diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzcgqp" "b/data_all_eng_slimpj/shuffled/split2/finalzzcgqp" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzcgqp" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nHigher-spin (HS) gauge theory describes interacting systems of massless fields of all spins\n(for reviews see e.g. \\cite{{Vasiliev:Golfandmem},Review4}).\n Effects of HS gauge theories are anticipated to play a role at ultra high energies of\nPlanck scale \\cite{Vasiliev:2016xui}.\nTheories of this class play a role in various contexts from holography \\cite{Klebanov:2002ja} to cosmology\n\\cite{Barv}. HS theory\ndiffers from usual local field theories because it contains\ninfinite tower of gauge fields of all spins and the number of space-time derivatives increases with the spins\nof fields in the vertex \\cite{Bengtsson:1983pd,Berends:1984wp,Fradkin:1987ks,Fradkin:1991iy}.\nHowever one may ask for spin-locality \\cite{Vasiliev:2016xui,Gelfond:2017wrh,4a1,4a2} which implies\nspace-time locality in the lowest orders of perturbation theory \\cite{4a1}. Even though details of the precise relation between spin-locality and space-time locality in higher orders of perturbation theory have not been yet elaborated, from the form of equations it is clear that spin-locality constraint provides one of the best tools to minimize the space-time non-locality. Moreover demanding spin-locality one actually fixes functional space for possible field redefinitions that is highly important for\nthe predictability of the theory.\n\n\nA useful way of description of HS dynamics is provided by the\ngenerating Vasiliev system of HS equations \\cite{more}. The latter {contains a free complex parameter\n $\\eta$. Solving the generating system order by order one obtains vertices\nproportional to various powers of $\\eta$ and $\\bar{\\eta}$. In the recent paper \\cite{4a3},\n$\\eta^2$ and $\\bar{\\eta}^2$ vertices were obtained in the sector of equations for zero-form fields,\ncontaining, in particular, a part of the $\\phi^4$ vertex for the scalar field $\\phi$\n in the theory. Though being seemingly $Z$-dependent, in \\cite{4a3} these vertices were written\n in the $Z$-dominated form which implies their spin-locality by virtue of $Z$-dominance\n Lemma of \\cite{2a1}. In this paper we obtain explicit $Z$-independent spin-local form for\n the vertex $\\Upsilon^{\\eta\\eta}_{\\go CCC}$ starting from the $Z$-dominated expression of \\cite{4a3}.\n The label $\\go CCC$ refers to the $\\go CCC$-ordered part of the vertex\n where $\\go$ and $C$ denote gauge one-form and field strength zero-form HS fields valued\n in arbitrary associative algebra in which case the order of the product factors in $\\go CCC$ matters.\n}\n\n\n\n\n\n\nThere are several ways to study the issue of (non)locality in HS gauge theory. One is reconstruction\nthe vertices from the boundary by the holographic prescription based on the Klebanov-Polyakov\n conjecture \\cite{Klebanov:2002ja} (see also\n\\cite{Sezgin:2002rt}, \\cite{SS}). Alternatively,\none can analyze vertices directly in the bulk starting from the generating equations\nof \\cite{more}. The latter approach developed in \\cite{4a1,4a2,4a3,2a1,2a2}\nis free from any holographic duality assumptions but demands careful choice of the\nhomotopy scheme to determine the choice of field variables compatible with spin-locality of the vertices.\nThe issue of (non)locality of HS gauge theories was also considered in\n\\cite{Fotopoulos:2010ay} and \\cite{David:2020ptn} with somewhat opposite conclusions.\n\n\nFrom the holographic point of view the vertex that contains $\\phi^4$ was argued to be essentially\nnon-local \\cite{Sleight:2017pcz} or at least should have non-locality of very specific form presented\nin \\cite{Ponomarev:2017qab}. On the other hand, the holomorphic, \\ie\n$\\eta^2$ and antiholomorphic $\\bar\\eta^2$ vertices, where $\\eta$ is a complex parameter in\nthe HS equations, were recently obtained in \\cite{4a3} where they were\nshown to be spin-local by virtue of $Z$-dominance lemma of \\cite{2a1}.\nThe computation was done directly in the bulk starting from the non-linear HS system of \\cite{more}.\n\n\nIn this formalism HS fields are described by one-forms\n$\\omega (Y;K|x) $ and zero-forms $C(Y;K|x)$ where $x$ are space-time coordinates while\n$Y_A=(y_\\ga,\\by_{\\dot \\ga})$ are auxiliary spinor variables.\nBoth dotted and undotted indices are two-component, $\\ga, {\\dot{\\ga}=1,2}$, while $K=(k,\\bar k)$ are outer\nKlein operators satisfying $k*k= \\bar{k}* \\bar{k}=1$\\,,\n \\bee\\label{Klein}&&\n\\lbrace k,y^\\ga\\rbrace_\\ast=\\lbrace k, z^\\ga \\rbrace_\\ast=\n\\lbrace \\bar{k},\\bar{y}^{\\dot{\\ga}}\\rbrace_\\ast=\\lbrace \\bar{k},\\bar{z}^{\\dot{\\ga}}\n\\rbrace_\\ast=\\lbrace k,\\theta^\\ga\\rbrace_\\ast=\\lbrace \\bar{k},\\bar{\\theta}^{\\dot{\\ga}}\\rbrace_\\ast=0,\n\\\\\\nn&&[k,\\bar{y}^{\\dot{\\ga}}]_\\ast=[ k, \\bar{z}^{\\dot{\\ga}}]_\\ast=\n[\\bar{k},y^\\ga]_\\ast=[ \\bar{k},z^\\ga]_\\ast=[k,\\bar{\\theta}^{\\dot{\\ga}}]_\\ast=[ \\bar{k},\\theta^\\ga]_\\ast=0\\,,\n\\eee\nwhere $\\theta $ and $\\bar \\theta $ are anticommuting spinors in the theory.\n\nSchematically, non-linear HS equations in the unfolded form read as\n\\begin{equation}\\label{oneform}\n\\dr_x \\go + \\go \\ast \\go=\\Upsilon(\\go,\\go,C)+\\Upsilon(\\go,\\go,C,C)+\\ldots,\n\\end{equation}\n\\begin{equation}\\label{zeroform}\n\\dr_x C+\\go \\ast C-C\\ast \\go=\\Upsilon(\\go,C,C)+\\Upsilon(\\go,C,C,C)+\\ldots.\n\\end{equation}\n\n\n\n\n\nAs recalled in Section \\ref{HSeq}, generating equations of \\cite{more} that reproduce the form of\nequations (\\ref{oneform}) and (\\ref{zeroform}) have a simple form as a result of\ndoubling of spinor variables,\nnamely $$\n\\go(Y;K|x)\\longrightarrow W(Z;Y;K|x)\\,,\\qquad\nC(Y;K|x) \\longrightarrow B(Z;Y;K|x). $$\nEquations\n\\eq{oneform} and \\eq{zeroform} result from the generating equations of \\cite{more} upon\norder by order reconstruction of $Z$-dependence (for more detail see Section \\ref{HSeq}).\nThe final form of equations (\\ref{oneform}) and (\\ref{zeroform}) turns out\nto be $Z$-independent as a consequence of consistency of the equations of \\cite{more}. This fact may not be\nmanifest however since the \\rhss of HS equations\nusually have the form of the sum of $Z$-dependent terms.\n\nHS equations have remarkable property \\cite{Vasiliev:1988sa} that they remain\nconsistent with the fields $W$ and $B$ valued in any associative algebra. For instance\n$W$ and $B$ can belong to the matrix algebra $Mat_n $ with any $n$. Since in that\ncase the components of $W$ and $B$ do not commute, different orderings of the fields\nshould be considered independently.\n(Mathematically, HS equations with this property correspond to $A_\\infty $ strong homotopy\n algebra introduced by Stasheff in \\cite{stash1},\\cite{stash2},\\cite{stash3}.)\nFor instance, holomorphic (\\ie $\\bar\\eta$-independent) vertices in the zero-form sector can be represented in the form\n\\be \\label{FieldOrdering}\n\\Upsilon^{\\eta }(\\go,C,C )=\\Upsilon^{ \\eta}_{\\go CC }+\\Upsilon^{ \\eta}_{C\\go C }+\\Upsilon^{ \\eta}_{CC\\go}\n\\,,\\quad\n\\Upsilon^{\\eta\\eta}(\\go,C,C,C)=\\Upsilon^{\\eta\\eta}_{\\go CCC}+\\Upsilon^{\\eta\\eta}_{C\\go CC}\n+\\Upsilon^{\\eta\\eta}_{CC\\go C}+\\Upsilon^{\\eta\\eta}_{CCC\\go }\\,,\\,\\,\\ldots\n\\ee\nwhere the subscripts of the vertices $\\Upsilon$ refer to the ordering of the product\nfactors.\n\n\nThe vertices obtained in \\cite{4a3} were shown to be\nspin-local due to the $Z$-dominance Lemma of \\cite{2a1} that identifies terms that\nmust drop from the \\rhss of HS equations together with the $Z$-dependence.\nRecall that spin-locality implies that the vertices are local in terms of spinor variables for\nany finite subset of fields of different spins \\cite{2a2} (for more detail on the notion of spin-locality see \\cite{2a2}).\n Analogous vertices in the one-form sector have been shown to be spin-local earlier in \\cite{4a2}.\n\n The main achievement of \\cite{4a3} consists of finding such solution of the generating\n system in the third order in $C$ that all spin-nonlocal terms containing infinite towers\n of derivatives in $y(\\bar y)$ between $C$-fields in the (anti)holomorphic\n in $\\eta(\\bar \\eta)$ sector do not contribute to\n $\\eta^2$ ($\\bar \\eta^2$) vertices by virtue of\n $Z$-dominance Lemma. Thus \\cite{4a3} gives spin-local expressions for the vertices\n $\\Upsilon^{\\eta\\eta}(\\go,C,C,C)$ which, however, have a form\n of a sum of a number of $Z$-dependent terms. To make spin-locality\n manifest one must remove the seeming Z-dependence from the vertex of \\cite{4a3}.\n Technically, this can be done with the help of partial integration and the Schouten identity.\n The aim of this paper is to show how this works in practice.\n\n\nSince the straightforward derivation presented in this paper\nis technically involved we confine ourselves to the\nparticular vertex $\\Upsilon^{\\eta\\eta}_{\\go CCC}$ \\eq{FieldOrdering}.\n{ Complexity of the calculations in this paper expresses complexity of the\nobtained vertex having no analogues in the literature. Indeed, this is explicitly calculated\nspin-local vertex\nof the third order in the equations, corresponding to the vertices\nof the fourth (and, in part, fifth) order for the fields of all spins.\nThe example described in the paper explains the formalism applicable to all other orderings\nof the fields in the vertex that are also computable. So, our results are most important from the general\n point of view\n highlighting a way for the computation of higher vertices in HS theory that may\n be important from various perspectives and, in the first place, for the analysis of HS holography.\nIt should be stressed that the results of \\cite{4a3} provided a sort of existence theorem for a spin-local\nvertex that was difficult to extract without developing specific tools like those developed in\nthis paper.\nIn particular, it is illustrated how the general statements\nlike $Z$-dominance Lemma work in practical computations. Let us stress that\nat the moment this is the only available approach allowing to compute explicit form\nof the spin-local vertices for all spins at higher orders.}\n\n\nThe rest of the paper is organized as follows. In Section \\ref{HSeq}, the necessary background on HS equations\nis presented with brief recollection on the procedure of derivation of vertices from the generating system.\n Section \\ref{SectionHplus}\nreviews the notion of the $\\Hp$ space as well as the justification for a computation modulo $\\Hp$.\nIn Section \\ref{Schema}, we present step-by-step scheme of computations performed in this paper.\n Section \\ref{Main} contains the final\nmanifestly spin-local expression for $\\Upsilon^{\\eta\\eta}_{\\go CCC}$ vertex.\nIn Sections \\ref{zlinear}\\,, \\ref{SecGTid}\\,, \\ref{uniform}\\,, \\ref{Eli0} and \\ref{proof}\ntechnical details of the steps\nsketched in Section \\ref{Schema} are presented. In particular, in Section \\ref{SecGTid}\nwe introduce important {\\it Generalised Triangle identity} which allows us to uniformize expressions\nfrom \\cite{4a3}.\n Conclusion section contains discussion of the obtained results. Appendices A, B, C and D contain\n technical detail on the steps listed in\nthe scheme of computation.\n Some useful formulas are collected in Appendix E.\n\n\\section{ Higher Spin equations}\n\\label{HSeq}\n\\subsection{Generating equations}\n\nSpin-$s$ HS fields are encoded in two generating functions, namely, the space-time one-form\n\\begin{equation}\n\\omega(y,\\bar{y},x)=\\dr x^\\mu \\go_\\mu(y,\\bar{y},x)=\\sum_{n,m} \\dr x^{\\mu} \\go_{\\mu} {}_{\\ga_1 \\ldots \\ga_n,\n \\dot{\\ga}_1 \\ldots \\dot{\\ga}_m}(x) y^{\\ga_1} \\ldots y^{\\ga_n} \\bar{y}^{\\dot{\\ga}_1}\n \\ldots \\bar{y}^{\\dot{\\ga}_m}\n\\q s=\\ff{2+m+n}{2} \\end{equation} and zero-form\n\\begin{equation}\nC(y,\\bar{y},x)=\\sum_{n,m} C_{\\ga_1 \\ldots \\ga_n,\n\\dot{\\ga}_1 \\ldots \\dot{\\ga}_m}(x) y^{\\ga_1} \\ldots y^{\\ga_n}\n\\bar{y}^{\\dot{\\ga}_1} \\ldots \\bar{y}^{\\dot{\\ga}_m}\n\\q s=\\ff{|m-n|}{2}.\\end{equation}\nwhere $\\ga=1,2$ and $\\dot{\\ga}=1,2$ are two-component spinor indices.\n Auxiliary commuting variables $y^\\ga$ and $\\bar{y}^{\\dot \\ga}$\n can be combined into an $\\mathfrak{sp}(4)$ spinor $Y^A=(y^\\ga,\\bar{y}^{\\dot{\\ga}})$, $A=1,..., 4$.\n\nThe vertices $\\Upsilon(\\go,\\go,C,C,\\ldots)$ \\eqref{oneform} and $\\Upsilon(\\go,C,C,\\ldots)$\n\\eqref{zeroform} result from\n the generating system of \\cite{more}\n\\begin{equation}\\label{HS1}\n\\dr_x W+W\\ast W=0,\n\\end{equation}\n\\begin{equation}\\label{HS2}\n\\dr_x S+W\\ast S+S\\ast W=0,\n\\end{equation}\n\\begin{equation}\\label{HS3}\n\\dr_x B+W\\ast B- B\\ast W=0,\n\\end{equation}\n\\begin{equation}\\label{HS4}\nS\\ast S=i(\\theta^A \\theta_A+\\eta B\\ast \\gga+\\bar{\\eta} B\\ast \\bar{\\gga}),\n\\end{equation}\n\\begin{equation}\\label{HS5}\nS\\ast B-B\\ast S=0.\n\\end{equation}\nApart from space-time coordinates $x$, the fields $W(Z;Y;K|x)$, $S(Z;Y;K|x)$ and $B(Z;Y;K|x)$\n depend on $Y^A$, $Z^A=(z^\\ga,\\bar{z}^{\\dot{\\ga}})$ and Klein operators $K=(k,\\bar{k})$\n\\eq{Klein}. $W$ is a space-time one-form, \\ie $W= dx^\\nu W_\\nu$\nwhile $S$ -field is a one-form in $Z$ spinor directions\n$\\theta^A=(\\theta^\\ga,\\bar{\\theta}^{\\dot{\\ga}})$,\\quad $\\lbrace\\theta^A, \\theta^B\\rbrace=0$, \\ie\n\\begin{equation}\nS(Z;Y;K)=\\theta^A S_A(Z;Y;K).\n\\end{equation}\n$B$ is a zero-form.\n\nStar product is defined as follows\n\\begin{equation}\\label{StarZY}\n(f\\ast g)(Z;Y;K)=\\frac{1}{(2\\pi)^4}\\int d^4 U \\, d^4 V e^{iU_A V^A}f(Z+U,Y+U;K)g(Z-V,Y+V;K).\n\\end{equation}\n Elements\n \\begin{equation}\n\\gga=\\theta^\\ga \\theta_\\ga e^{iz_\\ga y^\\ga}k\\mbox{\\qquad and\\qquad}\n\\bar{\\gga}=\\bar{\\theta}^{\\dot{\\ga}}\\bar{\\theta}_{\\dot{\\ga}}\ne^{i\\bar{z}_{\\dot{\\ga}}\\bar{y}^{\\dot{\\ga}}}\\bar{k}\n\\end{equation}\nare central because $\\theta^3=0$ since $\\theta_\\ga$ is a two-component anticommuting spinor.\n\\subsection{Perturbation theory}\nStarting with a particular solution of the form\n\\begin{equation}\\label{solution}\nB_0(Z;Y;K)=0\\q S_0(Z;Y;K)=\\theta^\\ga z_\\ga+\\bar{\\theta}^{\\dot{\\ga}}\\bar{z}_{\\dot{\\ga}}\\q\n W_0(Z;Y;K)=\\omega(Y;K)\\,,\n\\end{equation}\nwhich indeed solves \\eqref{HS1}-\\eqref{HS5} provided that $\\go(Y;K)$ satisfies zero-curvature condition,\n\\be\n\\dr \\go +\\go*\\go=0\\,,\n\\ee\n one develops perturbation theory. Starting from \\eqref{HS5} one finds\n\\begin{equation}\\label{1order}\n[S_0,B_1]_*=0.\n\\end{equation}\nFrom \\eqref{StarZY} one deduces that\n\\begin{equation}\n[Z_A,f(Z;Y;K)]_\\ast=-2i\\frac{\\p}{\\p Z^A} f(Z;Y;K).\n\\end{equation}\nHence, equation (\\ref{1order}) yields\n\\begin{equation}\n[S_0,B_1]=-2i \\theta^A \\frac{\\p}{\\p Z^A}B_1=-2i \\dr_Z B_1=0 \\; \\Longrightarrow\\; B_1(Z;Y;K)=C(Y;K).\n\\end{equation}\nThe $Z$-independent $C$-field that appears as the first-order part of $B$ is the same\n that enters equations \\eqref{oneform}, \\eqref{zeroform}. The perturbative procedure can be\n continued further leading to the equations of the form\n\\begin{equation}\n\\dr_Z \\Phi_{k+1}=J(\\Phi_k, \\Phi_{k-1},\\ldots)\\,,\n\\end{equation}\nwhere $\\Phi_k$ is either $W$, $S$ or $B$ field of the $k$-th order of perturbation theory,\n identified with the degree of $C$-field in the corresponding expression, \\ie\n\\bee\\nn&&\nW=\\go+W_1(\\go,C)+W_2(\\go,C,C)+\\ldots\\q S=S_0+S_1(C)+S_2(C,C)+\\ldots,\n\\\\&&\\nn B=C+B_2(C,C)+B_3(C,C,C)+\\ldots.\n\\eee\nTo obtain dynamical equations \\eqref{oneform}, \\eqref{zeroform} one should plug obtained\n solutions into equations \\eqref{HS1} and \\eqref{HS3}. For instance,\n \\eqref{HS3} up to the third order in $C$-field is\n\\begin{equation}\\label{B3EQ}\n\\dr_x C+[\\go,C]_\\ast=-\\dr_x B_2-[W_1,C]_\\ast-\\dr_x B_3-[W_1,B_2]_\\ast-[W_2,C]_\\ast+\\ldots\n\\end{equation}\nThough the fields $W_1$, $W_2$ and $B_2$, $B_3$ and hence various terms that enter (\\ref{B3EQ})\nare $Z$-dependent, equations \\eqref{HS1}-\\eqref{HS5} are designed in such a way that, as a consequence\nof their consistency, the sum of the terms on the \\rhs\nof (\\ref{B3EQ}) is $Z$-independent. To see this it suffices to apply $\\dr_Z$\nrealized as $\\ff{i}{2} [S_0\\,,\\quad ]_*$ to the \\rhs\nof (\\ref{B3EQ}) and make sure that it gives zero by virtue of already solved equations.\nFor more detail we refer the reader to the review \\cite{Review4}.\n\n\\section{ Subspace $\\Hp$ and $Z$-dominance lemma}\n\\label{SectionHplus}\n\n\\subsection{$\\Hp$}\n\nIn this Section the definition of the space $\\Hp$ \\cite{4a3} that plays a\ncrucial role in our computation is recollected.\nFunction $f(z,y\\vert \\theta)$ of the form\n\\begin{equation}\n\\label{class}\nf(z,y\\vert \\theta)=\\int_0^1 d\\mathcal{T}\\, e^{i\\mathcal{T}z_\\ga y^\\ga}\\phi\n\\left(\\mathcal{T}z,y\\vert \\mathcal{T} \\theta,\\mathcal{T}\\right)\\,\n\\end{equation}\n belongs to the space $\\Hp$ if there exists\n such a real $\\varepsilon>0$, that\n\\begin{equation}\\label{limit}\n\\lim_{\\mathcal{T}\\rightarrow 0}\\mathcal{T}^{1-\\varepsilon}\\phi(w,u\\vert\n\\theta,\\mathcal{T})=0\\,.\n\\end{equation}\nNote that this definition does not demand\nany specific behaviour of $\\phi$ at $\\mathcal{T}\\to1$ as was the case for the\n space $\\Sp^{+0}$ of \\cite{2a2}.\n\n\n In the sequel we use two main types of functions that obey \\eqref{limit}:\n\\begin{equation}\\label{kernels}\n\\phi_1(\\mathcal{T}z,y\\vert \\mathcal{T} \\theta,\n\\mathcal{T})=\\frac{\\mathcal{T}^{\\delta_1}}{\\mathcal{T}}\\widetilde{\\phi}_1(\\mathcal{T}z,y\\vert\n\\mathcal{T} \\theta)\\q \\phi_2(\\mathcal{T} z,y\\vert\n\\mathcal{T}\\theta,\n\\mathcal{T})=\\vartheta(\\mathcal{T}-\\delta_2)\\frac{1}{\\mathcal{T}}\\widetilde{\\phi}_2(\\mathcal{T}z,y\\vert\n\\mathcal{T} \\theta)\n\\end{equation}\nwith some $\\delta_{1,2}>0$. (Note that the second option with $\\delta_2>0$ can be\ninterpreted as the first one with arbitrary large $\\delta_1$. Here step-function is denoted as $\\vartheta$\nto distinguish it from the anticommuting variables $\\theta$.)\n\nSpace $\\Hp$ can be represented as the direct sum\n\\begin{equation}\n\\Hp=\\Hp_0 \\oplus \\Hp_1 \\oplus \\Hp_2\\,,\n\\end{equation}\nwhere $\\phi(w,u\\vert\\theta,\\mathcal{T})\\in\\Hp_p$ are degree-$p$ forms in $\\theta$ satisfying \\eqref{limit}.\n\n\nAll terms from $\\Hp$ on the \\rhs of HS field equations must vanish by $Z$-dominance Lemma \\cite{2a1}.\nFollowing \\cite{4a3} this can be understood as follows. All the expressions\nfrom \\eqref{B3EQ} have the form \\eqref{class} and the only way to obtain $Z$-independent non-vanishing\n expression is to bring the hidden $\\T$ dependence in $\\phi(\\T z,y\\vert \\T \\theta , {\\T})$\n to $\\delta(\\T)$. If a function contains an additional factor of\n$\\mathcal{T}^\\gvep$ or is isolated from $\\T=0$, it cannot contribute to the $Z$-independent\nanswer\n which is the content of $Z$-dominance Lemma \\cite{2a1}.\nThis just means that functions of the class $\\Hp_0$ cannot\ncontribute to the $Z$-independent equations \\eqref{zeroform}.\nApplication of this fact to locality is straightforward once this is\nshown that all terms containing\ninfinite towers of higher derivatives in the vertices of interest\nbelong to $\\Hp_0$ and, therefore, do not contribute to HS\nequations. This is what was in particular shown in \\cite{4a3}.\n\n\n\n\n\n\n\n\\subsection{Notation}\nAs in \\cite{4a3} we use \\textit{exponential} form for all the expressions below where by $\\go CCC$ we assume\n\\begin{equation}\n\\omega(\\mathsf{y}_\\go,\\bar{y})\\bar{\\ast}C(\\mathsf{y}_1,\\bar{y})\\bar{\\ast}C(\\mathsf{y}_2,\\bar{y})\\bar{\\ast}C(\\mathsf{y}_3,\\bar{y})\n\\end{equation}\nwith $\\bar{\\ast}$ denoting star-product with respect to $\\bar{y}$.\nDerivatives $\\p_\\go$ and $\\p_j$ act on auxiliary variables as follows\n\\begin{equation}\n\\p_{\\go\\ga}=\\frac{\\p}{\\p \\mathsf{y}_\\go^\\ga}\\q \\p_{j\\ga}=\\frac{\\p}{\\p \\mathsf{y}_j^\\ga}.\n\\end{equation}\nAfter all the derivatives in $\\mathsf{y}_\\go$ and $\\mathsf{y}_j$ are evaluated the latter are set to zero, \\ie\n\\begin{equation}\n\\mathsf{y}_\\go=\\mathsf{y}_j=0.\n\\end{equation}\nIn this paper we use the following notation of \\cite{4a3}:\n\\begin{equation}\nt_\\ga:=-i\\p_{\\go\\ga},\\;\\; p_{j\\ga}:=-i\\p_{j\\ga}\\q\n\\end{equation}\n \\be\\label{ro+}\n\\int d^n \\rho_+ :=\\int d\\rho_1 \\ldots d\\rho_n\\, \\vartheta(\\rho_1)\\ldots \\vartheta(\\rho_n)\\,.\n\\end{equation}\n\n\\subsection{Contribution to ${\\Upsilon}^{\\eta\\eta} _{\\go CCC}$ modulo $\\Hp$}\n\nThe $\\eta^2C^3$ vertex in the equations on the\nzero-forms $C$ resulting from equations of \\cite{more}\nis \\begin{equation}\\label{rightside} \\Upsilon^{\\eta\\eta}(\\go,C,C,C) =-\\left(\\dr_x B^{ \\eta \\eta }_3 + [\\go, B^{\\eta\\eta }_3]_* + [ {W}^\\eta_1, B^{\\eta }_2]_*\n +[ {W}^{ {\\eta}\\eta}_2, C]_*\n+\\dr_x B^{ \\eta }_2\\,\\right). \\end{equation}\n Recall, that, being $Z$-independent, ${\\Upsilon}^{\\eta\\eta} $ is a sum of $Z$-dependent terms\nthat makes its $Z$-independence implicit.\n\nAs explained in Introduction, ${\\Upsilon}^{\\eta\\eta}$ can be decomposed into parts\nwith different orderings of fields $\\go$ and $C$. In this paper we consider\n \\be\\label{projwccc}{\\Upsilon}^{\\eta\\eta}_{\\go C C C} := \\Upsilon^{\\eta\\eta}(\\go,C,C,C)\\Big|_{\\go CCC}\\,.\n \\ee\n Since the terms from $\\Hp$ do not contribute to the physical\nvertex such terms can be discarded. Following \\cite{4a3} equality up to terms from $\\Hp$\nreferred to as weak equality\nis denoted as $\\approx$ .\n\n We start with the following results of \\cite{4a3}:\n \\be\\label{rightsideU\n \\widehat{\\Upsilon}^{\\eta\\eta}_{\\go CCC}\n\\approx {\\Upsilon}^{\\eta\\eta}_{\\go C C C}=-\\Big(W_{1\\, \\go C}^\\eta \\ast B_2^{\\eta\\, loc}+ {W}_{2\\, \\go CC}^{\\eta\\eta}\\ast C+\n\\dr_x B^{\\eta\\, loc}_2\\big|_{\\go CCC}+\n\\omega\\ast {B}_3^{\\eta\\eta}+\\dr_x {B}_3^{\\eta\\eta}\\big|_{\\go CCC} \\Big)\n \\q\n\\ee where\n \\begin{multline}\\label{origW1B2}\nW_{1\\, \\go C}^\\eta \\ast B_2^{\\eta\\, loc}\\approx \\frac{\\eta^2}{4}\\int_0^1 d\\mathcal{T} \\T \\int_0^1 d\\gs\n \\int d^3\\rho_+ \\delta\\left(1-\\sum_{i=1}^3 \\rho_i\\right) \\frac{\\left(z_\\gga t^{ \\gga}\\right)\\big[z_\\ga y^\\ga+\\gs z_\\ga t^{\\ga}\\big]}{(\\rho_1+\\rho_2)}\n \\times\\\\\n\\times \\exp\\Big\\{i\\mathcal{T} z_\\ga y^\\ga+i(1-\\gs )t^{\\ga}\\p_{1\\ga}\n-i\\frac{\\rho_1\\gs }{\\rho_1+\\rho_2} t^{\\ga}p_{2\\ga}\n+i\\frac{\\rho_2\\gs }{\\rho_1+\\rho_2} t^{\\ga}p_{3\\ga} \\\\\n+i\\mathcal{T}z^\\ga\\Big(-(\\rho_1+\\rho_2+\\gs \\rho_3)t_{\\ga}-(\\rho_1+\\rho_2)p_{1 \\ga}\n+(\\rho_3-\\rho_1)p_{2 \\ga}+(\\rho_3+\\rho_2)p_{3\\ga}\\Big) \\\\\n+iy^\\ga\\Big(\\gs t_{\\ga}-\\frac{\\rho_1}{\\rho_1+\\rho_2}p_{2 \\ga}\n+\\frac{\\rho_2}{\\rho_1+\\rho_2}p_{3\\ga}\\Big)\\Big\\}\\go CCC\\,,\n\\end{multline}\n\n\\begin{multline}\\label{origW2C}\n {W}_{2\\, \\go CC}^{\\eta\\eta}\\ast C\\approx-\\frac{\\eta^2}{4}\\int_0^1 d\\mathcal{T}\\,\\T\n \\int d^4\\rho_+\\, \\delta\\left(1-\\sum_{i=1}^4 \\rho_i\\right)\n \\frac{\\rho_1 \\left(z_\\gga t^{\\gga}\\right)^2}{(\\rho_1+\\rho_2)(\\rho_3+\\rho_4)}\\times\\\\\n\\times \\exp\\Big\\{i\\mathcal{T}z_\\ga y^\\ga+i\\mathcal{T}z^\\ga\\Big((1-\\rho_2)t_{\\ga}\n-(\\rho_3+\\rho_4)p_{1\\ga}+(\\rho_1+\\rho_2)p_{2 \\ga}+p_{3 \\ga}\\Big)+i y^\\ga t_{\\ga} \\\\\n+\\frac{\\rho_1\\rho_3}{(\\rho_1+\\rho_2)(\\rho_3+\\rho_4)}\\left(i y^\\ga t_{ \\ga}\n+it^{ \\ga}p_{3\\ga}\\right)+i\\left(\\frac{(1-\\rho_4)\\rho_2}{\\rho_1+\\rho_2}\n+\\rho_4\\right)t^{\\ga}p_{1\\ga}-i\\frac{\\rho_4\\rho_1}{\\rho_3+\\rho_4}t^\\ga p_{2\\ga}\\Big\\} \\go CC C,\n\\end{multline}\n\n\\begin{multline}\\label{FFFFFFFFk}\n\\dr_x B^{\\eta\\, loc}_2\\big|_{\\go CCC}\\approx \\frac{\\eta^2}{4}\\int_0^1 d\\mathcal{T}\n\\int_0^1 d\\xi\\int d^3\\rho_+\\,\n \\delta\\left(1-\\sum_{i=1}^3\\rho_i\\right)\\left(z_\\ga y^\\ga\\right)\\Big[\\left(\\mathcal{T}z^\\ga\n -\\xi y^\\ga\\right)t_{ \\ga}\\Big]\\times\\\\\n\\times \\exp\\Big\\{i\\mathcal{T}z_\\ga y^\\ga+i(1-\\rho_2)t^\\ga p_{1\\ga}\n-i\\rho_2 t^\\ga p_{2\\ga} +i\\mathcal{T}z^\\ga\\Big(-(\\rho_1+\\rho_2)t_{\\ga}\n-\\rho_1 p_{1 \\ga}+(\\rho_2+\\rho_3)p_{2 \\ga}+p_{3\\ga}\\Big) \\\\\n+iy^\\ga\\Big(\\xi(\\rho_1+\\rho_2)t_{\\ga}+\\xi\\rho_1 p_{1 \\ga}\n-\\xi(\\rho_2+\\rho_3)p_{2 \\ga}+(1-\\xi)p_{3\\ga}\\Big) \\Big\\}\\go CCC\\,,\n\\end{multline}\n\n\\begin{multline}\\label{wB3modH+}\n\\omega\\ast {B}_3^{\\eta\\eta}\\approx-\\frac{\\eta^2}{4} \\int_0^1 d\\mathcal{T}\\, \\mathcal{T}\n \\int d^3 \\rho_+ \\delta\\left(1-\\sum_{i=1}^3 \\rho_i\\right) \\int_0^1 d\\xi\\, \\frac{\\rho_1\\,\n\\left[z_\\ga\\left(y^\\ga+t^\\ga\\right)\\right]^2 }{(\\rho_1+\\rho_2)(\\rho_1+\\rho_3)}\\times\\\\\n\\times\\exp\\Big\\{i\\mathcal{T}z_\\ga y^\\ga\n+i\\mathcal{T} z^\\ga\\Big(-t_{ \\ga}-(\\rho_1+\\rho_3)p_{1\\ga}+(\\rho_2-\\rho_3)p_{2\\ga}\n+(\\rho_1+\\rho_2)p_{3\\ga}\\Big)+iy^\\ga t_{\\ga}\\\\\n+i(1-\\xi)y^\\ga\\left(\\frac{\\rho_1}{\\rho_1+\\rho_2}p_{1\\ga}\n-\\frac{\\rho_2}{\\rho_1+\\rho_2}p_{2\\ga}\\right)\n+i\\xi\\, y^\\ga\\left(\\frac{\\rho_1}{\\rho_1+\\rho_3}p_{3\\ga}-\\frac{\\rho_3}{\\rho_1+\\rho_3}p_{2\\ga}\\right) \\\\\n+i\\frac{(1-\\xi)\\rho_1}{\\rho_1+\\rho_2}t^{\\ga}p_{1\\ga}\n-i\\left(\\frac{(1-\\xi)\\rho_2}{\\rho_1+\\rho_2}+\\frac{\\xi\\rho_3}{\\rho_1+\\rho_3}\\right) t^{\\ga}p_{2\\ga}\n+i\\frac{\\xi\\rho_1}{\\rho_1+\\rho_3}t^\\ga p_{3\\ga}\\Big\\} \\go CCC,\n\\end{multline}\n\n\n\\begin{multline}\\label{kuku5}\n\\dr_x {B}_3^{\\eta\\eta}\\big|_{\\go\nCCC}\\approx \\frac{\\eta^2}{4} \\int_0^1 d\\mathcal{T}\\, \\mathcal{T}\n\\int d^3 \\rho_+ \\delta\\left(1-\\sum_{i=1}^3 \\rho_i\\right) \\int_0^1\nd\\xi\\, \\frac{\\rho_1\\, (z_\\ga y^\\ga)^2\n }{(\\rho_1+\\rho_2)(\\rho_1+\\rho_3)}\\times\\\\\n\\times\\exp\\Big\\{i\\mathcal{T}z_\\ga y^\\ga+i\\mathcal{T} z^\\ga\n\\Big(-(\\rho_1+\\rho_3)(t_{\\ga}+p_{1\\ga})+(\\rho_2-\\rho_3)p_{2\\ga}\n+(\\rho_1+\\rho_2)p_{3\\ga}\\Big)+it^\\ga p_{1\\ga} \\\\\n+i(1-\\xi)y^\\ga\\left(\\frac{\\rho_1}{\\rho_1+\\rho_2}(t_{ \\ga}+p_{1\\ga})-\\frac{\\rho_2}{\\rho_1+\\rho_2}p_{2\\ga}\\right)+\\xi\\, y^\\ga\\left(\\frac{\\rho_1}{\\rho_1+\\rho_3}p_{3\\ga}-\\frac{\\rho_3}{\\rho_1+\\rho_3}p_{2\\ga}\\right)\\Big\\}\\go CCC.\n\\end{multline}\nThe sum of \\rhss of \\eq{origW1B2}-\\eq{kuku5} yields $\\widehat{\\Upsilon}^{\\eta\\eta}_{\\go CCC} (Z;Y) $.\n\n Note, that all terms on the \\rhss of \\eq{origW1B2}-\\eq{kuku5} contain no\n$p_j{}_\\ga p_i{}^\\ga$ contractions in the exponentials, hence being spin-local\n \\cite{4a3}. Thus $\\widehat{\\Upsilon}^{\\eta\\eta}_{\\go CCC} (Z;Y) $ is also spin-local.\n\n\nLet us emphasize that only the full expression for $\\Upsilon^{\\eta\\eta}_{\\go CCC}(Y) $ \\eq{projwccc}\nis $Z$-independent, while $\\widehat{\\Upsilon}^{\\eta\\eta}_{\\go CCC} (Z;Y) $ \\eqref{rightsideU}\nwith discarded terms in $\\Hp$ is not.\nThis does not allow one to find manifestly\n$Z$-independent expression for $ {\\Upsilon}^{\\eta\\eta}_{\\go CCC} $ by setting for instance\n $Z=0$ in Eqs.~\\eq{origW1B2}-\\eq{kuku5}.\n\n\n\n In this paper $Z$-dependence of $\\widehat{\\Upsilon}^{\\eta\\eta}_{\\go CCC}(Z;Y)$\nis eliminated modulo terms in $\\Hp$ by virtue of partial integration\n and the Schouten identity. As a result,\n $\n \\widehat{\\Upsilon}^{\\eta\\eta}_{\\go CCC}(Z;Y)\\approx \\widehat{\\widehat{\\Upsilon}} {\\,}^{\\eta\\eta}_{\\go CCC}(Y),$$\n where $\\widehat{\\widehat{\\Upsilon}} {\\,}^{\\eta\\eta}_{\\go CCC}(Y)$ is manifestly spin-local and $Z$-independent.\n Since $\\Hp_0$-terms do not contribute to the vertex by Z-dominance Lemma \\cite{2a1}\n\n $$\\Upsilon^{\\eta\\eta}_{\\go CCC}(Y)=\\widehat{\\widehat{\\Upsilon}} {\\,}^{\\eta\\eta}_{\\go CCC}(Y)\\,.$$\n\n\n Our goal is to find the manifest form of\n$\\widehat{\\widehat{\\Upsilon}} {\\,}^{\\eta\\eta}_{\\go CCC}(Y)$.\n\n \\section{Calculation scheme}\n\\label{Schema}\n\n\n\n\n\nThe calculation scheme is as follows.\n\n\\begin{itemize}\n\n\\item I. We start from the expression Eqs.~\\eq{origW1B2}-\\eq{kuku5} for the vertex obtained in \\cite{4a3}.\n\n\\bigskip\n\n\\item II. To $z$-linear pre-exponentials. \\\\\n Using partial integration and the Schouten identity\nwe transform Eqs.~\\eq{origW1B2}-\\eq{kuku5} to the form with $z$-linear pre-exponentials modulo\nweakly $Z$-independent\n (cohomology) terms.\nThese expressions are collected in Section \\ref{zlinear}, Eqs.~\\eq{RRwB3modH+}-\\eq{W2C3gr1}.\nThe respective cohomology terms being a part of the vertex\n$\\Upsilon^{\\eta\\eta}_{\\go CCC} $\nare presented in Section \\ref{Main}\\,.\n\n\\bigskip\n\\item\nIII. Uniformization.\\\\ We observe that the \\rhss of Eqs.~\\eq{RRwB3modH+}-\\eq{W2C3gr1}\ncan be re-written modulo cohomology and weakly zero terms in a form\nof integrals $\\int d\\Gamma$ over the same integration domain $\\II$\n\\begin{equation}\\label{comexp}\n \\int d\\Gamma \\, z_\\ga f^\\ga (y,t,p_1,p_2,p_3\\vert \\T,\\xi_i,\\rho_i)\\Ee\\, \\go CCC\\,,\n\\end{equation}\nwhere the integrand contains an overall exponential function $\\Ee$\n\\begin{equation}\n\\label{Ee}\\Ee= \\Ez E,\n\\end{equation}\n\\be\n\\label{expz}\n \\Ez:=\\exp i\\Big\\{\\T z_{\\ga}(y + \\Pz{})^{\\ga} \\Big\\}\n\\q\\ee\n \\bee\n \\label{Egx=}\n &&E:=\\exp i \\Big\\{\n - \\gx_2 \\ff{\\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3 )}\\,\\,\\big( y + \\Pz{}\\big)^\\ga y_{\\ga}\n\\\\ \\nn &&\n+ \\gx_1 \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)}\\big( y + \\Pz{}\\big)^\\ga\\tilde{t}{}_{\\ga}\n\\\\ \\nn &&+\\ff{ \\gr_3 }{(1-\\gr_1-\\gr_4 ) }\\,\\, ( p_3+p_2)^{\\ga}\n y_{\\ga}\n-\\ff{ \\gr_3 }{(1-\\gr_1-\\gr_4 )(1-\\gr_3 )}\\,\\, \\gr_1 {t}{}^{\\ga} y_{\\ga}\n \\\\ \\nn &&\n + \\ff{ \\gr_1 }{(1-\\gr_3)} (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}\n + p_3{}_{\\ga} y^{\\ga} +p_1{}_\\ga {t}{}^\\ga\n\\Big\\}\n\\,, \\eee\n\\bee &&\\label{tildet}\\tilde{t}{}=\\ff{\\gr_1}{\\gr_1+\\gr_4}{t}{}\\q\n\\\\ &&\\label{Pz5}\n\\Pz{}= \\PP + (1-\\gr_4){t}{}\\,,\n\\\\ &&\\label{PP=}\n\\PP =( 1-\\gr_1-\\gr_4)(p{}_1 +p_2) - (1-\\gr_3) (p_3+p_2)\\,, \\eee\n the integral over $\\II$ is denoted as\n \\begin{equation}\\label{dGamma}\n \\int d\\Gamma=\\int_0^1 d\\T\\int d^3 \\xi_+\\, \\delta\\left(1-\\sum_{i=1}^3 \\xi_i\\right)\n \\int d^4 \\rho_+ \\, \\delta\\left(1-\\sum_{j=1}^4 \\rho_j\\right)\\,.\n \\end{equation}\n\n\n\nEqs.~\\eq{RRwB3modH+}-\\eq{W2C3gr1} transformed to the form \\eq{comexp}\nare collected in Section \\ref{uniform}, Eqs.~\\eqref{F1}-\\eqref{F4}.\n\n\n\n\\item IV. Elimination of $\\gd$-functions.\\\\\nUsing partial integration and the Schouten identity we eliminate\nthe all factors of $\\gd(\\gr_i)$,\n $\\gd(\\gx_{1 })$ and $\\gd(\\gx_{ 2})$ from Eqs.~\\eqref{F1}-\\eqref{F4}. The result is presented in Section \\ref{Eli0}, Eqs.~\\eqref{rightsideUUNI==}-\\eq{FRest3}.\n\n\\item V. Final step.\\\\ Finally, we show in Section \\ref{proof} that a sum of\nthe \\rhss of Eqs.~\\eqref{FRest1}-\\eqref{FRest3}\n\n is $Z$-independent\n up to $\\Hp$.\n\\end{itemize}\n\nBy collecting all resulting $Z$-independent terms we finally\n obtain the manifest expression\n for vertex $\\Upsilon^{\\eta\\eta}_{ \\go CCC}$, being a sum of expressions \\eq{go B3modHcoh}-\\eq{ERRGTC}.\n\n\n\\section{Main result $\\Upsilon^{\\eta\\eta}_{\\go CCC}$}\n\\label{Main}\n\nHere\nthe final manifestly $Z$-independent $\\go CCC$ contribution to the equations is presented.\n\nVertex $\\Upsilon^{\\eta\\eta}_{\\go CCC}$ is\n \\be\\label{upsrES}\n\\Upsilon^{\\eta\\eta}_{\\go CCC}=\\sum_{j=1}^{11} J_j\\,\n\\ee\nwith $J_i$ given in Eqs.~\\eq{go B3modHcoh}-\\eq{ERRGTC}.\nNote that the integration\nregions may differ for different terms $J_j$\nin the vertex, depending on their genesis.\n\n\n\nFirstly we note that $B^{\\eta\\eta}_3$ \\eqref{B3modH=1406}, that contains a $Z$-independent part, generates cohomologies both from $\\go*B^{\\eta\\eta}_3$ and from $\\dr_x B^{\\eta\\eta}_3$,\n\\begin{equation}\\label{go B3modHcoh}\nJ_1= - \\ff{ \\eta^2 }{4 } \\int d\\Gamma\\, \\delta(\\xi_3)\\ff{\\gr_2}{(\\gr_2+\\gr_1 )(\\gr_2+\\gr_3)} \\gd(\\gr_4) E\\, \\go CCC,\n\\end{equation}\n \\begin{equation}\\label{dx B3modHcoh}\nJ_2= \\ff{ \\eta^2 }{4 } \\int d\\Gamma\\, \\delta(\\xi_3)\\ff{\\gr_2}{(\\gr_2+\\gr_4 )\n(\\gr_2+\\gr_3)} \\gd(\\gr_1) E\\, \\go CCC\\,.\n\\end{equation}\nRecall that $E$ and $d\\Gamma$ are defined in \\eq{Egx=} and \\eq{dGamma}, respectively.\n(Note, that, here and below, the integrands on the \\rhss of expressions for $J_i$ are $\\T$-independent, hence the factor of $\\int_0^1 d\\T$ in $d\\Gamma$ equals one.)\n\n\n\nOther\ncohomology terms are collected from \\eqref{FRest1}, \\eqref{FRest2}, \\eqref{FRest3},\n\\eq{lostcohomo}, \\eq{lostcohomo2}, \\eqref{D4}, \\eqref{D6}, \\eqref{D7}\nand \\eqref{ERRGT}, respectively,\n\n\\begin{multline}\\label{Result2}\nJ_3= -\\ff{i \\eta^2 }{4 } \\int d\\Gamma\\, \\delta(\\xi_3) \\ff{1}{(\\gr_2 +\\gr_3)(1-\\gr_3) }\n\\Big\\{\\gr_2 {t}{}^\\ga (p_1+p_2 ){} _\\ga \\big[\\overrightarrow{\\p}_{\\gr_2}-\\overrightarrow{\\p}_{\\gr_3}\\big] \\\\\n + \\gr_2 ( p_1{}+ p_2)^{\\ga} ( p_3{}+p_2)_{\\ga}\n \\big[\\overrightarrow{\\p}_{\\gr_4}-\\overrightarrow{\\p}_{\\gr_1}\\big] +\\gr_2 {t}{}^\\ga ( p_3{} +p_2{} )_\\ga \\big[\\overrightarrow{\\p}_{\\gr_2}-\\overrightarrow{\\p}_{\\gr_1}\\big]+\\ff{ \\gr_1+\\gr_4}{ (1-\\gr_3) } {t}{}^\\ga (p_1+p_2 ){} _\\ga\n \\Big\\}E\\, \\go C C C\\,,\n\\end{multline}\n\n\n\\begin{multline}\\label{Result3}\nJ_4=\\frac{i\\eta^2}{4}\\int d\\Gamma\\, \\frac{\\delta(\\xi_3)}{1-\\rho_3}\\Big(\n \n - \\ff{ \\gr_3}{(1-\\gr_1-\\gr_4 )^2(1-\\gr_3)}\n {t}{}^\\gga y_\\gga\n - \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)}\n {t}{}^\\gga y_\\gga\n [-\\overrightarrow{\\p}_{\\gr_1}+\\overrightarrow{\\p}_{\\gr_2}] \\\\\n - \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)}\n ( p_1{}+ p_2)^{\\gga} (y+\\tilde{t}{}) _{\\gga}\n [ \\overrightarrow{\\p}_{\\gr_4}-\\overrightarrow{\\p}_{\\gr_1}]\n \\Big)E\\go C C C\\,,\n\\end{multline}\n\n\\begin{multline}\\label{Result4}\nJ_5=-i \\ff{ \\eta^2 }{4 } \\int d\\Gamma\\,\\delta(\\xi_3)\\Big[1+ \\gx_1(\\overrightarrow{\\p}_{\\gx_1}-\\overrightarrow{\\p}_{\\gx_2})\\Big]\n \\Big\\{\n \\ff{ -\\gr_2 }{(1-\\gr_1-\\gr_4 )^2(1-\\gr_3) ( \\gr_1+\\gr_4 ) }\n (p_3{}^{\\ga}+p_2{}^{\\ga})^\\gga {t}{}_{\\gga} \\\\\n- \\ff{ \\gr_3 }{(1-\\gr_1-\\gr_4 )^2(1-\\gr_3 )^2}\\,\\, {t}{}^{\\ga} y_{\\ga}\n + \\ff{ 1}{(\\gr_2 +\\gr_3)(1-\\gr_3) ( \\gr_1+\\gr_4 ) } (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}\n \\Big\\} E\\, \\go C C C\\,,\n\\end{multline}\n\n\\begin{equation}\\label{Result5}\nJ_6=i\\ff{ \\eta^2 }{4 } \\int d\\Gamma\\,\\delta(\\xi_3) \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2(\\gr_1+\\gr_4)}\n ( p_1{}+ p_2)^{\\gga} ( {t}{}) _{\\gga}E \\go C C C\\,,\n\\end{equation\n\\begin{multline}\\label{Result6}\nJ_7=- \\frac{\\eta^2}{4}\\int d\\Gamma\\, \\delta(\\xi_3)\\, \\xi_1\n\\ff{ \\gr_2\\gr_2}{(\\gr_2 +\\gr_3)^3(1-\\gr_3)^3( \\gr_1+\\gr_4 ) }\\times\\\\\n\\times \\big( y+ (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} )+ (1 -\\gr_4 ){t}{} \\big)^\\gga\n\\big( y + \\tilde{t}{} \\big)_{\\gga}\n{t}{}^{\\ga} y_\\ga E \\, \\go CCC\\,,\n\\end{multline}\n\\begin{equation}\\label{Result1}\nJ_8=- \\ff{ \\eta^2 }{4 }\\int d\\Gamma \\, \\delta(\\rho_3)\n \\Big( \\gr_1\\gd(\\gx_3 )\n + \\Big[ i {\\gd(\\gr_4)} -( p_2{}_\\ga+ p_1{}_\\ga) {t}{}^{\\ga}\\Big]\n \\Big\\{ i \\gd(\\gx_3 )+\n \\tilde{t}{}^{\\gga} y_\\gga\\Big\\}\\Big)E\\, \\go CCC\\,,\n\\end{equation}\n\n\n\\begin{multline}\\label{goB3modH1406gr1C}\nJ_9= i\\eta^2\\chalf \\int d\\Gamma\\, \\delta(\\rho_1)\\delta(\\rho_4)\\delta(\\xi_3)\\exp \\Big\\{ -i\\gx_2 ( p_1+p_2+{t} - \\gr_2 (p_3+p_2))_{\\ga} (y )^{\\ga} \\\\\n-\\gx_1 ( y+ p_1+p_2 - \\gr_2 (p_3+p_2))_\\gga ( {t})^\\gga\n + ( 1-\\gr_2) (p_3+p_2) {}^\\gga y_\\gga\n + p_3{}_\\gga y^\\gga +{t}{}^\\gb p_1{}_\\gb \\Big\\}\\go CCC\\,,\n\\end{multline}\n\n\n\\begin{multline}\\label{dxB3modH1406gr1C}\nJ_{10}=-i\\eta^2\\chalf \\int d\\Gamma\\, \\delta(\\rho_4) \\gd(\\gx_1 )\\gd(\\gr_1)\n \\, \\exp i\\Big\\{-\\gx_2 ( y+ p_1+p_2+{t} - \\gr_2 (p_3+p_2))_{\\ga} (y )^{\\ga} \\\\\n+ ( 1-\\gr_2) (p_3+p_2) {}^\\gga y_\\gga +p_3{}_\\gga y^\\gga +{t}{}^\\gb p_1{}_\\gb \\Big\\}\\go CCC\\,,\n\\end{multline}\n\n\n\\begin{multline}\\label{ERRGTC}\nJ_{11}=\\frac{i \\eta^2}{4}\\int d\\Gamma \\, \\delta(\\rho_1)\\delta(\\rho_4) y^\\ga {t} {}_\\ga \\exp i\\Big\\{ (y+\\PP_0 +{t}){}^\\gga ( \\gx_1 {t}- \\gx_2 y)_\\gga + ( 1-\\gr_2) (p_3+p_2) {}^\\gga y_\\gga\n \\\\\n + p_3{}_\\gga y^\\gga +{t}{}^\\gb p_1{}_\\gb \\Big\\}\\go CCC\\,.\n\\end{multline}\n\n\nLet us emphasize, that neither exponential function $E$ \\eq{Egx=}\nnor the exponentials on the \\rhss of Eqs.~\\eq{goB3modH1406gr1C}-\\eq{ERRGTC}\ncontain $\\p_i{}_\\ga \\p_k{}^\\ga$ terms.\nHence, as anticipated, all $J_j$ are spin-local.\n\n\n One can see that though having poles in pre-exponentials these expressions are well defined.\n\\\\For instance a potentially dangerous factor on the \\rhs of \\eq{go B3modHcoh}\n is dominated by 1 as follows from the inequality\n$ {\\gr_2}-(\\gr_1+\\gr_2 )\n(\\gr_2+\\gr_3) =-\\gr_3\\gr_1\\le 0$\\, that holds due to the factor\nof $\\prod\\vartheta(\\gr_i)\\gd(1-\\sum\\gr_i )\\gd(\\gr_4)$.\nAnalogous simple reasoning applies to the \\rhs of \\eq{dx B3modHcoh}.\n\nThe case of \\eq{Result2}-\\eq{Result6} is a bit more tricky.\nBy partial integration one obtains from \\eq{Result2}-\\eq{Result4}\n\\bee\\label{RRult4+} &&\n J_3+J_4+J_5 = \\ff{i \\eta^2 }{4 } \\int d\\Gamma\\,\n\\delta(\\xi_3)\\ff{1 }{(\\gr_2 +\\gr_3)(1-\\gr_3) }\\Big\\{\n-\n\\gd({\\gr_3}) {t}{}^\\ga (p_1+p_2 ){} _\\ga\n\\\\&&\\nn\n + [ \\gd({\\gr_4})-\\gd({\\gr_1})]\\gr_2\n ( p_1{}+ p_2)^{\\ga} ( p_3{}+p_2)_{\\ga}\n + {t}{}^\\ga ( p_3{} +p_2{} )_\\ga -\\gd({\\gr_1})\\gr_2 {t}{}^\\ga ( p_3{} +p_2{} )_\\ga\n \\,\n\\\\&&\\nn\n -\\gd({\\gr_1}) \\ff{ \\gr_2}{ (1-\\gr_3)}\n {t}{}^\\gga y_\\gga\n + [ \\gd({\\gr_4})-\\gd({\\gr_1})] \\ff{ \\gr_2}{ (1-\\gr_3)}\n ( p_1{}+ p_2)^{\\gga} (y+\\tilde{t}{}) _{\\gga}\n \\\\ &&\\nn\n - \\gd({\\gx_2})\n\n \\Big(\n \\ff{ -\\gr_2 }{(\\gr_2 +\\gr_3)( \\gr_1+\\gr_4 ) }\n (p_3{}^{\\ga}+p_2{}^{\\ga})^\\gga {t}{}_{\\gga}\n \\\\&&\\nn\n- \\ff{ \\gr_3 }{(\\gr_2 +\\gr_3) (1-\\gr_3 ) }\\,\\, {t}{}^{\\ga} y_{\\ga}\n + \\ff{ 1}{ ( \\gr_1+\\gr_4 ) } (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}\n\\Big) \\Big\\} E\\, \\go C C C\\,.\n\\eee\nUsing that, due to the factor of $\\gd(1-\\sum \\gr_i)$,\nfor positive $\\gr_i $ it holds\n\\bee&&\\label{nopoles}\n \\ff{\\gr_2}{(\\gr_3+\\gr_2)(1-\\gr_3)}-1 =-\\ff{\\gr_3(1-(\\gr_3+\\gr_2 ))}{(\\gr_3+\\gr_2)(1-\\gr_3)}\\le 0\n \\q\\\\ \\label{nopoles3} && \\ff{ 1}{(\\gr_2 +\\gr_3)(1-\\gr_3) }\\,\\le\n\\ff{ 1}{(\\gr_2 +\\gr_3)(1-\\gr_3-\\gr_2) }=\n\\ff{ 1}{ ( \\gr_3+\\gr_2) }+\\ff{ 1}{ ( \\gr_1+\\gr_4 ) }\\,,\n\\eee\none can make sure that each of the expressions with poles in the pre-exponential in Eqs.~\\eq{Result5}, \\eq{Result6} and\n \\eq{RRult4+}\ncan be represented in the form of a sum of integrals with integrable pre-exponentials.\nFor instance, the potentially dangerous\nfactor in \\eq{Result6}, by virtue of \\eq{nopoles} and \\eq{nopoles3} satisfies\n\\be \\ff{ \\gr_2\\gr_2}{(\\gr_2 +\\gr_3)^3(1-\\gr_3)^3( \\gr_1+\\gr_4 ) }\\le\n \\ff{ 1}{ (1-\\gr_3)( \\gr_1+\\gr_4 ) }+\n \\ff{1}{(\\gr_3+\\gr_2) }\n +\\ff{ 1}{ ( \\gr_1+\\gr_4 ) }\\,.\\quad\\label{xx}\n\\ee\n Each of the terms on the \\rhs of Eq.~(\\ref{xx}) is integrable, because integration\n is over a three-dimensional compact area $\\sum\\gr_i=1$ in the positive quadrant.\nFor instance consider the first term. Swopping $\\gr_4\\leftrightarrow\\gr_2$ one has\n \\bee\n\\int d^4 \\gr_+ \\gd(1-\\sum_1^4 \\gr_i)\\ff{1}{(1-\\gr_3 ) ( \\gr_1+\\gr_2)}=\n\\int d^3 \\gr_+ \\vartheta(1-\\sum_1^3 \\gr_i)\\ff{1}{(1-\\gr_3 ) ( \\gr_1+\\gr_2)}=\\\\ \\nn\n-\\int_0^1 d \\gr_1 \\int_0^{1-\\gr_1} d \\gr_2\n \\ff{\\log( \\gr_1+\\gr_2)}{ ( \\gr_1+\\gr_2)}=\\half\\int_0^1 d \\gr_1 \\log^2( \\gr_1 )\\,,\n \\eee\n which is integrable.\n\n\n Analogously\nother seemingly dangerous factors can be shown to be harmless as well.\n\n\n\\section{To $z$-linear pre-exponentials}\n \\label{zlinear}\nStep II of the calculation scheme of Section \\ref{Schema} is to transform \\rhss of Eqs.~\\eq{origW1B2}-\\eq{kuku5} to $Z$-independent terms plus terms with linear in $z$ pre-exponentials\n(modulo $H^+$).\n\nTo this end, from \\eq{B3modH=1406}\none straightforwardly obtains that\n \\begin{multline} \\label{RRwB3modH+}\n\\omega \\ast {B}_3^{\\eta\\eta}\\approx J_1+ \\frac{\\eta^2}{4}\\int d\\Gamma\n\\frac{\\gd(\\xi_3)\\gd(\\gr_4)}{(1-\\gr_1)(1-\\gr_3)}\\Bigg[-\\gr_2 (z_\\ga (y^\\ga+t^\\ga))(p_{1\\gb}+p_{2\\gb})(p_2 {}^\\gb+p_3 {}^\\gb) \\\\\n+i\\Big[\\Big(\\gd(\\gr_1)+\\gd(\\gr_3)\\Big)(1-\\gr_1)(1-\\gr_3)-\\gd(\\xi_2)\\Big]\n z_\\ga\\Big((1-\\gr_1)(p_1 {}^\\ga+p_2 {}^\\ga)-(1-\\gr_3)(p_2 {}^\\ga+p_3 {}^\\ga)\\Big) \\\\\n+iz_\\ga (p_1 {}^\\ga+p_2 {}^\\ga)(1-\\gr_1)\\Big(\\gd(\\xi_2)-\\gd(\\xi_1)\\Big)\\Bigg]\n\\exp\\Big\\{i\\T z_\\ga\\big(y^\\ga+t^\\ga+(1-\\gr_1)(p_1 {}^\\ga+p_2 {}^\\ga)\n-(1-\\gr_3)(p_2 {}^\\ga+p_3 {}^\\ga)\\big) \\\\\n+\\frac{i(1-\\xi_1) \\gr_2}{\\gr_1+\\gr_2}(y^\\ga+t^\\ga) (p_{1\\ga}+p_{2\\ga})\n+\\frac{i\\xi_1 \\gr_2}{\\gr_2+\\gr_3}(y^\\ga+t^\\ga) (p_{2\\ga}+p_{3\\ga})-i(y^\\ga+t^\\ga) p_{2\\ga}\\Big\\}\\go CCC\\q\n\\end{multline}\nwhere $J_1$ is the cohomology term \\eq{go B3modHcoh}.\nAnalogously, \\begin{multline} \\label{RRdxB3modH+}\n\\dr_x {B}_3^{\\eta\\eta} \\approx J_2-\\frac{\\eta^2}{4}\\int d\\Gamma\n\\frac{\\gd(\\xi_3)\\gd(\\gr_4)}{(1-\\gr_1)(1-\\gr_3)}\n\\Bigg[-\\gr_2 (z_\\ga y^\\ga)(p_{1\\gb}+t_\\gb+p_{2\\gb})(p_2 {}^\\gb+p_3 {}^\\gb) \\\\\n+i\\Big[\\Big(\\gd(\\gr_1)+\\gd(\\gr_3)\\Big)(1-\\gr_1)(1-\\gr_3)-\\gd(\\xi_2)\\Big]\nz_\\ga\\Big((1-\\gr_1)(p_1 {}^\\ga+t^\\ga+p_2 {}^\\ga)-(1-\\gr_3)(p_2 {}^\\ga+p_3 {}^\\ga)\\Big) \\\\\n+iz_\\ga (p_1 {}^\\ga+t^\\ga+p_2 {}^\\ga)(1-\\gr_1)\\Big(\\gd(\\xi_2)-\\gd(\\xi_1)\\Big)\\Bigg]\n\\exp\\Big\\{i\\T z_\\ga\\big(y^\\ga+(1-\\gr_1)(p_1 {}^\\ga+t^\\ga+p_2 {}^\\ga)-(1-\\gr_3)(p_2 {}^\\ga+p_3 {}^\\ga)\\big)\n\\\\\n+\\frac{i(1-\\xi_1) \\gr_2}{\\gr_1+\\gr_2}y^\\ga (p_{1\\ga}+t_\\ga+p_{2\\ga})\n+\\frac{i\\xi_1 \\gr_2}{\\gr_2+\\gr_3}y^\\ga (p_{2\\ga}+p_{3\\ga})-iy^\\ga p_{2\\ga}+it^\\gb p_{1\\gb}\\Big\\}\\go CCC\n\\end{multline}\nwith $J_2$ \\eq{dx B3modHcoh}.\n\nUsing the Schouten identity and partial integration one obtains from Eqs.~\\eq{origW1B2}-\\eq{FFFFFFFFk}, respectively,\n \\begin{multline}\\label{RW1B2BP=}\nW_{1 \\, \\go C}^\\eta \\ast B_2^\\eta\\approx \\frac{\\eta^2}{4}\\int_0^1 d\\T\n\\int_0^1 d\\tau\\int_0^1 d\\gs_1 \\int_0^1 d\\gs_2\\Bigg[i(z_\\ga t^\\ga)\\gd(1-\\tau) \\\\\n+\\frac{z_\\ga(p_2 {}^\\ga+p_3 {}^\\ga)}{1-\\gt}\\Big(i\\big(\\gd(\\gs_1)-\\gd(1-\\gs_1)\\big)\n-\\big[y^\\ga+p_1 {}^\\ga +p_2 {}^\\ga-\\gs_2(p_2{}^\\ga+p_3 {}^\\ga)\\big]t_\\ga\\Big)\\Bigg]\\exp\\Big\\{i\\T z_\\ga y^\\ga \\\\\n+i\\T z_\\ga\\Big(\\tau(p_1 {}^\\ga +p_2 {}^\\ga)-((1-\\tau)+\\gs_2\\tau)(p_2 {}^\\ga +p_3 {}^\\ga)\n+\\big(\\gs_1+\\tau(1-\\gs_1)\\big)t^\\ga\\Big)+it^\\ga p_{1\\ga} \\\\\n+i\\gs_1\\big[y^\\ga+p_1 {}^\\ga +p_2 {}^\\ga-\\gs_2(p_2{}^\\ga+p_3 {}^\\ga)\\big]t_\\ga\n-i\\Big(\\gs_2 p_3 {}^\\ga-(1-\\gs_2)p_2 {}^\\ga\\Big)y_\\ga\\Big\\}\\go CCC\\,,\n\\end{multline}\n\\begin{multline}\\label{W2C3gr1}\nW_{2\\, \\go CC}^{\\eta\\eta}\\ast C\\approx -\\frac{i\\eta^2}{4}\n\\int d\\Gamma\\, \\gd(\\xi_3)\\gd(\\gr_3)\\frac{(z_\\gga t^\\gga)}{\\gr_1+\\gr_4}\n\\Big[-\\gr_1\\big( \\gd(\\gr_4)+i t^\\ga(p_{1\\ga}+p_{2\\ga})\\big)+\\xi_1\\gd(\\xi_2)\\Big]\\times\\\\\n\\times \\exp\\Big\\{i\\T z_\\ga y^\\ga+i\\T z_\\ga\n\\Big((1-\\gr_1-\\gr_4)(p_1 {}^\\ga+p_2 {}^\\ga)-(1-\\gr_3)(p_2{}^\\ga+p_3 {}^\\ga)+(1-\\gr_4)t^\\ga\\Big) \\\\\n+iy^\\ga\\left(\\frac{\\xi_1 \\gr_1}{1-\\gr_2}t_\\ga+p_{3\\ga}\\right)\n+i\\left(1-\\gr_1-\\frac{\\xi_1 \\gr_1\\gr_2}{1-\\gr_2}\\right)t^\\ga p_{1\\ga}-i(1-\\xi_1)\\gr_1 t^\\ga p_{2\\ga}\n+i\\frac{\\xi_1 \\gr_1}{1-\\gr_2}t^\\ga p_{3\\ga} \\Big\\}\\go CCC\\,,\n\\end{multline}\n\\begin{multline}\\label{FFFFFFFFk=}\n\\dr_x B_2^\\eta\\approx\\frac{i\\eta^2}{4} \\int d\\Gamma\\, \\gd(\\xi_3)\\gd(\\gr_4)\\, (z_\\ga y^\\ga)\n\\Big[it^\\gga(p_{1\\gga}+p_{2\\gga})+\\gd(\\gr_4)- \\gd(\\gr_1) \\Big]\\times\\\\\n\\times \\exp\\Big\\{i\\T z_\\ga y^\\ga\n+i\\T z_\\ga\\big((1-\\gr_1-\\gr_4)(p_1 {}^\\ga+ p_2 {}^\\ga)-(1-\\gr_3)(p_2 {}^\\ga +p_3 {}^\\ga)+(1-\\gr_4)t^\\ga\\big) \\\\\n+i(1-\\gr_2)t^\\gb p_{1\\gb}-i\\gr_2 t^\\gb p_{2\\gb}\n+i\\xi_2 y^\\ga \\Big((\\gr_1+\\gr_2)t_\\ga+\\gr_2 p_{1\\ga}-(1-\\gr_2)p_{2\\ga}-p_{3\\ga}\\Big)\n+iy^\\ga p_{3\\ga} \\Big\\}\\go CCC.\n\\end{multline}\n\n\n\\section{Generalised Triangle identity}\n\\label{SecGTid}\n\n\n Here a useful identity playing the key role in our computations is introduced.\n\n For any $F(x,y)$\nconsider\n\\bee\\label{GTH+F}\n &&I= \\int_{[0,1]} {d\n\\gt\\,}\\int d^3\n\\gx_+\n \\gd(1-\\gx_1-\\gx_2-\\gx_3 ) \\\\ \\nn&&\n z^\\gga \\Big[ (a_2-a_1)_\\gga \\gd(\\gx_3)+ (a_3-a_2)_\\gga \\gd(\\gx_1)\n+ (a_1-a_3)_\\gga \\gd(\\gx_2)\\Big] F \\big(\n \\gt z_\\gb P^\\gb\\,, ( -\\gx_1 a_1-\\gx_2 a_2-\\gx_3 a_3)_\\ga P^\\ga \\big)\\,\n\\eee\n with arbitrary $\\gt, \\gx$- independent $P$ and $a_i$.\n\nLet $G(x,y)$ be a solution to differential equation\n\\be\\label{difvim}\n\\ff{\\p}{\\p x} G(x,y)= \\ff{\\p}{\\p y}F (x,y)\\,. \\ee\nHence\n\\bee\\label{GTHF0}\n&& I = \\int_{[0,1]} {d\n\\gt\\,}\\int d^3\n\\gx_+ \\gd(1-\\gx_1-\\gx_2-\\gx_3 ) \\\\ \\nn&&\n (a_1-a_3)^\\ga(a_3-a_2)_\\ga\n \\overrightarrow{\\p}_\\gt G \\big(\n \\gt z_\\gb P^\\gb \\,, (-\\gx_1 a_1-\\gx_2 a_2-\\gx_3 a_3)_\\ga P^\\ga \\big). \\eee\nNote that there is a factor of $(a_1-a_3)^\\ga(a_3-a_2)_\\ga$ equal to the area\nof triangle spanned\nby the vectors $a_1\\,,a_2\\,, a_3$ on the \\rhs of \\eq{GTHF0}.\n\nThis identity is closely related to identity (3.24) of \\cite{4a1}, that, in turn, expresses\n{\\it triangle identity} of \\cite{Vasiliev:1989xz}.\nHence, \\eq{GTHF0} will be referred to as\n{\\it Generalised Triangle identity} or {\\it GT identity}.\n\nNote that,\n for appropriate $G$ partial integration on the \\rhs of \\eq{GTHF0}\n in $\\gt$ gives $z$-independent (cohomology) term plus $\\mathcal{H} ^+$-term. Namely,\n\\bee\\label{GTHF0pi}\n&& I = - \\int d^3\n\\gx_+ \\gd(1-\\gx_1-\\gx_2-\\gx_3 ) \\\\ \\nn&&\n (a_1-a_3)^\\ga(a_3-a_2)_\\ga\n G \\big(\n 0\\,, (-\\gx_1 a_1-\\gx_2 a_2-\\gx_3 a_3)_\\ga P^\\ga \\big)\n \\\\ \\nn&&+ \\int d^3_+\n\\gx \\gd(1-\\gx_1-\\gx_2-\\gx_3 ) \\\\ \\nn&&\n (a_1-a_3)^\\ga(a_3-a_2)_\\ga\n G \\big(\n z_\\gb P^\\gb \\,, (-\\gx_1 a_1-\\gx_2 a_2-\\gx_3 a_3)_\\ga P^\\ga \\big)\n . \\eee\nThe second term on the \\rhs belongs to $\\Hp$ if $G$ is of the form \\eq{class} satisfying \\eq{limit}.\n\n\n\n\n\n\nTo prove GT identity let us perform\n partial integration on the \\rhs of \\eq{GTH+F} with respect to $\\gx_i$. This yields\n\\bee\\label{GTH+F=}\n && I= \\int_{[0,1]} {d\n\\gt\\,}\\int {d^3\n\\gx_+\\,} \\gd(1-\\gx_1-\\gx_2-\\gx_3 ) \\\\ \\nn&&\n \\Big[\n z^\\gga (a_3-a_2)_\\gga P^\\ga a_1{}_\\ga\n+z^\\gga (a_1-a_3){}_\\gga P^\\ga a_2{}_\\ga\n+z^\\gga (a_2-a_1){}_\\gga P^\\ga a_3{}_\\ga\n\\Big]\\times\\\\ \\nn&& \\ff{\\p}{\\p y} F \\big(\n \\gt z_\\ga P^\\ga \\,,\\,\\,-(\\gx_1 a_1+\\gx_2 a_2+\\gx_3 a_3)_\\ga P^\\ga \\big)\\,. \\eee\nThe Schouten identity yields\n\\bee\n \\Big[\nz^\\gga a_1{}_\\gga P^\\ga(a_3-a_2)_\\ga\n+z^\\gga a_2{}_\\gga P^\\ga(a_1-a_3)_\\ga\n+z^\\gga a_3{}_\\gga P^\\ga(a_2-a_1)_\\ga\n\\Big]\n=\\\\ \\nn\n\\Big[z^\\gga P_\\gga \\big\\{\n a_1{}^\\ga(a_3-a_2)_\\ga\n+ a_2^\\ga(a_1-a_3)_\\ga\n+ a_3^\\ga(a_2-a_1)_\\ga\\big\\}\n\\\\ \\nn\n+z^\\gga (a_3-a_2)_\\gga P^\\ga a_1{}_\\ga\n+z^\\gga (a_1-a_3){}_\\gga P^\\ga a_2{}_\\ga\n+z^\\gga (a_2-a_1){}_\\gga P^\\ga a_3{}_\\ga\n\\Big].\n\\eee\nOne can observe that\n\\bee \\Big[z^\\gga (a_3-a_2)_\\gga P^\\ga a_1{}_\\ga\n+z^\\gga (a_1-a_3){}_\\gga P^\\ga a_2{}_\\ga\n+z^\\gga (a_2-a_1){}_\\gga P^\\ga a_3{}_\\ga\n\\Big]=\\\\ \\nn\n- \\Big[\nz^\\gga a_1{}_\\gga P^\\ga(a_3-a_2)_\\ga\n+z^\\gga a_2{}_\\gga P^\\ga(a_1-a_3)_\\ga\n+z^\\gga a_3{}_\\gga P^\\ga(a_2-a_1)_\\ga\n\\Big]\\q\n\\eee\n whence it follows \\eq{GTHF0}.\n\nA useful particular case of GT identity is that with $F(x,y) =f(x+y)$, namely\n \\bee\\label{GTH+==0}\n && \\int_{[0,1]} {d\n\\gt\\,}\\int {d^3\n\\gx_+\\,} \\gd(1-\\gx_1-\\gx_2-\\gx_3 ) z^\\gga \\Big[ (a_2-a_1)_\\gga \\gd(\\gx_3) \\\\ \\nn&&\n+ (a_3-a_2)_\\gga \\gd(\\gx_1)\n+ (a_1-a_3)_\\gga \\gd(\\gx_2)\\Big] f\\big(\n (\\gt z-\\gx_1 a_1-\\gx_2 a_2-\\gx_3 a_3)_\\ga P^\\ga \\big) \\quad\\\\ \\nn\n&& = - \\int_{[0,1]} {d \\gt\\,}\\int {d^3\n\\gx_+\\,} \\gd(1-\\gx_1-\\gx_2-\\gx_3 ) \\\\ \\nn&&\n (a_1-a_3)^\\ga(a_3-a_2)_\\ga\n \\overrightarrow{\\p}_\\gt f \\big(\n (\\gt z-\\gx_1 a_1-\\gx_2 a_2-\\gx_3 a_3)_\\ga P^\\ga \\big) \\,. \\eee\n\n\n\n\n\n \\section{Uniformization}\n\\label{uniform}\n\n\nStep III of Section \\ref{Schema} is to uniformize the \\rhs's of\n Eqs.~\\eq{RRwB3modH+}-\\eq{FFFFFFFFk=} putting them into the form \\eq{comexp}, where GT identity \\eq{GTH+F} plays an important role.\nDetails of uniformization are given in Appendix B\n (p. \\pageref{Auniform}).\n\n\n\n As a result, Eq.~\\eq{rightsideU} yields\n \\begin{equation} \\label{rightsideUUNI=}\n\\widehat{\\Upsilon}^{\\eta\\eta}_{\\go CCC}\\Big|_{\\text{mod}\\, cohomology}\\approx\n\\sum_{j=1}^4 F_j\n\\end{equation}\nwith $F_j$ presented in \\eq{F1}-\\eq{F4}.\n\n Note that different terms of $F_j$ will be considered separately in what is follows.\n For the future convenience the underbraced terms are re-numerated,\n being denoted as $F_{j,k}$, where $j$ refers to $F_j$ while $k$ refers to the\n respective underbraced term in the expression for $F_j$.\nFor instance, $F_1=F_{1,1}+F_{1,2}+F_{1,3}+F_{1,4}$, {\\it{etc}}.\n\n\\begin{multline}\\label{F1}\n-\\go\\ast B_3^{\\eta\\eta}\\Big|_{mod\\, \\delta(\\rho_1)\\&\\delta(\\T)}\\approx F_1 :=-\\frac{\\eta^2}{4}\\int d\\Gamma\\, \\frac{\\delta(\\xi_3)\\delta(\\rho_4)}{(1-\\rho_1-\\rho_4)(1-\\rho_3)}\\Big[\\underbrace{\\rho_2 (z_\\beta \\PP^\\beta)(p_{1\\ga}+p_{2\\ga})(p_2 {}^\\ga+p_3 {}^\\ga)}_1 \\\\\n+\\underbrace{ i\\delta(\\rho_3)(1-\\rho_1-\\rho_4)(1-\\rho_3) (z_\\ga \\PP^\\ga)}_2 +\\underbrace{-i\\xi_1\\delta(\\xi_2)(z_\\ga \\PP^\\ga)}_3 \\\\\n+\\underbrace{i(1-\\rho_1-\\rho_4)z_\\ga(p_1 {}^\\ga+p_2 {}^\\ga)\\Big(\\delta(\\xi_2)-\\delta(\\xi_1)\\Big)}_4\\Big]\n\\mathcal{E}\\go CCC\n\\,,\\end{multline}\n\n\\begin{multline}\\label{F2}\n-\\dr_x B^{\\eta\\eta}_3\\Big| _{mod\\, \\delta(\\rho_1)\\&\\delta(\\T)}\\approx F_2 :=+\\frac{\\eta^2}{4}\\int d\\Gamma\\, \\frac{\\delta(\\xi_3)\\delta(\\rho_1)}{(1-\\rho_1-\\rho_4)(1-\\rho_3)}\\Big[\\underbrace{\\rho_2 (z_\\beta \\PP^\\beta)(p_{1\\ga}+p_{2\\ga})(p_2 {}^\\ga+p_3 {}^\\ga)}_1 \\\\\n+\\underbrace{ \\rho_2(1-\\rho_4) (z_\\beta t^\\beta)t_\\ga(p_2 {}^\\ga+p_3 {}^\\ga)}_2 +\\underbrace{ \\rho_2(1-\\rho_4) (z_\\beta t^\\beta)(p_{1\\ga}+p_{2\\ga})(p_2 {}^\\ga+p_3 {}^\\ga)}_3+\\underbrace{\\rho_2 (z_\\beta \\PP^\\beta)t_\\ga(p_2 {}^\\ga+p_3 {}^\\ga)}_4 \\\\\n+\\underbrace{ i\\delta(\\rho_3)(1-\\rho_1-\\rho_4)(1-\\rho_3)(z_\\ga \\Pz^\\ga)}_5+\n \\underbrace{-i\\xi_1\\delta(\\xi_2)(z_\\ga \\PP^\\ga)}_6+\\underbrace{-i\\xi_1\\delta(\\xi_2)(1-\\rho_4)(z_\\ga t^\\ga)}_7 \\\\\n+\\underbrace{ i(1-\\rho_1-\\rho_4)z_\\ga(p_1 {}^\\ga+p_2 {}^\\ga)\n\\Big(\\delta(\\xi_2)-\\delta(\\xi_1)\\Big)}_8 +\n\\underbrace{ i(1-\\rho_1-\\rho_4)z_\\ga t^\\ga\\Big(\\delta(\\xi_2)-\\delta(\\xi_1)\\Big)}_9\\Big]\\mathcal{E}\\go CCC\n\\,,\\end{multline}\n\n\n\n\n\\begin{multline}\\label{F3}\n-\\dr_xB_2^\\eta-W_{2\\, \\go CC}^{\\eta\\eta}\\ast C\\Big|_{mod\\, \\delta(\\T)} \\approx\nF_3:=-\\frac{\\eta^2}{4}\\int d\\Gamma\\delta(\\rho_3)\\delta(\\xi_3)\\Bigg[\n\\underbrace{ i\\delta(\\rho_1)(z_\\ga \\Pz^\\ga)}_1+\n\\underbrace{-\\frac{i(z_\\ga t^\\ga)\\, \\xi_1\\delta(\\xi_2)}{\\rho_1+\\rho_4}}_2 \\\\\n+\\underbrace{ t^\\ga(p_{1\\ga}+p_{2\\ga})z_\\gga\\PP^\\gga}_3\n+\\underbrace{ i\\delta(\\rho_4)z_\\ga (-\\PP^\\ga)}_4 +\n\\underbrace{ t^\\gga(p_{1\\gga}+p_{2\\gga})z_\\ga t^\\ga\\left((1-\\rho_4)\n-\\frac{\\rho_1}{\\rho_1+\\rho_4}\\right)}_5\\Bigg]\\mathcal{E}\\, \\go CCC\n\\,,\\end{multline}\n\n\n\\begin{multline}\\label{F4}\n-(\\dr_x B_3^{\\eta\\eta}+\\go \\ast B_3^{\\eta\\eta})\\Big|_{\\delta(\\rho_1)}\n\\Big|_{mod\\, \\delta(\\T)}-W_{1\\, \\go C}^{\\eta}\\ast B_2^{\\eta\\, loc}\\approx\nF_4:=-\\frac{\\eta^2}{4}\\int d\\Gamma\\, \\frac{\\delta(\\xi_3)\\delta(\\xi_2)\\, z_\\ga(p_2 {}^\\ga+p_3 {}^\\ga)}\n{(\\rho_2+\\rho_3)(\\rho_1+\\rho_4)}\\times\\\\\n\\times \\left(\\underbrace{i\\Big(\\delta(\\rho_1)-\\delta(\\rho_4)\\Big)\\Ee}_1+\n\\underbrace{ i\\Ez\\left(\\frac{\\p}{\\p \\rho_1}-\\frac{\\p}{\\p \\rho_4}\\right)E}_2\\right) \\go CCC.\n\\end{multline}\n\n Note that\n\\be\nF_{1,2}+F_{3,4}=0,\n\\ee\n\\be\nF_{2,5}+F_{3,1}=0.\n\\ee\n\n\nLet us emphasise that, by virtue \\eq{EEgx14=}, each $F_j$ is of the form \\eq{comexp} as expected.\n\nNote that during uniformizing procedure the vertices\n\\eq{Result1} -\\eq{ERRGTC} are obtained in Appendix B (p. \\pageref{Auniform}).\n\n\n \\section{Eliminating $\\gd(\\gr_j)$ and $\\gd(\\gx_j)$. Result}\n\\label{Eli0}\n\nThe fourth step of Section \\ref{Schema} is to eliminate all\n$\\delta(\\rho_i)$\\,,\n$\\delta(\\xi_1)$ and $\\delta(\\xi_2)$ from the pre-exponentials on the \\rhss\nof Eqs.~\\eq{F1}-\\eq{F4}.\n\nMore precisely, using partial integration, the Schouten identity and\n{ Generalised Triangle identity} \\eq{GTHF0}, taking into account Eqs.~\\eq{tildet}-\\eq{PP=} one finds\n that Eq.~\\eq{rightsideUUNI=} yields\n\\begin{equation} \\label{rightsideUUNI==}\n\\big(\\widehat{\\Upsilon}^{\\eta\\eta}_{\\go CCC} - G_1-G_2-G_3\\big)\\big|_{\\ls\\mod cohomology }\\approx 0\n \\q\n\\end{equation}\nwhere\n\\begin{multline}\\label{FRest1}\nG_1 := J_3+\\frac{\\eta^2}{4}\n\\int d\\Gamma\\, \\delta(\\xi_3) z_\\gga\\Bigg\\{\n (y^\\gga+\\widetilde{t}^\\gga) \\frac{\\rho_2\\, t^\\ga (p_{1\\ga}+p_{2\\ga})}{(1-\\rho_1-\\rho_4)(1-\\rho_3)}\\Ez\\Bigg[\\frac{\\p}{\\p \\rho_2}-\\frac{\\p}{\\p \\rho_3}\\Bigg]E \\\\\n+ (y^\\gga+\\widetilde{t}^\\gga) \\frac{\\rho_2\\, (p_1 {}^\\ga+p_2 {}^\\ga)(p_{2\\ga}+p_{3\\ga})}{(1-\\rho_1-\\rho_4)(1-\\rho_3)}\\Ez \\Bigg[\\frac{\\p}{\\p \\rho_4}-\\frac{\\p}{\\p \\rho_1}\\Bigg]E \\\\\n+ (y^\\gga+\\tilde{t}^\\gga)\n\\frac{\\rho_2\\, t^\\ga(p_{2\\ga}+p_{3\\ga})}{(1-\\rho_1-\\rho_4)(1-\\rho_3)}\n\\Ez\\Bigg[\\frac{\\p}{\\p \\rho_2}-\\frac{\\p}{\\p \\rho_1}\\Bigg]E\n+ (y^\\gga+\\tilde{t}^\\gga)\n\\frac{(\\rho_1+\\rho_4) t^\\ga (p_{1\\ga}+p_{2\\ga})}{(1-\\rho_1-\\rho_4)(1-\\rho_3)}\\Ee \\\\\n+ (y^\\gga+\\tilde{t}^\\gga)\\frac{\\rho_3\\, t^\\ga (p_{2\\ga}+p_{3\\ga})}{(1-\\rho_1-\\rho_4)^2 (1-\\rho_3)}\n\\Ee\n+\\frac{\\rho_2\\, t^\\gga (p_2 {}^\\ga+p_3 {}^\\ga)(p_{1\\ga}+p_{2\\ga}\n+t_\\ga-\\tilde{t}_\\ga)}{(1-\\rho_1-\\rho_4)(1-\\rho_3)(\\rho_1+\\rho_4)}\\Ee\\Bigg\\}\\go CCC\n\\q\\end{multline}\n \\begin{multline}\\label{FRest2}\nG_2 := J_4\n +\\frac{\\eta^2}{4}\\int d\\Gamma\\, \\frac{\\delta(\\xi_3)}{1-\\rho_3}\\,z^\\ga\n \\Bigg\\{ \\frac{\\rho_3 (y_\\ga+\\tilde{t}_\\ga)t^\\gga(y_\\gga+\\Pz_\\gga) }{(1-\\rho_1-\\rho_4)^2(1-\\rho_3)}\n \\Ee \\\\\n-\\frac{\\rho_2\\rho_4\\, t_\\ga (y^\\gga+\\Pz^\\gga)t_\\gga }\n{(1-\\rho_1-\\rho_4)(1-\\rho_3)(\\rho_1+\\rho_4)^2}\\Ee-\\frac{\\rho_2\\,\n (y_\\ga+\\tilde{t}_\\ga) t^\\gga(p_{1\\gga}+p_{2\\gga}) }{(1-\\rho_1-\\rho_4)(1-\\rho_3)}\\Ee \\\\\n-\\frac{\\rho_2\\, (p_1 {}_\\ga +p_2 {}_\\ga )(y^\\gga+\\Pz^\\gga)t_\\gga}{(1-\\rho_1-\\rho_4)(\\rho_1+\\rho_4)(1-\\rho_3)}\n \\Ee\n+\\Ez\\frac{\\rho_2\\, t^\\gga (y_\\gga+\\Pz_\\gga)(y_\\ga+\\tilde{t}_\\ga)\n }\n{(1-\\rho_1-\\rho_4)(1-\\rho_3)}\\Bigg[ \\frac{\\p}{\\p \\rho_1}-\\frac{\\p}{\\p \\rho_2}\\Bigg]E \\\\\n+\\Ez \\frac{\\rho_2\\, (y_\\ga+\\tilde{t}_\\ga) (p_1 {}^\\gga+p_2 {}^\\gga)(y_\\gga+\\Pz_\\gga)\n }{(1-\\rho_1-\\rho_4)(1-\\rho_3)}\\Bigg[\\frac{\\p}{\\p \\rho_1}\n-\\frac{\\p}{\\p \\rho_4}\\Bigg]E\\Bigg\\}\\go CCC\\q\n\\end{multline}\n \\begin{multline}\\label{FRest3}\nG_{ 3\n := J_5 + \\frac{\\eta^2}{4}\n\\int d\\Gamma\\, \\delta(\\xi_3) \\Bigg(1+\\xi_1\\Bigg[\\frac{\\p}{\\p \\xi_1}\n-\\frac{\\p}{\\p \\xi_2}\\Bigg]\\Bigg)\\times\\\\\\times\n z_\\ga\n\\Bigg\\{\\frac{\\rho_2\\, t^\\ga(p_2 {}^\\gga+p_3 {}^\\gga)(y_\\gga+\\tilde{t}_\\gga)}\n{(1-\\rho_1-\\rho_4)^2 (1-\\rho_3)(\\rho_1+\\rho_4)}\n + \\frac{-\\rho_2\\, t^\\ga (\\tilde{t}^\\gga+y^\\gga)(y_\\gga+\\Pz_\\gga)\n }{(1-\\rho_1-\\rho_4)^2 (1-\\rho_3)^2 (\\rho_1+\\rho_4)}\n\\\\+\\frac{-\\rho_3\\, (y^\\ga+\\tilde{t}^\\ga) (t^\\gga y_\\gga)}\n{(1-\\rho_1-\\rho_4)^2 (1-\\rho_3)^2}+\\frac{ (y^\\ga+\\tilde{t}^\\ga)\n(p_1 {}^\\gga+p_2 {}^\\gga)t_\\gga}{(1-\\rho_1-\\rho_4)(1-\\rho_3)^2} \\Bigg\\}\\Ee \\, \\go CCC \\q\n\\end{multline}\nwith $J_3$, $J_4$ and $J_5$ being the cohomology terms \\eq{Result2}, \\eq{Result3} and \\eq{Result4}, respectively.\n(Details of the derivation are presented in Appendix C (p.\\pageref{AppD}).)\n\nNote that schematically\n \\begin{equation}\\label{NoDistrib}\n G_1+G_2+G_3 = \\int d\\Gamma\\, \\delta(\\xi_3)\n z_\\ga g ^\\ga(y,t,p_1,p_2,p_3\\vert \\rho ,\\xi ) \\Ee \\, \\go CCC\\,+ J_3+J_4+ J_5\\q\n\\end{equation}\n as expected . Let us stress that $g^\\ga(y,t,p_1,p_2,p_3\\vert \\rho ,\\xi)$ on the \\rhs of \\eq{NoDistrib} is\n free from a distributional behaviour.\n\n\n\\section{Final step of calculation}\n\\label{proof}\n\n\n Here this is shown that the sum of\n the \\rhss of Eqs.~\\eqref{FRest1}-\\eqref{FRest3} gives a $Z$-independent\n cohomology term up to terms in $\\Hp$.\n\n More in detail, the expression $ G_1+G_2+G_3 $\n of the form \\eq{NoDistrib} consists of two types of\nterms with the pre-exponential of degree four and six in $z, y,t,p_1,p_2,p_3$, respectively.\nThat with degree-four pre-exponential separately equals a $Z$-independent\n cohomology term up to terms in $\\Hp$. This is considered in Section \\ref{DVOJNYE}.\nThe term with degree-six pre-exponential is considered in Section \\ref{TROJNYE}.\nAs a result of these calculations $J_6$ \\eq{Result5} and $ J_7$ \\eq{Result6} are obtained.\n\n \\subsection{Degree-four pre-exponential}\n \\label{DVOJNYE} Consider the sum of expressions with $z$-dependent degree-four\n pre-exponential\nfrom Eqs.~ \\eqref{FRest1}, \\eqref{FRest2} and \\eq{FRest3}, denoting it as $S_4$.\n Partial integration yields\n \\bee\\label{lostcohomo}&&S_4\\approx J_6 +\\ff{ \\eta^2 }{4 }\\int d\\Gamma \\, \\delta(\\xi_3)\\\n \\Big[\n \\ff{\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3)(\\gr_1+\\gr_4) }\n {t}{}^\\ga z _\\ga ( p_3+p_2)^{\\gga}( {t}-\\tilde{t}\n ){}_{\\gga} \\\\ \\nn &&\n + \\ff{ \\gr_2 \\gr_4}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2(\\gr_1+\\gr_4)^2}\n {t}{}^\\gga z_\\gga \\big(y + \\Pz{}\\big)^\\ga {t}{} _{\\ga}\n \\\\ \\nn&&\n +\n \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2(\\gr_1+\\gr_4)}\n ( p_1{}+ p_2)^{\\gga} \\big(y + (1-\\gr_4){t}{}\n \\big) _{\\gga} z^\\ga {t}{}_{\\ga}\n \\\\ \\nn &&\n+ \\ff{ \\gr_2}{(1- \\gr_1 -\\gr_4)^2(1-\\gr_3)(\\gr_1+\\gr_4)} {t}{}^{\\ga}z_{\\ga}\n \\, ( p_3+p_2)^{\\gga} (y+\\tilde{t}{})_{\\gga}\n \\\\ \\nn &&\n \n + \\ff{ \\gr_2}{(1- \\gr_1 -\\gr_4)^2(1-\\gr_3)^2( \\gr_1+\\gr_4 ) }\n\\big( -\\Pz{}+ \\tilde{t}{} \\big)^\\gga \\big( y + \\tilde{t}{} \\big)_{\\gga} z^\\ga {t}{}_{\\ga}\n \\Big] \\Ee\\go CCC\\q\n\\eee\nwhere the cohomology term $J_6$ is given in \\eq{Result5}\\,.\nIt is not hard to see that the\nintegrand of the remaining term is zero by virtue of the Schouten identity.\n\n\\subsection{Degree-six pre-exponential}\n\\label{TROJNYE}\n\nTerms of this type either appear in \\eqref{FRest1}, \\eqref{FRest2} via differentiation\n in $\\gr_j$ or in \\eqref{FRest3} via differentiation in $\\gx_j$.\nDenoting a sum of these terms as $S_6$ we obtain\n \\bee\\label{SUM3} &&S_6= +\\ff{ \\eta^2 }{4 }\\int d\\Gamma \\, \\delta(\\xi_3) \n \\Big\\{\n \\Ez (y+ \\tilde{t}{} )^{\\gga}z_{\\gga}\\ff{\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3) }{t}{}^\\ga\n (p_1+p_2 ){} _\\ga \\Big[\n (\\overrightarrow{\\p}_{\\gr_2}-\\overrightarrow{\\p}_{\\gr_3})E \\Big]\\qquad\n \\\\ \\nn&&\n + \\Ez\n \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n \\Big[\n \n ( p_1{}+ p_2)^{\\gga} \\big(y + (1 -\\gr_4 ){t}{}\\big)_{\\gga} z_\\ga(y+\\tilde{t}{})^{\\ga}\n \\Big]\n [ \\overrightarrow{\\p}_{\\gr_4}-\\overrightarrow{\\p}_{\\gr_1}] E\n \\\\ \\nn&&\n + \\Ez\n \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga z_\\ga(y+\\tilde{t}{})^{\\ga}\n [\\overrightarrow{\\p}_{\\gr_2}-\\overrightarrow{\\p}_{\\gr_1} ]E\n \\\\\\nn &&\n \n+i \\gx_1\\Big[\n + \\Big\\{\n +\\ff{ \\gr_2\\gr_2}{(1- \\gr_1 -\\gr_4)^3(1-\\gr_3)^3( \\gr_1+\\gr_4 ) }\n\\big( y+ (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} )+ (1 -\\gr_4 ){t}{} \\big)^\\gga \\big( y + \\tilde{t}{} \\big)_{\\gga} z_\\ga {t}{}^{\\ga}\n \\\\ \\nn &&\n-\\ff{ \\gr_3\\gr_2}{(1- \\gr_1 -\\gr_4)^3(1-\\gr_3)^3 }\n {\\big( y + \\tilde{t}{} \\big)^\\gga z_{\\gga} {t}{}^{\\ga} y_{\\ga}}\n \\\\ \\nn &&\n + \\ff{ \\gr_2}{(1- \\gr_1 -\\gr_4)^2(1-\\gr_3)^3 } \\big( y\n + \\tilde{t}{} \\big)^\\gga z_{\\gga} { (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}}\n \\Big\\} \\Ee\\Big]\\time\n \\big(y + \\Pz{} \\big)^\\ga\n (y+\\tilde{t}{})_{\\ga}\\Big\\} \\go CCC\n \\eee\nRecall that the integral measure $\\dr \\Gamma$\\eq{dGamma} contains the factor of $ \\gd(1-\\sum_1^3 \\gx_i)$.\nHence taking into account the factor of $\\gd(\\gx_3)$ on the \\rhs of \\eq{SUM3} the\n dependence on $\\gx_2,\\gx_3$ can be eliminated\nby the substitution $\\gx_2\\to 1-\\gx_1$, $\\gx_3\\to 0$. Then we consider\n separately the terms that contain and do not contain $\\xi_1$ in the pre-exponentials.\nAs shown in Appendix D, those with $\\gx_1$-proportional pre-exponentials give $J_7$ \\eq{Result6} up to $\\Hp$,\nwhile those with\n $\\gx_1 $-independent pre-exponentials give zero up to $\\Hp$.\n\n\n\n\\section{Conclusion}\n\nIn this paper starting from $Z$-dominated expression obtained in \\cite{4a3} the manifestly\nspin-local holomorphic vertex $\\Upsilon^{\\eta\\eta}_{\\go CCC}$\nin the equation \\eqref{zeroform}\n is obtained for the $\\go CCC$ ordering.\nBesides evaluation the expression for the vertex,\nour analysis illustrates how $Z$-dominance implies spin-locality.\n\n\nOne of the main technical difficulties towards $Z$-independent expression was uniformization,\nthat is bringing\nthe exponential factors to the same form, for all contributions\n\\eqref{origW1B2}-\\eqref{kuku5} with the least amount of new integration parameters\npossible. Practically, some part of the uniformization procedure heavily used\nthe Generalized Triangle identity of Section \\ref{SecGTid} playing important role in our analysis.\n\n\n\nLet us stress that spin-locality of the vertices\n obtained in \\cite{4a3} follows from $Z$-dominance Lemma.\n However the evaluation the explicit spin-local vertex\n$\\Upsilon^{\\eta^2}_{\\go CCC}$ achieved in this\npaper is technically involved. To derive explicit form of other spin-local vertices\nin this and higher orders a more elegant approach to this problem is\nhighly desirable.\n\n\n\\section*{Acknowledgments}\n\nWe would like to thank Mikhail Vasiliev for fruitful discussions and useful comments on\nthe manuscript.\nWe acknowledge a partial support from the Russian Basic\nResearch Foundation Grant No 20-02-00208.\n The work of OG is partially supported by the FGU FNC SRISA RAS (theme 0065-2019-0007).\n\n\n\n\\newcounter{appendix}\n\\setcounter{appendix}{1}\n\\renewcommand{\\theequation}{\\Alph{appendix}.\\arabic{equation}}\n\\addtocounter{section}{1} \\setcounter{equation}{0}\n \\renewcommand{\\thesection}{\\Alph{appendix}.}\n \\addcontentsline{toc}{section}{\\,\\,\\,\\,\\,\\,\\,Appendix A: $B_3^{\\eta\\eta}$}\n\n\n\n\n\n \\section*{Appendix A: $B_3^{\\eta\\eta}$}\n\\label{AppC}\n\n$B_3^{\\eta\\eta}$ modulo $\\Hp$ terms from \\cite{4a3} is given by\n\\be\n{B}_3^{\\eta\\eta}\\approx-\\frac{\\eta^2}{4} \\int d\\Gamma \\delta(\\xi_3) \\delta(\\rho_4)\\frac{\\T\\rho_2 (z_\\ga y^\\ga)^2}{(\\rho_1+\\rho_2)(\\rho_2+\\rho_3)}\n \\exp\\big(\\KE \\big)CCC\n\\q\\ee\n where $d\\Gamma$ is defined in \\eq{dGamma},\n\\begin{equation}\n\\KE=i\\T z_\\ga\\left(y^\\ga+\\PP_0^\\ga\\right)\n+\\frac{i(1-\\xi_1) \\rho_2}{\\rho_1+\\rho_2}y^\\ga (p_{1\\ga}+p_{2\\ga})\n+\\frac{i\\xi_1 \\rho_2}{\\rho_2+\\rho_3}y^\\ga (p_{2\\ga}+p_{3\\ga})-iy^\\ga p_{2\\ga}\\q\n\\end{equation}\n \\begin{equation}\n\\PP_0=(1-\\rho_1)(p_1+p_2)-(1-\\rho_3)(p_2+p_3).\n\\end{equation}\nPerforming partial integration with respect to $\\T$ twice we obtain\n\\be\\label{B31}\n{B}_3^{\\eta\\eta}\\approx\\frac{\\eta^2}{4} \\int d\\Gamma\\frac{\\delta(\\xi_3)\\delta(\\rho_4)\\rho_2}{(1-\\rho_3)(1-\\rho_1)}\n \\Big[\\delta(\\T)+iz_\\ga \\PP_0^\\ga+iz_\\ga \\PP_0^\\ga\n\\Big(1+i\\T z_\\ga \\PP_0^\\ga\\Big)\\Big] \\exp\\big( \\KE\\big)CCC\n\\,.\n\\ee Noticing that\n\\begin{equation}\n\\frac{\\p}{\\p \\rho_1 } \\KE\n=-i\\T z_\\ga (p_1 {}^\\ga+p_2 {}^\\ga)-i\\frac{(1-\\xi_1)\\rho_2}{(\\rho_1+\\rho_2)^2}y^\\ga(p_{1\\ga}+p_{2\\ga}),\n\\end{equation}\n\\begin{equation}\n\\frac{\\p}{\\p \\rho_3} \\KE=\\\\\n=i\\T z_\\ga (p_2 {}^\\ga + p_3 {}^\\ga)-i\\frac{\\xi_1 \\rho_2}{(\\rho_2+\\rho_3)^2}\ny^\\ga (p_{2\\ga}+p_{3\\ga})\n\\end{equation}\nand\nperforming partial integration with respect to $\\rho_1$ and $\\rho_3$ we obtain\n\\begin{multline}\n{B}_3^{\\eta\\eta}\\approx\\frac{i\\eta^2}{4}\\int d\\Gamma\n\\frac{\\delta(\\xi_3)\\delta(\\rho_4)}{(1-\\rho_3)(1-\\rho_1)}\n\\Bigg[ {-i\\rho_2\n\\delta(\\T)}\n + \\, z_\\ga \\PP_0^\\ga \\big({(1-\\rho_3)(1-\\rho_1)}\\left(\\delta(\\rho_1)+\\delta(\\rho_3)\\right)\n-1\\big) \\\\\n- { i\\,\\rho_2 z_\\ga\n\\PP_0^\\ga} \\left( \\xi_2 \\frac{y^\\ga(p_{1\\ga}+p_{2\\ga})}{(\\rho_1+\\rho_2)}\n+ \\xi_1 \\frac{y^\\ga(p_{2\\ga}+p_{3\\ga})}{(\\rho_2+\\rho_3)}\\right)\\Bigg]\\exp\\big(\\KE \\big) CCC.\n\\end{multline}\nObserving that\n\\begin{equation}\n\\frac{\\p \\KE}{\\p \\xi_1}=\\frac{i\\rho_2}{\\rho_2+\\rho_3} y^\\ga (p_{2\\ga}+p_{3\\ga})-\\frac{i\\rho_2}{\\rho_1+\\rho_2} y^\\ga (p_{1\\ga}+p_{2\\ga})\n\\end{equation}\nand using the Schouten identity\n\\begin{equation}\nz_\\ga (p_2 {}^\\ga+p_3 {}^\\ga) y^\\beta (p_{1\\beta}+p_{2\\beta})=z_\\ga y^\\ga (p_2 {}^\\beta +p_3 {}^\\beta)(p_{1\\beta}+p_{2\\beta})+z_\\ga (p_1 {}^\\ga+p_2 {}^\\ga) y^\\beta(p_{2\\beta}+p_{3\\beta})\n\\end{equation}\n after partial integration with respect to $\\xi_1$ we obtain\n\\begin{multline}\\label{B3modH=1406}\n{B}_3^{\\eta\\eta}\\approx\\frac{i\\eta^2}{4}\\int d\\Gamma\n\\frac{\\delta(\\xi_3)\\delta(\\rho_4)}{(1-\\rho_3)(1-\\rho_1)}\n\\Bigg[ {-i\\rho_2\n\\delta(\\T)}+ {z_\\ga(p_1 {}^\\ga+p_2 {}^\\ga)(1-\\rho_1)} \\Big(\\delta(\\xi_2)-\\delta(\\xi_1)\\Big)\n\\\\\n+ z_\\ga \\PP_0^\\ga \\Big[(1-\\rho_1 )(1-\\rho_3)\\Big(\\delta(\\rho_1)\n+\\delta(\\rho_3)\\Big)-\\delta(\\xi_2) \\xi_1\\Big]\n+i\\rho_2 z_\\ga y^\\ga (p_{1\\beta}+p_{2\\beta})(p_2 {}^\\beta+p_3 {}^\\beta)\n\\Bigg]\\exp\\big(\\KE \\big) CCC.\n\\end{multline}\nThe $\\delta(\\T)$-proportional term gives rise to $J_1$ \\eq{go B3modHcoh} and $J_2$ \\eq{dx B3modHcoh}.\n\n\n\n\n\n\n\\addtocounter{appendix}{1}\n\\renewcommand{\\theequation}{\\Alph{appendix}.\\arabic{equation}}\n\\addtocounter{section}{1} \\setcounter{equation}{0}\n \\addcontentsline{toc}{section}{\\,\\,\\,\\,\\,\\,\\,Appendix B: Details of uniformization}\n\n\n\n\n\n\\section*{Appendix B: Uniformization Detail }\n\n\\label{Auniform}\nHere some details of the transformation of\nintegrands \\eqref{RRwB3modH+}--\\eqref{FFFFFFFFk=}\\, to the form \\eq{comexp} are presented.\n\nUniformization can be easily achieved for Eqs.~\\eq{RRwB3modH+} and \\eq{RRdxB3modH+} modulo $\\gd(\\gr_1)$-proportional terms.\n Indeed, eliminating $\\gd(\\gr_1)$-proportional term from the \\rhs of \\eq{RRwB3modH+}, adding an integration parameter\n $ \\gr_4 $ and a factor of $\\gd(\\gr_4 )$,\none obtains \\eqref{F1}.\nAnalogously, eliminating $\\gd(\\gr_1)$-proportional term from the \\rhs \\eq{RRdxB3modH+},\nadding an integration parameter\n $ \\gr_4 $,\nswapping $ \\gr_1\\leftrightarrow \\gr_4$ and then adding\n a factor of $\\gd(\\gr_1 )$\none obtains \\eqref{F2}.\n\n\n\n\n\nTo transform integrands of Eqs.~\\eq{W2C3gr1} and \\eq{FFFFFFFFk=}, as well as\n$\\gd(\\gr_1)$-proportional terms of the integrands of Eqs.~\\eq{RRwB3modH+} and \\eq{RRdxB3modH+},\nto the\n form \\eq{comexp}\n GT identity \\eq{GTH+F} is used in Sections { B.1} and { B.2}.\n\n\n \\subsection{ $d_x B_2 {} + W_2 * C $}\n \\label{GTdxB2+}\n\n\n\nNoticing that the exponential of \\eqref{W2C3gr1} coincides with $\\Ee$ at $\\xi_2=0$, while the exponential of \\eqref{FFFFFFFFk=} coincides with $\\Ee$ \\eqref{Ee}\n at $\\xi_1=0$,\n one can easily make sure, that\n only\n the $\\gd(\\gx_2)$-proportional term of\n\\eq{W2C3gr1} and the $\\gd(\\gr_1)$-proportional term of \\eq{FFFFFFFFk=} have the desired\n form \\eq{comexp}.\n\nUsing that $\\Ee$ \\eqref{Ee} does not depend on $\\gx_3$, swapping $\\gx_3 \\leftrightarrow \\gx_1$ in\nthe remaining part of \\eq{FFFFFFFFk=}, then swapping $\\gx_3 \\leftrightarrow \\gx_2$\nin the remaining part of \\eq{W2C3gr1}, one then can apply GT identity \\eqref{GTH+==0} to the sum of the\ntwo obtained\n terms .\n As a result, Eqs.~\\eqref{W2C3gr1}, \\eqref{FFFFFFFFk=}\nyield\n \\bee\\label{D4}&&\n\\dr_x B_2^{\\eta\\, loc}+W_{2\\, \\go CC}^{\\eta\\eta}\\ast C\\approx\\frac{\\eta^2}{4}\\int d\\Gamma\\, \\delta(\\rho_3)\\delta(\\xi_3)\n\\Big[-i\\frac{(z_\\ga t^\\ga)}{\\rho_1+\\rho_4}\n\\delta(\\xi_2)-i(\\underline{z_\\ga y^\\ga}) \\delta(\\rho_1)\\Big] \\mathcal{E} \\go CCC\\qquad\\\\\\nn&&+\n\\frac{\\eta^2}{4}\n\\int d\\Gamma\\, \\delta(\\rho_3)\\Big[i\\delta(\\rho_4)-t^\\gga(p_{1\\gga}+p_{2\\gga})\\Big]\n\\Big\\{\\delta(\\T) \\widetilde{t}^\\ga y_\\ga\n + \\delta(\\xi_3)\n (z_\\ga \\widetilde{t}^\\ga+\\underline{z_\\ga y^\\ga})\\Big\\}\\mathcal{E} \\go CCC\\q\n\\eee\nwhere the terms in the second row of formula \\eq{D4} result from applying $GT$ -identity.\nRewriting the underlined part as the result of differentiation with respect to $\\T$ and\nperforming partial integration one obtains Eq.~\\eqref{F3} plus the cohomology term $J_8$\n \\eqref{Result1}.\n\n\n\n\\subsection{ $(\\dr_x B^{\\eta\\eta}_3{}+\\go* B^{\\eta\\eta}_3)|_{\\gd(\\gr_1)}+W^\\eta_{1\\, \\go C}*B^{\\eta\\, loc}_2$}\n\n \\label{GTB3des}\n\nUniformization of the sum of $\\gd(\\gr_1)-$proportional terms on the \\rhss of \\eq{RRdxB3modH+} and \\eq{RRwB3modH+}\nis done with the help of $GT$ identity \\eqref{GTH+==0} as follows.\nDenoting \\be\n\\widetilde{P}= y+ p_1+p_2+{t} - \\gr_2 (p_3+p_2)\n\\ee\none can see that partial integration in $\\T$ yields\n\\begin{multline}\\label{D6}\n\\dr_x {B}^{\\eta\\eta}_3 \\bigg|_{\\delta(\\rho_1)}\\approx-\\frac{i\\eta^2}{4}\\int d\\Gamma\\,\n\\delta(\\rho_4)\\delta(\\rho_1)\\delta(\\xi_1) \\Big[i\\delta(\\T)-z_\\ga y^\\ga\\Big]\n\\exp\\Big\\{i\\T z_\\ga \\widetilde{P}^\\ga-i\\xi_2 \\widetilde{P}^\\ga y_\\ga \\\\\n+i(1-\\rho_2)(p_2 {}^\\ga +p_3 {}^\\ga)y_\\ga+ip_{3\\ga} y^\\ga+it^\\beta p_{1\\beta} \\Big\\}\\go CCC,\n\\end{multline}\n\\begin{multline}\\label{D7}\n\\go\\ast {B}^{\\eta\\eta}_3\\bigg|_{\\delta(\\rho_1)}\\approx \\frac{i\\eta^2}{4}\\int d\\Gamma\\, \\delta(\\rho_4)\\delta(\\rho_1)\\delta(\\xi_3)\\Big[i\\delta(\\T)-z_\\ga(y^\\ga+t^\\ga)\\Big]\\exp\\Big\\{i\\T z_\\ga \\widetilde{P}^\\ga-i\\xi_2 \\widetilde{P}^\\ga y_\\ga \\\\\n+i\\xi_1 \\widetilde{P}^\\ga t_\\ga+i(1-\\rho_2)(p_2 {}^\\ga +p_3 {}^\\ga)y_\\ga+ip_{3\\ga} y^\\ga+it^\\beta p_{1\\beta} \\Big\\}\\go CCC\\,.\n\\end{multline}\n\n\n\n\n\n\n\n\n\nThe sum of \\eqref{D6} and \\eqref{D7} gives\n\\begin{multline}\n\\Big(\\dr_x {B}^{\\eta\\eta}_3+\\omega \\ast B_3^{\\eta\\eta}\\Big)\\bigg|_{\\gd(\\rho_1)} \\approx\n \\frac{i\\eta^2}{4}\\int d\\Gamma\\,\\gd(\\rho_4)\\gd(\\rho_1)\\Big[z_\\gga(-t^\\gga-y^\\ga)\\gd(\\xi_3)+z_\\gga y^\\gga\\gd(\\xi_1)+z_\\gga t^\\gga\\gd(\\xi_2)\\Big]\\times\\\\\n\\times \\exp\\Big\\{i\\T z_\\ga \\widetilde{P}^\\ga-i\\xi_2 \\widetilde{P}^\\ga y_\\ga\n+i\\xi_1 \\widetilde{P}^\\ga t_\\ga+i(1-\\rho_2)(p_2 {}^\\ga +p_3 {}^\\ga)y_\\ga+ip_{3\\ga} y^\\ga+it^\\gb p_{1\\gb} \\Big\\} \\go CCC \\\\\n-\\frac{i\\eta^2}{4}\\int d\\Gamma\\, \\gd(\\rho_4)\\gd(\\rho_1)\n(z_\\gga t^\\gga)\\gd(\\xi_2)\\exp\\Big\\{i\\T z_\\ga \\widetilde{P}^\\ga-i\\xi_2 \\widetilde{P}^\\ga y_\\ga\n+i\\xi_1 \\widetilde{P}^\\ga t_\\ga+i(1-\\rho_2)(p_2 {}^\\ga +p_3 {}^\\ga)y_\\ga \\\\\n+ip_{3\\ga} y^\\ga+it^\\gb p_{1\\gb}\\Big\\}\\go CCC \\,+ J_9+J_{10}\n \\label{ERRGT}\\end{multline}\n with $J_9$ \\eq{goB3modH1406gr1C} and $J_{10}$ \\eq{dxB3modH1406gr1C}.\nBy virtue of GT identity \\eqref{GTH+==0} the first term weakly equals $J_{11}$ \\eq{ERRGTC}.\n Finally, Eq.~\\eq{ERRGT} yields\n\\begin{multline} \\label{dB3+wB3}\n\\Big(\\dr_x {B}^{\\eta\\eta}_3+\\omega \\ast B_3^{\\eta\\eta}\\Big)\\bigg|_{\\gd(\\rho_1)} \\approx\n-\\frac{i\\eta^2}{4}\\int d\\Gamma\\, \\gd(\\rho_4)\\gd(\\rho_1) (z_\\gga t^\\gga)\\gd(\\xi_2)\\exp\\Big\\{i\\T z_\\ga \\widetilde{P}^\\ga-i\\xi_2 \\widetilde{P}^\\ga y_\\ga\n \\\\\n+i\\xi_1 \\widetilde{P}^\\ga t_\\ga+i(1-\\rho_2)(p_2 {}^\\ga +p_3 {}^\\ga)y_\\ga+ip_{3\\ga} y^\\ga+it^\\gb p_{1\\gb}\\Big\\}\n\\go CCC\\, + J_9+J_{10}+J_{11}.\n\\end{multline}\nConsider $W_{1 \\go C}^\\eta \\ast B_2^{\\eta\\, loc}$ \\eq{origW1B2}.\nThis is convenient to change integration variables,\nmoving from the integration over simplex to integration over square. As a result\n \\begin{multline}\\label{W1B2mod1}\nW_{1 \\go C}^\\eta \\ast B_2^{\\eta\\, loc}\\approx \\frac{\\eta^2}{4}\\int_0^1 d\\T\\, \\T \\int d^2 \\tau_+\\, \\gd(1-\\tau_1-\\tau_2)\\int_0^1 d\\sigma_1 \\int_0^1 d\\sigma_2\\, (z_\\ga t^\\ga)\\times \\\\\n\\Big[z_\\ga y^\\ga+\\sigma_1 z_\\ga t^\\ga\\Big]\\exp\\Big\\{i\\T z_\\ga y^\\ga+i(1-\\sigma_2)\\sigma_1 t_\\ga p_1 {}^\\ga+i\\sigma_1\\sigma_2 t^\\ga p_{3\\ga}+i(1-\\sigma_1)t^\\ga p_{1\\ga} \\\\\n+i\\T z_\\ga \\Big((\\tau_1+\\tau_2 \\sigma_1)t^\\ga+\\tau_1 p_1 {}^\\ga-(\\tau_2-\\tau_1(1-\\sigma_2))p_2 {}^\\ga-(\\tau_2+\\sigma_2\\tau_1)p_3 {}^\\ga\\Big)+i\\sigma_1 y^\\ga t_\\ga \\\\\n-i(1-\\sigma_2)y^\\ga p_{2\\ga}+i\\sigma_2 y^\\ga p_{3\\ga}+i\\sigma_2 y^\\ga p_{3\\ga} \\Big\\}\\go CCC.\n\\end{multline} Partial integration with respect to $\\T$\nyields\\begin{multline}\nW_{1 \\go C}^\\eta \\ast B_2^{\\eta\\, loc}\\approx-\\frac{\\eta^2}{4}\\int_0^1 d\\T \\int d^2 \\tau_+\\,\n\\gd(1-\\tau_1-\\tau_2)\\int_0^1 d\\sigma_1 \\int_0^1 d\\sigma_2\\, (z_\\ga t^\\ga)\\times \\\\\n\\Big[\\T z_\\ga \\Big(\\tau_1(p_1 {}^\\ga+p_2 {}^\\ga)-(\\tau_2+\\sigma_2 \\tau_1)(p_2 {}^\\ga +p_3 {}^\\ga)\\Big)\n-i\\T \\tau_1 (1-\\sigma_1) z_\\ga t^\\ga\\Big]\\,\\exp(\\KEE)\\,\\,\\go CCC\\q\n\\end{multline}where\n\\begin{multline}\\label{tildeEe}\n\\KEE=i\\T z_\\ga y^\\ga+it^\\gb p_{1\\gb}+i\\sigma_1\\Big(y^\\ga t_\\ga+(p_1 {}^\\ga+p_2 {}^\\ga)t_\\ga\n-\\sigma_2(p_2 {}^\\ga+p_3 {}^\\ga)t_\\ga\\Big)-i\\big(\\sigma_2 p_3 {}^\\ga-(1-\\sigma_2)p_2 {}^\\ga\\big)y_\\ga \\\\\n+i\\T z_\\ga \\Big(\\tau_1(p_1 {}^\\ga +p_2 {}^\\ga)-(\\tau_2+\\sigma_2\\tau_1)(p_2 {}^\\ga+p_3 {}^\\ga)\n+(\\sigma_1+\\tau_1(1-\\sigma_1))t^\\ga\\Big).\n\\end{multline}\nBy virtue of evident formulas\n\\bee\\nn&&\n\\tau_1 \\left(\\frac{\\p}{\\p \\tau_1}-\\frac{\\p}{\\p \\tau_2}\\right)\n\\KEE=i\\T z_\\ga \\Big(\\tau_1(p_1 +p_2 {} )+\\big[(\\tau_1+\\tau_2)-(\\tau_2+\\sigma_2\\tau_1)\\big]\n(p_2 {} +p_3 {} )+\\tau_1(1-\\sigma_1)t \\Big){}^\\ga \\q\n\\\\ \\nn&&\\frac{\\p}{\\p \\sigma_1}\\KEE=\ni\\T (1-\\tau_1)z_\\ga t^\\ga+i\\Big(y^\\ga+p_1 {}^\\ga+p_2 {}^\\ga-\\sigma_2 (p_2 {}^\\ga+p_3 {}^\\ga)\\Big)t_\\ga,\n\\eee\n Eq.~\\eqref{W1B2mod1} acquires the form\n\\begin{multline}\nW_{1 \\go C}^\\eta \\ast B_2^{\\eta\\, loc}\\approx\\frac{\\eta^2}{4}\\int_0^1 d\\T\\int d^2\n\\tau_+\\gd(1-\\tau_1-\\tau_2)\\int_0^1 d\\sigma_1 \\int_0^1 d\\sigma_2\\bigg[iz_\\ga t^\\ga \\tau_1\n\\left(\\frac{\\p}{\\p \\tau_1}-\\frac{\\p}{\\p \\tau_2}\\right) \\\\\n-\\frac{z_\\ga (p_2 {}^\\ga+ p_3 {}^\\ga)}{1-\\tau_1}\\left(i\\frac{\\p}{\\p \\sigma_1}\n+\\Big(y^\\ga+p_1 {}^\\ga+p_2 {}^\\ga-\\sigma_2(p_2 {}^\\ga+p_3 {}^\\ga)\\Big)t_\\ga\\right)+iz_\\ga t^\\ga\\bigg]\n\\exp(\\KEE )\\go CCC.\n\\end{multline}\nAfter partial integrations in $\\tau_1$,$\\tau_2$ and $\\sigma_1$ one obtains\n\\bee&&\\label{underlC10}\nW_{1 \\go C}^\\eta \\ast B_2^{\\eta\\, loc}\\approx\\frac{\\eta^2}{4}\\int_0^1\nd\\T\\int d^2 \\tau_+\\gd(1-\\tau_1-\\tau_2)\\int_0^1 d\\sigma_1 \\int_0^1 d\\sigma_2\n\\bigg[\\underline{iz_\\ga t^\\ga \\gd(\\tau_2)} \\\\\n&&\\nn+\\frac{z_\\ga (p_2 {}^\\ga+ p_3 {}^\\ga)}{1-\\tau_1}\\left(i\\big(\\gd(\\sigma_1)-\\gd(1-\\sigma_1)\\big)\n-\\Big(y^\\ga+p_1 {}^\\ga+p_2 {}^\\ga-\\sigma_2(p_2 {}^\\ga+p_3 {}^\\ga)\\Big)t_\\ga\\right)\\bigg]\n\\exp (\\KEE )\\go CCC\\,.\n\\eee\nAfter a simple change of integration variables the underlined term on the \\rhs of Eq.~\\eq{underlC10}\n cancels the \\rhs of Eq.~\\eqref{dB3+wB3}. Performing integration with respect to $\\gt_2$\n in the remaining part of \\eq{underlC10},\nafter the following change of the integration variables\n\\bee\\nn&&\n\\int_0^1 d\\sigma_1\\int_0^1 d\\gt_1 \\int_0^1 d\\sigma_2\\, f(\\sigma_1,1-\\sigma_1,\\gt_1,\\sigma_2)\\\\\\nn&&\n=\\int d^4 \\rho_+\\, \\delta\\left(1-\\sum_{j=1}^4 \\rho_j\\right)\\frac{1}{(\\gr_2+\\gr_3)(1-\\gr_2-\\gr_3)}\nf\\left(\\frac{\\gr_1}{1-\\gr_2-\\gr_3},\\frac{\\rho_4}{1-\\gr_2-\\gr_3},\\gr_2+\\gr_3,\\frac{\\gr_2}{\\gr_2+\\gr_3}\\right)\n\\,, \\eee\n $\\exp(\\KEE)$ \\eq{tildeEe} acquires the form $\\Ee$ \\eq{Ee}. As a result, the sum of Eq.~\\eq{underlC10} and Eq.~\\eqref{dB3+wB3} by virtue Eq.~\\eqref{EEgx14=}\nyields Eq.~\\eqref{F4}.\n\n\n\n\\addtocounter{appendix}{1}\n\\renewcommand{\\theequation}{\\Alph{appendix}.\\arabic{equation}}\n\\addtocounter{section}{1} \\setcounter{equation}{0}\n\\setcounter{subsection}{0}\n \\addcontentsline{toc}{section}{\\,\\,\\,\\,\\,\\,\\,Appendix C: Eliminating $\\gd(\\gr_j)$ and $\\gd(\\gx_j)$}\n \\renewcommand{\\thesubsection}{\\Alph{appendix}.\\arabic{subsection}}\n\n\\section*{Appendix C: Eliminating $\\gd(\\gr_j)$ and $\\gd(\\gx_j)$}\n\\label{AppD}\nTo eliminate $\\gd(\\gr_j)$ and $\\gd(\\gx_j)$ from of the \\rhss of Eqs.~\\eqref{F1}, \\eqref{F2}\nthis is convenient to group similar pre-exponential terms as in Sections \\ref{Eli1} -\\ref{Eli4}.\n \\subsection{Terms proportional to $( p_1{}+ p_2)^{\\ga} ( p_3{}+p_2)_{\\ga}$}\n\\label{Eli1}\n\n Consider\n $F_{1,1}+F_{2,1}$ of \\eqref{F1} and \\eqref{F2}, respectively.\n Partial integration with respect to $\\rho_1$ and $\\rho_4$ yields\n \n\\begin{multline}\\label{F11+F21b}\nF_{1,1}+F_{2,1}\\approx - \\frac{\\eta^2}{4}\\int d\\Gamma\\frac{\\delta(\\xi_3)\\rho_2}{(1-\\rho_1-\\rho_4)(1-\\rho_3)}(p_1 {}^\\ga+p_2 {}^\\ga)(p_{2\\ga}+p_{3\\ga}) \\times \\\\\n\\times (z_\\gga \\PP^\\gga)\\left(\\frac{\\p}{\\p \\rho_4}\n-\\frac{\\p}{\\p \\rho_1}\\right)\\Ee \\go CCC.\n\\end{multline}\nBy direct calculation, Eq.~\\eqref{F11+F21b} gives\n\\begin{multline}\nF_{1,1}+F_{2,1}\\approx -\\frac{\\eta^2}{4}\\int d\\Gamma\\frac{\\delta(\\xi_3)\\rho_2}{(1-\\rho_1-\\rho_4)(1-\\rho_3)}(p_1 {}^\\ga+p_2 {}^\\ga)(p_{2\\ga}+p_{3\\ga})\\times\\\\\n\\Bigg[\\Ez \\left(\\frac{\\p}{\\p \\rho_4}-\n\\frac{\\p}{\\p \\rho_1}\\right) (z_\\gga \\PP^\\gga) E+(z_\\gga \\PP^\\gga)\n\\T (z_\\ga t^\\ga)\\mathcal{E} \\Bigg]\\go CCC\\,.\n\\end{multline}\nBy virtue of the Schouten identity\n\\begin{equation}\n z_\\ga t^\\ga (p_1+p_2 )^\\gga ( p_3{} +p_2{} )_\\gga=\n t^\\ga(p_1+p_2 ){} _\\ga z^\\gga ( p_3{} +p_2{} )_\\gga+ t^\\ga ( p_3{} +p_2{} )_\\ga (p_1+p_2 )^\\gga z _\\gga\n\\end{equation}\nand its consequence\n\\begin{multline}\\label{SchCons}\n z_\\ga t^\\ga (p_1+p_2 )^\\gga ( p_3{} +p_2{} )_\\gga \\E\n=t^\\ga(p_{1 }+p_{2 })_\\ga\\left[i\\left(\\frac{\\overleftarrow{\\p}}{\\p \\rho_2}\n-\\frac{\\overleftarrow{\\p}}{\\p \\rho_3}\\right)\\Ez E+i\\Ez\n\\left(\\frac{\\p}{\\p \\rho_2}-\\frac{\\p}{\\p \\rho_3}\\right)E\\right] \\\\\n+t^\\ga (p_{2 }+p_{3 })_\\ga\\left[i\\left(\\frac{\\overleftarrow{\\p} }\n{\\p \\rho_2}-\\frac{\\overleftarrow{\\p}}{\\p \\rho_1}\\right)\\Ez E\n+i\\Ez \\left(\\frac{\\p}{\\p \\rho_2}-\\frac{\\p}{\\p \\rho_1}\\right) E\\right]\\,\n\\end{multline}\nEq.~\\eqref{F11+F21b} yields\n\\begin{multline}\\label{F11+F21}\nF_{1,1}+F_{2,1}\\approx + \\frac{\\eta^2}{4}\\int d\\Gamma\\, \\delta(\\xi_3) \\Bigg\\{\\ff{(z_{\\gga}\\PP ^{\\gga})\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3) }\\times\\\\\n\\times\\Bigg(( p_1{}+ p_2)^{\\ga} ( p_3{}+p_2)_{\\ga}\n \\Ez \\Bigg[\\frac{\\p}{\\p \\rho_1}-\\frac{\\p}{\\p \\rho_4}\\Bigg] E\n +t^\\ga (p_1+p_2 ){} _\\ga \\Bigg[ \\underline{ \\gd(\\gr_3)} \\Ee\n - \\Ez\\Bigg(\\frac{\\p}{\\p \\rho_2}-\\frac{\\p}{\\p \\rho_3}\\Bigg)E \\Bigg] \\\\\n+t^\\ga ( p_3{} +p_2{} )_\\ga \\Bigg[ \\underline{ \\gd(\\gr_1)}\\Ee\n - \\Ez \\Bigg(\\frac{\\p}{\\p \\rho_2}-\\frac{\\p}{\\p \\rho_1}\\Bigg)E \\Bigg]\n\\Bigg)+\\ff{\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3) }\n\\Big( t^\\ga z _\\ga ( p_3+p_2)^{\\gga}(p_1+p_2 ){}_{\\gga} \\Ee\n \\Big)\\\\\n+(z_{\\gga}\\PP ^{\\gga})\\Bigg(- \\ff{1-\\gr_3-\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3)^2 } t^\\ga (p_1+p_2 ){} _\\ga\n \\Ee- \\ff{1- \\gr_1 -\\gr_4-\\gr_2}{(1- \\gr_1 -\\gr_4)^2(1-\\gr_3) }t^\\ga ( p_3{} +p_2{} )_\\ga\n \\Ee \\Bigg)\\Bigg\\}\\go CCC\\,.\n\\end{multline}\nOne can see that $\\delta(\\rho_1) $- and\n$\\delta(\\rho_3) $-proportional terms on the \\rhs of \\eq{F11+F21} (the underlined ones)\ncancel terms $F_{2,4}$ \\eqref{F2} and $F_{3,3}$ \\eqref{F3}, respectively.\n\n\n\n\n\n\n\\subsection{Term proportional to $t^\\ga(p_{1\\ga}+p_{2\\ga})$}\nConsider term $F_{3,5}$ of $F_{3 }$ \\eqref{F3}. By virtue of the following identity\n\\begin{equation}\n\\frac{\\rho_2}{(\\rho_2+\\rho_3)(1-\\rho_3)}\\left(\\delta(\\rho_3)-\\delta(\\rho_2)\\right)=1\n\\end{equation}\n\\begin{multline}\nF_{3,5} \\approx -\\frac{\\eta^2}{4}\\int d\\Gamma\\, \\frac{\\delta(\\xi_3)\\rho_2}{(\\rho_2+\\rho_3)(1-\\rho_3)}\\Big(\\delta(\\rho_3)-\\delta(\\rho_2)\\Big)\n\\\\ \\Big[ ( p_2{}_\\ga+ p_1{}_\\ga) t^{\\ga}\n (z_{\\gga}t^\\gga)\\Big( (1-\\gr_4) -\\ff{\\gr_1}{(\\gr_1+\\gr_4)} \\Big)\\Ee \\Big]\\go CCC.\n\\end{multline}\nPartial integrations along with the Schouten identity\n\\begin{equation}\nt^\\ga(p_{1\\ga}+p_{2\\ga} ) ( p_3{}^\\gga +p_2{}^\\gga ) z_\\gga\n= - \\underline{t^\\ga z _\\ga} (p_1{}^\\gga+p_2{}^\\gga ) ( p_{2\\gga}+p_{3\\gga})\n+ t^\\ga ( p_{3\\ga} +p_{2\\ga} ) \\underline{ (p_1+p_2 )^\\gga z _\\gga}\n\\end{equation}\nand realization of the underlined terms as derivative of $\\Ez$\n along with further partial integration\nyields\n\\begin{multline}\\label{F35=}\nF_{3,5}\\approx -\\frac{\\eta^2}{4}\\int d\\Gamma\\, \\delta(\\xi_3)\\Bigg[\\ff{ \\gr_4}{ (1-\\gr_3)^2} \\Big( ( p_2{}_\\ga+ p_1{}_\\ga) t^{\\ga} z_{\\gga}t^\\gga \\Big)\\Ee \\\\\n+\\ff{\\gr_2\\gr_4}{(\\gr_1+\\gr_4)(1-\\gr_3)} \\Big( ( p_2{}_\\ga+ p_1{}_\\ga) t^{\\ga}z_{\\gga}t^\\gga \\Big)\\Ez\\left[\\frac{\\p}{\\p \\rho_2}-\\frac{\\p}{\\p \\rho_3}\\right]E+ \\ff{\\gr_2\\gr_4}{(\\gr_1+\\gr_4)(1-\\gr_3)} (z_{\\ga}t^\\ga) \\times\\\\\n\\times\\Bigg( - (p_1+p_2 )^\\gga ( p_3{} +p_2{} )_\\gga \\Ez \\Bigg[\\frac{\\p}{\\p \\rho_1}-\\frac{\\p}{\\p \\rho_4}\\Bigg]E\n- \\underline{\\gd(\\gr_1)} (p_1+p_2 )^\\gga ( p_3{} +p_2{} )_\\gga\\Ee\\\\\n - t^\\ga( ( p_3{} +p_2{} )_\\ga )\\Ez \\Bigg[\\frac{\\p}{\\p \\rho_1}-\\frac{\\p}{\\p \\rho_2}\\Bigg]E\n- \\underline{\\gd(\\gr_1)}t^\\ga( ( p_3{} +p_2{} )_\\ga )\\Ee\\Bigg)\\\\\n+(z_{\\ga}t^\\ga)\\Bigg(\n\\ff{ \\gr_2}{ (1-\\gr_3) (\\gr_1+\\gr_4)} (p_1+p_2 )^\\gga ( p_3{} +p_2{} )_\\gga\\Ee\n + \\ff{\\gr_4 }{(\\gr_1+\\gr_4)^2} t^\\ga( ( p_3{} +p_2{} )_\\ga )\\Ee\n\\Bigg) \\Bigg]\\go CCC.\n\\end{multline}\nOne can see that the sum of the underlined $\\delta(\\rho_1)$-proportional terms cancel $F_{2,2}+F_{2,3}$ of \\eqref{F2}.\n\n\n\n\n\n\n\n\\subsection{Sum of $( p_1{}+ p_2)^{\\ga} ( p_3{}+p_2)_{\\ga}$-proportional and\n $t^\\ga(p_{1\\ga}+p_{2\\ga})$--proportional terms}\n\nSumming up $F_{1,1}+F_{2,1} $ \\eqref{F11+F21},\n $F_{3,3}$ \\eqref{F3},\n$F_{3,5}$ \\eqref{F35=} and $F_{2,2}+F_{2,3}+F_{2,4}$ \\eqref{F2}, then performing partial integrations\nand using the following simple identities\n\\begin{equation}\n(1-\\gr_4) -\\ff{\\gr_1}{(\\gr_1+\\gr_4)}=\n \\ff{\\gr_4( \\gr_2+\\gr_3)}{(\\gr_1+\\gr_4)},\n\\end{equation}\n\\begin{equation}\n- \\ff{\\gr_4 }{(\\gr_1+\\gr_4)^2} + \\ff{\\gr_4}{( \\gr_1+\\gr_4)}\n\\ff{ \\gr_3}{(1- \\gr_1 -\\gr_4) (1-\\gr_3) }\n= \\ff{-\\gr_2\\gr_4 }{(\\gr_1+\\gr_4)^2(1- \\gr_1 -\\gr_4) (1-\\gr_3)}\\q\n\\end{equation}\none obtains by virtue of Eqs.~\\eq{tildet}-\\eq{PP=} \n\\be \\label{FRest1=}\n F_{1,1}+F_{2,1}+F_{2,4}+F_{3,3}+F_{3,5}+F_{2,2}+F_{2,3}=G_1 \\ee\n with $G_1$ \\eq{FRest1}.\n\n\n \\subsection{Terms proportional to $\\delta(\\xi_1)-\\delta(\\xi_2)$}\n\\label{Eli3}\n\n\n\nConsider a sum of\n$F_{1,4}$ \\eqref{F1} and $F_{2,8}$ \\eqref{F2}.\nPerforming partial integrations with respect to $\\rho_1$ and $\\rho_4$, then applying the Schouten identity\n one obtains\n\\begin{multline}\\label{F14+F28}\n F_{1,4}+F_{2,8}\\approx -\\frac{\\eta^2}{4}\\int d\\Gamma \\, \\delta(\\xi_3)\n \\Bigg[\\frac{\\p}{\\p \\rho_1}-\\frac{\\p}{\\p \\rho_4}\\Bigg]\\frac{i z_\\ga (p_1 {}^\\ga+p_2 {}^\\ga)}{1-\\rho_3}\n \\Big(\\delta(\\xi_2)-\\delta(\\xi_1)\\Big)\\Ee \\,\\go CCC=\\\\\n=-\\frac{\\eta^2}{4}\\int d\\Gamma \\,\n\\delta(\\xi_3)\\Big(\\delta(\\xi_2)-\\delta(\\xi_1)\\Big)\\Bigg\\{\\frac{i\\, z_\\gga t^\\gga}{(1-\\rho_3)}\n\\Bigg(\\Ez\\Bigg[\\frac{\\p}{\\p \\rho_1}-\\frac{\\p}{\\p \\rho_2}\\Bigg]E+\\Big(\\underline{\\delta(\\rho_1)}\n-\\underline{\\underline{\\delta(\\rho_2)}}\\Big)\\Ee\\Bigg) \\\\\n+\\frac{i\\, z_\\ga (p_1 {}^\\ga+p_2 {}^\\ga)}{(1-\\rho_3)}\\Ez\n\\Bigg[\\frac{\\p}{\\p \\rho_1}-\\frac{\\p}{\\p \\rho_4}\\Bigg]E\\Bigg\\}\\go CCC.\n\\end{multline}\n\nThe underlined $ \\gd(\\gr_1)$-proportional term compensates $F_{2,9}$ of \\eqref{F2}.\nThe double underlined $\\gd(\\gr_2)$-proportional term vanishes due to the factor of $(\\gd(\\gx_2)-\\gd(\\gx_1))$\nwhich after partial integrations in $\\xi_1$ and $\\xi_2$ produces an expression\nproportional to $\\rho_2$.\n\n\nSumming up $F_{1,4}+F_{2,8}$ \\eq{F14+F28} and $F_{2,9}$ \\eqref{F2},\nperforming partial integrations with respect to $\\gx$ and $\\T$ along with the Schouten identity\none obtains \\be \\label{FRest2G}\n F_{1,4}+F_{2,8}+F_{2,9}\\approx G_2\n\\ee\nwith $G_2$ \\eq{FRest2}.\n\n\n \\subsection{Terms proportional to $\\xi_1 \\delta(\\xi_2)$ }\n\\label{Eli4}\n\nConsider a sum of $F_{1,3}$ \\eqref{F1}, $F_{2,6}$ \\eqref{F2} and $F_{4,1}$ \\eqref{F4}.\n\\bee\nF_{1,3}+F_{2,6}+F_{4,1}\\approx \\frac{i\\eta^2}{4}\n\\int d\\Gamma\\,\\ff{ \\delta(\\xi_3)\\delta(\\xi_2)[\\delta(\\rho_1)-\\delta(\\rho_4)]}{(\\rho_2+\\rho_3)} z_\\ga\n\\bigg\\{\\frac{\\PP^\\ga\n}{(1-\\rho_3)}\n- \\frac{\\xi_1 \\,(p_2 {}^\\ga+p_3 {}^\\ga)}{\n(\\rho_1+\\rho_4)}\n\\bigg\\}\\Ee\\, \\go CCC.\\quad\n\\eee\nPartial integration yields\n\\begin{multline}\\label{FRest2_5-}\nF_{1,3}+F_{2,6}+F_{4,1}\\approx\\frac{i\\eta^2}{4}\\int d\\Gamma\\,\n\\gd(\\xi_3)\\gd(\\xi_2)\\xi_1 \\Bigg\\{z_\\ga t^\\ga\\Bigg[\\frac{1}{\\rho_1+\\rho_4}\n\\bigg(\\Ez\\bigg[\\frac{\\p}{\\p \\rho_2}-\\frac{\\p}{\\p \\rho_3}\\bigg]E+\\Big[\\underline{\\gd(\\rho_2)}\n-\\gd(\\rho_3)\\Big]\\Ee\\bigg)\n \\\\\n+\\frac{1}{1-\\rho_3}\\bigg(\\Ez \\bigg[\\frac{\\p}{\\p \\rho_1}-\\frac{\\p}{\\p \\rho_2}\\bigg]E\n+\\Big[\\gd(\\rho_1)-\\underline{\\gd(\\rho_2)}\\Big]\\Ee\\bigg)\\Bigg] \\\\\n+\\bigg[\\frac{z_\\ga(p_2 {}^\\ga+p_3 {}^\\ga)}{\\rho_1+\\rho_4}\n+\\frac{z_\\ga(p_1 {}^\\ga+p_2 {}^\\ga)}{1-\\rho_3}\\bigg]\\Ez\n\\bigg[\\frac{\\p}{\\p \\rho_1}-\\frac{\\p}{\\p \\rho_4}\\bigg]E\\Bigg\\}\\go CCC\n\\,.\\end{multline}\nOne can see that the underlined $\\gd({\\gr_2})$-proportional terms vanish\ndue to the factor of $\\gd(1-\\sum\\gr_i)$ \\eq{dGamma}, while $\\gd({\\gr_1})$-proportional term compensates\n $F_{2,7}$\n\\eqref{F2} and $\\gd({\\gr_3})$-proportional term\ncompensates $F_{3,2}$ \\eqref{F3}.\n\nSumming up $F_{2,7}$ \\eqref{F2}, $F_{3,2}$ \\eqref{F2}, $F_{4,2}$ and\n$F_{1,3}+F_{2,6}+F_{4,1}$ \\eqref{F4},\nand then\nperforming partial integration in $\\T$ one obtains by virtue of the Schouten\nidentity\n\\begin{multline}\\label{G3=}\nF_{1,3}+F_{2,6}+F_{4,1}+F_{2,7}+F_{3,2}+F_{4,2}\\approx G_3:=\\frac{\\eta^2}{4}\\int d\\Gamma\\, \\delta(\\xi_3)\\delta(\\xi_2)\\times\\\\\n\\times\\Bigg\\{\\frac{\\rho_2\\, (z_\\ga t^\\ga)(p_2 {}^\\gga+p_3 {}^\\gga)(y_\\gga\n+\\tilde{t}_\\gga)}{(1-\\rho_1-\\rho_4)^2 (1-\\rho_3)(\\rho_1+\\rho_4)}\n+ \\frac{\\rho_2\\, \\Big[(\\tilde{t}^\\gga+y^\\gga)(y_\\gga+\\Pz_\\gga) (z^\\ga t_\\ga)+i\\delta(\\T)\nt_\\gga(\\tilde{t}^\\gga-\\Pz^\\gga)\\Big]}{(1-\\rho_1-\\rho_4)^2 (1-\\rho_3)^2 (\\rho_1+\\rho_4)} \\\\\n+\\frac{\\rho_3\\, \\big[i\\delta(\\T)-z_\\gga (y^\\gga+\\tilde{t}^\\gga)\\big]\n(t^\\ga y_\\ga)}{(1-\\rho_1-\\rho_4)^2 (1-\\rho_3)^2}\n+\\frac{\\big[-i\\delta(\\T)+z_\\gga (y^\\gga+\\tilde{t}^\\gga)\\big]\n(p_1 {}^\\ga+p_2 {}^\\ga)t_\\ga}{(1-\\rho_1-\\rho_4)(1-\\rho_3)^2} \\Bigg\\}\\Ee\\go CCC\\,.\n\\end{multline}\nSince by the partial integration procedure $\n \\gx_1\\gd(\\gx_2)\\equiv {1}+ \\gx_1(\\p_{\\gx_1}-\\p_{\\gx_2})$,\n \\eq{G3=} yields $G_{ 3}$ \\eq{FRest3}.\n\n\n\n\n\\addtocounter{appendix}{1}\n\\renewcommand{\\theequation}{\\Alph{appendix}.\\arabic{equation}}\n\\addtocounter{section}{1}\n\\setcounter{equation}{0}\n \\addcontentsline{toc}{section}{\\,\\,\\,\\,\\,\\,\\,Appendix D: Details of the final step of the calculation}\n \\renewcommand{\\thesubsection}{\\Alph{appendix}.\\arabic{subsection}}\n\\setcounter{subsection}{0}\n \\renewcommand{\\thesection}{\\Alph{appendix}}\n\n\\section*{Appendix D: Details of the final step of the calculation}\n\\label{AppE}\n\nBy virtue of Eqs.~\\eq{EEgx14=}-\\eq{Egx=21e}, Eq.~\\eq{SUM3} yields \\bee&&\n\n\\label{SUM=} S_6\n = + i \\ff{ \\eta^2 }{4 }\\int d\\Gamma \\, \\delta(\\xi_3) \\\\ \\nn&&\n \\Big\\{\n \n+(y+ \\tilde{t}{} )^{\\gga}z_{\\gga}\\ff{\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3) }{t}{}^\\ga (p_1+p_2 ){} _\\ga\n\\gx_1 \\ff{1-\\gr_3-\\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3 )^2}\\,\\, \\Pz{}^\\ga y_{\\ga}\n\\\\ \\nn&&\n+(y+ \\tilde{t}{} )^{\\gga}z_{\\gga}\\ff{\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3) }{t}{}^\\ga (p_1+p_2 ){} _\\ga\n \\gx_1 \\ff{1-\\gr_3-\\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3 )^2}\\big(y + \\Pz{} \\big)^\\ga\\tilde{t}{}_{\\ga}\n\\\\ \\nn&&\n+(y+ \\tilde{t}{} )^{\\gga}z_{\\gga}\\ff{\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3) }{t}{}^\\ga (p_1+p_2 ){} _\\ga\n (-) \\gx_1 \\ff{\\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3 )}\\,(p_3{}^{\\ga}+p_2{}^{\\ga}) y_{\\ga}\n\\\\ \\nn&&\n-(y+ \\tilde{t}{} )^{\\gga}z_{\\gga}\\ff{\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3) }{t}{}^\\ga (p_1+p_2 ){} _\\ga\n \\gx_1 \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)}(p_3{} +p_2{})^{\\gb}\\tilde{t}{}_{\\gb}\n+ \n\\eee\n\\bee \\nn&&\n+(y+ \\tilde{t}{} )^{\\gga}z_{\\gga}\\ff{\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3) }{t}{}^\\ga (p_1+p_2 ){} _\\ga\n(-) \\ff{ \\gr_1+\\gr_4}{ (1-\\gr_3 )^2}\\,\\,( (p{}_1 +p_2) )^\\ga y_{\\ga}\n \\\\ \\nn&&\n+(y+ \\tilde{t}{} )^{\\gga}z_{\\gga}\\ff{\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3) }{t}{}^\\ga (p_1+p_2 ){} _\\ga\n (-) \\ff{ \\gr_4 }{ (1-\\gr_3 )^2}\\,\\,{t}{} ^\\ga y_{\\ga}\n\\\\ \\nn&&\n+(y+ \\tilde{t}{} )^{\\gga}z_{\\gga}\\ff{\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3) }{t}{}^\\ga (p_1+p_2 ){} _\\ga\n (-)\\ff{ \\gr_1 }{(1-\\gr_3)^2} (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}\\\\ \\nn&&\n \\\\ \\nn&&\n \n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n ( p_1{}+ p_2)^{\\gga} \\big(y + (1 -\\gr_4 ){t}{}\\big)_{\\gga} z_\\ga(y+\\tilde{t}{})^{\\ga}\n (-) \\gx_1 \\ff{\\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3 )}\\,\\, {t}{}^\\ga y_{\\ga}\n \\\\ \\nn&&\n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n ( p_1{}+ p_2)^{\\gga} \\big(y + (1 -\\gr_4 ){t}{}\\big)_{\\gga} z_\\ga(y+\\tilde{t}{})^{\\ga}\n\\eee\\bee \\nn&&\\times (-) \\gx_1 \\ff{ \\gr_2 }{(1-\\gr_1-\\gr_4 )(1-\\gr_3)( \\gr_1+\\gr_4 ) }\\big(\n y^\\ga+ \\Pz{}^\\ga \\big) {t}{}_{\\ga}\n \\\\ \\nn&&\n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n ( p_1{}+ p_2)^{\\gga} \\big(y + (1 -\\gr_4 ){t}{}\\big)_{\\gga} z_\\ga(y+\\tilde{t}{})^{\\ga}\n \\ff{1}{ (1-\\gr_3 )}\\,\\, {t}{}^\\ga y_{\\ga}\\\\ \\nn&&\n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n ( p_1{}+ p_2)^{\\gga} \\big(y + (1 -\\gr_4 ){t}{}\\big)_{\\gga} z_\\ga(y+\\tilde{t}{})^{\\ga}\n (-) \\ff{ 1 }{(1-\\gr_3)} (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}\n \n+ \\eee\\bee \\nn&&\n \n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga z_\\ga(y+\\tilde{t}{})^{\\ga}\n \\gx_1 \\ff{\\gr_3}{(1-\\gr_1-\\gr_4 )^2 }\\,\\,( - ( p_3+p_2) )^\\ga y_{\\ga}\n \\\\\\nn &&\n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga z_\\ga(y+\\tilde{t}{})^{\\ga}\n \\gx_1 \\ff{ \\gr_3}{(1-\\gr_1-\\gr_4 )^2 }\n\\big( ( - ( p_3+p_2) )^\\ga \\big)\\tilde{t}{}_{\\ga}\n\\\\\\nn &&\n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga z_\\ga(y+\\tilde{t}{})^{\\ga}\n\\\\ \\nn&&\\times \\gx_1 \\ff{\\gr_3\\gr_4}{(1-\\gr_1-\\gr_4 ) (1-\\gr_3 )(\\gr_1+\\gr_4)}\\,\\, {t}{} ^\\ga y_{\\ga}\n \\\\\\nn &&\n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga z_\\ga(y+\\tilde{t}{})^{\\ga}\n \\gx_1 \\ff{1}{ (1-\\gr_3 )}\\,(p_1 +p_2 )^{\\ga} y_{\\ga}\n \\\\\\nn &&\n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga z_\\ga(y+\\tilde{t}{})^{\\ga}\n \\gx_1 \\ff{ 1}{ (1-\\gr_3)}(p_1 +p_2 )^{\\ga}\\tilde{t}{}_{\\ga}\n \\eee\\bee\\nn &&\n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga z_\\ga(y+\\tilde{t}{})^{\\ga}\n\\\\ \\nn&&\\times (-) \\gx_1 \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)} \\ff{\\gr_4}{(\\gr_1+\\gr_4)^2}\n\\big(y^\\ga+ \\Pz{}^\\ga \\big){t}{}_{\\ga}\n \\\\\\nn &&\n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga z_\\ga(y+\\tilde{t}{})^{\\ga}\n(-) \\ff{ 1}{ (1-\\gr_3 )}\\,(p_1 +p_2 )^{\\ga} y_{\\ga}\n \\\\\\nn &&\n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga z_\\ga(y+\\tilde{t}{})^{\\ga}\n (-) \\ff{ 1 }{(1-\\gr_3)} (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}\n + \\eee \\bee\\nn &&\n \n+ \\gx_1\\Big[\n \\ff{ \\gr_2\\gr_2}{(1- \\gr_1 -\\gr_4)^3(1-\\gr_3)^3( \\gr_1+\\gr_4 ) }\n\\big( y+ (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} )+ (1 -\\gr_4 ){t}{} \\big)^\\gga\n\\big( y + \\tilde{t}{} \\big)_{\\gga} z_\\ga {t}{}^{\\ga}\n \\\\ \\nn &&\n-\\ff{ \\gr_3\\gr_2}{(1- \\gr_1 -\\gr_4)^3(1-\\gr_3)^3 }\n {\\big( y + \\tilde{t}{} \\big)^\\gga z_{\\gga} {t}{}^{\\ga} y_{\\ga}}\n \\\\ \\nn &&\n + \\ff{ \\gr_2}{(1- \\gr_1 -\\gr_4)^2(1-\\gr_3)^3 } \\big( y\n + \\tilde{t}{} \\big)^\\gga z_{\\gga} { (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}}\n \\Big]\n \\big(y + \\Pz{} \\big)^\\gb\n (y+\\tilde{t}{})_{\\gb}\\Big\\} \\Ee\\go CCC\\,.\n \\eee\n Terms from the \\rhs of \\eq{SUM=} with $\\gx$-independent pre-exponentials are considered in Section \\ref{NEgx},\n while those with $\\gx_1$-proportional pre-exponentials are considered in Section \\ref{gx}.\n\n\\subsection{ $\\gx_1$-independent pre-exponentials }\n\\label{NEgx}\n Here we consider only pre-exponentials, omitting for brevity integrals, integral measures {\\it etc} of \\eq{SUM=}.\nBy virtue of the Schouten identity taking into account that $\\sum \\gr_i=1$\nEq.~\\eq{SUM=} yields\n\\bee\\nn && Integrand(S_6)\\Big|_{\\mod \\gx}=(y+ \\tilde{t}{} )^{\\gn}z_{\\gn}\\Big\\\n-\n\\ff{\\gr_2(\\gr_1+\\gr_4)}{(1- \\gr_1 -\\gr_4)(1-\\gr_3)^3 }{t}{}^\\ga (p_1+p_2 ){} _\\ga\n \\,\\,( (p{}_1 +p_2) )^\\ga y_{\\ga}\n\\qquad \\\\ \\nn&&\n- \\ff{\\gr_2 \\gr_4 }{(1- \\gr_1 -\\gr_4)(1-\\gr_3)^3 }{t}{}^\\ga (p_1+p_2 ){} _\\ga\n \\,{t}{} ^\\ga y_{\\ga}\n\\\\ \\nn&&\n- \\ff{\\gr_2\\gr_1 }{(1- \\gr_1 -\\gr_4)(1-\\gr_3)^3 }{t}{}^\\ga (p_1+p_2 ){} _\\ga\n (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}\n \\\\ \\nn&\n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^3}\n ( p_1{}+ p_2)^{\\gga} \\big(y + (1 -\\gr_4 ){t}{}\\big)_{\\gga}\n \\,\\, {t}{}^\\ga y_{\\ga}\\eee\\bee \\nn&&\n - \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^3}\n ( p_1{}+ p_2)^{\\gga} \\big(y + (1 -\\gr_4 ){t}{}\\big)_{\\gga}\n (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}\n \\\\ \\nn&&\n - \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^3}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga \\,(p_1 +p_2 )^{\\ga} y_{\\ga}\n \\\\\\nn &&\n - \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^3}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga\n (p_1 +p_2 )^{\\ga}{t}{}_{\\ga} \\Big\\}\\Ee\\go CCC \n \n\\nn \\\\\n\\nn &&=(y+ \\tilde{t}{} )^{\\gn}z_{\\gn}\\ff{\\gr_2 }{(1- \\gr_1 -\\gr_4)(1-\\gr_3)^3 }\\Big\\\n \\gr_1{t}{}^\\ga (p_1+p_2 ){} _\\ga\n (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}\n +\n ( p_1{}+ p_2)^{\\gga} y _{\\gga}\n \\,\\, {t}{}^\\ga y_{\\ga}\n \\\\ \\nn&&\n -\n ( p_1{}+ p_2)^{\\gga} (1 -\\gr_4 ){t}{}_{\\gga}\n (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}\n \\\\ \\nn&&\n -\n {t}{}^\\gga y _\\gga \\,(p_1 +p_2 )^{\\ga} y_{\\ga}\n - {t}{}^\\gga (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ) _\\gga\n (p_1 +p_2 )^{\\ga}{t}{}_{\\ga} \\Big\\}\\Ee\\go CCC\n \\\\ \\nn\n &&=(y+ \\tilde{t}{} )^{\\gn}z_{\\gn}\\ff{\\gr_2 }{(1- \\gr_1 -\\gr_4)(1-\\gr_3)^3 }\\Big\\\n - \\gr_1{t}{}^\\ga (p_1+p_2 ){} _\\ga\n (p_1{}^{\\ga}+p_2{}^{\\gb}){t}{}_{\\gb}\n \\\\ \\nn&&\n - ( p_1{}+ p_2)^{\\gga} (1 -\\gr_4 ){t}{}_{\\gga}\n (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}\n -\n {t}{}^\\gga (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ) _\\gga\n (p_1 +p_2 )^{\\ga}{t}{}_{\\ga} \\Big\\}\\Ee\\go CCC\\equiv 0 .\\eee\n\\subsection{ $\\gx_1$-proportional pre-exponentials}\n\\label{gx}\n\\bee\\label{lostcohomo2}&&\n S_6\\, \\Big|_{\\gx_1 }\n = J_7 + i \\ff{ \\eta^2 }{4 }\\int d\\Gamma \\delta(\\xi_3) \\\\ \\nn&&\\Big\\{\n \n(y+ \\tilde{t}{} )^{\\gga}z_{\\gga}\\ff{\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3) }{t}{}^\\ga (p_1+p_2 ){} _\\ga\n\\gx_1 \\ff{1-\\gr_3-\\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3 )^2}\\,\\, \\Pz{}^\\ga y_{\\ga}\n\\\\ \\nn&&\n+(y+ \\tilde{t}{} )^{\\gga}z_{\\gga}\\ff{\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3) }{t}{}^\\ga (p_1+p_2 ){} _\\ga\n \\gx_1 \\ff{1-\\gr_3-\\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3 )^2}\\big(y + \\Pz{} \\big)^\\ga\\tilde{t}{}_{\\ga}\n\\\\ \\nn&&\n-(y+ \\tilde{t}{} )^{\\gga}z_{\\gga}\\ff{\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3) }{t}{}^\\ga (p_1+p_2 ){} _\\ga\n \\gx_1 \\ff{\\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3 )}\\,(p_3{}^{\\ga}+p_2{}^{\\ga}) y_{\\ga}\n\\\\ \\nn&&\n-(y+ \\tilde{t}{} )^{\\gga}z_{\\gga}\\ff{\\gr_2}{(1- \\gr_1 -\\gr_4)(1-\\gr_3) }{t}{}^\\ga (p_1+p_2 ){} _\\ga\n \\gx_1 \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)}(p_3{} +p_2{})^{\\gb}\\tilde{t}{}_{\\gb}\n \\\\ \\nn&&\n - \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n ( p_1{}+ p_2)^{\\gga} \\big(y + (1 -\\gr_4 ){t}{}\\big)_{\\gga} z_\\ga(y+\\tilde{t}{})^{\\ga}\n \\gx_1 \\ff{\\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3 )}\\,\\, {t}{}^\\ga y_{\\ga}\n \\\\ \\nn&&\n - \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n ( p_1{}+ p_2)^{\\gga} \\big(y + (1 -\\gr_4 ){t}{}\\big)_{\\gga} z_\\ga(y+\\tilde{t}{})^{\\ga}\n \\gx_1 \\ff{ \\gr_2 }{(1-\\gr_1-\\gr_4 )(1-\\gr_3)( \\gr_1+\\gr_4 ) }\n \\\\\\nn &&\\times\n \\big( y^\\ga+ \\Pz{}^\\ga \\big) {t}{}_{\\ga}\n \n \\\\ \\ls\\ls\\ls\\nn&&\n \n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga z_\\ga(y+\\tilde{t}{})^{\\ga}\n \\gx_1 \\ff{\\gr_3}{(1-\\gr_1-\\gr_4 )^2 }\\,\\,( - ( p_3+p_2) )^\\ga y_{\\ga}\n \\\\\\nn &&\n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga z_\\ga(y+\\tilde{t}{})^{\\ga}\n \\gx_1 \\ff{ \\gr_3}{(1-\\gr_1-\\gr_4 )^2 }\n\\big( ( - ( p_3+p_2) )^\\ga \\big)\\tilde{t}{}_{\\ga}\n\\\\\\nn &&\n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga z_\\ga(y+\\tilde{t}{})^{\\ga}\n \\\\\\nn &&\\times\n \\gx_1 \\ff{\\gr_3\\gr_4}{(1-\\gr_1-\\gr_4 ) (1-\\gr_3 )(\\gr_1+\\gr_4)}\\,\\, {t}{} ^\\ga y_{\\ga}\n \\\\\\nn &&\n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga z_\\ga(y+\\tilde{t}{})^{\\ga}\n \\gx_1 \\ff{1}{ (1-\\gr_3 )}\\,(p_1 +p_2 )^{\\ga} y_{\\ga}\n \\\\\\nn &&\n + \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga z_\\ga(y+\\tilde{t}{})^{\\ga}\n \\gx_1 \\ff{ 1}{ (1-\\gr_3)}(p_1 +p_2 )^{\\ga}\\tilde{t}{}_{\\ga}\n \\\\\\nn &&\n - \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga z_\\ga(y+\\tilde{t}{})^{\\ga}\n \\\\\\nn &&\\times\n \\gx_1 \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)} \\ff{\\gr_4}{(\\gr_1+\\gr_4)^2}\n\\big(y^\\ga+ \\Pz{}^\\ga \\big){t}{}_{\\ga}\n \n \\\\ \\nn &&\n \n+ \\gx_1\\Big[ \n \\ff{ \\gr_2\\gr_2}{(1- \\gr_1 -\\gr_4)^3(1-\\gr_3)^3( \\gr_1+\\gr_4 ) }\n\\big( y+ (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} )+ (1 -\\gr_4 ){t}{} \\big)^\\gga\n\\big( y + \\tilde{t}{} \\big)_{\\gga}\\\\ \\nn &&\\times\\Big\\{\n{t}{}^{\\ga}\\big(y + \\Pz{} \\big)_\\ga z^\\gs (y+\\tilde{t}{})_{\\gs}\n\\Big\\}\n-\\ff{ \\gr_3\\gr_2}{(1- \\gr_1 -\\gr_4)^3(1-\\gr_3)^3 }\n {\\big( y + \\tilde{t}{} \\big)^\\gga z_{\\gga} {t}{}^{\\ga} y_{\\ga}}\n \\big(y + \\Pz{} \\big)^\\gs\n (y+\\tilde{t}{})_{\\gs} \\\\ \\nn &&\n + \\ff{ \\gr_2}{(1- \\gr_1 -\\gr_4)^2(1-\\gr_3)^3 } \\big( y\n + \\tilde{t}{} \\big)^\\gga z_{\\gga} \\big(y + \\Pz{} \\big)^\\gs\n (y+\\tilde{t}{})_{\\gs} { (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}}\n \\Big]\n\\Big\\}\\Ee\\go CCC\\q\n \\eee\nwhere $J_7$ is the cohomology term \\eq{Result6}\\,.\nThis yields\n\\bee\\label{Endofgx}&&\n S_6\\, \\Big|_{\\gx_1 } \\approx J_7 + i \\ff{ \\eta^2 }{4 }\\int d\\Gamma \\delta(\\xi_3)\n \\ff{\\gr_2 }{(1- \\gr_1 -\\gr_4)^2(1-\\gr_3)^2 }\n \\\\ \\nn&&\\gx_1(y+ \\tilde{t}{} )^{\\gga}z_{\\gga} \\Big\\{\n \\ff{ (1-\\gr_3-\\gr_2)}{ (1-\\gr_3) }{t}{}^\\ga (p_1+p_2 ){} _\\ga\n {\\big(y + \\Pz{} \\big)^\\gb(y+ \\tilde{t}{} )_{\\gb}}\n \\\\ \\nn&&\n- \\gr _2 {t}{}^\\ga (p_1+p_2 ){} _\\ga\n (p_3{}^{\\ga}+p_2{}^{\\gb})(y+ \\tilde{t}{} )_{\\gb}\n - \\ff{\\gr _2}{ (1-\\gr_3) }\n ( p_1{}+ p_2)^{\\gga} \\big(y + (1 -\\gr_4 ){t}{}\\big)_{\\gga}\n \\,\\, {t}{}^\\ga y_{\\ga}\n \\\\ \\nn&&\n - \\ff{\\gr _2}{ (1-\\gr_3) ( \\gr_1+\\gr_4 )}\n ( p_1{}+ p_2)^{\\gga} \\big(y + (1 -\\gr_4 ){t}{}\\big)_{\\gga}\n \\big(y+ \\Pz{} \\big)^\\ga {t}{}_{\\ga}\n \\\\ \\nn&&\n - \\ff{ \\gr_3}{(1-\\gr_1-\\gr_4 )^2 }\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga\n ( p_3+p_2)^\\ga (y+ \\tilde{t}{} )_{\\ga}\n\\\\\\nn &&\n + \\ff{ \\gr_3\\gr_4}{ (1-\\gr_3) (\\gr_1+\\gr_4)}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga\n \\, {t}{} ^\\ga y_{\\ga}\n \\\\\\nn &&\n + \\ff{ (1-\\gr_1-\\gr_4 )}{ (1-\\gr_3) }\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga\n (p_1 +p_2 )^{\\ga}(y+ \\tilde{t}{} )_{\\ga}\n \\\\\\nn &&\n - \\ff{ \\gr _2\\gr_4}{ (1-\\gr_3) (\\gr_1+\\gr_4)^2}\n {t}{}^\\gga \\big(y + (1-\\gr_1-\\gr_4 )( p_1{} +p_2{} ){}\\big)_\\gga\n \\big(y^\\ga+ \\Pz{}^\\ga \\big){t}{}_{\\ga}\n \\\\\\nn &&\n +\\ff{ \\gr_2}{(1- \\gr_1 -\\gr_4) (1-\\gr_3) ( \\gr_1+\\gr_4 ) }\n\\big( y+ (1 -\\gr_4 ){t}{} \\big)^\\gga\n\\big( y + \\tilde{t}{} \\big)_{\\gga}\n{t}{}_{\\ga}\\big(y + \\Pz{} \\big)^\\ga\n\\\\ \\nn &&\n +\\ff{ \\gr_2 }{ (1-\\gr_3) ( \\gr_1+\\gr_4 ) }\n ( p_1{} +p_2{} )^\\gga\n\\big( y + \\tilde{t}{} \\big)_{\\gga}\n {t}{}_{\\ga}\\big(y + \\Pz{} \\big)^\\ga\n \\\\ \\nn &&\n-\\ff{ \\gr_3 }{(1- \\gr_1 -\\gr_4) (1-\\gr_3) }\n {t}{}^{\\ga} y_{\\ga}\n \\big(y + \\Pz{} \\big)^\\gs\n (y+\\tilde{t}{})_{\\gs} \\\\ \\nn &&\n + \\ff{ 1}{ (1-\\gr_3) } \\big(y + \\Pz{} \\big)^\\gs\n (y+\\tilde{t}{})_{\\gs} { (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}}\n\\Big\\}\\Ee\\go CCC\\equiv J_7\n \\eee\nsince, using the Schouten identity, one can see that\nthe pre-exponential of the integrand on the \\rhs of \\eq{Endofgx} equals zero.\n\\addtocounter{appendix}{1}\n\\renewcommand{\\theequation}{\\Alph{appendix}.\\arabic{equation}}\n\\addtocounter{section}{1} \\setcounter{equation}{0}\n \\addcontentsline{toc}{section}{\\,\\,\\,\\,\\,\\,\\,Appendix E: Useful formulas}\n \\renewcommand{\\thesubsection}{\\Alph{appendix}.\\arabic{subsection}}\n\n\n\\section*{Appendix E: Useful formulas}\n\\label{AppG}\n\nFrom \\eq{Egx=}\n one has\n \\bee\\label{EEgx14=}&&\\left(\\frac{\\p}{\\p \\rho_1}-\\frac{\\p}{\\p \\rho_4}\\right) E= i\\Big\\{\n \\gx_1 \\ff{\\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3 )}\\,\\, {t}{}^\\ga y_{\\ga}\n\\\\ \\nn &&+ \\gx_1 \\ff{ \\gr_2 }{(1-\\gr_1-\\gr_4 )(1-\\gr_3)( \\gr_1+\\gr_4 ) }\\big( y + \\Pz{}\\big)^\\ga {t}{}_{\\ga}\n \\\\ \\nn &&\n + \\ff{ 1 }{(1-\\gr_3)} (y+p_1{}^{\\ga}+p_2{}^{\\ga}){t}{}_{\\ga}\n \\Big\\}E \\eee\n \\bee\\label{EEgx23}&&\\left(\\frac{\\p}{\\p \\rho_2}-\\frac{\\p}{\\p \\rho_3}\\right) E= i \\Big\\{\n \\gx_1 \\ff{1-\\gr_3-\\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3 )^2}\n \\big( y + \\Pz{}\\big)^\\ga(y+\\tilde{t}{})_{\\ga}\n\\\\ \\nn&& - \\gx_1 \\ff{\\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3 )}\\,(p_3{}^{\\ga}+p_2{}^{\\ga}) (y+\\tilde{t}{})_{\\ga}\n\n - \\ff{ \\gr_1+\\gr_4}{ (1-\\gr_3 )^2}\\,\\,( (p{}_1 +p_2) )^\\ga y_{\\ga}\n\\\\ \\nn && - \\ff{ \\gr_4 }{ (1-\\gr_3 )^2}\\,\\,{t}{} ^\\ga y_{\\ga}\n\n - \\ff{ \\gr_1 }{(1-\\gr_3)^2} (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}\n \\Big\\}\n E \\,, \\eee\n\\bee\\label{Egx=21e}&&\\left(\\frac{\\p}{\\p \\rho_2}-\\frac{\\p}{\\p \\rho_1}\\right) E = i \\Big\\{\n \\gx_1 \\ff{ -\\gr_3}{(1-\\gr_1-\\gr_4 )^2 }\n ( p_3+p_2 )^\\ga (y+\\tilde{t}{})_{\\ga}\n\\\\ \\nn&&\n + \\gx_1 \\ff{\\gr_3\\gr_4}{(1-\\gr_1-\\gr_4 ) (1-\\gr_3 )(\\gr_1+\\gr_4)}\\,\\, {t}{} ^\\ga y_{\\ga}\n + \\gx_1 \\ff{ 1}{ (1-\\gr_3)}(p_1 +p_2 )^{\\ga}(y+\\tilde{t}{})_{\\ga}\n\\\\ \\nn &&- \\gx_1 \\ff{ \\gr_2}{(1-\\gr_1-\\gr_4 )(1-\\gr_3)} \\ff{\\gr_4}{(\\gr_1+\\gr_4)^2}\n\\big( y + \\Pz{}\\big)^\\ga{t}{}_{\\ga}\n \\\\ \\nn &&\n\\\\ \\nn&&\n- \\ff{ 1}{ (1-\\gr_3 )}\\,(p_1 +p_2 )^{\\ga} y_{\\ga}\n- \\ff{ 1 }{(1-\\gr_3)} (p_1 +p_2 )^{\\ga}{t}{}_{\\ga}\n \\Big\\}\n E \\, . \\eee\n\n \\addcontentsline{toc}{section}{\\,\\,\\,\\,\\,\\,\\,References}\n\n\n\\section*{}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nNeurons are morphological structures: they have dendritic branches on which most inputs are received and an axonal tree through which the output signal is communicated with other neurons. In this light, neuronal computations can be seen as the integration of synaptic inputs along the dendrites up to the axon initial segment where an output signal is generated. Hence a key role in neuronal computation is taken by the exact shape and composition of dendrites. Indeed, it is known that the neuronal response is shaped by the precise location and activation pattern of synapses \\citep{Branco2010, Torben-Nielsen2010, Gidon2012} and by the expression and distribution of (voltage-gated) ion-channels \\citep{Migliore2002, Magee1999, Torben-Nielsen2010,Spruston2008}. \n\nDespite this proven importance, dendritic processing is usually ignored in network simulation \\citep{Gewaltig2007,Brette2007,Richert2011}, but see \\citep{Markram2006} for an exception. One reason is the computational cost associated with multi-compartmental simulations: a costs that, at the level of the model neuron, scales with the morphological complexity of the dendritic arborization. Related is the conceptual cost associated with building detailed single-neuron models \\citep{Hay2013} with the spatial distribution of conductances across the membrane and localized non-linearities. The key is to capture the somatic voltage in response to synaptic inputs on the dendrites. Is there an alternative to multi-compartmental models to simulate the effects of dendrites on synaptic potentials, without large computational overhead?\n\nTo this end, two strategies are commonly adopted in the literature. The first consist of performing a morphological reduction by reducing the number of dendritic segments while attempting to capture crucial characteristics of dendritic processing \\citep{Traub2005,Kellems2010}. A second strategy is to by-pass multiple (dendritic) compartments altogether by using point-neurons and fit voltage-kernels that matches the dendritic signal transformation shaping the voltage waveform caused by a synaptic input at the soma \\citep{Jolivet2004, Gutig2006}. The fitted \\citep{Jolivet2004} or learned \\citep{Gutig2006} kernel is then simply added to the somatic membrane potential. While this strategy is computationally efficient and some temporal effects of dendritic processing can be captured, it is a rather crude approximation of what dendritic integration stands for and elementary features of dendritic processing, such as local interaction between inputs, are impossible to achieve. \n\nIn this work we present a true alternative based on applying the Green's function formalism to cable theory. This way we can exactly compute the effect of synaptic inputs located in the dendrites on the somatic membrane potential \\citep{Koch1985}. By design we thus compute the linear transfer function between the site of the synaptic inputs and the soma. The main advantage of this approach is that the effect of synaptic inputs along a dendrite on the somatic membrane potential can be calculated analytically. Consequently, simulations in our model are independent of the morphological complexity and a full reduction to a point-neuron can be used, as the entire effect of the morphology is captured in a transfer function. This property sets our approach apart from existing methods to model dendrites implicitly: the approach based on the equivalent cable works only with geometrically tightly constrained morphologies \\citep{Ohme1998}, while, as in \\citep{VanPelt1992} all branch points of a dendritic tree have to be modeled explicitly. Because we capture arbitrary dendritic morphologies by means of transfer functions, our synapse model is able to use dendrite-specific mechanism of computation, such as delay lines (as \\citep{Gutig2006}) but also local non-linearities due to membrane saturation. Hence, we can capture fundamental features of dendritic integration by directly deriving the Green's function from dendritic cable theory. \n\nWe implemented our synapse model in the Python programming language as a proof of principle, and validated it by evaluating its correctness and execution times on two tasks. First, we show that a morphology-less point-neuron equipped with the proposed synapse model can exploit differential dendritic processing to perform an input-order detection task \\citep{Agmon-Snir1998}. We show that both for passive models and models with active currents in the soma, the agreement with a reference \\textsc{neuron} simulation \\citep{Carnevale2006} is seamless. Second, we show that the proposed neuron model is capable of accurate temporal integration of multiple synaptic inputs, a result for which knowledge of the precise neuronal morphology in relation to the synaptic locations is imperative. To this end, we construct a point-neuron model mimicking the dendritic processing in the dendrites of a Layer 5 pyramidal cell. Again, we demonstrate that the agreement with a reference \\textsc{neuron} simulation is seamless. By providing this example, we demonstrate that our proposed approach is highly suitable for the common scenarios to investigate dendritic processing. In such scenarios, the somatic response to a limited number of synapses located in the dendrites is measured while changing the dendritic properties.\n\n\\section{Synapse model based on the Green's function formalism}\\label{sec:methods}\n\nThe core rationale of this work is the simplification of a passive neuron model by analytically computing the transfer function between synapses and the soma. Solving the cable equation for dendrites is not new, and several ways are documented \\citep{Koch1985, Butz1974, Norman1972}. The application of the cable equation to simplify arbitrarily morphologically extended multi-compartmental models to a point-neuron is, however, new.\n\nBy solving the cable equation, we thus substitute the effects of an electrical waveform traveling down a dendrite by a so-called pulse-response kernel. Conceptually, we think of the neural response to a spike input as being characterized by three functions: the conductance profile of the synapse, the pulse-response kernel at the synapse and the pulse-response transfer kernel between the input location and the soma to mimic the actual dendritic propagation. The first function is chosen by the modeller: common examples are the alpha function, the double exponential or the single decaying exponential \\citep{Rotter1999,Giugliano2000,Carnevale2006}. The second function captures the decay of the voltage at the synapse given a pulse input, and thus allows for a computation of the synaptic driving force, whereas the third function allows for the computation of the response at the soma, given the synaptic profile, driving force, and dendritic profile.\n\nMore formally, we write $g(t)$ for the synaptic conductance profile, $G_{\\text{syn}}(t)$ for the pulse response kernel at the synapse and $G_{\\text{som}}(t)$ for the pulse response kernel between synapse and soma. Then, given a presynaptic spiketrain $\\{ t_s \\}$ and a synaptic reversal potential $E_r$, the somatic response of the neuron is characterized by:\n\\begin{eqnarray}\\label{eq:intro}\n\\begin{aligned}\ng(t) & = F(\\mathbf{a}(t)), \\hspace{4mm} \\frac{\\mathrm{d}\\mathbf{a}}{\\mathrm{d}t}(t) = H(\\mathbf{a}(t),\\{t_s\\})\\\\\nV_\\text{syn}(t) & = \\int_{-\\infty}^{t} \\mathrm{d}k \\ G_{\\text{syn}}(t-k) \\ g(k) \\ (V_\\text{syn}(k)-E_r) \\\\\nV_\\text{som}(t) & = \\int_{-\\infty}^{t} \\mathrm{d}k \\ G_{\\text{som}}(t-k) \\ g(k) \\ (V_\\text{syn}(k)-E_r),\n\\end{aligned}\n\\end{eqnarray}\nwhere $E_r$ is the synaptic reversal potential, $F(.)$ and $H(.)$ depend on the type of synapse chosen and $\\mathbf{a}$ denotes the set of synaptic parameters required to generate the conductance profile $g(t)$. Our task is to compute $G_{\\text{syn}}(t)$ and $G_{\\text{som}}(t)$. We will show that these functions follow from the Green's function formalism.\n\n\\subsection{The neuron model in time and frequency domains}\n\n\\subsubsection{Time domain}\n\nHere, we assume a morphological neuron models with passive dendritic segments. Each segment, labeled $d = 1,\\hdots,N$, is modeled as a passive cylinder of constant radius $a_d$ and length $L_d$. It is assumed that all segments have an equal membrane conductance $g_m$, reversal potential $E$, intracellular axial resistance $r_a$ and membrane capacitance $c_m$. By convention we label the locations along a dendrite by $x$, with $x=0$ and $x=L_d$ denoting the proximal and distal end of the dendrite, respectively. Then, in accordance with cable theory, the voltage in a segment $d$ follows from solving the partial differential equation \\citep{Tuckwell1988Introduction}:\n\\begin{eqnarray}\\label{eq:cable}\n\\begin{aligned}\n\\frac{\\pi a_d^2}{r_a}\\frac{\\mathrm{\\partial}^2 V_d}{\\mathrm{\\partial}x^2}(x,t) \\ - \\ 2\\pi a_d g_m V_d(x,t) \\ - 2\\pi a_d c_m \\frac{\\mathrm{\\partial} V_d}{\\mathrm{\\partial}t}(x,t) \\ = \\ I_d(x,t),\n\\end{aligned}\n\\end{eqnarray}\nwhere $I_d(x,t)$ represents the input current in branch $d$, at time $t$ and at location $x$. We assume that the dendritic segments are linked together by boundary conditions that follow from the requirement that the membrane potential is continuous and the longitudinal currents (denoted by $I_{ld}$) conserved:\n\\begin{eqnarray}\n\\begin{aligned}\nV_{d}(L_{d}, t) & = V_{i}(0,t), \\hspace{4mm} i \\in \\mathcal{C}(d) \\\\\nI_{ld}(L_{d}, t) & = \\sum_{i \\in \\mathcal{C}(d)} I_{li}(0,t)\n\\end{aligned}\n\\end{eqnarray}\nwhere $\\mathcal{C}(d)$ denotes the set of all child segments of segment $d$. The longitudonal currents are given by:\n\\begin{equation}\nI_{ld}(x,t) = \\frac{\\pi a_d^2}{r_a}\\frac{\\mathrm{\\partial} V_d}{\\mathrm{\\partial}x}(x,t). \n\\end{equation}\nDifferent dendritic branches originating at the soma are joined together by the lumped-soma boundary condition, which implies for the somatic voltage $V_{\\text{som}}(t)$:\n\\begin{equation} \\label{eq:lsb1}\nV_{\\text{som}}(t) = V_d(0,t) \\hspace{3mm} \\forall d \\in \\mathcal{C}(\\text{soma})\n\\end{equation}\nand\n\\begin{equation}\\label{eq:lsb2}\n\\sum_{d=1}^{\\mathcal{C}(\\text{soma})} I_{ld}(0,t) = I_{\\text{som}}(V_{\\text{som}}(t)) + C_{\\text{som}} \\frac{\\mathrm{\\partial} V_{\\text{som}}}{\\mathrm{\\partial}t}(t),\n\\end{equation}\nwith $I_{\\text{som}}$ denoting the transmembrane currents in the soma, that can be either passive or active. Note that, for all further calculations, we will treat $I_{\\text{som}}(V_{\\text{som}}(t))$ as an external input current, and apply the Green's function formalism only on a soma with a capacitive current.\nFor segments that have no children (i.e., the leafs of the tree structure), the sealed end boundary condition is used at the distal end:\n\\begin{equation}\nI_{ld}(L_d,t) = 0 \\hspace{3mm} \\forall d.\n\\end{equation}\n\n\\subsubsection{Frequency domain}\n\nFourrier-transforming this system of equations allows for the time-derivatives to be written as complex multiplications, for which analytic \\citep{Butz1974} or semi-analytic \\citep{Koch1985} solutions can be computed. Doing so transforms equation \\eqref{eq:cable} into:\n\\begin{equation}\\label{eq:freqcable}\n\\frac{\\mathrm{\\partial}^2 V_d}{\\mathrm{\\partial}x^2}(0,\\omega) - \\gamma _d(\\omega)^2 V_d(x,\\omega) = I_d(x,\\omega)\n\\end{equation}\nwhere $\\omega$ is now a complex number and $\\gamma_d(\\omega)$ is the frequency-dependent space constant, given by\n\\begin{equation}\n\\gamma_d(\\omega) = \\sqrt{\\frac{z_{ad}}{z_{md}(\\omega)}}\n\\end{equation}\nwith $z_{ad} = \\frac{r_a}{\\pi a_d^2}$ the dendritic axial impedance and $z_{md} = \\frac{1}{2 \\pi a_d (i c_m \\omega + g_m)}$ the membrane impedance in branch $d$. The lumped soma boundary conditions \\eqref{eq:lsb1} and \\eqref{eq:lsb2} become\n\\begin{equation} \nV_{\\text{som}}(\\omega) = V_d(0,\\omega) \\hspace{3mm} \\forall d\n\\end{equation}\nand\n\\begin{equation}\n\\sum_{d=1}^N I_{ld}(0,\\omega) = \\sum_{d=1}^N \\frac{1}{z_{ad}}\\frac{\\mathrm{\\partial} V_d}{\\mathrm{\\partial}x}(0,\\omega) = \\frac{1}{Z_{\\text{som}}(\\omega)}V_{\\text{som}}(\\omega),\n\\end{equation}\nwhere\n\\begin{equation}\nZ_{\\text{som}}(\\omega) = \\frac{1}{i C_{\\text{som}} \\omega}\n\\end{equation}\nis the somatic impedance. The sealed-end boundary conditions are:\n\\begin{equation}\\label{eq:bcfreq}\nI_{ld}(L_d,\\omega) = \\frac{1}{Z_L} V_d(L_d,\\omega) = 0\n\\end{equation} \nwith sealed-end impedance $Z_L = \\infty$.\n\n\\subsection{Morphological simplification by applying Green's function}\nHere we will describe the Green's function formalism formally in the time domain to explain the main principles. In the next paragraph we will then turn back to the frequency-domain to compute the actual solution. For the argument we consider a general current input $I_d(x,t)$. In the case of dynamic synapses, such a current input is obtained from the synaptic conductances by the Ohmic relation:\n\\begin{equation}\\label{eq:current}\nI_{d}(x,t) = g(t)(E_r-V_d(x,t))\n\\end{equation}\nor, in the case of active channels, from the ion channel dynamics\nThe cable equation \\eqref{eq:cable} can be written formally as: \n\\begin{equation}\\label{eq:operator}\n\\hat{L}_d V_d(x,t) = I_d(x,t)\n\\end{equation}\nwhere $\\hat{L}_d = \\frac{\\pi a_d^2}{r_a}\\frac{\\mathrm{\\partial}^2 }{\\mathrm{\\partial}x^2} - 2\\pi a_d g_m - 2\\pi a_d c_m \\frac{\\mathrm{\\partial}}{\\mathrm{\\partial}t}$ is a linear operator\\footnote{Note that formally, the operator $\\hat{L}_d$ depends on $x$ explicitly in a discontinuous way: for for $0x_i$ follows from interchanging $x$ and $x_i$ in \\eqref{eq:gf}.\nTo compute the effect of a synaptic input on the driving force in other branches (denoted by $d'$), we first use equation \\eqref{eq:gf} (corresponding to rule III of \\citep{Koch1985}) to obtain the pulse-voltage response in the frequency domain at the soma. Then, to compute the pulse voltage response in the branch where the driving force needs to be known, we use the following identity:\n\\begin{equation}\nG_{dd'}(x,x',\\omega) = \\frac{G_{d'd'}(x,0,\\omega) G_{dd}(0,x',\\omega)}{G(\\text{soma}, \\text{soma}, \\omega)},\n\\end{equation}\ncorresponding to rule IV of \\citep{Koch1985}.\n\n\n\\subsubsection{Transforming the Green's function to the time domain}\n\nGiven the conventions we assumed when transforming the original equation, the inverse Fourier transform has following form:\n\\begin{equation}\\label{eq:transint}\nG(x,x_i,t) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty}\\mathrm{d}\\omega \\ G(x,x_i,\\omega) \\ e^{i \\omega t}.\n\\end{equation}\nIf the Green's function in the time-domain rises continuously from zero, which is generally the case if $x \\neq x_i$, it can be approximated with negligible error by the standard technique for evaluating Fourier integrals with the fast-Fourier transform (FFT) algorithm \\citep{Press2007Numerical}: we choose a sufficiently large interval $[-\\omega_m,\\omega_m]$ (where $G(x,x_i,\\pm \\omega_m)$ is practically 0), divide it in $M=2^n$ pieces of with $\\Delta \\omega = \\frac{2\\omega_m}{M}$ and approximate the integral by a discrete sum:\n\\begin{equation}\\label{eq:tranform}\nG(x,x_i,t) = \\frac{1}{2\\pi} \\sum_{j=0}^{M-1}G(x,x_i,\\omega_j)e^{i\\omega_j t},\n\\end{equation}\nwhere $\\omega_j = -\\omega_m + j \\Delta \\omega$. The choice of discretization step then fixes the timestep $\\Delta t = \\frac{2\\pi}{M \\Delta \\omega}$. Upon evaluating the Green's function in the time-domain at $t_l = l \\Delta t, \\ l=0,\\hdots,\\frac{M}{2}-1$, expression \\eqref{eq:tranform} can be written in a form that is suitable for the fast Fourier transform algorithm:\n\\begin{equation}\nG(x,x_i,t_l) = \\frac{\\Delta \\omega}{2\\pi} e^{-i \\omega_m t_l} \\sum_{j=0}^{M-1}G(x,x_i,\\omega_j)e^{i\\frac{2\\pi}{M}jl},\n\\end{equation}\nand hence:\n\\begin{equation}\nG(x,x_i,t_l) = \\frac{M \\Delta \\omega}{2\\pi} e^{-i \\omega_m t_l} \\text{FFT}(G(x,x_i,\\omega_j))_l\n\\end{equation}\nThe situation is different if we consider the Green's function at the input location ($x = x_i$). There, the function rises discontinuously from zero at $t=0$, which causes the spectrum in the frequency-domain to have non-vanishing values at arbitrary high frequencies. Hence, the effect of integrating over a finite interval $[-\\omega_m,\\omega_m]$ will be non-negligible. Formally, this truncation can be interpreted as multiplying the original function with a window function $H(\\omega)$ that is 1 in the interval $[-\\omega_m,\\omega_m]$ and 0 elsewhere, resulting in a time-domain function that is a convolution of the real function and the transform of the window:\n\\begin{equation}\n\\begin{aligned}\n& \\tilde{G}(\\omega) = G(x_i,x_i,\\omega)H(\\omega) \\hspace{4mm} \\\\\n& \\hspace{4mm} \\Longrightarrow \\hspace{4mm} \\tilde{G}(t) = \\int_{-\\infty}^{\\infty} G(x_i,x_i,\\tau)H(t-\\tau).\n\\end{aligned}\n\\end{equation}\nFor the rectangular window, the transform $H(t)$ has significant amplitude components for $t\\neq 0$, an unwanted property that will cause the Green's function to have spurious oscillations, a phenomenon that is known as spectral leakage \\citep{Blackman1958}. This problem can be solved by chosing a different window function, which is 1 at the center of the spectrum and drops continuously to zero at $-\\omega_m$ and $\\omega_m$. For this work we found that the Hanning window,\n\\begin{equation}\nH(\\omega) = \\frac{1}{2}\\left(1+\\cos \\left( \\frac{\\pi \\omega}{\\omega_m} \\right) \\right),\n\\end{equation} \ngave accurate results for $t\\neq 0$. For $t=0$, the amplitude is slightly underestimated as a consequence of the truncation of the spectrum, whereas for $t$ very close to, but larger than $0$, the amplitude is slightly overestimated. However, these errors only cause discrepancy in a very small window ($<\\unit[0.1]{ms}$) and thus have negligible effect on the neural dynamics.\n\n\\section{Model implementation \\& Validation}\n\n\\subsection{Synapse model implementation}\n\nWe implemented a prototype of the synapse model discussed above in two stages. First, after specifying the morphology and the synapse locations, the Green's Function is evaluated at the locations that are needed to solve the system, thus yielding a set of pulse response kernels. As modern high-level languages can handle vectorization very efficiently, these functions can be evaluated for a large set of frequencies $\\omega$ quickly, thus allowing for great accuracy. Second, we implemented a model neuron that uses these Green's functions, sampled at the desired temporal accuracy. Then, given a set of synaptic parameters, the somatic membrane potential is computed by integrating the Volterra-equations \\eqref{eq:greenssynapse} and \\eqref{eq:greenssoma} \\citep{Press2007Numerical}.\n\n\n\\begin{table}[htb]\n\\centering\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline \n\\multicolumn{5}{|c|}{Physiology} \\\\\n\\hline \n$C_m$ & \\multicolumn{4}{c|}{\\unit[1]{$\\mu F\/cm^2$}} \\\\\n$g_m$ & \\multicolumn{4}{c|}{\\unit[0.02]{$mS\/cm^2$}} \\\\\n$r_a$ & \\multicolumn{4}{c|}{\\unit[100]{$\\Omega cm$}} \\\\\n$E_l$ & \\multicolumn{4}{c|}{\\unit[-65]{mV}} \\\\\n\\hline\n\\multicolumn{5}{|c|}{Morphology} \\\\\n\\hline\nSoma length & \\multicolumn{4}{c|}{\\unit[25]{$\\mu m$}} \\\\\nSoma diam & \\multicolumn{4}{c|}{\\unit[25]{$\\mu m$}} \\\\\n\\hline\n& \\multicolumn{2}{|c|}{Fig~\\ref{fig:input_order}B} & \\multicolumn{2}{c|}{Fig~\\ref{fig:input_order}C} \\\\\n\\hline\n& dend 1 & dend 2 & dend 1 & dend 2 \\\\ \\cline{2-5}\n$L_d$ & \\unit[950]{$\\mu$m} & \\unit[450]{$\\mu$m} & \\unit[900]{$\\mu$m} & \\unit[500]{$\\mu$m} \\\\\n$a_d$ & \\unit[0.25]{$\\mu$m} & \\unit[0.5]{$\\mu$m} & \\unit[0.5]{$\\mu$m} & \\unit[1]{$\\mu$m}\\\\\n\\hline\n\\multicolumn{5}{|c|}{Synapses} \\\\\n\\hline\n& syn 1 & syn 2 & syn 1 & syn 2 \\\\ \\cline{2-5}\n$E_r$ & \\unit[0]{mV} & \\unit[0]{mV} & \\unit[0]{mV} & \\unit[0]{mV} \\\\\n$\\tau$ & \\unit[1.5]{ms} & \\unit[1.5]{ms} & \\unit[1.5]{ms} & \\unit[1.5]{ms} \\\\\n$\\overline{g}$ & \\unit[5]{nS} & \\unit[2]{nS} & \\unit[20]{nS} & \\unit[9]{nS} \\\\\n\\hline\n\\end{tabular}\n\\caption{Model neuron parameters. The multi-compartmental model explicitly simulates the dendritic structure, while the point-neuron is equipped with our model synapse based on Green's functions and implicitly simulates the dendritic structure.}\n\\label{table:parameters}\n\\end{table}\n\n\n\\subsection{Multi-compartmental and point-neuron model}\n\nTo compare the performance between a multi-compartmental model and a point-neuron model using the proposed synapse model, we created two comparable neuron models. In the multi-compartmental model, the dendrites are modeled explicitly using \\textsc{neuron} \\citep{Carnevale2006}, while in the point-neuron model the dendrites are omitted and dendritic processing is carried out implicitly by the new synapse model. The properties of both model neurons are listed in Table~\\ref{table:parameters}. Evidently, the implicit model has no real morphology and the parameters related to the geometry are used to instantiate the synapse model.\n\n\\subsection{Input-order detection with differential dendritic filtering}\\label{sec:iodetect}\n\n\\begin{figure*}[htb!]\n \\centering\n \\includegraphics[width=0.9\\textwidth]{fig1_.pdf} \n \\caption{Comparison between a reference multi-compartmental model and a point-neuron model equipped with the new synapse model implicetely simulating dendritic processing. A: Both model neurons performed the input-order detection task: The neuron has to respond as strong as possible to the temporal activation 1 $\\rightarrow$ 2 and as weak as possible to the reverse temporal order. B: The input-order dectection task for a completely passive neuron. Left and right panels contain the somatic membrane potential when the synapses were activated in the preferred ($1 \\rightarrow 2$) and null ($2 \\rightarrow 1$) temporal order respectively. Colored lines represent the voltage in the point-neuron model and the black dashed line depicts the \\textsc{neuron} trace for comparison. As a reference the waveform when only the first synapse is activated is also shown (left: 1 and right: 2). Vertical dashed-dotted lines denote the spikes arriving at synapse 1 and 2 (left) or 2 and 1 (right). (C) Same as (B), but now the soma contained active HH-currents. \n }\n \\label{fig:input_order}\n\\end{figure*}\n\nTo show the applicability of the new type of model synapse, we use it to perform input-order detection: Suppose a neuron with two dendrites and one synapse (or one group of synapses) on either dendrite (shown in figure~\\ref{fig:input_order}A). In the input-order task, the neuron has to generate a strong response to the temporal activation of the synapses $1 \\rightarrow 2$, while generating a weak response to the reversed temporal activation $2 \\rightarrow 1$. This behavior is achieved by differential dendritic filtering and can thus not be achieved in a straight-forward way by a single-compartmental model. \n\nWe compared the implicit point-neuron model equipped with the new synapse model to the explicit multi-compartmental model in the input-order detection task. The results are illustrated in figure~\\ref{fig:input_order}B. Somatic membrane voltages are shown for the point-neuron model and the multi-compartmental model, after synapse activation in the preferred (left) and null temporal order (right). Because the traces are nearly identical, this result validates our approach and the implementation of the synapse model based on the Green's function solution to the cable theory. \n\n\\subsection{Voltage-gated active currents}\n\nThe most prominent non-linear neuronal response is the action potential. Since it is possible in our synapse model to include any non-linear conductance mechanism, as long as it is spatially restricted to a point-like location, we built a prototype containing the $\\text{Na}^+$ and $\\text{K}^+$ conductances required to generated action potentials. By computing the kernels needed to run the upgraded point-neuron model in the input-order detection task and by adjusting the synaptic weights, we yielded a point-neuron model able to generate a spike in response to the preferred activation pattern, while remaining silent in response to the reversed temporal activation. Note that the active somatic currents shorten the timescale of the neuron's response compared to the passive model. The timscale of the t-axis was scaled accordingly. In order to validate these outcomes, we again built an equivalent multi-compartmental model in \\textsc{neuron} in which we inserted the same $\\text{Na}^+$ and $\\text{K}^+$ conductances into the soma. The multi-compartmental model generated identical results, as shown in Figure~\\ref{fig:input_order}C. Thus, in principle we can include conductance descriptions to obtain hallmark neuronal non-linearities.\n\n\\subsection{Multiple synapse interactions}\n\n\\begin{figure*}[htb!]\n \\centering\n \\includegraphics[width=0.9\\textwidth]{fig2_.pdf}\n \\caption{Comparison between the ``implicit'' (red lines) and ``explicit'' (black lines) model neurons of a pyramidal cell stimulated by Poisson spiketrains. A: The neuron morphology together with the synapse locations. B: The membrane potential traces at the soma, for the input locations shown in panel A (red dots). C: Comparison of the runtime versus the number of input locations. For few input locations, our prototype python code outperforms the \\textsc{neuron} code.}\n \\label{fig:spiketrain}\n\\end{figure*}\n\nWe then checked the correctness of the integrative properties of our implicit point-neuron model by stimulating it with realistic spiketrains at multiple synapses. To that end we added five synapses to a model of a Layer 5 pyramidal neuron equipped with a experimentally reconstructed morphology. The morphology wad retrieved from the NeuroMorpho.org repository \\citep{Ascoli2007} and originally published in \\citep{Wang2002}. We stimulated each synapse with Poisson spike trains of rate \\unit[10]{Hz}. The result is shown in Figure~\\ref{fig:spiketrain}. Again, we compared the implicit model's membrane potential traces to the traces obtained from a multi-compartmental model. The agreement is excellent, as can be seen in Figure~\\ref{fig:spiketrain}B, which also validates our approach when processing inputs from multiple, interacting synapses. \n\n\\subsection{Runtime}\nWe established that the ``implicit'' model neuron equipped with our new synapse model generated near-identical voltage traces as a reference multi-compartmental model. Next we compared the run-time of our implementation to the gold standard in multi-compartmental modeling, the \\textsc{neuron} software \\citep{Carnevale2006}. To this end we simulated a detailed multi-compartmental model (Figure~\\ref{fig:spiketrain}) in \\textsc{neuron} as well as with our approach, for increasing numbers of input locations. For each of those numbers we ran three simulations of 1 second of simulated time at an integration step of 0.1 ms (10 kHz). Because in our approach the execution time is independent of the morphological complexity but rather scales with the number of input locations, it is expected that for a low number of input locations, applying our model will be much faster. As shown in Figure~\\ref{fig:spiketrain} C, for two input locations, our approach runs 20 times faster than \\textsc{neuron}, while at 13 input locations the execution time is equal. Keeping in mind \\textit{i}) that our implementation is done in Python, and \\textit{ii}) that often synapses can be grouped together \\citep{Pissadaki2010} we consider this a good outcome.\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nWe presented a bridge between single-compartment and multi-compartmental neuron models by creating a synapse model that analytically computes the dendritic processing between the synaptic input locations and the soma. We then demonstrated that point-neuron models equipped with this new synapse model could flawlessly perform the input-order detection computation; a neuronal computation exploiting differential dendritic processing \\citep{Agmon-Snir1998}. Thus, the new synapse model can be used to introduce computations to point-neurons that previously only belonged to the realm of multi-compartmental neuron models, with a computational cost that does not depend on the morphological complexity. \n\nThen the question arises when it would be advisable to use our synapse model over the standard tools. Although a quantitative comparison should be treated with care due to the different implementation languages, we still found that our Python-prototype was much faster than the optimized, C++-based \\textsc{neuron}-simulation when the number of input locations was low. This, together with the fact that the computational cost of our model does not depend on morphological complexity, then defines the use case for our model. In scenarios where the number of input locations is low, as is the case in some (invertebrate) cells \\citep{Bullock1965} and as in many \\emph{in-silico} scenarios, only few Volterra equations have to be integrated. There our model represents considerable computational advantage. This arguments also holds when more complex neuron types are considered: while cortical neurons receive often as many as 10000 synapses, many of those can be grouped together. To a good approximation, small dendritic branches act as single units, both in terms of short-term input integration \\citep{Poirazi2003, London2005} as in terms of long-term plasticity related processes \\citep{Govindarajan2011}. Thus, one could group all synapses in a small branch together and then compute the Green's function for that group of synapses as a hole. Such a grouping would drastically reduce the number of Volterra equations to be integrated and hence enhance performance accordingly.\n\n\nWe assumed that the PSP waveform is transformed only in a passive manner on its way to the soma. In reality, this might sound like a drastic simplification as non-linearity is often cited as a hallmark of neuronal computation, not in the least to generate output spikes. How can we evaluate our synapse model in the light of non-linear computations?\n\nNon-linearities in neural response can occur in two ways. First, at the synapse level a non-linear response can be generated principally through the recruitment of NMDA receptors during repetitive synaptic activation \\citep{Branco2010}. As we assume the evolution in time of the synaptic conductance to be of a known shape, we could -in principle- also mimic a non-linear synaptic conductance by using a more specific description of the synaptic conductance evolution.\n\nSecond, non-linearities can arise from voltage-gated conductances in neuronal membranes, that are often distributed non-uniformly along the dendrite \\citep{Larkum1999, Angelo2007, Mathews2010}. The distributed nature of voltage-gated conductances leads to the view that dendritic processing is non-linear, and shaped by these conductances and their spatial distributions. Recent work actually challenges this view as it is known that in some behavioral regimes, dendrites act linearly \\citep{Ulrich2002, Schoen2012}. Since our Green's function approach relies only on the assumption of linearity, it is not intrinsically restricted to passive dendrites. Ion channels distributed along a dendrite can be linearized \\citep{Mauro1970}, and thus yield a quasi-active cable \\citep{Koch1998}. We anticipate that such a linearization procedure can be plugged into our synapse model, so that the linear (but active) properties of the membrane are captured in the Green's function, yielding accurate and efficient simulations of dendrites that reside in their linear regime. Also, in some cases the actual distribution of voltage-gated conductances along the dendrite does not seem to have any effect as long as the time constant for activation is slower than the spread of voltage itself, which makes the actual location of the voltage-gated conductance irrelevant \\citep{Angelo2007}. Thus, in those cases were the spread of voltage is faster than the activation of the conductance, dendrites can act in a passive way, as long as the appropriate non-linearity is introduced at one or a few point-like locations. This can be introduced easily in our synapse model (see Figure~\\ref{fig:input_order}C, with the soma as point-like location with active currents).\n\nWhile dealing with neuronal non-linearities the focus is often on supra-linear responses to inputs, despite the fact that sub-linear responses are also intrinsically non-linear. Moreover, recently it has been shown both in theory and experiment that sub-linear response are used by neurons \\citep{Vervaeke2012,Abrahamsson2012}. Even in passive dendrites, sub-linear responses can be generated when the dendrite locally saturates: due to high input resistance the local voltage response to an input can reach the reversal potential of the membrane. At that moment the driving force disappears and a sub-linear response is generated to inputs. This sort of sub-linear response can be generated in conductance-based models with realistic morphologies. Because we implicitly model dendritic morphology, our synapse model is capable of generating these sub-linear responses.\n\nIn conclusion, we presented a new synapse model that computes the PSP waveforms as if they were subject to dendritic processing without the need to explicitly simulate the dendrites themselves. With this synapse model comes the ability to simulate dendritic processing at a low computational complexity, that allows it's incorporation in large scale models of neural networks. We thus made a first step to bridge single and multi-compartmental modeling.\n\n\\subsubsection*{Acknowledgements}\nWe thank Marc-Oliver Gewaltig for comments on the manuscript and Moritz Deger for helpful discussion. This work was supported by the BrainScaleS EU FET-proactive FP7 grant.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \nA key step in characterising the behaviour of a system is the identification of the relevant degrees of freedom. This is exemplified by Landau's theory of the Fermi liquid \\cite{landau1957,landau1959}, which offers a general description of metallic states in terms of weakly interacting fermions, degrees of freedom obeying the canonical anti-commutation relations\n\\begin{equation}\\label{can_f}\n\\{{\\bm c}_{\\sigma},{\\bm c}^\\dagger_{\\sigma'}\\}=\\delta_{\\sigma\\s'}.\n\\end{equation}\nThese account not just for the long-wavelength phenomenology, but also the electronic band structure, and underlie powerful techniques such as density functional theory which provide a detailed description of a wide variety of materials \\cite{gross2013density}. \n\n\nSome of the most interesting materials have however resisted a description within this framework.\nChief among these are the cuprates, whose puzzling behaviour has provided the central challenge in the field of condensed matter for three decades \\cite{BednorzMuller86,ANDERSON_1987,Keimer_rev}. \nBeyond having some of the highest known superconducting transition temperatures, they exhibit a Mott transition, a pseudogap regime displaying a landscape of intertwined orders \\cite{Keimer_rev,Fradkin_2015}, and a strange metal regime which appears to defy a quasi-particle description \\cite{marginalFL}. Other notable examples include iron pnictides and chalcogenides \\cite{Si_2016}, heavy-fermion compounds \\cite{Gegenwart_2008}, and organic charge-transfer salts \\cite{Powell_2011}.\n\n An important question is whether canonical degrees of freedom, bosons and fermions, are sufficient to account for such behaviour \\cite{ANDERSON_1987}.\nA quantum degree of freedom is specified by the algebra it obeys, which for bosons and fermions has a schematic form $[{\\bm a},{\\bm a}]\\sim1$. \n Here we argue that strongly correlated electrons are instead governed by degrees of freedom which obey a non-canonical Lie algebra, i.e.~an algebra of the form $[{\\bm a},{\\bm a}]\\sim{\\bm a}$. The bracket again reduces the order of operators, but by one, as opposed to two in the canonical case. \n The challenge then is to control the growth of correlations generated by the Hamiltonian through $[{\\bm H},{\\bm a}]$. \n \n \n \nIn one dimension it is well understood how algebraic structures govern the behaviour of correlated electrons, through the formalism of algebraic Bethe ansatz \\cite{Faddeev_2016,EKS,Hbook,HS1}. \nThis is specialised to one dimension however, owing to enhanced symmetries resulting from the constrained geometry \\cite{ZAMOLODCHIKOV1979253}. \nNumerous efforts have been made to exploit Lie algebraic structures in higher dimensions \\cite{Wiegmann_1988,Forster_1989,Chaichian_1991,Kochetov_1996,Coleman_2002,Anderson08,Avella_2011,Ramires}, most specifically through the formalism of Hubbard operators \\cite{Hubbard4,Vedyaev_1984,RuckensteinSR,Izyumov_1990,Ovchinnikov_2004,Izyumov_2005,PhysRevB.70.205112}, but a controlled theoretical framework has so far remained elusive. \n A significant advancement has however recently been made by Shastry \\cite{Shastry_2011,Shastry_2013}, who has developed a perturbative scheme for gaining control over certain non-canonical degrees of freedom, assuming there exists a suitable expansion parameter.\n \n\nIn this work we readdress the question of how to characterise the behaviour of interacting electrons. As the electron has an inherent fermionic nature, we argue that graded Lie algebras provide the natural language for the task. We consider the two such algebras relevant for the electronic degree of freedom: $\\alg{su}(1|1)\\otimes\\alg{su}(1|1)$ and ${\\su(2|2)}$.\nThe first is the algebra of canonical fermions, Eq.~\\eqref{can_f}, which underlies the Fermi liquid description of interacting electrons. The second is closely related to the algebra of Hubbard operators, and we will exploit it to obtain a distinct controlled description of interacting electrons. In particular, we will consider an exceptional central extension of ${\\su(2|2)}$, introduced by Beisert \\cite{Beisert07,Beisert08}, which naturally provides a parameter for the use of Shastry's perturbative scheme. \n\n\n\n\nWe focus on the simplest setting where the novel features of this new controlled description can clearly be seen. \nWe will not attempt to explicitly model any given system, but instead frame our discussion around two overarching themes: the Luttinger sum rule and the Mott metal-insulator transition. \n\nThe Luttinger sum rule states that the volume of the region enclosed by the Fermi surface is directly proportional to the electron density, and independent of interactions.\nIt is proven to be valid for a Fermi liquid in the sense of Landau \\cite{Luttinger_1960}, but there is strong evidence that it is violated in certain strongly correlated systems, such as the cuprates in the pseudogap regime \\cite{Doiron_Leyraud_2007,Badoux_2016}. We explicitly demonstrate that $\\alg{su}(2|2)$ degrees of freedom account for a violation of the Luttinger sum rule, and thus characterise an electronic state of matter which is not a Fermi liquid. \n\nA Mott metal-insulator transition occurs when electronic correlations induce the opening of a gap within an electronic band, signifying a failure of band theory. \nThis phenomenon has played a pivotal role in the study of strongly correlated electrons, but remains incompletely understood \\cite{Imada_1998,LNWrev}. \nIt directly conflicts with the Luttinger sum rule, which implies that a partially filled band has a non-trivial Fermi surface and so is metallic. \nA controlled description consistent with Fermi liquid behaviour is however provided by dynamical mean-field theory \\cite{Metzner_1989,DMFT}, which is exact in the limit of infinite dimensions. \nHere the localisation of electronic quasi-particles is driven by the divergence of their effective mass, as previously described by Brinkman--Rice \\cite{PhysRevB.2.4302}. In contrast, we demonstrate that $\\alg{su}(2|2)$ degrees of freedom result in a splitting in two of the electronic band, each carrying a fraction of the electron's spectral weight. These bands violate the Luttinger sum rule, and a Mott transition naturally occurs when the two bands separate. In the language of the seminal review \\cite{Imada_1998}, this can be understood as a carrier-number-vanishing transition as opposed to a mass-diverging transition. \n We thus offer a controlled framework for characterising Mott transitions in materials, such as the cuprates, where\nthe carrier number vanishes as the transition is approached \\cite{Ando1,Ando2}.\n\n\nThe paper is structured as follows. In Sec.~\\ref{sec:dof} we consider a general lattice model of interacting electrons, and demonstrate that it can be expressed through the generators of either $\\alg{su}(1|1)\\otimes\\alg{su}(1|1)$ or ${\\su(2|2)}$. We interpret these as two ways to characterise the electronic degree of freedom. In Sec.~\\ref{sec:GF} we derive a controlled framework for organising the growth of correlations in the ${\\su(2|2)}$ regime. That is, we obtain a series of successive approximations for the electronic Green's function, which mirrors the self-energy expansion for the canonical regime. In Sec.~\\ref{sec:approx} we examine the leading approximation and find that it captures a splitting of the electronic band. We demonstrate that the Luttinger sum rule is violated, and we observe a Mott transition of carrier-number-vanishing type. Section~\\ref{sec:disc} is a discussion, where we provide further context to our results and offer some perspectives. We conclude in Sec.~\\ref{sec:conc}.\n\n\nThere are five appendices: \\ref{app:su22} reviews the graded Lie algebra ${\\su(2|2)}$, \\ref{app:params} provides explicit expressions for constants and parameters, \\ref{app:canGF} reviews the Green's function analysis for the case of a canonical fermion, \\ref{app:sch} presents a schematic overview of the Green's function analysis for non-canonical ${\\su(2|2)}$, and \\ref{app:2nd} contains the second order contributions to the ${\\su(2|2)}$ self-energy and adaptive spectral weight. \n\n\n\n\n\\section{Electronic degrees of freedom}\\label{sec:dof}\n\nWe wish to address the question of how to characterise behaviour resulting from electronic correlations. Let us consider a lattice with four states per site\n\\begin{equation}\\label{4states}\n\\ket{{\\mathlarger{\\mathlarger{\\circ}}}}=\\ket{0},~~~\\ket{\\downarrow}={\\bm c}^\\dagger_{{\\mathsmaller{\\downarrow}}}\\ket{0},~~~\\ket{\\uparrow}={\\bm c}^\\dagger_{{\\mathsmaller{\\uparrow}}}\\ket{0},~~~\\ket{{\\mathlarger{\\mathlarger{\\bullet}}}}={\\bm c}^\\dagger_{{\\mathsmaller{\\downarrow}}} {\\bm c}^\\dagger_{{\\mathsmaller{\\uparrow}}}\\ket{0},\n\\end{equation} \nwhich provides the Hilbert space for a single-orbital tight-binding model. We disregard disorder and lattice vibrations, focusing solely on electronic interactions. The simpler case of just the two states $\\{\\ket{\\downarrow},\\ket{\\uparrow}\\}$ at each site is relatively well understood in terms of the spin degree of freedom, governed by the Lie algebra $\\alg{su}(2)$ \\cite{Holstein_1940,Dyson_1956}. The complication in the present case is the fermionic nature of the electron, which induces a graded structure between $\\{\\ket{\\downarrow},\\ket{\\uparrow}\\}$ and $\\{\\ket{{\\mathlarger{\\mathlarger{\\circ}}}},\\ket{{\\mathlarger{\\mathlarger{\\bullet}}}}\\}$.\n\n\n\n\nFor concreteness we focus on a Hamiltonian which encompasses both the Hubbard and {$t$-$J$} models,\n\\begin{equation}\\label{eq:ham}\n{\\bm H}=\\sum_{\\braket{i,j}}{\\bm T}_{ij} + J \\sum_{\\braket{i,j}} \\vec{{\\bm s}}_i\\cdot \\vec{{\\bm s}}_j+U \\sum_i {\\bm V}^H_i -2\\mu\\sum_i \\mathlarger{\\bm \\eta}_i^z,\n\\end{equation}\non a $d$-dimensional hypercubic lattice.\nThe Heisenberg spin interaction is expressed through the local spin operators \n\\begin{equation}\\label{eq:spin}\n{\\bm s}^z=\\frac{1}{2}({\\bm n}_{{\\mathsmaller{\\uparrow}}}-{\\bm n}_{{\\mathsmaller{\\downarrow}}}),~~{\\bm s}^+={\\bm c}^\\dagger_{{\\mathsmaller{\\uparrow}}} {\\bm c}_{{\\mathsmaller{\\downarrow}}},~~{\\bm s}^-={\\bm c}^\\dagger_{{\\mathsmaller{\\downarrow}}} {\\bm c}_{{\\mathsmaller{\\uparrow}}},\n\\end{equation}\nwhich obey $ [{\\bm s}^z,{\\bm s}^\\pm]=\\pm{\\bm s}^\\pm$ and $[{\\bm s}^+,{\\bm s}^-]=2{\\bm s}^z$, and generate $\\alg{su}(2)$ rotations between the local spin doublet $\\{\\ket{\\downarrow},\\ket{\\uparrow}\\}$. In addition it is useful to introduce the corresponding local charge operators\n\\begin{equation}\\label{eq:charge}\n \\mathlarger{\\bm \\eta}^z=\\frac{1}{2}({\\bm n}_{{\\mathsmaller{\\uparrow}}}+{\\bm n}_{{\\mathsmaller{\\downarrow}}}-1), ~~\\mathlarger{\\bm \\eta}^+ ={\\bm c}^\\dagger_{{\\mathsmaller{\\downarrow}}} {\\bm c}^\\dagger_{{\\mathsmaller{\\uparrow}}},~~ \\mathlarger{\\bm \\eta}^- = {\\bm c}_{{\\mathsmaller{\\uparrow}}} {\\bm c}_{{\\mathsmaller{\\downarrow}}},\n\\end{equation}\nwhich obey $[\\mathlarger{\\bm \\eta}^z,\\mathlarger{\\bm \\eta}^\\pm]=\\pm\\mathlarger{\\bm \\eta}^\\pm$ and $[\\mathlarger{\\bm \\eta}^+,\\mathlarger{\\bm \\eta}^-]=2\\mathlarger{\\bm \\eta}^z$, and generate $\\alg{su}(2)$ rotations between the local charge doublet $\\{\\ket{{\\mathlarger{\\mathlarger{\\circ}}}},\\ket{{\\mathlarger{\\mathlarger{\\bullet}}}}\\}$.\nWe choose the Hubbard interaction \n\\begin{equation}\n{\\bm V}^H=({\\bm n}_{{\\mathsmaller{\\uparrow}}}-1\/2)({\\bm n}_{{\\mathsmaller{\\downarrow}}}-1\/2),\n\\end{equation}\n to be of a particle-hole symmetric form, and the chemical potential $\\mu$ couples to the charge density.\n\n\nWe take the kinetic term to be of a general correlated form \n\\begin{equation}\\label{eq:CH}\n{\\bm T}_{ij} =t(1-\\lambda) {\\bm T}^{\\circ}_{ij}+t(1+\\lambda){\\bm T}^{\\bullet}_{ij}+t_\\pm ({\\bm T}^+_{ij}+{\\bm T}^-_{ij}),\n\\end{equation}\nwhere the three parameters $t$, $\\lambda$, $t_\\pm$ decouple the terms \n\\begin{equation}\n\\begin{split}\n{\\bm T}^{\\circ}_{ij} &=- \\sum_{\\sigma={\\mathsmaller{\\downarrow}},{\\mathsmaller{\\uparrow}}} \\big({\\bm c}^\\dagger_{i\\sigma} {\\bm c}_{j\\sigma} + {\\bm c}^\\dagger_{j\\sigma} {\\bm c}_{i\\sigma}\\big)\\bar{\\bm n}_{i{\\bar{\\sigma}}}\\bar{\\bm n}_{j{\\bar{\\sigma}}},\\\\\n{\\bm T}^{\\bullet}_{ij} &=- \\sum_{\\sigma={\\mathsmaller{\\downarrow}},{\\mathsmaller{\\uparrow}}} \\big({\\bm c}^\\dagger_{i\\sigma} {\\bm c}_{j\\sigma} + {\\bm c}^\\dagger_{j\\sigma} {\\bm c}_{i\\sigma}\\big){\\bm n}_{i{\\bar{\\sigma}}}{\\bm n}_{j{\\bar{\\sigma}}},\\\\\n{\\bm T}^+_{ij}&=- \\sum_{\\sigma={\\mathsmaller{\\downarrow}},{\\mathsmaller{\\uparrow}}} \\big({\\bm c}^\\dagger_{i\\sigma} {\\bm c}_{j\\sigma}{\\bm n}_{i{\\bar{\\sigma}}}\\bar{\\bm n}_{j{\\bar{\\sigma}}} + {\\bm c}^\\dagger_{j\\sigma} {\\bm c}_{i\\sigma}\\bar{\\bm n}_{i{\\bar{\\sigma}}}{\\bm n}_{j{\\bar{\\sigma}}}\\big),\\\\\n{\\bm T}^-_{ij}&=- \\sum_{\\sigma={\\mathsmaller{\\downarrow}},{\\mathsmaller{\\uparrow}}} \\big({\\bm c}^\\dagger_{i\\sigma} {\\bm c}_{j\\sigma}\\bar{\\bm n}_{i{\\bar{\\sigma}}}{\\bm n}_{j{\\bar{\\sigma}}} + {\\bm c}^\\dagger_{j\\sigma} {\\bm c}_{i\\sigma}{\\bm n}_{i{\\bar{\\sigma}}}\\bar{\\bm n}_{j{\\bar{\\sigma}}}\\big),\n\\end{split}\n\\end{equation}\nwith ${\\bar{\\sigma}}=-\\sigma$ and $\\bar{\\bm n}_{\\sigma}=1-{\\bm n}_{\\sigma}$. This allows for distinct hopping amplitudes depending on the occupancy of the two sites involved by electrons of the opposite spin, see Fig.~\\ref{fig_chop}.\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.55\\columnwidth]{fig_chop.pdf}\n\\caption{\\label{fig_chop}\nCorrelated hopping is when the hopping amplitude depends on how the two sites are occupied by electrons of the opposite spin. Here we illustrate the four possibilities for a hopping spin-up electron (the final two of which are hermitian conjugate). We argue that decoupling these amplitudes from the uncorrelated limit may induce a splitting of the electron.\n}\n\\end{figure}\nCorrelated hopping is an important interaction in, for example, charge-transfer insulators \\cite{Fujimori_1984,Zaanen_1985}, a family of materials which includes the cuprates, when described by an effective single-orbital lattice model that eliminates the low-lying ligand $p$ orbital degree of freedom \\cite{Zhang_1988,Micnas89,MarsiglioHirsch,Sim_n_1993}.\nIn addition, it has recently been shown that correlated hopping can be induced as an effective interaction of ultracold atoms in periodically driven optical lattice setups \\cite{Rapp12,Liberto2014}.\nThe {$t$-$J$} model corresponds to an extreme form of correlated hopping $\\lambda=-1$, $t_\\pm=0$, which disallows hopping processes involving doubly occupied sites. While the Hubbard and {$t$-$J$} models are often regarded as good minimal models for characterising strong correlation effects, we will see that a rich and useful structure arises by considering this more general model which encompasses them both. \n\n\nConventional band theory is founded upon having a kinetic term that is bilinear in $\\Ocd_{\\s}$, a feature that is lost when there is correlated hopping. We can however re-express the kinetic term through the generators of a different algebra as follows\n\\begin{equation}\\label{eq:CHQ}\n{\\bm T}_{ij}=-\\sum_{\\sigma={\\mathsmaller{\\downarrow}},{\\mathsmaller{\\uparrow}}}\\sum_{\\nu={\\circ},{\\bullet}} t_\\nu\n\t\\big({\\bm q}^\\dagger_{i\\sigma\\nu} {\\bm q}_{j\\sigma\\nu} + {\\bm q}^\\dagger_{j\\sigma\\nu} {\\bm q}_{i\\sigma\\nu}\\big),\n\\end{equation}\nwhich is now bilinear in \n\\begin{equation}\\label{Q0s}\n\\begin{split}\n{\\bm q}^\\dagger_{\\sigma{\\circ}} &=\\frac{1+\\kappa}{2}{\\bm c}_{{\\bar{\\sigma}}} - \\kappa {\\bm n}_{\\sigma}{\\bm c}_{{\\bar{\\sigma}}},\\\\\n{\\bm q}^\\dagger_{\\sigma{\\bullet}} &={\\bar{\\sigma}}\\Big(\\frac{1-\\kappa}{2}{\\bm c}^\\dagger_{\\sigma} +\\kappa {\\bm n}_{{\\bar{\\sigma}}}{\\bm c}^\\dagger_{\\sigma}\\Big) ,\n\\end{split}\n\\end{equation}\nwith hopping parameters given by\n\\begin{equation}\\label{eq:CHparam}\nt_\\nu =\n\\Big(\\frac{2\\nu }{1+\\kappa^2}+\\frac{\\lambda}{\\kappa}\\Big)t,\n\\quad \\kappa=\\sqrt{\\frac{t-t_\\pm}{t+t_\\pm}},\n\\end{equation}\nwhere $\\sigma$ takes values $-1,1$ for $\\sigma=\\downarrow,\\uparrow$, and $\\nu$ takes values $-1,1$ for $\\nu={\\mathlarger{\\mathlarger{\\circ}}},{\\mathlarger{\\mathlarger{\\bullet}}}$ respectively. The $\\OQd_{\\s\\nu}$ are the fermionic generators of the graded Lie algebra ${\\su(2|2)}$ \\cite{Beisert07,Beisert08,HS1}, summarised in Appendix \\ref{app:su22}. \nTheir anti-commutation relations are \n\\begin{equation}\\label{eq:su22}\n\\begin{split}\n& \\{ {\\bm q}_{\\sigma\\nu}, {\\bm q}^\\dagger_{\\sigma\\nu}\\}= \\frac{1+\\kappa^2}{4}+\\kappa (\\nu \\mathlarger{\\bm \\eta}^z - \\sigma {\\bm s}^z),\\\\\n& \\{{\\bm q}_{{\\mathsmaller{\\downarrow}}\\nu}, {\\bm q}^\\dagger_{{\\mathsmaller{\\uparrow}}\\nu}\\}=\\kappa{ {\\bm s}}^+, ~~~~~~~ \n\t\\{ {\\bm q}_{\\sigma{\\circ}},{\\bm q}^\\dagger_{\\sigma{\\bullet}}\\}= \\kappa{ \\mathlarger{\\bm \\eta}}^+,\\\\\n& \\{ {\\bm q}_{{\\mathsmaller{\\uparrow}}\\nu}, {\\bm q}^\\dagger_{{\\mathsmaller{\\downarrow}}\\nu}\\}= \\kappa{ {\\bm s}}^-, ~~~~~~~\n\t\\{{\\bm q}_{\\sigma{\\bullet}}, {\\bm q}^\\dagger_{\\sigma{\\circ}}\\}= \\kappa{\\mathlarger{\\bm \\eta}}^-,\\\\\n& \\{ {\\bm q}_{\\sigma\\nu}, {\\bm q}_{\\sigma'\\nu'}\\}=\\{ {\\bm q}^\\dagger_{\\sigma\\nu}, {\\bm q}^\\dagger_{\\sigma'\\nu'}\\}=\\frac{1-\\kappa^2}{4}\\epsilon_{\\sigma' \\sigma}\t\t\\epsilon_{\\nu \\nu'} ,\n\\end{split}\n\\end{equation}\nwith $\\epsilon_{{\\mathsmaller{\\downarrow}}{\\mathsmaller{\\uparrow}}}=-\\epsilon_{{\\mathsmaller{\\uparrow}}{\\mathsmaller{\\downarrow}}}=\\epsilon_{{\\circ}{\\bullet}}=-\\epsilon_{{\\bullet}{\\circ}}=1$. They provide a non-canonical symmetry of the electronic degree of freedom, one that interplays with spin and charge.\nThe inversion of Eqs.~\\eqref{Q0s} takes a linear form\n\\begin{equation}\\label{inv_rels}\n{\\bm c}^\\dagger_{{\\mathsmaller{\\downarrow}}} ={\\bm q}_{{\\mathsmaller{\\uparrow}}{\\circ}}+{\\bm q}^\\dagger_{{\\mathsmaller{\\downarrow}}{\\bullet}},\\quad \n{\\bm c}^\\dagger_{{\\mathsmaller{\\uparrow}}} ={\\bm q}_{{\\mathsmaller{\\downarrow}}{\\circ}}-{\\bm q}^\\dagger_{{\\mathsmaller{\\uparrow}}{\\bullet}},\n\\end{equation}\nand we refer to this as a splitting of the electron, as opposed to `fractionalisation' which takes a product form.\n\n\nWhile graded Lie algebras are not commonly referred to by name in the physics literature, they are frequently used. Indeed, the canonical fermion algebra $\\{{\\bm c},{\\bm c}^\\dagger\\}=1$ is the graded Lie algebra $\\alg{su}(1|1)$. This is extended to $\\alg{u}(1|1)$ by adding ${\\bm n} = {\\bm c}^\\dagger {\\bm c}$, obeying $[{\\bm n} ,{\\bm c}^\\dagger]={\\bm c}^\\dagger$, $[{\\bm n} ,{\\bm c}]=-{\\bm c}$. The canonical algebra of Eq.~\\eqref{can_f} is $\\alg{su}(1|1)\\otimes\\alg{su}(1|1)$. This offers one way to characterise the electronic degree of freedom, which can be viewed as grouping the four electronic states as\n\\begin{equation}\n\\{ \\ket{{\\mathlarger{\\mathlarger{\\circ}}}};\\ket{\\downarrow}\\}\\otimes \\{ \\ket{{\\mathlarger{\\mathlarger{\\circ}}}};\\ket{\\uparrow}\\}.\n\\end{equation}\nThis canonical algebra underlies the Fermi liquid description of correlated matter.\n\n\nThe graded Lie algebra ${\\su(2|2)}$ offers an alternative way to characterise the electronic degree of freedom. Here it is useful to view the four states grouped as\n\\begin{equation}\n\\{ \\ket{\\downarrow},\\ket{\\uparrow}; \\ket{{\\mathlarger{\\mathlarger{\\circ}}}},\\ket{{\\mathlarger{\\mathlarger{\\bullet}}}}\\}.\n\\end{equation}\nThe algebra contains $\\alg{su}(2)$ spin generators $\\vec{{\\bm S}}$ acting on the first pair, $\\alg{su}(2)$ charge generators $\\vec{\\mathlarger{{\\bm \\eta}}}$ acting on the second pair, and fermionic generators $\\OQd_{\\s\\nu}$ which act between the two pairs. The anti-commutation relations of the $\\OQd_{\\s\\nu}$ are not canonical, but instead yield the generators $\\vec{{\\bm S}}$ and $\\vec{\\mathlarger{{\\bm \\eta}}}$ through Eqs.~\\eqref{eq:su22}.\nThe algebra can be extended to ${\\alg{u}(2|2)}$ by adding \n${\\bm \\theta}=\\kappa {\\bm V}^H=\\frac{\\kappa}{3}(\\vec{\\mathlarger{\\bm \\eta}}\\cdot\\vec{\\mathlarger{\\bm \\eta}}-\\vec{{\\bm s}}\\cdot\\vec{{\\bm s}})$, \nwhich obeys\n\\begin{equation}\\label{VHQ0}\n\\begin{split}\n\\lbrack {\\bm \\theta}, {\\bm q}^\\dagger_{\\sigma\\nu} \\rbrack &= \\frac{1+\\kappa^2}{4 } {\\bm q}^\\dagger_{\\sigma\\nu} +\\frac{1-\\kappa^2}{4 } \\epsilon_{\\sigma\\s'}\\epsilon_{\\nu\\nu'} {\\bm q}_{\\sigma'\\nu'},\\\\\n\\lbrack {\\bm \\theta}, {\\bm q}_{\\sigma\\nu} \\rbrack &= -\\frac{1+\\kappa^2}{4 } {\\bm q}_{\\sigma\\nu} -\\frac{1-\\kappa^2}{4 } \\epsilon_{\\sigma\\s'}\\epsilon_{\\nu\\nu'} {\\bm q}^\\dagger_{\\sigma'\\nu'},\n\\end{split}\n\\end{equation}\nand commutes with the spin and charge generators. This linear action of ${\\bm \\theta}$ has the consequence that the parameter $U$ plays a role akin to an additional chemical potential for the $\\OQd_{\\s\\nu}$ degrees of freedom, controlling their splitting. \nFor $\\kappa=1$, the algebra ${\\alg{u}(2|2)}$ is closely related to the Hubbard algebra \\cite{Hubbard4}, see Appendix~\\ref{app:su22}. The appearance of $\\kappa$ in the algebra formally corresponds to an exceptional central extension \\cite{Beisert07,Beisert08}. It has the role of suppressing the spin and charge generators in the anti-commutation relations Eqs.~\\eqref{eq:su22} for small $\\kappa$. We will exploit this to gain perturbative control over the growth of correlations. As $\\kappa\\to0$ the $\\OQd_{\\s\\nu}$ collapse pairwise onto the $\\Ocd_{\\s}$, the anti-commutation relations reduce to canonical relations of Eq.~\\eqref{can_f}, the kinetic term becomes uncorrelated, and ${\\bm \\theta}$ vanishes.\n\n\nWe thus see there are two possibilities for characterising the electronic degree of freedom: $\\alg{su}(1|1)\\otimes\\alg{su}(1|1)$ and ${\\su(2|2)}$. Both are graded algebras, which inherently take into account the grading of the four states of Eq.~\\eqref{4states}. The graded Lie algebras have been classified \\cite{kac1977lie}, and there do not appear to be other independent possibilities relevant for the single-orbital electronic problem.\n\n\nLet us emphasise that we will not consider to what extent these algebras provide explicit symmetries of a system. Instead we will examine how they govern the underlying degrees of freedom, \ni.e.~how they organise correlations. There is no fine tuning in this approach. \n\nThe canonical degree of freedom governs the Fermi liquid description of electronic matter. In the next two sections we will show \nthat ${\\su(2|2)}$ degrees of freedom underlie a controlled description of an alternative strongly correlated regime.\n\n\n\n\n\n\n\\section{Green's function analysis}\\label{sec:GF}\n\n\n\nIn the previous section we have identified two ways to characterise the electronic degree of freedom. We now demonstrate that they each offer a means to systematically organise the electronic correlations of an interacting system.\n\n\nWe focus our effort on obtaining the electronic Green's function. \nLet us first review how the imaginary-time formalism provides access to the retarded and advanced Green's functions\n\\begin{equation}\\label{GRA}\n\\begin{split}\nG^{\\mathrm{ret}}_{ij\\sigma}(t) &= - i \\Theta(t) \\braket{ \\{{\\bm c}_{i\\sigma}(t), {\\bm c}^\\dagger_{j\\sigma}(0)\\} } ,\\\\\nG^{\\mathrm{adv}}_{ij\\sigma}(t) &= i \\Theta(-t) \\braket{ \\{{\\bm c}_{i\\sigma}(t), {\\bm c}^\\dagger_{j\\sigma}(0)\\} },\n\\end{split}\n\\end{equation}\nwith $ \\Theta$ the Heaviside function.\nWe start with the imaginary-time thermal Green's function \n\\begin{equation}\\label{thGFcan}\n\\begin{split}\n{\\mathcal G_{ij\\sigma}}(\\tau)= &- \\braket{{\\bm c}_{i\\sigma}(\\tau) {\\bm c}^\\dagger_{j\\sigma}(0)}\\\\\n=&-\\frac{1}{\\mathcal Z}\\Tr \\Big(e^{-\\beta {\\bm H}}\\mathcal T\\big[{\\bm c}_{i\\sigma}(\\tau) {\\bm c}^\\dagger_{j\\sigma}(0)\\big]\\Big),\n\\end{split}\n\\end{equation}\nwhere $\\mathcal Z=\\Tr e^{-\\beta {\\bm H}}$, $\\beta$ is inverse temperature, ${\\bm a}(\\tau)= e^{\\tau{\\bm H}}{\\bm a} e^{-\\tau{\\bm H}}$, and $\\mathcal T$ is the $\\tau$-ordering operator which is antisymmetric under interchange of fermionic operators\n\\begin{equation}\n\\mathcal T\\big[{\\bm c}_{i\\sigma}(\\tau) {\\bm c}^\\dagger_{j\\sigma}(0)\\big] = \\Theta(\\tau) {\\bm c}_{i\\sigma}(\\tau) {\\bm c}^\\dagger_{j\\sigma}(0) - \\Theta(-\\tau) {\\bm c}^\\dagger_{j\\sigma}(0) {\\bm c}_{i\\sigma}(\\tau).\n\\end{equation}\nTaking the $\\tau$-derivative yields the equation of motion\n\\begin{equation}\\label{elEoM}\n\\partial_{\\tau} \\mathcal G_{ij\\sigma}(\\tau) = - \\delta(\\tau)\\delta_{ij}-\\braket{ [{\\bm H},{\\bm c}_{i\\sigma}(\\tau)] {\\bm c}^\\dagger_{j\\sigma}(0)}.\n\\end{equation}\nThe advantage over the real time equation of motion is the anti-periodic boundary condition \n$\\mathcal G_{ij\\sigma}(\\beta) = - \\mathcal G_{ij\\sigma}(0)$, which follows from the cyclicity of the trace and antisymmetry of $\\mathcal T$. The Fourier transform\n\\begin{equation}\\label{FT}\n\\mathcal G_{p\\sigma}(i\\omega_n) = \\frac{1}{\\mathcal V}\\sum_{i,j}\\int_0^{\\beta} d\\tau e^{\\mathrm i \\omega_n\\tau-\\mathrm i p(i-j)} \\mathcal G_{ij\\sigma}(\\tau),\n\\end{equation}\nis then defined at the Matsubara frequencies $\\omega_n=(2n+1)\\frac{\\pi}{\\beta}$, with $n\\in\\mathbb Z$, and $\\mathcal V$ is the total number of lattice sites. We define $G_{p\\sigma}(\\omega)$ by analytically continuing $\\mathcal G_{p\\sigma}(\\omega)$ to all non-real $\\omega$, provided it satisfies the causality condition that it has no singularities in this region. The retarded and advanced Green's functions are then obtained as\n\\begin{equation}\nG^{\\mathrm{ret}}_{p\\sigma}(\\omega)=G_{p\\sigma}(\\omega+\\mathrm i 0^+),\\quad G^{\\mathrm{adv}}_{p\\sigma}(\\omega)=G_{p\\sigma}(\\omega-\\mathrm i 0^+).\n\\end{equation}\n\n\nIt appears that the challenge of computing the Green's function revolves around solving the equation of motion, Eq.~\\eqref{elEoM}. For example if ${\\bm H}$ is bilinear in $\\Ocd_{\\s}$, say \n ${\\bm H}=- \\sum_{i,j,\\sigma} t_{ij} {\\bm c}^\\dagger_{i\\sigma} {\\bm c}_{j\\sigma}- \\mu\\sum_{i,\\sigma} {\\bm n}_{i\\sigma}$, then the equation of motion takes the form\n\\begin{equation}\\label{GF0can}\n\\begin{split}\n\\sum_k \\Big[ \\delta_{ik} \\big( &-\\partial_\\tau + \\mu\\big)+ t_{ik} \\Big] \\mathcal G_{kj\\sigma}(\\tau) = \\delta(\\tau)\\delta_{ij},\n\\end{split}\n\\end{equation}\nwhich upon Fourier transformation becomes \n\\begin{equation}\n(i\\omega_n+\\mu-\\varepsilon_p)\\mathcal G_{p\\sigma}(i\\omega_n) = 1,\n\\end{equation}\n with dispersion relation $\\varepsilon_p = -\\frac{1}{\\mathcal V}\\sum_{i,j} t_{ij} e^{\\mathrm i p(i-j)}$. Inverting, and analytically continuing $\\mathcal G_{p\\sigma}(\\omega)$ to all non-real $\\omega$, results in the non-interacting Green's function\n\\begin{equation}\nG_{p\\sigma}(\\omega) = \\frac{1}{\\omega+\\mu-\\varepsilon_p}.\n\\end{equation}\nThe Hamiltonian of Eq.~\\eqref{eq:ham} is not bilinear in $\\Ocd_{\\s}$ however. It contains both biquadratic and bicubic terms, and these induce correlations in the system. \n\n\nOne way to proceed is to investigate how the growth of correlations is controlled by Eq.~\\eqref{elEoM}, with a perturbative treatment of the interactions. This leads to the canonical description of correlated electrons which underlies the Fermi liquid \\cite{Abrikosov,kadanoff1962quantum}. We review this in Appendix~\\ref{app:canGF} for the case of spinless fermions. Our subsequent analysis parallels the discussion there, and the reader may find it useful to contrast the two. \n\n\nWe now however take an alternative route, and consider the Green's functions of the ${\\su(2|2)}$ degrees of freedom, e.g.~$\\braket{{\\bm q}_{i\\sigma\\nu}(\\tau) {\\bm q}^\\dagger_{j\\sigma'\\nu'}(0)}$. We will use their equation of motion to \ngain control of correlations, employing the Green's function factorisation technique recently pioneered by Shastry \\cite{Shastry_2011,Shastry_2013}. As the splitting of Eqs.~\\eqref{inv_rels} is linear, the electronic Green's functions $\\mathcal G_{ij\\sigma}(\\tau)$ are immediately reobtained through linear combinations of the ${\\su(2|2)}$ Green's functions. In this way we gain access to a regime of strongly correlated behaviour. \n\n\nWe will continue our analysis in an explicit manner. While this obscures the presentation to a certain extent, it has the benefit of avoiding ambiguity. We complement this with Appendix~\\ref{app:sch} which contains a schematic summary of the derivation.\n\n\n\nIt is useful to introduce some simplifying notations. We collect the fermionic generators as\n\\begin{equation}\n{\\bm \\psi}_i^\\alpha = \\left(\\begin{array}{cccccccc}\n{\\bm q}^\\dagger_{i{\\mathsmaller{\\uparrow}}{\\circ}}&{\\bm q}_{i{\\mathsmaller{\\downarrow}}{\\bullet}}& {\\bm q}^\\dagger_{i{\\mathsmaller{\\downarrow}}{\\circ}}&{\\bm q}_{i{\\mathsmaller{\\uparrow}}{\\bullet}}&\n {\\bm q}_{i{\\mathsmaller{\\uparrow}}{\\circ}}&{\\bm q}^\\dagger_{i{\\mathsmaller{\\downarrow}}{\\bullet}}&{\\bm q}_{i{\\mathsmaller{\\downarrow}}{\\circ}}&{\\bm q}^\\dagger_{i{\\mathsmaller{\\uparrow}}{\\bullet}}\n \\end{array}\\right),\n \\end{equation}\nwith greek indices, and the bosonic generators as\n\\begin{equation}\n {\\bm \\phi}_i^a = \\left(\\begin{array}{cccccc}\n {\\bm s}_i^z&{\\bm s}_i^-&{\\bm s}_i^+ &\\mathlarger{\\bm \\eta}_i^z&\\mathlarger{\\bm \\eta}_i^-&\\mathlarger{\\bm \\eta}_i^+ \n\\end{array}\\right),\n\\end{equation} \nwith latin indices. The ${\\su(2|2)}$ algebra is then compactly expressed as \n\\begin{equation}\\label{su22alg}\n\\begin{split}\n\\{ {\\bm \\psi}_i^\\alpha,{\\bm \\psi}_j^\\beta\\} & = \\delta_{ij} \\big( f^{\\alpha\\beta}{}_I + f^{\\alpha\\beta}{}_a {\\bm \\phi}_i^a\\big),\\\\\n [ {\\bm \\phi}_i^a,{\\bm \\psi}_j^\\beta] &= \\delta_{ij}f^{a \\beta}{}_\\gamma {\\bm \\psi}_i^\\gamma,\\quad\\\\\n [ {\\bm \\phi}_i^a,{\\bm \\phi}_j^b] &= \\delta_{ij}f^{ab}{}_c {\\bm \\phi}_i^c,\n \\end{split}\n\\end{equation}\n and the extension to ${\\alg{u}(2|2)}$ is given by\n\\begin{equation}\n [ {\\bm \\theta}_i,{\\bm \\psi}_j^\\alpha] = \\delta_{ij}f^{\\Theta \\alpha}{}_\\beta {\\bm \\psi}_i^\\beta,\\quad [ {\\bm \\theta}_i,{\\bm \\phi}_j^a] =0.\n\\end{equation}\nSummation over repeated algebraic indices is implied, and we collect the structure constants $f$ in Appendix~\\ref{app:params}. \n\n\nWe now consider the Hamiltonian \n\\begin{equation}\\label{hamTTR}\n\\begin{split}\n{\\bm H} =& -\\frac{1}{2}\\sum_{i,j} t_{ij,\\alpha\\beta} {\\bm \\psi}_i^\\alpha {\\bm \\psi}_{j}^\\beta \n\t+\\frac{1}{2}\\sum_{i,j} V_{ij,ab} {\\bm \\phi}_i^a {\\bm \\phi}_j^b\\\\\n\t & \\qquad - \\mu_{a} \\sum_i {\\bm \\phi}_i^a+\\tilde{U}\\sum_i {\\bm \\theta}_i,\n\\end{split}\n\\end{equation}\nwith hopping and interaction parameters obeying $t_{ii,\\alpha\\beta}=0$, $t_{ji,\\alpha\\beta}=t_{ij,\\alpha\\beta} $, $t_{ij,\\beta\\alpha}=-t_{ij,\\alpha\\beta} $ and $V_{ii,ab}=0$, $V_{ji,ab}=V_{ij,ab}$, $V_{ij,ba}=V_{ij,ab}$, chemical potentials $\\mu_a=( h~0~0~2\\mu~0~0)$, and $\\tilde{U}=U\/\\kappa$. This model is extremely general, as the sixteen generators $\\{8\\times {\\bm \\psi},6\\times {\\bm \\phi}, 1,{\\bm \\theta}\\}$ provide a complete basis for the local operators at each site.\nThis reflects the wide range of applicability of our approach, though we remind that it is important for the model to have correlated hopping. We include the specific hopping and interaction parameters corresponding to the Hamiltonian of Eq.~\\eqref{eq:ham} in Appendix~\\ref{app:params}. \n\nTo introduce the Green's function of the ${\\bm q}$ it is useful to first set a matrix structure via \n\\begin{equation}\\label{eq:metric}\n{\\bm \\psi}_{i\\alpha} = {\\bm \\psi}_i^\\beta K_{\\beta\\alpha}= \\big({\\bm \\psi}_i^\\alpha\\big)^\\dagger,\n\\end{equation}\ndefining a metric $K$, presented explicitly in Appendix~\\ref{app:params}. \nOur object of study is then the matrix Green's function \n\\begin{equation}\n\\mathcal G_{ij}{}^\\alpha_\\beta (\\tau,\\tau')= -{\\braket{{\\bm \\psi}_{i}^\\alpha(\\tau) {\\bm \\psi}_{j\\beta}(\\tau')}}.\n\\end{equation}\nAs highlighted above, the electronic Green's function is directly obtained from linear combinations of these, via Eqs.~\\eqref{inv_rels},\n\\begin{equation}\\label{GeGQ}\n\\begin{split}\n\\mathcal G_{ij{\\mathsmaller{\\downarrow}}}(\\tau) &= {\\mathcal G_{ij}}^1_1(\\tau)+{\\mathcal G_{ij}}^1_2(\\tau)+{\\mathcal G_{ij}}^2_1(\\tau)+{\\mathcal G_{ij}}^2_2(\\tau),\\\\\n\\mathcal G_{ij{\\mathsmaller{\\uparrow}}}(\\tau) &= {\\mathcal G_{ij}}^3_3(\\tau)-{\\mathcal G_{ij}}^3_4(\\tau)-{\\mathcal G_{ij}}^4_3(\\tau)+{\\mathcal G_{ij}}^4_4(\\tau),\n\\end{split}\n\\end{equation}\nwith $\\mathcal G_{ij}{}^\\alpha_\\beta (\\tau)=\\mathcal G_{ij}{}^\\alpha_\\beta (\\tau,0)$.\nIn addition, as the bosonic generators $\\vec{{\\bm S}}$ and $\\vec{\\mathlarger{{\\bm \\eta}}}$ are quadratic in ${\\bm c}$, see Eqs.~\\eqref{eq:spin} and \\eqref{eq:charge}, we can also use Eqs.~\\eqref{inv_rels} to obtain\n\\begin{equation}\\label{GtoT}\n\\begin{split}\n\\braket{{\\bm \\phi}_i^a(\\tau)} &= \\varphi^a{}^\\alpha_\\beta \\mathcal G_{ii}{}^\\beta_\\alpha (\\tau,\\tau^+),\n\\end{split}\n\\end{equation}\nwith coefficients $\\varphi^a{}^\\alpha_\\beta$ which are independent of $\\kappa$, presented explicitly in Appendix~\\ref{app:params}.\n\n\nAlthough the Hamiltonian is at most bilinear in the generators of ${\\su(2|2)}$, correlations are nevertheless induced as a result of the non-canonical nature of the algebra. \nTo handle these we incorporate sources for the ${\\bm \\phi}$ into the imaginary-time thermal expectation value as follows\n\\begin{equation}\\label{ev_source}\n \\braket{ \\mathcal O(\\tau_1,\\tau_2,\\ldots)} = \\frac{\\Tr \\Big( e^{-\\beta {\\bm H}} \\mathcal T \\big[e^{\\int_0^\\beta d\\tau \\mathcal S(\\tau)} \\mathcal O(\\tau_1,\\tau_2,\\ldots) \\big] \\Big)}{\\Tr \\big( e^{-\\beta H} \\mathcal T [e^{\\int_0^\\beta d\\tau \\mathcal S(\\tau)}] \\big)},\n\\end{equation}\nwith $\\mathcal S(\\tau) = \\sum_i J_{ia}(\\tau) {\\bm \\phi}^a_i(\\tau)$, and we consider all $\\tau$ to take values on the interval $(0,\\beta)$.\nThe source term breaks translational invariance in both time and space, providing a means of organising correlations by trading bosonic correlations for their variations through \n\\begin{equation}\n\\begin{split}\n\\nabla_i^a(\\tau)\\braket{\\mathcal O(\\tau_1,\\tau_2,\\ldots,\\tau_n)}= &\\braket{{\\bm \\phi}^a_i(\\tau)\\mathcal O(\\tau_1,\\tau_2,\\ldots,\\tau_n)}\\\\\n\t& -\\braket{{\\bm \\phi}^a_i(\\tau)}\\braket{\\mathcal O(\\tau_1,\\tau_2,\\ldots,\\tau_n)},\n\\end{split}\n\\end{equation}\nwhere $\\nabla_i^a(\\tau) = \\frac{{\\delta} }{ {\\delta} J_{ia}(\\tau^+)}$ denotes the functional derivative, and $\\tau^+=\\tau+0^+$ incorporates an infinitesimal regulator which ensures a consistent ordering when $\\tau$ is one of the $\\tau_1,\\tau_2,\\ldots,\\tau_n$. At the end of the computation the sources will be set to zero without difficulty, restoring translational invariance. \n\n\n\nAs for the electronic Green's function, there is again the anti-periodic boundary condition \n\\begin{equation}\n\\mathcal G_{ij}{}^\\alpha_\\beta(\\beta,\\tau) = - \\mathcal G_{ij}{}^\\alpha_\\beta(0,\\tau).\n\\end{equation}\nThe equation of motion \n\\begin{equation}\n\\begin{split}\n\\partial_{\\tau} \\mathcal G_{ij}{}^\\alpha_\\beta(\\tau,\\tau') = -\\delta(\\tau-\\tau')\\braket{\\{{\\bm \\psi}_{i}^\\alpha(\\tau), {\\bm \\psi}_{j\\beta}(\\tau)\\}}&\\\\\n\t + \\braket{[\\mathcal S(\\tau), {\\bm \\psi}^\\alpha_i(\\tau)]{\\bm \\psi}_{j\\beta}(\\tau')}&\\\\\n\t - \\braket{ [{\\bm H},{\\bm \\psi}^\\alpha_i(\\tau)]{\\bm \\psi}_{j\\beta}(\\tau') }&,\n\\end{split}\n\\end{equation}\npicks up an additional contribution from the source term, a consequence of the $\\tau$-ordering operator. \nThe first two terms are straightforwardly evaluated from Eqs.~\\eqref{su22alg}\n\\begin{equation}\n\\begin{split}\n\\braket{\\{{\\bm \\psi}_{i}^\\alpha(\\tau), {\\bm \\psi}_{j\\beta}(\\tau)\\}} &=\\delta_{ij}\\big( f^{\\alpha\\gamma}{}_I + f^{\\alpha\\gamma}{}_a \\braket{{\\bm \\phi}_i^a(\\tau)}\n\t\\big)K_{\\gamma\\beta},\\\\\n\\braket{[\\mathcal S(\\tau), {\\bm \\psi}^\\alpha_i(\\tau)]{\\bm \\psi}_{j\\beta}(\\tau')} &= - f^{a\\alpha}{}_\\gamma J_{ia}(\\tau) \\mathcal G_{ij}{}^\\gamma_\\beta(\\tau,\\tau').\n\\end{split}\n\\end{equation}\n\\begin{widetext}\\noindent\nThe commutator in the final term is\n\\begin{equation}\n\\begin{split}\n\\lbrack{\\bm H},{\\bm \\psi}^\\alpha_i\\rbrack =\n\t\\sum_{l}\\big[ f^{\\alpha\\delta}{}_I t_{il,\\delta\\gamma} {\\bm \\psi}_l^\\gamma\n\t+ f^{\\alpha\\delta}{}_a t_{il,\\delta\\gamma} {\\bm \\phi}_i^a {\\bm \\psi}_l^\\gamma\\big]\n\t+ \\sum_{l}f^{a\\alpha}{}_\\gamma V_{il,ab}{\\bm \\phi}_l^b {\\bm \\psi}_i^\\gamma \n\t- \\mu_a f^{a\\alpha}{}_\\gamma {\\bm \\psi}_i^\\gamma + \\tilde{U} f^{\\Theta\\alpha}{}_\\gamma {\\bm \\psi}_i^\\gamma,\n\\end{split}\n\\end{equation}\nand, recasting the bosonic correlations as variations of the sources, we obtain\n\\begin{equation}\n\\begin{split}\n\\braket{\\lbrack{\\bm H},{\\bm \\psi}^\\alpha_i(\\tau)\\rbrack {\\bm \\psi}_{j\\beta}(\\tau')}=& (\\mu_a f^{a\\alpha}{}_\\gamma - \\tilde{U} f^{\\Theta\\alpha}{}_\\gamma) \\mathcal G_{ij}{}^\\gamma_\\beta(\\tau,\\tau') \n\t -\\sum_l f^{a\\alpha}{}_\\gamma V_{il,ab} \\big(\\braket{{\\bm \\phi}_l^b(\\tau)}\n\t\t+ \\nabla_l^b(\\tau) \\big) \\mathcal G_{ij}{}^\\gamma_\\beta(\\tau,\\tau')\\\\\n\t&- \\sum_l f^{\\alpha\\delta}{}_I t_{il,\\delta\\gamma} \\mathcal G_{lj}{}^\\gamma_\\beta(\\tau,\\tau')\n\t- \\sum_l f^{\\alpha\\delta}{}_a t_{il,\\delta\\gamma} \\big(\n\t\t\\braket{{\\bm \\phi}_i^a(\\tau)}+\\nabla_i^a(\\tau)\\big) \\mathcal G_{lj}{}^\\gamma_\\beta(\\tau,\\tau').\n\\end{split}\n\\end{equation} \nCollecting these expressions, the equation of motion takes the form\n\\begin{equation}\\label{GFeqn}\n\\begin{split}\n\\sum_k \\Big[ \n\\delta_{ik}\\Big(-\\delta^\\alpha_\\gamma \\partial_{\\tau} -f^{a\\alpha}{}_\\gamma J_{ia}(\\tau) \n - \\mu_a f^{a\\alpha}{}_\\gamma + \\tilde{U} f^{\\Theta\\alpha}{}_\\gamma + \\sum_l f^{a\\alpha}{}_\\gamma V_{il,ab} \\big(\\braket{{\\bm \\phi}_l^b(\\tau)}\n\t\t + \\nabla_l^b(\\tau) \\big)\\Big) ~~~~~~~~~~~~~~~ &\\\\\n+ f^{\\alpha\\delta}{}_I t_{ik,\\delta\\gamma}\n\t+f^{\\alpha\\delta}{}_a t_{ik,\\delta\\gamma} \\big( \\braket{{\\bm \\phi}_i^a(\\tau)} \n\t + \\nabla_i^a(\\tau) \\big)\\Big] \\mathcal G_{kj}{}^\\gamma_\\beta(\\tau,\\tau')&\\\\\n= \\delta(\\tau-\\tau')\\delta_{ij}\\big(f^{\\alpha\\gamma}{}_I+ f^{\\alpha\\gamma}{}_a\\braket{{\\bm \\phi}^a_i(\\tau)}\\big) K_{\\gamma\\beta}&.\n\\end{split}\n\\end{equation}\n\nWe want to obtain solutions to this equation. Its analogue in the canonical case is Eq.~\\eqref{canGFeqn}, to which it has a very similar structure. The primary complication of the non-canonical degree of freedom is the appearance of $\\braket{{\\bm \\phi}}$ on the right-hand side, which indicates that the spectral weight of the Green's function is dressed by correlations. Here it depends explicitly on $\\mathcal G$ through Eq.~\\eqref{GtoT}. A technique for overcoming this difficulty has been pioneered by Shastry \\cite{Shastry_2011,Shastry_2013}: the trick is to factorise $\\mathcal G$ into its numerator and denominator, and obtain a coupled controlled description of both \\cite{Shastry11_Anatomy}. In practice we write \\footnote{The asymmetry in this factorisation $\\mathcal G=\\mathpzc g\\mathpzc w$ results from considering the equation of motion $\\partial_\\tau \\mathcal G(\\tau,\\tau')$. Alternatively we could consider $\\partial_{\\tau'} \\mathcal G(\\tau,\\tau')$, and then factorise the Green's function as $\\mathcal G=\\mathpzc w\\mathpzc g$.}\n\\begin{equation}\\label{Shansatz}\n\\mathcal G_{ij}{}^\\alpha_\\beta(\\tau,\\tau') = \\sum_l \\int_0^\\beta d\\tau'' \\mathpzc g_{il}{}^\\alpha_\\gamma(\\tau,\\tau'') \\mathpzc w_{lj}{}^\\gamma_\\beta(\\tau'',\\tau').\n\\end{equation}\nThe functional derivative in Eq.~\\eqref{GFeqn} then gives two contributions\n\\begin{equation}\n\\nabla_l(\\tau'') \\mathcal G_{ij}{}^\\alpha_\\beta(\\tau,\\tau') = \\sum_k \\int_0^\\beta d\\tau''' \\Big[ \\Big(\\nabla_l(\\tau'')\\mathpzc g_{ik}{}^\\alpha_\\gamma(\\tau,\\tau''')\\Big) \\mathpzc w_{kj}{}^\\gamma_\\beta(\\tau''',\\tau')+ \\mathpzc g_{ik}{}^\\alpha_\\gamma(\\tau,\\tau''') \\Big(\\nabla_l(\\tau'')\\mathpzc w_{kj}{}^\\gamma_\\beta(\\tau''',\\tau')\\Big)\\Big].\n\\end{equation}\nSubstituting these into Eq.~\\eqref{GFeqn}, and bringing the terms with $\\nabla\\mathpzc w$ to the right-hand side, \npermits a factorisation of the equation of motion. \nSetting\n\\begin{equation}\\label{eq:cW}\n\\begin{split}\n \\mathpzc w_{ij}{}^\\alpha_\\beta(\\tau,\\tau') = \\delta(\\tau-\\tau')\\delta_{ij}\\big(f^{\\alpha\\gamma}{}_I+ f^{\\alpha\\gamma}{}_a\\braket{{\\bm \\phi}^a_i(\\tau)}\\big)K_{\\gamma\\beta} -\\sum_{k,l}\\int_0^\\beta d\\tau'' \\Big( \n &f^{\\alpha\\delta}{}_a t_{il,\\delta\\epsilon} \\mathpzc g_{lk}{}^\\epsilon_\\gamma(\\tau,\\tau'') \\nabla^a_i(\\tau)\\mathpzc w_{kj}{}^\\gamma_\\beta(\\tau'',\\tau') \\\\\n &+f^{a\\alpha}{}_\\delta V_{il,ab} \\mathpzc g_{ik}{}^\\delta_\\gamma(\\tau,\\tau'') \\nabla^b_l(\\tau)\\mathpzc w_{kj}{}^{\\gamma}_\\beta(\\tau'',\\tau') \\Big),\n\\end{split}\n \\end{equation}\n fixes the ratio between the two factors in Eq.~\\eqref{Shansatz},\nwith the remainder satisfying \n\\begin{equation}\\label{eq:gT}\n\\begin{split}\n\\sum_k \\Big[ \n\\delta_{ik}\\Big(-\\delta^\\alpha_\\gamma \\partial_{\\tau} -f^{a\\alpha}{}_\\gamma J_{ia}(\\tau) \n - \\mu_a f^{a\\alpha}{}_\\gamma + \\tilde{U} f^{\\Theta\\alpha}{}_\\gamma + \\sum_l f^{a\\alpha}{}_\\gamma V_{il,ab} \\big(\\braket{{\\bm \\phi}_l^b(\\tau)}\n\t\t + \\nabla_l^b(\\tau) \\big)\\Big)& \\\\\n ~~~~~~~~~~~~~~~~~~~~~~~~~~~ + f^{\\alpha\\delta}{}_I t_{ik,\\delta\\gamma}\n\t+ f^{\\alpha\\delta}{}_a t_{ik,\\delta\\gamma} \\big( \\braket{{\\bm \\phi}_i^a(\\tau)} \n\t + \\nabla_i^a(\\tau) \\big)&\\Big] \\mathpzc g_{kj}{}^\\gamma_\\beta(\\tau,\\tau')\t= \\delta(\\tau-\\tau')\\delta_{ij}\\delta^\\alpha_\\beta.\n\\end{split}\n\\end{equation}\nThese two coupled equations are an exact rewriting of the equation of motion Eq.~\\eqref{GFeqn}.\nWe call $\\mathpzc g$ the canonised Green's function and $\\mathpzc w$ the spectral weight.\n\n\n\nWe proceed by introducing two functionals $\\Sigma[\\mathpzc g,\\mathpzc w]$ and $ \\cW[\\mathpzc g,\\mathpzc w]$ of the full $\\mathpzc g$ and $\\mathpzc w$ as follows \\footnote{Refs.~\\cite{Shastry_2011,Shastry_2013} treats these as functionals of $\\mathpzc g$ only, corresponding to a perturbative expansion of $\\mathpzc w$.}. We \ndefine the self-energy $\\Sigma$ through\n\\begin{equation}\\label{gTI}\n\\mathpzc g^{-1}_{ij}{}^\\alpha_\\beta(\\tau,\\tau') = \\mathpzc g_{0,ij}^{-1}{}^\\alpha_\\beta(\\tau,\\tau') - \\Sigma_{ij}{}^\\alpha_\\beta(\\tau,\\tau'),\n\\end{equation}\nwhere $\\mathpzc g_0$ satisfies\n\\begin{equation}\n \\Big[ \n\\delta_{ik}\\big(-\\delta^\\alpha_\\gamma \\partial_{\\tau} -f^{a\\alpha}{}_\\gamma J_{ia}(\\tau) \n - \\mu_a f^{a\\alpha}{}_\\gamma + \\tilde{U} f^{\\Theta\\alpha}{}_\\gamma\\big) + f^{\\alpha\\delta}{}_I t_{ik,\\delta\\gamma}\n \t\t\\Big] \\mathpzc g_{0,kj}{}^\\gamma_\\beta(\\tau,\\tau')\\\\\n\t= \\delta(\\tau-\\tau')\\delta_{ij}\\delta^\\alpha_\\beta,\n\\end{equation}\nand the adaptive spectral weight $\\cW$ through\n\\begin{equation}\\label{wT}\n\\mathpzc w_{ij}{}^\\alpha_\\beta(\\tau,\\tau') = \\mathpzc w_{0,ij}{}^\\alpha_\\beta(\\tau,\\tau') + \\cW_{ij}{}^\\alpha_\\beta(\\tau,\\tau'),\n\\end{equation}\nwith\n\\begin{equation}\n\\mathpzc w_{0,ij}{}^\\alpha_\\beta(\\tau,\\tau')= \\delta(\\tau-\\tau')\\delta_{ij}f^{\\alpha\\gamma}{}_I K_{\\gamma\\beta}.\n\\end{equation}\nWe obtain a closed equation for $\\Sigma$ by convolving Eq.~\\eqref{eq:gT} on the right with $\\mathpzc g^{-1}$, which gives\n\\begin{equation}\\label{Sg0}\n\\begin{split}\n\\Sigma_{ij}{}^\\alpha_\\beta(\\tau,\\tau') =& - \\delta(\\tau-\\tau')\\Big(\n\tf^{\\alpha\\gamma}{}_a t_{ij,\\gamma\\beta} \\braket{{\\bm \\phi}_i^a(\\tau)} +\n\t\\delta_{ij} \\sum_l f^{a\\alpha}{}_\\beta V_{il,ab} \\braket{{\\bm \\phi}_l^b(\\tau)}\n\t\\Big)\\\\\n&- \\delta(\\tau-\\tau')\\Big(\n\t\\delta_{ij}\\sum_l f^{\\alpha\\delta}{}_a t_{il,\\delta\\epsilon} \\mathpzc g_{li}{}^\\epsilon_\\gamma(\\tau,\\tau^+)f^{a\\gamma}{}_\\beta\n \t+f^{a\\alpha}{}_\\delta V_{ij,ab} \\mathpzc g_{ij}{}^\\delta_\\gamma(\\tau,\\tau^+)f^{b\\gamma}{}_\\beta \n\t\\Big)\\\\\n&-\\sum_{k,l}\\int_0^\\beta d\\tau'' \\Big(\n\tf^{\\alpha\\delta}{}_a t_{il,\\delta\\epsilon} \\mathpzc g_{lk}{}^\\epsilon_\\gamma(\\tau,\\tau'') \\nabla_i^a(\\tau)\\Sigma_{kj}{}^{\\gamma}_\\beta(\\tau'',\\tau')\n\t+f^{a\\alpha}{}_\\delta V_{il,ab} \\mathpzc g_{ik}{}^\\delta_\\gamma(\\tau,\\tau'') \\nabla_l^b(\\tau)\\Sigma_{kj}{}^{\\gamma}_\\beta(\\tau'',\\tau')\n\t\\Big),\n\\end{split}\n\\end{equation}\nupon using $(\\nabla \\mathpzc g)\\mathpzc g^{-1}=-\\mathpzc g\\nabla\\mathpzc g^{-1}= -\\mathpzc g\\nabla\\mathpzc g_0^{-1} + \\mathpzc g\\nabla\\Sigma$, with\n\\begin{equation}\n\\nabla_l^a(\\tau'')\\mathpzc g_{0,ij}^{-1}{}^\\alpha_\\beta(\\tau,\\tau')=-\\delta(\\tau-\\tau')\\delta(\\tau-\\tau''-0^+)\\delta_{ij}\\delta_{il}f^{a\\alpha}{}_\\beta.\n\\end{equation}\nA closed equation for $\\cW$ follows directly from Eq.~\\eqref{eq:cW},\n\\begin{equation}\\label{SW0}\n\\begin{split}\n \\cW_{ij}{}^\\alpha_\\beta(\\tau,\\tau') = \\delta(\\tau-\\tau')\\delta_{ij} f^{\\alpha\\gamma}{}_aK_{\\gamma\\beta} \\braket{{\\bm \\phi}^a_i(\\tau)} -\\sum_{k,l}\\int_0^\\beta d\\tau'' \\Big( \n &f^{\\alpha\\delta}{}_a t_{il,\\delta\\epsilon} \\mathpzc g_{lk}{}^\\epsilon_\\gamma(\\tau,\\tau'') \\nabla^a_i(\\tau)\\cW_{kj}{}^\\gamma_\\beta(\\tau'',\\tau') \\\\\n &+f^{a\\alpha}{}_\\delta V_{il,ab} \\mathpzc g_{ik}{}^\\delta_\\gamma(\\tau,\\tau'') \\nabla^b_l(\\tau)\\cW_{kj}{}^\\gamma_\\beta(\\tau'',\\tau')\\Big).\n\\end{split}\n \\end{equation}\n\nEquations \\eqref{Sg0} and \\eqref{SW0} are exact. We now obtain successive approximate solutions with a perturbative expansion in $\\kappa$. We introduce rescaled parameters $\n\\tilde{f}^{\\alpha\\beta}{}_a= f^{\\alpha\\beta}{}_a\/\\kappa$, $\\tilde{V}_{ij,ab} = V_{ij,ab}\/\\kappa$,\nso that $t_{ij,\\alpha\\beta}$, $\\tilde{V}_{ij,ab}$, $f^{a\\alpha}{}_\\beta$ and $\\tilde{f}^{\\alpha\\beta}{}_a$ are all independent of $\\kappa$, and write $\\Sigma = \\sum_{s=0}^\\infty \\kappa^s [\\Sigma]_s$ and $\\cW = \\sum_{s=0}^\\infty \\kappa^s [\\cW]_s$. The\nleading contributions are\n\\begin{equation}\\label{SW1}\n\\begin{split}\n\\lbrack \\Sigma_{ij}{}^\\alpha_\\beta(\\tau,\\tau') \\rbrack_1= &-\\delta(\\tau-\\tau')\\sum_{k}\\int_0^\\beta d\\tau'' \\Big(\n\t\\tilde{f}^{\\alpha\\gamma}{}_a t_{ij,\\gamma\\beta} \\varphi^a{}^\\rho_\\sigma \\mathpzc g_{ik}{}^\\sigma_\\lambda(\\tau,\\tau'') \\mathpzc w_{ki}{}^\\lambda_\\rho(\\tau'',\\tau^+)\\\\\n\t&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +\\delta_{ij} \\sum_l f^{a\\alpha}{}_\\beta \\tilde{V}_{il,ab} \n\t\t\\varphi^b{}^\\rho_\\sigma \\mathpzc g_{lk}{}^\\sigma_\\lambda(\\tau,\\tau'') \\mathpzc w_{kl}{}^\\lambda_\\rho(\\tau'',\\tau^+)\n\t\\Big),\\\\\n&- \\delta(\\tau-\\tau')\\Big(\n\t\\delta_{ij}\\sum_l \\tilde{f}^{\\alpha\\delta}{}_a t_{il,\\delta\\epsilon} \\mathpzc g_{li}{}^\\epsilon_\\gamma(\\tau,\\tau^+)f^{a\\gamma}{}_\\beta\n \t+f^{a\\alpha}{}_\\delta \\tilde{V}_{ij,ab} \\mathpzc g_{ij}{}^\\delta_\\gamma(\\tau,\\tau^+)f^{b\\gamma}{}_\\beta \n\t\\Big)\\\\\n[ \\cW_{ij}{}^\\alpha_\\beta(\\tau,\\tau') ]_1=& \\delta(\\tau-\\tau')\\delta_{ij}\\sum_{k}\\int_0^\\beta d\\tau'' \n\t \\tilde{f}^{\\alpha\\gamma}{}_a K_{\\gamma\\beta} \\varphi^a{}^\\rho_\\sigma \\mathpzc g_{ik}{}^\\sigma_\\lambda(\\tau,\\tau'') \\mathpzc w_{ki}{}^\\lambda_\\rho(\\tau'',\\tau^+).\n\\end{split}\n\\end{equation}\nHigher order terms are then obtained recursively through\n\\begin{equation}\\label{SgWrec}\n\\begin{split}\n\\lbrack\\Sigma_{ij}{}^\\alpha_\\beta(\\tau,\\tau')\\rbrack_{s+1} & = -\\sum_{k,l}\\int_0^\\beta d\\tau'' \\Big(\n\t\\tilde{f}^{\\alpha\\delta}{}_a t_{il,\\delta\\epsilon} \\mathpzc g_{lk}{}^\\epsilon_\\gamma(\\tau,\\tau'') \\nabla_i^a(\\tau)[\\Sigma_{kj}{}^{\\gamma}_\\beta(\\tau'',\\tau')]_s\n\t+f^{a\\alpha}{}_\\delta \\tilde{V}_{il,ab} \\mathpzc g_{ik}{}^\\delta_\\gamma(\\tau,\\tau'') \\nabla_l^b(\\tau)[\\Sigma_{kj}{}^{\\gamma}_\\beta(\\tau'',\\tau')]_s\n\t\\Big),\\\\\n[\\cW_{ij}{}^\\alpha_\\beta(\\tau,\\tau')]_{s+1} &= -\\sum_{k,l}\\int_0^\\beta d\\tau'' \\Big( \n\t\\tilde{f}^{\\alpha\\delta}{}_a t_{il,\\delta\\epsilon} \\mathpzc g_{lk}{}^\\epsilon_\\gamma(\\tau,\\tau'') \\nabla^a_i(\\tau)[\\cW_{kj}{}^\\gamma_\\beta(\\tau'',\\tau')]_s \t\n\t+f^{a\\alpha}{}_\\delta \\tilde{V}_{il,ab} \\mathpzc g_{ik}{}^\\delta_\\gamma(\\tau,\\tau'') \\nabla^b_l(\\tau)[\\cW_{kj}{}^\\gamma_\\beta(\\tau'',\\tau')]_s\n\t\\Big).\n\\end{split}\n\\end{equation}\nThese depend on the sources only through $\\mathpzc g$ and $\\mathpzc w$, and at each order we need use only the leading contributions from\n\\begin{equation}\n\\nabla_l^a(\\tau'') \\mathpzc g_{ij}{}^\\alpha_\\beta(\\tau,\\tau') = \\mathpzc g_{il}{}^\\alpha_\\gamma(\\tau,\\tau'') f^{a\\gamma}{}_\\delta \\mathpzc g_{lj}{}^\\delta_\\beta(\\tau'',\\tau')+\\mathcal O(\\kappa),\\quad \\nabla_l^a(\\tau'') \\mathpzc w_{ij}{}^\\alpha_\\beta(\\tau,\\tau') = 0+\\mathcal O(\\kappa),\n\\end{equation}\n\n\\newpage\n\\end{widetext} \\noindent\nwhere here we have suppressed the infinitesimal regulator. In this way we can systematically construct the functionals $\\Sigma[\\mathpzc g,\\mathpzc w]$ and $ \\cW[\\mathpzc g,\\mathpzc w]$ to any desired order. We provide the second order contributions explicitly in Appendix~\\ref{app:2nd}. \n\n\n\n\n\nWe have thus succeeded in our goal. We have obtained a series of successive approximations for the Green's function, mirroring the self-energy expansion of the canonical case. Let us summarise. Upon expanding $\\Sigma[\\mathpzc g,\\mathpzc w]$ and $\\cW[\\mathpzc g,\\mathpzc w]$ to some desired order, the zero source limit is straightforwardly taken as $J$ enters only through $\\mathpzc g_0$. Equations~\\eqref{gTI} and \\eqref{wT} then provide a set of coupled self-consistent equations for $\\mathpzc g$ and $\\mathpzc w$. The solutions can be combined to give $\\mathcal G$, and the electronic Green's function is in turn obtained from Eqs.~\\eqref{GeGQ}.\n\nThe simplest approximation is to take $\\mathcal G=\\mathpzc g_0 \\mathpzc w_0$. We will examine this in the following section, and find that it captures an essential feature of ${\\su(2|2)}$ degrees of freedom: a splitting of the electronic dispersion. The next approximation is to take just the first order contributions to the self-energy and adaptive spectral weight from Eqs.~\\eqref{SW1}. This is the analogue of the Hartree-Fock approximation for the canonical case, see Eq.~\\eqref{HF}, and likewise captures static correlations. The effects of collisions can be examined by including the second order contributions of Eqs.~\\eqref{SW2}. \n\n\n\n\n \n\n\n\n\\section{A controlled approximation} \\label{sec:approx}\n\n\nIn the previous section we have derived a systematic framework for characterising interacting electrons with ${\\su(2|2)}$ degrees of freedom. We now take the simplest approximation, $\\mathcal G=\\mathpzc g_0 \\mathpzc w_0$, and investigate the resulting electronic Green's function. The unexpanded $\\mathpzc g_0$ and $\\mathpzc w_0$ contain explicit dependence on $\\kappa$ through the structure constants $f^{\\alpha\\beta}{}_I$ and $f^{\\Theta\\alpha}{}_\\beta$, expressed in Appendix~\\ref{app:params}. That is, we are not setting $\\kappa=0$, but rather are truncating the expansions of $\\Sigma$ and $\\cW$ at the zeroth order.\nThe full dependence on the Hubbard interaction, as well as some of the hopping correlations, enter already here. The affect of the approximation is to suppress all spin and charge correlations. In particular, the Heisenberg spin-exchange interaction does not contribute at this order. \n\n\n\nFirst we obtain the matrix Green's function of the ${\\bm q}$. Setting the sources to zero, and recombining $\\mathpzc g_0$ and $\\mathpzc w_0$, the equation of motion\nin this approximation becomes\n\\begin{widetext}\n\\begin{equation}\n\\begin{split}\n\\sum_k \\Big[ \n\\delta_{ik}\\big(-\\delta^\\alpha_\\gamma \\partial_{\\tau} \n - \\mu_a f^{a\\alpha}{}_\\gamma + \\tilde{U} f^{\\Theta\\alpha}{}_\\gamma \\big) \n+ f^{\\alpha\\delta}{}_I t_{ik,\\delta\\gamma}\n\t\\Big] \\mathcal G_{kj}{}^\\gamma_\\beta(\\tau,\\tau')\n= \\delta(\\tau-\\tau')\\delta_{ij} f^{\\alpha\\gamma}{}_I K_{\\gamma\\beta}.\n\\end{split}\n\\end{equation}\n\\end{widetext}\n\\noindent\nIt is sufficient to restrict the greek indices to run over $\\{1,2,3,4\\}$.\nFourier transforming, performing matrix inversion, and analytically continuing to all non-real $\\omega$, we obtain\n\\begin{equation}\n{G_p}^\\alpha_\\beta(\\omega)= \n \\left(\\begin{array}{cccc}\n {\\mathsf g}^- & {\\mathsf h} & 0 & 0 \\\\\n {\\mathsf h} & {\\mathsf g}^+ & 0 & 0 \\\\\n 0 & 0 & {\\mathsf g}^- & -{\\mathsf h} \\\\\n 0 & 0 & -{\\mathsf h} & {\\mathsf g}^+ \\\\\n \\end{array}\\right),\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{split}\n{\\mathsf g}^\\pm&=\\frac{(1+\\kappa^2)(\\omega+\\mu) -\\frac{2\\kappa ^2 \\varepsilon_p}{1+\\kappa^2} \\pm\\kappa^2 (\\tilde{U}+\\tilde{\\lambda} \\varepsilon_p)}{4\\big(\\omega+\\mu -\\frac{\\varepsilon_p}{1+\\kappa^2}\\big) \\big(\\omega+\\mu -\\frac{\\kappa ^2\\varepsilon_p}{1+\\kappa^2} \\big)-\\kappa^2(\\tilde{U}+\\tilde{\\lambda} \\varepsilon_p)^2},\\\\\n{\\mathsf h}&=\\frac{(1-\\kappa ^2) (\\omega+\\mu )}{4\\big(\\omega+\\mu - \\frac{\\varepsilon_p}{1+\\kappa^2} \\big) \\big(\\omega+\\mu - \\frac{\\kappa^2 \\varepsilon_p}{1+\\kappa^2} \\big)-\\kappa^2 (\\tilde{U}+\\tilde{\\lambda} \\varepsilon_p)^2},\n\\end{split}\n\\end{equation}\nwith non-interacting dispersion $\\varepsilon_p =- \\frac{t}{\\mathcal V} \\sum_{\\braket{i,j}} e^{\\mathrm i p(i-j)}$, and $\\tilde{\\lambda}=\\lambda\/\\kappa$. \n \n\n\nThe electronic Green's function can now be immediately obtained via Eqs.~\\eqref{GeGQ}, yielding\n\\begin{equation}\\label{eqG}\nG_{p\\sigma}(\\omega) = \\frac{1}{\\omega +\\mu - \\frac{\\varepsilon_p}{1+\\kappa^2} - \\frac{\\kappa^2}{4} \\frac{(\\tilde{U}+\\tilde{\\lambda} \\varepsilon_p)^2}{\\omega +\\mu -\\frac{\\kappa^2 \\varepsilon_p}{1+\\kappa^2}}}.\n\\end{equation}\nWe choose $t$, $\\kappa$, $\\tilde{U}$ and $\\tilde{\\lambda}$ to parametrise the model, and ascribe the following roles: $t$ controls the strength of dispersion, $\\kappa$ controls the strength of correlations, $\\tilde{U}$ controls the band splitting, and $\\tilde{\\lambda}$ controls asymmetry. They are related to the original parameters of the model by \n\\begin{equation}\\label{chparams}\n\\kappa= \\sqrt{\\frac{t-t_\\pm}{t+t_\\pm}},~~\\tilde{U}=\\frac{U}{\\kappa},~~\\tilde{\\lambda}=\\frac{\\lambda}{\\kappa}.\n\\end{equation}\nWhile it may be tempting to view the term with prefactor $\\frac{\\kappa^2}{4}$ in the denominator as a self-energy, we suggest this would be a misinterpretation of the degrees of freedom. This is clarified by rewriting Eq.~\\eqref{eqG} as \n\\begin{equation}\\label{eqGS}\nG_{p\\sigma}(\\omega) = \\frac{a_{p{\\circ}} }{\\omega+\\mu - \\omega_{p{\\circ}}} + \\frac{a_{p{\\bullet}}}{\\omega+\\mu - \\omega_{p{\\bullet}}},\n\\end{equation}\nwhich makes manifest the splitting of Eq.~\\eqref{inv_rels}. \nThere are now two dispersive bands, which we label with $\\nu={\\mathlarger{\\mathlarger{\\circ}}},{\\mathlarger{\\mathlarger{\\bullet}}}$ as follows\n\\begin{equation}\n\\omega_{p\\nu} = \\frac{\\varepsilon_p}{2} +\\frac{\\nu}{2} \\sqrt{\\Big(\\frac{1-\\kappa^2}{1+\\kappa^2}\\Big)^2\\varepsilon_p^2+\\kappa^2 \\big(\\tilde{U}+\\tilde{\\lambda} \\varepsilon_p\\big)^2},\n\\end{equation}\nand the electronic spectral weight is split between them\n\\begin{equation}\na_{p\\nu} = \\frac{1}{2}+\\frac{\\nu}{2}\\frac{\\frac{1-\\kappa^2}{1+\\kappa^2}\\varepsilon_p}{\\sqrt{\\big(\\frac{1-\\kappa^2}{1+\\kappa^2}\\big)^2\\varepsilon_p^2+\\kappa^2 \\big(\\tilde{U}+\\tilde{\\lambda} \\varepsilon_p\\big)^2}},\n\\end{equation}\nwith $a_{p{\\circ}}+a_{p{\\bullet}}=1$.\nThis is in sharp contrast with the canonical perspective, i.e.~conventional band theory, where the entire electronic spectral weight is locked together in a single band. \n\n\n\n\n\\begin{figure}[!]\n\\centering\n\\includegraphics[width=0.99\\columnwidth]{plot_BS.pdf}\n\\caption{\\label{plot_BS}\nThe splitting of the electronic dispersion, exemplified on a square lattice. \nWe focus on the symmetric case $\\tilde{\\lambda}=0$, and $J$ does not contribute at this order of approximation.\nThe left panel shows the electronic density of states (DOS), $\\sum_\\sigma\\int_{BZ}\\frac{d^2p}{(2\\pi)^2} A_{p\\sigma}(\\omega)$, with the contributions of the ${\\mathlarger{\\mathlarger{\\circ}}}$ (blue, lower) and ${\\mathlarger{\\mathlarger{\\bullet}}}$ (red, upper) bands distinguished. \nThe central panel shows the band structure, an intensity plot of the electronic spectral function (with Lorentzian broadening of $10^{-3}$), along the $\\Gamma$-$X$-$M$-$\\Gamma$ high-symmetry path in the Brillouin zone. The right panel shows the spectral weights $a_{p{\\circ}}$ and $a_{p{\\bullet}}$, which are momentum independent at this order of approximation. \nThe horizontal lines (a)-(d) indicate slices along which the spectral function is plotted in Fig.~\\ref{plot_FS}. Four examples (i)-(iv) of couplings $\\kappa$ and $\\tilde{U}$ are presented: (i) on leaving the non-interacting point the band structure splits with the introduction of weak and flat dispersion near $\\omega=0$, and the 2d Van Hove singularity of the DOS also splits in two. (ii) when $\\tilde{U}$ is increased above $\\tilde{U}_M=\\frac{8}{1+\\kappa^2}$ the two bands separate and a Mott gap opens. (iii) as the strength of correlated hopping is amplified the two bands overlap significantly for $\\tilde{U}<\\tilde{U}_M$. (iv) as $\\kappa$ approaches 1 the two bands decouple, each with half the weight of an electron, and $\\tilde{U}$ behaves as an additional chemical potential that shifts the bands oppositely in $\\omega$.\n}\n\\end{figure}\n\n\n\nA Mott metal-insulator transition takes place when a gap opens between the two bands, i.e.~when $ \\max_p \\omega_{p{\\circ}} = \\min_p \\omega_{p{\\bullet}}$. For $\\tilde{\\lambda}=0$ and $W=2\\max_p \\varepsilon_p=-2\\min_p \\varepsilon_p$, the transition occurs at $\\tilde{U}_M= \\frac{W}{1+\\kappa^2}$. The nature of the transition bears a close resemblance to a band insulator transition, but we emphasise the essential role of electronic correlations is reflected in the splitting of the electronic spectral weight across the gap. \nThis differs from the Brinkmann--Rice description of the Mott transition as the spectral weight $a_{p\\nu}$ does not go continuously to zero as the gap is opened \\cite{PhysRevB.2.4302}, and from the doublon-holon binding description \\cite{doublon-holon} as there is no rearrangement of the degrees of freedom coincident with the Mott transition. \nThe splitting of the electronic band is reminiscent of the foundational work of Hubbard \\cite{Hubbard1,Hubbard3}, though our approach is very different from the large-$U$ perspective taken there.\n\n\nIt is illustrative to plot the electronic spectral function\n\\begin{equation}\n\\begin{split}\nA_{p\\sigma}(\\omega) &= - \\frac{1}{\\pi} \\im G^\\mathrm{ret}_{p\\sigma}(\\omega)\\\\\n&=a_{p{\\circ}} \\delta(\\omega - \\omega_{p{\\circ}}) + a_{p{\\bullet}}\\delta(\\omega - \\omega_{p{\\bullet}}).\n\\end{split}\n\\end{equation}\n We focus on the example of nearest-neighbour hopping on the square lattice, with dispersion relation $\\varepsilon_p=-2 \\cos p_x -2\\cos p_y$, setting $t=1$. In Fig.~\\ref{plot_BS} we plot the frequency dependence of the spectral function along the $\\Gamma$-$X$-$M$-$\\Gamma$ high-symmetry path in the Brillouin zone for a choice of values of $\\kappa$ and $\\tilde{U}$, with $\\tilde{\\lambda}=0$. We also set $\\mu=0$, but as $\\mu$ enters Eq.~\\eqref{eqGS} solely as a shift of $\\omega$, the results for non-zero chemical potential correspond to translating the plots vertically in $\\omega$. The figure helps to visualise how the two bands emerge from a single band in the non-interacting limit, via hybridisation with an additional band carrying vanishing spectral weight. \nAs the interactions are increased the two bands separate, and for $\\tilde{U}>\\tilde{U}_M$ a Mott gap is observed, with the vanishing of the carrier density at the transition evident through the density of states. \n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{plot_FS.pdf}\n\\caption{\\label{plot_FS}\nPlots of the electronic spectral function throughout the 2d Brillouin zone on the slices (a)-(d) indicated in Fig.~\\ref{plot_BS}. The violation of the Luttinger sum rule can be seen by contrasting between (a) and (b), both of which correspond to below half-filling: while less than half the Brillouin zone is enclosed in (a), this is clearly not the case in (b). In (c) and (d) the appearance of two surfaces is in sharp contrast to conventional band theory.\n}\n\\end{figure}\n\nFigure~\\ref{plot_FS} displays cross sections of Fig.~\\ref{plot_BS}, \nshowing the spectral function throughout the 2d Brillouin zone for a choice of $\\mu$, $\\kappa$ and $\\tilde{U}$. \nThis reveals surfaces which violate the Luttinger sum rule \\footnote{The generalised sense of Luttinger's theorem argued in Ref.~\\cite{Dzyaloshinskii_2003} is also violated, with the exception of the particle-hole symmetric case (i.e. here when $\\tilde{\\lambda}=\\mu=0$) for which it has been proven to be true \\cite{PhysRevB.75.104503,PhysRevB.96.085124}.}, clearly evidenced by contrasting Figs.~\\ref{plot_FS}.(a) and \\ref{plot_FS}.(b).\nThis is indeed reasonable as the sum rule relies on the existence of the Luttinger-Ward functional, which is tied to canonical characterisation of interactions \\cite{LuttingerWard_1960,Luttinger_1960}.\nThe violation can be understood as a consequence of the non-canonical nature of the ${\\su(2|2)}$ degrees of freedom, for which the non-trivial spectral weight unties the link between electron density and Luttinger volume.\n\n\nIn summary, we have found that ${\\su(2|2)}$ degrees of freedom govern a regime of behaviour which is fundamentally distinct from a Fermi liquid. \n\n\n\n\n\n\\section{Discussion} \\label{sec:disc}\n\nThe standard way to characterise the behaviour of interacting electrons is through perturbation theory from the non-interacting limit, built upon canonical degrees of freedom \\cite{Abrikosov}. This logic is supported both by Landau's arguments on the robustness of the Fermi liquid \\cite{landau1957,landau1959}, and Shankar's renormalisation group analysis \\cite{Shankar_1994}. The approach has had great success, it underlies our understanding of a wide variety of materials.\n\n\nHere we have identified a distinct way to characterise the electronic degree of freedom, and have demonstrated that it permits a description of a regime of behaviour different from the Fermi liquid. \nWe have cast the electronic problem through the generators of the graded Lie algebra ${\\su(2|2)}$, and have shown how this provides a way to systematically organise the effects of electronic correlations. We have focused on the leading contribution, which reveals a splitting in two of the electronic band, see Fig.~\\ref{plot_BS}. \nThe Luttinger sum rule is violated, and a carrier-number-vanishing Mott metal-insulator transition is exhibited.\nThis reveals a scenario beyond Shankar's analysis, as that is formulated with canonical fermion coherent states which lack the freedom to capture the splitting of Eq.~\\eqref{inv_rels}.\n\n\nThe canonical description is expected to capture metallic behaviour for $\\kappa\\ll U$, i.e.~when any correlations in hopping are weak. We expect that ${\\su(2|2)}$ degrees of freedom may govern behaviour when the parameters $\\tilde{U}$, $\\tilde{\\lambda}$ of Eq.~\\eqref{chparams} are $\\mathcal O(1)$, in particular for $U\\sim \\kappa$ at small $\\kappa$. We represent this schematically in Fig.~\\ref{fig_kU}. There is an argument to be made that the two regimes extend to either side of the point $\\kappa=1$, $U=\\infty$. On the one hand, one can consider departing from the degenerate atomic limit through a continuous unitary transformation organised in powers of $t\/U$ \\cite{Stein_1997}, \nbut this breaks down when $t_\\pm\\sim t\/U$, i.e.~close to $\\kappa=1$.\nOn the other, the parametrisation of Eq.~\\eqref{eq:CHparam} is \ndiscontinuous at $t=t_\\pm=0$, and equivalence with \\eqref{eq:CH} requires that $t$ is not taken to zero first, i.e.~to approach the atomic limit keeping $\\kappa\\sim1$.\nThese singular behaviours can be attributed to the fact that the Hubbard interaction commutes with correlated hopping when $\\kappa=1$. \nSee Ref.~\\cite{Hidden_structure} for a closely related discussion. Let us also comment here that the framework pursued by Shastry sets $U=\\infty$ from the outset, and this plays an important role throughout his analysis \\cite{Shastry_2011,Shastry_2013}. \n\n\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{fig_kU2.pdf}\n\\caption{\\label{fig_kU}\nA schematic depiction of how metallic behaviour may be governed by either canonical or ${\\su(2|2)}$ degrees of freedom in different regions of parameter space.\nHere correlated hopping, controlled by $\\kappa$, and onsite repulsion, controlled by $U$, compete to organise the electronic degree of freedom in distinct ways. While the ${\\su(2|2)}$ regime is restricted to small $U$ for small $\\kappa$, it may extend to large $U$ when $\\kappa$ is $\\mathcal O(1)$. We speculate on the nature of the `transition' between the two regimes towards the end of the Discussion.}\n\\end{figure}\n\n\n\nAn important question is whether there exist materials whose behaviour is governed by ${\\su(2|2)}$?\nWe consider the pseudogap regime found in the cuprates to be an ideal candidate, \nas it is a metallic state with a distinct non-Fermi liquid character \\cite{Timusk_1999,Hashimoto_2014,Fradkin_2015}.\nThe cuprates are charge-transfer insulators \\cite{Fujimori_1984,Zaanen_1985}, and their electronic structure suggests they should admit an effective single-orbital description with the eliminated low-lying ligand $p$ orbital inducing significant correlations in the hopping amplitudes \\cite{Zhang_1988,Micnas89,MarsiglioHirsch,Sim_n_1993}. \nQuantum oscillation experiments indicate a clear violation of the Luttinger sum rule as the pseudogap regime is entered, and furthermore that the carrier density vanishes as the Mott transition is approached, see e.g.~Fig.~4.b of Ref.~\\cite{Badoux_2016}.\n \n \n \nA next step is to go beyond the leading approximation upon which we focused in Sec.~\\ref{sec:approx}. Indeed, this is important to fully characterise the ${\\su(2|2)}$ regime of behaviour. Incorporating the first order contributions to the self-energy and adaptive spectral weight from Eqs.~\\eqref{SW1} will capture the leading contributions of static spin and charge correlations. This is the analogue of the Hartree-Fock approximation for the canonical case. In the context of cuprate physics it would be interesting to investigate if the phenomenological Yang--Rice--Zhang ansatz \\cite{YRZ,YRZ_rev}, which bears a similar form to Eq.~\\eqref{eqG}, can be justified in this way.\nAnother direction is to establish the thermodynamic properties of the regime. \nWe hope such studies will clarify the relevance of ${\\su(2|2)}$ degrees of freedom for characterising the behaviour of strongly correlated materials.\n \n \n The details of the underlying lattice have not played an important role in our analysis. In practice, it is good to have translational invariance as Eqs.~\\eqref{SgWrec} generate a local expansion, which is most conveniently handled in momentum space. \n\n\n\nA special case is when the lattice is a one-dimensional chain. Here the low-energy degrees of freedom are generically spin-charge separated \\cite{HaldaneLL,GiamarchiLL}. Thus we do not expect ${\\su(2|2)}$ degrees of freedom to govern behaviour there, just as canonical fermions do not govern behaviour away from the non-interacting limit \\cite{DL74}. \nInstead, degrees of freedom in one dimension are truly interacting. They can be characterised by their behaviour at integrable limits, where scattering becomes completely elastic but remains non-trivial \\cite{ZAMOLODCHIKOV1979253}, \nallowing for a complete description of the energy spectrum in terms of stable particles \\cite{Bethe31,TakBook}. The classification of such integrable models is understood within the framework of algebraic Bethe ansatz \\cite{Faddeev_2016}. It is noteworthy that the primary integrable models relevant for interacting electrons \\cite{Hbook,EKS,AlcarazBariev,HS1} descend from an R-matrix governed by the exceptional central extension of ${\\su(2|2)}$ symmetry we use here \\cite{Beisert07,HS1}, or a q-deformation thereof \\cite{BeisertKoroteev}. Indeed, the present work was greatly motivated by a combined study of these models \\cite{Hidden_structure}.\n\n\n\nAnother important case is that of infinite dimensions, i.e. when the coordination number of the lattice diverges. While the notion of local degrees of freedom disappears in this limit, dynamical correlations can survive. The frequency dependent electronic Green's function of the Hubbard model can be determined in an exact way here through dynamical mean-field theory \\cite{Metzner_1989,DMFT}. There exist works which incorporate correlated hopping into the formalism \\cite{PhysRevB.67.075101,StanescuKotliar,PEREPELITSKY2013283}, but unfortunately we have not found an explicit study of the effect of correlated hopping on the electronic spectral function. We hope that this may be achieved, as it will provide a complementary controlled perspective on our description of the Mott transition.\n\n\nThe splitting of the electron in Eq.~\\eqref{inv_rels} admits an interpretation in terms of slave particles. Slave bosons $\\Obd_{\\s}$ and fermions $\\Ofd_{\\nu}$ fractionalise the canonical fermion generators as \n\\begin{equation}\\label{cansp}\n{\\bm c}^\\dagger_{\\sigma} ={\\bm b}^\\dagger_{\\sigma} {\\bm f}_{{\\circ}} + \\epsilon_{\\sigma\\s'} {\\bm f}^\\dagger_{{\\bullet}} {\\bm b}_{\\sigma'},\n\\end{equation}\nor alternatively by interchanging $\\Obd_{\\s} \\leftrightarrow \\Ofd_{\\nu}$ \\cite{Barnes_1976,Coleman_1984,Arovas_1988,Yoshioka_1989}.\nThey are often invoked to characterise strongly correlated electrons \\cite{Senthil_2003,LNWrev}. \nThe $\\OQd_{\\s\\nu}$ of ${\\su(2|2)}$ can be viewed as a decoupling of the two contributions to Eq.~\\eqref{cansp} as follows \n\\begin{equation}\n{\\bm q}^\\dagger_{\\sigma\\nu}=\\frac{1+\\kappa}{2}{\\bm f}^\\dagger_{\\nu} {\\bm b}_{{\\bar{\\sigma}}}+\\frac{1-\\kappa}{2}\\epsilon_{{\\bar{\\sigma}}\\sigma'}\\epsilon_{\\nu\\nu'} {\\bm b}^\\dagger_{\\sigma'} {\\bm f}_{\\nu'}.\n\\end{equation}\nDescriptions of correlated matter where deconfined slave particles govern the behaviour require emergent gauge fields \\cite{BaskaranAnderson}. This is not the case with the $\\OQd_{\\s\\nu}$ however, which can be viewed as binding the $\\Obd_{\\s}$ and $\\Ofd_{\\nu}$ to gauge invariant degrees of freedom. Such a binding has been considered from a phenomenological perspective in the context of cuprate physics \\cite{PhysRevLett.76.503,PhysRevB.71.172509}.\n\n\n\n\n\nFinally we offer a more general perspective. We have argued that there exist two distinct regimes of electronic behaviour, governed either by canonical $\\alg{su}(1|1)\\otimes\\alg{su}(1|1)$ or non-canonical ${\\su(2|2)}$ degrees of freedom, which are Fermi liquid and non-Fermi liquid respectively, see Fig.~\\ref{fig_kU}. This raises the question: what happens in between? A phase transition in a conventional sense does not seem possible, as there is no clear notion of order parameter. Instead, each regime may be characterised by a quasi-particle description, where correlations are controlled in perturbative manner by distinct sets of degrees of freedom. While it is possible to connect the two regimes in a controlled way through the non-interacting point, this is highly singular due to the enhanced symmetry there, see Fig.~\\ref{plot_BS}.(i). Instead, we suggest that connecting the two regimes along a generic path requires the breakdown of a quasi-particle description in between. This mirrors a previous proposal in an identical setting in one dimension \\cite{Hidden_structure}. \n\n\nMore specifically, the robustness of the Fermi liquid owes to the fact that the lifetimes of the electronic quasi-particles scale as $(\\omega-\\varepsilon_F)^{-2}$, guaranteeing their stability in the vicinity of the Fermi surface. A `transition' may however occur if correlations shrink the domain over which this scaling is valid to zero. That is, the Fermi liquid may be destroyed by `coherence closing', while the spectrum remains gapless. Such a quantum chaotic regime would permit a rearrangement of the spectrum, allowing in turn for a rearrangement of the electronic degree of freedom.\n\n\nAgain, the cuprates offer a prime candidate for identifying such behaviour in a material setting. They exhibit a `strange metal' regime, lying between the pseudogap and Fermi liquid regimes in their phase diagram, where the featureless linear in temperature resistivity has defied a quasi-particle interpretation \\cite{Martin_1990,Chien_1991,Hussey_2011}. \nOur description of `coherence closing' is consistent with the phenomenological marginal Fermi liquid description of this regime \\cite{marginalFL}. \nEstablishing the necessity for a breakdown of a quasi-particle description in this way would provide a fresh starting point for understanding the anomalous behaviour there. \n\nFrom the perspective of either set of degrees of freedom, the intermediate regime is where correlations grow out of control. Characterising such behaviour requires an alternative framework, not built upon \nunderlying degrees of freedom. An intriguing possibility is holographic duality, which offers a controlled description through the semi-classical regime of a dual gravity theory \\cite{zaanen2015holographic,hartnoll2016holographic}. \n\n\n\n\n\n\n\n\n\\section{Conclusion} \\label{sec:conc}\n\n\nCharacterising the behaviour of interacting electrons is an outstanding challenge, despite many decades of effort. \nHere we have offered a novel approach, based around characterising the electronic degree of freedom.\n\nWe have argued that strong electronic correlations are governed by the graded Lie algebra ${\\su(2|2)}$, as opposed to the canonical fermion algebra which underlies the Fermi liquid. \nWe have derived a controlled description by obtaining a series of successive approximations for the electronic Green's function, mirroring the self-energy expansion of the canonical case. \nFocusing on the leading approximation, we found a splitting in two of the electronic band, a violation of the Luttinger sum rule, and a Mott transition when the split bands separate.\n\n\nMuch work is required to further characterise this non-Fermi liquid regime.\nUltimately, we hope this will lead to efficient techniques for understanding materials whose behaviour is driven by strong electronic correlations.\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\nWe thank Jean-S\\'ebastien Caux, Philippe Corboz, Sergey Frolov, Mark Golden, Enej Ilievski, Jasper van Wezel and Jan Zaanen for useful discussions. Support from the Foundation for Fundamental Research on Matter (FOM) and the Netherlands Organization for Scientific Research (NWO) is gratefully acknowledged.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nInspired by the ground-breaking results coming from the Atacama Large\n(sub)Millimeter Array, and the Jansky Very Large Array, the\nastronomical community is considering a future large area radio array\noptimized to perform imaging of thermal emission down to\nmilliarcsecond scales. Currently designated the `Next Generation Very\nLarge Array,' such an array would entail ten times the effective\ncollecting area of the JVLA and ALMA, operating from 1GHz to 115GHz,\nwith ten times longer baselines (300km) providing mas-resolution, plus\na dense core on km-scales for high surface brightness imaging. Such an\narray bridges the gap between ALMA, a superb submillimeter array, and\nthe future Square Kilometer Array phase 1 (SKA-1), optimized for few\ncentimeter and longer wavelengths. The ngVLA opens unique new\nparameter space in the imaging of thermal emission from cosmic objects\nranging from protoplanetary disks to distant galaxies, as well as\nunprecedented broad band continuum polarimetric imaging of non-thermal\nprocesses. \n\nWe are considering the current VLA site as a possible location, in the\nhigh desert plains of the Southwest USA. At over 2000m elevation, this\nregion provides good observing conditions for the frequencies under\nconsideration, including reasonable phase stability and opacity at 3mm\nover a substantial fraction of the year (see JVLA and ngVLA memos by\nOwen 2015, Clark 2015, Carilli 2015, Butler 2002).\n\nOver the last year, the astronomical community has been considering\npotential science programs that would drive the design of a future\nlarge area facility operating in this wavelength range. These goals\nare described in a series of reports published as part of the ngVLA\nmemo series, and can be found in the ngVLA memo series:\n\n\\begin{center}\n{\\url{http:\/\/library.nrao.edu\/ngvla.shtml}}\n\\end{center}\n\n\\begin{itemize}\n\n\\item Isella et al., 2015, 'Cradle of Life' (ngVLA Memo 6)\n\n\\item Leroy et al., 2015, 'Galaxy Ecosystems' (ngVLA Memo 7)\n\n\\item Casey et al. 2015, 'Galaxy Assembly through Cosmic Time' (ngVLA Memo 8)\n\n\\item Bower et al. 2015, 'Time Domain, Cosmology, Physics' (ngVLA Memo 9)\n\n\\end{itemize}\n\nThe white papers will be expanded with new ideas, and more detailed\nanalyses, as the project progresses (eg. a white paper on\nmagneto-plasma processes on scales from the Sun to clusters of\ngalaxies is currently in preparation). In the coming months, the\nproject will initiate mechanisms to further expand the ngVLA science\nprogram, through continued community leadership.\n\nSuch a facility will have broad impact on many of the paramount\nquestions in modern astronomy. The science working groups are in the\nprocess of identifying a number of key science programs that push the\nrequirements of the telscope. Three exciting programs that have\ncome to the fore thus far, and {\\sl that can only be done with the\nngVLA}, include:\n\n\\begin{itemize}\n\n\\item {\\bf Imaging the 'terrestrial-zone' of planet formation in\nprotoplanetary disks}: Probing dust gaps on 1AU scales at the distance\nof the nearest major star forming regions (Taurus and Ophiucus\ndistance $\\sim$ 130pc) requires baselines 10 times that of the JVLA,\nwith a sensitivity adequate to reach a few K brightness at 1cm\nwavelength and 9mas resolution. Note that these inner regions of\nprotoplanetry disks are optically thick at shorter wavelengths (see\nsection 4.1). The ngVLA will image the gap-structures indicating\nplanet formation on solar-system scales, determine the growth of\ngrains from dust to pebbles to planets, and image accretion onto the\nproto-planets themselves.\n\n\\item {\\bf ISM and star formation physics on scales from GMCs down to\ncloud cores throughout the local super-cluster}: a centrally condensed\nantenna distribution on scales of a few km (perhaps up to 50\\%\nof the total collecting area), is required for wide field, high\nsurface brightness (mK) sensitivity. The ngVLA covers the spectral\nrange richest in the ground state transitions of the most important\nmolecules in astrochemistry and astrobiology, as well as key thermal\nand non-thermal continuum emission process relating to star\nformation. The ngVLA will perform wide field imaging of line and\ncontinuum emission on scales from GMCs (100pc) down to clump\/cores\n(few pc) in galaxies out to the Virgo Cluster.\n\n\\item {\\bf A complete census of the cold molecular gas fueling the\nstar formation history of the Universe back to the first galaxies:}\noctave bandwidth at $\\sim 1$cm wavelength, is required for large\ncosmic volume surveys of low order CO emission from distant galaxies\n(the fundamental tracer of total gas mass), as well as for dense gas\ntracers such as HCN and HCO+. The spatial resolution and sensitivity\nwill also be adequate to image gas dynamics on sub-kpc scales and detect\nmolecular gas masses down to dwarf galaxies.\n\n\\end{itemize}\n\nIn this summary paper, we present a general description of the\nproject, basic design goals for sensitivity and resolution, and the\nunique observational parameter space opened by such a revolutionary\nfacility. We emphasize that the ngVLA is a project under\ndevelopment. While the broad parameter space is reasonably well\ndelineated, there are many issues to explore, ranging from element\ndiameter to the number of frequency bands to the detailed array\nconfiguration, including consideration of VLBI-length baselines (see\nsection 2.2). The science white papers are identifying the primary\nscience use cases that will dictate the ultimate design of the\ntelescope, in concert with the goal of minimization of construction\nand operations costs. The requirements will mature with time, informed\nby ALMA, the JVLA, the imminent JWST and thirty meter-class optical\ntelescopes, and others.\n\n\\section{Telescope specifications}\n\n\\subsection{Basic array}\n\nIn Table 1 we summarize the initial telescope specifications for the\nngVLA. As a first pass, we present numbers for an 18m diameter\nantenna, although the range from 12m to 25m is being considered. A\nkey design goal is good antenna performance at higher frequency,\neg. at least 75\\% efficiency at 30GHz. The nominal frequency range of\n1GHz to 115GHz is also under discussion. The bandwidths quoted are\npredominantly 2:1, or less, although broader bandwidths are being\ninvestigated. Receiver temperatures are based on ALMA and VLA\nexperience. We emphasize that these specifications are a first pass at\ndefining the facility, and that this should be considered an evolving\nstudy.\n\nBrightness sensitivity for an array is critically dependent on the\narray configuration. We are assuming an array of 300 antennas in this\ncurrent configuration. The ngVLA has the competing desires of both\ngood point source sensitivity at full resolution for few hundred km\nbaselines, and good surface brightness sensitivity on scales\napproaching the primary beam size. Clark \\& Brisken (2015) explore\ndifferent array configurations that might provide a reasonable\ncompromise through judicious weighting of the visibilities for a given\napplication (see eg. Lal et al. 2010 for similar studies for the\nSKA). It is important to recognize the fact that for any given\nobservation, from full resolution imaging of small fields, to imaging\nstructure on scales approaching that of the primary beam, some\ncompromise will have to be accepted. \n\nFor the numbers in Table 1, we have used the Clark\/Conway\nconfigurations described in ngVLA memos 2 and 3. Very briefly, this\narray entails a series of concentric 'fat-ring' configurations out to\na maximum baseline of 300km, plus about 20\\% of the area in a compact\ncore in the inner 300m. The configuration will be a primary area for\ninvestigation in the coming years. We have investigated different\nBriggs weighting schemes for specific science applications, and find\nthat the Clark\/Conway configuration provides a reasonable starting\ncompromise for further calculation (see notes to Table 1).\n\n\\begin{table}\n\\footnotesize\n\\caption{Next Generation VLA nominal parameters}\n\\label{tlab}\n\\begin{tabular}{lccccc}\\hline\n~ & 2GHz & 10GHz & 30GHz & 80GHz & 100GHz \\\\ \n\\hline\nField of View FWHM (18m$^a$) arcmin & 29 & 5.9 & 2 & 0.6 & 0.51 \\\\\nAperture Efficiency (\\%) & 65 & 80 & 75 & 40 & 30 \\\\\nA$_{eff}^b$ x$10^4$ m$^2$ & 5.1 & 6.2 & 5.9 & 3.1 & 2.3 \\\\\nT$_{sys}^c$ K & 29 & 34 & 45 & 70 & 80 \\\\\nBandwidth$^d$ GHz & 2 & 8 & 20 & 30 & 30 \\\\\nContinuum rms$^e$ 1hr, $\\mu$Jy bm$^{-1}$ & 0.93 & 0.45 & 0.39 & 0.96 & 1.48 \\\\\nLine rms 1hr, 10 km s$^{-1}$, $\\mu$Jy bm$^{-1}$ & 221 & 70 & 57 & 100 & 130 \\\\\nResolution$^f$ FWHM milliarcsec & 140 & 28 & 9.2 & 3.5 & 2.8 \\\\\nT$_B^g$ rms continuum 1hr K & 14 & 7 & 6 & 15 & 23 \\\\\nLine$^h$ rms 1hr, $1\"$, 10 km s$^{-1}$, $\\mu$Jy bm$^{-1}$ & 340 & 140 & 240 & 860 & -- \\\\\nT$_B^i$ rms line, 1hr, $1\"$, 10 km s$^{-1}$, K & 100 & 1.8 & 0.32 & 0.17 & -- \\\\\n\\hline\n\\vspace{0.1cm}\n\\end{tabular}\n$^a$Under investigation: antenna diameters from 12m to 25m are being considered. \\\\ \n$^b$300 x 18m antennas with given efficiency. \\\\\n$^c$Current performance of JVLA below 50GHz. Above 70GHz we assume the T$_{sys}$ =60K value\nfor ALMA at 86GHz, increased by 15\\% and 25\\%, respectively, due to \nincreased sky contribution at 2200m. \\\\\n$^d$Under investigation. For much wider bandwidths, system temperatures are \nlikely to be larger. \\\\\n$^e$Noise in 1hour for given continuum bandwidth for a Clark\/Conway configuration \n(ngVLA memo 2 and 3) scaled to a maximum baseline of 300km,\nusing Briggs weighting with R=0. Using R=1 decreases the noise by a factor 0.87, \nand using R=-1 increases the noise by a factor 2.5. \\\\\n$^f$Synthesized beam for a Clark\/Conway configuration scaled to a\nmaximum baseline of 300km, using Briggs weighting with R=0. For R=1, the beam size increases\nby a factor 1.36, and for R=-1 the beam size decreases by a factor 0.63. \\\\\n$^g$Continuum brightness temperature corresponding to point source sensitivity (row 6) and resolution of Clark\/Conway configuration, using Briggs weighting with R = 0 (row 8). \\\\\n$^h$Line rms in 1hr, 10 km s$^{-1}$, after tapering to $1\"$ resolution for the Clark\/Conway configuration. \\\\\n$^i$Line brightness temperature rms in 1hr, 10 km s$^{-1}$, after tapering to $1\"$ resolution for the Clark\/Conway configuration. \\\\\n\\end{table}\n\n\\subsection{VLBI implementation}\n\nThe science white papers present a number of compelling VLBI\nastrometric science programs made possible by the increased\nsensitivity of the ngVLA. These include: Local Group cosmology\nthrough measurements of proper motions of nearby galaxies, delineation\nof the full spiral structure of the Milky Way, and measuring the\nmasses of supermassive black holes and H$_0$.\n\nThe exact implementation of interferometry with the\nngVLA on baselines longer than the nominal 300km array remains under\ninvestigation. These astrometric programs require excellent\nsensitivity per baseline, but may not require dense coverage of the UV\nplane, since high dynamic range imaging may not be required.\n\nOne possible implementation would be to use the ngVLA as an\nultra-sensitive, anchoring instrument, in concert with radio \ntelescopes across the globe. Such a model would parallel the\nplanned implementation for submm VLBI, which employs the\nultra-sensitive phased ALMA, plus single dish submm telescopes around\nthe globe, to perform high priority science programs, such as imaging\nthe event horizons of supermassive black holes (Akiyama et\nal. 2015). A second possibility would be to include out-lying stations\nwithin the ngVLA construction plan itself, perhaps comprising up to\n20\\% of the total area out to trans-continental baselines. The cost,\npracticability, and performance of different options for VLBI will be\nstudied in the coming year.\n\n\\section{New Parameter Space}\n\nFigure 1 shows one slice through the parameter space covered by the\nngVLA: resolution versus frequency, along with other existing and\nplanned facilities. The maximum baselines of the ngVLA imply a\nresolution of better than 10mas at 1cm. As we shall see below, coupled\nwith the high sensitivity of the array, this resolution provides a\nunique window into the formation of planets in disks on scales of our\nown Solar system at the distance of the nearest active star forming\nregions, eg. Taurus and Ophiucus.\n\nFigure 2 shows a second slice through parameter space: effective\ncollecting area versus frequency. In this case, we have not included\nmuch higher and lower frequencies, eg. the SKA-1 will extended\nto much lower frequency (below 100MHz, including SKA-Low), while ALMA\nextends up to almost a THz.\n\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[scale=0.32]{angularresfreq_oct20.png}\n\\end{center}\n\\caption{\\footnotesize \\em{Spatial resolution versus frequency set by the \nmaximum baselines of the ngVLA, and\nother existing and planned facilities across a broad range of\nwavelengths. }}\n\\end{figure}\n\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[scale=0.32]{area_freq_log_oct19.png}\n\\end{center}\n\\caption{\\footnotesize \\em{Effective collecting area versus frequency for the ngVLA,\nand other existing or planned facilities operating in a comparable\nfrequency range. We have not included much higher and lower\nfrequencies, eg. the SKA-1 will extended to below 100MHz\n(including SKA-Low), while ALMA extends up to close to a THz. }}\n\\end{figure}\n\nGiven the collecting area and reasonable receiver performance (Table\n1), the ngVLA will achieve sub-$\\mu$Jy sensitivity in the continuum in\n1 hour at 1cm (30GHz). This implies that, at 1cm, the ngVLA will\nobtain 6K brightness temperature sensitivity with 9mas resolution in\njust 1 hour!\n\nWe note that there are other aspects of telescope phase space that are\nrelevant, including field of view and mapping speed, configuration and\nsurface brightness sensitivity, bandwidth, T$_{sys}$, etc... Given\nthe early stage in the design, we have presented the two principle and\nsimplest design goals, namely, maximum spatial resolution and total\neffective collecting area. A deeper consideration of parameter space\nwill depend on the primary science drivers that emerge in the coming\nyears.\n\n\\section{Science Examples}\n\nIn the following, we highlight some of the science that is enabled by\nsuch a revolutionary facility. These three areas are among the high\npriority goals identified by the science working groups, and in\nparticular, these are the goals that have been best quantified to\ndate. We note that the most important science from such a\nrevolutionary facility is difficult to predict, and perhaps the most\nimportant aspect of the science analysis is simply the large volume of\nunique parameter space opened by the ngVLA (Figs 1 and 2).\n\n\\subsection{Imaging terrestrial-zone planet formation}\n\nWith the discovery of thousands of extrasolar planets, and the first\nhigh resolution images of protoplanetary disks with ALMA, the field of\nextrasolar planets and planet formation has gone from rudimentary\nstudies, to a dominant field in astrophysics, in less than a\ndecade. This remarkable progress promises to continue, as ALMA comes\ninto full operation, and with future space missions targetting planet\ndetection, such as the High Definition Space Telescope, for which the\nprimary science goals are direct imaging of terrestrial planets and\nthe search for atmospheric bio-signatures.\n\nThe first high resolution images from ALMA of the protoplanetary disk\nin HL Tau are clearly game-changing (Brogan et al. 2015). The ALMA\nimages show a dust disk out to 100AU radius, with a series of gaps at\nradii ranging from 13 AU to 80AU. These gaps may correspond to\nthe formation zones of planets. Coupled with JVLA imaging at longer\nwavelengths, these HL Tau images usher in a new era in the study of\nplanet formation.\n\nWhile revolutionary, there are limitations to the current\ncapabilities of ALMA and the JVLA in the study of protoplanetary\ndisks. First, for ALMA, the inner 10AU of protoplanetary disks like HL\nTau become optically thick at wavelengths of 3mm and shorter. Second,\nfor the JVLA, the sensitivity and spatial resolution are insufficient\nto image the terrestrial-zone of planet formation at the longer\nwavelengths where the disks become optically thin.\n\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{PPdisk_2.png}\n\\end{center}\n\\caption{\\footnotesize \\em{Models and images of a $\\sim 1$Myr old protoplanetary\ndisk, comparable to HL Tau, at a distance of 130pc. This 'minimum mass\nsolar nebula disk' has a mass of 0.1M$_\\odot$ orbiting a 1 M$_\\odot$ star.\nThe model includes the formation of a Jupiter\nmass planet at 13AU radius, and Saturn at 6AU.\nThe left frame shows the model emission at 100GHz, the center frame\nshows the 25GHz model, and the right shows the ngVLA image for a\n100hour observation at 25GHz with 10mas resolution. The noise \nin the ngVLA image is 0.1$\\mu$Jy, corresponding to 1K at 10mas resolution. }}\n\\end{figure}\n\nThe ngVLA solves both of these problems, through ultra-high sensitivity\nin the 0.3cm to 3cm range, with milliarcsecond resolution. Figure 3\nshows a simulation of the ability of the ngVLA to probe the previously\ninaccessible scales of 1AU to 10AU. This simulation involves an\nHL-Tau like protoplanetary disk, including the formation of a Jupiter\nmass planet at 13AU radius, and Saturn at 6AU. Note that the inner\nring caused by Saturn is optically thick at 3mm. However, this inner\ngap is easily visible at 25GHz, and well imaged by the\nngVLA. Moreover, the ngVLA will have the sensitivity and resolution to\nimage circum-planetary disks, ie. the formation of planets themselves\nvia accretion. In parallel, the ngVLA covers the optimum frequency\nrange to study pre-biotic molecules, including rudimentary amino acids\nsuch as glycine (see Isella et al. 2015 for more details).\n\n{\\sl Next Generation Synergy:} The High Definition Space\nTelescope has made its highest priority goals the direct imaging of\nterrestrial-zone planets, and detection of atmospheric biosignatures.\nThe ngVLA provides a perfect evolutionary compliment to the HDST\ngoals, through unparalleled imaging of terrestrial zone planet\nformation, and the study of pre-biotic molecules.\n\n\\subsection{The dense gas history of the Universe}\n\nUsing deep fields at optical through radio wavelengths, the evolution\nof cosmic star formation and the build up of stellar mass have been\ndetermined in exquisite detail, from the epoch of first light (cosmic\nreionization, $z > 7$), through the peak epoch of cosmic star\nformation ('epoch of galaxy assembly', $z \\sim 1$ to 3), to the\npresent day (Madau \\& Dickinson 2014). However, these studies reveal\nonly one aspect of the baryonic evolution of galaxies, namely, the\nstars. What is currently less well understood, but equally important,\nis the cosmic evolution of the cool, molecular gas out of which stars\nform. Initial in-roads into the study of the cool gas content of\ngalaxies has been made using the JVLA, GBT, Plateau de Bure, and now\nALMA. These initial studies have shown a profound change in the\nbaryonic content of star forming galaxies out to the epoch of galaxy\nassembly: the gas baryon fraction (the gas to stellar mass ratio)\nincreases from less than 10\\% nearby, to unity, or larger, at $z \\sim\n2$ to 3 (Genzel et al. 2015, Carilli \\& Walter 2013). \nThis profound change in galaxy properties with redshift is\nlikely the root-cause of the evolution of the cosmic star formation\nrate.\n\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{dncc_fig2.png}\n\\end{center}\n\\caption{\\footnotesize \\em{\nLeft: A model of the integrated CO 1-0 emission from a\nmassive z=2 galaxy from the cosmological zoom simulations of Narayanan\net al. (2015). The total SFR $= 150$ M$_\\odot$ year$^{-1}$, and the stellar\nmass = $4\\times 10^{11}$ M$_\\odot$. The native resolution (pixel size)\nis 30mas, and the peak brightness temperature is 14K. The fainter\nregions have T$_B \\ge 0.1$K. Right: the ngVLA image of\nthe field assuming a 8 x 5hour synthesis using only antennas within a\n15km radius (about 50\\% of the full array for the Clark\/Conway\nconfiguration, and using Briggs weighting\nwith R=0.5. The rms noise is 5$\\mu$Jy beam$^{-1}$, and the beam size\nis $0.11\"$. One tick mark = $1\"$. The peak surface brightness is 0.18\nmJy beam$^{-1}$. \n}}\n\\end{figure}\n\nHowever, studies of the gas mass in early galaxies, typically using\nthe low order transitions of CO, remain severely sensitivity limited,\nrequiring long observations even for the more massive galaxies.\nThe sensitivity and resolution of the ngVLA opens a new window on the\ngas properties of early galaxies, through efficient large cosmic\nvolume surveys for low order CO emission, and detailed imaging of gas\nin galaxies to sub-kpc scales (see Casey et al. 2015). The ngVLA will\ndetect CO emission from tens to hundreds of galaxies per hour in\nsurveys in the 20GHz to 40GHz range. In parallel, imaging of the gas\ndynamics will allow for an empirical calibration of the CO luminosity\nto gas mass conversion factor at high redshift.\n\nFigure 4 shows a simulation of the CO 1-0 emission from a \nmassive z=2 galaxy from the cosmological zoom simulations of Narayanan\net al. (2015), plus the ngVLA simulated image. The ngVLA reaches an\nrms noise of 5$\\mu$Jy beam$^{-1}$ (over 9MHz bandwidth and 40hours),\nand the beam size is $0.11\"$ = 0.9kpc at z=2, only using antennas\nwithin 15km radius of the array center. The ngVLA can detect the large\nscale gas distribution, including tidal structures, streamers,\nsatellite galaxies, and possible accretion. Note that the rms\nsensitivity of the ngVLA image corresponds to an H$_2$ mass\nlimit of $3.3\\times 10^8$ ($\\alpha$\/4) M$_\\odot$. Further, the ngVLA\nhas the resolution to image the gas dynamics on scales approaching\nGMCs. For comparison, the JVLA in a similar integration time would\nonly detect the brightest two knots at the very center of galaxy,\nwhile emission from the high order transitions imaged by ALMA misses\nthe extended, low excitation, diffuse gas in the system. \n\n{\\sl Next Generation Synergy:} With new facilities such as\nthirty-meter class optical telescopes, the JWST, and ALMA, study of\nthe stars, ionized gas, and dust during the peak epochs of galaxy\nformation, will continue to accelerate. The ngVLA sensitivity and\nresolution in the 0.3cm to 3cm window is the required complement to\nsuch studies, through observation of the cool gas out of which stars\nform throughout the Cosmos.\n\n\\subsection{Ultra-sensitive, wide field imaging}\n\nScience working group 2 (`Galaxy ecosystems'; Leroy etal. 2015)\nemphasized the extraordinary mapping speed of the ngVLA in line and\ncontinuum, for study of the gas and star formation in the nearby\nUniverse. The frequency range of the ngVLA covers, simultaneously,\nmultiple continuum emission mechanisms, from synchrotron, to\nfree-free, to cold (or spinning) dust. These mechanisms are key\ndiagnostics of star formation, cosmic rays, magnetic fields, and other\nimportant ISM properties. This range also covers low order and maser\ntransitions of most astrochemically important molecules, such as CO,\nHCN, HCO$^+$, NH$_3$, H$_2$O, CS...\n \nFigure 5 shows an ngVLA simulation of the thermal free-free emission\nin the 30GHz band from a star forming galaxy at 27Mpc distance, with a\nmoderate star formation rate of 4 M$_\\odot$ year$^{-1}$. The ngVLA\nwill image the free-free emission with a sensitivity adequate to\ndetect an HII region associated with a single O7.5 main sequence star\nat the distance of the Virgo cluster! In general, the combination of\nspectral and spatial resolution will allow for decomposition\nof the myriad spectral lines, and various continuum emission\nmechanisms, on scales down to a few parsecs at the distance of Virgo,\nthereby enabling Local-Group-type science throughout the local\nsupercluster.\n\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{FF.png}\n\\end{center}\n\\caption{\\footnotesize \\em{Left: a model for the thermal free-free\nemission from NGC 5713 at a distance of 27Mpc with \na SFR = 4 M$_\\odot$ year$^{-1}$. The model was estimated from H$\\alpha$ imaging\nat a native resolution of 2$\"$. The peak brightness temperature is \n150mK, and the fainter knots are about 1mK. Right: The ngVLA image\nfor 10hrs integration, with a bandwidth of 20GHz, centered at 30GHz. \nThe rms is 0.1$\\mu$Jy beam$^{-1}$. \nNote that the ngVLA image has been restored with a beam of \n$0.5\"$. }}\n\\end{figure}\n\n\\subsection{Exploring the Time Domain}\n\nThe ngVLA is being designed for optimal exploitation of the time\ndomain. Fast triggered response modes on minute timescales will be\nstandard practice. Commensal searches for ultra-fast transients, such\nas Fast Radio Bursts or SETI signals, will also be incorporated into\nthe design. And monitoring of slow transients, from novae to AGN, will\nbe possible at unprecedented sensitivities, bandwidths, and angular\nresolutions. The 2cm and shorter capabilities will be complimentary to\nthe SKA-1 at longer wavelengths, in particular for the broad band\nphenomena typical of fast and slow transients.\n\nThe broad band coverage and extreme sensitivity of the ngVLA provides\na powerful tool to search for, and characterize, the early time\nemission from processes ranging from gravity wave EM counter-parts to\ntidal disruption events around supermassive black holes as well as\nprobing through the dense interstellar fog in search of Galactic\nCenter pulsars. The system will also provide unique insights into\nvariable radio emission associated with 'exo-space weather,' such as\nstellar winds, flares, and aurorae. Moreover, many transient phenomena\npeak earlier, and brighter, at higher frequencies, and full spectral\ncoverage to high frequency is required for accurate calorimetry. Full\npolarization information will also be available, as a key diagnostic\non the physical emission mechanism and propagation effects. \n\n\\vspace{0.5cm}\n\nWe invite the reader to investigate the science programs in more\ndetail in the working group reports, as well as to participate in the\npublic forums and meetings in the on-going development of the ngVLA\nscience case.\n\n\\section*{Acknowledgments}\n\nThe National Radio Astronomy Observatory is a facility of the National\nScience Foundation operated under cooperative agreement by Associated\nUniversities, Inc.\n\n\\vskip 0.2in\n\n\\noindent{\\sl References}\n\n\\noindent Akiyama, K. et al. 2015, ApJ, 807, 150\n\n\\noindent Bower, G. et al. 2015, {\\sl Next Generation VLA memo. No. 9}\n\n\\noindent Brogan, C. et al. 2015, ApJ, 808, L3\n\n\\noindent Butler, B. 2002, VLA Test Memo 232\n\n\\noindent Carilli, C. 2015, {\\sl Next Generation VLA memo. No. 1}\n\n\\noindent Carilli, C. \\& Walter, F. 2013, ARAA, 51, 105\n\n\\noindent Casey, C. et al. 2015, {\\sl Next Generation VLA memo. No. 8}\n\n\\noindent Clark, B. \\& Brisken, W. 2015, {\\sl Next Generation VLA memo. No. 3}\n\n\\noindent Clark, B. 2015, {\\sl Next Generation VLA memo. No. 2}\n\n\\noindent Genzel, R. et al. 2015, ApJ, 800, 20\n\n\\noindent Isella, A. et al. 2015, {\\sl Next Generation VLA memo. No. 6}\n\n\\noindent Lal, D., Lobanov, A., Jimenez-Monferrer, S. 2011, \nSKA Design Studies Technical Memo 107\n\n\\noindent Madau, P. \\& Dickinson, M. 2014, ARAA, 52, 415\n\n\\noindent Leroy, E. et al. 2015, {\\sl Next Generation VLA memo. No. 7}\n\n\\noindent Narayanan, D. et al. 2015, Nature, 525, 496\n\n\\noindent Owen, F. 2015, {\\sl Next Generation VLA memo. No. 4}\n\n\\end{document}\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction.} \nThe Fourier coefficients $a(F,n)$ of a cusp form $F$ of integral weight $k$ for\nthe group $\\Gamma_0(M)$ are bounded from above by\n$\\sigma_0(n)n^\\frac{k-1}{2}$ if $F$ is a primitive form (also called\nnormalized newform) by the famous Ramanujan-Petersson-Deligne\nbound. For applications one often needs bounds for an arbitrary cusp\nform which is a linear combination of old and new forms. Such bounds\nhave first been given in special cases in \\cite{fomenko1,fomenko2}.\nThe first step for this is the construction of an explicit\northogonal basis for the space $S_k(\\Gamma_0(M), \\chi)$. \nStarting from\nthe usual basis of translates of primitive forms and using the well\nknown fact that translates of\ndifferent primitive forms are pairwise orthogonal, one is left with\nthe task to orthogonalize the translates of the same primitive\nform, in particular, one has to compute their Petersson scalar products.\n\nChoie and Kohnen in \\cite{choie-kohnen} and Iwaniec, Luo, Sarnak in\n\\cite{iwa_luo_sar} cover arbitrary integral weights, square free\nlevel and trivial character, using Rankin $L$-functions for the\ncomputation of the Petersson products of translates of a primitive form. By the same\nmethod, Rouymi \\cite{rouymi} treated prime power level and trivial\ncharacter. His approach was generalized to arbitrary levels and\ntrivial character by Ng Ming Ho in his unpublished master thesis \n\\cite{ngmingho}. Blomer and Mili\\'{c}i\\'{c} in \\cite{blomer_milicic} treat\nMaa\\ss forms and holomorphic modular forms for arbitrary level and\ntrivial character by the same method.\n\nIn this note we investigate the case of arbitrary level and arbitrary\ncharacter with a rather elementary \napproach. \nIn order to compute the Petersson product of two translates of the\nsame primitive form we use the trace operator sending a form of level $M$ to a\nform of level $N$ dividing $M$. Together with the well known fact that the $p$-th Hecke\noperator on forms of level $N$ can be obtained by first translating the argument by a factor\n$p$ and then applying the trace operator from level $Np$ down to level\n$N$ this allows us to express the scalar products quite easily in\nterms of Hecke eigenvalues of the underlying primitive form. The formulas we get and the relations between\nthe Hecke eigenvalues $\\lambda_f(1,p^j)$ of a primitive form $f$ for\nvarying $j$ imply then that each element of the orthogonal basis\nobtained by the Gram-Schmidt procedure involves\nonly very few of the translates of its underlying primitive form. For\nforms of half integral weight our approach works in essentially the\nsame way as far as the computation of the Petersson product of a Hecke\neigenform with its translates is concerned. Since the theory of\nnewforms is in this case completely known only for the Kohnen plus\nspace in square free level, it is however not clear how large the part\nof the space of all cusp forms of a given arbitrary level is that is\ncovered by our result.\n\nWe then use in the integral weight case the orthogonal basis to obtain \nan explicit bound for the Fourier coefficient $a(F,n)$ of an arbitrary\ncusp form $F$ in terms of the Petersson norm $\\langle\nF,F \\rangle$ and the level $M$.\n\nIn applications to the theory of integral quadratic forms it is\nusually possible to compute or at least bound $\\langle F,F \\rangle$ for the cusp form\n$F$ at hand (the difference between a genus theta series and a theta\nseries), so that our result is directly applicable to such problems;\nthis will be worked out separately.\n\nAn estimate for the Fourier coefficients in the half integral weight\ncase could in principle be obtained in the same way as in the integral\nweight case discussed above as long as one has an explicit bound for\nthe Fourier coefficients with square free index of a Hecke\neigenform. Unfortunately most of the known estimates (see\n\\cite[Appendix 2]{blomer_michel_mao} involve\nconstants which are not explicitly known, and we prefer not to discuss\nthis possibility in detail in the present paper.\n\nThis article is an extension of work from the master thesis of the\nsecond named author at Universit\u00e4t des Saarlandes, 2014.\n\nAfter the first version of this article was posted in the matharxiv Ng\nMing Ho sent us his master thesis, from which we also learnt of the\nprevious work of Iwaniec, Luo and Sarnak and of Rouymi. \nWe thank Ng Ming Ho for providing this information to us.\n \\section{Trace operator and scalar products} \nLet $N\\mid M$ be integers and let $\\chi$ be a Dirichlet character modulo $N$; we denote the Dirichlet\ncharacter modulo $M$ induced by it by $\\chi$ as well. We have induced characters on the groups \n$\\Gamma_0(N)$, $\\Gamma_0(M)$ given by\n \\begin{equation*} \n\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\longmapsto \\chi(d) \n \\end{equation*}\nas usual and denote these again by $\\chi$.\n\nFor an integer $k$ we denote by\n$M_k(\\Gamma_0(N),\\chi),S_k(\\Gamma_0(N),\\chi)$ the spaces of modular\nforms respectively cusp forms of weight $k$ and character $\\chi$ for\nthe group $\\Gamma_0(N)$. On $S_k(\\Gamma_0(N),\\chi)$ we consider the\nPetersson inner product given by\n\\begin{equation*}\n\\langle f,g\\rangle:=\\langle f,g\\rangle_{\\Gamma_0(N)}:=\\frac{1}{(SL_2({\\mathbb Z}):\\Gamma_0(N))}\\int_{\\mathcal F}f(x+iy)\\overline{g(x+iy)}y^{k-2}dxdy, \n\\end{equation*}\nwhere ${\\mathcal F}$ is a fundamental domain for the action of\n$\\Gamma_0(N)$ on the upper half plane $H\\subseteq {\\mathbb C}$ by fractional\nlinear transformations. The normalization chosen implies that for\n$N\\mid M$ and $f,g \\in S_k(\\Gamma_0(N),\\chi)\\subseteq\nS_k(\\Gamma_0(M),\\chi)$ we have $\\langle f,g\n\\rangle_{\\Gamma_0(N)}=\\langle f,g \\rangle_{\\Gamma_0(M)}$.\n\nFor $\\gamma=\\bigl(\n\\begin{smallmatrix}\n a&b\\\\c&d\n\\end{smallmatrix}\\bigr)\n\\in GL_2({\\mathbb R})$ with $\\det(\\gamma)>0$ we write as usual\n$f\\vert_k\\gamma(z)=\\det(\\gamma)^{k\/2}(cz+d)^{-k}f(\\frac{az+b}{cz+d})$.\n \n\\medskip\nWe define trace operators as in \\cite{kume,bfsp}:\n \\newpage\n \\begin{definition}\nFor $N\\mid M$ and $\\chi$ as above we put for $f \\in M_k(\\Gamma_0(M),\\chi)$:\n \\begin{equation*} \n f|_k {\\rm tr}_N^M = \\frac{1}{(\\Gamma_0(N):\\Gamma_0(M))} \\sum_{i} \\overline{\\chi(\\alpha_i)} f\\vert_k \\alpha_i,\n \\end{equation*}\nwhere $\\Gamma_0(N) = {\\underset{i}{\\stackrel{\\cdot}{\\bigcup}}} \\,\\Gamma_0(M) \\alpha_i$ is a disjoint coset\ndecomposition.\n \\end{definition}\n\n \\begin{lemma}\nThe definition above is independent of the choice of coset representatives. \nOne has $f|_k {\\rm tr}_N^M \\in M_k(\\Gamma_0(N),\\chi)$ and $f|_k {\\rm tr}_N^M \\in S_k(\\Gamma_0(N),\\chi)$ if $f$ is\ncuspidal.\n \\end{lemma}\n\n \\begin{proof} \nThis is a routine calculation, see e.g. \\cite[Prop. 2.1]{bfsp}.\n \\end{proof}\n\n \\begin{lemma} \\label{trace-skp}\nWith notations as above one has for $f \\in S_k(\\Gamma_0(N),\\chi)$, $g \\in S_k(\\Gamma_0(M),\\chi)$:\n \\begin{equation*} \n \\langle f,g \\rangle = \\langle f,g~|_k ~{\\rm tr}_N^M \\rangle,\n \\end{equation*}\nwhere the Petersson product on the left hand side is with respect to\n$\\Gamma_0(M)$ and that on the right hand side is with respect to $\\Gamma_0(N)$.\n \\end{lemma}\n\n \\begin{proof}\nOne has for $\\alpha_i \\in \\Gamma_0(N)$:\n\\begin{align*}\n \\langle f,\\overline{\\chi}(\\alpha_i)g|_k \\alpha_i \\rangle & = \\langle \\chi(\\alpha_i)f|_k \\alpha_i^{-1}),g \\rangle\\\\\n &= \\langle f,g\\rangle ,\n \\end{align*}\nwhich implies the assertion.\n \\end{proof}\n\n \\begin{definition}\n Let $\\gcd(\\ell,N) = 1$\n \\begin{itemize} \n \\item[a)] With $\\delta_{\\ell} := \\begin{pmatrix} \\ell & 0 \\\\ 0 & 1 \\end{pmatrix} \\in {\\rm GL}_2^+(\\mathbb Q)$ we put\n \\begin{equation*} \n f|_k V_{\\ell}(z) := f(\\ell z) = \\ell^{-k\/2} f|_k \\delta_{\\ell} (z)\n \\end{equation*}\nfor $f \\in M_k(\\Gamma_0(N),\\chi)$.\n \\item[b)] For $\\ell \\mid m$ we denote by $T_N(\\ell,m)$ the Hecke operator given by the double coset $\\Gamma_0(N) \\begin{pmatrix} \\ell & 0 \\\\ 0 & m\\end{pmatrix} \\Gamma_0(N)$.\n \\item[c)] For $\\ell\\mid m$ we denote by $T^{\\ast}_N(m,\\ell)$ the Hecke operator given by the double coset \n$\\Gamma_0(N) \\begin{pmatrix} m & 0\\\\ 0 & \\ell \\end{pmatrix} \\Gamma_0(N)$.\n \\end{itemize}\n \\end{definition}\n\n \\begin{bem}\n \\begin{itemize}\n \\item[a)] It is well-known (see \\cite[\\S 4.5]{Miy}) that on spaces of cusp forms of level $N$ the\noperator $T^{\\ast}(m,\\ell)$ is adjoint to $T(m,\\ell)$ with respect to the Petersson inner product.\n \\item[b)] As usual we write\n \\begin{equation*} \n T_N(n) = \\sum_{\\ell m = n}T_N(\\ell,m),\\: T_N^{\\ast}(n) = \\sum_{\\ell m = n} T_N^{\\ast} (m,\\ell).\n \\end{equation*}\n \\end{itemize}\n \\end{bem}\n\n \\begin{lemma} \\label{trace-hecke} \nLet $f \\in S_k(\\Gamma_0(N),\\chi)$ and $d \\in \\mathbb N$. Then\n \\begin{equation*} \n (\\Gamma_0(N):\\Gamma_0(Nd))(f|_k V_d)~|_k~{\\rm tr}_N^{Nd} = \\frac{1}{d^{k-1}} f|_k T_N^{\\ast}(d,1).\n \\end{equation*}\n \\end{lemma}\n\n \\begin{proof}\nPutting $\\Gamma_0(N) = {\\underset{i}{\\stackrel{\\cdot}{\\bigcup}}} \\Gamma_0(Nd)\\alpha_i$ we have \n(using $\\delta_d^{-1} \\Gamma_0(N){\\delta_d} \\cap \\Gamma_0(N) =\n\\Gamma_0(Nd)$ and the proof of Prop. 3.1 of \\cite{shimura_book}):\n\n \\begin{align*}\n f|_k T_N^{\\ast}(d,1)& = d^{k-1} \\sum_i \\overline{\\chi(\\alpha_i)} (d^{-k\/2}f|_k \\delta_d)|_k \\alpha_i\\\\\n & = d^{k-1}\\sum_i \\overline{\\chi(\\alpha_i)} (f|_k V_d)|_k \\alpha_i\\\\\n &=(\\Gamma_0(N):\\Gamma_0(Nd)) d^{k-1}(f|_k V_d)|_k {\\rm tr}_N^{Nd}.\n \\end{align*}\n \\end{proof}\n\n \\begin{theorem}\\label{gram_matrix_theorem}\nLet $f \\in S_k(\\Gamma_0(N),\\chi)$ be a primitive form, let $m,n \\in \\mathbb N$ with $\\gcd(m,n) = d$. \nThen\n \\begin{equation*}\n\\langle f|_k V_m,f|_kV_n\\rangle = \\frac{\\lambda(1,\\frac{n}{d})\\overline{\\lambda(1,\\frac{m}{d})}}\n{ (\\frac{mn}{d})^k \\underset{p|\\frac{mn}{d^2} \\atop p\\nmid N}{\\prod} (1+\\frac{1}{p})} \\langle f,f \\rangle ,\n \\end{equation*}\nwhere we denote by $\\lambda(1,\\frac{n}{d})$ the $T(1,\\frac{n}{d})$-eigenvalue of $f$ (and analogously\nfor $\\lambda(1,\\frac{m}{d})$).\n \\end{theorem}\n\n \\begin{proof}\n Since we have\n \\begin{align*}\n \\langle f|_k V_m, f|_k V_n \\rangle &= \\langle f|_k V_{m\/d} |_k V_d,f|_k V_{n\/d}|_k V_d \\rangle\\\\\n &=d^{-k} \\langle f|_k V_{m\/d}|_k \\delta_d,f|_k V_{n\/d}|_k \\delta_d \\rangle \\\\\n &= d^{-k} \\langle f|_k V_{m\/d},f|_k V_{n\/d}\\rangle,\n \\end{align*}\n\nwe can restrict attention the case\n \\begin{equation*}\nd= \\gcd(m,n) = 1.\n \\end{equation*}\nIn that case we have \n\n\\begin{align*}\n & \\langle f|_k V_m,f|_k V_n \\rangle = \\langle f|_k V_m,f|_k V_n ~|~ {\\rm tr}_{mN}^{mnN} \\rangle\\\\\n&= \\frac{1}{(\\Gamma_0(mN):\\Gamma_0(mnN))} \\frac{1}{n^{k-1}} \\langle f|_k V_m, f|_k T_{mN}^{\\ast} (n,1) \\rangle,\n \\end{align*}\nwhere we used Lemma \\ref{trace-skp} and Lemma \\ref{trace-hecke}.\n \\medskip\nWe split $n$ as $n=\\tilde{n}n'$ with $\\gcd(\\tilde{n},N)=1$ and $n'|N^{\\infty}$ (i.e., $n'$ is divisible \nonly by primes dividing $N$) and have \n\n\\begin{align*}\n T_{mN}^{\\ast} (n,1) &= T_{mN}^{\\ast}(\\tilde{n},1) T_{mN}^{\\ast}(n',1)\\\\\nf|_k T_{mN}^{\\ast}(\\tilde{n},1) &= f|_kT_N^{\\ast}(\\tilde{n},1)\\\\\n&= \\overline{\\lambda(1,\\tilde{n})} f\n \\end{align*}\nsince $T_N^{\\ast}(\\tilde{n},1)$ is adjoint to $T_N^{\\ast}(1,\\tilde{n})$; in the same way we see \n\n \\begin{align*}\n f|_k T_{mN}^{\\ast} (n',1)&= f|_k T_N^{\\ast}(n',1)\\\\\n &=\\overline{\\lambda(1,n')}f.\n \\end{align*}\nThis gives us\n\n \\begin{align*}\n \\langle f|_k V_m,f|_k V_n\\rangle &= \\frac{1}{n^{k-1}(\\Gamma_0(mN): \\Gamma_0(mnN))} \\cdot\n \\lambda(1,\\tilde{n}) \\lambda (1,n') \\langle f|_k V_m,f \\rangle\\\\\n &= \\frac{\\lambda(1,n)}{n^{k-1}(\\Gamma_0(mN):\\Gamma_0(mnN))} \\overline{\\langle f,f|_k V_m \\rangle}.\n \\end{align*}\nIn particular, we get\n \\begin{equation*}\n \\langle f,f|_k V_m\\rangle = \\frac{\\lambda(1,m)}{(\\Gamma_0(N):\\Gamma_0(mN))m^{k-1}}\n \\overline{\\langle f,f \\rangle},\n \\end{equation*}\nand thus (computing the group index in the denominator )\n\n \\begin{align*}\n \\langle f|_k V_m,f|_k V_n\\rangle &= \\frac{\\lambda(1,n)\\overline{\\lambda(1,m)}}{(mn)^{k-1}(\\Gamma_0(N):\\Gamma_0(mN))}\n \\langle f,f \\rangle\\\\\n &= \\frac{\\lambda(1,n) \\overline{\\lambda(1,m)}}{(mn)^k \\underset{p|mn\n \\atop p\\nmid N}{\\prod} (1+\\frac{1}{p})}\n \\langle f,f \\rangle\n \\end{align*}\nas asserted.\n \\end{proof}\n\n \\section{Orthogonal bases for spaces of cusp forms}\nThe formulas for the Petersson products derived in the previous\nsection allow to construct an orthogonal basis by Gram Schmidt\northogonalization. As we learnt from Ng Ming Ho after version one of\nthis article was posted, this has been done for trivial character in\n\\cite{rouymi} for prime power level \nand in \\cite{ngmingho} for general level. For the sake of completeness\nand since \\cite{ngmingho} is at present not published we give here our\nversion of it.\n\n\\smallskip\nWe recall first the well-known fact (see e.g. \\cite[Lemma 4.6.9]{Miy} that the space $S_k(\\Gamma_0(M),\\chi)$ \nhas a basis consisting of the $f|_{V_{\\ell}}$, where $f$ runs over the primitive forms (normalized Hecke\neigenforms) of levels $N\\mid M$ where $N$ is divisible by the conductor of $\\chi$, and where $\\ell$ is\na positive integer such that $\\ell N$ divides $M$. We will call this basis the basis of translates of\nnewforms.\n\n \\begin{lemma}\\label{product_decomposition-Lemma}\nLet $f\\in S_k(\\Gamma_0(N,\\chi))$ be a primitive form, let $m_1,m'_1,m_2,m'_2$ be positive integers with\n$ \\gcd(m_1m'_1,m_2m'_2)=1$\nput $\\tilde{f} = \\frac{f}{\\sqrt{\\langle f,f \\rangle}}$. Then\n \\begin{align*}\n \\langle \\tilde{f}|_k V_{m_1},\\tilde{f}|_k V_{m'_1} \\rangle \\cdot \\langle \\tilde{f}|_k V_{m_2},\\tilde{f}|_k V_{m'_2} \n \\rangle \\\\\n = \\langle \\tilde{f}|_k V_{m_1m_2},\\tilde{f}|_k V_{m'_1m'_2} \\rangle\\,.\n \\end{align*}\n \\end{lemma}\n\n \\begin{proof}\nThis follows directly from the theorem above.\n \\medskip\n\\end{proof}\nIt is well-known that for primitive forms $f \\not= g$ all translates of $f$ by some $V_{m'}$\nare orthogonal to all translates of $g$ by some $V_{m'}$. Our lemma above shows that for a primitive form\n$f \\in S_k(\\Gamma_0(N),\\chi)$ for some $N\\mid M$ the space of translates of $f$ in $S_k(\\Gamma_0(M),\\chi)$ is \nisometric (with respect to Petersson norms) to the tensor products of the spaces $W_{p_i}^{(f)}$ for the\n$p_i|\\frac{M}{N}$ consisting of $p_i$-power-translates of $f$. An isometry is given by the unique linear\nmap sending \n \\begin{equation*}\n \\tilde{f}|_{V_{p_1}^{r_1}} \\otimes \\cdots \\otimes \\tilde{f}|_{V_{p_z}^{r_z}} \\mbox{ to } \n \\tilde{f}|_{V_{p_1}^{r_1} \\cdots p_z^{n_z}},\n \\end{equation*}\nwhere $\\displaystyle \\tilde{f} = \\frac{f}{\\sqrt{\\langle f,f \\rangle}}$.\n \\medskip\n\nTo construct an orthogonal basis for $S_k(\\Gamma_0(M),\\chi)$ it suffices therefore to do that for\neach space $W_{p_i}$.\n \n\n \\begin{theorem}\\label{ogbasis_prime}\nLet $f \\in S_k(\\Gamma_0(N),\\chi)$ be a primitive form, put $\\tilde{f}=\\frac{f}{\\sqrt{\\langle f,f \\rangle}}$,\nlet $p$ be a prime number, $r \\in \\mathbb N$, let $W_p(f)$ be the space generated by\n$f,f|_{V_p},\\ldots,f_{V_{p^r}}$.\n \\medskip\n\n \\begin{itemize}\n \\item[a)] If $p\\mid N$ the space $W_p(f)$ has an orthogonal basis consisting of\n \\begin{equation*}\n g_0 = \\tilde{f}, \\quad g_j = p^{jk\/2}(\\tilde{f}|_{V_{p^j}} - \\frac{\\overline{\\lambda(1,p)}}{p^k} \n \\tilde{f}|_{V_{p^{j-1}}} )\n \\text{ for } 1 \\leq j \\leq r\n \\end{equation*}\nwith \n\\begin{eqnarray*}\n\\langle g_0,g_0 \\rangle& =& 1,\\\\\n\\langle g_j,g_j \\rangle& =&1-\\frac{|\\lambda(1,p)|^2}{p^k}\n\\text{ for }1 \\leq j \\leq r.\n\\end{eqnarray*}\n \\item[b)] If $p\\nmid N$ the space $W_p(f)$ has an orthogonal basis consisting of\n \\begin{eqnarray*}\n g_0 &=& \\tilde{f},\\\\\n g_1&=& p^{k\/2} \\tilde{f}|_k V_p - \\frac{\\overline{\\lambda(1,p)}}\n{p^{k\/2}(1+\\frac{1}{p})} \\tilde{f},\\\\\ng_j &=& p^{jk\/2}(\\tilde{f}|_k V_{p^j} - \\frac{\\overline{\\lambda(1,p)}}{p^k}\n \\tilde{f}|_k V_{p^{j-1}}+ \\frac{\\overline{\\chi(p)}}{p^{k+1}}\n \\tilde{f} |_k V_{p^{j-2}}) \\text{ for }2 \\leq j \\leq r\n \\end{eqnarray*}\nfor $2 \\leq j \\leq r$, with \n\\begin{eqnarray*}\n\\langle g_0,g_0 \\rangle &=&1,\\\\\n\\langle g_1,g_1 \\rangle &=& 1-\\frac{|\\lambda(1,p)|^2}\n {p^k(1+\\frac{1}{p})^2},\\\\\n\\langle g_j,g_j \\rangle &=& (1-\\frac{1}{p^2})(1- \\frac{|\\lambda(1,p)|^2}{p^k(1+\\frac{1}{p})^2})\n\\text{ for } 2 \\leq j \\leq r. \n\\end{eqnarray*}\n \\end{itemize}\n \\end{theorem}\n \n \\begin{proof}\na) In the case $p|N$ we have by Theorem \\ref{gram_matrix_theorem}\nfor $0 \\leq i \\leq j \\leq r$:\n \\begin{align*}\n \\langle \\tilde{f}|_k V_{p^i}, \\tilde{f}|_k V_{p^j} \\rangle &= p^{-ik} \\langle \\tilde{f},\\tilde{f}|_k V_{p^{j-i}} \\rangle\\\\\n &= p^{-ik} \\frac{\\lambda(1,p^{j-i})}{p^{(j-i)k}}\\\\\n &=p^{-jk}\\lambda(1,p^{j-i})\n \\end{align*}\nThis gives for $1 \\leq j \\leq r$\n\n \\begin{align*}\n \\langle g_0,g_j \\rangle &= p^{jk\/2} \\langle \\tilde{f},\\tilde{f}|_k V_{p^j} - \\frac{\\lambda(1,p)}{p^k}\n \\tilde{f}|_k V_{p^{j-1}}\\rangle\\\\\n&= p^{jk\/2} (\\frac{\\overline{\\lambda(1,p^j)}}{p^{jk}} - \\frac{\\overline{\\lambda(1,p)} \\overline{\\lambda(1,p^{j-1})}}\n {p^k p^{(j-1)k}})\\\\\n &= 0\\, ,\n\\end{align*}\nbecause of $\\lambda(1,p) \\lambda(1,p^{j-1}) = \\lambda(1,p^j)$ for $p|M$.\n \\medskip\n\nSimilarly, we see for $1 \\leq i < j \\leq r$ \n\n \\begin{align*}\n \\langle g_i,g_j \\rangle &= p^{(i+j)k\/2} \\langle \\tilde{f}|_k V_{p^i} - \\frac{\\overline{\\lambda(1,p)}}{p^k} \n \\tilde{f}|_k V_{p^{i-1}},\\tilde{f}|_k V_{p^j}-\\frac{\\overline{\\lambda(1,p)}}{p^k} \\tilde{f}|_k V_{p^{j-1}} \\rangle\\\\\n =& p^{(i+j)k\/2}(p^{-jk} \\lambda(1,p^{j-i})+ \\frac{|\\lambda(1,p)|^2}{p^{2k}} p^{-(j-1)k} \\lambda(1,p^{j-1})\\\\\n &- \\frac{\\overline{\\lambda(1,p)}}{p^k} p^{-jk} \\lambda(1,p^{j-i+1})-\n \\frac{\\lambda(1,p)}{p^k} p^{-(j-1)k} \\lambda(1,p^{j-i-1}))\\\\ \n =& 0\n \\end{align*}\nbecause of $\\lambda(1,p) \\lambda(1,p^{j-i-1}) = \\lambda(1,p^{j-1})$ and\n$\\overline{\\lambda(1,p)} \\lambda(1,p^{j-1+1}) = \\overline{\\lambda(1,p)} \\lambda(1,p) \\lambda(1,p^{j-i})\n= |\\lambda(1,p)|^2 \\lambda(1,p^{j-i})$.\n \\medskip\n\nFinally we have for $1 \\leq j \\leq r$\n\n \\begin{align*}\n \\langle g_j,g_j \\rangle& = p^{jk} \\langle \\tilde{f}|_k V_{p^j}-\\frac{\\overline{\\lambda(1,q)}}{p^k}\n \\tilde{f} |_kV_{p^{j-1}}, \\tilde{f}|_k V_{p^j} - \\frac{\\lambda(1,p)}{p^k} \\tilde{f}|_k V_{p^j-1} \\rangle\\\\\n & = p^{jk}(p^{-jk}+ \\frac{p^{-(j-1)k}}{p^{2k}} |\\lambda(1,p)|^2-\\frac{\\overline{\\lambda(1,p)} p^{-jk}}{p^k}\n \\lambda(1,p) -\\frac{\\lambda(1,p)}{p^k} p^{-jk} \\overline{\\lambda(1,p)}) \\\\\n & = (1- \\frac{|\\lambda(1,p)|^2}{p^k})\n \\end{align*}\nb) Consider now the case $p \\nmid N$. From Theorem \\ref{gram_matrix_theorem} we have for $0 \\leq i < j \\leq r$\n \\begin{equation*}\n \\langle \\tilde{f}|_k V_{p^i},\\tilde{f}|_k V_{p^j} \\rangle = \\frac{\\lambda(1,p^{j-i})}{p^{jk}(1+\\frac{1}{p})}\n\\end{equation*}\nand \n \\begin{equation*}\n\\langle \\tilde{f}|_k V_{p^j}, \\tilde{f}|_k V_{p^j}\\rangle = \\frac{1}{p^{jk}}.\n \\end{equation*}\nFrom \\cite[Lemma 4.5.7]{Miy} we have $\\lambda(1,p^2) = \\lambda(1,p)^2-(p+1)p^{k-2}\\chi(p)$ and \n$\\lambda(1,p^j) = \\lambda(1,p)\\lambda(1,p^{j-1})-p^{k-1}\\chi(p) T(1,p^{j-2})$ for $j \\geq 3$.\n\\medskip\n\nThis gives us first\n\n \\begin{align*}\n \\langle g_0,g_1 \\rangle &= p^{k\/2} \\langle \\tilde{f},\\tilde{f}|_k V_p \\rangle - \\frac{\\lambda(1,p)}\n {p^{k\/2}(1+\\frac{1}{p})}\\\\\n&=0\\,.\n \\end{align*}\nFor $i \\geq 1$ we get\n\n \\begin{align*}\np^{-(i+1)k\/2} \\langle \\tilde{f}|_k V_{p^i},g_{i+1} \\rangle =& \\langle \\tilde{f}|_k V_{p^i},\\tilde{f}|_k V_{p^{i+1}} \\rangle\n - \\frac{-\\lambda(1,p)} {\\langle \\tilde{f}|_k V_{p^i},\\tilde{f}|_k V_{p^i} \\rangle} \\\\ \n & \\quad +\\frac{\\chi(p)}{p^{k+1}} \\langle \\tilde{f}|_k V_{p^i},\\tilde{f}|_k V_{p^{i-1}} \\rangle \\\\\n =& \\frac{\\lambda(1,p)}{p^{(i+1)k}(1+{\\frac{1}{p}})} -\\frac{\\lambda(1,p)}{p^k} \\frac{1}{p^{ik}} +\n \\frac{\\chi(p)}{p^{k+1}} \\frac{\\overline{\\lambda(1+p)}}{p^{ik}(1+\\frac{1}{p})}\\\\\n =& 0\n\\end{align*}\n(using $\\chi(p) \\overline{\\lambda(1,p)}=\\lambda(1,p)$, see \\cite[Theorem 4.5.4]{Miy}).\n\n\\medskip\nFor $0 \\leq i < j \\leq r$ with $j \\geq 2+i$ we obtain\n\n \\begin{align*}\np^{-jk\/2} \\langle \\tilde{f}|_k V_{p^i},g_j \\rangle = &\\langle \\tilde{f}|_k V_{p^i}, \\tilde{f}|_k V_{p^j} \\rangle \n -\\frac{\\lambda(1,p)}{p^k} \\langle \\tilde{f}|_k V_{p^i}, \\tilde{f}|_k V_{p^{j-1}} \\rangle \\\\\n &\\quad + \\frac{\\chi(p)}{p^{k+1}} \\langle \\tilde{f}|_k V_{p^i},\\tilde{f}|_k V_{p^{j-2}} \\rangle \\\\\n =& \\frac{\\lambda(1,p^{j-i})}{p^{jk}(1+\\frac{1}{p})} - \\frac{\\lambda(1,p)\\lambda(1,p^{j-i-1})}\n {p^k p^{(j-1)k}(1+\\frac{1}{p})}\n + \\frac{\\chi(p) \\lambda(1,p^{j-i-2})}{p^{(j-2)k}(1+\\frac{1}{p})}\\\\\n =&0 \\,.\n \\end{align*}\nTaken together we see that the $g_i$ form an orthogonal basis, it remains to compute the $\\langle g_i,g_i\\rangle$.\\\\\nFor this, $\\langle g_0,g_0\\rangle = 1$ is clear.\n\n \\medskip\nNext, we have \n\n \\begin{align*}\n\\langle g_1,g_1\\rangle &= \\langle g_1,p^{k\/2} \\tilde{f}|_k V_p \\rangle \\\\\n&= p^k \\langle \\tilde{f}|_k V_p,\\tilde{f}|_k V_p\\rangle -p^{k\/2} \\cdot\n \\frac{\\overline{\\lambda(1,p)}}{p^{k\/2}(1+\\frac{1}{p})} \\langle \\tilde{f},\\tilde{f}|_k V_p \\rangle\\\\\n&= 1-\\frac{\\overline{\\lambda(1,p)}}{(1+\\frac{1}{p})} \\cdot \\frac{\\lambda(1,p)}{p^k}\\\\\n&= 1-\\frac{|\\lambda(1,p)|^2}{p^k(1+\\frac{1}{p})}.\n \\end{align*}\nFor $j \\geq 2 $ we see\n\n \\begin{align*}\n\\langle g_j,g_j \\rangle =& \\langle g_j,p^{jk\/2} \\tilde{f}|_k V_{p^j} \\rangle \\\\\n =&p^{jk} \\langle \\tilde{f}|_k V_{p^j}, \\tilde{f}|_k V_{p^j} \\rangle \n - \\frac{\\overline{\\lambda(1,p)}}{p^k} p^{jk} \\langle \\tilde{f}|_k V_{p^{j-1}},\\tilde{f}|_k V_{p^j} \\rangle \\\\\n &+ \\quad \\frac{\\overline{\\chi(p)}p^{jk}} {\\langle \\tilde{f}|_k V_{p^{j-2}},\\tilde{f}|_k V_{p^j} \\rangle} \\\\\n =& 1-\\frac{|\\lambda(1,p)|^2}{p^k(1+\\frac{1}{p})} + \\frac{\\overline{\\chi(p)} \\lambda(1,p^2)}\n {p^{k+1}(1+\\frac{1}{p})}\n\n\n\n\n\n \\end{align*}\nUsing again $\\lambda(1,p^2) = \\lambda(1,p)^2-(p+1)p^{k-2}\\chi(p)$ and\n$\\chi(p) \\overline{\\lambda(1,p)}=\\lambda(1,p)$ we obtain the assertion.\n\\end{proof}\n\\section{Half integral weights}\nFor positive integers $\\kappa,N$ we denote by $M_{k}(4N, \\chi)$\nthe space of holomorphic modular forms of weight $k=\\kappa+\\frac{1}{2}$ and\ncharacter $\\chi$ for the group $\\Gamma_0(4N)$. For the relevant\ndefinitions and notations see \\cite{shimura_halfintegral}.\nIn particular, we denote by ${\\mathfrak G}$ the covering group of\n$GL_2^+({\\mathbb R})$ defined there and by $\\gamma \\mapsto \\gamma^*$ the\nembedding of $\\Gamma_0(4)$ into ${\\mathfrak G}$ with image\n$\\Delta_0(4)$. We can extend this embedding by putting $\\bigl(\n\\begin{smallmatrix}\n 1&0\\\\0&m^2\n\\end{smallmatrix}\\bigr)^*=\\bigl(\\bigl(\n\\begin{smallmatrix}\n 1&0\\\\0&m^2\n\\end{smallmatrix}\\bigr), m^{\\frac{1}{2}}\\bigr)$ and \n $\\bigl(\n\\begin{smallmatrix}\n m^2&0\\\\0&1\n\\end{smallmatrix}\\bigr)^*=\\bigl(\\bigl(\n\\begin{smallmatrix}\n m^2&0\\\\0&1\n\\end{smallmatrix}\\bigr), m^{-\\frac{1}{2}}\\bigr)$ and $(\\gamma_1 \\alpha\n \\gamma_2)^*=\\gamma_1^* \\alpha^* \\gamma_2^*$ for $\\gamma_1,\\gamma_2\n \\in \\Gamma_0(4)$ and $\\alpha$ one of the above matrices of\n determinant $m^2$ . In the sequel we will omit the superscript $*$\n if this can cause no confusion.\n\nWe also use the\naction of double cosets of integral matrices of non zero square\ndeterminant on half integral weight modular forms of level\n$4N$ as defined there. In particular we\nhave associated to the double coset with respect to $\\Delta_0(4N)$ of $\\bigl(\\bigl(\n\\begin{smallmatrix}\n 1&0\\\\0&m^2\n\\end{smallmatrix}\\bigr),m^{\\frac{1}{2}} \\bigr)$\nthe Hecke\noperators $T_{4N}(1,m^2)$ which for $m\\mid 4N$ coincide with the the operators $U(m^2) $\nsending $\\sum_n a_f(n)e(nz)$ to $\\sum_n a_f(nm^2)e(nz)$. By\nconsidering a modular form of level $4N$ as a form of level\n${\\rm lcm}(m,4N)$ we can let $U(m^2)$ act on forms of any level divisible\nby $4$. \nThe operator $T^*_{4N}(m^2,1)$ associated to the double coset of $\\bigl(\n\\begin{smallmatrix}\n m^2&0\\\\0&1\n\\end{smallmatrix}\\bigr)^*$ is adjoint to $T_{4N}(1,m^2)$ with respect\nto the Petersson product and coincides with it if one has\n$\\gcd(m,4N)=1$, we write then as usual $T_{4N}(m^2)$.\nFor $N$ dividing $M$ we have as in the integral weight case a trace\noperator ${\\rm tr}^M_N$ from $M_{k}(4M, \\chi)$ to $M_{k}(4N, \\chi)$\nsending cusp forms to cusp forms and satisfying for cusp forms $f,g$\n\\begin{equation*} \n \\langle f,g \\rangle = \\langle f,g~|_k ~{\\rm tr}_N^M \\rangle,\n \\end{equation*}\nwhere the Petersson product on the left hand side is with respect to\n$\\Gamma_0(M)$ and that on the right hand side is with respect to $\\Gamma_0(N)$.\n \n\nIn the theory of half integral weight modular forms there are two\ndifferent methods used for the definition of oldforms, namely using\nthe operator $V_{d^2}$ as in the integral weight case (but with square\ndeterminant), raising the level by a factor $d^2$, and using the\noperator $U(p^2)$ for a prime not dividing the level, raising the\nlevel by a factor $p$.\nWe start with the first method.\n\\begin{proposition}\n\\label{trace-hecke_halfintegral} \nLet $k=\\kappa+\\frac{1}{2}$ be half integral,\nlet $f \\in S_k(\\Gamma_0(N),\\chi)$ and $d \\in \\mathbb N$. Then\n \\begin{equation*} \n (\\Gamma_0(N):\\Gamma_0(Nd^2))(f|_k V_{d^2})~|_k~{\\rm tr}_N^{Nd^2} = \\frac{1}{d^{2(k-1)}} f|_k T_N^{\\ast}(d^2,1).\n \\end{equation*} \nIn particular, if $p$ is a prime with $p\\nmid 4N$ and $f$ is an\neigenform of the Hecke operator $T(p^2)$ with eigenvalue $\\lambda_p$,\nwe have \n \\begin{equation*} \n (p^2+p)(f|_k V_{p^2})~|_k~{\\rm tr}_N^{Np^2} =\n \\frac{ \\lambda_p}{p^{2(k-1)}} f\n \\end{equation*} \nand \n\\begin{equation*}\n \\langle f, f|_k V_{p^2} \\rangle =\\frac{\n \\lambda_p}{(p^2+p)p^{2(k-1)}}\\langle f,f\\rangle.\n\\end{equation*}\n\n\\end{proposition}\n\\begin{proof}\n This is proven in the same way as Lemma \\ref{trace-hecke}. Notice\n that in the case of half integral weight we can only use shift\n operators $V_{d^2}$ and Hecke operators $T_N^{\\ast}(d^2,1)$ with\n squares $d^2$.\n\\end{proof}\n\\begin{proposition}\n\\label{trace-hecke_halfintegral} \nLet $k=\\kappa+\\frac{1}{2}$ be half integral,\nlet $f \\in S_k(\\Gamma_0(N),\\chi)$ and $p\\nmid 4N$ be a prime.\n\nThen \n\\begin{equation*}\n f \\mid_k U(p^2)|_k {\\rm tr}^{Np}_N=p^2 f|_kT(p^2).\n\\end{equation*}\nIn particular, if $f$ is an\neigenform of the Hecke operator $T(p^2)$ with eigenvalue $\\lambda_p$,\nwe have \n \\begin{equation*}\n\\langle f,f|U(p^2) \\rangle =p^2 \\lambda_p \\langle f, f\\rangle.\n \\end{equation*} \n\\end{proposition}\n\\begin{proof}\n With $\\alpha_b=\\bigl(\n \\begin{smallmatrix}\n 1&b\\\\0&p^2\n \\end{smallmatrix}\\bigr)$ we have (see \\cite{shimura_halfintegral})\n \\begin{equation*}\n f|_kU(p^2)=f|_k\\Gamma_0(4N)\\alpha_0\\Gamma_0(4Np)=(p^2)^{\\frac{k}{2}-1}\\sum_{b=0}^{p^2-1}f|_k\\alpha_b^*.\n \\end{equation*}\nMoreover, we have\n$\\Gamma_0(4N)\\alpha_0\\Gamma_0(4Np)=\\cup_b\\Gamma_0(4N)\\alpha_b$, and by\nSection 3.1 of \\cite{shimura_book},\n$\\Gamma_0(4N)\\alpha_0\\Gamma_0(4Np)\\Gamma_0(4Np)1_2\\Gamma_0(4N)=(p+1)p^2\\Gamma_0(4N)\\alpha_0\\Gamma_0(4N)$.\nFrom this the first assertion follows, and the second one follows in\nthe same way as in the integral weight case, using Lemma\n\\ref{trace-skp}, which is valid for half integral weight too. \n\\end{proof}\nAs mentioned in the introduction, because of the lack of a\nsatisfactory theory of oldforms and newforms in the half integral\nweight case we finish the investigation of this case here without\ntrying to find good orthogonal bases for the space of all cusp forms.\n \\section{Fourier coefficients of cusp forms}\nFor the rest of this paper we concentrate again on the case of modular\nforms of integral weight $k$. \n \\begin{theorem}\nThe space $S_k(\\Gamma_0(M),\\chi)$ has an orthonormal basis $(h_1,\\ldots,h_d)$, where each \n$h_i$ is an eigenform of all Hecke operators $T(p)$ for $p\\nmid M$ and where the Fourier coefficients\n$a(h_i,n)$ satisfy\n \\begin{equation*}\n|a(h_i,n)| \\leq 2 \\sqrt{\\pi} e^{2\\pi}\\sigma_0(n) n^{\\frac{k-1}{2}} \\cdot M^{\\frac{1}{2}}\\cdot \\prod_{p|M} \\frac{(1+\\frac{1}{p})^3}{\\sqrt{1-\\frac{1}{p^4}}}.\n \\end{equation*}\n \\end{theorem}\n \n \\begin{proof}\nWe write $g_j = \\phi_{p,j}(\\tilde{f})$ for the basis vectors $g_j \\in W_p(f)$ constructed in Theorem\n\\ref{ogbasis_prime} and view $\\phi_{p,j}$ as an operator transforming\na modular form $g$ into the expression on the right hand side (with\n$g$ in place of $\\tilde{f}$) of the definition of $g_j$. Obviously,\nthese operators commute. As noticed\nafter Lemma \\ref{product_decomposition-Lemma} the space\n$S_k(\\Gamma_0(M),\\chi)$ \nhas then an orthogonal basis consisting of the $(\\prod_{p|M} \\phi_{p,j_p})(\\tilde{f})$, where $f$ runs over the primitive\nforms of levels $N_f\\mid M$ in $S_k(\\Gamma_0(M),\\chi)$ and $j_p \\geq\n0$ over the integers satisfying $N_fp^{j_p}\\mid M$.\n \\medskip\n\nExamining the Proof of Theorem \\ref{ogbasis_prime} we see that the Petersson norm of $(\\prod_{i} \\phi_{p_i,j_{p_i}})\n(\\tilde{f})$ is equal to the product over $i$ of the norms of the $\\phi_{p_i,j_{p_i}}(\\tilde{f})$, which were computed in that \ntheorem.\n \\medskip\n\nAnalogously, we can decompose the computation of a bound for the Fourier coefficients of $(\\prod_{i} \\phi_{p_i,j_{p_i}})\n(\\tilde{f})$ into the computation of such a bound for each $\\phi_{p_i,j_{p_i}}(\\tilde{f})$. Looking at the $g_j$\nagain, we have for $p|N_f$ (using $|a(f,n)| \\leq\n\\sigma_0(n)n^{\\frac{(k-1)}{2}}$ and $\\vert \\lambda(1,p)\\vert \\le\np^{\\frac{k-1}{2}}$ for primitive forms $f$ and $p\\mid N_f$) \n \\begin{align*}\n \\langle f,f \\rangle^{\\frac{1}{2}} |a(g_0,n)| \\leq &\\sigma_0(n) n^{\\frac{k-1}{2}} \\mbox{ and}\\\\\n \\langle f,f \\rangle^{\\frac{1}{2}} |a(g_j,n)| \\leq & p^{\\frac{jk}{2}} \\sigma_0(\\frac{n}{p^j})(\\frac{n}{p^j})^{\\frac{k-1}{2}}\\\\\n & +p^{\\frac{jk}{2}} p^{-\\frac{(k+1)}{2}} \\sigma_0(\\frac{n}{p^{j-1}})(\\frac{n}{p^{j-1}})^{\\frac{k-1}{2}}\n \\end{align*}\nfor $j \\geq 1$, where the terms involving\n$\\frac{n}{p^j},\\frac{n}{p^{j-1}}$ appear only if the respective\nquotient is integral. \nThis gives $\\langle f,f \\rangle^{\\frac{1}{2}} |a(g_j,n)| \\leq\n\\sigma_0(n) n^{\\frac{k-1}{2}} p^{\\frac{j}{2}} (1+\\frac{1}{p}) $ for\n$j\\ge 1$, and we see that this estimate holds indeed for all $j$.\n\\medskip\n\nFor $p\\nmid N$ we obtain (with $|\\lambda(1,p)| \\leq\n2p^{\\frac{k-1}{2}}$ for $p\\nmid N_f$):\n \\begin{align*}\n\\langle f,f \\rangle^{\\frac{1}{2}} |a(g_0,n)| \\leq & \\sigma_0(n) n^{\\frac{k-1}{2}}\\\\\n\\langle f,f \\rangle^{\\frac{1}{2}} |a(g_1,n)| \\leq & p^{\\frac{k}{2}} \\sigma_0(\\frac{n}{p})(\\frac{n}{p})^{\\frac{k-1}{2}}\\\\\n & + 2\\sigma_0(n)n^{\\frac{k-1}{2}} \\cdot \\frac{p^{\\frac{k-1}{2}}}{p^{\\frac{k}{2}}(1+\\frac{1}{p})}\\\\\n \\leq & \\sigma_0(n) n^{\\frac{k-1}{2}} p^{\\frac{1}{2}}(1+\\frac{2}{p(1+\\frac{1}{p})})\n \\end{align*}\nand for $ j \\geq 2$\n \\begin{align*}\n \\langle f,f\\rangle^{\\frac{1}{2}} |a(g_j,n)| \\leq & p^{\\frac{jk}{2}} (\\sigma_0(\\frac{n}{p^j})(\\frac{n}{p^j})^{\\frac{k-1}{2}} +\n 2 \\cdot \\frac{p^{\\frac{k-1}{2}}}{p^k} \\cdot \\sigma_0(\\frac{n}{p^{j-1}})(\\frac{n}{p^{j-1}})^{\\frac{k-1}{2}}\\\\\n & +\\frac{1}{p^{k+1}} \\sigma_0( \\frac{n}{p^{j-2}})(\\frac{n}{p^{j-2}})^{\\frac{k-1}{2}})\\\\\n \\leq & \\sigma_0(n) n^{\\frac{k-1}{2}} p^{\\frac{j}{2}} (1+\\frac{1}{p})^2,\n \\end{align*}\nand we see that the latter bound holds for all $j$.\n \\medskip\n\nFinally, to estimate $\\langle f,f \\rangle $ for the primitive form\n$f$ from below we choose the fundamental domain\n${\\mathcal F}$ so that it contains $\\{x+iy \\in H\\mid \\vert x \\vert\n<\\frac{1}{2}, y>1\\}$, use $a(f,1)=1$ and get as in \n\\cite{fomenko1} \n \\begin{equation*}\n \\langle f,f \\rangle \\geq ( 4\\pi e^{4\\pi}N_f \\cdot \\prod_{p|N_f} (1+\\frac{1}{p}))^{-1} \n \\end{equation*}\nfrom the trivial bound \n$\\int_{\\mathcal F}\\vert f(x+iy)\\vert^2 y^{k-2}dxdy\\ge \\int_1^\\infty\n\\exp(-4\\pi y)dy$.\n\nImprovements on this are possible by \\cite{Go-Ho-Li, Ho-Lo} but have been made effective so\nfar only in few cases, see \\cite{rouse}. \nAt least if the conductor $M_\\chi$ of the character $\\chi$ is small\ncompared to $M$\nthese don't give much for our present purpose because of the \nadditional factors coming from oldforms which we computed above. \n\n\\medskip\nPutting things together and comparing the bounds in the cases $p\\mid N$ and\n$p \\nmid N$ , we arrive for $h$ equal to the quotient of one of the\n$\\prod_{p|M} \\phi_{p,j_p}(\\tilde{f})$ by its Petersson norm at the\ncommon bound\n \\begin{equation*}\n |a(h,n)| \\leq 2\\sqrt{\\pi} e^{2 \\pi} \\sigma_0(n) n^{\\frac{k-1}{2}} M^{\\frac{1}{2}} \\prod_{p|M} \\frac{(1+\\frac{1}{p})^3}\n {\\sqrt{1-\\frac{1}{p^4}}}\n \\end{equation*}\nfor both cases as asserted.\n \\end{proof}\n\n \\begin{theorem}\\label{fourier_estimate}\nLet $F\\in S_k(\\Gamma_0(M),\\chi)$. Then the Fourier coefficients $a(F,n)$ satisfy\n \\begin{equation*}\n |a(F,n)| \\leq 2\\sqrt{\\pi} e^{2 \\pi}\\sqrt{\\langle F,F \\rangle} \\cdot (\\dim S_k(\\Gamma_0(M),\\chi))^{\\frac{1}{2}} \\cdot \\sigma_0(n) n^{\\frac{k-1}{2}}\n M^{\\frac{1}{2}} \\cdot \\prod_{p|M} \\frac{(1+\\frac{1}{p})^3}{\\sqrt{1-\\frac{1}{p^4}}}.\n \\end{equation*}\n \\end{theorem}\n\n \\begin{proof}\nThis follows immediately from the previous theorem, using the Cauchy-Schwarz inequality.\n \\end{proof}\n\n \\begin{remark}\n \\begin{enumerate}\n\\item As indicated above it should be possible to improve on the factor\n $M^{\\frac{1}{2}}$ in the bound for $a(F,n)$ if the conductor $M_\\chi$ of the character $\\chi$ is equal to\n $M$ or at least relatively\n large compared to $M$ by using an effective\n version of the bound for the Petersson norm of a primitive form from \\cite{Go-Ho-Li, Ho-Lo} .\n \\item \nFor $\\gcd(n,M) = 1$ we obtain the better estimate\n \\begin{equation*}\n |a(h_i,n)| \\leq 2\\sqrt{\\pi} e^{2 \\pi}\\sigma_0(n) n^{\\frac{k-1}{2}} \\cdot M^{\\frac{1}{2}} \\prod_{p|M} \\frac{(1+\\frac{1}{p})}\n {\\sqrt{1-\\frac{1}{p^4}}}\n \\end{equation*}\nin Theorem \\ref{fourier_estimate} and hence\n \\begin{equation*}\n|a(F,n)| \\leq 2\\sqrt{\\pi} e^{2 \\pi}\\sqrt{\\langle F,F \\rangle} \\cdot (\\dim S_k(\\Gamma_0(M),\\chi))^{\\frac{1}{2}} \\cdot \\sigma_0(n) n^{\\frac{k-1}{2}}\n M^{\\frac{1}{2}} \\cdot \\prod_{p|M} \\frac{(1+\\frac{1}{p})}{\\sqrt{1-\\frac{1}{p^4}}}.\n \\end{equation*}\n \\end{enumerate}\n \\end{remark}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}