FIAN/TD/21-2020

Manifest Form of the Spin-Local Higher-Spin Vertex
$\Upsilon^{\eta\eta}_{\omega CCC}$

O.A. Gelfond1,2 and A.V. Korybut1

1 I.E. Tamm Department of Theoretical Physics, Lebedev Physical Institute,
Leninsky prospect 53, 119991, Moscow, Russia

2 Federal State Institution ”Scientific Research Institute for System Analysis
of the Russian Academy of Science”,
Nakhimovsky prospect 36-1, 117218, Moscow, Russia

<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Vasiliev generating system of higher-spin equations allowing to reconstruct
nonlinear vertices of field equations for higher-spin gauge fields contains a
free complex parameter $\eta$. Solving the generating system order by order
one obtains physical vertices proportional to various powers of $\eta$ and
$\bar{\eta}$. Recently $\eta^{2}$ and $\bar{\eta}^{2}$ vertices in the zero-
form sector were presented in [1] in the $Z$-dominated form implying their
spin-locality by virtue of $Z$-dominance Lemma of [2]. However the vertex of
[1] had the form of a sum of spin-local terms dependent on the auxiliary
spinor variable $Z$ in the theory modulo so-called $Z$-dominated terms,
providing a sort of existence theorem rather than explicit form of the vertex.
The aim of this paper is to elaborate an approach allowing to systematically
account for the effect of $Z$-dominated terms on the final $Z$-independent
form of the vertex needed for any practical analysis. Namely, in this paper we
obtain explicit $Z$-independent spin-local form for the vertex
$\Upsilon^{\eta\eta}_{\omega CCC}$ for its $\omega CCC$-ordered part where
$\omega$ and $C$ denote gauge one-form and field strength zero-form higher-
spin fields valued in an arbitrary associative algebra in which case the order
of product factors in the vertex matters. The developed formalism is based on
the Generalized Triangle identity derived in the paper and is applicable to
all other orderings of the fields in the vertex.

###### Contents

  1. 1 Introduction
  2. 2  Higher Spin equations
    1. 2.1 Generating equations
    2. 2.2 Perturbation theory
  3. 3  Subspace ${\mathcal{H}}^{+}$ and $Z$-dominance lemma
    1. 3.1 ${\mathcal{H}}^{+}$
    2. 3.2 Notation
    3. 3.3 Contribution to ${\Upsilon}^{\eta\eta}_{\omega CCC}$ modulo ${\mathcal{H}}^{+}$
  4. 4 Calculation scheme
  5. 5 Main result $\Upsilon^{\eta\eta}_{\omega CCC}$
  6. 6 To $z$-linear pre-exponentials
  7. 7 Generalised Triangle identity
  8. 8 Uniformization
  9. 9 Eliminating $\delta(\rho_{j})$ and $\delta(\xi_{j})$. Result
  10. 10 Final step of calculation
    1. 10.1 Degree-four pre-exponential
    2. 10.2 Degree-six pre-exponential
  11. 11 Conclusion
  12. Acknowledgments
  13. Appendix A: $B_{3}^{\eta\eta}$
  14. 13.1  $d_{x}B_{2}{}+W_{2}*C$
  15. 13.2  $({{\rm d}}_{x}B^{\eta\eta}_{3}{}+\omega*B^{\eta\eta}_{3})|_{\delta(\rho_{1})}+W^{\eta}_{1\,\omega C}*B^{\eta\,loc}_{2}$
  16. C.1 Terms proportional to $(p_{1}{}+p_{2})^{\alpha}(p_{3}{}+p_{2})_{\alpha}$
  17. C.2 Term proportional to $t^{\alpha}(p_{1\alpha}+p_{2\alpha})$
  18. C.3 Sum of $(p_{1}{}+p_{2})^{\alpha}(p_{3}{}+p_{2})_{\alpha}$-proportional and $t^{\alpha}(p_{1\alpha}+p_{2\alpha})$–proportional terms
  19. C.4 Terms proportional to $\delta(\xi_{1})-\delta(\xi_{2})$
  20. C.5 Terms proportional to $\xi_{1}\delta(\xi_{2})$
  21. D.1  $\xi_{1}$-independent pre-exponentials
  22. D.2  $\xi_{1}$-proportional pre-exponentials
  23. Appendix E: Useful formulas

##  1 Introduction

Higher-spin (HS) gauge theory describes interacting systems of massless fields
of all spins (for reviews see e.g. [3, 4]). Effects of HS gauge theories are
anticipated to play a role at ultra high energies of Planck scale [5].
Theories of this class play a role in various contexts from holography [6] to
cosmology [7]. HS theory differs from usual local field theories because it
contains infinite tower of gauge fields of all spins and the number of space-
time derivatives increases with the spins of fields in the vertex [8, 9, 10,
11]. However one may ask for spin-locality [5, 12, 13, 14] which implies
space-time locality in the lowest orders of perturbation theory [13]. 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 the predictability of the theory.

A useful way of description of HS dynamics is provided by the generating
Vasiliev system of HS equations [15]. The latter contains a free complex
parameter $\eta$. Solving the generating system order by order one obtains
vertices proportional to various powers of $\eta$ and $\bar{\eta}$. In the
recent paper [1], $\eta^{2}$ and $\bar{\eta}^{2}$ vertices were obtained in
the sector of equations for zero-form fields, containing, in particular, a
part of the $\phi^{4}$ vertex for the scalar field $\phi$ in the theory.
Though being seemingly $Z$-dependent, in [1] these vertices were written in
the $Z$-dominated form which implies their spin-locality by virtue of
$Z$-dominance Lemma of [2]. In this paper we obtain explicit $Z$-independent
spin-local form for the vertex $\Upsilon^{\eta\eta}_{\omega CCC}$ starting
from the $Z$-dominated expression of [1]. The label $\omega CCC$ refers to the
$\omega CCC$-ordered part of the vertex where $\omega$ and $C$ denote gauge
one-form and field strength zero-form HS fields valued in arbitrary
associative algebra in which case the order of the product factors in $\omega
CCC$ matters.

There are several ways to study the issue of (non)locality in HS gauge theory.
One is reconstruction the vertices from the boundary by the holographic
prescription based on the Klebanov-Polyakov conjecture [6] (see also [16],
[17]). Alternatively, one can analyze vertices directly in the bulk starting
from the generating equations of [15]. The latter approach developed in [13,
14, 1, 2, 18] is free from any holographic duality assumptions but demands
careful choice of the homotopy scheme to determine the choice of field
variables compatible with spin-locality of the vertices. The issue of
(non)locality of HS gauge theories was also considered in [19] and [20] with
somewhat opposite conclusions.

From the holographic point of view the vertex that contains $\phi^{4}$ was
argued to be essentially non-local [21] or at least should have non-locality
of very specific form presented in [22]. On the other hand, the holomorphic,
i.e., $\eta^{2}$ and antiholomorphic $\bar{\eta}^{2}$ vertices, where $\eta$
is a complex parameter in the HS equations, were recently obtained in [1]
where they were shown to be spin-local by virtue of $Z$-dominance lemma of
[2]. The computation was done directly in the bulk starting from the non-
linear HS system of [15].

In this formalism HS fields are described by one-forms $\omega(Y;K|x)$ and
zero-forms $C(Y;K|x)$ where $x$ are space-time coordinates while
$Y_{A}=(y_{\alpha},\bar{y}_{\dot{\alpha}})$ are auxiliary spinor variables.
Both dotted and undotted indices are two-component,
$\alpha,{\dot{\alpha}=1,2}$, while $K=(k,\bar{k})$ are outer Klein operators
satisfying $k*k=\bar{k}*\bar{k}=1$ ,

$\displaystyle\\{k,y^{\alpha}\\}_{\ast}=\\{k,z^{\alpha}\\}_{\ast}=\\{\bar{k},\bar{y}^{\dot{\alpha}}\\}_{\ast}=\\{\bar{k},\bar{z}^{\dot{\alpha}}\\}_{\ast}=\\{k,\theta^{\alpha}\\}_{\ast}=\\{\bar{k},\bar{\theta}^{\dot{\alpha}}\\}_{\ast}=0,$
(1.1)
$\displaystyle[k,\bar{y}^{\dot{\alpha}}]_{\ast}=[k,\bar{z}^{\dot{\alpha}}]_{\ast}=[\bar{k},y^{\alpha}]_{\ast}=[\bar{k},z^{\alpha}]_{\ast}=[k,\bar{\theta}^{\dot{\alpha}}]_{\ast}=[\bar{k},\theta^{\alpha}]_{\ast}=0\,,$

where $\theta$ and $\bar{\theta}$ are anticommuting spinors in the theory.

Schematically, non-linear HS equations in the unfolded form read as

${{\rm
d}}_{x}\omega+\omega\ast\omega=\Upsilon(\omega,\omega,C)+\Upsilon(\omega,\omega,C,C)+\ldots,$
(1.2) ${{\rm d}}_{x}C+\omega\ast
C-C\ast\omega=\Upsilon(\omega,C,C)+\Upsilon(\omega,C,C,C)+\ldots.$ (1.3)

As recalled in Section 2, generating equations of [15] that reproduce the form
of equations (1.2) and (1.3) have a simple form as a result of doubling of
spinor variables, namely

$\omega(Y;K|x)\longrightarrow W(Z;Y;K|x)\,,\qquad C(Y;K|x)\longrightarrow
B(Z;Y;K|x).$

Equations (1.2) and (1.3) result from the generating equations of [15] upon
order by order reconstruction of $Z$-dependence (for more detail see Section
2). The final form of equations (1.2) and (1.3) turns out to be
$Z$-independent as a consequence of consistency of the equations of [15]. This
fact may not be manifest however since the r.h.s.’s of HS equations usually
have the form of the sum of $Z$-dependent terms.

HS equations have remarkable property [23] that they remain consistent with
the fields $W$ and $B$ valued in any associative algebra. For instance $W$ and
$B$ can belong to the matrix algebra $Mat_{n}$ with any $n$. Since in that
case the components of $W$ and $B$ do not commute, different orderings of the
fields should be considered independently. (Mathematically, HS equations with
this property correspond to $A_{\infty}$ strong homotopy algebra introduced by
Stasheff in [24],[25],[26].) For instance, holomorphic (i.e.,
$\bar{\eta}$-independent) vertices in the zero-form sector can be represented
in the form

$\Upsilon^{\eta}(\omega,C,C)=\Upsilon^{\eta}_{\omega
CC}+\Upsilon^{\eta}_{C\omega
C}+\Upsilon^{\eta}_{CC\omega}\,,\quad\Upsilon^{\eta\eta}(\omega,C,C,C)=\Upsilon^{\eta\eta}_{\omega
CCC}+\Upsilon^{\eta\eta}_{C\omega CC}+\Upsilon^{\eta\eta}_{CC\omega
C}+\Upsilon^{\eta\eta}_{CCC\omega}\,,\,\,\ldots$ (1.4)

where the subscripts of the vertices $\Upsilon$ refer to the ordering of the
product factors.

The vertices obtained in [1] were shown to be spin-local due to the
$Z$-dominance Lemma of [2] that identifies terms that must drop from the
r.h.s.’s of HS equations together with the $Z$-dependence. Recall that spin-
locality implies that the vertices are local in terms of spinor variables for
any finite subset of fields of different spins [18] (for more detail on the
notion of spin-locality see [18]). Analogous vertices in the one-form sector
have been shown to be spin-local earlier in [14].

The main achievement of [1] consists of finding such solution of the
generating system in the third order in $C$ that all spin-nonlocal terms
containing infinite towers of derivatives in $y(\bar{y})$ between $C$-fields
in the (anti)holomorphic in $\eta(\bar{\eta})$ sector do not contribute to
$\eta^{2}$ ($\bar{\eta}^{2}$) vertices by virtue of $Z$-dominance Lemma. Thus
[1] gives spin-local expressions for the vertices
$\Upsilon^{\eta\eta}(\omega,C,C,C)$ which, however, have a form of a sum of a
number of $Z$-dependent terms. To make spin-locality manifest one must remove
the seeming Z-dependence from the vertex of [1]. Technically, this can be done
with the help of partial integration and the Schouten identity. The aim of
this paper is to show how this works in practice.

Since the straightforward derivation presented in this paper is technically
involved we confine ourselves to the particular vertex
$\Upsilon^{\eta\eta}_{\omega CCC}$ (1.4). Complexity of the calculations in
this paper expresses complexity of the obtained vertex having no analogues in
the literature. Indeed, this is explicitly calculated spin-local vertex of the
third order in the equations, corresponding to the vertices of the fourth
(and, in part, fifth) order for the fields of all spins. The example described
in the paper explains the formalism applicable to all other orderings of the
fields in the vertex that are also computable. So, our results are most
important from the general point of view highlighting a way for the
computation of higher vertices in HS theory that may be important from various
perspectives and, in the first place, for the analysis of HS holography. It
should be stressed that the results of [1] provided a sort of existence
theorem for a spin-local vertex that was difficult to extract without
developing specific tools like those developed in this paper. In particular,
it is illustrated how the general statements like $Z$-dominance Lemma work in
practical computations. Let us stress that at the moment this is the only
available approach allowing to compute explicit form of the spin-local
vertices for all spins at higher orders.

The rest of the paper is organized as follows. In Section 2, the necessary
background on HS equations is presented with brief recollection on the
procedure of derivation of vertices from the generating system. Section 3
reviews the notion of the ${\mathcal{H}}^{+}$ space as well as the
justification for a computation modulo ${\mathcal{H}}^{+}$. In Section 4, we
present step-by-step scheme of computations performed in this paper. Section 5
contains the final manifestly spin-local expression for
$\Upsilon^{\eta\eta}_{\omega CCC}$ vertex. In Sections 6 , 7 , 8 , 9 and 10
technical details of the steps sketched in Section 4 are presented. In
particular, in Section 7 we introduce important Generalised Triangle identity
which allows us to uniformize expressions from [1]. Conclusion section
contains discussion of the obtained results. Appendices A, B, C and D contain
technical detail on the steps listed in the scheme of computation. Some useful
formulas are collected in Appendix E.

##  2  Higher Spin equations

###  2.1 Generating equations

Spin-$s$ HS fields are encoded in two generating functions, namely, the space-
time one-form

$\omega(y,\bar{y},x)={{\rm d}}x^{\mu}\omega_{\mu}(y,\bar{y},x)=\sum_{n,m}{{\rm
d}}x^{\mu}\omega_{\mu}{}_{\alpha_{1}\ldots\alpha_{n},\dot{\alpha}_{1}\ldots\dot{\alpha}_{m}}(x)y^{\alpha_{1}}\ldots
y^{\alpha_{n}}\bar{y}^{\dot{\alpha}_{1}}\ldots\bar{y}^{\dot{\alpha}_{m}}\,,\qquad
s=\frac{2+m+n}{2}$ (2.1)

and zero-form

$C(y,\bar{y},x)=\sum_{n,m}C_{\alpha_{1}\ldots\alpha_{n},\dot{\alpha}_{1}\ldots\dot{\alpha}_{m}}(x)y^{\alpha_{1}}\ldots
y^{\alpha_{n}}\bar{y}^{\dot{\alpha}_{1}}\ldots\bar{y}^{\dot{\alpha}_{m}}\,,\qquad
s=\frac{|m-n|}{2}.$ (2.2)

where $\alpha=1,2$ and $\dot{\alpha}=1,2$ are two-component spinor indices.
Auxiliary commuting variables $y^{\alpha}$ and $\bar{y}^{\dot{\alpha}}$ can be
combined into an $\mathfrak{sp}(4)$ spinor
$Y^{A}=(y^{\alpha},\bar{y}^{\dot{\alpha}})$, $A=1,...,4$.

The vertices $\Upsilon(\omega,\omega,C,C,\ldots)$ (1.2) and
$\Upsilon(\omega,C,C,\ldots)$ (1.3) result from the generating system of [15]

${{\rm d}}_{x}W+W\ast W=0,$ (2.3) ${{\rm d}}_{x}S+W\ast S+S\ast W=0,$ (2.4)
${{\rm d}}_{x}B+W\ast B-B\ast W=0,$ (2.5) $S\ast S=i(\theta^{A}\theta_{A}+\eta
B\ast\gamma+\bar{\eta}B\ast\bar{\gamma}),$ (2.6) $S\ast B-B\ast S=0.$ (2.7)

Apart from space-time coordinates $x$, the fields $W(Z;Y;K|x)$, $S(Z;Y;K|x)$
and $B(Z;Y;K|x)$ depend on $Y^{A}$,
$Z^{A}=(z^{\alpha},\bar{z}^{\dot{\alpha}})$ and Klein operators
$K=(k,\bar{k})$ (1.1). $W$ is a space-time one-form, i.e., $W=dx^{\nu}W_{\nu}$
while $S$ -field is a one-form in $Z$ spinor directions
$\theta^{A}=(\theta^{\alpha},\bar{\theta}^{\dot{\alpha}})$,
$\\{\theta^{A},\theta^{B}\\}=0$, i.e.,

$S(Z;Y;K)=\theta^{A}S_{A}(Z;Y;K).$ (2.8)

$B$ is a zero-form.

Star product is defined as follows

$(f\ast g)(Z;Y;K)=\frac{1}{(2\pi)^{4}}\int
d^{4}U\,d^{4}Ve^{iU_{A}V^{A}}f(Z+U,Y+U;K)g(Z-V,Y+V;K).$ (2.9)

Elements

$\gamma=\theta^{\alpha}\theta_{\alpha}e^{iz_{\alpha}y^{\alpha}}k\mbox{\qquad
and\qquad}\bar{\gamma}=\bar{\theta}^{\dot{\alpha}}\bar{\theta}_{\dot{\alpha}}e^{i\bar{z}_{\dot{\alpha}}\bar{y}^{\dot{\alpha}}}\bar{k}$
(2.10)

are central because $\theta^{3}=0$ since $\theta_{\alpha}$ is a two-component
anticommuting spinor.

###  2.2 Perturbation theory

Starting with a particular solution of the form

$B_{0}(Z;Y;K)=0\,,\qquad
S_{0}(Z;Y;K)=\theta^{\alpha}z_{\alpha}+\bar{\theta}^{\dot{\alpha}}\bar{z}_{\dot{\alpha}}\,,\qquad
W_{0}(Z;Y;K)=\omega(Y;K)\,,$ (2.11)

which indeed solves (2.3)-(2.7) provided that $\omega(Y;K)$ satisfies zero-
curvature condition,

${{\rm d}}\omega+\omega*\omega=0\,,$ (2.12)

one develops perturbation theory. Starting from (2.7) one finds

$[S_{0},B_{1}]_{*}=0.$ (2.13)

From (2.9) one deduces that

$[Z_{A},f(Z;Y;K)]_{\ast}=-2i\frac{\partial}{\partial Z^{A}}f(Z;Y;K).$ (2.14)

Hence, equation (2.13) yields

$[S_{0},B_{1}]=-2i\theta^{A}\frac{\partial}{\partial Z^{A}}B_{1}=-2i{{\rm
d}}_{Z}B_{1}=0\;\Longrightarrow\;B_{1}(Z;Y;K)=C(Y;K).$ (2.15)

The $Z$-independent $C$-field that appears as the first-order part of $B$ is
the same that enters equations (1.2), (1.3). The perturbative procedure can be
continued further leading to the equations of the form

${{\rm d}}_{Z}\Phi_{k+1}=J(\Phi_{k},\Phi_{k-1},\ldots)\,,$ (2.16)

where $\Phi_{k}$ is either $W$, $S$ or $B$ field of the $k$-th order of
perturbation theory, identified with the degree of $C$-field in the
corresponding expression, i.e.,

$\displaystyle W=\omega+W_{1}(\omega,C)+W_{2}(\omega,C,C)+\ldots\,,\qquad
S=S_{0}+S_{1}(C)+S_{2}(C,C)+\ldots,$ $\displaystyle
B=C+B_{2}(C,C)+B_{3}(C,C,C)+\ldots.$

To obtain dynamical equations (1.2), (1.3) one should plug obtained solutions
into equations (2.3) and (2.5). For instance, (2.5) up to the third order in
$C$-field is

${{\rm d}}_{x}C+[\omega,C]_{\ast}=-{{\rm d}}_{x}B_{2}-[W_{1},C]_{\ast}-{{\rm
d}}_{x}B_{3}-[W_{1},B_{2}]_{\ast}-[W_{2},C]_{\ast}+\ldots$ (2.17)

Though the fields $W_{1}$, $W_{2}$ and $B_{2}$, $B_{3}$ and hence various
terms that enter (2.17) are $Z$-dependent, equations (2.3)-(2.7) are designed
in such a way that, as a consequence of their consistency, the sum of the
terms on the r.h.s. of (2.17) is $Z$-independent. To see this it suffices to
apply ${{\rm d}}_{Z}$ realized as $\frac{i}{2}[S_{0}\,,\quad]_{*}$ to the
r.h.s. of (2.17) and make sure that it gives zero by virtue of already solved
equations. For more detail we refer the reader to the review [4].

##  3  Subspace ${\mathcal{H}}^{+}$ and $Z$-dominance lemma

###  3.1 ${\mathcal{H}}^{+}$

In this Section the definition of the space ${\mathcal{H}}^{+}$ [1] that plays
a crucial role in our computation is recollected. Function $f(z,y|\theta)$ of
the form

$f(z,y|\theta)=\int_{0}^{1}d\mathcal{T}\,e^{i\mathcal{T}z_{\alpha}y^{\alpha}}\phi\left(\mathcal{T}z,y|\mathcal{T}\theta,\mathcal{T}\right)\,$
(3.1)

belongs to the space ${\mathcal{H}}^{+}$ if there exists such a real
$\varepsilon>0$, that

$\lim_{\mathcal{T}\rightarrow
0}\mathcal{T}^{1-\varepsilon}\phi(w,u|\theta,\mathcal{T})=0\,.$ (3.2)

Note that this definition does not demand any specific behaviour of $\phi$ at
$\mathcal{T}\to 1$ as was the case for the space ${\mathcal{H}}^{+0}$ of [18].

In the sequel we use two main types of functions that obey (3.2):

$\phi_{1}(\mathcal{T}z,y|\mathcal{T}\theta,\mathcal{T})=\frac{\mathcal{T}^{\delta_{1}}}{\mathcal{T}}\widetilde{\phi}_{1}(\mathcal{T}z,y|\mathcal{T}\theta)\,,\qquad\phi_{2}(\mathcal{T}z,y|\mathcal{T}\theta,\mathcal{T})=\vartheta(\mathcal{T}-\delta_{2})\frac{1}{\mathcal{T}}\widetilde{\phi}_{2}(\mathcal{T}z,y|\mathcal{T}\theta)$
(3.3)

with some $\delta_{1,2}>0$. (Note that the second option with $\delta_{2}>0$
can be interpreted as the first one with arbitrary large $\delta_{1}$. Here
step-function is denoted as $\vartheta$ to distinguish it from the
anticommuting variables $\theta$.)

Space ${\mathcal{H}}^{+}$ can be represented as the direct sum

${\mathcal{H}}^{+}={\mathcal{H}}^{+}_{0}\oplus{\mathcal{H}}^{+}_{1}\oplus{\mathcal{H}}^{+}_{2}\,,$
(3.4)

where $\phi(w,u|\theta,\mathcal{T})\in{\mathcal{H}}^{+}_{p}$ are degree-$p$
forms in $\theta$ satisfying (3.2).

All terms from ${\mathcal{H}}^{+}$ on the r.h.s. of HS field equations must
vanish by $Z$-dominance Lemma [2]. Following [1] this can be understood as
follows. All the expressions from (2.17) have the form (3.1) and the only way
to obtain $Z$-independent non-vanishing expression is to bring the hidden
$\mathcal{T}$ dependence in
$\phi(\mathcal{T}z,y|\mathcal{T}\theta,{\mathcal{T}})$ to
$\delta(\mathcal{T})$. If a function contains an additional factor of
$\mathcal{T}^{\varepsilon}$ or is isolated from $\mathcal{T}=0$, it cannot
contribute to the $Z$-independent answer which is the content of $Z$-dominance
Lemma [2]. This just means that functions of the class ${\mathcal{H}}^{+}_{0}$
cannot contribute to the $Z$-independent equations (1.3). Application of this
fact to locality is straightforward once this is shown that all terms
containing infinite towers of higher derivatives in the vertices of interest
belong to ${\mathcal{H}}^{+}_{0}$ and, therefore, do not contribute to HS
equations. This is what was in particular shown in [1].

###  3.2 Notation

As in [1] we use exponential form for all the expressions below where by
$\omega CCC$ we assume

$\omega(\mathsf{y}_{\omega},\bar{y})\bar{\ast}C(\mathsf{y}_{1},\bar{y})\bar{\ast}C(\mathsf{y}_{2},\bar{y})\bar{\ast}C(\mathsf{y}_{3},\bar{y})$
(3.5)

with $\bar{\ast}$ denoting star-product with respect to $\bar{y}$. Derivatives
$\partial_{\omega}$ and $\partial_{j}$ act on auxiliary variables as follows

$\partial_{\omega\alpha}=\frac{\partial}{\partial\mathsf{y}_{\omega}^{\alpha}}\,,\qquad\partial_{j\alpha}=\frac{\partial}{\partial\mathsf{y}_{j}^{\alpha}}.$
(3.6)

After all the derivatives in $\mathsf{y}_{\omega}$ and $\mathsf{y}_{j}$ are
evaluated the latter are set to zero, i.e.,

$\mathsf{y}_{\omega}=\mathsf{y}_{j}=0.$ (3.7)

In this paper we use the following notation of [1]:

$t_{\alpha}:=-i\partial_{\omega\alpha},\;\;p_{j\alpha}:=-i\partial_{j\alpha}\,,\qquad$
(3.8) $\int d^{n}\rho_{+}:=\int d\rho_{1}\ldots
d\rho_{n}\,\vartheta(\rho_{1})\ldots\vartheta(\rho_{n})\,.$ (3.9)

###  3.3 Contribution to ${\Upsilon}^{\eta\eta}_{\omega CCC}$ modulo
${\mathcal{H}}^{+}$

The $\eta^{2}C^{3}$ vertex in the equations on the zero-forms $C$ resulting
from equations of [15] is

$\Upsilon^{\eta\eta}(\omega,C,C,C)=-\left({{\rm
d}}_{x}B^{\eta\eta}_{3}+[\omega,B^{\eta\eta}_{3}]_{*}+[{W}^{\eta}_{1},B^{\eta}_{2}]_{*}+[{W}^{{\eta}\eta}_{2},C]_{*}+{{\rm
d}}_{x}B^{\eta}_{2}\,\right).$ (3.10)

Recall, that, being $Z$-independent, ${\Upsilon}^{\eta\eta}$ is a sum of
$Z$-dependent terms that makes its $Z$-independence implicit.

As explained in Introduction, ${\Upsilon}^{\eta\eta}$ can be decomposed into
parts with different orderings of fields $\omega$ and $C$. In this paper we
consider

${\Upsilon}^{\eta\eta}_{\omega
CCC}:=\Upsilon^{\eta\eta}(\omega,C,C,C)\Big{|}_{\omega CCC}\,.$ (3.11)

Since the terms from ${\mathcal{H}}^{+}$ do not contribute to the physical
vertex such terms can be discarded. Following [1] equality up to terms from
${\mathcal{H}}^{+}$ referred to as weak equality is denoted as $\approx$ .

We start with the following results of [1]:

$\widehat{\Upsilon}^{\eta\eta}_{\omega
CCC}\approx{\Upsilon}^{\eta\eta}_{\omega CCC}=-\Big{(}W_{1\,\omega
C}^{\eta}\ast B_{2}^{\eta\,loc}+{W}_{2\,\omega CC}^{\eta\eta}\ast C+{{\rm
d}}_{x}B^{\eta\,loc}_{2}\big{|}_{\omega
CCC}+\omega\ast{B}_{3}^{\eta\eta}+{{\rm
d}}_{x}{B}_{3}^{\eta\eta}\big{|}_{\omega CCC}\Big{)}\,,\qquad$ (3.12)

where

$W_{1\,\omega C}^{\eta}\ast
B_{2}^{\eta\,loc}\approx\frac{\eta^{2}}{4}\int_{0}^{1}d\mathcal{T}\mathcal{T}\int_{0}^{1}d\sigma\int
d^{3}\rho_{+}\delta\left(1-\sum_{i=1}^{3}\rho_{i}\right)\frac{\left(z_{\gamma}t^{\gamma}\right)\big{[}z_{\alpha}y^{\alpha}+\sigma
z_{\alpha}t^{\alpha}\big{]}}{(\rho_{1}+\rho_{2})}\times\\\
\times\exp\Big{\\{}i\mathcal{T}z_{\alpha}y^{\alpha}+i(1-\sigma)t^{\alpha}\partial_{1\alpha}-i\frac{\rho_{1}\sigma}{\rho_{1}+\rho_{2}}t^{\alpha}p_{2\alpha}+i\frac{\rho_{2}\sigma}{\rho_{1}+\rho_{2}}t^{\alpha}p_{3\alpha}\\\
+i\mathcal{T}z^{\alpha}\Big{(}-(\rho_{1}+\rho_{2}+\sigma\rho_{3})t_{\alpha}-(\rho_{1}+\rho_{2})p_{1\alpha}+(\rho_{3}-\rho_{1})p_{2\alpha}+(\rho_{3}+\rho_{2})p_{3\alpha}\Big{)}\\\
+iy^{\alpha}\Big{(}\sigma
t_{\alpha}-\frac{\rho_{1}}{\rho_{1}+\rho_{2}}p_{2\alpha}+\frac{\rho_{2}}{\rho_{1}+\rho_{2}}p_{3\alpha}\Big{)}\Big{\\}}\omega
CCC\,,$ (3.13)

${W}_{2\,\omega CC}^{\eta\eta}\ast
C\approx-\frac{\eta^{2}}{4}\int_{0}^{1}d\mathcal{T}\,\mathcal{T}\int
d^{4}\rho_{+}\,\delta\left(1-\sum_{i=1}^{4}\rho_{i}\right)\frac{\rho_{1}\left(z_{\gamma}t^{\gamma}\right)^{2}}{(\rho_{1}+\rho_{2})(\rho_{3}+\rho_{4})}\times\\\
\times\exp\Big{\\{}i\mathcal{T}z_{\alpha}y^{\alpha}+i\mathcal{T}z^{\alpha}\Big{(}(1-\rho_{2})t_{\alpha}-(\rho_{3}+\rho_{4})p_{1\alpha}+(\rho_{1}+\rho_{2})p_{2\alpha}+p_{3\alpha}\Big{)}+iy^{\alpha}t_{\alpha}\\\
+\frac{\rho_{1}\rho_{3}}{(\rho_{1}+\rho_{2})(\rho_{3}+\rho_{4})}\left(iy^{\alpha}t_{\alpha}+it^{\alpha}p_{3\alpha}\right)+i\left(\frac{(1-\rho_{4})\rho_{2}}{\rho_{1}+\rho_{2}}+\rho_{4}\right)t^{\alpha}p_{1\alpha}-i\frac{\rho_{4}\rho_{1}}{\rho_{3}+\rho_{4}}t^{\alpha}p_{2\alpha}\Big{\\}}\omega
CCC,$ (3.14)

${{\rm d}}_{x}B^{\eta\,loc}_{2}\big{|}_{\omega
CCC}\approx\frac{\eta^{2}}{4}\int_{0}^{1}d\mathcal{T}\int_{0}^{1}d\xi\int
d^{3}\rho_{+}\,\delta\left(1-\sum_{i=1}^{3}\rho_{i}\right)\left(z_{\alpha}y^{\alpha}\right)\Big{[}\left(\mathcal{T}z^{\alpha}-\xi
y^{\alpha}\right)t_{\alpha}\Big{]}\times\\\
\times\exp\Big{\\{}i\mathcal{T}z_{\alpha}y^{\alpha}+i(1-\rho_{2})t^{\alpha}p_{1\alpha}-i\rho_{2}t^{\alpha}p_{2\alpha}+i\mathcal{T}z^{\alpha}\Big{(}-(\rho_{1}+\rho_{2})t_{\alpha}-\rho_{1}p_{1\alpha}+(\rho_{2}+\rho_{3})p_{2\alpha}+p_{3\alpha}\Big{)}\\\
+iy^{\alpha}\Big{(}\xi(\rho_{1}+\rho_{2})t_{\alpha}+\xi\rho_{1}p_{1\alpha}-\xi(\rho_{2}+\rho_{3})p_{2\alpha}+(1-\xi)p_{3\alpha}\Big{)}\Big{\\}}\omega
CCC\,,$ (3.15)

$\omega\ast{B}_{3}^{\eta\eta}\approx-\frac{\eta^{2}}{4}\int_{0}^{1}d\mathcal{T}\,\mathcal{T}\int
d^{3}\rho_{+}\delta\left(1-\sum_{i=1}^{3}\rho_{i}\right)\int_{0}^{1}d\xi\,\frac{\rho_{1}\,\left[z_{\alpha}\left(y^{\alpha}+t^{\alpha}\right)\right]^{2}}{(\rho_{1}+\rho_{2})(\rho_{1}+\rho_{3})}\times\\\
\times\exp\Big{\\{}i\mathcal{T}z_{\alpha}y^{\alpha}+i\mathcal{T}z^{\alpha}\Big{(}-t_{\alpha}-(\rho_{1}+\rho_{3})p_{1\alpha}+(\rho_{2}-\rho_{3})p_{2\alpha}+(\rho_{1}+\rho_{2})p_{3\alpha}\Big{)}+iy^{\alpha}t_{\alpha}\\\
+i(1-\xi)y^{\alpha}\left(\frac{\rho_{1}}{\rho_{1}+\rho_{2}}p_{1\alpha}-\frac{\rho_{2}}{\rho_{1}+\rho_{2}}p_{2\alpha}\right)+i\xi\,y^{\alpha}\left(\frac{\rho_{1}}{\rho_{1}+\rho_{3}}p_{3\alpha}-\frac{\rho_{3}}{\rho_{1}+\rho_{3}}p_{2\alpha}\right)\\\
+i\frac{(1-\xi)\rho_{1}}{\rho_{1}+\rho_{2}}t^{\alpha}p_{1\alpha}-i\left(\frac{(1-\xi)\rho_{2}}{\rho_{1}+\rho_{2}}+\frac{\xi\rho_{3}}{\rho_{1}+\rho_{3}}\right)t^{\alpha}p_{2\alpha}+i\frac{\xi\rho_{1}}{\rho_{1}+\rho_{3}}t^{\alpha}p_{3\alpha}\Big{\\}}\omega
CCC,$ (3.16)

${{\rm d}}_{x}{B}_{3}^{\eta\eta}\big{|}_{\omega
CCC}\approx\frac{\eta^{2}}{4}\int_{0}^{1}d\mathcal{T}\,\mathcal{T}\int
d^{3}\rho_{+}\delta\left(1-\sum_{i=1}^{3}\rho_{i}\right)\int_{0}^{1}d\xi\,\frac{\rho_{1}\,(z_{\alpha}y^{\alpha})^{2}}{(\rho_{1}+\rho_{2})(\rho_{1}+\rho_{3})}\times\\\
\times\exp\Big{\\{}i\mathcal{T}z_{\alpha}y^{\alpha}+i\mathcal{T}z^{\alpha}\Big{(}-(\rho_{1}+\rho_{3})(t_{\alpha}+p_{1\alpha})+(\rho_{2}-\rho_{3})p_{2\alpha}+(\rho_{1}+\rho_{2})p_{3\alpha}\Big{)}+it^{\alpha}p_{1\alpha}\\\
+i(1-\xi)y^{\alpha}\left(\frac{\rho_{1}}{\rho_{1}+\rho_{2}}(t_{\alpha}+p_{1\alpha})-\frac{\rho_{2}}{\rho_{1}+\rho_{2}}p_{2\alpha}\right)+\xi\,y^{\alpha}\left(\frac{\rho_{1}}{\rho_{1}+\rho_{3}}p_{3\alpha}-\frac{\rho_{3}}{\rho_{1}+\rho_{3}}p_{2\alpha}\right)\Big{\\}}\omega
CCC.$ (3.17)

The sum of r.h.s.’s of (3.13)-(3.17) yields
$\widehat{\Upsilon}^{\eta\eta}_{\omega CCC}(Z;Y)$.

Note, that all terms on the r.h.s.’s of (3.13)-(3.17) contain no
$p_{j}{}_{\alpha}p_{i}{}^{\alpha}$ contractions in the exponentials, hence
being spin-local [1]. Thus $\widehat{\Upsilon}^{\eta\eta}_{\omega CCC}(Z;Y)$
is also spin-local.

Let us emphasize that only the full expression for
$\Upsilon^{\eta\eta}_{\omega CCC}(Y)$ (3.11) is $Z$-independent, while
$\widehat{\Upsilon}^{\eta\eta}_{\omega CCC}(Z;Y)$ (3.12) with discarded terms
in ${\mathcal{H}}^{+}$ is not. This does not allow one to find manifestly
$Z$-independent expression for ${\Upsilon}^{\eta\eta}_{\omega CCC}$ by setting
for instance $Z=0$ in Eqs. (3.13)-(3.17).

In this paper $Z$-dependence of $\widehat{\Upsilon}^{\eta\eta}_{\omega
CCC}(Z;Y)$ is eliminated modulo terms in ${\mathcal{H}}^{+}$ by virtue of
partial integration and the Schouten identity. As a result,

$\widehat{\Upsilon}^{\eta\eta}_{\omega
CCC}(Z;Y)\approx\widehat{\widehat{\Upsilon}}{\,}^{\eta\eta}_{\omega CCC}(Y),$

where $\widehat{\widehat{\Upsilon}}{\,}^{\eta\eta}_{\omega CCC}(Y)$ is
manifestly spin-local and $Z$-independent. Since ${\mathcal{H}}^{+}_{0}$-terms
do not contribute to the vertex by Z-dominance Lemma [2]

$\Upsilon^{\eta\eta}_{\omega
CCC}(Y)=\widehat{\widehat{\Upsilon}}{\,}^{\eta\eta}_{\omega CCC}(Y)\,.$

Our goal is to find the manifest form of
$\widehat{\widehat{\Upsilon}}{\,}^{\eta\eta}_{\omega CCC}(Y)$.

##  4 Calculation scheme

The calculation scheme is as follows.

  * •

I. We start from the expression Eqs. (3.13)-(3.17) for the vertex obtained in
[1].

  * •

II. To $z$-linear pre-exponentials.
Using partial integration and the Schouten identity we transform Eqs.
(3.13)-(3.17) to the form with $z$-linear pre-exponentials modulo weakly
$Z$-independent (cohomology) terms. These expressions are collected in Section
6, Eqs. (6.1)-(6.4). The respective cohomology terms being a part of the
vertex $\Upsilon^{\eta\eta}_{\omega CCC}$ are presented in Section 5 .

  * •

III. Uniformization.
We observe that the r.h.s.’s of Eqs. (6.1)-(6.4) can be re-written modulo
cohomology and weakly zero terms in a form of integrals $\int d\Gamma$ over
the same integration domain ${\cal I}$

$\int
d\Gamma\,z_{\alpha}f^{\alpha}(y,t,p_{1},p_{2},p_{3}|\mathcal{T},\xi_{i},\rho_{i}){\cal
E}\,\omega CCC\,,$ (4.1)

where the integrand contains an overall exponential function ${\cal E}$

${\cal E}={E_{z}{\,}}E,$ (4.2) ${E_{z}{\,}}:=\exp
i\Big{\\{}\mathcal{T}z_{\alpha}(y+\mathbb{P}{})^{\alpha}\Big{\\}}\,,\qquad$
(4.3) $\displaystyle E:=\exp
i\Big{\\{}-\xi_{2}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\,\,\big{(}y+\mathbb{P}{}\big{)}^{\alpha}y_{\alpha}$
(4.4)
$\displaystyle+\xi_{1}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\big{(}y+\mathbb{P}{}\big{)}^{\alpha}\tilde{t}{}_{\alpha}$
$\displaystyle+\frac{\rho_{3}}{(1-\rho_{1}-\rho_{4})}\,\,(p_{3}+p_{2})^{\alpha}y_{\alpha}-\frac{\rho_{3}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\,\,\rho_{1}{t}{}^{\alpha}y_{\alpha}$
$\displaystyle+\frac{\rho_{1}}{(1-\rho_{3})}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}+p_{3}{}_{\alpha}y^{\alpha}+p_{1}{}_{\alpha}{t}{}^{\alpha}\Big{\\}}\,,$
$\displaystyle\tilde{t}{}=\frac{\rho_{1}}{\rho_{1}+\rho_{4}}{t}{}\,,\qquad$
(4.5) $\displaystyle\mathbb{P}{}={\cal P}+(1-\rho_{4}){t}{}\,,$ (4.6)
$\displaystyle{\cal
P}=(1-\rho_{1}-\rho_{4})(p{}_{1}+p_{2})-(1-\rho_{3})(p_{3}+p_{2})\,,$ (4.7)

the integral over ${\cal I}$ is denoted as

$\int d\Gamma=\int_{0}^{1}d\mathcal{T}\int
d^{3}\xi_{+}\,\delta\left(1-\sum_{i=1}^{3}\xi_{i}\right)\int
d^{4}\rho_{+}\,\delta\left(1-\sum_{j=1}^{4}\rho_{j}\right)\,.$ (4.8)

Eqs. (6.1)-(6.4) transformed to the form (4.1) are collected in Section 8,
Eqs. (8.2)-(8.5).

  * •

IV. Elimination of $\delta$-functions.
Using partial integration and the Schouten identity we eliminate the all
factors of $\delta(\rho_{i})$, $\delta(\xi_{1})$ and $\delta(\xi_{2})$ from
Eqs. (8.2)-(8.5). The result is presented in Section 9, Eqs. (9.1)-(9.4).

  * •

V. Final step.
Finally, we show in Section 10 that a sum of the r.h.s.’s of Eqs. (9.2)-(9.4)
is $Z$-independent up to ${\mathcal{H}}^{+}$.

By collecting all resulting $Z$-independent terms we finally obtain the
manifest expression for vertex $\Upsilon^{\eta\eta}_{\omega CCC}$, being a sum
of expressions (5.2)-(5.12).

##  5 Main result $\Upsilon^{\eta\eta}_{\omega CCC}$

Here the final manifestly $Z$-independent $\omega CCC$ contribution to the
equations is presented.

Vertex $\Upsilon^{\eta\eta}_{\omega CCC}$ is

$\Upsilon^{\eta\eta}_{\omega CCC}=\sum_{j=1}^{11}J_{j}\,$ (5.1)

with $J_{i}$ given in Eqs. (5.2)-(5.12). Note that the integration regions may
differ for different terms $J_{j}$ in the vertex, depending on their genesis.

Firstly we note that $B^{\eta\eta}_{3}$ (A.10), that contains a
$Z$-independent part, generates cohomologies both from
$\omega*B^{\eta\eta}_{3}$ and from ${{\rm d}}_{x}B^{\eta\eta}_{3}$,

$J_{1}=-\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\frac{\rho_{2}}{(\rho_{2}+\rho_{1})(\rho_{2}+\rho_{3})}\delta(\rho_{4})E\,\omega
CCC,$ (5.2) $J_{2}=\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\frac{\rho_{2}}{(\rho_{2}+\rho_{4})(\rho_{2}+\rho_{3})}\delta(\rho_{1})E\,\omega
CCC\,.$ (5.3)

Recall that $E$ and $d\Gamma$ are defined in (4.4) and (4.8), respectively.
(Note, that, here and below, the integrands on the r.h.s.’s of expressions for
$J_{i}$ are $\mathcal{T}$-independent, hence the factor of
$\int_{0}^{1}d\mathcal{T}$ in $d\Gamma$ equals one.)

Other cohomology terms are collected from (9.2), (9.3), (9.4), (10.1), (D.2),
(B.1), (B.3), (B.4) and (B.5), respectively,

$J_{3}=-\frac{i\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\frac{1}{(\rho_{2}+\rho_{3})(1-\rho_{3})}\Big{\\{}\rho_{2}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}\big{[}\overrightarrow{\partial}_{\rho_{2}}-\overrightarrow{\partial}_{\rho_{3}}\big{]}\\\
+\rho_{2}(p_{1}{}+p_{2})^{\alpha}(p_{3}{}+p_{2})_{\alpha}\big{[}\overrightarrow{\partial}_{\rho_{4}}-\overrightarrow{\partial}_{\rho_{1}}\big{]}+\rho_{2}{t}{}^{\alpha}(p_{3}{}+p_{2}{})_{\alpha}\big{[}\overrightarrow{\partial}_{\rho_{2}}-\overrightarrow{\partial}_{\rho_{1}}\big{]}+\frac{\rho_{1}+\rho_{4}}{(1-\rho_{3})}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}\Big{\\}}E\,\omega
CCC\,,$ (5.4)

$J_{4}=\frac{i\eta^{2}}{4}\int
d\Gamma\,\frac{\delta(\xi_{3})}{1-\rho_{3}}\Big{(}-\frac{\rho_{3}}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})}{t}{}^{\gamma}y_{\gamma}-\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{t}{}^{\gamma}y_{\gamma}[-\overrightarrow{\partial}_{\rho_{1}}+\overrightarrow{\partial}_{\rho_{2}}]\\\
-\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}(p_{1}{}+p_{2})^{\gamma}(y+\tilde{t}{})_{\gamma}[\overrightarrow{\partial}_{\rho_{4}}-\overrightarrow{\partial}_{\rho_{1}}]\Big{)}E\omega
CCC\,,$ (5.5)

$J_{5}=-i\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\Big{[}1+\xi_{1}(\overrightarrow{\partial}_{\xi_{1}}-\overrightarrow{\partial}_{\xi_{2}})\Big{]}\Big{\\{}\frac{-\rho_{2}}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})(\rho_{1}+\rho_{4})}(p_{3}{}^{\alpha}+p_{2}{}^{\alpha})^{\gamma}{t}{}_{\gamma}\\\
-\frac{\rho_{3}}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})^{2}}\,\,{t}{}^{\alpha}y_{\alpha}+\frac{1}{(\rho_{2}+\rho_{3})(1-\rho_{3})(\rho_{1}+\rho_{4})}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}\Big{\\}}E\,\omega
CCC\,,$ (5.6)

$J_{6}=i\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}(\rho_{1}+\rho_{4})}(p_{1}{}+p_{2})^{\gamma}({t}{})_{\gamma}E\omega
CCC\,,$ (5.7) $J_{7}=-\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\,\xi_{1}\frac{\rho_{2}\rho_{2}}{(\rho_{2}+\rho_{3})^{3}(1-\rho_{3})^{3}(\rho_{1}+\rho_{4})}\times\\\
\times\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{})+(1-\rho_{4}){t}{}\big{)}^{\gamma}\big{(}y+\tilde{t}{}\big{)}_{\gamma}{t}{}^{\alpha}y_{\alpha}E\,\omega
CCC\,,$ (5.8) $J_{8}=-\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\rho_{3})\Big{(}\rho_{1}\delta(\xi_{3})+\Big{[}i{\delta(\rho_{4})}-(p_{2}{}_{\alpha}+p_{1}{}_{\alpha}){t}{}^{\alpha}\Big{]}\Big{\\{}i\delta(\xi_{3})+\tilde{t}{}^{\gamma}y_{\gamma}\Big{\\}}\Big{)}E\,\omega
CCC\,,$ (5.9)

$J_{9}=i\eta^{2}\frac{1}{4}\int
d\Gamma\,\delta(\rho_{1})\delta(\rho_{4})\delta(\xi_{3})\exp\Big{\\{}-i\xi_{2}(p_{1}+p_{2}+{t}-\rho_{2}(p_{3}+p_{2}))_{\alpha}(y)^{\alpha}\\\
-\xi_{1}(y+p_{1}+p_{2}-\rho_{2}(p_{3}+p_{2}))_{\gamma}({t})^{\gamma}+(1-\rho_{2})(p_{3}+p_{2}){}^{\gamma}y_{\gamma}+p_{3}{}_{\gamma}y^{\gamma}+{t}{}^{\beta}p_{1}{}_{\beta}\Big{\\}}\omega
CCC\,,$ (5.10)

$J_{10}=-i\eta^{2}\frac{1}{4}\int
d\Gamma\,\delta(\rho_{4})\delta(\xi_{1})\delta(\rho_{1})\,\exp
i\Big{\\{}-\xi_{2}(y+p_{1}+p_{2}+{t}-\rho_{2}(p_{3}+p_{2}))_{\alpha}(y)^{\alpha}\\\
+(1-\rho_{2})(p_{3}+p_{2}){}^{\gamma}y_{\gamma}+p_{3}{}_{\gamma}y^{\gamma}+{t}{}^{\beta}p_{1}{}_{\beta}\Big{\\}}\omega
CCC\,,$ (5.11)

$J_{11}=\frac{i\eta^{2}}{4}\int
d\Gamma\,\delta(\rho_{1})\delta(\rho_{4})y^{\alpha}{t}{}_{\alpha}\exp
i\Big{\\{}(y+{\cal
P}_{0}+{t}){}^{\gamma}(\xi_{1}{t}-\xi_{2}y)_{\gamma}+(1-\rho_{2})(p_{3}+p_{2}){}^{\gamma}y_{\gamma}\\\
+p_{3}{}_{\gamma}y^{\gamma}+{t}{}^{\beta}p_{1}{}_{\beta}\Big{\\}}\omega
CCC\,.$ (5.12)

Let us emphasize, that neither exponential function $E$ (4.4) nor the
exponentials on the r.h.s.’s of Eqs. (5.10)-(5.12) contain
$\partial_{i}{}_{\alpha}\partial_{k}{}^{\alpha}$ terms. Hence, as anticipated,
all $J_{j}$ are spin-local.

One can see that though having poles in pre-exponentials these expressions are
well defined.
For instance a potentially dangerous factor on the r.h.s. of (5.2) is
dominated by 1 as follows from the inequality
${\rho_{2}}-(\rho_{1}+\rho_{2})(\rho_{2}+\rho_{3})=-\rho_{3}\rho_{1}\leq 0$
that holds due to the factor of
$\prod\vartheta(\rho_{i})\delta(1-\sum\rho_{i})\delta(\rho_{4})$. Analogous
simple reasoning applies to the r.h.s. of (5.3).

The case of (5.4)-(5.8) is a bit more tricky. By partial integration one
obtains from (5.4)-(5.6)

$\displaystyle J_{3}+J_{4}+J_{5}=\frac{i\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\frac{1}{(\rho_{2}+\rho_{3})(1-\rho_{3})}\Big{\\{}-\delta({\rho_{3}}){t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}$
(5.13)
$\displaystyle+[\delta({\rho_{4}})-\delta({\rho_{1}})]\rho_{2}(p_{1}{}+p_{2})^{\alpha}(p_{3}{}+p_{2})_{\alpha}+{t}{}^{\alpha}(p_{3}{}+p_{2}{})_{\alpha}-\delta({\rho_{1}})\rho_{2}{t}{}^{\alpha}(p_{3}{}+p_{2}{})_{\alpha}\,$
$\displaystyle-\delta({\rho_{1}})\frac{\rho_{2}}{(1-\rho_{3})}{t}{}^{\gamma}y_{\gamma}+[\delta({\rho_{4}})-\delta({\rho_{1}})]\frac{\rho_{2}}{(1-\rho_{3})}(p_{1}{}+p_{2})^{\gamma}(y+\tilde{t}{})_{\gamma}$
$\displaystyle-\delta({\xi_{2}})\Big{(}\frac{-\rho_{2}}{(\rho_{2}+\rho_{3})(\rho_{1}+\rho_{4})}(p_{3}{}^{\alpha}+p_{2}{}^{\alpha})^{\gamma}{t}{}_{\gamma}$
$\displaystyle-\frac{\rho_{3}}{(\rho_{2}+\rho_{3})(1-\rho_{3})}\,\,{t}{}^{\alpha}y_{\alpha}+\frac{1}{(\rho_{1}+\rho_{4})}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}\Big{)}\Big{\\}}E\,\omega
CCC\,.$

Using that, due to the factor of $\delta(1-\sum\rho_{i})$, for positive
$\rho_{i}$ it holds

$\displaystyle\frac{\rho_{2}}{(\rho_{3}+\rho_{2})(1-\rho_{3})}-1=-\frac{\rho_{3}(1-(\rho_{3}+\rho_{2}))}{(\rho_{3}+\rho_{2})(1-\rho_{3})}\leq
0\,,\qquad$ (5.14)
$\displaystyle\frac{1}{(\rho_{2}+\rho_{3})(1-\rho_{3})}\,\leq\frac{1}{(\rho_{2}+\rho_{3})(1-\rho_{3}-\rho_{2})}=\frac{1}{(\rho_{3}+\rho_{2})}+\frac{1}{(\rho_{1}+\rho_{4})}\,,$
(5.15)

one can make sure that each of the expressions with poles in the pre-
exponential in Eqs. (5.7), (5.8) and (5.13) can be represented in the form of
a sum of integrals with integrable pre-exponentials. For instance, the
potentially dangerous factor in (5.8), by virtue of (5.14) and (5.15)
satisfies

$\frac{\rho_{2}\rho_{2}}{(\rho_{2}+\rho_{3})^{3}(1-\rho_{3})^{3}(\rho_{1}+\rho_{4})}\leq\frac{1}{(1-\rho_{3})(\rho_{1}+\rho_{4})}+\frac{1}{(\rho_{3}+\rho_{2})}+\frac{1}{(\rho_{1}+\rho_{4})}\,.\quad$
(5.16)

Each of the terms on the r.h.s. of Eq. (5.16) is integrable, because
integration is over a three-dimensional compact area $\sum\rho_{i}=1$ in the
positive quadrant. For instance consider the first term. Swopping
$\rho_{4}\leftrightarrow\rho_{2}$ one has

$\displaystyle\int
d^{4}\rho_{+}\delta(1-\sum_{1}^{4}\rho_{i})\frac{1}{(1-\rho_{3})(\rho_{1}+\rho_{2})}=\int
d^{3}\rho_{+}\vartheta(1-\sum_{1}^{3}\rho_{i})\frac{1}{(1-\rho_{3})(\rho_{1}+\rho_{2})}=$
(5.17)
$\displaystyle-\int_{0}^{1}d\rho_{1}\int_{0}^{1-\rho_{1}}d\rho_{2}\frac{\log(\rho_{1}+\rho_{2})}{(\rho_{1}+\rho_{2})}=\frac{1}{2}\int_{0}^{1}d\rho_{1}\log^{2}(\rho_{1})\,,$

which is integrable.

Analogously other seemingly dangerous factors can be shown to be harmless as
well.

##  6 To $z$-linear pre-exponentials

Step II of the calculation scheme of Section 4 is to transform r.h.s.’s of
Eqs. (3.13)-(3.17) to $Z$-independent terms plus terms with linear in $z$ pre-
exponentials (modulo $H^{+}$).

To this end, from (A.10) one straightforwardly obtains that

$\omega\ast{B}_{3}^{\eta\eta}\approx J_{1}+\frac{\eta^{2}}{4}\int
d\Gamma\frac{\delta(\xi_{3})\delta(\rho_{4})}{(1-\rho_{1})(1-\rho_{3})}\Bigg{[}-\rho_{2}(z_{\alpha}(y^{\alpha}+t^{\alpha}))(p_{1\beta}+p_{2\beta})(p_{2}{}^{\beta}+p_{3}{}^{\beta})\\\
+i\Big{[}\Big{(}\delta(\rho_{1})+\delta(\rho_{3})\Big{)}(1-\rho_{1})(1-\rho_{3})-\delta(\xi_{2})\Big{]}z_{\alpha}\Big{(}(1-\rho_{1})(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})-(1-\rho_{3})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})\Big{)}\\\
+iz_{\alpha}(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})(1-\rho_{1})\Big{(}\delta(\xi_{2})-\delta(\xi_{1})\Big{)}\Bigg{]}\exp\Big{\\{}i\mathcal{T}z_{\alpha}\big{(}y^{\alpha}+t^{\alpha}+(1-\rho_{1})(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})-(1-\rho_{3})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})\big{)}\\\
+\frac{i(1-\xi_{1})\rho_{2}}{\rho_{1}+\rho_{2}}(y^{\alpha}+t^{\alpha})(p_{1\alpha}+p_{2\alpha})+\frac{i\xi_{1}\rho_{2}}{\rho_{2}+\rho_{3}}(y^{\alpha}+t^{\alpha})(p_{2\alpha}+p_{3\alpha})-i(y^{\alpha}+t^{\alpha})p_{2\alpha}\Big{\\}}\omega
CCC\,,\qquad$ (6.1)

where $J_{1}$ is the cohomology term (5.2). Analogously,

${{\rm d}}_{x}{B}_{3}^{\eta\eta}\approx J_{2}-\frac{\eta^{2}}{4}\int
d\Gamma\frac{\delta(\xi_{3})\delta(\rho_{4})}{(1-\rho_{1})(1-\rho_{3})}\Bigg{[}-\rho_{2}(z_{\alpha}y^{\alpha})(p_{1\beta}+t_{\beta}+p_{2\beta})(p_{2}{}^{\beta}+p_{3}{}^{\beta})\\\
+i\Big{[}\Big{(}\delta(\rho_{1})+\delta(\rho_{3})\Big{)}(1-\rho_{1})(1-\rho_{3})-\delta(\xi_{2})\Big{]}z_{\alpha}\Big{(}(1-\rho_{1})(p_{1}{}^{\alpha}+t^{\alpha}+p_{2}{}^{\alpha})-(1-\rho_{3})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})\Big{)}\\\
+iz_{\alpha}(p_{1}{}^{\alpha}+t^{\alpha}+p_{2}{}^{\alpha})(1-\rho_{1})\Big{(}\delta(\xi_{2})-\delta(\xi_{1})\Big{)}\Bigg{]}\exp\Big{\\{}i\mathcal{T}z_{\alpha}\big{(}y^{\alpha}+(1-\rho_{1})(p_{1}{}^{\alpha}+t^{\alpha}+p_{2}{}^{\alpha})-(1-\rho_{3})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})\big{)}\\\
+\frac{i(1-\xi_{1})\rho_{2}}{\rho_{1}+\rho_{2}}y^{\alpha}(p_{1\alpha}+t_{\alpha}+p_{2\alpha})+\frac{i\xi_{1}\rho_{2}}{\rho_{2}+\rho_{3}}y^{\alpha}(p_{2\alpha}+p_{3\alpha})-iy^{\alpha}p_{2\alpha}+it^{\beta}p_{1\beta}\Big{\\}}\omega
CCC$ (6.2)

with $J_{2}$ (5.3).

Using the Schouten identity and partial integration one obtains from Eqs.
(3.13)-(3.15), respectively,

$W_{1\,\omega C}^{\eta}\ast
B_{2}^{\eta}\approx\frac{\eta^{2}}{4}\int_{0}^{1}d\mathcal{T}\int_{0}^{1}d\tau\int_{0}^{1}d\sigma_{1}\int_{0}^{1}d\sigma_{2}\Bigg{[}i(z_{\alpha}t^{\alpha})\delta(1-\tau)\\\
+\frac{z_{\alpha}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})}{1-\tau}\Big{(}i\big{(}\delta(\sigma_{1})-\delta(1-\sigma_{1})\big{)}-\big{[}y^{\alpha}+p_{1}{}^{\alpha}+p_{2}{}^{\alpha}-\sigma_{2}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})\big{]}t_{\alpha}\Big{)}\Bigg{]}\exp\Big{\\{}i\mathcal{T}z_{\alpha}y^{\alpha}\\\
+i\mathcal{T}z_{\alpha}\Big{(}\tau(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})-((1-\tau)+\sigma_{2}\tau)(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})+\big{(}\sigma_{1}+\tau(1-\sigma_{1})\big{)}t^{\alpha}\Big{)}+it^{\alpha}p_{1\alpha}\\\
+i\sigma_{1}\big{[}y^{\alpha}+p_{1}{}^{\alpha}+p_{2}{}^{\alpha}-\sigma_{2}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})\big{]}t_{\alpha}-i\Big{(}\sigma_{2}p_{3}{}^{\alpha}-(1-\sigma_{2})p_{2}{}^{\alpha}\Big{)}y_{\alpha}\Big{\\}}\omega
CCC\,,$ (6.3) $W_{2\,\omega CC}^{\eta\eta}\ast
C\approx-\frac{i\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\delta(\rho_{3})\frac{(z_{\gamma}t^{\gamma})}{\rho_{1}+\rho_{4}}\Big{[}-\rho_{1}\big{(}\delta(\rho_{4})+it^{\alpha}(p_{1\alpha}+p_{2\alpha})\big{)}+\xi_{1}\delta(\xi_{2})\Big{]}\times\\\
\times\exp\Big{\\{}i\mathcal{T}z_{\alpha}y^{\alpha}+i\mathcal{T}z_{\alpha}\Big{(}(1-\rho_{1}-\rho_{4})(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})-(1-\rho_{3})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})+(1-\rho_{4})t^{\alpha}\Big{)}\\\
+iy^{\alpha}\left(\frac{\xi_{1}\rho_{1}}{1-\rho_{2}}t_{\alpha}+p_{3\alpha}\right)+i\left(1-\rho_{1}-\frac{\xi_{1}\rho_{1}\rho_{2}}{1-\rho_{2}}\right)t^{\alpha}p_{1\alpha}-i(1-\xi_{1})\rho_{1}t^{\alpha}p_{2\alpha}+i\frac{\xi_{1}\rho_{1}}{1-\rho_{2}}t^{\alpha}p_{3\alpha}\Big{\\}}\omega
CCC\,,$ (6.4) ${{\rm d}}_{x}B_{2}^{\eta}\approx\frac{i\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\delta(\rho_{4})\,(z_{\alpha}y^{\alpha})\Big{[}it^{\gamma}(p_{1\gamma}+p_{2\gamma})+\delta(\rho_{4})-\delta(\rho_{1})\Big{]}\times\\\
\times\exp\Big{\\{}i\mathcal{T}z_{\alpha}y^{\alpha}+i\mathcal{T}z_{\alpha}\big{(}(1-\rho_{1}-\rho_{4})(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})-(1-\rho_{3})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})+(1-\rho_{4})t^{\alpha}\big{)}\\\
+i(1-\rho_{2})t^{\beta}p_{1\beta}-i\rho_{2}t^{\beta}p_{2\beta}+i\xi_{2}y^{\alpha}\Big{(}(\rho_{1}+\rho_{2})t_{\alpha}+\rho_{2}p_{1\alpha}-(1-\rho_{2})p_{2\alpha}-p_{3\alpha}\Big{)}+iy^{\alpha}p_{3\alpha}\Big{\\}}\omega
CCC.$ (6.5)

##  7 Generalised Triangle identity

Here a useful identity playing the key role in our computations is introduced.

For any $F(x,y)$ consider

$\displaystyle I=\int_{[0,1]}{d\tau\,}\int
d^{3}\xi_{+}\delta(1-\xi_{1}-\xi_{2}-\xi_{3})$ (7.1) $\displaystyle
z^{\gamma}\Big{[}(a_{2}-a_{1})_{\gamma}\delta(\xi_{3})+(a_{3}-a_{2})_{\gamma}\delta(\xi_{1})+(a_{1}-a_{3})_{\gamma}\delta(\xi_{2})\Big{]}F\big{(}\tau
z_{\beta}P^{\beta}\,,(-\xi_{1}a_{1}-\xi_{2}a_{2}-\xi_{3}a_{3})_{\alpha}P^{\alpha}\big{)}\,$

with arbitrary $\tau,\xi$\- independent $P$ and $a_{i}$.

Let $G(x,y)$ be a solution to differential equation

$\frac{\partial}{\partial x}G(x,y)=\frac{\partial}{\partial y}F(x,y)\,.$ (7.2)

Hence

$\displaystyle I=\int_{[0,1]}{d\tau\,}\int
d^{3}\xi_{+}\delta(1-\xi_{1}-\xi_{2}-\xi_{3})$ (7.3)
$\displaystyle(a_{1}-a_{3})^{\alpha}(a_{3}-a_{2})_{\alpha}\overrightarrow{\partial}_{\tau}G\big{(}\tau
z_{\beta}P^{\beta}\,,(-\xi_{1}a_{1}-\xi_{2}a_{2}-\xi_{3}a_{3})_{\alpha}P^{\alpha}\big{)}.$

Note that there is a factor of $(a_{1}-a_{3})^{\alpha}(a_{3}-a_{2})_{\alpha}$
equal to the area of triangle spanned by the vectors $a_{1}\,,a_{2}\,,a_{3}$
on the r.h.s. of (7.3).

This identity is closely related to identity (3.24) of [13], that, in turn,
expresses triangle identity of [27]. Hence, (7.3) will be referred to as
Generalised Triangle identity or GT identity.

Note that, for appropriate $G$ partial integration on the r.h.s. of (7.3) in
$\tau$ gives $z$-independent (cohomology) term plus $\mathcal{H}^{+}$-term.
Namely,

$\displaystyle I=-\int d^{3}\xi_{+}\delta(1-\xi_{1}-\xi_{2}-\xi_{3})$ (7.4)
$\displaystyle(a_{1}-a_{3})^{\alpha}(a_{3}-a_{2})_{\alpha}G\big{(}0\,,(-\xi_{1}a_{1}-\xi_{2}a_{2}-\xi_{3}a_{3})_{\alpha}P^{\alpha}\big{)}$
$\displaystyle+\int d^{3}_{+}\xi\delta(1-\xi_{1}-\xi_{2}-\xi_{3})$
$\displaystyle(a_{1}-a_{3})^{\alpha}(a_{3}-a_{2})_{\alpha}G\big{(}z_{\beta}P^{\beta}\,,(-\xi_{1}a_{1}-\xi_{2}a_{2}-\xi_{3}a_{3})_{\alpha}P^{\alpha}\big{)}.$

The second term on the r.h.s. belongs to ${\mathcal{H}}^{+}$ if $G$ is of the
form (3.1) satisfying (3.2).

To prove GT identity let us perform partial integration on the r.h.s. of (7.1)
with respect to $\xi_{i}$. This yields

$\displaystyle
I=\int_{[0,1]}{d\tau\,}\int{d^{3}\xi_{+}\,}\delta(1-\xi_{1}-\xi_{2}-\xi_{3})$
(7.5)
$\displaystyle\Big{[}z^{\gamma}(a_{3}-a_{2})_{\gamma}P^{\alpha}a_{1}{}_{\alpha}+z^{\gamma}(a_{1}-a_{3}){}_{\gamma}P^{\alpha}a_{2}{}_{\alpha}+z^{\gamma}(a_{2}-a_{1}){}_{\gamma}P^{\alpha}a_{3}{}_{\alpha}\Big{]}\times$
$\displaystyle\frac{\partial}{\partial y}F\big{(}\tau
z_{\alpha}P^{\alpha}\,,\,\,-(\xi_{1}a_{1}+\xi_{2}a_{2}+\xi_{3}a_{3})_{\alpha}P^{\alpha}\big{)}\,.$

The Schouten identity yields

$\displaystyle\Big{[}z^{\gamma}a_{1}{}_{\gamma}P^{\alpha}(a_{3}-a_{2})_{\alpha}+z^{\gamma}a_{2}{}_{\gamma}P^{\alpha}(a_{1}-a_{3})_{\alpha}+z^{\gamma}a_{3}{}_{\gamma}P^{\alpha}(a_{2}-a_{1})_{\alpha}\Big{]}=$
(7.6)
$\displaystyle\Big{[}z^{\gamma}P_{\gamma}\big{\\{}a_{1}{}^{\alpha}(a_{3}-a_{2})_{\alpha}+a_{2}^{\alpha}(a_{1}-a_{3})_{\alpha}+a_{3}^{\alpha}(a_{2}-a_{1})_{\alpha}\big{\\}}$
$\displaystyle+z^{\gamma}(a_{3}-a_{2})_{\gamma}P^{\alpha}a_{1}{}_{\alpha}+z^{\gamma}(a_{1}-a_{3}){}_{\gamma}P^{\alpha}a_{2}{}_{\alpha}+z^{\gamma}(a_{2}-a_{1}){}_{\gamma}P^{\alpha}a_{3}{}_{\alpha}\Big{]}.$

One can observe that

$\displaystyle\Big{[}z^{\gamma}(a_{3}-a_{2})_{\gamma}P^{\alpha}a_{1}{}_{\alpha}+z^{\gamma}(a_{1}-a_{3}){}_{\gamma}P^{\alpha}a_{2}{}_{\alpha}+z^{\gamma}(a_{2}-a_{1}){}_{\gamma}P^{\alpha}a_{3}{}_{\alpha}\Big{]}=$
(7.7)
$\displaystyle-\Big{[}z^{\gamma}a_{1}{}_{\gamma}P^{\alpha}(a_{3}-a_{2})_{\alpha}+z^{\gamma}a_{2}{}_{\gamma}P^{\alpha}(a_{1}-a_{3})_{\alpha}+z^{\gamma}a_{3}{}_{\gamma}P^{\alpha}(a_{2}-a_{1})_{\alpha}\Big{]}\,,\qquad$

whence it follows (7.3).

A useful particular case of GT identity is that with $F(x,y)=f(x+y)$, namely

$\displaystyle\int_{[0,1]}{d\tau\,}\int{d^{3}\xi_{+}\,}\delta(1-\xi_{1}-\xi_{2}-\xi_{3})z^{\gamma}\Big{[}(a_{2}-a_{1})_{\gamma}\delta(\xi_{3})$
(7.8)
$\displaystyle+(a_{3}-a_{2})_{\gamma}\delta(\xi_{1})+(a_{1}-a_{3})_{\gamma}\delta(\xi_{2})\Big{]}f\big{(}(\tau
z-\xi_{1}a_{1}-\xi_{2}a_{2}-\xi_{3}a_{3})_{\alpha}P^{\alpha}\big{)}\quad$
$\displaystyle=-\int_{[0,1]}{d\tau\,}\int{d^{3}\xi_{+}\,}\delta(1-\xi_{1}-\xi_{2}-\xi_{3})$
$\displaystyle(a_{1}-a_{3})^{\alpha}(a_{3}-a_{2})_{\alpha}\overrightarrow{\partial}_{\tau}f\big{(}(\tau
z-\xi_{1}a_{1}-\xi_{2}a_{2}-\xi_{3}a_{3})_{\alpha}P^{\alpha}\big{)}\,.$

##  8 Uniformization

Step III of Section 4 is to uniformize the r.h.s. ’s of Eqs. (6.1)-(6.5)
putting them into the form (4.1), where GT identity (7.1) plays an important
role. Details of uniformization are given in Appendix B (p. Appendix B:
Uniformization Detail).

As a result, Eq. (3.12) yields

$\widehat{\Upsilon}^{\eta\eta}_{\omega
CCC}\Big{|}_{\text{mod}\,cohomology}\approx\sum_{j=1}^{4}F_{j}$ (8.1)

with $F_{j}$ presented in (8.2)-(8.5).

Note that different terms of $F_{j}$ will be considered separately in what is
follows. For the future convenience the underbraced terms are re-numerated,
being denoted as $F_{j,k}$, where $j$ refers to $F_{j}$ while $k$ refers to
the respective underbraced term in the expression for $F_{j}$. For instance,
$F_{1}=F_{1,1}+F_{1,2}+F_{1,3}+F_{1,4}$, etc.

$-\omega\ast
B_{3}^{\eta\eta}\Big{|}_{mod\,\delta(\rho_{1})\&\delta(\mathcal{T})}\approx
F_{1}:=-\frac{\eta^{2}}{4}\int
d\Gamma\,\frac{\delta(\xi_{3})\delta(\rho_{4})}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\Big{[}\underbrace{\rho_{2}(z_{\beta}{\cal
P}^{\beta})(p_{1\alpha}+p_{2\alpha})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})}_{1}\\\
+\underbrace{i\delta(\rho_{3})(1-\rho_{1}-\rho_{4})(1-\rho_{3})(z_{\alpha}{\cal
P}^{\alpha})}_{2}+\underbrace{-i\xi_{1}\delta(\xi_{2})(z_{\alpha}{\cal
P}^{\alpha})}_{3}\\\
+\underbrace{i(1-\rho_{1}-\rho_{4})z_{\alpha}(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})\Big{(}\delta(\xi_{2})-\delta(\xi_{1})\Big{)}}_{4}\Big{]}\mathcal{E}\omega
CCC\,,$ (8.2)

$-{{\rm
d}}_{x}B^{\eta\eta}_{3}\Big{|}_{mod\,\delta(\rho_{1})\&\delta(\mathcal{T})}\approx
F_{2}:=+\frac{\eta^{2}}{4}\int
d\Gamma\,\frac{\delta(\xi_{3})\delta(\rho_{1})}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\Big{[}\underbrace{\rho_{2}(z_{\beta}{\cal
P}^{\beta})(p_{1\alpha}+p_{2\alpha})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})}_{1}\\\
+\underbrace{\rho_{2}(1-\rho_{4})(z_{\beta}t^{\beta})t_{\alpha}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})}_{2}+\underbrace{\rho_{2}(1-\rho_{4})(z_{\beta}t^{\beta})(p_{1\alpha}+p_{2\alpha})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})}_{3}+\underbrace{\rho_{2}(z_{\beta}{\cal
P}^{\beta})t_{\alpha}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})}_{4}\\\
+\underbrace{i\delta(\rho_{3})(1-\rho_{1}-\rho_{4})(1-\rho_{3})(z_{\alpha}\mathbb{P}^{\alpha})}_{5}+\underbrace{-i\xi_{1}\delta(\xi_{2})(z_{\alpha}{\cal
P}^{\alpha})}_{6}+\underbrace{-i\xi_{1}\delta(\xi_{2})(1-\rho_{4})(z_{\alpha}t^{\alpha})}_{7}\\\
+\underbrace{i(1-\rho_{1}-\rho_{4})z_{\alpha}(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})\Big{(}\delta(\xi_{2})-\delta(\xi_{1})\Big{)}}_{8}+\underbrace{i(1-\rho_{1}-\rho_{4})z_{\alpha}t^{\alpha}\Big{(}\delta(\xi_{2})-\delta(\xi_{1})\Big{)}}_{9}\Big{]}\mathcal{E}\omega
CCC\,,$ (8.3)

$-{{\rm d}}_{x}B_{2}^{\eta}-W_{2\,\omega CC}^{\eta\eta}\ast
C\Big{|}_{mod\,\delta(\mathcal{T})}\approx F_{3}:=-\frac{\eta^{2}}{4}\int
d\Gamma\delta(\rho_{3})\delta(\xi_{3})\Bigg{[}\underbrace{i\delta(\rho_{1})(z_{\alpha}\mathbb{P}^{\alpha})}_{1}+\underbrace{-\frac{i(z_{\alpha}t^{\alpha})\,\xi_{1}\delta(\xi_{2})}{\rho_{1}+\rho_{4}}}_{2}\\\
+\underbrace{t^{\alpha}(p_{1\alpha}+p_{2\alpha})z_{\gamma}{\cal
P}^{\gamma}}_{3}+\underbrace{i\delta(\rho_{4})z_{\alpha}(-{\cal
P}^{\alpha})}_{4}+\underbrace{t^{\gamma}(p_{1\gamma}+p_{2\gamma})z_{\alpha}t^{\alpha}\left((1-\rho_{4})-\frac{\rho_{1}}{\rho_{1}+\rho_{4}}\right)}_{5}\Bigg{]}\mathcal{E}\,\omega
CCC\,,$ (8.4)

$-({{\rm d}}_{x}B_{3}^{\eta\eta}+\omega\ast
B_{3}^{\eta\eta})\Big{|}_{\delta(\rho_{1})}\Big{|}_{mod\,\delta(\mathcal{T})}-W_{1\,\omega
C}^{\eta}\ast B_{2}^{\eta\,loc}\approx F_{4}:=-\frac{\eta^{2}}{4}\int
d\Gamma\,\frac{\delta(\xi_{3})\delta(\xi_{2})\,z_{\alpha}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})}{(\rho_{2}+\rho_{3})(\rho_{1}+\rho_{4})}\times\\\
\times\left(\underbrace{i\Big{(}\delta(\rho_{1})-\delta(\rho_{4})\Big{)}{\cal
E}}_{1}+\underbrace{i{E_{z}{\,}}\left(\frac{\partial}{\partial\rho_{1}}-\frac{\partial}{\partial\rho_{4}}\right)E}_{2}\right)\omega
CCC.$ (8.5)

Note that

$F_{1,2}+F_{3,4}=0,$ (8.6) $F_{2,5}+F_{3,1}=0.$ (8.7)

Let us emphasise that, by virtue (E.1), each $F_{j}$ is of the form (4.1) as
expected.

Note that during uniformizing procedure the vertices (5.9) -(5.12) are
obtained in Appendix B (p. Appendix B: Uniformization Detail).

##  9 Eliminating $\delta(\rho_{j})$ and $\delta(\xi_{j})$. Result

The fourth step of Section 4 is to eliminate all $\delta(\rho_{i})$ ,
$\delta(\xi_{1})$ and $\delta(\xi_{2})$ from the pre-exponentials on the
r.h.s.’s of Eqs. (8.2)-(8.5).

More precisely, using partial integration, the Schouten identity and
Generalised Triangle identity (7.3), taking into account Eqs. (4.5)-(4.7) one
finds that Eq. (8.1) yields

$\big{(}\widehat{\Upsilon}^{\eta\eta}_{\omega
CCC}-G_{1}-G_{2}-G_{3}\big{)}\big{|}_{\\!\\!\\!\\!\\!\\!\mod
cohomology}\approx 0\,,\qquad$ (9.1)

where

$G_{1}:=J_{3}+\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})z_{\gamma}\Bigg{\\{}(y^{\gamma}+\widetilde{t}^{\gamma})\frac{\rho_{2}\,t^{\alpha}(p_{1\alpha}+p_{2\alpha})}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{E_{z}{\,}}\Bigg{[}\frac{\partial}{\partial\rho_{2}}-\frac{\partial}{\partial\rho_{3}}\Bigg{]}E\\\
+(y^{\gamma}+\widetilde{t}^{\gamma})\frac{\rho_{2}\,(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})(p_{2\alpha}+p_{3\alpha})}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{E_{z}{\,}}\Bigg{[}\frac{\partial}{\partial\rho_{4}}-\frac{\partial}{\partial\rho_{1}}\Bigg{]}E\\\
+(y^{\gamma}+\tilde{t}^{\gamma})\frac{\rho_{2}\,t^{\alpha}(p_{2\alpha}+p_{3\alpha})}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{E_{z}{\,}}\Bigg{[}\frac{\partial}{\partial\rho_{2}}-\frac{\partial}{\partial\rho_{1}}\Bigg{]}E+(y^{\gamma}+\tilde{t}^{\gamma})\frac{(\rho_{1}+\rho_{4})t^{\alpha}(p_{1\alpha}+p_{2\alpha})}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{\cal
E}\\\
+(y^{\gamma}+\tilde{t}^{\gamma})\frac{\rho_{3}\,t^{\alpha}(p_{2\alpha}+p_{3\alpha})}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})}{\cal
E}+\frac{\rho_{2}\,t^{\gamma}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})(p_{1\alpha}+p_{2\alpha}+t_{\alpha}-\tilde{t}_{\alpha})}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})(\rho_{1}+\rho_{4})}{\cal
E}\Bigg{\\}}\omega CCC\,,\qquad$ (9.2) $G_{2}:=J_{4}+\frac{\eta^{2}}{4}\int
d\Gamma\,\frac{\delta(\xi_{3})}{1-\rho_{3}}\,z^{\alpha}\Bigg{\\{}\frac{\rho_{3}(y_{\alpha}+\tilde{t}_{\alpha})t^{\gamma}(y_{\gamma}+\mathbb{P}_{\gamma})}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})}{\cal
E}\\\
-\frac{\rho_{2}\rho_{4}\,t_{\alpha}(y^{\gamma}+\mathbb{P}^{\gamma})t_{\gamma}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})(\rho_{1}+\rho_{4})^{2}}{\cal
E}-\frac{\rho_{2}\,(y_{\alpha}+\tilde{t}_{\alpha})t^{\gamma}(p_{1\gamma}+p_{2\gamma})}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{\cal
E}\\\
-\frac{\rho_{2}\,(p_{1}{}_{\alpha}+p_{2}{}_{\alpha})(y^{\gamma}+\mathbb{P}^{\gamma})t_{\gamma}}{(1-\rho_{1}-\rho_{4})(\rho_{1}+\rho_{4})(1-\rho_{3})}{\cal
E}+{E_{z}{\,}}\frac{\rho_{2}\,t^{\gamma}(y_{\gamma}+\mathbb{P}_{\gamma})(y_{\alpha}+\tilde{t}_{\alpha})}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\Bigg{[}\frac{\partial}{\partial\rho_{1}}-\frac{\partial}{\partial\rho_{2}}\Bigg{]}E\\\
+{E_{z}{\,}}\frac{\rho_{2}\,(y_{\alpha}+\tilde{t}_{\alpha})(p_{1}{}^{\gamma}+p_{2}{}^{\gamma})(y_{\gamma}+\mathbb{P}_{\gamma})}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\Bigg{[}\frac{\partial}{\partial\rho_{1}}-\frac{\partial}{\partial\rho_{4}}\Bigg{]}E\Bigg{\\}}\omega
CCC\,,\qquad$ (9.3) $G_{3}:=J_{5}+\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\Bigg{(}1+\xi_{1}\Bigg{[}\frac{\partial}{\partial\xi_{1}}-\frac{\partial}{\partial\xi_{2}}\Bigg{]}\Bigg{)}\times\\\
\times
z_{\alpha}\Bigg{\\{}\frac{\rho_{2}\,t^{\alpha}(p_{2}{}^{\gamma}+p_{3}{}^{\gamma})(y_{\gamma}+\tilde{t}_{\gamma})}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})(\rho_{1}+\rho_{4})}+\frac{-\rho_{2}\,t^{\alpha}(\tilde{t}^{\gamma}+y^{\gamma})(y_{\gamma}+\mathbb{P}_{\gamma})}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})^{2}(\rho_{1}+\rho_{4})}\\\
+\frac{-\rho_{3}\,(y^{\alpha}+\tilde{t}^{\alpha})(t^{\gamma}y_{\gamma})}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})^{2}}+\frac{(y^{\alpha}+\tilde{t}^{\alpha})(p_{1}{}^{\gamma}+p_{2}{}^{\gamma})t_{\gamma}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}\Bigg{\\}}{\cal
E}\,\omega CCC\,,\qquad$ (9.4)

with $J_{3}$, $J_{4}$ and $J_{5}$ being the cohomology terms (5.4), (5.5) and
(5.6), respectively. (Details of the derivation are presented in Appendix C
(p.Appendix C: Eliminating $\delta(\rho_{j})$ and $\delta(\xi_{j})$).)

Note that schematically

$G_{1}+G_{2}+G_{3}=\int
d\Gamma\,\delta(\xi_{3})z_{\alpha}g^{\alpha}(y,t,p_{1},p_{2},p_{3}|\rho,\xi){\cal
E}\,\omega CCC\,+J_{3}+J_{4}+J_{5}\,,\qquad$ (9.5)

as expected . Let us stress that $g^{\alpha}(y,t,p_{1},p_{2},p_{3}|\rho,\xi)$
on the r.h.s. of (9.5) is free from a distributional behaviour.

##  10 Final step of calculation

Here this is shown that the sum of the r.h.s.’s of Eqs. (9.2)-(9.4) gives a
$Z$-independent cohomology term up to terms in ${\mathcal{H}}^{+}$.

More in detail, the expression $G_{1}+G_{2}+G_{3}$ of the form (9.5) consists
of two types of terms with the pre-exponential of degree four and six in
$z,y,t,p_{1},p_{2},p_{3}$, respectively. That with degree-four pre-exponential
separately equals a $Z$-independent cohomology term up to terms in
${\mathcal{H}}^{+}$. This is considered in Section 10.1. The term with degree-
six pre-exponential is considered in Section 10.2. As a result of these
calculations $J_{6}$ (5.7) and $J_{7}$ (5.8) are obtained.

###  10.1 Degree-four pre-exponential

Consider the sum of expressions with $z$-dependent degree-four pre-exponential
from Eqs. (9.2), (9.3) and (9.4), denoting it as $S_{4}$. Partial integration
yields

$\displaystyle S_{4}\approx J_{6}+\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\,\Big{[}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})(\rho_{1}+\rho_{4})}{t}{}^{\alpha}z_{\alpha}(p_{3}+p_{2})^{\gamma}({t}-\tilde{t}){}_{\gamma}$
(10.1)
$\displaystyle+\frac{\rho_{2}\rho_{4}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}(\rho_{1}+\rho_{4})^{2}}{t}{}^{\gamma}z_{\gamma}\big{(}y+\mathbb{P}{}\big{)}^{\alpha}{t}{}_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}(\rho_{1}+\rho_{4})}(p_{1}{}+p_{2})^{\gamma}\big{(}y+(1-\rho_{4}){t}{}\big{)}_{\gamma}z^{\alpha}{t}{}_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})(\rho_{1}+\rho_{4})}{t}{}^{\alpha}z_{\alpha}\,(p_{3}+p_{2})^{\gamma}(y+\tilde{t}{})_{\gamma}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})^{2}(\rho_{1}+\rho_{4})}\big{(}-\mathbb{P}{}+\tilde{t}{}\big{)}^{\gamma}\big{(}y+\tilde{t}{}\big{)}_{\gamma}z^{\alpha}{t}{}_{\alpha}\Big{]}{\cal
E}\omega CCC\,,\qquad$

where the cohomology term $J_{6}$ is given in (5.7) . It is not hard to see
that the integrand of the remaining term is zero by virtue of the Schouten
identity.

###  10.2 Degree-six pre-exponential

Terms of this type either appear in (9.2), (9.3) via differentiation in
$\rho_{j}$ or in (9.4) via differentiation in $\xi_{j}$. Denoting a sum of
these terms as $S_{6}$ we obtain

$\displaystyle S_{6}=+\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\Big{\\{}{E_{z}{\,}}(y+\tilde{t}{})^{\gamma}z_{\gamma}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}\Big{[}(\overrightarrow{\partial}_{\rho_{2}}-\overrightarrow{\partial}_{\rho_{3}})E\Big{]}\qquad$
(10.2)
$\displaystyle+{E_{z}{\,}}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}\Big{[}(p_{1}{}+p_{2})^{\gamma}\big{(}y+(1-\rho_{4}){t}{}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}\Big{]}[\overrightarrow{\partial}_{\rho_{4}}-\overrightarrow{\partial}_{\rho_{1}}]E$
$\displaystyle+{E_{z}{\,}}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}[\overrightarrow{\partial}_{\rho_{2}}-\overrightarrow{\partial}_{\rho_{1}}]E$
$\displaystyle+i\xi_{1}\Big{[}+\Big{\\{}+\frac{\rho_{2}\rho_{2}}{(1-\rho_{1}-\rho_{4})^{3}(1-\rho_{3})^{3}(\rho_{1}+\rho_{4})}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{})+(1-\rho_{4}){t}{}\big{)}^{\gamma}\big{(}y+\tilde{t}{}\big{)}_{\gamma}z_{\alpha}{t}{}^{\alpha}$
$\displaystyle-\frac{\rho_{3}\rho_{2}}{(1-\rho_{1}-\rho_{4})^{3}(1-\rho_{3})^{3}}{\big{(}y+\tilde{t}{}\big{)}^{\gamma}z_{\gamma}{t}{}^{\alpha}y_{\alpha}}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})^{3}}\big{(}y+\tilde{t}{}\big{)}^{\gamma}z_{\gamma}{(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}}\Big{\\}}{\cal
E}\Big{]}\times\big{(}y+\mathbb{P}{}\big{)}^{\alpha}(y+\tilde{t}{})_{\alpha}\Big{\\}}\omega
CCC$

Recall that the integral measure ${{\rm d}}\Gamma$(4.8) contains the factor of
$\delta(1-\sum_{1}^{3}\xi_{i})$. Hence taking into account the factor of
$\delta(\xi_{3})$ on the r.h.s. of (10.2) the dependence on $\xi_{2},\xi_{3}$
can be eliminated by the substitution $\xi_{2}\to 1-\xi_{1}$, $\xi_{3}\to 0$.
Then we consider separately the terms that contain and do not contain
$\xi_{1}$ in the pre-exponentials. As shown in Appendix D, those with
$\xi_{1}$-proportional pre-exponentials give $J_{7}$ (5.8) up to
${\mathcal{H}}^{+}$, while those with $\xi_{1}$-independent pre-exponentials
give zero up to ${\mathcal{H}}^{+}$.

##  11 Conclusion

In this paper starting from $Z$-dominated expression obtained in [1] the
manifestly spin-local holomorphic vertex $\Upsilon^{\eta\eta}_{\omega CCC}$ in
the equation (1.3) is obtained for the $\omega CCC$ ordering. Besides
evaluation the expression for the vertex, our analysis illustrates how
$Z$-dominance implies spin-locality.

One of the main technical difficulties towards $Z$-independent expression was
uniformization, that is bringing the exponential factors to the same form, for
all contributions (3.13)-(3.17) with the least amount of new integration
parameters possible. Practically, some part of the uniformization procedure
heavily used the Generalized Triangle identity of Section 7 playing important
role in our analysis.

Let us stress that spin-locality of the vertices obtained in [1] follows from
$Z$-dominance Lemma. However the evaluation the explicit spin-local vertex
$\Upsilon^{\eta^{2}}_{\omega CCC}$ achieved in this paper is technically
involved. To derive explicit form of other spin-local vertices in this and
higher orders a more elegant approach to this problem is highly desirable.

## Acknowledgments

We would like to thank Mikhail Vasiliev for fruitful discussions and useful
comments on the manuscript. We acknowledge a partial support from the Russian
Basic Research Foundation Grant No 20-02-00208. The work of OG is partially
supported by the FGU FNC SRISA RAS (theme 0065-2019-0007).

## Appendix A: $B_{3}^{\eta\eta}$

$B_{3}^{\eta\eta}$ modulo ${\mathcal{H}}^{+}$ terms from [1] is given by

${B}_{3}^{\eta\eta}\approx-\frac{\eta^{2}}{4}\int
d\Gamma\delta(\xi_{3})\delta(\rho_{4})\frac{\mathcal{T}\rho_{2}(z_{\alpha}y^{\alpha})^{2}}{(\rho_{1}+\rho_{2})(\rho_{2}+\rho_{3})}\exp\big{(}{\mathsf{F}}\big{)}CCC\,,\qquad$
(A.1)

where $d\Gamma$ is defined in (4.8),

${\mathsf{F}}=i\mathcal{T}z_{\alpha}\left(y^{\alpha}+{\cal
P}_{0}^{\alpha}\right)+\frac{i(1-\xi_{1})\rho_{2}}{\rho_{1}+\rho_{2}}y^{\alpha}(p_{1\alpha}+p_{2\alpha})+\frac{i\xi_{1}\rho_{2}}{\rho_{2}+\rho_{3}}y^{\alpha}(p_{2\alpha}+p_{3\alpha})-iy^{\alpha}p_{2\alpha}\,,\qquad$
(A.2) ${\cal P}_{0}=(1-\rho_{1})(p_{1}+p_{2})-(1-\rho_{3})(p_{2}+p_{3}).$
(A.3)

Performing partial integration with respect to $\mathcal{T}$ twice we obtain

${B}_{3}^{\eta\eta}\approx\frac{\eta^{2}}{4}\int
d\Gamma\frac{\delta(\xi_{3})\delta(\rho_{4})\rho_{2}}{(1-\rho_{3})(1-\rho_{1})}\Big{[}\delta(\mathcal{T})+iz_{\alpha}{\cal
P}_{0}^{\alpha}+iz_{\alpha}{\cal
P}_{0}^{\alpha}\Big{(}1+i\mathcal{T}z_{\alpha}{\cal
P}_{0}^{\alpha}\Big{)}\Big{]}\exp\big{(}{\mathsf{F}}\big{)}CCC\,.$ (A.4)

Noticing that

$\frac{\partial}{\partial\rho_{1}}{\mathsf{F}}=-i\mathcal{T}z_{\alpha}(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})-i\frac{(1-\xi_{1})\rho_{2}}{(\rho_{1}+\rho_{2})^{2}}y^{\alpha}(p_{1\alpha}+p_{2\alpha}),$
(A.5) $\frac{\partial}{\partial\rho_{3}}{\mathsf{F}}=\\\
=i\mathcal{T}z_{\alpha}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})-i\frac{\xi_{1}\rho_{2}}{(\rho_{2}+\rho_{3})^{2}}y^{\alpha}(p_{2\alpha}+p_{3\alpha})$
(A.6)

and performing partial integration with respect to $\rho_{1}$ and $\rho_{3}$
we obtain

${B}_{3}^{\eta\eta}\approx\frac{i\eta^{2}}{4}\int
d\Gamma\frac{\delta(\xi_{3})\delta(\rho_{4})}{(1-\rho_{3})(1-\rho_{1})}\Bigg{[}{-i\rho_{2}\delta(\mathcal{T})}+\,z_{\alpha}{\cal
P}_{0}^{\alpha}\big{(}{(1-\rho_{3})(1-\rho_{1})}\left(\delta(\rho_{1})+\delta(\rho_{3})\right)-1\big{)}\\\
-{i\,\rho_{2}z_{\alpha}{\cal
P}_{0}^{\alpha}}\left(\xi_{2}\frac{y^{\alpha}(p_{1\alpha}+p_{2\alpha})}{(\rho_{1}+\rho_{2})}+\xi_{1}\frac{y^{\alpha}(p_{2\alpha}+p_{3\alpha})}{(\rho_{2}+\rho_{3})}\right)\Bigg{]}\exp\big{(}{\mathsf{F}}\big{)}CCC.$
(A.7)

Observing that

$\frac{\partial{\mathsf{F}}}{\partial\xi_{1}}=\frac{i\rho_{2}}{\rho_{2}+\rho_{3}}y^{\alpha}(p_{2\alpha}+p_{3\alpha})-\frac{i\rho_{2}}{\rho_{1}+\rho_{2}}y^{\alpha}(p_{1\alpha}+p_{2\alpha})$
(A.8)

and using the Schouten identity

$z_{\alpha}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})y^{\beta}(p_{1\beta}+p_{2\beta})=z_{\alpha}y^{\alpha}(p_{2}{}^{\beta}+p_{3}{}^{\beta})(p_{1\beta}+p_{2\beta})+z_{\alpha}(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})y^{\beta}(p_{2\beta}+p_{3\beta})$
(A.9)

after partial integration with respect to $\xi_{1}$ we obtain

${B}_{3}^{\eta\eta}\approx\frac{i\eta^{2}}{4}\int
d\Gamma\frac{\delta(\xi_{3})\delta(\rho_{4})}{(1-\rho_{3})(1-\rho_{1})}\Bigg{[}{-i\rho_{2}\delta(\mathcal{T})}+{z_{\alpha}(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})(1-\rho_{1})}\Big{(}\delta(\xi_{2})-\delta(\xi_{1})\Big{)}\\\
+z_{\alpha}{\cal
P}_{0}^{\alpha}\Big{[}(1-\rho_{1})(1-\rho_{3})\Big{(}\delta(\rho_{1})+\delta(\rho_{3})\Big{)}-\delta(\xi_{2})\xi_{1}\Big{]}+i\rho_{2}z_{\alpha}y^{\alpha}(p_{1\beta}+p_{2\beta})(p_{2}{}^{\beta}+p_{3}{}^{\beta})\Bigg{]}\exp\big{(}{\mathsf{F}}\big{)}CCC.$
(A.10)

The $\delta(\mathcal{T})$-proportional term gives rise to $J_{1}$ (5.2) and
$J_{2}$ (5.3).

## Appendix B: Uniformization Detail

Here some details of the transformation of integrands (6.1)–(6.5) to the form
(4.1) are presented.

Uniformization can be easily achieved for Eqs. (6.1) and (6.2) modulo
$\delta(\rho_{1})$-proportional terms. Indeed, eliminating
$\delta(\rho_{1})$-proportional term from the r.h.s. of (6.1), adding an
integration parameter $\rho_{4}$ and a factor of $\delta(\rho_{4})$, one
obtains (8.2). Analogously, eliminating $\delta(\rho_{1})$-proportional term
from the r.h.s. (6.2), adding an integration parameter $\rho_{4}$, swapping
$\rho_{1}\leftrightarrow\rho_{4}$ and then adding a factor of
$\delta(\rho_{1})$ one obtains (8.3).

To transform integrands of Eqs. (6.4) and (6.5), as well as
$\delta(\rho_{1})$-proportional terms of the integrands of Eqs. (6.1) and
(6.2), to the form (4.1) GT identity (7.1) is used in Sections B.1 and B.2.

###  13.1  $d_{x}B_{2}{}+W_{2}*C$

Noticing that the exponential of (6.4) coincides with ${\cal E}$ at
$\xi_{2}=0$, while the exponential of (6.5) coincides with ${\cal E}$ (4.2) at
$\xi_{1}=0$, one can easily make sure, that only the
$\delta(\xi_{2})$-proportional term of (6.4) and the
$\delta(\rho_{1})$-proportional term of (6.5) have the desired form (4.1).

Using that ${\cal E}$ (4.2) does not depend on $\xi_{3}$, swapping
$\xi_{3}\leftrightarrow\xi_{1}$ in the remaining part of (6.5), then swapping
$\xi_{3}\leftrightarrow\xi_{2}$ in the remaining part of (6.4), one then can
apply GT identity (7.8) to the sum of the two obtained terms . As a result,
Eqs. (6.4), (6.5) yield

$\displaystyle{{\rm d}}_{x}B_{2}^{\eta\,loc}+W_{2\,\omega CC}^{\eta\eta}\ast
C\approx\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\rho_{3})\delta(\xi_{3})\Big{[}-i\frac{(z_{\alpha}t^{\alpha})}{\rho_{1}+\rho_{4}}\delta(\xi_{2})-i(\underline{z_{\alpha}y^{\alpha}})\delta(\rho_{1})\Big{]}\mathcal{E}\omega
CCC\qquad$ (B.1) $\displaystyle+\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\rho_{3})\Big{[}i\delta(\rho_{4})-t^{\gamma}(p_{1\gamma}+p_{2\gamma})\Big{]}\Big{\\{}\delta(\mathcal{T})\widetilde{t}^{\alpha}y_{\alpha}+\delta(\xi_{3})(z_{\alpha}\widetilde{t}^{\alpha}+\underline{z_{\alpha}y^{\alpha}})\Big{\\}}\mathcal{E}\omega
CCC\,,\qquad$

where the terms in the second row of formula (B.1) result from applying $GT$
-identity. Rewriting the underlined part as the result of differentiation with
respect to $\mathcal{T}$ and performing partial integration one obtains Eq.
(8.4) plus the cohomology term $J_{8}$ (5.9).

###  13.2  $({{\rm
d}}_{x}B^{\eta\eta}_{3}{}+\omega*B^{\eta\eta}_{3})|_{\delta(\rho_{1})}+W^{\eta}_{1\,\omega
C}*B^{\eta\,loc}_{2}$

Uniformization of the sum of $\delta(\rho_{1})-$proportional terms on the
r.h.s.’s of (6.2) and (6.1) is done with the help of $GT$ identity (7.8) as
follows. Denoting

$\widetilde{P}=y+p_{1}+p_{2}+{t}-\rho_{2}(p_{3}+p_{2})$ (B.2)

one can see that partial integration in $\mathcal{T}$ yields

${{\rm
d}}_{x}{B}^{\eta\eta}_{3}\bigg{|}_{\delta(\rho_{1})}\approx-\frac{i\eta^{2}}{4}\int
d\Gamma\,\delta(\rho_{4})\delta(\rho_{1})\delta(\xi_{1})\Big{[}i\delta(\mathcal{T})-z_{\alpha}y^{\alpha}\Big{]}\exp\Big{\\{}i\mathcal{T}z_{\alpha}\widetilde{P}^{\alpha}-i\xi_{2}\widetilde{P}^{\alpha}y_{\alpha}\\\
+i(1-\rho_{2})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})y_{\alpha}+ip_{3\alpha}y^{\alpha}+it^{\beta}p_{1\beta}\Big{\\}}\omega
CCC,$ (B.3)
$\omega\ast{B}^{\eta\eta}_{3}\bigg{|}_{\delta(\rho_{1})}\approx\frac{i\eta^{2}}{4}\int
d\Gamma\,\delta(\rho_{4})\delta(\rho_{1})\delta(\xi_{3})\Big{[}i\delta(\mathcal{T})-z_{\alpha}(y^{\alpha}+t^{\alpha})\Big{]}\exp\Big{\\{}i\mathcal{T}z_{\alpha}\widetilde{P}^{\alpha}-i\xi_{2}\widetilde{P}^{\alpha}y_{\alpha}\\\
+i\xi_{1}\widetilde{P}^{\alpha}t_{\alpha}+i(1-\rho_{2})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})y_{\alpha}+ip_{3\alpha}y^{\alpha}+it^{\beta}p_{1\beta}\Big{\\}}\omega
CCC\,.$ (B.4)

The sum of (B.3) and (B.4) gives

$\Big{(}{{\rm d}}_{x}{B}^{\eta\eta}_{3}+\omega\ast
B_{3}^{\eta\eta}\Big{)}\bigg{|}_{\delta(\rho_{1})}\approx\frac{i\eta^{2}}{4}\int
d\Gamma\,\delta(\rho_{4})\delta(\rho_{1})\Big{[}z_{\gamma}(-t^{\gamma}-y^{\alpha})\delta(\xi_{3})+z_{\gamma}y^{\gamma}\delta(\xi_{1})+z_{\gamma}t^{\gamma}\delta(\xi_{2})\Big{]}\times\\\
\times\exp\Big{\\{}i\mathcal{T}z_{\alpha}\widetilde{P}^{\alpha}-i\xi_{2}\widetilde{P}^{\alpha}y_{\alpha}+i\xi_{1}\widetilde{P}^{\alpha}t_{\alpha}+i(1-\rho_{2})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})y_{\alpha}+ip_{3\alpha}y^{\alpha}+it^{\beta}p_{1\beta}\Big{\\}}\omega
CCC\\\ -\frac{i\eta^{2}}{4}\int
d\Gamma\,\delta(\rho_{4})\delta(\rho_{1})(z_{\gamma}t^{\gamma})\delta(\xi_{2})\exp\Big{\\{}i\mathcal{T}z_{\alpha}\widetilde{P}^{\alpha}-i\xi_{2}\widetilde{P}^{\alpha}y_{\alpha}+i\xi_{1}\widetilde{P}^{\alpha}t_{\alpha}+i(1-\rho_{2})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})y_{\alpha}\\\
+ip_{3\alpha}y^{\alpha}+it^{\beta}p_{1\beta}\Big{\\}}\omega
CCC\,+J_{9}+J_{10}$ (B.5)

with $J_{9}$ (5.10) and $J_{10}$ (5.11). By virtue of GT identity (7.8) the
first term weakly equals $J_{11}$ (5.12). Finally, Eq. (B.5) yields

$\Big{(}{{\rm d}}_{x}{B}^{\eta\eta}_{3}+\omega\ast
B_{3}^{\eta\eta}\Big{)}\bigg{|}_{\delta(\rho_{1})}\approx-\frac{i\eta^{2}}{4}\int
d\Gamma\,\delta(\rho_{4})\delta(\rho_{1})(z_{\gamma}t^{\gamma})\delta(\xi_{2})\exp\Big{\\{}i\mathcal{T}z_{\alpha}\widetilde{P}^{\alpha}-i\xi_{2}\widetilde{P}^{\alpha}y_{\alpha}\\\
+i\xi_{1}\widetilde{P}^{\alpha}t_{\alpha}+i(1-\rho_{2})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})y_{\alpha}+ip_{3\alpha}y^{\alpha}+it^{\beta}p_{1\beta}\Big{\\}}\omega
CCC\,+J_{9}+J_{10}+J_{11}.$ (B.6)

Consider $W_{1\omega C}^{\eta}\ast B_{2}^{\eta\,loc}$ (3.13). This is
convenient to change integration variables, moving from the integration over
simplex to integration over square. As a result

$W_{1\omega C}^{\eta}\ast
B_{2}^{\eta\,loc}\approx\frac{\eta^{2}}{4}\int_{0}^{1}d\mathcal{T}\,\mathcal{T}\int
d^{2}\tau_{+}\,\delta(1-\tau_{1}-\tau_{2})\int_{0}^{1}d\sigma_{1}\int_{0}^{1}d\sigma_{2}\,(z_{\alpha}t^{\alpha})\times\\\
\Big{[}z_{\alpha}y^{\alpha}+\sigma_{1}z_{\alpha}t^{\alpha}\Big{]}\exp\Big{\\{}i\mathcal{T}z_{\alpha}y^{\alpha}+i(1-\sigma_{2})\sigma_{1}t_{\alpha}p_{1}{}^{\alpha}+i\sigma_{1}\sigma_{2}t^{\alpha}p_{3\alpha}+i(1-\sigma_{1})t^{\alpha}p_{1\alpha}\\\
+i\mathcal{T}z_{\alpha}\Big{(}(\tau_{1}+\tau_{2}\sigma_{1})t^{\alpha}+\tau_{1}p_{1}{}^{\alpha}-(\tau_{2}-\tau_{1}(1-\sigma_{2}))p_{2}{}^{\alpha}-(\tau_{2}+\sigma_{2}\tau_{1})p_{3}{}^{\alpha}\Big{)}+i\sigma_{1}y^{\alpha}t_{\alpha}\\\
-i(1-\sigma_{2})y^{\alpha}p_{2\alpha}+i\sigma_{2}y^{\alpha}p_{3\alpha}+i\sigma_{2}y^{\alpha}p_{3\alpha}\Big{\\}}\omega
CCC.$ (B.7)

Partial integration with respect to $\mathcal{T}$ yields

$W_{1\omega C}^{\eta}\ast
B_{2}^{\eta\,loc}\approx-\frac{\eta^{2}}{4}\int_{0}^{1}d\mathcal{T}\int
d^{2}\tau_{+}\,\delta(1-\tau_{1}-\tau_{2})\int_{0}^{1}d\sigma_{1}\int_{0}^{1}d\sigma_{2}\,(z_{\alpha}t^{\alpha})\times\\\
\Big{[}\mathcal{T}z_{\alpha}\Big{(}\tau_{1}(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})-(\tau_{2}+\sigma_{2}\tau_{1})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})\Big{)}-i\mathcal{T}\tau_{1}(1-\sigma_{1})z_{\alpha}t^{\alpha}\Big{]}\,\exp({\mathbb{F}})\,\,\omega
CCC\,,\qquad$ (B.8)

where

${\mathbb{F}}=i\mathcal{T}z_{\alpha}y^{\alpha}+it^{\beta}p_{1\beta}+i\sigma_{1}\Big{(}y^{\alpha}t_{\alpha}+(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})t_{\alpha}-\sigma_{2}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})t_{\alpha}\Big{)}-i\big{(}\sigma_{2}p_{3}{}^{\alpha}-(1-\sigma_{2})p_{2}{}^{\alpha}\big{)}y_{\alpha}\\\
+i\mathcal{T}z_{\alpha}\Big{(}\tau_{1}(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})-(\tau_{2}+\sigma_{2}\tau_{1})(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})+(\sigma_{1}+\tau_{1}(1-\sigma_{1}))t^{\alpha}\Big{)}.$
(B.9)

By virtue of evident formulas

$\displaystyle\tau_{1}\left(\frac{\partial}{\partial\tau_{1}}-\frac{\partial}{\partial\tau_{2}}\right){\mathbb{F}}=i\mathcal{T}z_{\alpha}\Big{(}\tau_{1}(p_{1}+p_{2}{})+\big{[}(\tau_{1}+\tau_{2})-(\tau_{2}+\sigma_{2}\tau_{1})\big{]}(p_{2}{}+p_{3}{})+\tau_{1}(1-\sigma_{1})t\Big{)}{}^{\alpha}\,,\qquad$
$\displaystyle\frac{\partial}{\partial\sigma_{1}}{\mathbb{F}}=i\mathcal{T}(1-\tau_{1})z_{\alpha}t^{\alpha}+i\Big{(}y^{\alpha}+p_{1}{}^{\alpha}+p_{2}{}^{\alpha}-\sigma_{2}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})\Big{)}t_{\alpha},$

Eq. (B.7) acquires the form

$W_{1\omega C}^{\eta}\ast
B_{2}^{\eta\,loc}\approx\frac{\eta^{2}}{4}\int_{0}^{1}d\mathcal{T}\int
d^{2}\tau_{+}\delta(1-\tau_{1}-\tau_{2})\int_{0}^{1}d\sigma_{1}\int_{0}^{1}d\sigma_{2}\bigg{[}iz_{\alpha}t^{\alpha}\tau_{1}\left(\frac{\partial}{\partial\tau_{1}}-\frac{\partial}{\partial\tau_{2}}\right)\\\
-\frac{z_{\alpha}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})}{1-\tau_{1}}\left(i\frac{\partial}{\partial\sigma_{1}}+\Big{(}y^{\alpha}+p_{1}{}^{\alpha}+p_{2}{}^{\alpha}-\sigma_{2}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})\Big{)}t_{\alpha}\right)+iz_{\alpha}t^{\alpha}\bigg{]}\exp({\mathbb{F}})\omega
CCC.$ (B.10)

After partial integrations in $\tau_{1}$,$\tau_{2}$ and $\sigma_{1}$ one
obtains

$\displaystyle W_{1\omega C}^{\eta}\ast
B_{2}^{\eta\,loc}\approx\frac{\eta^{2}}{4}\int_{0}^{1}d\mathcal{T}\int
d^{2}\tau_{+}\delta(1-\tau_{1}-\tau_{2})\int_{0}^{1}d\sigma_{1}\int_{0}^{1}d\sigma_{2}\bigg{[}\underline{iz_{\alpha}t^{\alpha}\delta(\tau_{2})}$
(B.11)
$\displaystyle+\frac{z_{\alpha}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})}{1-\tau_{1}}\left(i\big{(}\delta(\sigma_{1})-\delta(1-\sigma_{1})\big{)}-\Big{(}y^{\alpha}+p_{1}{}^{\alpha}+p_{2}{}^{\alpha}-\sigma_{2}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})\Big{)}t_{\alpha}\right)\bigg{]}\exp({\mathbb{F}})\omega
CCC\,.$

After a simple change of integration variables the underlined term on the
r.h.s. of Eq. (B.11) cancels the r.h.s. of Eq. (B.6). Performing integration
with respect to $\tau_{2}$ in the remaining part of (B.11), after the
following change of the integration variables

$\displaystyle\int_{0}^{1}d\sigma_{1}\int_{0}^{1}d\tau_{1}\int_{0}^{1}d\sigma_{2}\,f(\sigma_{1},1-\sigma_{1},\tau_{1},\sigma_{2})$
$\displaystyle=\int
d^{4}\rho_{+}\,\delta\left(1-\sum_{j=1}^{4}\rho_{j}\right)\frac{1}{(\rho_{2}+\rho_{3})(1-\rho_{2}-\rho_{3})}f\left(\frac{\rho_{1}}{1-\rho_{2}-\rho_{3}},\frac{\rho_{4}}{1-\rho_{2}-\rho_{3}},\rho_{2}+\rho_{3},\frac{\rho_{2}}{\rho_{2}+\rho_{3}}\right)\,,$

$\exp({\mathbb{F}})$ (B.9) acquires the form ${\cal E}$ (4.2). As a result,
the sum of Eq. (B.11) and Eq. (B.6) by virtue Eq. (E.1) yields Eq. (8.5).

## Appendix C: Eliminating $\delta(\rho_{j})$ and $\delta(\xi_{j})$

To eliminate $\delta(\rho_{j})$ and $\delta(\xi_{j})$ from of the r.h.s.’s of
Eqs. (8.2), (8.3) this is convenient to group similar pre-exponential terms as
in Sections C.1 -C.5.

###  C.1 Terms proportional to
$(p_{1}{}+p_{2})^{\alpha}(p_{3}{}+p_{2})_{\alpha}$

Consider $F_{1,1}+F_{2,1}$ of (8.2) and (8.3), respectively. Partial
integration with respect to $\rho_{1}$ and $\rho_{4}$ yields

$F_{1,1}+F_{2,1}\approx-\frac{\eta^{2}}{4}\int
d\Gamma\frac{\delta(\xi_{3})\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})(p_{2\alpha}+p_{3\alpha})\times\\\
\times(z_{\gamma}{\cal
P}^{\gamma})\left(\frac{\partial}{\partial\rho_{4}}-\frac{\partial}{\partial\rho_{1}}\right){\cal
E}\omega CCC.$ (C.1)

By direct calculation, Eq. (C.1) gives

$F_{1,1}+F_{2,1}\approx-\frac{\eta^{2}}{4}\int
d\Gamma\frac{\delta(\xi_{3})\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})(p_{2\alpha}+p_{3\alpha})\times\\\
\Bigg{[}{E_{z}{\,}}\left(\frac{\partial}{\partial\rho_{4}}-\frac{\partial}{\partial\rho_{1}}\right)(z_{\gamma}{\cal
P}^{\gamma})E+(z_{\gamma}{\cal
P}^{\gamma})\mathcal{T}(z_{\alpha}t^{\alpha})\mathcal{E}\Bigg{]}\omega CCC\,.$
(C.2)

By virtue of the Schouten identity

$z_{\alpha}t^{\alpha}(p_{1}+p_{2})^{\gamma}(p_{3}{}+p_{2}{})_{\gamma}=t^{\alpha}(p_{1}+p_{2}){}_{\alpha}z^{\gamma}(p_{3}{}+p_{2}{})_{\gamma}+t^{\alpha}(p_{3}{}+p_{2}{})_{\alpha}(p_{1}+p_{2})^{\gamma}z_{\gamma}$
(C.3)

and its consequence

$z_{\alpha}t^{\alpha}(p_{1}+p_{2})^{\gamma}(p_{3}{}+p_{2}{})_{\gamma}{\cal
E}=t^{\alpha}(p_{1}+p_{2})_{\alpha}\left[i\left(\frac{\overleftarrow{\partial}}{\partial\rho_{2}}-\frac{\overleftarrow{\partial}}{\partial\rho_{3}}\right){E_{z}{\,}}E+i{E_{z}{\,}}\left(\frac{\partial}{\partial\rho_{2}}-\frac{\partial}{\partial\rho_{3}}\right)E\right]\\\
+t^{\alpha}(p_{2}+p_{3})_{\alpha}\left[i\left(\frac{\overleftarrow{\partial}}{\partial\rho_{2}}-\frac{\overleftarrow{\partial}}{\partial\rho_{1}}\right){E_{z}{\,}}E+i{E_{z}{\,}}\left(\frac{\partial}{\partial\rho_{2}}-\frac{\partial}{\partial\rho_{1}}\right)E\right]\,$
(C.4)

Eq. (C.1) yields

$F_{1,1}+F_{2,1}\approx+\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\Bigg{\\{}\frac{(z_{\gamma}{\cal
P}^{\gamma})\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\times\\\
\times\Bigg{(}(p_{1}{}+p_{2})^{\alpha}(p_{3}{}+p_{2})_{\alpha}{E_{z}{\,}}\Bigg{[}\frac{\partial}{\partial\rho_{1}}-\frac{\partial}{\partial\rho_{4}}\Bigg{]}E+t^{\alpha}(p_{1}+p_{2}){}_{\alpha}\Bigg{[}\underline{\delta(\rho_{3})}{\cal
E}-{E_{z}{\,}}\Bigg{(}\frac{\partial}{\partial\rho_{2}}-\frac{\partial}{\partial\rho_{3}}\Bigg{)}E\Bigg{]}\\\
+t^{\alpha}(p_{3}{}+p_{2}{})_{\alpha}\Bigg{[}\underline{\delta(\rho_{1})}{\cal
E}-{E_{z}{\,}}\Bigg{(}\frac{\partial}{\partial\rho_{2}}-\frac{\partial}{\partial\rho_{1}}\Bigg{)}E\Bigg{]}\Bigg{)}+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\Big{(}t^{\alpha}z_{\alpha}(p_{3}+p_{2})^{\gamma}(p_{1}+p_{2}){}_{\gamma}{\cal
E}\Big{)}\\\ +(z_{\gamma}{\cal
P}^{\gamma})\Bigg{(}-\frac{1-\rho_{3}-\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}t^{\alpha}(p_{1}+p_{2}){}_{\alpha}{\cal
E}-\frac{1-\rho_{1}-\rho_{4}-\rho_{2}}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})}t^{\alpha}(p_{3}{}+p_{2}{})_{\alpha}{\cal
E}\Bigg{)}\Bigg{\\}}\omega CCC\,.$ (C.5)

One can see that $\delta(\rho_{1})$\- and $\delta(\rho_{3})$-proportional
terms on the r.h.s. of (C.5) (the underlined ones) cancel terms $F_{2,4}$
(8.3) and $F_{3,3}$ (8.4), respectively.

###  C.2 Term proportional to $t^{\alpha}(p_{1\alpha}+p_{2\alpha})$

Consider term $F_{3,5}$ of $F_{3}$ (8.4). By virtue of the following identity

$\frac{\rho_{2}}{(\rho_{2}+\rho_{3})(1-\rho_{3})}\left(\delta(\rho_{3})-\delta(\rho_{2})\right)=1$
(C.6) $F_{3,5}\approx-\frac{\eta^{2}}{4}\int
d\Gamma\,\frac{\delta(\xi_{3})\rho_{2}}{(\rho_{2}+\rho_{3})(1-\rho_{3})}\Big{(}\delta(\rho_{3})-\delta(\rho_{2})\Big{)}\\\
\Big{[}(p_{2}{}_{\alpha}+p_{1}{}_{\alpha})t^{\alpha}(z_{\gamma}t^{\gamma})\Big{(}(1-\rho_{4})-\frac{\rho_{1}}{(\rho_{1}+\rho_{4})}\Big{)}{\cal
E}\Big{]}\omega CCC.$ (C.7)

Partial integrations along with the Schouten identity

$t^{\alpha}(p_{1\alpha}+p_{2\alpha})(p_{3}{}^{\gamma}+p_{2}{}^{\gamma})z_{\gamma}=-\underline{t^{\alpha}z_{\alpha}}(p_{1}{}^{\gamma}+p_{2}{}^{\gamma})(p_{2\gamma}+p_{3\gamma})+t^{\alpha}(p_{3\alpha}+p_{2\alpha})\underline{(p_{1}+p_{2})^{\gamma}z_{\gamma}}$
(C.8)

and realization of the underlined terms as derivative of ${E_{z}{\,}}$ along
with further partial integration yields

$F_{3,5}\approx-\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\Bigg{[}\frac{\rho_{4}}{(1-\rho_{3})^{2}}\Big{(}(p_{2}{}_{\alpha}+p_{1}{}_{\alpha})t^{\alpha}z_{\gamma}t^{\gamma}\Big{)}{\cal
E}\\\
+\frac{\rho_{2}\rho_{4}}{(\rho_{1}+\rho_{4})(1-\rho_{3})}\Big{(}(p_{2}{}_{\alpha}+p_{1}{}_{\alpha})t^{\alpha}z_{\gamma}t^{\gamma}\Big{)}{E_{z}{\,}}\left[\frac{\partial}{\partial\rho_{2}}-\frac{\partial}{\partial\rho_{3}}\right]E+\frac{\rho_{2}\rho_{4}}{(\rho_{1}+\rho_{4})(1-\rho_{3})}(z_{\alpha}t^{\alpha})\times\\\
\times\Bigg{(}-(p_{1}+p_{2})^{\gamma}(p_{3}{}+p_{2}{})_{\gamma}{E_{z}{\,}}\Bigg{[}\frac{\partial}{\partial\rho_{1}}-\frac{\partial}{\partial\rho_{4}}\Bigg{]}E-\underline{\delta(\rho_{1})}(p_{1}+p_{2})^{\gamma}(p_{3}{}+p_{2}{})_{\gamma}{\cal
E}\\\
-t^{\alpha}((p_{3}{}+p_{2}{})_{\alpha}){E_{z}{\,}}\Bigg{[}\frac{\partial}{\partial\rho_{1}}-\frac{\partial}{\partial\rho_{2}}\Bigg{]}E-\underline{\delta(\rho_{1})}t^{\alpha}((p_{3}{}+p_{2}{})_{\alpha}){\cal
E}\Bigg{)}\\\
+(z_{\alpha}t^{\alpha})\Bigg{(}\frac{\rho_{2}}{(1-\rho_{3})(\rho_{1}+\rho_{4})}(p_{1}+p_{2})^{\gamma}(p_{3}{}+p_{2}{})_{\gamma}{\cal
E}+\frac{\rho_{4}}{(\rho_{1}+\rho_{4})^{2}}t^{\alpha}((p_{3}{}+p_{2}{})_{\alpha}){\cal
E}\Bigg{)}\Bigg{]}\omega CCC.$ (C.9)

One can see that the sum of the underlined $\delta(\rho_{1})$-proportional
terms cancel $F_{2,2}+F_{2,3}$ of (8.3).

###  C.3 Sum of
$(p_{1}{}+p_{2})^{\alpha}(p_{3}{}+p_{2})_{\alpha}$-proportional and
$t^{\alpha}(p_{1\alpha}+p_{2\alpha})$–proportional terms

Summing up $F_{1,1}+F_{2,1}$ (C.5), $F_{3,3}$ (8.4), $F_{3,5}$ (C.9) and
$F_{2,2}+F_{2,3}+F_{2,4}$ (8.3), then performing partial integrations and
using the following simple identities

$(1-\rho_{4})-\frac{\rho_{1}}{(\rho_{1}+\rho_{4})}=\frac{\rho_{4}(\rho_{2}+\rho_{3})}{(\rho_{1}+\rho_{4})},$
(C.10)
$-\frac{\rho_{4}}{(\rho_{1}+\rho_{4})^{2}}+\frac{\rho_{4}}{(\rho_{1}+\rho_{4})}\frac{\rho_{3}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}=\frac{-\rho_{2}\rho_{4}}{(\rho_{1}+\rho_{4})^{2}(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\,,\qquad$
(C.11)

one obtains by virtue of Eqs. (4.5)-(4.7)

$F_{1,1}+F_{2,1}+F_{2,4}+F_{3,3}+F_{3,5}+F_{2,2}+F_{2,3}=G_{1}$ (C.12)

with $G_{1}$ (9.2).

###  C.4 Terms proportional to $\delta(\xi_{1})-\delta(\xi_{2})$

Consider a sum of $F_{1,4}$ (8.2) and $F_{2,8}$ (8.3). Performing partial
integrations with respect to $\rho_{1}$ and $\rho_{4}$, then applying the
Schouten identity one obtains

$F_{1,4}+F_{2,8}\approx-\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\Bigg{[}\frac{\partial}{\partial\rho_{1}}-\frac{\partial}{\partial\rho_{4}}\Bigg{]}\frac{iz_{\alpha}(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})}{1-\rho_{3}}\Big{(}\delta(\xi_{2})-\delta(\xi_{1})\Big{)}{\cal
E}\,\omega CCC=\\\ =-\frac{\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\Big{(}\delta(\xi_{2})-\delta(\xi_{1})\Big{)}\Bigg{\\{}\frac{i\,z_{\gamma}t^{\gamma}}{(1-\rho_{3})}\Bigg{(}{E_{z}{\,}}\Bigg{[}\frac{\partial}{\partial\rho_{1}}-\frac{\partial}{\partial\rho_{2}}\Bigg{]}E+\Big{(}\underline{\delta(\rho_{1})}-\underline{\underline{\delta(\rho_{2})}}\Big{)}{\cal
E}\Bigg{)}\\\
+\frac{i\,z_{\alpha}(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})}{(1-\rho_{3})}{E_{z}{\,}}\Bigg{[}\frac{\partial}{\partial\rho_{1}}-\frac{\partial}{\partial\rho_{4}}\Bigg{]}E\Bigg{\\}}\omega
CCC.$ (C.13)

The underlined $\delta(\rho_{1})$-proportional term compensates $F_{2,9}$ of
(8.3). The double underlined $\delta(\rho_{2})$-proportional term vanishes due
to the factor of $(\delta(\xi_{2})-\delta(\xi_{1}))$ which after partial
integrations in $\xi_{1}$ and $\xi_{2}$ produces an expression proportional to
$\rho_{2}$.

Summing up $F_{1,4}+F_{2,8}$ (C.13) and $F_{2,9}$ (8.3), performing partial
integrations with respect to $\xi$ and $\mathcal{T}$ along with the Schouten
identity one obtains

$F_{1,4}+F_{2,8}+F_{2,9}\approx G_{2}$ (C.14)

with $G_{2}$ (9.3).

###  C.5 Terms proportional to $\xi_{1}\delta(\xi_{2})$

Consider a sum of $F_{1,3}$ (8.2), $F_{2,6}$ (8.3) and $F_{4,1}$ (8.5).

$\displaystyle F_{1,3}+F_{2,6}+F_{4,1}\approx\frac{i\eta^{2}}{4}\int
d\Gamma\,\frac{\delta(\xi_{3})\delta(\xi_{2})[\delta(\rho_{1})-\delta(\rho_{4})]}{(\rho_{2}+\rho_{3})}z_{\alpha}\bigg{\\{}\frac{{\cal
P}^{\alpha}}{(1-\rho_{3})}-\frac{\xi_{1}\,(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})}{(\rho_{1}+\rho_{4})}\bigg{\\}}{\cal
E}\,\omega CCC.\quad$ (C.15)

Partial integration yields

$F_{1,3}+F_{2,6}+F_{4,1}\approx\frac{i\eta^{2}}{4}\int
d\Gamma\,\delta(\xi_{3})\delta(\xi_{2})\xi_{1}\Bigg{\\{}z_{\alpha}t^{\alpha}\Bigg{[}\frac{1}{\rho_{1}+\rho_{4}}\bigg{(}{E_{z}{\,}}\bigg{[}\frac{\partial}{\partial\rho_{2}}-\frac{\partial}{\partial\rho_{3}}\bigg{]}E+\Big{[}\underline{\delta(\rho_{2})}-\delta(\rho_{3})\Big{]}{\cal
E}\bigg{)}\\\
+\frac{1}{1-\rho_{3}}\bigg{(}{E_{z}{\,}}\bigg{[}\frac{\partial}{\partial\rho_{1}}-\frac{\partial}{\partial\rho_{2}}\bigg{]}E+\Big{[}\delta(\rho_{1})-\underline{\delta(\rho_{2})}\Big{]}{\cal
E}\bigg{)}\Bigg{]}\\\
+\bigg{[}\frac{z_{\alpha}(p_{2}{}^{\alpha}+p_{3}{}^{\alpha})}{\rho_{1}+\rho_{4}}+\frac{z_{\alpha}(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})}{1-\rho_{3}}\bigg{]}{E_{z}{\,}}\bigg{[}\frac{\partial}{\partial\rho_{1}}-\frac{\partial}{\partial\rho_{4}}\bigg{]}E\Bigg{\\}}\omega
CCC\,.$ (C.16)

One can see that the underlined $\delta({\rho_{2}})$-proportional terms vanish
due to the factor of $\delta(1-\sum\rho_{i})$ (4.8), while
$\delta({\rho_{1}})$-proportional term compensates $F_{2,7}$ (8.3) and
$\delta({\rho_{3}})$-proportional term compensates $F_{3,2}$ (8.4).

Summing up $F_{2,7}$ (8.3), $F_{3,2}$ (8.3), $F_{4,2}$ and
$F_{1,3}+F_{2,6}+F_{4,1}$ (8.5), and then performing partial integration in
$\mathcal{T}$ one obtains by virtue of the Schouten identity

$F_{1,3}+F_{2,6}+F_{4,1}+F_{2,7}+F_{3,2}+F_{4,2}\approx
G_{3}:=\frac{\eta^{2}}{4}\int d\Gamma\,\delta(\xi_{3})\delta(\xi_{2})\times\\\
\times\Bigg{\\{}\frac{\rho_{2}\,(z_{\alpha}t^{\alpha})(p_{2}{}^{\gamma}+p_{3}{}^{\gamma})(y_{\gamma}+\tilde{t}_{\gamma})}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})(\rho_{1}+\rho_{4})}+\frac{\rho_{2}\,\Big{[}(\tilde{t}^{\gamma}+y^{\gamma})(y_{\gamma}+\mathbb{P}_{\gamma})(z^{\alpha}t_{\alpha})+i\delta(\mathcal{T})t_{\gamma}(\tilde{t}^{\gamma}-\mathbb{P}^{\gamma})\Big{]}}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})^{2}(\rho_{1}+\rho_{4})}\\\
+\frac{\rho_{3}\,\big{[}i\delta(\mathcal{T})-z_{\gamma}(y^{\gamma}+\tilde{t}^{\gamma})\big{]}(t^{\alpha}y_{\alpha})}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})^{2}}+\frac{\big{[}-i\delta(\mathcal{T})+z_{\gamma}(y^{\gamma}+\tilde{t}^{\gamma})\big{]}(p_{1}{}^{\alpha}+p_{2}{}^{\alpha})t_{\alpha}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}\Bigg{\\}}{\cal
E}\omega CCC\,.$ (C.17)

Since by the partial integration procedure
$\xi_{1}\delta(\xi_{2})\equiv{1}+\xi_{1}(\partial_{\xi_{1}}-\partial_{\xi_{2}})$,
(C.17) yields $G_{3}$ (9.4).

## Appendix D: Details of the final step of the calculation

By virtue of Eqs. (E.1)-(E.3), Eq. (10.2) yields

$\displaystyle S_{6}=+i\frac{\eta^{2}}{4}\int d\Gamma\,\delta(\xi_{3})$ (D.1)
$\displaystyle\Big{\\{}+(y+\tilde{t}{})^{\gamma}z_{\gamma}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}\xi_{1}\frac{1-\rho_{3}-\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}\,\,\mathbb{P}{}^{\alpha}y_{\alpha}$
$\displaystyle+(y+\tilde{t}{})^{\gamma}z_{\gamma}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}\xi_{1}\frac{1-\rho_{3}-\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}\big{(}y+\mathbb{P}{}\big{)}^{\alpha}\tilde{t}{}_{\alpha}$
$\displaystyle+(y+\tilde{t}{})^{\gamma}z_{\gamma}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}(-)\xi_{1}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\,(p_{3}{}^{\alpha}+p_{2}{}^{\alpha})y_{\alpha}$
$\displaystyle-(y+\tilde{t}{})^{\gamma}z_{\gamma}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}\xi_{1}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}(p_{3}{}+p_{2}{})^{\beta}\tilde{t}{}_{\beta}+$
$\displaystyle+(y+\tilde{t}{})^{\gamma}z_{\gamma}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}(-)\frac{\rho_{1}+\rho_{4}}{(1-\rho_{3})^{2}}\,\,((p{}_{1}+p_{2}))^{\alpha}y_{\alpha}$
$\displaystyle+(y+\tilde{t}{})^{\gamma}z_{\gamma}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}(-)\frac{\rho_{4}}{(1-\rho_{3})^{2}}\,\,{t}{}^{\alpha}y_{\alpha}$
$\displaystyle+(y+\tilde{t}{})^{\gamma}z_{\gamma}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}(-)\frac{\rho_{1}}{(1-\rho_{3})^{2}}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}(p_{1}{}+p_{2})^{\gamma}\big{(}y+(1-\rho_{4}){t}{}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}(-)\xi_{1}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\,\,{t}{}^{\alpha}y_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}(p_{1}{}+p_{2})^{\gamma}\big{(}y+(1-\rho_{4}){t}{}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}$
$\displaystyle\times(-)\xi_{1}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})(\rho_{1}+\rho_{4})}\big{(}y^{\alpha}+\mathbb{P}{}^{\alpha}\big{)}{t}{}_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}(p_{1}{}+p_{2})^{\gamma}\big{(}y+(1-\rho_{4}){t}{}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}\frac{1}{(1-\rho_{3})}\,\,{t}{}^{\alpha}y_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}(p_{1}{}+p_{2})^{\gamma}\big{(}y+(1-\rho_{4}){t}{}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}(-)\frac{1}{(1-\rho_{3})}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}+$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}\xi_{1}\frac{\rho_{3}}{(1-\rho_{1}-\rho_{4})^{2}}\,\,(-(p_{3}+p_{2}))^{\alpha}y_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}\xi_{1}\frac{\rho_{3}}{(1-\rho_{1}-\rho_{4})^{2}}\big{(}(-(p_{3}+p_{2}))^{\alpha}\big{)}\tilde{t}{}_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}$
$\displaystyle\times\xi_{1}\frac{\rho_{3}\rho_{4}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})(\rho_{1}+\rho_{4})}\,\,{t}{}^{\alpha}y_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}\xi_{1}\frac{1}{(1-\rho_{3})}\,(p_{1}+p_{2})^{\alpha}y_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}\xi_{1}\frac{1}{(1-\rho_{3})}(p_{1}+p_{2})^{\alpha}\tilde{t}{}_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}$
$\displaystyle\times(-)\xi_{1}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\frac{\rho_{4}}{(\rho_{1}+\rho_{4})^{2}}\big{(}y^{\alpha}+\mathbb{P}{}^{\alpha}\big{)}{t}{}_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}(-)\frac{1}{(1-\rho_{3})}\,(p_{1}+p_{2})^{\alpha}y_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}(-)\frac{1}{(1-\rho_{3})}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}+$
$\displaystyle+\xi_{1}\Big{[}\frac{\rho_{2}\rho_{2}}{(1-\rho_{1}-\rho_{4})^{3}(1-\rho_{3})^{3}(\rho_{1}+\rho_{4})}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{})+(1-\rho_{4}){t}{}\big{)}^{\gamma}\big{(}y+\tilde{t}{}\big{)}_{\gamma}z_{\alpha}{t}{}^{\alpha}$
$\displaystyle-\frac{\rho_{3}\rho_{2}}{(1-\rho_{1}-\rho_{4})^{3}(1-\rho_{3})^{3}}{\big{(}y+\tilde{t}{}\big{)}^{\gamma}z_{\gamma}{t}{}^{\alpha}y_{\alpha}}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})^{3}}\big{(}y+\tilde{t}{}\big{)}^{\gamma}z_{\gamma}{(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}}\Big{]}\big{(}y+\mathbb{P}{}\big{)}^{\beta}(y+\tilde{t}{})_{\beta}\Big{\\}}{\cal
E}\omega CCC\,.$

Terms from the r.h.s. of (D.1) with $\xi$-independent pre-exponentials are
considered in Section D.1, while those with $\xi_{1}$-proportional pre-
exponentials are considered in Section D.2.

###  D.1  $\xi_{1}$-independent pre-exponentials

Here we consider only pre-exponentials, omitting for brevity integrals,
integral measures etc of (D.1). By virtue of the Schouten identity taking into
account that $\sum\rho_{i}=1$ Eq. (D.1) yields

$\displaystyle
Integrand(S_{6})\Big{|}_{\mod\xi}=(y+\tilde{t}{})^{\nu}z_{\nu}\Big{\\{}-\frac{\rho_{2}(\rho_{1}+\rho_{4})}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{3}}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}\,\,((p{}_{1}+p_{2}))^{\alpha}y_{\alpha}\qquad$
$\displaystyle-\frac{\rho_{2}\rho_{4}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{3}}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}\,{t}{}^{\alpha}y_{\alpha}$
$\displaystyle-\frac{\rho_{2}\rho_{1}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{3}}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{3}}(p_{1}{}+p_{2})^{\gamma}\big{(}y+(1-\rho_{4}){t}{}\big{)}_{\gamma}\,\,{t}{}^{\alpha}y_{\alpha}$
$\displaystyle-\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{3}}(p_{1}{}+p_{2})^{\gamma}\big{(}y+(1-\rho_{4}){t}{}\big{)}_{\gamma}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}$
$\displaystyle-\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{3}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}\,(p_{1}+p_{2})^{\alpha}y_{\alpha}$
$\displaystyle-\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{3}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}\Big{\\}}{\cal
E}\omega CCC=$
$\displaystyle=(y+\tilde{t}{})^{\nu}z_{\nu}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{3}}\Big{\\{}\rho_{1}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}+(p_{1}{}+p_{2})^{\gamma}y_{\gamma}\,\,{t}{}^{\alpha}y_{\alpha}$
$\displaystyle-(p_{1}{}+p_{2})^{\gamma}(1-\rho_{4}){t}{}_{\gamma}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}$
$\displaystyle-{t}{}^{\gamma}y_{\gamma}\,(p_{1}+p_{2})^{\alpha}y_{\alpha}-{t}{}^{\gamma}(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{})_{\gamma}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}\Big{\\}}{\cal
E}\omega CCC$
$\displaystyle=(y+\tilde{t}{})^{\nu}z_{\nu}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{3}}\Big{\\{}-\rho_{1}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}(p_{1}{}^{\alpha}+p_{2}{}^{\beta}){t}{}_{\beta}$
$\displaystyle-(p_{1}{}+p_{2})^{\gamma}(1-\rho_{4}){t}{}_{\gamma}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}-{t}{}^{\gamma}(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{})_{\gamma}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}\Big{\\}}{\cal
E}\omega CCC\equiv 0.$

###  D.2  $\xi_{1}$-proportional pre-exponentials

$\displaystyle S_{6}\,\Big{|}_{\xi_{1}}=J_{7}+i\frac{\eta^{2}}{4}\int
d\Gamma\delta(\xi_{3})$ (D.2)
$\displaystyle\Big{\\{}(y+\tilde{t}{})^{\gamma}z_{\gamma}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}\xi_{1}\frac{1-\rho_{3}-\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}\,\,\mathbb{P}{}^{\alpha}y_{\alpha}$
$\displaystyle+(y+\tilde{t}{})^{\gamma}z_{\gamma}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}\xi_{1}\frac{1-\rho_{3}-\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}\big{(}y+\mathbb{P}{}\big{)}^{\alpha}\tilde{t}{}_{\alpha}$
$\displaystyle-(y+\tilde{t}{})^{\gamma}z_{\gamma}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}\xi_{1}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\,(p_{3}{}^{\alpha}+p_{2}{}^{\alpha})y_{\alpha}$
$\displaystyle-(y+\tilde{t}{})^{\gamma}z_{\gamma}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}\xi_{1}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}(p_{3}{}+p_{2}{})^{\beta}\tilde{t}{}_{\beta}$
$\displaystyle-\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}(p_{1}{}+p_{2})^{\gamma}\big{(}y+(1-\rho_{4}){t}{}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}\xi_{1}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\,\,{t}{}^{\alpha}y_{\alpha}$
$\displaystyle-\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}(p_{1}{}+p_{2})^{\gamma}\big{(}y+(1-\rho_{4}){t}{}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}\xi_{1}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})(\rho_{1}+\rho_{4})}$
$\displaystyle\times\big{(}y^{\alpha}+\mathbb{P}{}^{\alpha}\big{)}{t}{}_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}\xi_{1}\frac{\rho_{3}}{(1-\rho_{1}-\rho_{4})^{2}}\,\,(-(p_{3}+p_{2}))^{\alpha}y_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}\xi_{1}\frac{\rho_{3}}{(1-\rho_{1}-\rho_{4})^{2}}\big{(}(-(p_{3}+p_{2}))^{\alpha}\big{)}\tilde{t}{}_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}$
$\displaystyle\times\xi_{1}\frac{\rho_{3}\rho_{4}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})(\rho_{1}+\rho_{4})}\,\,{t}{}^{\alpha}y_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}\xi_{1}\frac{1}{(1-\rho_{3})}\,(p_{1}+p_{2})^{\alpha}y_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}\xi_{1}\frac{1}{(1-\rho_{3})}(p_{1}+p_{2})^{\alpha}\tilde{t}{}_{\alpha}$
$\displaystyle-\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}z_{\alpha}(y+\tilde{t}{})^{\alpha}$
$\displaystyle\times\xi_{1}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\frac{\rho_{4}}{(\rho_{1}+\rho_{4})^{2}}\big{(}y^{\alpha}+\mathbb{P}{}^{\alpha}\big{)}{t}{}_{\alpha}$
$\displaystyle+\xi_{1}\Big{[}\frac{\rho_{2}\rho_{2}}{(1-\rho_{1}-\rho_{4})^{3}(1-\rho_{3})^{3}(\rho_{1}+\rho_{4})}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{})+(1-\rho_{4}){t}{}\big{)}^{\gamma}\big{(}y+\tilde{t}{}\big{)}_{\gamma}$
$\displaystyle\times\Big{\\{}{t}{}^{\alpha}\big{(}y+\mathbb{P}{}\big{)}_{\alpha}z^{\sigma}(y+\tilde{t}{})_{\sigma}\Big{\\}}-\frac{\rho_{3}\rho_{2}}{(1-\rho_{1}-\rho_{4})^{3}(1-\rho_{3})^{3}}{\big{(}y+\tilde{t}{}\big{)}^{\gamma}z_{\gamma}{t}{}^{\alpha}y_{\alpha}}\big{(}y+\mathbb{P}{}\big{)}^{\sigma}(y+\tilde{t}{})_{\sigma}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})^{3}}\big{(}y+\tilde{t}{}\big{)}^{\gamma}z_{\gamma}\big{(}y+\mathbb{P}{}\big{)}^{\sigma}(y+\tilde{t}{})_{\sigma}{(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}}\Big{]}\Big{\\}}{\cal
E}\omega CCC\,,\qquad$

where $J_{7}$ is the cohomology term (5.8) . This yields

$\displaystyle S_{6}\,\Big{|}_{\xi_{1}}\approx J_{7}+i\frac{\eta^{2}}{4}\int
d\Gamma\delta(\xi_{3})\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})^{2}(1-\rho_{3})^{2}}$
(D.3)
$\displaystyle\xi_{1}(y+\tilde{t}{})^{\gamma}z_{\gamma}\Big{\\{}\frac{(1-\rho_{3}-\rho_{2})}{(1-\rho_{3})}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}{\big{(}y+\mathbb{P}{}\big{)}^{\beta}(y+\tilde{t}{})_{\beta}}$
$\displaystyle-\rho_{2}{t}{}^{\alpha}(p_{1}+p_{2}){}_{\alpha}(p_{3}{}^{\alpha}+p_{2}{}^{\beta})(y+\tilde{t}{})_{\beta}-\frac{\rho_{2}}{(1-\rho_{3})}(p_{1}{}+p_{2})^{\gamma}\big{(}y+(1-\rho_{4}){t}{}\big{)}_{\gamma}\,\,{t}{}^{\alpha}y_{\alpha}$
$\displaystyle-\frac{\rho_{2}}{(1-\rho_{3})(\rho_{1}+\rho_{4})}(p_{1}{}+p_{2})^{\gamma}\big{(}y+(1-\rho_{4}){t}{}\big{)}_{\gamma}\big{(}y+\mathbb{P}{}\big{)}^{\alpha}{t}{}_{\alpha}$
$\displaystyle-\frac{\rho_{3}}{(1-\rho_{1}-\rho_{4})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}(p_{3}+p_{2})^{\alpha}(y+\tilde{t}{})_{\alpha}$
$\displaystyle+\frac{\rho_{3}\rho_{4}}{(1-\rho_{3})(\rho_{1}+\rho_{4})}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}\,{t}{}^{\alpha}y_{\alpha}$
$\displaystyle+\frac{(1-\rho_{1}-\rho_{4})}{(1-\rho_{3})}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}(p_{1}+p_{2})^{\alpha}(y+\tilde{t}{})_{\alpha}$
$\displaystyle-\frac{\rho_{2}\rho_{4}}{(1-\rho_{3})(\rho_{1}+\rho_{4})^{2}}{t}{}^{\gamma}\big{(}y+(1-\rho_{1}-\rho_{4})(p_{1}{}+p_{2}{}){}\big{)}_{\gamma}\big{(}y^{\alpha}+\mathbb{P}{}^{\alpha}\big{)}{t}{}_{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})(\rho_{1}+\rho_{4})}\big{(}y+(1-\rho_{4}){t}{}\big{)}^{\gamma}\big{(}y+\tilde{t}{}\big{)}_{\gamma}{t}{}_{\alpha}\big{(}y+\mathbb{P}{}\big{)}^{\alpha}$
$\displaystyle+\frac{\rho_{2}}{(1-\rho_{3})(\rho_{1}+\rho_{4})}(p_{1}{}+p_{2}{})^{\gamma}\big{(}y+\tilde{t}{}\big{)}_{\gamma}{t}{}_{\alpha}\big{(}y+\mathbb{P}{}\big{)}^{\alpha}$
$\displaystyle-\frac{\rho_{3}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}{t}{}^{\alpha}y_{\alpha}\big{(}y+\mathbb{P}{}\big{)}^{\sigma}(y+\tilde{t}{})_{\sigma}$
$\displaystyle+\frac{1}{(1-\rho_{3})}\big{(}y+\mathbb{P}{}\big{)}^{\sigma}(y+\tilde{t}{})_{\sigma}{(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}}\Big{\\}}{\cal
E}\omega CCC\equiv J_{7}$

since, using the Schouten identity, one can see that the pre-exponential of
the integrand on the r.h.s. of (D.3) equals zero.

## Appendix E: Useful formulas

From (4.4) one has

$\displaystyle\left(\frac{\partial}{\partial\rho_{1}}-\frac{\partial}{\partial\rho_{4}}\right)E=i\Big{\\{}\xi_{1}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\,\,{t}{}^{\alpha}y_{\alpha}$
(E.1)
$\displaystyle+\xi_{1}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})(\rho_{1}+\rho_{4})}\big{(}y+\mathbb{P}{}\big{)}^{\alpha}{t}{}_{\alpha}$
$\displaystyle+\frac{1}{(1-\rho_{3})}(y+p_{1}{}^{\alpha}+p_{2}{}^{\alpha}){t}{}_{\alpha}\Big{\\}}E$
$\displaystyle\left(\frac{\partial}{\partial\rho_{2}}-\frac{\partial}{\partial\rho_{3}}\right)E=i\Big{\\{}\xi_{1}\frac{1-\rho_{3}-\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})^{2}}\big{(}y+\mathbb{P}{}\big{)}^{\alpha}(y+\tilde{t}{})_{\alpha}$
(E.2)
$\displaystyle-\xi_{1}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\,(p_{3}{}^{\alpha}+p_{2}{}^{\alpha})(y+\tilde{t}{})_{\alpha}-\frac{\rho_{1}+\rho_{4}}{(1-\rho_{3})^{2}}\,\,((p{}_{1}+p_{2}))^{\alpha}y_{\alpha}$
$\displaystyle-\frac{\rho_{4}}{(1-\rho_{3})^{2}}\,\,{t}{}^{\alpha}y_{\alpha}-\frac{\rho_{1}}{(1-\rho_{3})^{2}}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}\Big{\\}}E\,,$
$\displaystyle\left(\frac{\partial}{\partial\rho_{2}}-\frac{\partial}{\partial\rho_{1}}\right)E=i\Big{\\{}\xi_{1}\frac{-\rho_{3}}{(1-\rho_{1}-\rho_{4})^{2}}(p_{3}+p_{2})^{\alpha}(y+\tilde{t}{})_{\alpha}$
(E.3)
$\displaystyle+\xi_{1}\frac{\rho_{3}\rho_{4}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})(\rho_{1}+\rho_{4})}\,\,{t}{}^{\alpha}y_{\alpha}+\xi_{1}\frac{1}{(1-\rho_{3})}(p_{1}+p_{2})^{\alpha}(y+\tilde{t}{})_{\alpha}$
$\displaystyle-\xi_{1}\frac{\rho_{2}}{(1-\rho_{1}-\rho_{4})(1-\rho_{3})}\frac{\rho_{4}}{(\rho_{1}+\rho_{4})^{2}}\big{(}y+\mathbb{P}{}\big{)}^{\alpha}{t}{}_{\alpha}$
$\displaystyle-\frac{1}{(1-\rho_{3})}\,(p_{1}+p_{2})^{\alpha}y_{\alpha}-\frac{1}{(1-\rho_{3})}(p_{1}+p_{2})^{\alpha}{t}{}_{\alpha}\Big{\\}}E\,.$

##

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