1 Introduction

Higher-spin (HS) gauge theory describes interacting systems of massless fields of all spins (for reviews see e.g. [1, 2]). Effects of HS gauge theories are anticipated to play a role at ultra high energies of Planck scale [3]. Theories of this class play a role in various contexts from holography [4] to cosmology [5]. 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 [6,7,8,9]. However one may ask for spin-locality [3, 10,11,12] which implies space-time locality in the lowest orders of perturbation theory [11]. 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 [13]. 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 [14], \(\eta ^2\) 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 [14] these vertices were written in the Z-dominated form which implies their spin-locality by virtue of Z-dominance Lemma of [15]. In this paper we obtain explicit Z-independent spin-local form for the vertex \(\varUpsilon ^{\eta \eta }_{\omega CCC}\) starting from the Z-dominated expression of [14]. 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 [4] (see also [16, 17]). Alternatively, one can analyze vertices directly in the bulk starting from the generating equations of [13]. The latter approach developed in [11, 12, 14, 15, 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, 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 [14] where they were shown to be spin-local by virtue of Z-dominance lemma of [15]. The computation was done directly in the bulk starting from the non-linear HS system of [13].

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 \(k*k= \bar{k}* \bar{k}=1\),

$$\begin{aligned} \lbrace k,y^\alpha \rbrace _*= & {} \lbrace k, z^\alpha \rbrace _*= \lbrace \bar{k},\bar{y}^{\dot{\alpha }}\rbrace _*=\lbrace \bar{k},\bar{z}^{\dot{\alpha }} \rbrace _*\nonumber \\= & {} \lbrace k,\theta ^\alpha \rbrace _*=\lbrace \bar{k},{\bar{\theta }}^{\dot{\alpha }}\rbrace _*=0, \nonumber \\ {[}k,\bar{y}^{\dot{\alpha }}]_*= & {} [ k, \bar{z}^{\dot{\alpha }}]_*= [\bar{k},y^\alpha ]_*=[ \bar{k},z^\alpha ]_*\nonumber \\= & {} [k,{\bar{\theta }}^{\dot{\alpha }}]_*=[ \bar{k},\theta ^\alpha ]_*=0, \end{aligned}$$
(1.1)

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

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

$$\begin{aligned}&\!\!\!{\mathrm{d}}_x \omega + \omega *\omega =\varUpsilon (\omega ,\omega ,C)+\varUpsilon (\omega ,\omega ,C,C)+\cdots , \end{aligned}$$
(1.2)
$$\begin{aligned}&\!\!\!{\mathrm{d}}_x C+\omega *C-C*\omega =\varUpsilon (\omega ,C,C)+\varUpsilon (\omega ,C,C,C)+\cdots .\nonumber \\ \end{aligned}$$
(1.3)

As recalled in Sect. 2, generating equations of [13] that reproduce the form of Eqs. (1.2) and (1.3) have a simple form as a result of doubling of spinor variables, namely

$$\begin{aligned} \omega (Y;K|x)\longrightarrow & {} W(Z;Y;K|x),\\ C(Y;K|x)\longrightarrow & {} B(Z;Y;K|x). \end{aligned}$$

Equations (1.2) and (1.3) result from the generating equations of [13] upon order by order reconstruction of Z-dependence (for more detail see Sect. 2). The final form of Eqs. (1.2) and (1.3) turns out to be Z-independent as a consequence of consistency of the equations of [13]. 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

$$\begin{aligned}&\varUpsilon ^{\eta }(\omega ,C,C )=\varUpsilon ^{ \eta }_{\omega CC }+\varUpsilon ^{ \eta }_{C\omega C }+\varUpsilon ^{ \eta }_{CC\omega } ,\nonumber \\&\varUpsilon ^{\eta \eta }(\omega ,C,C,C)=\varUpsilon ^{\eta \eta }_{\omega CCC}\!+\!\varUpsilon ^{\eta \eta }_{C\omega CC} \!+\!\varUpsilon ^{\eta \eta }_{CC\omega C}\!+\!\varUpsilon ^{\eta \eta }_{CCC\omega }\!,\nonumber \\&\ldots \end{aligned}$$
(1.4)

where the subscripts of the vertices \(\varUpsilon \) refer to the ordering of the product factors.

The vertices obtained in [14] were shown to be spin-local due to the Z-dominance Lemma of [15] 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 [12].

The main achievement of [14] 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 [14] gives spin-local expressions for the vertices \(\varUpsilon ^{\eta \eta }(\omega ,C,C,C)\) 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 [14]. 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 \(\varUpsilon ^{\eta \eta }_{\omega CCC}\) (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 [14] 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 Sect. 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 Sect. 4, we present step-by-step scheme of computations performed in this paper. Section 5 contains the final manifestly spin-local expression for \(\varUpsilon ^{\eta \eta }_{\omega CCC}\) vertex. In Sects. 6–10 technical details of the steps sketched in Sect. 4 are presented. In particular, in Sect. 7 we introduce important Generalised Triangle identity which allows us to uniformize expressions from [14]. 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

$$\begin{aligned}&\omega (y,\bar{y},x)={\mathrm{d}}x^\mu \omega _\mu (y,\bar{y},x)\nonumber \\&\quad =\sum _{n,m} {\mathrm{d}}x^{\mu } \omega _{\mu } {}_{\alpha _1 \ldots \alpha _n, \dot{\alpha }_1 \ldots \dot{\alpha }_m}(x) y^{\alpha _1} \ldots y^{\alpha _n} \bar{y}^{\dot{\alpha }_1} \ldots \bar{y}^{\dot{\alpha }_m}\nonumber \\ \end{aligned}$$
(2.1)

with \( s=\frac{2+m+n}{2} \) and zero-form

$$\begin{aligned} \!\!C(y,\bar{y},x)=\sum _{n,m}\! C_{\alpha _1 \ldots \alpha _n, \dot{\alpha }_1 \ldots \dot{\alpha }_m}(x) y^{\alpha _1}\!\! \!\ldots y^{\alpha _n} \bar{y}^{\dot{\alpha }_1}\!\! \!\ldots \bar{y}^{\dot{\alpha }_m} \end{aligned}$$
(2.2)

with \(s=\frac{|m-n|}{2}\), 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 \({\mathfrak {s}}{\mathfrak {p}}(4)\) spinor \(Y^A=(y^\alpha ,\bar{y}^{\dot{\alpha }})\), \(A=1,\ldots , 4\).

The vertices \(\varUpsilon (\omega ,\omega ,C,C,\ldots )\) (1.2) and \(\varUpsilon (\omega ,C,C,\ldots )\) (1.3) result from the generating system of [13]

$$\begin{aligned}&{\mathrm{d}}_x W+W*W=0, \end{aligned}$$
(2.3)
$$\begin{aligned}&{\mathrm{d}}_x S+W*S+S*W=0, \end{aligned}$$
(2.4)
$$\begin{aligned}&{\mathrm{d}}_x B+W*B- B*W=0, \end{aligned}$$
(2.5)
$$\begin{aligned}&S*S=i(\theta ^A \theta _A+\eta B*\gamma +{\bar{\eta }} B*{\bar{\gamma }}), \end{aligned}$$
(2.6)
$$\begin{aligned}&S*B-B*S=0. \end{aligned}$$
(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 \(\theta ^A=(\theta ^\alpha ,{\bar{\theta }}^{\dot{\alpha }})\),   \(\lbrace \theta ^A, \theta ^B\rbrace =0\), i.e.,

$$\begin{aligned} S(Z;Y;K)=\theta ^A S_A(Z;Y;K). \end{aligned}$$
(2.8)

B is a zero-form.

Star product is defined as follows

$$\begin{aligned}&\!\!\!(f*g)(Z;Y;K)=\frac{1}{(2\pi )^4}\!\int d^4 U \, d^4 V e^{iU_A V^A}\nonumber \\&\!\!\!\quad \times f(Z+U,Y+U;K)g(Z-V,Y+V;K). \end{aligned}$$
(2.9)

Elements

$$\begin{aligned} \gamma =\theta ^\alpha \theta _\alpha e^{iz_\alpha y^\alpha }k \quad \hbox {and}\quad {\bar{\gamma }}={\bar{\theta }}^{\dot{\alpha }}{\bar{\theta }}_{\dot{\alpha }} e^{i\bar{z}_{\dot{\alpha }}\bar{y}^{\dot{\alpha }}}\bar{k} \end{aligned}$$
(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

$$\begin{aligned}&\!\!\!B_0(Z;Y;K)=0,\quad S_0(Z;Y;K)=\theta ^\alpha z_\alpha +{\bar{\theta }}^{\dot{\alpha }}\bar{z}_{\dot{\alpha }},\quad \nonumber \\&\!\!\!W_0(Z;Y;K)=\omega (Y;K), \end{aligned}$$
(2.11)

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

$$\begin{aligned} {\mathrm{d}}\omega +\omega *\omega =0, \end{aligned}$$
(2.12)

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

$$\begin{aligned}{}[S_0,B_1]_*=0. \end{aligned}$$
(2.13)

From (2.9) one deduces that

$$\begin{aligned}{}[Z_A,f(Z;Y;K)]_*=-2i\frac{\partial }{\partial Z^A} f(Z;Y;K). \end{aligned}$$
(2.14)

Hence, Eq. (2.13) yields

$$\begin{aligned}&[S_0,B_1]=-2i \theta ^A \frac{\partial }{\partial Z^A}B_1=-2i {\mathrm{d}}_Z B_1=0 \nonumber \\&\quad \Longrightarrow B_1(Z;Y;K)=C(Y;K). \end{aligned}$$
(2.15)

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

$$\begin{aligned} {\mathrm{d}}_Z \varPhi _{k+1}=J(\varPhi _k, \varPhi _{k-1},\ldots ), \end{aligned}$$
(2.16)

where \(\varPhi _k\) 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.,

$$\begin{aligned} W= & {} \omega +W_1(\omega ,C)+W_2(\omega ,C,C)+\cdots ,\quad \\ S= & {} S_0+S_1(C)+S_2(C,C)+\cdots ,\\ B= & {} C+B_2(C,C)+B_3(C,C,C)+\cdots . \end{aligned}$$

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

$$\begin{aligned}&{\mathrm{d}}_x C+[\omega ,C]_*=-{\mathrm{d}}_x B_2-[W_1,C]_*\nonumber \\&\quad -{\mathrm{d}}_x B_3-[W_1,B_2]_*-[W_2,C]_*+\cdots \end{aligned}$$
(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, Eqs. (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 \({\mathrm{d}}_Z\) 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 [2].

3 Subspace \({\mathcal H}^{+}\) and Z-dominance lemma

3.1 \({\mathcal H}^{+}\)

In this section the definition of the space \({\mathcal H}^{+}\) [14] that plays a crucial role in our computation is recollected. Function \(f(z,y\vert \theta )\) of the form

$$\begin{aligned} f(z,y\vert \theta )=\!\int _0^1 d{\mathcal {T}}\, e^{i{\mathcal {T}}z_\alpha y^\alpha }\phi \left( {\mathcal {T}}z,y\vert {\mathcal {T}} \theta ,{\mathcal {T}}\right) \, \end{aligned}$$
(3.1)

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

$$\begin{aligned} \lim _{{\mathcal {T}}\rightarrow 0}{\mathcal {T}}^{1-\varepsilon }\phi (w,u\vert \theta ,{\mathcal {T}})=0. \end{aligned}$$
(3.2)

Note that this definition does not demand any specific behaviour of \(\phi \) at \({\mathcal {T}}\rightarrow 1\) 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):

$$\begin{aligned}&\phi _1({\mathcal {T}}z,y\vert {\mathcal {T}} \theta , {\mathcal {T}})=\frac{{\mathcal {T}}^{\delta _1}}{{\mathcal {T}}}{\widetilde{\phi }}_1({\mathcal {T}}z,y\vert {\mathcal {T}} \theta ),\quad \nonumber \\&\phi _2({\mathcal {T}} z,y\vert {\mathcal {T}}\theta , {\mathcal {T}})=\vartheta ({\mathcal {T}}-\delta _2)\frac{1}{{\mathcal {T}}}{\widetilde{\phi }}_2({\mathcal {T}}z,y\vert {\mathcal {T}} \theta ) \end{aligned}$$
(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

$$\begin{aligned} {\mathcal H}^{+}={\mathcal H}^{+}_0 \oplus {\mathcal H}^{+}_1 \oplus {\mathcal H}^{+}_2, \end{aligned}$$
(3.4)

where \(\phi (w,u\vert \theta ,{\mathcal {T}})\in {\mathcal H}^{+}_p\) 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 [15]. Following [14] 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 \(\phi ({\mathcal {T}}z,y\vert {\mathcal {T}}\theta , {{\mathcal {T}}})\) 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 [15]. 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 [14].

3.2 Notation

As in [14] we use exponential form for all the expressions below where by \(\omega CCC\) we assume

$$\begin{aligned} \omega ({\mathsf {y}}_\omega ,\bar{y}){\bar{*}}C({\mathsf {y}}_1,\bar{y}){\bar{*}}C({\mathsf {y}}_2,\bar{y}){\bar{*}}C({\mathsf {y}}_3,\bar{y}) \end{aligned}$$
(3.5)

with \({\bar{*}}\) denoting star-product with respect to \(\bar{y}\). Derivatives \(\partial _\omega \) and \(\partial _j\) act on auxiliary variables as follows

$$\begin{aligned} \partial _{\omega \alpha }=\frac{\partial }{\partial {\mathsf {y}}_\omega ^\alpha },\quad \partial _{j\alpha }=\frac{\partial }{\partial {\mathsf {y}}_j^\alpha }. \end{aligned}$$
(3.6)

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

$$\begin{aligned} {\mathsf {y}}_\omega ={\mathsf {y}}_j=0. \end{aligned}$$
(3.7)

In this paper we use the following notation of [14]:

$$\begin{aligned}&t_\alpha :=-i\partial _{\omega \alpha },\quad p_{j\alpha }:=-i\partial _{j\alpha },\quad \end{aligned}$$
(3.8)
$$\begin{aligned}&\!\int d^n \rho _+ :=\!\int d\rho _1 \ldots d\rho _n\, \vartheta (\rho _1)\ldots \vartheta (\rho _n). \end{aligned}$$
(3.9)

3.3 Contribution to \({\varUpsilon }^{\eta \eta } _{\omega CCC}\) modulo \({\mathcal H}^{+}\)

The \(\eta ^2C^3\) vertex in the equations on the zero-forms C resulting from equations of [13] is

$$\begin{aligned}&\!\!\!\varUpsilon ^{\eta \eta }(\omega ,C,C,C) = -\left( {\mathrm{d}}_x B^{ \eta \eta }_3 + [\omega , B^{\eta \eta }_3]_* \right. \nonumber \\&\!\!\!\quad \left. +\, [ {W}^\eta _1, B^{\eta }_2]_* +[ {W}^{ {\eta }\eta }_2, C]_* +{\mathrm{d}}_x B^{ \eta }_2\,\right) . \end{aligned}$$
(3.10)

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

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

$$\begin{aligned} {\varUpsilon }^{\eta \eta }_{\omega C C C} := \varUpsilon ^{\eta \eta }(\omega ,C,C,C)\Big |_{\omega CCC}. \end{aligned}$$
(3.11)

Since the terms from \({\mathcal H}^{+}\) do not contribute to the physical vertex such terms can be discarded. Following [14] equality up to terms from \({\mathcal H}^{+}\) referred to as weak equality is denoted as \(\approx \) .

We start with the following results of [14]:

$$\begin{aligned}&{\widehat{\varUpsilon }}^{\eta \eta }_{\omega CCC} \approx {\varUpsilon }^{\eta \eta }_{\omega C C C}=-\Big (W_{1\, \omega C}^\eta *B_2^{\eta \, loc} + \omega *{B}_3^{\eta \eta } \nonumber \\&\quad + {W}_{2\, \omega CC}^{\eta \eta }*C + {\mathrm{d}}_x B^{\eta \, loc}_2\big |_{\omega CCC} +{\mathrm{d}}_x {B}_3^{\eta \eta }\big |_{\omega CCC} \Big ) ,\nonumber \\ \end{aligned}$$
(3.12)

where

$$\begin{aligned}&W_{1\, \omega C}^\eta *B_2^{\eta \, loc}\approx \frac{\eta ^2}{4}\!\int _0^1 d{\mathcal {T}} {\mathcal {T}}\!\int _0^1 d\sigma \!\int d^3\rho _+ \nonumber \\&\quad \times \delta \left( 1-\sum _{i=1}^3 \rho _i\right) \frac{ (z_\gamma t^{ \gamma } \big [z_\alpha y^\alpha +\sigma z_\alpha t^{\alpha }\big ]}{(\rho _1+\rho _2)} \nonumber \\&\quad \times \exp \Big \{i{\mathcal {T}} z_\alpha y^\alpha +i(1-\sigma )t^{\alpha }\partial _{1\alpha } -i\frac{\rho _1\sigma }{\rho _1+\rho _2} t^{\alpha }p_{2\alpha } \nonumber \\&\quad +i\frac{\rho _2\sigma }{\rho _1 +\rho _2} t^{\alpha }p_{3\alpha } +i{\mathcal {T}}z^\alpha \Big (-(\rho _1+\rho _2+\sigma \rho _3)t_{\alpha }\nonumber \\&\quad -(\rho _1+\rho _2)p_{1 \alpha } +(\rho _3-\rho _1)p_{2 \alpha }+(\rho _3+\rho _2)p_{3\alpha }\Big ) \nonumber \\&\quad +iy^\alpha \Big (\sigma t_{\alpha }-\frac{\rho _1}{\rho _1+\rho _2}p_{2 \alpha } +\frac{\rho _2}{\rho _1+\rho _2}p_{3\alpha }\Big )\Big \}\omega CCC, \end{aligned}$$
(3.13)
$$\begin{aligned}&{W}_{2\, \omega CC}^{\eta \eta }*C\approx -\frac{\eta ^2}{4}\!\int _0^1 d{\mathcal {T}}\,{\mathcal {T}}\!\int d^4\rho _+\, \delta \left( 1-\sum _{i=1}^4 \rho _i\right) \nonumber \\&\quad \times \frac{\rho _1 \left( z_\gamma t^{\gamma }\right) ^2}{(\rho _1+\rho _2)(\rho _3+\rho _4)} \exp \Big \{i{\mathcal {T}}z_\alpha y^\alpha +i{\mathcal {T}}z^\alpha \Big ((1-\rho _2)t_{\alpha } \nonumber \\&\quad -(\rho _3+\rho _4)p_{1\alpha }+(\rho _1+\rho _2)p_{2 \alpha }+p_{3 \alpha }\Big )+i y^\alpha t_{\alpha } \nonumber \\&\quad +\frac{\rho _1\rho _3}{(\rho _1+\rho _2)(\rho _3+\rho _4)}\left( i y^\alpha t_{ \alpha } +it^{ \alpha }p_{3\alpha }\right) \nonumber \\&\quad +i\left( \frac{(1-\rho _4)\rho _2}{\rho _1+\rho _2} +\rho _4\right) t^{\alpha }p_{1\alpha }-i\frac{\rho _4\rho _1}{\rho _3+\rho _4}t^\alpha p_{2\alpha }\Big \} \omega CC C,\nonumber \\ \end{aligned}$$
(3.14)
$$\begin{aligned}&{\mathrm{d}}_x B^{\eta \, loc}_2\big |_{\omega CCC}\approx \frac{\eta ^2}{4}\!\int _0^1 d{\mathcal {T}} \!\int _0^1 d\xi \!\int d^3\rho _+\,\nonumber \\&\quad \times \delta \left( 1-\sum _{i=1}^3\rho _i\right) z_\alpha y^\alpha \Big [\left( {\mathcal {T}}z^\alpha -\xi y^\alpha \right) t_{ \alpha }\Big ]\nonumber \\&\quad \times \exp \Big \{i{\mathcal {T}}z_\alpha y^\alpha +i(1-\rho _2)t^\alpha p_{1\alpha } -i\rho _2 t^\alpha p_{2\alpha } \nonumber \\&\quad -i{\mathcal {T}}z^\alpha \Big ( (\rho _1+\rho _2)t_{\alpha } +\rho _1 p_{1 \alpha }-(\rho _2+\rho _3)p_{2 \alpha }-p_{3\alpha }\Big ) \nonumber \\&\quad +iy^\alpha \Big (\xi (\rho _1+\rho _2)t_{\alpha }+\xi \rho _1 p_{1 \alpha } -\xi (\rho _2+\rho _3)p_{2 \alpha }\nonumber \\&\quad +(1-\xi )p_{3\alpha }\Big ) \Big \}\omega CCC, \end{aligned}$$
(3.15)
$$\begin{aligned}&\omega *{B}_3^{\eta \eta }\approx -\frac{\eta ^2}{4} \!\int _0^1 d{\mathcal {T}}\, {\mathcal {T}} \!\int d^3 \rho _+ \nonumber \\&\quad \times \delta \left( 1-\sum _{i=1}^3 \rho _i\right) \!\int _0^1 d\xi \, \frac{\rho _1\, \left[ z_\alpha \left( y^\alpha +t^\alpha \right) \right] ^2 }{(\rho _1+\rho _2)(\rho _1+\rho _3)} \exp \Big \{i{\mathcal {T}}z_\alpha y^\alpha \nonumber \\&\quad +i{\mathcal {T}} z^\alpha \Big (-t_{ \alpha }-(\rho _1+\rho _3)p_{1\alpha }+(\rho _2-\rho _3)p_{2\alpha }\nonumber \\&\quad +(\rho _1+\rho _2)p_{3\alpha }\Big ) +iy^\alpha t_{\alpha }\nonumber \\&\quad +i(1-\xi )y^\alpha \left( \frac{\rho _1}{\rho _1+\rho _2}p_{1\alpha } -\frac{\rho _2}{\rho _1+\rho _2}p_{2\alpha }\right) \nonumber \\&\quad +i\xi \, y^\alpha \left( \frac{\rho _1}{\rho _1+\rho _3}p_{3\alpha }-\frac{\rho _3}{\rho _1+\rho _3}p_{2\alpha }\right) \nonumber \\&\quad +i\frac{(1-\xi )\rho _1}{\rho _1+\rho _2}t^{\alpha }p_{1\alpha } -i\left( \frac{(1-\xi )\rho _2}{\rho _1+\rho _2}+\frac{\xi \rho _3}{\rho _1+\rho _3}\right) t^{\alpha }p_{2\alpha } \nonumber \\&\quad +i\frac{\xi \rho _1}{\rho _1+\rho _3}t^\alpha p_{3\alpha }\Big \} \omega CCC, \end{aligned}$$
(3.16)
$$\begin{aligned}&{\mathrm{d}}_x {B}_3^{\eta \eta }\big |_{\omega CCC}\approx \frac{\eta ^2}{4} \!\int _0^1 d{\mathcal {T}}\, {\mathcal {T}} \!\int d^3 \rho _+ \delta \left( 1-\sum _{i=1}^3 \rho _i\right) \nonumber \\&\quad \times \!\int _0^1 d\xi \, \frac{\rho _1\, (z_\alpha y^\alpha )^2 }{(\rho _1+\rho _2)(\rho _1+\rho _3)}\exp \Big \{i{\mathcal {T}}z_\alpha y^\alpha \nonumber \\&\quad +i{\mathcal {T}} z^\alpha \Big (-(\rho _1+\rho _3)(t_{\alpha }+p_{1\alpha })+(\rho _2-\rho _3)p_{2\alpha }\nonumber \\&\quad +(\rho _1+\rho _2)p_{3\alpha }\Big )\nonumber \\&\quad +it^\alpha p_{1\alpha } +i(1-\xi )y^\alpha \left( \frac{\rho _1}{\rho _1+\rho _2} (t_{ \alpha }+p_{1\alpha })-\frac{\rho _2}{\rho _1+\rho _2}p_{2\alpha }\right) \nonumber \\&\quad +\xi \, y^\alpha \left( \frac{\rho _1}{\rho _1+\rho _3}p_{3\alpha }-\frac{\rho _3}{\rho _1+\rho _3}p_{2\alpha }\right) \Big \}\omega CCC. \end{aligned}$$
(3.17)

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

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 [14]. Thus \({\widehat{\varUpsilon }}^{\eta \eta }_{\omega CCC} (Z;Y) \) is also spin-local.

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

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

$$\begin{aligned} {\widehat{\varUpsilon }}^{\eta \eta }_{\omega CCC}(Z;Y)\approx \widehat{{\widehat{\varUpsilon }}} {\,}^{\eta \eta }_{\omega CCC}(Y), \end{aligned}$$

where \(\widehat{{\widehat{\varUpsilon }}} {\,}^{\eta \eta }_{\omega CCC}(Y)\) is manifestly spin-local and Z-independent. Since \({\mathcal H}^{+}_0\)-terms do not contribute to the vertex by Z-dominance Lemma [15]

$$\begin{aligned} \varUpsilon ^{\eta \eta }_{\omega CCC}(Y)=\widehat{{\widehat{\varUpsilon }}} {\,}^{\eta \eta }_{\omega CCC}(Y). \end{aligned}$$

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

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 [14].

  • 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 Sect. 6, Eqs. (6.1)–(6.4). The respective cohomology terms being a part of the vertex \(\varUpsilon ^{\eta \eta }_{\omega CCC} \) are presented in Sect. 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 \(\int d\varGamma \) over the same integration domain \({{\mathcal {I}}}\)

    $$\begin{aligned} \!\int d\varGamma \, z_\alpha f^\alpha (y,t,p_1,p_2,p_3\vert {\mathcal {T}},\xi _i,\rho _i){{\mathcal {E}}}\, \omega CCC, \end{aligned}$$
    (4.1)

    where the integrand contains an overall exponential function \({{\mathcal {E}}}\)

    $$\begin{aligned} {{\mathcal {E}}}= & {} {E_z{\,}}E, \end{aligned}$$
    (4.2)
    $$\begin{aligned} {E_z{\,}}:= & {} \exp i\Big \{{\mathcal {T}}z_{\alpha }(y + \mathbb {P}{})^{\alpha } \Big \} \end{aligned}$$
    (4.3)
    $$\begin{aligned} \!\!\!\!\!\!E:= & {} \exp i \Big \{ \xi _2 \frac{-\rho _2}{(1-\rho _1-\rho _4 )(1-\rho _3 )} \big ( y + \mathbb {P}{}\big )^\alpha y_{\alpha }\end{aligned}$$
    (4.4)
    $$\begin{aligned}&+ \xi _1 \frac{ \rho _2}{(1-\rho _1-\rho _4 )(1-\rho _3)}\big ( y + \mathbb {P}{}\big )^\alpha {\tilde{t}}{}_{\alpha } \nonumber \\&+\frac{ \rho _3 }{(1\!-\!\rho _1\!-\!\rho _4 ) }\,\, ( p_3+p_2)^{\alpha } y_{\alpha }\nonumber \\&-\frac{ \rho _3 }{(1\!-\!\rho _1\!-\!\rho _4 )(1\!-\!\rho _3 )}\,\, \rho _1 {t}{}^{\alpha } y_{\alpha } \end{aligned}$$
    (4.5)
    $$\begin{aligned}&+ \frac{ \rho _1 }{(1\!-\!\rho _3)} (p_1{}^{\alpha }+p_2{}^{\alpha }){t}{}_{\alpha } + p_3{}_{\alpha } y^{\alpha } +p_1{}_\alpha {t}{}^\alpha \Big \} , \nonumber \\ {\tilde{t}}{}= & {} \frac{\rho _1}{\rho _1+\rho _4}{t}{},\quad \end{aligned}$$
    (4.6)
    $$\begin{aligned} \mathbb {P}{}= & {} {{\mathcal {P}}}+ (1-\rho _4){t}{}, \end{aligned}$$
    (4.7)
    $$\begin{aligned} {{\mathcal {P}}}= & {} ( 1\!-\!\rho _1\!-\!\rho _4)(p{}_1 +p_2) \!-\! (1\!-\!\rho _3) (p_3+p_2), \end{aligned}$$
    (4.8)

    the integral over \({{\mathcal {I}}}\) is denoted as

    $$\begin{aligned}&\int d\varGamma =\!\int _0^1 d{\mathcal {T}}\!\int d^3 \xi _+\, \delta \left( 1-\sum _{i=1}^3 \xi _i\right) \nonumber \\&\quad \times \int d^4 \rho _+ \, \delta \left( 1-\sum _{j=1}^4 \rho _j\right) . \end{aligned}$$
    (4.9)

    Equations (6.1)–(6.4) transformed to the form (4.1) are collected in Sect. 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 Sect. 9, Eqs. (9.1)–(9.4).

  • V. Final step.

    Finally, we show in Sect. 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 \(\varUpsilon ^{\eta \eta }_{ \omega CCC}\), being a sum of expressions (5.2)–(5.12).

5 Main result \(\varUpsilon ^{\eta \eta }_{\omega CCC}\)

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

Vertex \(\varUpsilon ^{\eta \eta }_{\omega CCC}\) is

$$\begin{aligned} \varUpsilon ^{\eta \eta }_{\omega CCC}=\sum _{j=1}^{11} J_j\, \end{aligned}$$
(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 \({\mathrm{d}}_x B^{\eta \eta }_3\),

$$\begin{aligned} J_1= & {} - \frac{ \eta ^2 }{4 } \!\int d\varGamma \, \frac{\delta (\xi _3)\rho _2}{(\rho _2+\rho _1 )(\rho _2+\rho _3)} \delta (\rho _4) E\, \omega CCC, \end{aligned}$$
(5.2)
$$\begin{aligned} J_2= & {} \frac{ \eta ^2 }{4 } \!\int d\varGamma \, \frac{ \delta (\xi _3)\rho _2}{(\rho _2+\rho _4 ) (\rho _2+\rho _3)} \delta (\rho _1) E\, \omega CCC. \end{aligned}$$
(5.3)

Recall that E and \(d\varGamma \) are defined in (4.44.5) and (4.9), 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 \(d\varGamma \) 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,

$$\begin{aligned} J_3= & {} -\frac{i \eta ^2 }{4 } \!\int d\varGamma \, \delta (\xi _3) \frac{1}{(\rho _2 +\rho _3)(1-\rho _3) } \nonumber \\&\times \Big \{\rho _2 {t}{}^\alpha (p_1+p_2 ){} _\alpha \big [\overrightarrow{\partial }_{\rho _2}-\overrightarrow{\partial }_{\rho _3}\big ] \nonumber \\&+ \rho _2 ( p_1{}+ p_2)^{\alpha } ( p_3{}+p_2)_{\alpha } \big [\overrightarrow{\partial }_{\rho _4}-\overrightarrow{\partial }_{\rho _1}\big ] \nonumber \\&+\rho _2 {t}{}^\alpha ( p_3{} +p_2{} )_\alpha \big [\overrightarrow{\partial }_{\rho _2}-\overrightarrow{\partial }_{\rho _1}\big ]\nonumber \\&+\frac{ \rho _1+\rho _4}{ (1-\rho _3) } {t}{}^\alpha (p_1+p_2 ){} _\alpha \Big \}E\, \omega C C C, \end{aligned}$$
(5.4)
$$\begin{aligned} J_4= & {} \frac{i\eta ^2}{4}\!\int d\varGamma \, \frac{\delta (\xi _3)}{1-\rho _3} \Big ( - \frac{ \rho _3}{(1-\rho _1-\rho _4 )^2(1-\rho _3)} {t}{}^\gamma y_\gamma \nonumber \\&- \frac{ \rho _2}{(1-\rho _1-\rho _4 )(1-\rho _3)} {t}{}^\gamma y_\gamma [-\overrightarrow{\partial }_{\rho _1}+\overrightarrow{\partial }_{\rho _2}] \nonumber \\&- \frac{ \rho _2}{(1-\rho _1-\rho _4 )(1-\rho _3)} ( p_1{}+ p_2)^{\gamma } (y+{\tilde{t}}{}) _{\gamma } [ \overrightarrow{\partial }_{\rho _4}-\overrightarrow{\partial }_{\rho _1}] \Big )\nonumber \\&\times E\omega C C C, \end{aligned}$$
(5.5)
$$\begin{aligned} J_5= & {} -i \frac{ \eta ^2 }{4 } \!\int d\varGamma \,\delta (\xi _3) \Big [1+ \xi _1(\overrightarrow{\partial }_{\xi _1}-\overrightarrow{\partial }_{\xi _2})\Big ] \nonumber \\&\times \Big \{ \frac{ -\rho _2 }{(1-\rho _1-\rho _4 )^2(1-\rho _3) ( \rho _1+\rho _4 ) } (p_3{}^{\alpha }+p_2{}^{\alpha })^\gamma {t}{}_{\gamma } \nonumber \\&- \frac{ \rho _3 }{(1-\rho _1-\rho _4 )^2(1-\rho _3 )^2}\,\, {t}{}^{\alpha } y_{\alpha } \nonumber \\&+ \frac{ 1}{(\rho _2 +\rho _3)(1-\rho _3) ( \rho _1+\rho _4 ) } (p_1{}^{\alpha }+p_2{}^{\alpha }){t}{}_{\alpha } \Big \} E\, \omega C C C, \end{aligned}$$
(5.6)
$$\begin{aligned} J_6= & {} i\frac{ \eta ^2 }{4 } \!\int d\varGamma \,\delta (\xi _3) \frac{ \rho _2}{(1-\rho _1-\rho _4 )(1-\rho _3)^2(\rho _1+\rho _4)} \nonumber \\&\times ( p_1{}+ p_2)^{\gamma } ( {t}{}) _{\gamma }E \omega C C C, \end{aligned}$$
(5.7)
$$\begin{aligned} J_7= & {} - \frac{\eta ^2}{4}\!\int d\varGamma \, \delta (\xi _3)\, \xi _1 \frac{ \rho _2\rho _2}{(\rho _2 +\rho _3)^3(1-\rho _3)^3( \rho _1+\rho _4 ) } \nonumber \\&\times \big ( y+ (1-\rho _1-\rho _4 )( p_1{} +p_2{} )+ (1 -\rho _4 ){t}{} \big )^\gamma \big ( y + {\tilde{t}}{} \big )_{\gamma }\nonumber \\&\times {t}{}^{\alpha } y_\alpha E \, \omega CCC, \end{aligned}$$
(5.8)
$$\begin{aligned} J_8= & {} - \frac{ \eta ^2 }{4 }\!\int d\varGamma \, \delta (\rho _3) \Big ( \rho _1\delta (\xi _3 ) \nonumber \\&+ \Big [ i {\delta (\rho _4)} -( p_2{}_\alpha + p_1{}_\alpha ) {t}{}^{\alpha }\Big ] \Big \{ i \delta (\xi _3 )+ {\tilde{t}}{}^{\gamma } y_\gamma \Big \}\Big )\nonumber \\&\times E\, \omega CCC, \end{aligned}$$
(5.9)
$$\begin{aligned} J_9= & {} i\eta ^2\frac{1}{4}\!\int d\varGamma \, \delta (\rho _1)\delta (\rho _4)\delta (\xi _3) \exp \Big \{ -i\xi _2 ( p_1+p_2+{t}\nonumber \\&- \rho _2 (p_3+p_2))_{\alpha } (y )^{\alpha } -\xi _1 ( y+ p_1+p_2 - \rho _2 (p_3+p_2))_\gamma ( {t})^\gamma \nonumber \\&+ ( 1-\rho _2) (p_3+p_2) {}^\gamma y_\gamma + p_3{}_\gamma y^\gamma +{t}{}^\beta p_1{}_\beta \Big \}\nonumber \\&\times \omega CCC, \end{aligned}$$
(5.10)
$$\begin{aligned} J_{10}= & {} -i\eta ^2\frac{1}{4}\!\int d\varGamma \, \delta (\rho _4) \delta (\xi _1 )\delta (\rho _1) \, \exp i\Big \{-\xi _2 ( y+ p_1+p_2\nonumber \\&+{t} - \rho _2 (p_3+p_2))_{\alpha } (y )^{\alpha } + ( 1-\rho _2) (p_3+p_2) {}^\gamma y_\gamma \nonumber \\&+p_3{}_\gamma y^\gamma +{t}{}^\beta p_1{}_\beta \Big \} \omega CCC, \end{aligned}$$
(5.11)
$$\begin{aligned} J_{11}= & {} \frac{i \eta ^2}{4}\!\int d\varGamma \, \delta (\rho _1)\delta (\rho _4) y^\alpha {t} {}_\alpha \exp i\Big \{p_3{}_\gamma y^\gamma +{t}{}^\beta p_1{}_\beta \nonumber \\&+ (y+{{\mathcal {P}}}_0 +{t}){}^\gamma ( \xi _1 {t}- \xi _2 y)_\gamma + ( 1-\rho _2) (p_3+p_2) {}^\gamma y_\gamma \Big \}\nonumber \\&\times \omega CCC. \end{aligned}$$
(5.12)

Let us emphasize, that neither exponential function E (4.44.5) 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 \( {\rho _2}-(\rho _1+\rho _2 ) (\rho _2+\rho _3) =-\rho _3\rho _1\le 0\)  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)

$$\begin{aligned}&J_3+J_4+J_5 = \frac{i \eta ^2 }{4 } \!\int d\varGamma \, \delta (\xi _3)\frac{1 }{(\rho _2 +\rho _3)(1-\rho _3) }\nonumber \\&\quad \times \Big \{ - \delta ({\rho _3}) {t}{}^\alpha (p_1+p_2 ){} _\alpha -\delta ({\rho _1}) \frac{ \rho _2}{ (1-\rho _3)} {t}{}^\gamma y_\gamma \nonumber \\&\quad + [ \delta ({\rho _4})-\delta ({\rho _1})]\rho _2 ( p_1{}+ p_2)^{\alpha } ( p_3{}+p_2)_{\alpha } \nonumber \\&\quad + {t}{}^\alpha ( p_3{} +p_2{} )_\alpha -\delta ({\rho _1})\rho _2 {t}{}^\alpha ( p_3{} +p_2{} )_\alpha \, \nonumber \\&\quad + [ \delta ({\rho _4})-\delta ({\rho _1})] \frac{ \rho _2}{ (1-\rho _3)} ( p_1{}+ p_2)^{\gamma } (y+{\tilde{t}}{}) _{\gamma } \nonumber \\&\quad - \delta ({\xi _2}) \Big ( \frac{ -\rho _2 }{(\rho _2 +\rho _3)( \rho _1+\rho _4 ) } (p_3{}^{\alpha }+p_2{}^{\alpha })^\gamma {t}{}_{\gamma } \nonumber \\&\quad - \frac{ \rho _3 }{(\rho _2 +\rho _3) (1-\rho _3 ) }\,\, {t}{}^{\alpha } y_{\alpha } + \frac{ 1}{ ( \rho _1+\rho _4 ) } (p_1{}^{\alpha }+p_2{}^{\alpha }){t}{}_{\alpha } \Big ) \Big \}\nonumber \\&\quad \times E\, \omega C C C. \end{aligned}$$
(5.13)

Using that, due to the factor of \(\delta (1-\sum \rho _i)\), for positive \(\rho _i \) it holds

$$\begin{aligned} \frac{\rho _2}{(\rho _3+\rho _2)(1-\rho _3)}\!-1\!= & {} \frac{-\rho _3(1- \rho _3-\rho _2 )}{(\rho _3+\rho _2)(1-\rho _3)}\le 0 ,\quad \end{aligned}$$
(5.14)
$$\begin{aligned} \frac{ 1}{(\rho _2 +\rho _3)(1-\rho _3) }\,\le & {} \frac{ 1}{(\rho _2 +\rho _3)(1-\rho _3-\rho _2) }\nonumber \\= & {} \frac{ 1}{ ( \rho _3+\rho _2) }+\frac{ 1}{ ( \rho _1+\rho _4 ) }, \end{aligned}$$
(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

$$\begin{aligned}&\frac{ \rho _2\rho _2}{(\rho _2 +\rho _3)^3(1-\rho _3)^3( \rho _1+\rho _4 ) } \nonumber \\&\quad \le \frac{ 1}{ (1-\rho _3)( \rho _1+\rho _4 ) }+ \frac{1}{(\rho _3+\rho _2) } +\frac{ 1}{ ( \rho _1+\rho _4 ) }.\quad \end{aligned}$$
(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

$$\begin{aligned}&\!\int d^4 \rho _+ \delta (1-\sum _1^4 \rho _i)\frac{1}{(1-\rho _3 ) ( \rho _1+\rho _2)} \nonumber \\&\quad =\!\int d^3 \rho _+ \vartheta (1-\sum _1^3 \rho _i)\frac{1}{(1-\rho _3 ) ( \rho _1+\rho _2)}\nonumber \\&\quad = -\!\int _0^1 d \rho _1 \!\int _0^{1-\rho _1} d \rho _2 \frac{\log ( \rho _1+\rho _2)}{ ( \rho _1+\rho _2)}=\frac{1}{2}\!\int _0^1 d \rho _1 \log ^2( \rho _1 ),\nonumber \\ \end{aligned}$$
(5.17)

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 Sect. 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

$$\begin{aligned}&\omega *{B}_3^{\eta \eta }\approx J_1+ \frac{\eta ^2}{4}\!\int d\varGamma \frac{\delta (\xi _3)\delta (\rho _4)}{(1-\rho _1)(1-\rho _3)}\nonumber \\&\quad \times \Bigg [-\rho _2 z_\alpha (y+t )^\alpha (p_{1\beta }+p_{2\beta })(p_2 {}^\beta +p_3 {}^\beta ) \nonumber \\&\quad +i\Big [\Big (\delta (\rho _1)+\delta (\rho _3)\Big )(1-\rho _1)(1-\rho _3)-\delta (\xi _2)\Big ] \nonumber \\&\quad \times z_\alpha \Big ((1-\rho _1)(p_1 {}^\alpha +p_2 {}^\alpha )-(1-\rho _3)(p_2 {}^\alpha +p_3 {}^\alpha )\Big ) \nonumber \\&\quad +iz_\alpha (p_1 {}^\alpha +p_2 {}^\alpha )(1-\rho _1)\Big (\delta (\xi _2)-\delta (\xi _1)\Big )\Bigg ] \exp \Big \{i{\mathcal {T}}z_\alpha \big (y^\alpha \nonumber \\&\quad +t^\alpha +(1-\rho _1)(p_1 {}^\alpha +p_2 {}^\alpha ) -(1-\rho _3)(p_2 {}^\alpha +p_3 {}^\alpha )\big ) \nonumber \\&\quad +\frac{i(1-\xi _1) \rho _2}{\rho _1+\rho _2}(y^\alpha +t^\alpha ) (p_{1\alpha }+p_{2\alpha }) \nonumber \\&\quad +\frac{i\xi _1 \rho _2}{\rho _2+\rho _3}(y^\alpha +t^\alpha ) (p_{2\alpha }+p_{3\alpha })-i(y^\alpha +t^\alpha ) p_{2\alpha }\Big \}\nonumber \\&\quad \times \omega CCC,\quad \end{aligned}$$
(6.1)

where \(J_1\) is the cohomology term (5.2). Analogously,

$$\begin{aligned}&{\mathrm{d}}_x {B}_3^{\eta \eta } \approx J_2-\frac{\eta ^2}{4}\!\int d\varGamma \frac{\delta (\xi _3)\delta (\rho _4)}{(1-\rho _1)(1-\rho _3)} \nonumber \\&\quad \times \Bigg [-\rho _2 (z_\alpha y^\alpha )(p_{1\beta }+t_\beta +p_{2\beta })(p_2 {}^\beta +p_3 {}^\beta ) \nonumber \\&\quad +i\Big [\Big (\delta (\rho _1)+\delta (\rho _3)\Big )(1-\rho _1)(1-\rho _3)-\delta (\xi _2)\Big ] \nonumber \\&\quad \times z_\alpha \Big ((1-\rho _1)(p_1 {}^\alpha +t^\alpha +p_2 {}^\alpha )-(1-\rho _3)(p_2 {}^\alpha +p_3 {}^\alpha )\Big ) \nonumber \\&\quad +iz_\alpha (p_1 {}^\alpha +t^\alpha +p_2 {}^\alpha )(1-\rho _1)\Big (\delta (\xi _2)-\delta (\xi _1)\Big )\Bigg ] \exp \Big \{i{\mathcal {T}}z_\alpha \big (y^\alpha \nonumber \\&\quad +(1-\rho _1)(p_1 {}^\alpha +t^\alpha +p_2 {}^\alpha )-(1-\rho _3)(p_2 {}^\alpha +p_3 {}^\alpha )\big ) \nonumber \\&\quad +\frac{i(1-\xi _1) \rho _2}{\rho _1+\rho _2}y^\alpha (p_{1\alpha }+t_\alpha +p_{2\alpha }) +\frac{i\xi _1 \rho _2}{\rho _2+\rho _3}y^\alpha (p_{2\alpha }+p_{3\alpha })\nonumber \\&\quad -iy^\alpha p_{2\alpha }+it^\beta p_{1\beta }\Big \}\omega CCC \end{aligned}$$
(6.2)

with \(J_2\) (5.3). Using the Schouten identity and partial integration one obtains from Eqs. (3.13)–(3.15), respectively,

$$\begin{aligned}&W_{1 \, \omega C}^\eta *B_2^\eta \approx \frac{\eta ^2}{4}\!\int _0^1 d{\mathcal {T}}\!\int _0^1 d\tau \!\int _0^1 d\sigma _1 \!\int _0^1 d\sigma _2 \nonumber \\&\quad \times \Bigg [i z_\alpha t^\alpha \delta (1-\tau ) +\frac{z_\alpha (p_2 {}^\alpha +p_3 {}^\alpha )}{1-\tau }\Big (i\big (\delta (\sigma _1)-\delta (1-\sigma _1)\big ) \nonumber \\&\quad -\big [y^\alpha +p_1 {}^\alpha +p_2 {}^\alpha -\sigma _2(p_2{}^\alpha +p_3 {}^\alpha )\big ]t_\alpha \Big )\Bigg ] \exp \Big \{i{\mathcal {T}}z_\alpha y^\alpha \nonumber \\&\quad +i{\mathcal {T}}z_\alpha \Big (\tau (p_1 {}^\alpha +p_2 {}^\alpha )-((1-\tau )+\sigma _2\tau )(p_2 {}^\alpha +p_3 {}^\alpha ) \nonumber \\&\quad +\big (\sigma _1+\tau (1-\sigma _1)\big )t^\alpha \Big )+it^\alpha p_{1\alpha } \nonumber \\&\quad +i\sigma _1\big [y^\alpha +p_1 {}^\alpha +p_2 {}^\alpha -\sigma _2(p_2{}^\alpha +p_3 {}^\alpha )\big ]t_\alpha \nonumber \\&\quad -i\Big (\sigma _2 p_3 {}^\alpha -(1-\sigma _2)p_2 {}^\alpha \Big )y_\alpha \Big \}\omega CCC, \end{aligned}$$
(6.3)
$$\begin{aligned}&W_{2\, \omega CC}^{\eta \eta }*C\approx -\frac{i\eta ^2}{4} \!\int d\varGamma \, \delta (\xi _3)\delta (\rho _3)\frac{(z_\gamma t^\gamma )}{\rho _1+\rho _4}\nonumber \\&\quad \times \Big [-\rho _1\big ( \delta (\rho _4)+i t^\alpha (p_{1\alpha }+p_{2\alpha })\big )\nonumber \\&\quad +\xi _1\delta (\xi _2)\Big ] \exp \Big \{i{\mathcal {T}}z_\alpha y^\alpha \nonumber \\&\quad +i{\mathcal {T}}z_\alpha \Big ((1-\rho _1-\rho _4)(p_1 {}^\alpha +p_2 {}^\alpha )-(1-\rho _3)(p_2{}^\alpha +p_3 {}^\alpha )+(1-\rho _4)t^\alpha \Big ) \nonumber \\&\quad +iy^\alpha \left( \frac{\xi _1 \rho _1}{1-\rho _2}t_\alpha +p_{3\alpha }\right) +i\left( 1-\rho _1-\frac{\xi _1 \rho _1\rho _2}{1-\rho _2}\right) t^\alpha p_{1\alpha } \nonumber \\&\quad -i(1-\xi _1)\rho _1 t^\alpha p_{2\alpha } +i\frac{\xi _1 \rho _1}{1-\rho _2}t^\alpha p_{3\alpha } \Big \}\omega CCC, \end{aligned}$$
(6.4)
$$\begin{aligned}&{\mathrm{d}}_x B_2^\eta \approx \frac{i\eta ^2}{4} \!\int d\varGamma \, \delta (\xi _3)\delta (\rho _4)\, (z_\alpha y^\alpha )\nonumber \\&\quad \times \Big [it^\gamma (p_{1\gamma }+p_{2\gamma })+\delta (\rho _4)- \delta (\rho _1) \Big ]\exp \Big \{i{\mathcal {T}}z_\alpha y^\alpha \nonumber \\&\quad +i{\mathcal {T}}z_\alpha \big ((1-\rho _1-\rho _4)(p_1 {}^\alpha + p_2 {}^\alpha )-(1-\rho _3)(p_2 {}^\alpha +p_3 {}^\alpha ) \nonumber \\&\quad +(1-\rho _4)t^\alpha \big ) +i(1-\rho _2)t^\beta p_{1\beta }-i\rho _2 t^\beta p_{2\beta } +i\xi _2 y^\alpha \Big ((\rho _1+\rho _2)t_\alpha \nonumber \\&\quad +\rho _2 p_{1\alpha }-(1-\rho _2)p_{2\alpha }-p_{3\alpha }\Big ) +iy^\alpha p_{3\alpha } \Big \}\nonumber \\&\quad \times \omega CCC. \end{aligned}$$
(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

$$\begin{aligned}&\!\!\!I= \!\int _{[0,1]} {d \tau \,}\!\int d^3 \xi _+ \delta (1-\xi _1-\xi _2-\xi _3 ) \nonumber \\&\!\!\!\quad \times z^\gamma \Big [ (a_2\!-\!a_1)_\gamma \delta (\xi _3)\!+\! (a_3\!-\!a_2)_\gamma \delta (\xi _1) + (a_1\!-\!a_3)_\gamma \delta (\xi _2)\Big ]\nonumber \\&\!\!\!\quad \times F \big ( \tau z_\beta P^\beta , ( -\xi _1 a_1-\xi _2 a_2-\xi _3 a_3)_\alpha P^\alpha \big )\, \end{aligned}$$
(7.1)

with arbitrary \(\tau , \xi \)- independent P and \(a_i\).

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

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

Hence

$$\begin{aligned}&\! I \! =\!\! \!\int \limits _{[0,1]}\!\!\! {d \tau }\!\!\int \!\! d^3 \xi _+ \delta (1\!-\!\xi _1\!-\!\xi _2\!-\!\xi _3 ) (a_1\!-\!a_3)^\alpha (a_3\!-\!a_2)_\alpha \nonumber \\&\quad \overrightarrow{\partial }_\tau G \big ( \tau z_\beta P^\beta , (-\xi _1 a_1-\xi _2 a_2-\xi _3 a_3)_\alpha P^\alpha \big ). \end{aligned}$$
(7.3)

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 [11], 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,

$$\begin{aligned} I= & {} - \!\int \!\! d^3 \xi _+ \delta (1\!-\!\xi _1\!-\!\xi _2\!-\!\xi _3 ) (a_1\!-\!a_3)^\alpha (a_3\!-\!a_2)_\alpha \nonumber \\&\times G \big ( 0, (-\xi _1 a_1-\xi _2 a_2-\xi _3 a_3)_\alpha P^\alpha \big ) \nonumber \\&+ \!\int \!\! d^3 \xi _+ \delta (1\!-\!\xi _1\!-\!\xi _2\!-\!\xi _3 ) (a_1\!-\!a_3)^\alpha (a_3\!-\!a_2)_\alpha \nonumber \\&\times G \big ( z_\beta P^\beta , (-\xi _1 a_1-\xi _2 a_2-\xi _3 a_3)_\alpha P^\alpha \big ) . \end{aligned}$$
(7.4)

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

$$\begin{aligned} I= & {} \int _{[0,1]} {d \tau \,}\!\int {d^3 \xi _+\,} \delta (1-\xi _1-\xi _2-\xi _3 ) \nonumber \\&\times \Big [ z^\gamma (a_3\!-\!a_2)_\gamma P^\alpha a_1{}_\alpha +z^\gamma (a_1\!-\!a_3){}_\gamma P^\alpha a_2{}_\alpha \nonumber \\&+z^\gamma (a_2\!-\!a_1){}_\gamma P^\alpha a_3{}_\alpha \Big ]\nonumber \\&\times \frac{\partial }{\partial y} F \big ( \tau z_\alpha P^\alpha ,\,\,-(\xi _1 a_1+\xi _2 a_2+\xi _3 a_3)_\alpha P^\alpha \big ). \end{aligned}$$
(7.5)

The Schouten identity yields

$$\begin{aligned}&\Big [ z^\gamma a_1{}_\gamma P^\alpha (a_3-a_2)_\alpha +z^\gamma a_2{}_\gamma P^\alpha (a_1-a_3)_\alpha \nonumber \\&\quad +z^\gamma a_3{}_\gamma P^\alpha (a_2-a_1)_\alpha \Big ] = \Big [z^\gamma P_\gamma \big \{ a_1{}^\alpha (a_3-a_2)_\alpha \nonumber \\&\quad + a_2^\alpha (a_1-a_3)_\alpha + a_3^\alpha (a_2-a_1)_\alpha \big \} +z^\gamma (a_3-a_2)_\gamma P^\alpha a_1{}_\alpha \nonumber \\&\quad +z^\gamma (a_1-a_3){}_\gamma P^\alpha a_2{}_\alpha +z^\gamma (a_2-a_1){}_\gamma P^\alpha a_3{}_\alpha \Big ]. \end{aligned}$$
(7.6)

One can observe that

$$\begin{aligned}&\Big [z^\gamma (a_3-a_2)_\gamma P^\alpha a_1{}_\alpha +z^\gamma (a_1-a_3){}_\gamma P^\alpha a_2{}_\alpha \nonumber \\&\quad +z^\gamma (a_2-a_1){}_\gamma P^\alpha a_3{}_\alpha \Big ]= - \Big [ z^\gamma a_1{}_\gamma P^\alpha (a_3-a_2)_\alpha \nonumber \\&\quad +z^\gamma a_2{}_\gamma P^\alpha (a_1-a_3)_\alpha +z^\gamma a_3{}_\gamma P^\alpha (a_2-a_1)_\alpha \Big ],\quad \end{aligned}$$
(7.7)

whence it follows (7.3).

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

$$\begin{aligned}&\!\int _{[0,1]} {d \tau \,}\!\int {d^3 \xi _+\,} \delta (1\!-\!\xi _1\!-\!\xi _2\!-\!\xi _3 ) z^\gamma \nonumber \\&\qquad \times \Big [ (a_2\!-\!a_1)_\gamma \delta (\xi _3)+ (a_3\!-\!a_2)_\gamma \delta (\xi _1) + (a_1\!-\!a_3)_\gamma \delta (\xi _2)\Big ]\nonumber \\&\qquad \times f\big ( (\tau z\!-\!\xi _1 a_1\!-\!\xi _2 a_2\!-\!\xi _3 a_3)_\alpha P^\alpha \big ) \nonumber \\&\quad = \!-\! \!\int \limits _{[0,1]}\!\! {d \tau \,}\!\int {d^3 \xi _+\,} \delta (1\!-\!\xi _1\!-\!\xi _2\!-\!\xi _3 ) (a_1\!-\!a_3)^\alpha (a_3\!-\!a_2)_\alpha \nonumber \\&\qquad \overrightarrow{\partial }_\tau f \big ( (\tau z\!-\!\xi _1 a_1\!-\!\xi _2 a_2\!-\!\xi _3 a_3)_\alpha P^\alpha \big ) . \end{aligned}$$
(7.8)

8 Uniformization

Step III of Sect. 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. 13).

As a result, Eq. (3.12) yields

$$\begin{aligned} {\widehat{\varUpsilon }}^{\eta \eta }_{\omega CCC}\Big |_{\text {mod}\, cohomology}\approx \sum _{j=1}^4 F_j \end{aligned}$$
(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.

$$ \begin{aligned}&-\omega *B_3^{\eta \eta }\Big |_{mod\, \delta (\rho _1) \& \delta ({\mathcal {T}})}\approx F_1 : \nonumber \\&\quad =-\frac{\eta ^2}{4}\!\int d\varGamma \, \frac{\delta (\xi _3) \delta (\rho _4)}{(1-\rho _1-\rho _4)(1-\rho _3)}\nonumber \\&\qquad \times \Big [\underbrace{\rho _2 (z_\beta {{\mathcal {P}}}^\beta )(p_{1\alpha }+p_{2\alpha })(p_2 {}^\alpha +p_3 {}^\alpha )}_1 \nonumber \\&\qquad +\underbrace{ i\delta (\rho _3)(1-\rho _1-\rho _4)(1-\rho _3) (z_\alpha {{\mathcal {P}}}^\alpha )}_2 \nonumber \\&\qquad +\underbrace{-i\xi _1\delta (\xi _2)(z_\alpha {{\mathcal {P}}}^\alpha )}_3 \nonumber \\&\qquad +\underbrace{i(1-\rho _1-\rho _4)z_\alpha (p_1 {}^\alpha +p_2 {}^\alpha ) \Big (\delta (\xi _2)-\delta (\xi _1)\Big )}_4\Big ] \nonumber \\&\qquad \times \mathcal {E}\omega CCC, \end{aligned}$$
(8.2)
$$ \begin{aligned}&-{\mathrm{d}}_x B^{\eta \eta }_3\Big | _{mod\, \delta (\rho _1) \& \delta ({\mathcal {T}})}\approx F_2 :=+\frac{\eta ^2}{4}\!\int d\varGamma \, \nonumber \\&\quad \times \frac{\delta (\xi _3)\delta (\rho _1)}{(1\!-\!\rho _1\!-\!\rho _4)(1\!-\!\rho _3)} \Big [\underbrace{\rho _2 (z_\beta {{\mathcal {P}}}^\beta )(p_{1 }+p_{2 })_\alpha (p_2 {}+p_3 {} )^\alpha }_1 \nonumber \\&\quad +\underbrace{ \rho _2(1-\rho _4) (z_\beta t^\beta )t_\alpha (p_2 {}^\alpha +p_3 {}^\alpha )}_2 \nonumber \\&\quad +\underbrace{ \rho _2(1-\rho _4) (z_\beta t^\beta )(p_{1\alpha }+p_{2\alpha })(p_2 {}^\alpha +p_3 {}^\alpha )}_3 \nonumber \\&\quad +\underbrace{\rho _2 (z_\beta {{\mathcal {P}}}^\beta )t_\alpha (p_2 {}^\alpha +p_3 {}^\alpha )}_4 \nonumber \\&\quad +\underbrace{ i\delta (\rho _3)(1-\rho _1-\rho _4)(1-\rho _3)(z_\alpha \mathbb {P}^\alpha )}_5+ \underbrace{-i\xi _1\delta (\xi _2)(z_\alpha {{\mathcal {P}}}^\alpha )}_6 \nonumber \\&\quad +\underbrace{-i\xi _1\delta (\xi _2)(1-\rho _4) z_\alpha t^\alpha }_7 \nonumber \\&\quad +\underbrace{ i(1-\rho _1-\rho _4)z_\alpha (p_1 {}^\alpha +p_2 {}^\alpha ) \Big (\delta (\xi _2)-\delta (\xi _1)\Big )}_8 \nonumber \\&\quad + \underbrace{ i(1-\rho _1-\rho _4)z_\alpha t^\alpha \Big (\delta (\xi _2)-\delta (\xi _1)\Big )}_9\Big ] \mathcal {E}\omega CCC , \end{aligned}$$
(8.3)
$$\begin{aligned}&-{\mathrm{d}}_xB_2^\eta -W_{2\, \omega CC}^{\eta \eta }*C\Big |_{mod\, \delta ({\mathcal {T}})} \approx F_3:\nonumber \\&\quad =-\frac{\eta ^2}{4}\!\int d\varGamma \delta (\rho _3)\delta (\xi _3)\Bigg [ \underbrace{ i\delta (\rho _1)(z_\alpha \mathbb {P}^\alpha )}_1 + \underbrace{\frac{-i z_\alpha t^\alpha \, \xi _1\delta (\xi _2)}{\rho _1+\rho _4}}_2 \nonumber \\&\quad +\underbrace{ t^\alpha (p_{1\alpha }+p_{2\alpha })z_\gamma {{\mathcal {P}}}^\gamma }_3 +\underbrace{ -i\delta (\rho _4)z_\alpha {{\mathcal {P}}}^\alpha }_4 \nonumber \\&\quad + \underbrace{ t^\gamma (p_{1\gamma }+p_{2\gamma })z_\alpha t^\alpha \left( (1-\rho _4) -\frac{\rho _1}{\rho _1+\rho _4}\right) }_5\Bigg ]\nonumber \\&\quad \times \mathcal {E}\, \omega CCC , \end{aligned}$$
(8.4)
$$\begin{aligned}&-({\mathrm{d}}_x B_3^{\eta \eta }+\omega *B_3^{\eta \eta })\Big |_{\delta (\rho _1)} \Big |_{mod\, \delta ({\mathcal {T}})}-W_{1\, \omega C}^{\eta }*B_2^{\eta \, loc}\approx F_4:\nonumber \\&\quad =-\frac{\eta ^2}{4}\!\int d\varGamma \, \frac{\delta (\xi _3)\delta (\xi _2)\, z_\alpha (p_2 {}^\alpha +p_3 {}^\alpha )}{(\rho _2+\rho _3)(\rho _1+\rho _4)}\times \nonumber \\&\quad \times \left( \underbrace{i\Big (\delta (\rho _1)-\delta (\rho _4)\Big ){{\mathcal {E}}}}_1+ \underbrace{ i{E_z{\,}}\left( \frac{\partial }{\partial \rho _1}-\frac{\partial }{\partial \rho _4}\right) E}_2\right) \nonumber \\&\quad \times \omega CCC. \end{aligned}$$
(8.5)

Note that

$$\begin{aligned}&F_{1,2}+F_{3,4}=0, \end{aligned}$$
(8.6)
$$\begin{aligned}&F_{2,5}+F_{3,1}=0. \end{aligned}$$
(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. 13).

9 Eliminating \(\delta (\rho _j)\) and \(\delta (\xi _j)\): result

The fourth step of Sect. 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. (4.6)–(4.8) one finds that Eq. (8.1) yields

$$\begin{aligned} \big ({\widehat{\varUpsilon }}^{\eta \eta }_{\omega CCC} - G_1-G_2-G_3\big )\big |_{\!\!\!\!\!\!\mod cohomology }\approx 0 ,\quad \end{aligned}$$
(9.1)

where

$$\begin{aligned}&G_1 := J_3+\frac{\eta ^2}{4} \!\int d\varGamma \, \delta (\xi _3) z_\gamma \Bigg \{ \frac{(y +\widetilde{t})^\gamma \rho _2\, t^\alpha (p_{1 }+p_{2 })_\alpha }{(1-\rho _1-\rho _4)(1-\rho _3)} \nonumber \\&\quad \times {E_z{\,}}\Bigg [\frac{\partial }{\partial \rho _2}-\frac{\partial }{\partial \rho _3}\Bigg ]E \nonumber \\&\quad + (y^\gamma +\widetilde{t}^\gamma ) \frac{\rho _2\, (p_1 {}^\alpha +p_2 {}^\alpha )(p_{2\alpha }+p_{3\alpha })}{(1-\rho _1-\rho _4)(1-\rho _3)}{E_z{\,}}\Bigg [\frac{\partial }{\partial \rho _4}-\frac{\partial }{\partial \rho _1}\Bigg ]E \nonumber \\&\quad + (y +{\tilde{t}})^\gamma \frac{\rho _2\, t^\alpha (p_{2\alpha }+p_{3\alpha })}{(1-\rho _1-\rho _4)(1-\rho _3)} \nonumber \\&\quad \times {E_z{\,}}\Bigg [\frac{\partial }{\partial \rho _2}-\frac{\partial }{\partial \rho _1}\Bigg ]E + (y +{\tilde{t}})^\gamma \frac{(\rho _1+\rho _4) t^\alpha (p_{1\alpha }+p_{2\alpha })}{(1-\rho _1-\rho _4)(1-\rho _3)}{{\mathcal {E}}}\nonumber \\&\quad + (y +{\tilde{t}})^\gamma \frac{\rho _3\, t^\alpha (p_{2\alpha }+p_{3\alpha })}{(1-\rho _1-\rho _4)^2 (1-\rho _3)} {{\mathcal {E}}}\nonumber \\&\quad +\frac{\rho _2 t^\gamma (p_2 {}+p_3 {} )^\alpha (p_{1 }+p_{2 } +t -{\tilde{t}})_\alpha }{(1-\rho _1-\rho _4)(1-\rho _3)(\rho _1+\rho _4)}{{\mathcal {E}}}\Bigg \} \nonumber \\&\quad \times \omega CCC \end{aligned}$$
(9.2)
$$\begin{aligned}&G_2 := J_4 +\frac{\eta ^2}{4}\!\int d\varGamma \, \frac{\delta (\xi _3)}{1-\rho _3}\,z^\alpha \Bigg \{ \frac{\rho _3 (y_\alpha +{\tilde{t}}_\alpha )t^\gamma (y_\gamma +\mathbb {P}_\gamma ) }{(1-\rho _1-\rho _4)^2(1-\rho _3)} {{\mathcal {E}}}\nonumber \\&\quad -\frac{\rho _2\rho _4\, t_\alpha (y^\gamma +\mathbb {P}^\gamma )t_\gamma }{(1-\rho _1-\rho _4)(1-\rho _3)(\rho _1+\rho _4)^2}{{\mathcal {E}}}\nonumber \\&\quad -\frac{\rho _2\, (y_\alpha +{\tilde{t}}_\alpha ) t^\gamma (p_{1\gamma }+p_{2\gamma }) }{(1-\rho _1-\rho _4)(1-\rho _3)}{{\mathcal {E}}}\nonumber \\&\quad -\frac{\rho _2\, (p_1 {}_\alpha +p_2 {}_\alpha )(y^\gamma +\mathbb {P}^\gamma )t_\gamma }{(1-\rho _1-\rho _4)(\rho _1+\rho _4)(1-\rho _3)} {{\mathcal {E}}}\nonumber \\&\quad +{E_z{\,}}\frac{\rho _2\, t^\gamma (y_\gamma +\mathbb {P}_\gamma )(y_\alpha +{\tilde{t}}_\alpha ) }{(1-\rho _1-\rho _4)(1-\rho _3)}\Bigg [ \frac{\partial }{\partial \rho _1}-\frac{\partial }{\partial \rho _2}\Bigg ]E \nonumber \\&\quad +{E_z{\,}}\frac{\rho _2\, (y_\alpha +{\tilde{t}}_\alpha ) (p_1 {}^\gamma +p_2 {}^\gamma )(y_\gamma +\mathbb {P}_\gamma ) }{(1-\rho _1-\rho _4)(1-\rho _3)}\nonumber \\&\quad \times \Bigg [\frac{\partial }{\partial \rho _1} -\frac{\partial }{\partial \rho _4}\Bigg ]E\Bigg \}\omega CCC,\quad \end{aligned}$$
(9.3)
$$\begin{aligned}&G_{ 3} := J_5 + \frac{\eta ^2}{4} \!\int d\varGamma \, \delta (\xi _3) \Bigg (1+\xi _1\Bigg [\frac{\partial }{\partial \xi _1} -\frac{\partial }{\partial \xi _2}\Bigg ]\Bigg )\nonumber \\&\quad \times z_\alpha \Bigg \{\frac{\rho _2\, t^\alpha (p_2 {}^\gamma +p_3 {}^\gamma )(y_\gamma +{\tilde{t}}_\gamma )}{(1-\rho _1-\rho _4)^2 (1-\rho _3)(\rho _1+\rho _4)} \nonumber \\&\quad + \frac{-\rho _2\, t^\alpha (y +{\tilde{t}})^\gamma (y_\gamma +\mathbb {P}_\gamma ) }{(1-\rho _1-\rho _4)^2 (1-\rho _3)^2 (\rho _1+\rho _4)} \nonumber \\&\quad +\frac{\!-\rho _3 (y^\alpha +{\tilde{t}}^\alpha ) (t^\gamma y_\gamma )}{(1-\rho _1-\rho _4)^2 (1-\rho _3)^2}\nonumber \\&\quad +\frac{ (y^\alpha +{\tilde{t}}^\alpha ) (p_1 {}^\gamma +p_2 {}^\gamma )t_\gamma }{(1-\rho _1-\rho _4)(1-\rho _3)^2} \Bigg \}{{\mathcal {E}}}\, \omega CCC ,\quad \end{aligned}$$
(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. 15).)

Note that schematically

$$\begin{aligned}&G_1+G_2+G_3 =J_3+J_4+ J_5 \nonumber \\&\quad +\int d\varGamma \, \delta (\xi _3) z_\alpha g ^\alpha (y,t,p_1,p_2,p_3\vert \rho ,\xi ) {{\mathcal {E}}}\, \omega CCC ,\quad \nonumber \\ \end{aligned}$$
(9.5)

as expected . Let us stress that \(g^\alpha (y,t,p_1,p_2,p_3\vert \rho ,\xi )\) 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 Sect. 10.1. The term with degree-six pre-exponential is considered in Sect. 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)–(9.4), denoting it as \(S_4\). Partial integration yields

$$\begin{aligned}&S_4\approx J_6 +\frac{ \eta ^2 }{4 }\!\int d\varGamma \, \delta (\xi _3)\, \Big [ \frac{\rho _2{t}{}^\alpha z _\alpha ( p_3+p_2)^{\gamma }( {t}\!-\!{\tilde{t}} ){}_{\gamma }}{(1\!-\! \rho _1 \!-\!\rho _4)(1\!-\!\rho _3)(\rho _1+\rho _4) } \nonumber \\&\quad + \frac{ \rho _2 \rho _4 {t}{}^\gamma z_\gamma \big (y + \mathbb {P}{}\big )^\alpha {t}{} _{\alpha }}{(1-\rho _1-\rho _4 )(1-\rho _3)^2(\rho _1+\rho _4)^2} \nonumber \\&\quad + \frac{ \rho _2 ( p_1{}+ p_2)^{\gamma } \big (y + (1-\rho _4){t}{} \big ) _{\gamma } z^\alpha t _{\alpha }}{(1-\rho _1-\rho _4 )(1-\rho _3)^2(\rho _1+\rho _4)} \nonumber \\&\quad + \frac{ \rho _2 {t}{}^{\gamma }z_{\gamma } \, ( p_3+p_2)^{\alpha } (y+{\tilde{t}}{})_{\alpha }}{(1- \rho _1 -\rho _4)^2(1-\rho _3)(\rho _1+\rho _4)} \nonumber \\&\quad + \frac{ \rho _2 \big ( -\mathbb {P}{}+ {\tilde{t}}{} \big )^\gamma \big ( y + {\tilde{t}}{} \big )_{\gamma } z^\alpha {t}{}_{\alpha }}{(1- \rho _1 -\rho _4)^2(1-\rho _3)^2( \rho _1+\rho _4 ) } \Big ] \nonumber \\&\quad \times {{\mathcal {E}}}\omega CCC,\quad \end{aligned}$$
(10.1)

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

$$\begin{aligned}&S_6= \frac{ \eta ^2 }{4 }\!\int d\varGamma \, \delta (\xi _3) \Big \{{E_z{\,}}\frac{\rho _2 (y+ {\tilde{t}}{} )^{\gamma }z_{\gamma }{t}{}^\alpha ( (p_1+p_2 ){} _\alpha ) }{(1- \rho _1 -\rho _4)(1-\rho _3) }\nonumber \\&\quad \times \Big [ (\overrightarrow{\partial }_{\rho _2}-\overrightarrow{\partial }_{\rho _3})E \Big ]\quad \nonumber \\&\quad + {E_z{\,}}\frac{ \rho _2 ( p_1{}+ p_2)^{\gamma } \big (y + (1 -\rho _4 ){t}{}\big )_{\gamma } z_\alpha (y+{\tilde{t}}{})^{\alpha }}{(1-\rho _1-\rho _4 )(1-\rho _3)^2} \nonumber \\&\quad \times [ \overrightarrow{\partial }_{\rho _4}-\overrightarrow{\partial }_{\rho _1}] E \nonumber \\&\quad + {E_z{\,}}\frac{ \rho _2 {t}{}^\gamma \big (y + (1-\rho _1-\rho _4 )( p_1{} +p_2{} ){}\big )_\gamma z_\alpha (y+{\tilde{t}}{})^{\alpha }}{(1-\rho _1-\rho _4 )(1-\rho _3)^2} \nonumber \\&\quad \times [\overrightarrow{\partial }_{\rho _2}\!-\!\overrightarrow{\partial }_{\rho _1} ]E +i \xi _1 \Big [ \frac{ \rho _2^2 z_\alpha {t}{}^{\alpha }}{(1\!-\! \rho _1 \!-\!\rho _4)^3(1\!-\!\rho _3)^3( \rho _1\!+\!\rho _4 ) } \nonumber \\&\quad \times \big ( y+ (1\!-\!\rho _1\!-\!\rho _4 )( p_1{} +p_2{} )+ (1 \!-\!\rho _4 ){t}{} \big )^\gamma \big ( y + {\tilde{t}}{} \big )_{\gamma } \nonumber \\&\quad -\frac{ \rho _3\rho _2}{(1- \rho _1 -\rho _4)^3(1-\rho _3)^3 } {\big ( y + {\tilde{t}}{} \big )^\gamma z_{\gamma } {t}{}^{\alpha } y_{\alpha }} \nonumber \\&\quad + \frac{ \rho _2 \big ( y + {\tilde{t}}{} \big )^\gamma z_{\gamma } { (p_1{}^{\alpha }+p_2{}^{\alpha }){t}{}_{\alpha }}}{(1- \rho _1 -\rho _4)^2(1-\rho _3)^3 } \Big ] \nonumber \\&\quad \times {{\mathcal {E}}}\big (y + \mathbb {P}{} \big )^\alpha (y+{\tilde{t}}{})_{\alpha }\Big \} \omega CCC \end{aligned}$$
(10.2)

Recall that the integral measure \({\mathrm{d}}\varGamma \) (4.9) 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 \(\xi _2\rightarrow 1-\xi _1\), \(\xi _3\rightarrow 0\). 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 [14] the manifestly spin-local holomorphic vertex \(\varUpsilon ^{\eta \eta }_{\omega CCC}\) in the Eq. (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 Sect. 7 playing important role in our analysis.

Let us stress that spin-locality of the vertices obtained in [14] follows from Z-dominance Lemma. However the evaluation the explicit spin-local vertex \(\varUpsilon ^{\eta ^2}_{\omega CCC}\) 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.