}{m}}=x^{\\frac{%\np}{m}-1}$. In this case, the martingale $M_{t}=\\left\\vert W_{t}\\right\\vert\n^{m}-A_{m,t}$ is such that $\\left\\langle M\\right\\rangle\n_{t}=m^{2}\\int_{0}^{t}W_{s}^{2m-2}ds$, $t\\geq 0$, and $\\Phi \\left(\nx,z\\right) =-\\int_{z}^{x}y\\phi ^{\\prime }\\left( y\\right) dy=-\\left( \\frac{p}{%\nm}-1\\right) \\int_{z}^{x}y^{\\frac{p}{m}-1}dy=\\frac{m-p}{p}\\left( x^{\\frac{p}{m%\n}}-z^{\\frac{p}{m}}\\right) $. Also, for every $T>\\varepsilon >0$%\n\\begin{eqnarray}\n\\mathbb{E[}\\int_{\\varepsilon }^{T}\\phi \\left( X_{s}^{\\ast }\\right)\n^{2}d\\left\\langle M\\right\\rangle _{s}] &=&m^{2}\\mathbb{E[}\\int_{\\varepsilon\n}^{T}\\left( W_{s}^{\\ast }\\right) ^{2p-2m}W_{s}^{2m-2}ds]  \\notag \\\\\n&\\leq &m^{2}\\mathbb{E[}\\int_{\\varepsilon }^{T}\\left( W_{s}^{\\ast }\\right)\n^{2p-2}ds]=m^{2}\\mathbb{E[}\\left( W_{1}^{\\ast }\\right) ^{2p-2}\\mathbb{]}%\n\\int_{\\varepsilon }^{T}s^{\\frac{p}{2}-1}ds\\text{,}  \\label{jjj}\n\\end{eqnarray}%\nso that $\\phi $ verifies (\\ref{int}) and (\\ref{f2}). Relations (\\ref{f1})\nand (\\ref{f4}) are trivially satisfied. To see that (\\ref{f3}) holds, use\nthe relations\n\\begin{eqnarray*}\n\\mathbb{E}\\left\\{ \\left\\vert \\phi \\left( X_{\\varepsilon }^{\\ast }\\right)\n\\left( X_{\\varepsilon }-A_{\\varepsilon }\\right) \\right\\vert \\right\\} &=&%\n\\mathbb{E\\{}\\left\\vert \\left( W_{\\varepsilon }^{\\ast }\\right) ^{p-m}\\left[\n\\left\\vert W_{\\varepsilon }\\right\\vert ^{m}-A_{m,\\varepsilon }\\right]\n\\right\\vert \\} \\\\\n&=&\\mathbb{E}\\left\\{ \\left\\vert \\left( W_{\\varepsilon }^{\\ast }\\right)\n^{p-m}M_{\\varepsilon }\\right\\vert \\right\\} \\leq \\mathbb{E}\\left\\{ \\left(\nW_{\\varepsilon }^{\\ast }\\right) ^{2p-2m}\\right\\} ^{1\/2}\\mathbb{E}\\left\\{\n\\left\\langle M\\right\\rangle _{\\varepsilon }\\right\\} ^{1\/2} \\\\\n&=&m\\mathbb{E}\\left\\{ W_{1}^{2m-2}\\right\\} ^{1\/2}\\mathbb{E}\\left\\{ \\left(\nW_{1}^{\\ast }\\right) ^{2p-2m}\\right\\} ^{1\/2}\\varepsilon ^{\\frac{p}{2}-\\frac{m%\n}{2}}\\left( \\int_{0}^{\\varepsilon }s^{m-1}ds\\right) ^{1\/2} \\\\\n&\\rightarrow &0\\text{, \\ as }\\varepsilon \\downarrow 0\\text{.}\n\\end{eqnarray*}%\nFrom Point 1 of Proposition \\ref{P : epsilon}, we therefore deduce that the\nprocess $Z\\left( t\\right) $ defined as $Z\\left( 0\\right) =0$ and, for $t>0$,%\n\\begin{eqnarray}\nZ\\left( t\\right) &=&\\phi \\left( \\left( W_{t}^{\\ast }\\right) ^{m}\\right)\n\\left[ \\left\\vert W_{t}\\right\\vert ^{m}-A_{m,t}\\right] +\\alpha \\Phi \\left(\n\\left( W_{t}^{\\ast }\\right) ^{m},0\\right)  \\label{g} \\\\\n&=&\\left( W_{t}^{\\ast }\\right) ^{p-m}\\left[ \\left\\vert W_{t}\\right\\vert\n^{m}-A_{m,t}\\right] +\\alpha \\frac{m-p}{p}\\left( W_{t}^{\\ast }\\right) ^{p}%\n\\text{,}  \\label{gg}\n\\end{eqnarray}%\nis a $\\mathcal{F}_{t}^{W}$-submartingale for every $\\alpha \\geq 1$. By\nwriting $c=\\alpha \\frac{m-p}{p}$ in the previous expression, and by using\nthe fact that $\\frac{m-p}{p}\\geq 0$ by assumption, we deduce immediately\nthat $J_{t}\\left( m,c;p\\right) $ is a submartingale for every $c\\geq \\frac{%\nm-p}{p}$. Now suppose $c<\\frac{m-p}{p}$. One can use formulae (\\ref{a}), (%\n\\ref{g}) and (\\ref{gg}) to prove that\n\\begin{eqnarray*}\nJ_{t}\\left( m,c;p\\right) &=&\\int_{0}^{t}\\phi (X_{s}^{\\ast\n})dM_{s}+\\int_{0}^{t}[-A_{m,s}\\phi ^{\\prime }\\left( (W_{s}^{\\ast\n})^{m}\\right) ]d(W_{s}^{\\ast })^{m}+\\left( \\alpha -1\\right) \\Phi \\left(\n\\left( W_{t}^{\\ast }\\right) ^{m},0\\right) \\\\\n&=&\\int_{0}^{t}(W_{s}^{\\ast })^{p-m}dM_{s} \\\\\n&&+\\left( \\frac{p}{m}-1\\right) \\int_{0}^{t}[\\left( 1-\\alpha \\right) \\left(\nW_{s}^{\\ast }\\right) ^{m}-A_{m,s}](W_{s}^{\\ast })^{p-2m}d(W_{s}^{\\ast })^{m}%\n\\text{,}\n\\end{eqnarray*}%\nwhere $1-\\alpha =1-pc\/(m-p)>0$. Note that $\\int_{0}^{t}(W_{s}^{\\ast\n})^{p-m}dM_{s}$ is a square-integrable martingale, due to (\\ref{jjj}). To\nconclude that, in this case, $J_{t}\\left( m,c;p\\right) $ cannot be a\nsubmartingale (nor a supermartingale), it is sufficient to observe that (for\nevery $m\\geq 1$ and every $\\alpha <1$) the paths of the finite variation\nprocess\n\\begin{equation*}\nt\\mapsto \\int_{0}^{t}[\\left( 1-\\alpha \\right) \\left( W_{s}^{\\ast }\\right)\n^{m}-A_{m,s}](W_{s}^{\\ast })^{p-2m}d(W_{s}^{\\ast })^{m}\\text{ }\n\\end{equation*}%\nare neither non-decreasing nor non-increasing, with $\\mathbb{P}$-probability\none.\n\nTo prove Point 2, one can argue in exactly the same way, and use Point 2 of\nProposition \\ref{P : epsilon} to obtain that the process $Z\\left( t\\right) $\ndefined as $Z\\left( 0\\right) =0$ and, for $t>0$,%\n\\begin{equation*}\nZ\\left( t\\right) =\\left( W_{t}^{\\ast }\\right) ^{p-m}\\left[ \\left\\vert\nW_{t}\\right\\vert ^{m}-A_{m,t}\\right] +\\alpha \\frac{m-p}{p}\\left( W_{t}^{\\ast\n}\\right) ^{p}\n\\end{equation*}%\nis a $\\mathcal{F}_{t}^{W}$-supermartingale for every $\\alpha \\geq 1$. By\nwriting once again $c=\\alpha \\frac{m-p}{p}$ in the previous expression, and\nsince $\\frac{m-p}{p}\\leq 0$, we immediately deduce that $J_{t}\\left(\nm,c;p\\right) $ is a supermartingale for every $c\\leq \\frac{m-p}{p}$. One can\nshow that $J_{t}\\left( m,c;p\\right) $ cannot be a supermartingale, whenever $%\nc>\\frac{m-p}{p}$, by using arguments analogous to those displayed in the\nlast part of the proof of Point 1.\n\\end{proof}\n\n\\bigskip\n\nThe following result is obtained by specializing Theorem \\ref{T : DS} to the\ncase $m=1$ (via Tanaka's formula).\n\n\\begin{corollary}\n\\label{C : Tanaka}Denote by $\\left\\{ \\ell _{t}:t\\geq 0\\right\\} $ the local\ntime at zero of the Brownian motion $W$. Then, the process\n\\begin{eqnarray*}\nJ_{t}\\left( p\\right) &=&\\left( W_{t}^{\\ast }\\right) ^{p-1}\\left[ \\left\\vert\nW_{t}\\right\\vert -\\ell _{t}\\right] +c\\left( W_{t}^{\\ast }\\right) ^{p}\\text{,\n}t>0\\text{,} \\\\\nJ_{0}\\left( p\\right) &=&0\\text{,}\n\\end{eqnarray*}%\nis such that: (i) for $p\\in (0,1]$, $J_{t}\\left( p\\right) $ is a $\\mathcal{F}%\n_{t}^{W}$-submartingale if, and only if, $c\\geq 1\/p-1$, and (ii) for $p\\in\n\\lbrack 1,+\\infty )$, $J_{t}\\left( p\\right) $ is a $\\mathcal{F}_{t}^{W}$%\n-supermartingale if, and only if, $c\\leq 1\/p-1.$\n\\end{corollary}\n\n\\section{Burkholder-Davis-Gundy (BDG) inequalities\\label{S : BDG}}\n\nWe reproduce an argument taken from \\cite[p. 314]{DS}, showing that the\nfirst part of Theorem \\ref{T : DS} can be used to obtain a strong version of\nthe BDG inequalities (see e.g. \\cite[Ch. IV, \\S 4]{RY}).\n\nFix $p\\in (0,2)$ and define $c=(2-p)\/p=2\/p-1.$ Since, according to the first\npart of Theorem \\ref{T : DS}, $Y_{t}=Y_{t}(c,p)$ is a $\\mathcal{F}_{t}^{W}$%\n-submartingale starting from zero, we deduce that, for every bounded and\nstrictly positive $\\mathcal{F}_{t}^{W}$-stopping time $\\tau $, one has $%\n\\mathbb{E}(Y_{\\tau })\\geq 0$. In particular, this yields%\n\\begin{equation}\n\\mathbb{E}\\left( \\frac{\\tau }{(W_{\\tau }^{\\ast })^{2-p}}\\right) \\leq \\frac{2%\n}{p}\\mathbb{E}\\left( (W_{\\tau }^{\\ast })^{p}\\right) \\text{.}  \\label{zio}\n\\end{equation}%\nFormula (\\ref{zio}), combined with an appropriate use of H\\\"{o}lder's\ninequality, entails finally that, for $0<p<2$,%\n\\begin{equation}\n\\mathbb{E}\\left( \\tau ^{\\frac{p}{2}}\\right) \\leq \\left[ \\frac{2}{p}\\mathbb{E}%\n\\left( (W_{\\tau }^{\\ast })^{p}\\right) \\right] ^{\\frac{p}{2}}\\left[ \\mathbb{E}%\n\\left( (W_{\\tau }^{\\ast })^{p}\\right) \\right] ^{\\frac{2-p}{2}}=\\left[ \\frac{2%\n}{p}\\right] ^{\\frac{p}{2}}\\mathbb{E}\\left( (W_{\\tau }^{\\ast })^{p}\\right)\n\\text{.}  \\label{gi}\n\\end{equation}\n\nOf course, relation (\\ref{gi}) extends to general stopping times $\\tau $\n(not necessarily bounded) by monotone convergence (\\textit{via} the\nincreasing sequence $\\left\\{ \\tau \\wedge n:n\\geq 1\\right\\} $).\n\n\\bigskip\n\n\\textbf{Remark. }Let $\\left\\{ \\mathfrak{A}_{n}:n\\geq 0\\right\\} $ be a\ndiscrete filtration of the reference $\\sigma $-field $\\mathfrak{A}$, and\nconsider a $\\mathfrak{A}_{n}$-adapted sequence of measurable random elements\n$\\left\\{ f_{n}:n\\geq 0\\right\\} $ with values in a Banach space $\\mathbf{B}$.\nWe assume that $f_{n}$ is a martingale, i.e. that, for every $n$, $\\mathbb{E}%\n\\left[ f_{n}-f_{n-1}\\mid \\mathfrak{A}_{n-1}\\right] =\\mathbb{E}\\left[\nd_{n}\\mid \\mathfrak{A}_{n-1}\\right] =0$, where $d_{n}:=f_{n}-f_{n-1}$. We\nnote\n\\begin{equation*}\nS_{n}\\left( f\\right) =\\sqrt{\\sum_{k=0}^{n}\\left\\vert d_{k}\\right\\vert ^{2}}%\n\\text{ \\ \\ and \\ \\ }f_{n}^{\\ast }=\\sup_{0\\leq m\\leq n}\\left\\vert\nf_{m}\\right\\vert ,\n\\end{equation*}%\nand write $S\\left( f\\right) $ and $f^{\\ast }$, respectively, to indicate the\npointwise limits of $S_{n}\\left( f\\right) $ and $f_{n}^{\\ast }$, as $%\nn\\rightarrow +\\infty .$ In \\cite{Burkhj}, D.L.\\ Burkholder proved that\n\\begin{equation}\n\\mathbb{E}\\left( S\\left( f\\right) \\right) \\leq \\sqrt{3}\\mathbb{E}\\left(\nf^{\\ast }\\right) \\text{,}  \\label{BI}\n\\end{equation}%\nwhere $\\sqrt{3}$ is the best possible constant, in the sense that for every $%\n\\eta \\in (0,\\sqrt{3})$ there exists a Banach space-valued martingale $%\nf_{\\left( \\eta \\right) }$ such that $\\mathbb{E}\\left( S\\left( f_{\\left( \\eta\n\\right) }\\right) \\right) >\\eta \\mathbb{E}\\left( f_{\\left( \\eta \\right)\n}^{\\ast }\\right) $. As observed in \\cite{DS}, Burkholder's inequality (\\ref%\n{BI}) should be compared with (\\ref{gi}) for $p=1$, which yields\nthe relation\n $\\mathbb{E}\\left( \\tau ^{1\/2}\\right) \\leq \\sqrt{2}\\mathbb{E}%\n(W_{\\tau }^{\\ast })$ for every stopping time $\\tau $. This shows\nthat in such a framework, involving uniquely continuous\nmartingales, the constant $\\sqrt{3}$ is no longer optimal.\n\n\\section{Balayage \\label{S : Bal}}\n\nKeep the assumptions and notation of Section \\ref{S : Gen} and Theorem \\ref%\n{Th : general}, fix $\\varepsilon >0$ and consider a finite variation\nfunction $\\psi :(0,+\\infty )\\mapsto \\mathbb{R}.$ In this section we focus on\nthe formula%\n\\begin{equation}\n\\psi \\left( X_{t}^{\\ast }\\right) \\left( X_{t}-A_{t}\\right) -\\psi \\left(\nX_{\\varepsilon }^{\\ast }\\right) \\left( X_{\\varepsilon }-A_{\\varepsilon\n}\\right) =\\int_{\\varepsilon }^{t}\\psi (X_{s}^{\\ast })d\\left(\nX_{s}-A_{s}\\right) +\\int_{\\varepsilon }^{t}\\left( X_{s}^{\\ast }-A_{s}\\right)\nd\\psi (X_{s}^{\\ast }),  \\label{d}\n\\end{equation}%\nwhere $\\varepsilon >0$. Note that by choosing $\\psi =\\phi $ in (\\ref{d}),\nwhere $\\phi \\in C^{1}$ is monotone, one recovers formula (\\ref{uyu}), which\nwas crucial in the proof Theorem \\ref{Th : general}. We shall now show that (%\n\\ref{d}) can be obtained by means of the \\textsl{balayage formulae} proved\nin \\cite{MY}.\\textit{\\ }\n\nTo see this, let $U=\\left\\{ U_{t}:t\\geq 0\\right\\} $ be a continuous $%\n\\mathcal{F}_{t}$-semimartingale issued from zero. For every $t>0$ we define\nthe random time\n\\begin{equation}\n\\sigma \\left( t\\right) =\\sup \\left\\{ s<t:U_{s}=0\\right\\} \\text{.}\n\\label{tauti}\n\\end{equation}%\nThe following result is a particular case of \\cite[Th. 1]{MY}.\n\n\\bigskip\n\n\\begin{proposition}[Balayage Formula]\nConsider a stochastic process $\\left\\{ K_{t}:t>0\\right\\} $ such that the\nrestriction $\\left\\{ K_{t}:t\\geq \\varepsilon \\right\\} $ is locally bounded\nand $\\mathcal{F}_{t}$-predictable on $\\left[ \\varepsilon ,+\\infty \\right) $\nfor every $\\varepsilon >0$. Then, for every fixed $\\varepsilon >0$, the\nprocess $K_{\\sigma \\left( t\\right) }$, $t\\geq \\varepsilon $, is locally\nbounded and $\\mathcal{F}_{t}$-predictable, and moreover%\n\\begin{equation}\nU_{t}K_{\\sigma \\left( t\\right) }=U_{\\varepsilon }K_{\\sigma \\left(\n\\varepsilon \\right) }+\\int_{\\varepsilon }^{t}K_{\\sigma \\left( s\\right)\n}dU_{s}\\text{.}  \\label{balabala}\n\\end{equation}\n\\end{proposition}\n\n\\bigskip\n\nTo see how (\\ref{d}) can be recovered from (\\ref{balabala}), set $%\nU_{t}=X_{t}-X_{t}^{\\ast }$ and $K_{t}=\\psi \\left( X_{t}^{\\ast }\\right) $.\nThen, $K_{t}=K_{\\sigma \\left( t\\right) }=\\psi (X_{\\sigma \\left( t\\right)\n}^{\\ast })$ by construction, where $\\sigma \\left( t\\right) $ is defined as\nin (\\ref{tauti}). As a consequence, (\\ref{balabala}) gives\n\\begin{equation*}\n\\psi \\left( X_{t}^{\\ast }\\right) \\left( X_{t}-X_{t}^{\\ast }\\right) =\\psi\n\\left( X_{\\varepsilon }^{\\ast }\\right) \\left( X_{\\varepsilon\n}-X_{\\varepsilon }^{\\ast }\\right) +\\int_{\\varepsilon }^{t}\\psi (X_{s}^{\\ast\n})d\\left( X_{s}-X_{s}^{\\ast }\\right) \\text{.}\n\\end{equation*}%\nFinally, a standard integration by parts applied to $\\psi \\left( X_{t}^{\\ast\n}\\right) \\left( X_{t}^{\\ast }-A_{t}\\right) $ yields\n\\begin{eqnarray*}\n\\psi \\left( X_{t}^{\\ast }\\right) \\left( X_{t}-A_{t}\\right) &=&\\psi \\left(\nX_{t}^{\\ast }\\right) \\left( X_{t}-X_{t}^{\\ast }\\right) +\\psi \\left(\nX_{t}^{\\ast }\\right) \\left( X_{t}^{\\ast }-A_{t}\\right) \\\\\n&=&\\psi \\left( X_{\\varepsilon }^{\\ast }\\right) \\left( X_{\\varepsilon\n}-X_{\\varepsilon }^{\\ast }\\right) +\\int_{\\varepsilon }^{t}\\psi (X_{s}^{\\ast\n})d\\left( X_{s}-X_{s}^{\\ast }\\right) \\\\\n&&+\\psi \\left( X_{\\varepsilon }^{\\ast }\\right) \\left( X_{\\varepsilon }^{\\ast\n}-A_{\\varepsilon }\\right) +\\int_{\\varepsilon }^{t}\\psi (X_{s}^{\\ast\n})d\\left( X_{s}^{\\ast }-A_{s}\\right) \\\\\n&&+\\int_{\\varepsilon }^{t}\\left( X_{s}^{\\ast }-A_{s}\\right) d\\psi \\left(\nX_{s}^{\\ast }\\right) \\text{,}\n\\end{eqnarray*}%\nwhich is equivalent to (\\ref{d}).\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}