# Patterns of gauge symmetry in the background field method

A. C. Aguilar  University of Campinas - UNICAMP, Institute of Physics “Gleb
Wataghin”, 13083-859 Campinas, São Paulo, Brazil.  M. N. Ferreira  Department
of Theoretical Physics and IFIC,
University of Valencia and CSIC, E-46100, Valencia, Spain.  D. Ibañez
University Centre EDEM, Muelle de la Aduana, La Marina de Valencia, 46024,
Valencia, Spain.  B. M. Oliveira  University of Campinas - UNICAMP, Institute
of Physics “Gleb Wataghin”, 13083-859 Campinas, São Paulo, Brazil.  J.
Papavassiliou  Department of Theoretical Physics and IFIC,
University of Valencia and CSIC, E-46100, Valencia, Spain.

###### Abstract

The correlation functions of Yang-Mills theories formulated in the background
field method satisfy linear Slavnov-Taylor identities, which are naive
generalizations of simple tree level relations, with no deformations
originating from the ghost-sector of the theory. In recent years, a stronger
version of these identities has been found to hold at the level of the
background gluon self-energy, whose transversality is enforced separately for
each special block of diagrams contributing to the gluon Schwinger-Dyson
equation. In the present work we demonstrate by means of explicit calculations
that the same distinct realization of the Slavnov-Taylor identity persists in
the case of the background three-gluon vertex. The analysis is carried out at
the level of the exact Schwinger-Dyson equation for this vertex, with no
truncations or simplifying assumptions. The demonstration entails the
contraction of individual vertex diagrams by the relevant momentum, which
activates Slavnov-Taylor identities of vertices and multi-particle kernels
nested inside these graphs; the final result emerges by virtue of a multitude
of extensive cancellations, without the need of performing explicit
integrations. In addition, we point out that background Ward identities amount
to replacing derivatives of propagators by zero-momentum background-gluon
insertions, in exact analogy to standard properties of Abelian gauge theories.
Finally, certain potential applications of these results are briefly
discussed.

##  I Introduction

In recent years, the systematic exploration of Green’s (correlation) functions
has afforded important insights on the nonperturbative properties of non-
Abelian gauge theories, such as pure Yang-Mills theories and Quantum
Chromodynamics Roberts and Williams (1994); Alkofer and von Smekal (2001);
Fischer (2006); Roberts (2008); Binosi and Papavassiliou (2009); Binosi  _et
al._ (2015); Cloet and Roberts (2014); Aguilar  _et al._ (2016a); Binosi  _et
al._ (2016, 2017); Huber (2020); Papavassiliou (2022). This ongoing scrutiny
relies on continuum studies based on nonperturbative functional methods Maris
and Roberts (1997, 2003); Braun  _et al._ (2010); Eichmann  _et al._ (2009);
Cloet  _et al._ (2009); Boucaud  _et al._ (2008); Eichmann  _et al._ (2010);
Fischer  _et al._ (2009); Boucaud  _et al._ (2008); Dudal  _et al._ (2008);
Rodriguez-Quintero (2011); Tissier and Wschebor (2010); Pennington and Wilson
(2011); Huber  _et al._ (2012); Cloet and Roberts (2014); Fister and Pawlowski
(2013); Cyrol  _et al._ (2015); Boucaud  _et al._ (2008); Pawlowski  _et al._
(2004); Pawlowski (2007); Cyrol  _et al._ (2018a, b); Corell  _et al._ (2018);
Gao  _et al._ (2018); Blaizot  _et al._ (2021); Roberts (2020, 2020); Horak
_et al._ (2021); Gao  _et al._ (2021) carried out almost exclusively in the
linear covariant ($R_{\xi}$) gauges, where the Landau gauge is the preferred
choice, and on lattice simulations performed in the same gauge Cucchieri  _et
al._ (2006); Cucchieri and Mendes (2007); Bogolubsky  _et al._ (2007, 2009);
Cucchieri  _et al._ (2008); Cucchieri and Mendes (2010); Oliveira and Silva
(2009); Oliveira and Bicudo (2011); Maas (2013); Ayala  _et al._ (2012);
Oliveira and Silva (2012); Athenodorou  _et al._ (2016); Duarte  _et al._
(2016); Boucaud  _et al._ (2017, 2018); Aguilar  _et al._ (2021). However, the
background field method (BFM) DeWitt (1967); Honerkamp (1972); Kallosh (1974);
Kluberg-Stern and Zuber (1975); Arefeva  _et al._ (1974); Abbott (1981);
Weinberg (1980); Abbott (1982); Shore (1981); Abbott  _et al._ (1983) has also
been employed in several occasions, furnishing useful vantage points, and
exposing key properties of the theory that are normally distorted by standard
quantization procedures Aguilar and Papavassiliou (2006); Aguilar  _et al._
(2008, 2016a); Papavassiliou (2022).

The BFM is a powerful framework that enables the implementation of the gauge-
fixing procedure necessary for quantizing gauge theories without losing
explicit gauge invariance, in contradistinction to the conventional
quantization schemes DeWitt (1967); Honerkamp (1972); Kallosh (1974); Kluberg-
Stern and Zuber (1975); Arefeva  _et al._ (1974); Abbott (1981); Weinberg
(1980); Abbott (1982); Shore (1981); Abbott  _et al._ (1983). The starting
point of the BFM is the splitting of the gauge field $A_{\mu}^{a}$ appearing
in the classical action of the theory according to
$A_{\mu}^{a}=B_{\mu}^{a}+Q_{\mu}^{a}$, where $B_{\mu}^{a}$ and $Q_{\mu}^{a}$
are the background and quantum (fluctuating) fields, respectively. The quantum
field is the variable of integration in the generating functional $Z(J)$, and
external sources are coupled only to it, as $J\cdot Q$. The background field
does not enter in loops; it couples externally to Feynman diagrams, connecting
them with the asymptotic states to form S-matrix elements. Then, by virtue of
a special gauge-fixing condition, the resulting action (with the corresponding
ghost terms included) is no longer invariant under transformations of the
quantum field, but retains its invariance intact with respect to the
background field Abbott (1982); Binosi and Papavassiliou (2009).

A key consequence of the background gauge invariance of the action is that
Green’s functions involving the $B_{\mu}^{a}$ field satisfy ghost-free
Slavnov-Taylor identities (STIs), akin to the Takahashi identity known from
QED: the STIs are straightforward generalization of tree level relations,
receiving no ghost-related contributions after the inclusion of quantum
corrections. Instead, in the standard STIs Taylor (1971); Slavnov (1972) of
the $R_{\xi}$ gauges Fujikawa  _et al._ (1972), starting already at one-loop,
the ghost sector modifies these tree level relations non-trivially. To fix the
ideas with a simple example, the quark-gluon vertex with either a $B$ or a $Q$
gluon satisfies at tree level the simple identity (suppressing color)
$q^{\mu}\gamma_{\mu}=\not{q}=(-\not{p}-m)-(\not{r}-m)=S_{0}^{-1}(-\not{p})-S_{0}^{-1}(\not{r})$,
where $S_{0}$ is the tree level version of the quark propagator $S$. In the
case of a $B$ gluon, the all-order STI is obtained from the above tree level
identity by simply substituting $S_{0}\to S$, namely
$q^{\mu}\widehat{\Gamma}_{\mu}(q,r,p)=S^{-1}(-\not{p})-S^{-1}(\not{r})$.
Instead, in the case of a $Q$ gluon ($R_{\xi}$ gauges), the STI gets modified
by quantum corrections Taylor (1971); Slavnov (1972), which induce a
dependence on the ghost dressing function and the quark-ghost kernel Marciano
and Pagels (1978); Davydychev  _et al._ (2001); Aguilar and Papavassiliou
(2011); Aguilar  _et al._ (2017).

As was pointed in earlier studies, the STI satisfied by the background-gluon
self-energy, $\widehat{\Pi}_{\mu\nu}(q)$, namely the standard transversality
condition $q^{\mu}\widehat{\Pi}_{\mu\nu}(q)=0$, is implemented in a very
special way, which has been denominated “block-wise” Aguilar and Papavassiliou
(2006); Aguilar  _et al._ (2008, 2016a). Specifically, the diagrammatic
representation of the SDE governing $\widehat{\Pi}_{\mu\nu}(q)$ is composed by
four distinct blocks, namely one- and two-loop diagrams containing only
gluons, and one- and two-loop diagrams containing ghost fields, as shown in
Fig. 1. The block-wise realization of the STI in this case is the simple
statement that the transversality of $\widehat{\Pi}_{\mu\nu}(q)$ is enforced
independently for each of the four blocks. This is in sharp contrast to what
happens in the $R_{\xi}$ gauges, where, already at the one-loop perturbative
level, it is only the sum of gluon and ghost diagrams that is transverse
Itzykson and Zuber (1980); Peskin and Schroeder (1995). The proof of this
property is particularly simple; it proceeds by contracting the various
diagrams by $q^{\mu}$ from the side of the fully dressed vertices, thus
triggering the corresponding naive STIs of the BFM. It is important to
emphasize that the proof holds for any value of the gauge-fixing parameter
$\xi_{{\scriptscriptstyle Q}}$ used to define the propagators of the $Q$-type
gluons entering in the various loops.

The basic question that arises naturally in this context is whether the
special block-wise realization of the STI described above is particular to the
two-point function, or if it is a common feature of all Green’s functions
containing only $B$ gluons. In the present work we take a first step in the
exploration of this issue, and demonstrate that the same pattern persists in
the STI of the BFM vertex with three incoming background gluons, to be denoted
by $\widehat{\Gamma}_{\alpha\mu\nu}(q,r,p)$. This Abelian STI relates the
contraction $q^{\mu}\widehat{\Gamma}_{\alpha\mu\nu}(q,r,p)$ to the difference
$\widehat{\Pi}_{\mu\nu}(r)-\widehat{\Pi}_{\mu\nu}(p)$ Cornwall and
Papavassiliou (1989); Aguilar and Papavassiliou (2006); Binosi and
Papavassiliou (2009). It turns out that the diagrams comprising the SDE of
$\widehat{\Gamma}_{\alpha\mu\nu}(q,r,p)$ may also be classified into four
subsets in a way completely analogous to the case of
$\widehat{\Pi}_{\mu\nu}(q)$. Then, the contraction of each subset by $q^{\mu}$
generates the difference of the corresponding subsets of
$\widehat{\Pi}_{\mu\nu}$, confirming the block-wise realization of this STI.
We emphasize that, as in the case of the gluon self-energy, this property is
completely $\xi_{{\scriptscriptstyle Q}}$-independent.

An additional noteworthy aspect of the BFM Green’s functions is the Ward
identities (WIs) they satisfy, namely the relations that emerge when the
momentum that triggers the STIs is taken to vanish. For example, in the case
of the $\widehat{\Gamma}_{\mu}(q,r,p)$ mentioned above, a Taylor expansion of
the STI around $q=0$ and subsequent matching of terms linear in $q$ yields the
relation $\widehat{\Gamma}_{\mu}(0,-p,p)=\partial S^{-1}(\not{p})/\partial
p^{\mu}$, which is the precise equivalent of the text-book WI known from QED,
relating the photon-electron vertex with the electron propagator Itzykson and
Zuber (1980); Peskin and Schroeder (1995). In fact, exactly as happens in QED,
this WI admits a simple diagrammatic interpretation: the derivative of the
inverse quark propagator may be depicted as the insertion of a background
gluon carrying zero momentum. These observations may be straightforwardly
extended to the case of the three-gluon vertex
$\widehat{\Gamma}_{\alpha\mu\nu}(q,r,p)$, allowing for a completely analogous
pictorial representation of the corresponding WI. In fact, the block-wise
realization of the STI leads to a corresponding pattern for the WIs that
emerges from it: the derivative acting on any of the blocks of
$\widehat{\Pi}_{\mu\nu}(q)$ is identical to the diagrams comprising the
associated block of $\widehat{\Gamma}_{\alpha\mu\nu}(q,r,p)$, when the
corresponding momentum is set to zero. To the best of our knowledge, the
notions described above appear for the first time in the literature.

The article is organized as follows. In Sec. II we review certain pivotal
properties of the BFM, and explain the notion of the block-wise transversality
at the level of the SDE that governs the background gluon propagator. Sec. III
contains the main result of this work, namely the demonstration of the block-
wise realization of the STI for the case of the background three-gluon vertex.
Then, in Sec. IV we focus on the WIs of the BFM, their graphical
representation in terms of zero-momentum gluon insertions, and demonstrate the
block-wise realization of the three-gluon WI, for the operationally simplest
subset of graphs. In Sec. V we summarize our findings and discuss future
directions. Finally, in four Appendices we present complementary material that
facilitates the perusal of the article.

##  II General theoretical framework

In this section we highlight some of the significant features of the BFM
formalism that are relevant for the demonstrations that follow; for further
details the reader is referred to the extensive literature on the subject,
see, e.g., Abbott (1981); Abbott  _et al._ (1983); Binosi and Papavassiliou
(2009).

(i) The initial decomposition of the gauge field into $B_{\mu}^{a}$ and
$Q_{\mu}^{a}$ components increases considerably the number of Green’s
functions that can be defined, which may be classified into three broad
subsets: those with $B_{\mu}^{a}$ fields only, those with $Q_{\mu}^{a}$ fields
only (corresponding to the standard Green’s functions of the $R_{\xi}$
gauges), and mixed ones, with both $B_{\mu}^{a}$ and $Q_{\mu}^{a}$ fields. We
will occasionally denote Green’s functions according to the type of incoming
fields, such as “BB” for the case of the propagator connecting two background
fields, or “BBB” for the case of the three-gluon vertex connecting three such
fields.

(ii) The gluon propagator QQ that enters in the quantum loops will be denoted
by $\Delta_{\mu\nu}^{ab}(q)=-i\delta^{ab}\Delta_{\mu\nu}(q)$, with

$\displaystyle\Delta_{\mu\nu}(q)=P_{\mu\nu}(q)\Delta(q)+\xi_{{\scriptscriptstyle
Q}}\frac{q_{\mu}q_{\nu}}{q^{4}}\,,\qquad
P_{\mu\nu}(q)=g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\,,$ (1)

whose inverse is

$\displaystyle\Delta_{\mu}^{\nu}(q)\Delta_{\nu\rho}^{-1}(q)=g_{\mu\rho}\,;\quad\quad\Delta_{\nu\rho}^{-1}(q)=\Delta^{-1}(q)P_{\nu\rho}(q)+\xi_{{\scriptscriptstyle
Q}}^{-1}q_{\nu}q_{\rho}\,.$ (2)

The scalar function $\Delta(q)$ is related to the gluon self-energy
$\Pi_{\mu\nu}(q)=P_{\mu\nu}(q)\Pi(q)$ through $\Delta^{-1}(q)=q^{2}+i\Pi(q)$,
and $\xi_{{\scriptscriptstyle Q}}$ is the quantum gauge-fixing parameter. Note
that $\xi_{{\scriptscriptstyle Q}}$ enters also in the tree level expressions
of the vertices BQQ and BBQQ, given in Table 1.

(iii) In what follows we will use extensively a number of three- and four-
particle vertices, which we list here. In particular, the relevant three-
particle vertices are

$\displaystyle\Gamma_{\bar{c}^{m}c^{n}Q_{\mu}^{a}}(r,p,q)=-gf^{mna}\Gamma_{\mu}(r,p,q)\,,\qquad$
$\displaystyle\Gamma_{\\!Q_{\alpha}^{a}Q_{\mu}^{b}Q_{\nu}^{c}}(q,r,p)=gf^{abc}\widetilde{\Gamma}_{\alpha\mu\nu}(q,r,p)\,,$
$\displaystyle\Gamma_{\bar{c}^{m}c^{n}B_{\mu}^{a}}(r,p,q)=-gf^{mna}\widetilde{\Gamma}_{\mu}(r,p,q)\,,\qquad$
$\displaystyle\Gamma_{\\!B_{\alpha}^{a}Q_{\mu}^{b}Q_{\nu}^{c}}(q,r,p)=gf^{abc}\widetilde{\Gamma}_{\alpha\mu\nu}(q,r,p)\,,$
(3)

while the four-particle vertices are

$\displaystyle\Gamma_{B_{\mu}^{a}B_{\nu}^{b}\bar{c}^{m}c^{n}}(q,r,p,t)=-ig^{2}\widehat{\Gamma}_{\mu\nu}^{abmn}(q,r,p,t),$
$\displaystyle\Gamma_{\\!B_{\alpha}^{a}Q_{\beta}^{b}Q_{\mu}^{c}Q_{\nu}^{d}}(q,r,p,t)=-ig^{2}\widetilde{\Gamma}_{\alpha\beta\mu\nu}^{abcd}(q,r,p,t),$
$\displaystyle\Gamma_{B_{\mu}^{a}Q_{\nu}^{b}\bar{c}^{m}c^{n}}(q,r,p,t)=-ig^{2}\widetilde{\Gamma}_{\mu\nu}^{abmn}(q,r,p,t),$
$\displaystyle\Gamma_{\\!B_{\alpha}^{a}B_{\beta}^{b}Q_{\mu}^{b}Q_{\nu}^{c}}(q,r,p,t)=-ig^{2}\widehat{\Gamma}_{\alpha\beta\mu\nu}^{abcd}(q,r,p,t),$
(4)

where $g$ denotes the gauge coupling constant, and $f^{abc}$ are the SU(3)
structure constants.

(iv) A central quantity in our analysis is the background self-energy,
$\widehat{\Pi}_{\mu\nu}(q)$, related to the inverse background gluon
propagator $\widehat{\Delta}^{-1}_{\mu\nu}(q)$ by 111The definition of the BB
propagator $\widehat{\Delta}_{\mu\nu}(q)$ requires the addition to the action
of a supplementary gauge-fixing term, which introduces the “classical” gauge-
fixing parameter, $\xi_{{\scriptscriptstyle C}}$ Abbott (1981); Abbott  _et
al._ (1983); Binosi and Papavassiliou (2009). Note that this step is necessary
only when connecting the background gluon to external states in order to
construct S-matrix elements, and will be omitted here.

$\widehat{\Delta}^{-1}_{\mu\nu}(q)=q^{2}P_{\mu\nu}(q)+i\widehat{\Pi}_{\mu\nu}(q)\,.$
(5)

The gauge symmetry enforces the fundamental STI

$q^{\mu}\widehat{\Pi}_{\mu\nu}(q)=0\,,$ (6)

from which follows that
$\widehat{\Pi}_{\mu\nu}(q)=P_{\mu\nu}(q)\widehat{\Pi}(q)$, where
$\widehat{\Pi}(q)$ is a scalar function. Thus, Eq. (5) may be cast in the form

$\widehat{\Delta}^{-1}_{\mu\nu}(q)=P_{\mu\nu}(q)\left[q^{2}+i\widehat{\Pi}(q)\right]\,.$
(7)

The SDE that defines $\widehat{\Pi}_{\mu\nu}(q)$ is diagrammatically
represented in Fig. 1. Note the separation of the dressed Feynman diagrams
into the following four distinct groups:

(1) One-loop gluonic graphs, enclosed in the blue box; their total
contribution is denoted by $\widehat{\Pi}^{(1)}_{\mu\nu}(q)$.

(2) One-loop ghost graphs, enclosed in the orange box; their total
contribution is denoted by $\widehat{\Pi}^{(2)}_{\mu\nu}(q)$.

(3) Two-loop gluonic graphs, enclosed in the purple box; their total
contribution is denoted by $\widehat{\Pi}^{(3)}_{\mu\nu}(q)$.

(4) Two-loop ghost graphs, enclosed in the green box; their total contribution
is denoted by $\widehat{\Pi}^{(4)}_{\mu\nu}(q)$.

Figure 1: Diagrammatic representation of the self-energy
$\widehat{\Pi}_{\mu\nu}(q)$. The small gray circles at the end of gluon legs
indicate a background field. The orange and green circles represent
conventional fully dressed propagators and vertices, respectively, while the
blue circles represent fully dressed vertices with one background gluon.

One of the most exceptional properties of $\widehat{\Pi}_{\mu\nu}(q)$ is its
block-wise transversality Aguilar and Papavassiliou (2006); Binosi and
Papavassiliou (2008a, b). Specifically, the fundamental relation given in Eq.
(6) is realized in a very special way: each of the four subsets of diagrams in
Fig. 1 is individually transverse, i.e.,

$q^{\mu}\widehat{\Pi}^{(i)}_{\mu\nu}(q)=0\,,\qquad i=1,2,3,4\,.$ (8)

This particular result is a direct consequence of the Abelian STIs satisfied
by the fully dressed vertices entering in the diagrams comprising the
$\widehat{\Pi}^{(i)}_{\mu\nu}(q)$ Aguilar and Papavassiliou (2006); Binosi and
Papavassiliou (2008a, b), namely BQQ, ${\rm B\bar{c}c}$, BQQQ and ${\rm
BQ\bar{c}c}$, reported in Table 2 of the Appendix C.

(v) In order to elucidate with a simple example how this special
transversality is enforced at the diagrammatic level, we consider the case of
$\widehat{\Pi}^{(2)}_{\mu\nu}(q)$, whose diagrams are enclosed by the orange
box of Fig. 1.

The diagrams $(a_{{\scriptscriptstyle 3}})$ and $(a_{{\scriptscriptstyle 4}})$
are given by

$\displaystyle(a_{{\scriptscriptstyle 3}})_{\mu\nu}(q)$
$\displaystyle=\lambda\int_{k}(2k-q)_{\mu}D(k-q)D(k)\widetilde{\Gamma}_{\nu}(q-k,k,-q)\,,$
(9) $\displaystyle(a_{{\scriptscriptstyle 4}})_{\mu\nu}(q)$
$\displaystyle=2\lambda\,g_{\mu\nu}\int_{k}D(k)\,,$ (10)

where we have used the Feynman rules given in Eq. (95) and Eq. (99) of the
Appendix B, and factored out the trivial color structure $\delta^{ab}$ from
both expressions. In addition, we have defined

$\lambda:=g^{2}C_{\rm A}\,,$ (11)

where $C_{\rm A}$ is the Casimir eigenvalue of the adjoint representation [$N$
for SU($N$)]. Furthermore, we have introduced

$\int_{k}:=\frac{1}{(2\pi)^{4}}\\!\int_{-\infty}^{+\infty}\\!\\!\mathrm{d}^{4}k\,,$
(12)

where the use of a symmetry-preserving regularization scheme is implicitly
assumed.

We next contract graph $(a_{{\scriptscriptstyle 3}})_{\mu\nu}(q)$ by
$q^{\nu}$, thus triggering the STI satisfied by
$\widetilde{\Gamma}_{\nu}(q-k,k,-q)$, given in Eq. (101), to obtain

$\displaystyle q^{\nu}(a_{{\scriptscriptstyle 3}})_{\mu\nu}(q)$
$\displaystyle=$
$\displaystyle\lambda\int_{k}(2k-q)_{\mu}D(k-q)D(k)\left[D^{-1}(k-q)-D^{-1}(k)\right]$
(13) $\displaystyle=$
$\displaystyle\lambda\int_{k}(2k-q)_{\mu}\left[D(k)-D(k-q)\right]$
$\displaystyle=$ $\displaystyle-2\lambda\,q_{\mu}\int_{k}D(k)\,,$

which is exactly the negative of the contraction
$q^{\nu}(a_{{\scriptscriptstyle 4}})_{\mu\nu}(q)$. Hence,

$\displaystyle q^{\nu}\left[(a_{{\scriptscriptstyle
3}})_{\mu\nu}(q)+(a_{{\scriptscriptstyle
4}})_{\mu\nu}(q)\right]=q^{\nu}\widehat{\Pi}^{(2)}_{\mu\nu}(q)=0\,.$ (14)

We emphasize that the above strategy of contracting directly individual
diagrams and triggering the corresponding STIs will be followed unaltered in
the more complicated case of the three-gluon vertex treated in the next
section. Note finally that the entire demonstration leading to Eq. (8) is
carried out for a general value of the gauge-fixing parameter
$\xi_{{\scriptscriptstyle Q}}$ Aguilar and Papavassiliou (2006); Binosi and
Papavassiliou (2008a, b).

##  III Block-wise STI of the three-gluon vertex

In this section we demonstrate the block-wise realization of the STI satisfied
by the BBB three-gluon vertex.

###  III.1 General considerations

(i) The one-particle irreducible three-gluon vertex,
$\widehat{\bm{\Gamma}}_{\alpha\mu\nu}^{abc}(q,r,p)$, is defined from the
vacuum expectation value of the time ordered product of three background
gluons (in momentum space), as

$\langle
0|\,T\\!\left[{B}^{a}_{\alpha^{\prime}}(q)\,{B}^{b}_{\mu^{\prime}}(r)\,{B}^{c}_{\nu^{\prime}}(p)\right]\\!|0\rangle=g\,\widehat{\bm{\Gamma}}_{\alpha\mu\nu}^{abc}(q,r,p)\widehat{\Delta}^{\alpha}_{\alpha^{\prime}}(q)\widehat{\Delta}^{\mu}_{\mu^{\prime}}(r)\widehat{\Delta}^{\nu}_{\nu^{\prime}}(p)\,.$
(15)

The three-gluon vertex $\widehat{\bm{\Gamma}}_{\alpha\mu\nu}^{abc}(q,r,p)$ is
naturally cast in the form

$\widehat{\bm{\Gamma}}^{abc}_{\alpha\mu\nu}(q,r,p)=\widehat{\Gamma}^{(0)abc}_{\alpha\mu\nu}(q,r,p)+\widehat{\Gamma}^{abc}_{\alpha\mu\nu}(q,r,p)\,,$
(16)

where the tree level component
$\widehat{\Gamma}^{(0)abc}_{\alpha\mu\nu}(q,r,p)=f^{abc}\widehat{\Gamma}^{(0)}_{\alpha\mu\nu}(q,r,p)$
coincides with that of the conventional three-gluon vertex (QQQ), i.e.,

$\widehat{\Gamma}^{(0)}_{\alpha\mu\nu}(q,r,p)=\Gamma^{(0)}_{\alpha\mu\nu}(q,r,p)=(q-r)_{\nu}g_{\alpha\mu}+(r-p)_{\alpha}g_{\mu\nu}+(p-q)_{\mu}g_{\alpha\nu}\,,$
(17)

while $\widehat{\Gamma}^{abc}_{\alpha\mu\nu}(q,r,p)$ captures all quantum
corrections, both perturbative and nonperturbative.

Figure 2: The block-wise structure of the SDE which describes the three-gluon
vertex $\widehat{\bm{\Gamma}}^{abc}_{\alpha\mu\nu}(q,r,p)$ (BBB). The blue
circles represent full one-particle irreducible vertices, while the purple
ones the four- and five-point scattering kernels.

(ii) The SDE that defines $\widehat{\bm{\Gamma}}^{abc}_{\alpha\mu\nu}(q,r,p)$
is shown diagrammatically in Fig. 2, written with respect to the gluon that
carries momentum $q$; therefore, the corresponding vertices to which this leg
is attached are kept at tree level. The corresponding Feynman diagrams have
been classified in four blocks, applying the exact same criterion as in the
case of $\widehat{\Pi}_{\mu\nu}(q)$, and employing the same color code for the
individual boxes as in Fig. 1. Thus,

$\widehat{\Gamma}^{abc}_{\alpha\mu\nu}(q,r,p)=\sum_{i=1}^{4}\widehat{\Gamma}^{(i)abc}_{\alpha\mu\nu}(q,r,p)\,,$
(18)

where, as shown in Fig. 2, the four blocks are comprised by the diagrams
(suppressing indices)

$\widehat{\Gamma}^{(1)}=(b_{{\scriptscriptstyle 1}})+(b_{{\scriptscriptstyle
2}})+(b_{{\scriptscriptstyle
3}})\,,\qquad\widehat{\Gamma}^{(2)}=(c_{{\scriptscriptstyle
1}})+(c_{{\scriptscriptstyle 2}})+(c_{{\scriptscriptstyle 3}})\,,\qquad\
\widehat{\Gamma}^{(3)}=(d)\,,\qquad\widehat{\Gamma}^{(4)}=(e)\,.$ (19)

(iii) It is well-known that
$\widehat{\bm{\Gamma}}^{abc}_{\alpha\mu\nu}(q,r,p)$ satisfies the Abelian STI
Cornwall and Papavassiliou (1989); Aguilar and Papavassiliou (2006); Binosi
and Papavassiliou (2009)

$p^{\nu}\widehat{\bm{\Gamma}}^{abc}_{\alpha\mu\nu}(q,r,p)=f^{cae}\left[\widehat{\Delta}^{be}_{\alpha\mu}(r)\right]^{-1}-f^{bce}\left[\widehat{\Delta}^{ae}_{\alpha\mu}(q)\right]^{-1}\,,$
(20)

and cyclic permutations thereof. The STI of Eq. (20) may be obtained by means
of formal manipulations of the BFM generating functional, or simply from the
STI of the conventional QQQ vertex Marciano and Pagels (1978); Ball and Chiu
(1980); Davydychev  _et al._ (1996), by setting all ghost-related
contributions to their tree level values.

From Eq. (17) it is elementary to show that

$\displaystyle
p^{\nu}\widehat{\Gamma}^{(0)}_{\alpha\mu\nu}(q,r,p)=r^{2}P_{\alpha\mu}(r)-q^{2}P_{\alpha\mu}(q)\,.$
(21)

Then, from Eqs. (5), (16) and (20) follows that

$p^{\nu}\widehat{\Gamma}^{abc}_{\alpha\mu\nu}(q,r,p)=if^{cae}\widehat{\Pi}^{be}_{\alpha\mu}(r)-if^{bce}\widehat{\Pi}^{ae}_{\alpha\mu}(q)\,.$
(22)

(iv) The central observation of the present study is that, as happens in the
case of Eq. (6), the STI in Eq. (22) admits a block-wise realization.
Specifically, as we will demonstrate in this section,

$p^{\nu}\widehat{\Gamma}^{(i)abc}_{\alpha\mu\nu}(q,r,p)=if^{cae}\widehat{\Pi}^{(i)be}_{\alpha\mu}(r)-if^{bce}\widehat{\Pi}^{(i)ae}_{\alpha\mu}(q)\,,\qquad
i=1,2,3,4\,.$ (23)

In diagrammatic terms, Eq. (23) states that the contraction by $p^{\nu}$ of
the diagrams within a given block (color) in Fig. 2 generates the difference
between diagrams within the corresponding block (color) of Fig. 1. In fact,
the validity of Eq. (23) will be demonstrated by acting with $p^{\nu}$ on
vertex diagrams, and exploiting the STIs triggered by this contraction in
order to cast the result in the form of self-energy contributions.

(v) Note that the diagrams shown in Fig. 2 have a factor $g$ removed from
them, which cancels against the $g$ appearing in the definition of the three-
gluon vertex in Eq. (15). In addition a factor of $i$ will be factored out,
which will cancel against the explicit $i$ appearing on the r.h.s. of Eq.
(23). Thus, on the r.h.s. of all intermediate formulas will appear directly
the self-energy diagrams $(a_{i})$ of Fig. 1. Furthermore, with the exception
of Sec. IV, we will suppress the argument $(q,r,p)$ in all vertex graphs.

(vi) We introduce the definitions

$\displaystyle h_{1}^{abmn}=f^{abe}f^{mne}\,,\qquad
h_{2}^{abcde}=f^{abm}f^{cmn}f^{dne}\,.$ (24)

Note that $h_{1}^{abmn}=h_{1}^{banm}$ and $h_{2}^{abcde}=h_{2}^{baced}$.

(vii) We introduce the short-hand notation

$\displaystyle
R^{\alpha\beta}_{\mu}(r,p)\equiv\Delta^{\alpha\sigma}(r)\Delta^{\beta\rho}(p)\Gamma_{\mu\sigma\rho}(-r-p,r,p),\,$
$\displaystyle R_{\mu}(r,p)\equiv D(r)D(p)\Gamma_{\mu}(r,p,-r-p),$
$\displaystyle\widetilde{R}^{\alpha\beta}_{\mu}(r,p)\equiv\Delta^{\alpha\sigma}(r)\Delta^{\beta\rho}(p)\widetilde{\Gamma}_{\mu\sigma\rho}(-r-p,r,p),\,$
$\displaystyle\widetilde{R}_{\mu}(r,p)\equiv
D(r)D(p)\widetilde{\Gamma}_{\mu}(r,p,-r-p).$ (25)

The special relations

$\displaystyle\int_{k}\widetilde{R}^{\sigma\beta}_{\mu}(k,r-k)=\int_{k}\widetilde{R}_{\mu}(k,r-k)=0\,,$
(26)

will be employed in the analysis that follows. Their validity may be
established by appealing to the Bose symmetry of the BQQ vertex with respect
to its two quantum legs or the ghost-antighost symmetry of the ${\rm
B\bar{c}c}$ vertex, and the change of integration variable $r-k\to k$. An
alternative demonstration proceeds by noting that

$\int_{k}\widetilde{R}_{\mu}(k,r-k)=\frac{r_{\mu}}{r^{2}}I(r^{2})\,,$ (27)

with

$\displaystyle I(r^{2})=$
$\displaystyle\int_{k}D(k)D(r-k)\,r^{\mu}\widetilde{\Gamma}_{\mu}(k,r-k,-r)$
$\displaystyle=$
$\displaystyle\int_{k}D(k)D(r-k)\left[D^{-1}(k)-D^{-1}(r-k)\right]$
$\displaystyle=$ $\displaystyle\int_{k}\left[D(r-k)-D(k)\right]=0\,,$ (28)

where the STI of Eq. (101) was used. The first relation in Eq. (26) may be
proved in the exact same way, employing the STI of Eq. (100).

(viii) Lastly, from now on we adopt the convention that 1PI vertices
containing at least one background gluon will be represented diagrammatically
by a blue circle (see, e.g., the BBQQ vertex in diagram
$(b_{{\scriptscriptstyle 3,2}})$ of Fig. 3).

###  III.2 One-loop gluonic sector (first block)

We begin by considering the one-loop gluonic vertex graphs, namely the set
$\\{(b_{{\scriptscriptstyle 1}}),(b_{{\scriptscriptstyle
2}}),(b_{{\scriptscriptstyle 3}})\\}$, enclosed in the blue box of Fig. 2.

As a first step, we recognize that, by virtue of Eq. (26), diagrams
$(b_{{\scriptscriptstyle 1}})$ and $(b_{{\scriptscriptstyle 2}})$ vanish,

$\displaystyle(b_{{\scriptscriptstyle 1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{g^{2}}{2}f^{ced}\\!\\!\int_{k}\\!\widehat{\Gamma}_{\alpha\mu\beta\sigma}^{(0)abde}\widetilde{R}^{\sigma\beta}_{\nu}(-k,k-p)=0\,,$
$\displaystyle(b_{{\scriptscriptstyle 2}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{g^{2}}{2}f^{bed}\\!\\!\int_{k}\\!\widehat{\Gamma}_{\alpha\nu\beta\sigma}^{(0)acde}\widetilde{R}^{\sigma\beta}_{\mu}(-k,k-r)=0\,,$
(29)

since $\widehat{\Gamma}_{\alpha\mu\beta\sigma}^{(0)abde}$ is momentum-
independent, and may be pulled out of the integral sign.

Figure 3:  The two contributions which arise from diagram
$(b_{{\scriptscriptstyle 3}})$ of Fig. 2 after performing the skeleton
expansion of the four-gluon scattering kernel (purple blob).

Diagram $(b_{{\scriptscriptstyle 3}})$ has two contributions,

$\displaystyle(b_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}=(b_{{\scriptscriptstyle
3,1}})^{abc}_{\alpha\mu\nu}+(b_{{\scriptscriptstyle
3,2}})^{abc}_{\alpha\mu\nu}\,,$ (30)

given in Fig. 3, with

$\displaystyle(b_{{\scriptscriptstyle 3,1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{\lambda}{2}f^{abc}\\!\\!\int_{k}\\!\widetilde{\Gamma}_{\alpha\beta\sigma}^{(0)}(q,k-q,-k)\Delta^{\rho\sigma}(k)\widetilde{R}_{\mu}^{\lambda\beta}(k+p,q-k)\widetilde{\Gamma}_{\nu\rho\lambda}(p,k,-k-p)\,,$
$\displaystyle(b_{{\scriptscriptstyle 3,2}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{g^{2}}{2}f^{ade}\\!\\!\int_{k}\\!\widetilde{\Gamma}_{\alpha\beta\sigma}^{(0)}(q,k-q,-k)\Delta^{\beta\rho}(k-q)\Delta^{\sigma\lambda}(k)\widehat{\Gamma}^{cbde}_{\nu\mu\rho\lambda}(p,r,q-k,k)\,.$
(31)

The contraction $p^{\nu}(b_{{\scriptscriptstyle 3}})^{abc}_{\alpha\mu\nu}$ may
be evaluated using the STIs in Eqs. (100) and (2) and a moderate amount of
algebra, yielding

$\displaystyle p^{\nu}(b_{{\scriptscriptstyle 3}})^{abc}_{\alpha\mu\nu}=$
$\displaystyle-\frac{\lambda}{2}f^{abc}\\!\\!\int_{k}\\!\widetilde{\Gamma}_{\alpha\beta\sigma}^{(0)}(q,k-q,-k)\left[\widetilde{R}^{\sigma\beta}_{\mu}(k,q-k)+\widetilde{R}^{\sigma\beta}_{\mu}(-k-p,k-q)\right]\,.$
(32)

The first term in Eq. (32) is exactly $-f^{bce}(a_{{\scriptscriptstyle
1}})^{ae}_{\alpha\mu}(q)$. The second term, after $k\rightarrow-k-p$,
generates $f^{cae}(a_{{\scriptscriptstyle 1}})^{be}_{\alpha\mu}(r)$; note, in
particular, that

$\displaystyle\widetilde{\Gamma}_{\alpha\beta\sigma}^{(0)}(q,k-q,-k)\rightarrow-\widetilde{\Gamma}_{\alpha\beta\sigma}^{(0)}(r,k-r,-k)+2p_{\beta}g_{\alpha\sigma}-p_{\alpha}g_{\beta\sigma}-(\xi_{{\scriptscriptstyle
Q}}^{-1}+1)p_{\sigma}g_{\alpha\beta}\,,$ (33)

where, due to Eq. (26), the last three terms give vanishing contributions.
Thus, one arrives at

$\displaystyle p^{\nu}(b_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}=f^{cae}(a_{{\scriptscriptstyle
1}})^{be}_{\alpha\mu}(r)-f^{bce}(a_{{\scriptscriptstyle
1}})^{ae}_{\alpha\mu}(q)\,.$ (34)

At this point, we add and subtract on the r.h.s. of Eq. (34) the momentum-
independent seagull graph $(a_{{\scriptscriptstyle 2}})$, to obtain

$\displaystyle p^{\nu}(b_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}=f^{cae}[(a_{{\scriptscriptstyle
1}})+(a_{{\scriptscriptstyle
2}})]_{\alpha\mu}^{be}(r)-f^{bce}[(a_{{\scriptscriptstyle
1}})+(a_{{\scriptscriptstyle 2}})]_{\alpha\mu}^{ae}(q)\,,$ (35)

which, in view of Eq. (29), is precisely Eq. (23) for $i=1$.

###  III.3 One-loop ghost sector (second block)

Figure 4:  The three contributions which emerge from the diagram
$(c_{{\scriptscriptstyle 3}})$ shown in the Fig. 2, after implementing the
skeleton expansion of the four-point scattering kernel (purple blob) formed by
two background gluons with a ghost-antighost pair.

We next focus on the one-loop ghost graphs, forming the set
$\\{(c_{{\scriptscriptstyle 1}}),(c_{{\scriptscriptstyle
2}}),(c_{{\scriptscriptstyle 3}})\\}$, enclosed in the orange box of Fig. 2.
The demonstration that follows is completely analogous to that of the previous
subsection.

Due to Eq. (26), diagrams $(c_{{\scriptscriptstyle 1}})$ and
$(c_{{\scriptscriptstyle 2}})$ vanish,

$\displaystyle(c_{{\scriptscriptstyle 1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=$ $\displaystyle
g^{2}f^{edc}\\!\\!\int_{k}\\!\widehat{\Gamma}_{\alpha\mu}^{(0)abde}\widetilde{R}_{\nu}(-k,k-p)=0\,,$
$\displaystyle(c_{{\scriptscriptstyle 2}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=$ $\displaystyle
g^{2}f^{edb}\\!\\!\int_{k}\\!\widehat{\Gamma}_{\alpha\nu}^{(0)acde}\widetilde{R}_{\mu}(-k,k-r)=0\,,$
(36)

and we only need to consider the contraction of graph $(c_{{\scriptscriptstyle
3}})$. This diagram contains three contributions, depicted in Fig. 4,

$\displaystyle(c_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}=\sum_{j=1}^{3}(c_{{\scriptscriptstyle
3,j}})^{abc}_{\alpha\mu\nu}\,,$ (37)

with

$\displaystyle(c_{{\scriptscriptstyle 3,1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{\lambda}{2}f^{abc}\\!\\!\int_{k}\\!(2k-q)_{\alpha}D(k-q)\widetilde{R}_{\mu}(k,-k-r)\widetilde{\Gamma}_{\nu}(k+r,q-k,p)\,,$
$\displaystyle(c_{{\scriptscriptstyle 3,2}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{\lambda}{2}f^{abc}\\!\\!\int_{k}\\!(2k-q)_{\alpha}D(k-q)\widetilde{R}_{\mu}(-k-r,k)\widetilde{\Gamma}_{\nu}(q-k,k+r,p)\,,$
$\displaystyle(c_{{\scriptscriptstyle 3,3}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-g^{2}f^{eda}\\!\\!\int_{k}\\!(2k-q)_{\alpha}D(k)D(q-k)\widehat{\Gamma}_{\mu\nu}^{bcde}(r,p,q-k,k)\,.$
(38)

The contraction of diagram $(c_{{\scriptscriptstyle 3}})$ by the momentum
$p^{\nu}$ activates the STIs of Eqs. (101) and (2), and one obtains.

$\displaystyle p^{\nu}(c_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}=-\lambda
f^{abc}\\!\\!\int_{k}\\!(2k-q)_{\alpha}\\!\left[\widetilde{R}_{\mu}(k-q,-p-k)+\widetilde{R}_{\mu}(q-k,k)\right]\,.$
(39)

The second term of Eq. (39) is simply $-f^{bce}(a_{{\scriptscriptstyle
3}})^{ae}_{\alpha\mu}(q)$, while the first term, after the shift
$k\rightarrow-k-p$ and use of Eq. (26), furnishes
$f^{cae}(a_{{\scriptscriptstyle 3}})^{be}_{\alpha\mu}(r)$. Thus, we conclude
that

$\displaystyle p^{\nu}(c_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}=f^{cae}(a_{{\scriptscriptstyle
3}})^{be}_{\alpha\mu}(r)-f^{bce}(a_{{\scriptscriptstyle
3}})^{ae}_{\alpha\mu}(q)\,,$ (40)

which, after adding and subtracting the momentum-independent ($a_{4}$), is
tantamount to the validity of Eq. (23) for $i=2$.

###  III.4 Two-loop gluonic sector (third block)

We turn to the two-loop dressed gluonic contributions, contained within the
diagram $(d)$, enclosed by the purple box in Fig. 2. To that end, in Fig. 5 we
show the individual diagrams that emerge upon implementing the skeleton
expansion of the five-gluon kernel (purple circle) inside $(d)$. Note that all
vertices appearing in these graphs are one-particle irreducible.

Figure 5:  The seven contributions originating from diagram $(d)$ of the Fig.
2. This group splits into two subsets: the first composed by the diagrams
$\\{(d_{{\scriptscriptstyle 1,1}}),(d_{{\scriptscriptstyle 1,2}})\\}$ and the
second by $\\{(d_{{\scriptscriptstyle 2,1}}),(d_{{\scriptscriptstyle
2,2}}),(d_{{\scriptscriptstyle 2,3}}),(d_{{\scriptscriptstyle
2,4}}),(d_{{\scriptscriptstyle 2,5}})\\}$.

The diagrams in Fig. 5 are naturally organized into two subsets,

$\displaystyle(d)^{abc}_{\alpha\mu\nu}=(d_{{\scriptscriptstyle
1}})^{abc}_{\alpha\mu\nu}+(d_{{\scriptscriptstyle
2}})^{abc}_{\alpha\mu\nu}\,,$ (41)

with

$\displaystyle(d_{{\scriptscriptstyle
1}})^{abc}_{\alpha\mu\nu}=\sum_{j=1}^{2}(d_{{\scriptscriptstyle
1,j}})^{abc}_{\alpha\mu\nu}\,,\qquad(d_{{\scriptscriptstyle
2}})^{abc}_{\alpha\mu\nu}=\sum_{j=1}^{5}(d_{{\scriptscriptstyle
2,j}})^{abc}_{\alpha\mu\nu}\,,$ (42)

where

$\displaystyle(d_{{\scriptscriptstyle 1,1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{ig^{4}}{2}f^{cen}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\tau}(k)\Delta^{\sigma\eta}(s)\widetilde{R}_{\nu}^{\rho\lambda}(l,-p-l)\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bnmd}(r,l+p,s,k)\,,$
$\displaystyle(d_{{\scriptscriptstyle 1,2}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{ig^{4}}{6}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\tau}(k)\Delta^{\sigma\eta}(s)\Delta^{\rho\lambda}(l)\widehat{\Gamma}_{\nu\mu\lambda\eta\tau}^{cbemd}(p,r,l,s,k)\,,$
(43)

and

$\displaystyle(d_{{\scriptscriptstyle 2,1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{ig^{4}}{2}f^{ben}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\tau}(k)\Delta^{\sigma\eta}(s)\widetilde{R}_{\mu}^{\rho\lambda}(l,-r-l)\widetilde{\Gamma}_{\nu\lambda\eta\tau}^{cnmd}(p,l+r,s,k)\,,$
$\displaystyle(d_{{\scriptscriptstyle 2,2}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{ig^{4}}{2}h_{2}^{dmbec}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\rho\lambda}(l)R_{\tau}^{\sigma\beta}(s,k)\widetilde{R}_{\mu}^{\eta\tau}(l+p,q-l)\widetilde{\Gamma}_{\nu\lambda\eta}(p,l,-l-p)\,,$
$\displaystyle(d_{{\scriptscriptstyle 2,3}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{ig^{4}}{2}h_{2}^{dmceb}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\tau}(k)R_{\tau}^{\eta\sigma}(l-q,s)\widetilde{R}_{\mu}^{\rho\lambda}(l,-r-l)\widetilde{\Gamma}_{\nu\lambda\eta}(p,l+r,q-l)\,,$
$\displaystyle(d_{{\scriptscriptstyle 2,4}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{2}^{cdmbe}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\eta}(k)\widetilde{R}_{\mu}^{\rho\tau}(l,-r-l)R_{\tau}^{\sigma\lambda}(s,k+p)\widetilde{\Gamma}_{\nu\lambda\eta}(p,-k-p,k)\,,$
$\displaystyle(d_{{\scriptscriptstyle 2,5}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{ig^{4}}{2}f^{mdn}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\tau}(k)\Delta^{\rho\lambda}(l)R_{\tau}^{\eta\sigma}(l-q,s)\widehat{\Gamma}^{cben}_{\nu\mu\lambda\eta}(p,r,l,q-l)\,.$
(44)

This particular separation is motivated by the observation that the
contraction by $p^{\nu}$ of each subset generates a concrete term of the STI,
namely

$\displaystyle p^{\nu}(d_{{\scriptscriptstyle
1}})^{abc}_{\alpha\mu\nu}=f^{cae}(a_{{\scriptscriptstyle
5}})^{be}_{\alpha\mu}(r)-f^{bce}(a_{{\scriptscriptstyle
5}})^{ae}_{\alpha\mu}(q)\,,$ $\displaystyle p^{\nu}(d_{{\scriptscriptstyle
2}})^{abc}_{\alpha\mu\nu}=f^{cae}(a_{{\scriptscriptstyle
6}})^{be}_{\alpha\mu}(r)-f^{bce}(a_{{\scriptscriptstyle
6}})^{ae}_{\alpha\mu}(q)\,,$ (45)

where the self-energy diagrams $(a_{{\scriptscriptstyle 5}})$ and
$(a_{{\scriptscriptstyle 6}})$ (purple box in Fig. 1) are given by

$\displaystyle(a_{{\scriptscriptstyle 5}})^{ab}_{\alpha\mu}(q)$
$\displaystyle=-\frac{ig^{4}}{6}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\tau}(k)\Delta^{\sigma\eta}(s)\Delta^{\rho\lambda}(l)\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bemd}(-q,l,s,k)$
$\displaystyle=ig^{4}h_{1}^{adme}\int_{k}\int_{l}\Delta^{\beta\tau}(k)\Delta^{\eta}_{\beta}(s)\Delta^{\lambda}_{\alpha}(l)\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bemd}(-q,l,s,k)\,,$
$\displaystyle(a_{{\scriptscriptstyle 6}})^{ab}_{\alpha\mu}(q)$
$\displaystyle=\frac{ig^{4}}{2}h_{1}^{bedm}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\widetilde{R}_{\mu}^{\rho\lambda}(l,q-l)R_{\lambda}^{\sigma\beta}(s,k)\,,$
(46)

with $s=q-k-l$. In passing from the first to the second expression for
$(a_{{\scriptscriptstyle 5}})$, the explicit form of
$\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}$, given in Eq. (96), has
been used.

The contraction of the momentum $p^{\nu}$ with the above diagrams will
activate a series of STIs, which will furnish the desired structures, together
with a considerable number of terms that will cancel exactly among each other.
In what follows, we briefly outline how this calculation may be best
organized.

We begin with the diagrams comprising $(d_{{\scriptscriptstyle
1}})^{abc}_{\alpha\mu\nu}$. The action of $p^{\nu}$ on
$(d_{{\scriptscriptstyle 1,1}})^{abc}_{\alpha\mu\nu}$ leads to the contraction
$p^{\nu}\widetilde{R}_{\nu}^{\rho\lambda}(l,-p-l)$, triggering the STI of Eq.
(100), and yielding

$\displaystyle
p^{\nu}\widetilde{R}_{\nu}^{\rho\lambda}(l,-p-l)=\Delta^{\rho\alpha}(l)\Delta^{\lambda\beta}(l+p)\left[\Delta^{-1}_{\alpha\beta}(l+p)-\Delta^{-1}_{\alpha\beta}(l)\right]=\Delta^{\rho\lambda}(l)-\Delta^{\rho\lambda}(l+p)\,.$
(47)

The second term in the above expression will give

$\displaystyle p^{\nu}(d_{{\scriptscriptstyle 1,1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{ig^{4}}{2}f^{cen}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\tau}(k)\Delta^{\sigma\eta}(s)\Delta^{\rho\lambda}(l+p)\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bnmd}(r,l+p,s,k)+\cdots,$
(48)

where the ellipsis denotes terms that do not contribute to the r.h.s. of Eq.
(45). We now change the integration variables as $k\rightarrow-k$ and
$l\rightarrow-l-p$, and exploit Lorentz invariance to replace
$\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bnmd}(r,-l,-t,-k)\rightarrow\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bnmd}(-r,l,t,k)$.
Then, substituting the Feynman rule of Eq. (96) and using the Bose symmetry of
the vertices, we arrive at

$\displaystyle p^{\nu}(d_{{\scriptscriptstyle 1,1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}f^{cae}h_{1}^{edmn}\int_{k}\int_{l}\Delta^{\beta\tau}(k)\Delta^{\eta}_{\beta}(t)\Delta^{\lambda}_{\alpha}(l)\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bnmd}(-r,l,t,k)+\cdots\,,$
(49)

which is precisely $f^{cae}(a_{{\scriptscriptstyle 5}})^{be}_{\mu\nu}(r)$, in
the form given in the second line of Eq. (46).

Similarly, the contraction of $(d_{{\scriptscriptstyle
1,2}})^{abc}_{\alpha\mu\nu}$ by $p^{\nu}$ activates the STI for the five-point
function $\widehat{\Gamma}_{\nu\mu\lambda\eta\tau}^{cbemd}(p,r,l,s,k)$, given
in Eq. (106), namely

$\displaystyle
p^{\nu}\widehat{\Gamma}^{cbemd}_{\nu\mu\lambda\eta\tau}(p,r,l,s,k)$
$\displaystyle=f^{bcx}\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{xemd}(r+p,l,s,k)+f^{ecx}\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bxmd}(r,l+p,s,k)$
$\displaystyle+f^{mcx}\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bexd}(r,p,s+p,k)+f^{dcx}\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bemx}(r,p,s,k+p)\,.$
(50)

Note that only the first term on the r.h.s. of Eq. (50) contains the four-
point function with a $q$ entry, since $r+p=-q$. Thus, one gets

$\displaystyle p^{\nu}(d_{{\scriptscriptstyle
1,2}})^{abc}_{\alpha\mu\nu}=\frac{ig^{4}}{6}f^{bcx}\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}$
$\displaystyle\Delta^{\beta\tau}(k)\Delta^{\sigma\eta}(s)\Delta^{\rho\lambda}(l)\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{xemd}(-q,l,s,k)+\cdots\,,$
(51)

which is precisely $f^{bce}(a_{{\scriptscriptstyle 5}})^{ae}(q)$.

After appropriate changes in the integration variables and judicious use of
Bose symmetry, one may show that all remaining terms, denoted by the ellipses
in Eqs. (49) and (51), cancel against each other. Thus, one is left with the
first equation in Eq. (45).

A similar line of reasoning reveals that the term
$f^{cae}(a_{{\scriptscriptstyle 6}})^{be}_{\alpha\mu}(r)$ originates from the
contraction of $p^{\nu}$ with the diagrams $(d_{{\scriptscriptstyle 2,2}})$
and $(d_{{\scriptscriptstyle 2,4}})$ of the second group. Specifically, one
triggers the STI of Eq. (100) to obtain

$\displaystyle p^{\nu}\left[(d_{{\scriptscriptstyle
2,2}})\\!+\\!(d_{{\scriptscriptstyle
2,4}})\right]^{abc}_{\alpha\mu\nu}\\!=\frac{ig^{4}}{2}\left[h_{2}^{dmbec}\\!+\\!2h_{2}^{cdmbe}\right]\\!\\!\int_{k}\\!\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\widetilde{R}_{\mu}^{\rho\lambda}(l,r-l)R_{\lambda}^{\sigma\beta}(t,k)+\cdots\,.$
(52)

Using the Feynman rule given by Eq. (96) for the tree level vertex one gets
that

$\displaystyle\frac{ig^{4}}{2}\left[h_{2}^{dmbec}\\!+\\!2h_{2}^{cdmbe}\right]\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}=\frac{ig^{4}}{2}f^{cax}h_{1}^{bedm}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)xdme}+\cdots\,;$
(53)

the substitution of the first term into Eq. (52) gives precisely
$f^{cae}(a_{{\scriptscriptstyle 6}})^{be}_{\alpha\mu}(r)$, while the ellipsis
contains the terms that will cancel.

Finally, the contraction of $(d_{{\scriptscriptstyle 2,5}})$ by $p^{\nu}$
activates the STI of Eq. (2), giving

$\displaystyle p^{\nu}(d_{{\scriptscriptstyle 2,5}})^{abc}_{\alpha\mu\nu}=$
$\displaystyle-\frac{ig^{4}}{2}f^{bcx}h_{1}^{xedm}\\!\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\widetilde{R}_{\mu}^{\rho\lambda}(l,q-l)R_{\lambda}^{\sigma\beta}(s,k)+\cdots\,,$
(54)

which is exactly $-f^{bce}(a_{{\scriptscriptstyle 6}})^{ae}_{\alpha\mu}(q)$.

Once again, all the terms inside the ellipses in Eqs. (52) and (54), cancel
exactly against the terms coming from $p^{\nu}\left[(d_{\scriptscriptstyle
2,1})+(d_{\scriptscriptstyle 2,3})\right]_{\alpha\mu\nu}$, leading to the
second line in Eq. (45).

Thus, the above considerations demonstrate the validity of Eq. (23) for $i=3$.

###  III.5 Two-loop ghost sector (fourth block)

Finally, the two-loop dressed ghost graphs, given by the diagram $(e)$,
enclosed by the green box in Fig. 2, have twenty two contributions, depicted
in Fig. 6, which have been further separated into three subgroups as

$\displaystyle(e)^{abc}_{\alpha\mu\nu}=\sum_{i=1}^{3}(e_{{\scriptscriptstyle
i}})^{abc}_{\alpha\mu\nu}\,,$ (55)

with

$\displaystyle(e_{{\scriptscriptstyle
1}})^{abc}_{\alpha\mu\nu}=\sum_{j=1}^{4}(e_{{\scriptscriptstyle
1,j}})^{abc}_{\alpha\mu\nu}\,,\qquad(e_{{\scriptscriptstyle
2}})^{abc}_{\alpha\mu\nu}=\sum_{j=1}^{6}(e_{{\scriptscriptstyle
2,j}})^{abc}_{\alpha\mu\nu}\,,\qquad(e_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}=\sum_{j=1}^{12}(e_{{\scriptscriptstyle
3,j}})^{abc}_{\alpha\mu\nu}\,,$ (56)

such that

$\displaystyle p^{\nu}(e_{{\scriptscriptstyle 1}})^{abc}_{\alpha\mu\nu}\\!$
$\displaystyle=\\!f^{cae}(a_{{\scriptscriptstyle
7}})^{be}_{\alpha\mu}(r)\\!-\\!f^{bce}(a_{{\scriptscriptstyle
7}})^{ae}_{\alpha\mu}(q)\,,$ $\displaystyle p^{\nu}(e_{{\scriptscriptstyle
2}})^{abc}_{\alpha\mu\nu}\\!$ $\displaystyle=\\!f^{cae}(a_{{\scriptscriptstyle
8}})^{be}_{\alpha\mu}(r)\\!-\\!f^{bce}(a_{{\scriptscriptstyle
8}})^{ae}_{\alpha\mu}(q)\,,$ $\displaystyle p^{\nu}(e_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}\\!$
$\displaystyle=\\!f^{cae}\left[(a_{{\scriptscriptstyle
9}})\\!+\\!(a_{{\scriptscriptstyle
10}})\right]^{be}_{\alpha\mu}(r)\\!-\\!f^{bce}\left[(a_{{\scriptscriptstyle
9}})\\!+\\!(a_{{\scriptscriptstyle 10}})\right]^{ae}_{\alpha\mu}(q)\,.$ (57)

The expressions for all the diagrams in Fig. 6, together with the associated
self-energy graphs, are given in the Appendix D.

A close inspection of these expressions reveals that

$\displaystyle p^{\nu}(e_{{\scriptscriptstyle
1,4}})^{abc}_{\alpha\mu\nu}=-f^{bce}(a_{{\scriptscriptstyle
7}})^{ae}_{\alpha\mu}(q)+\cdots\,,\qquad\quad p^{\nu}(e_{{\scriptscriptstyle
3,6}})^{abc}_{\alpha\mu\nu}=-f^{bce}(a_{{\scriptscriptstyle
9}})^{ae}_{\alpha\mu}(q)+\cdots\,,$ $\displaystyle
p^{\nu}(e_{{\scriptscriptstyle
2,6}})^{abc}_{\alpha\mu\nu}=-f^{bce}(a_{{\scriptscriptstyle
8}})^{ae}_{\alpha\mu}(q)+\cdots\,,\qquad\quad p^{\nu}(e_{{\scriptscriptstyle
3,12}})^{abc}_{\alpha\mu\nu}=-f^{bce}(a_{{\scriptscriptstyle
10}})^{ae}_{\alpha\mu}(q)+\cdots\,,$ (58)

and

$\displaystyle p^{\nu}\left[(e_{{\scriptscriptstyle
1,1}})\\!+\\!(e_{{\scriptscriptstyle 1,2}})\\!+\\!(e_{{\scriptscriptstyle
1,3}})\right]^{abc}_{\alpha\mu\nu}=$ $\displaystyle
f^{cae}(a_{{\scriptscriptstyle 7}})^{be}_{\alpha\mu}(r)+\cdots\,,$
$\displaystyle p^{\nu}\left[(e_{{\scriptscriptstyle
2,2}})\\!+\\!(e_{{\scriptscriptstyle 2,4}})\right]^{abc}_{\alpha\mu\nu}=$
$\displaystyle f^{cae}(a_{{\scriptscriptstyle
8}})^{be}_{\alpha\mu}(r)+\cdots\,,$ $\displaystyle
p^{\nu}\left[(e_{{\scriptscriptstyle 3,3}})\\!+\\!(e_{{\scriptscriptstyle
3,4}})\\!+\\!(e_{{\scriptscriptstyle 3,9}})\\!+\\!(e_{{\scriptscriptstyle
3,10}})\right]^{abc}_{\alpha\mu\nu}=$ $\displaystyle
f^{cae}\left[(a_{{\scriptscriptstyle 9}})\\!+\\!(a_{{\scriptscriptstyle
10}})\right]^{be}_{\alpha\mu}(r)+\cdots\,.$ (59)

Figure 6: Contributions from the diagram $(e)$ in Fig. 2 after expanding the
five-point kernel represented by the purple blob. This group is organized in
the three subsets $(e_{{\scriptscriptstyle 1}})$, $(e_{{\scriptscriptstyle
2}})$, and $(e_{{\scriptscriptstyle 3}})$.

It is then a matter of straightforward algebra to demonstrate that all terms
contained in the ellipses of Eqs. (III.5) and (III.5) cancel against each
other and with the other diagrams, as

$\displaystyle p^{\nu}[(e_{{\scriptscriptstyle
2,1}})\\!+\\!(e_{{\scriptscriptstyle 2,3}})\\!+\\!(e_{{\scriptscriptstyle
2,5}})]^{abc}_{\alpha\mu\nu}+\cdots=0\,,$ $\displaystyle
p^{\nu}[(e_{{\scriptscriptstyle 3,1}})\\!+\\!(e_{{\scriptscriptstyle
3,2}})\\!+\\!(e_{{\scriptscriptstyle 3,3}})\\!+\\!(e_{{\scriptscriptstyle
3,5}})\\!+\\!(e_{{\scriptscriptstyle 3,7}})\\!+\\!(e_{{\scriptscriptstyle
3,8}})\\!+\\!(e_{{\scriptscriptstyle
3,11}})]^{abc}_{\alpha\mu\nu}+\cdots=0\,,$ (60)

leaving Eq. (57) as the final result. Thus, the validity of Eq. (23) for $i=4$
is confirmed.

The final conclusion drawn from the analysis presented in subsections III.2,
III.3, III.4, and III.5 is that the block-wise realization of the STI
announced in subsection III.1, holds. Notice, in fact, that the validity of
Eq. (23) has been demonstrated for an arbitrary value of the gauge-fixing
parameter $\xi_{{\scriptscriptstyle Q}}$.

##  IV Abelian Ward identities with Background gluons

In this section we derive Abelian WIs from the STIs satisfied by the BFM
vertices, and apply to them the text-book diagrammatic representation for the
WIs known from QED Itzykson and Zuber (1980). In addition, we demonstrate the
block-wise realization of the WI that connects the vertex
$\widehat{\Gamma}_{\alpha\mu\nu}(0,r,-r)$ with the derivative of
${\widehat{\Delta}}(r)$.

Figure 7: Diagrammatic representation of the WI for the ${\rm B\bar{c}c}$
vertex, where the derivative of the ghost propagator can be identified as the
insertion of a zero-momentum background gluon leg. Here we define the notation
of a perforated circle as being the derivative acting on the Green’s function.

As is well-known in the context of Abelian gauge theories, such as spinor or
scalar QED, the implementation of the limit $q\to 0$ of the Takahashi identity
gives rise to the corresponding WI. In order to fix the ideas consider the
latter theory, describing the interaction of a photon with a complex scalar,
where the full photon-scalar vertex $\Gamma_{\mu}(r,p,q)$ satisfies the
Abelian STI (Takahashi identity)

$q^{\mu}\Gamma_{\mu}(r,p,q)={\cal D}^{-1}(p)-{\cal D}^{-1}(r),$ (61)

with ${\cal D}(p)$ denoting the fully dressed propagator of the scalar field.
Then, the standard WI is determined by expanding both sides of Eq. (61) around
$q=0$, and equating the linear terms. Specifically, this procedure yields

$\Gamma_{\mu}(r,-r,0)=\frac{\partial\,{\cal D}^{-1}(r)}{\partial r^{\mu}}\,,$
(62)

or, equivalently,

$\frac{\partial\,{\cal D}(r)}{\partial r^{\mu}}=-{\cal
D}(r)\Gamma_{\mu}(r,-r,0){\cal D}(r)\,.$ (63)

The version of the WI given in Eq. (63) admits the text-book diagrammatic
interpretation: the derivative of the propagator ${\cal D}(r)$ is equivalent
to the insertion of a zero-momentum photon in it Itzykson and Zuber (1980).

It turns out that the Abelian STIs satisfied by the BFM three-point functions
give rise to WIs completely analogous to that of Eq. (63), which admit the
same diagrammatic interpretation given above, but now in terms of zero-
momentum insertions of a background gluon.

The simplest case is that of the ghost-gluon vertex
$\widetilde{\Gamma}_{\mu}(r,p,q)$, whose WI is identical to that of Eqs. (62)
and (63), after the replacement $\Gamma_{\mu}\to\widetilde{\Gamma}_{\mu}$ and
${\cal D}\to D$, i.e.,

$\widetilde{\Gamma}_{\mu}(r,-r,0)=\frac{\partial\,{D}^{-1}(r)}{\partial
r^{\mu}}\,\Longrightarrow\,\frac{\partial\,{D}(r)}{\partial
r^{\mu}}=-{D}(r)\widetilde{\Gamma}_{\mu}(r,-r,0){D}(r)\,;$ (64)

the corresponding diagrammatic representation is shown in Fig. 7.

Turning to the case of the BQQ vertex
$\widetilde{\Gamma}_{\alpha\mu\nu}(q,r,p)$, it is rather straightforward to
deduce from the STI of Eq. (100) the corresponding WI, namely

$\widetilde{\Gamma}_{\alpha\mu\nu}(0,-p,p)=-\frac{\partial\Delta^{-1}_{\mu\nu}(p)}{\partial
p^{\alpha}}\,\Longrightarrow\,\frac{\partial\Delta^{\mu\nu}(p)}{\partial
p^{\alpha}}=\Delta^{\mu\rho}(p)\,\widetilde{\Gamma}_{\alpha\rho\sigma}(0,-p,p)\,\Delta^{\nu\sigma}(p)\,;$
(65)

the last relation is diagrammatically depicted in Fig. 8. Note that the above
WI, when applied at tree level, reproduces from Eq. (2) the expression for
$\widetilde{\Gamma}_{\alpha\mu\nu}^{(0)}(q,r,p)$ given in Eq. (1), capturing
correctly its dependence on the gauge-fixing parameter
$\xi_{{\scriptscriptstyle Q}}$.

Figure 8: Diagrammatic representation of the WI for the BQQ vertex, where the
derivative of the gluon propagator can be identified as the insertion of a
zero-momentum background gluon leg.

We next focus our attention on the WIs satisfied by BFM vertices with more
than three incoming fields. As a concrete example, consider the vertex ${\rm
BB\bar{c}c}$; when contracted with respect to the momentum carried by one of
the background gluons, it satisfies the STI given by Eq. (2). Expanding both
sides of Eq. (2) around $q=0$, and using the Jacobi identity to eliminate the
zeroth order term on the r.h.s., we obtain the WI

$\widetilde{\Gamma}^{abmn}_{\mu\nu}(0,-p-t,p,t)=\left(f^{amx}f^{nbx}\frac{\partial}{\partial
p^{\mu}}+f^{anx}f^{bmx}\frac{\partial}{\partial
t^{\mu}}\right)\widetilde{\Gamma}_{\nu}(p,t,-p-t)\,.$ (66)

Exactly analogous expressions may be deduced for higher point Green’s
functions; for a formal derivation of the STI satisfied by a general vertex of
the form ${\rm BQ}^{n}$, see Appendix A.

Now we want to explore the block-wise realization of the WI of the BBB vertex
for the case of the one-loop ghost group, which satisfies the STI of Eq. (23)
for $i=2$, or, equivalently, Eq. (40). In the soft-gluon limit, we obtain
simply

$\displaystyle\widehat{\Gamma}^{(2)}_{\alpha\mu\nu}(q,-q,0)=\frac{\partial\widehat{\Pi}^{(2)}_{\alpha\mu}(q)}{\partial
q^{\nu}}\,,$ (67)

or, in terms of diagrams

$\displaystyle(c_{{\scriptscriptstyle
3}})_{\alpha\mu\nu}(q,-q,0)=\frac{\partial(a_{{\scriptscriptstyle
3}})_{\alpha\mu}(q)}{\partial q^{\nu}}\,.$ (68)

In arriving at Eq. (68) we have used that $(a_{{\scriptscriptstyle 4}})$ is
$q$-independent, and that, in the soft-gluon limit, $(c_{{\scriptscriptstyle
1}})=(c_{{\scriptscriptstyle 2}})=0$ (see Sec. III.3).

To prove Eq. (68), we first symmetrize the process of differentiation of
$(a_{{\scriptscriptstyle 3}})_{\alpha\mu}$ by shifting the loop momentum
($k\rightarrow u-k$, with $u=q/2$), to get

$\displaystyle\frac{\partial(a_{{\scriptscriptstyle
3}})_{\alpha\mu}(q)}{\partial q^{\nu}}=(a^{\prime}_{{\scriptscriptstyle
3,1}})_{\alpha\mu\nu}(q)+(a^{\prime}_{{\scriptscriptstyle
3,2}})_{\alpha\mu\nu}(q)+(a^{\prime}_{{\scriptscriptstyle
3,3}})_{\alpha\mu\nu}(q)\,,$ (69)

with

$\displaystyle(a^{\prime}_{{\scriptscriptstyle 3,1}})_{\alpha\mu\nu}(q)$
$\displaystyle=$
$\displaystyle-2\lambda\int_{k}k_{\alpha}\left[\frac{\partial}{\partial
q^{\nu}}D(k-u)\right]D(k+u)\widetilde{\Gamma}_{\mu}(k+u,u-k,-q)\,,$
$\displaystyle(a^{\prime}_{{\scriptscriptstyle 3,2}})_{\alpha\mu\nu}(q)$
$\displaystyle=$
$\displaystyle-2\lambda\int_{k}k_{\alpha}D(k-u)\left[\frac{\partial}{\partial
q^{\nu}}D(k+u)\right]\widetilde{\Gamma}_{\mu}(k+u,u-k,-q)\,,$
$\displaystyle(a^{\prime}_{{\scriptscriptstyle 3,3}})_{\alpha\mu\nu}(q)$
$\displaystyle=$
$\displaystyle-2\lambda\int_{k}k_{\alpha}D(k-u)D(k+u)\left[\frac{\partial}{\partial
q^{\nu}}\widetilde{\Gamma}_{\mu}(k+u,u-k,-q)\right]\,;$ (70)

the last three contributions are depicted graphically in the first line of
Fig. 9.

Next, for the terms $(a^{\prime}_{{\scriptscriptstyle
3,1}})_{\alpha\mu\nu}(q)$ and $(a^{\prime}_{{\scriptscriptstyle
3,2}})_{\alpha\mu\nu}(q)$ we use Eq. (64) to write

$\displaystyle(a^{\prime}_{{\scriptscriptstyle
3,1}})_{\alpha\mu\nu}(q)\\!=\\!-$
$\displaystyle\lambda\\!\\!\int_{k}k_{\alpha}\left[D(k\\!-\\!u)\widetilde{\Gamma}_{\nu}(k\\!-\\!u,u\\!-\\!k,0)D(k\\!-\\!u)\right]\\!D(k\\!+\\!u)\widetilde{\Gamma}_{\mu}(k\\!+\\!u,u\\!-\\!k,-q)\,,$
$\displaystyle(a^{\prime}_{{\scriptscriptstyle
3,2}})_{\alpha\mu\nu}(q)\\!=\\!-$
$\displaystyle\lambda\\!\\!\int_{k}k_{\alpha}D(k\\!-\\!u)\left[D(k\\!+\\!u)\widetilde{\Gamma}_{\nu}(k\\!+\\!u,-u\\!-\\!k,0)D(k\\!+\\!u)\right]\widetilde{\Gamma}_{\mu}(k\\!+\\!u,u\\!-\\!k,-q)\,.$
(71)

A direct comparison of these last expressions with the contributions to
$(c_{{\scriptscriptstyle 3}})_{\alpha\mu\nu}(q,-q,0)$ in Eq. (37) [for
$(q,r,p)\to(q,-q,0)$] allows one to establish that

$\displaystyle(a^{\prime}_{{\scriptscriptstyle
3,1}})_{\alpha\mu\nu}(q)=(c_{{\scriptscriptstyle
3,1}})_{\alpha\mu\nu}(q,-q,0)\,,\qquad(a^{\prime}_{{\scriptscriptstyle
3,2}})_{\alpha\mu\nu}(q)=(c_{{\scriptscriptstyle
3,2}})_{\alpha\mu\nu}(q,-q,0)\,.$ (72)

Figure 9: The diagrammatic representation of the differentiation of the graph
$(a_{{\scriptscriptstyle 3}})$ with respect to $q^{\nu}$. The effect of
differentiating the ghost propagator (dressed background ghost-gluon vertex)
is the insertion of a zero-momentum background gluon leg in the propagator
(vertex).

Consider finally the $(c_{{\scriptscriptstyle 3,3}})^{abc}_{\alpha\mu\nu}$ in
Eq. (III.3); setting $(q,r,p)\to(q,-q,0)$, shifting $k\rightarrow u-k$, and
employing Eq. (66), we get

$\displaystyle(c_{{\scriptscriptstyle 3,3}})^{abc}_{\alpha\mu\nu}(q,-q,0)$
$\displaystyle=2g^{2}f^{eda}\int_{k}k_{\alpha}D(k-u)D(k+u)\widehat{\Gamma}_{\mu\nu}^{bcde}(-q,0,k+u,u-k)\,,$
$\displaystyle=-\frac{\lambda}{2}\\!\\!\int_{k}\\!\\!k_{\alpha}D(k\\!-\\!u)D(k\\!+\\!u)\\!\\!\left(\\!\frac{\partial}{\partial(k+u)^{\nu}}\\!+\\!\frac{\partial}{\partial(u-k)^{\nu}}\\!\\!\right)\\!\widetilde{\Gamma}_{\mu}(k\\!+\\!u,u\\!-\\!k,-q)\,,$
$\displaystyle=-2\lambda\int_{k}k_{\alpha}D(k-u)D(k+u)\frac{\partial}{\partial
q^{\nu}}\widetilde{\Gamma}_{\mu}(k+u,u-k,-q)\,,$ (73)

and therefore

$\displaystyle(a^{\prime}_{{\scriptscriptstyle
3,3}})_{\alpha\mu\nu}(q)=(c_{{\scriptscriptstyle
3,3}})_{\alpha\mu\nu}(q,-q,0)\,,$ (74)

which completes the proof of Eq. (68). The interpretation of the previous
steps in terms of background-gluon insertions is given in the second line of
Fig. 9.

##  V Discussion and Conclusions

It has been known for some time Aguilar and Papavassiliou (2006) that the
transversality of the background self-energy is enforced in a special way,
namely independently for each one of the four subsets (blocks) of diagrams
comprising the corresponding SDE. In the present work we have shown that the
Abelian STI of the background three-gluon vertex is also realized according to
the exact same pattern, at the level of the corresponding SDE: the momentum
contraction of each subset of vertex diagrams generates the difference of the
corresponding self-energy subsets.

The demonstration of this property has been carried out at the level of the
fully dressed Feynman diagrams that comprise the relevant SDEs. In particular,
the contraction of all three-gluon vertex diagrams by the appropriate momentum
triggers STIs satisfied by the vertices and the kernels embedded in them,
giving rise to crucial rearrangements and cancellations, which are implemented
algebraically, with no need to resort to any integrations. Note that the
extensive reorganization of diagrams observed here has been first identified
in the context of the pinch technique Cornwall (1982); Papavassiliou (1990);
Pilaftsis (1997); Binosi and Papavassiliou (2002a, 2009), where the “gauge-
invariant” three-gluon vertex was first studied at the one-loop level Cornwall
and Papavassiliou (1989); Hashimoto  _et al._ (1994); Binger and Brodsky
(2006). Evidently, it would be particularly interesting to explore the origin
of the block-wise STIs at a formal level, and establish its validity by means
of the Batalin-Vilkovisky functional machinery Batalin and Vilkovisky (1977,
1983); Binosi and Papavassiliou (2002b, 2009); Binosi and Quadri (2012, 2013)
.

It is natural to conjecture that the STI of the background four-gluon vertex,
B4, given by Papavassiliou (1993); Hashimoto  _et al._ (1994)

$\displaystyle
q^{\mu}\widehat{\bm{\Gamma}}^{mnrs}_{\mu\alpha\beta\gamma}(q,r,p,t)$
$\displaystyle=$ $\displaystyle
f^{mse}f^{ern}\widehat{\bm{\Gamma}}_{\alpha\beta\gamma}(r,p,q+t)+f^{mne}f^{esr}\widehat{\bm{\Gamma}}_{\beta\gamma\alpha}(p,t,q+r)$
(75) $\displaystyle+$ $\displaystyle
f^{mre}f^{ens}\widehat{\bm{\Gamma}}_{\gamma\alpha\beta}(t,r,q+p)\,,$

is realized according to the same block-wise pattern described above. A
diagrammatic demonstration along the lines presented in this work appears to
be quite feasible, and would give further support to the notion that the STI
of any Bn-type of vertex is enforced in this characteristic manner.

Some of the results presented in Sec. IV may be used in order to explore the
numerical impact of certain truncations or approximations, in the spirit of
the recent study presented in Aguilar  _et al._ (2022a). For example, the
equality shown in Fig. 9 will be distorted if the vertex BB${\rm\bar{c}}\rm c$
were to be replaced by its tree level value, given by Eq. (99). The amount of
discrepancy induced between the two sides of this equation is a quantitative
indicator of the veracity of such an approximation.

Throughout the present analysis we have assumed that the BBB vertex does not
contain irregularities in the form of massless poles. However, as has been
shown in detail in a series of studies, the emergence of a dynamical gluon
mass Cornwall (1982) through the operation of the Schwinger mechanism
Schwinger (1962a, b) hinges on the inclusion of longitudinally coupled
massless poles in the fundamental vertices of the theory Jackiw and Johnson
(1973); Eichten and Feinberg (1974); Aguilar  _et al._ (2008, 2012); Ibañez
and Papavassiliou (2013); Aguilar  _et al._ (2016b); Eichmann  _et al._ (2021)
Quite importantly: (a) the STIs satisfied by the vertices are resolved with
the nontrivial participation of these poles, and (b) in the soft-gluon limit,
the associated WIs are displaced by an amount controlled by the corresponding
pole residues Aguilar  _et al._ (2016b, 2022b, 2022c). In particular, ongoing
research reveals that the STIs impose stringent conditions on the pole content
of the three-gluon vertex, which must, at the same time, be dynamically
realized. The treatment of this problem within the BFM (i.e., at the level of
the BBB rather than the QQQ vertex) eliminates structures originating from the
ghost-sector of the theory, which tend to complicate and obscure the
underlying physical picture. We expect that the completion of this study will
shed light on the question of how symmetry-induced constraints are dynamically
enforced at the level the corresponding SDEs.

##  VI Acknowledgments

The work of A. C. A. and B. M. O. are supported by the CNPq grants
307854/2019-1 and 141409/2021-5, respectively. A. C. A also acknowledges
financial support from project 464898/2014-5 (INCT-FNA). M. N. F. and J. P.
are supported by the Spanish MICINN grant PID2020-113334GB-I00. M. N. F.
acknowledges financial support from Generalitat Valenciana through contract
CIAPOS/2021/74. J. P. also acknowledges funding from the regional
Prometeo/2019/087 from the Generalitat Valenciana.

##  Appendix A Derivation of Abelian STIs

In this Appendix we employ the Batalin-Vilkovisky formalism Batalin and
Vilkovisky (1977, 1983); Binosi and Papavassiliou (2002b, 2009); Binosi and
Quadri (2012, 2013) to derive the Abelian STI satisfied by the generic vertex
${\rm BQ}^{n}$ when contracted by the momentum carried by the gluon $\rm B$.

We start with the WI functional, given by Binosi and Papavassiliou (2009)

$\displaystyle
W\\!=\\!\\!\int\\!\\!d^{4}x\\!\left[\delta_{\vartheta}Q^{x,\mu}(x)\frac{\delta\Gamma}{\delta
Q_{\mu}^{x}(x)}\\!+\\!\delta_{\vartheta}B^{x,\mu}(x)\frac{\delta\Gamma}{\delta
B_{\mu}^{x}(x)}\\!+\\!\delta_{\vartheta}c^{x}(x)\frac{\delta\Gamma}{\delta
c^{x}(x)}\\!+\\!\delta_{\vartheta}\bar{c}^{x}(x)\frac{\delta\Gamma}{\delta\bar{c}^{x}(x)}\right]\\!\\!=0\,,$
(76)

where $\vartheta^{a}$ are the local infinitesimal parameters which correspond
to the SU(3) generators $t^{a}$, and play the role of the ghost field.
$\Gamma$ in Eq. (76) is the “reduced” effective action, defined as the full
effective action without the gauge-fixing term Binosi and Papavassiliou
(2002b, 2009). Consequently, the Green’s functions obtained from $\Gamma$ will
be missing the corresponding gauge-dependent contribution at tree level.
Finally, the gauge transformations of the fields are given by

$\displaystyle\delta_{\vartheta}Q_{\mu}^{x}=gf^{xdc}Q_{\mu}^{d}\vartheta^{c}\,,\qquad$
$\displaystyle\delta_{\vartheta}B_{\mu}^{x}=\partial_{\mu}\vartheta^{x}+gf^{xdc}B_{\mu}^{d}\vartheta^{c}\,,$
(77)
$\displaystyle\delta_{\vartheta}c^{x}=-gf^{xdc}c^{d}\vartheta^{c}\,,\qquad$
$\displaystyle\delta_{\vartheta}\bar{c}^{x}=-gf^{xdc}\bar{c}^{d}\vartheta^{c}\,.$

To obtain the background Abelian STIs the first step is to differentiate the
functional $W$ with respect to the parameter $\vartheta^{a}(x)$, furnishing

$\displaystyle\frac{\delta
W}{\delta\vartheta^{a}(x)}=gf^{eda}Q_{\mu}^{d}(x)\Gamma_{Q_{\mu}^{e}}(x)+\partial_{\mu}\Gamma_{B_{\mu}^{a}}(x)=0\,,$
(78)

where we have already set to zero the vacuum expectation values (VEVs) of the
ghost, antighost, and background gluon fields222In the end of the procedure
all of the VEVs are set to zero. Since the ${\rm BQ}^{n}$ vertex has only one
external $\rm B$ and no external ghost and antighost fields these VEVs can be
set to zero from the outset.. Moreover, we introduce the shorthand notation
for functional derivatives

$\Gamma_{\phi_{1}\phi_{2}\ldots\phi_{n}}(x_{1},x_{2},\ldots,x_{n}):=\frac{\delta^{n}\Gamma}{\delta\phi_{1}(x_{1})\delta\phi_{2}(x_{2})\ldots\delta\phi_{n}(x_{n})}\,,$
(79)

where $\phi_{i}(x_{i})$ denotes a generic field.

The STIs of interest are then obtained by differentiating Eq. (78) $n$ times
with respect to the quantum gluon. Note, in particular, that the functional
derivatives of the term $\partial_{\mu}\Gamma_{B_{\mu}^{a}}(x)$ in Eq. (78)
generate divergences such as
$\partial_{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{{\scriptscriptstyle
1}}}^{b_{{\scriptscriptstyle 1}}}\ldots Q_{\nu_{{\scriptscriptstyle
n}}}^{b_{{\scriptscriptstyle n}}}}(x,y_{1},\ldots,y_{n})$ which, after Fourier
transformation, result in the typical l.h.s. of the Abelian STIs, i.e., a
Green’s function contracted with a background gluon momentum.

To fix the ideas, let us consider as two special cases the STIs for the BQ and
BQQ functions.

Differentiating Eq. (78) with respect to $Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}(y_{1})$ we obtain

$\displaystyle\frac{\delta^{2}W}{\delta Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}(y_{1})\delta\vartheta^{a}(x)}\\!\\!=$ $\displaystyle
gf^{eb_{{\scriptscriptstyle
1}}a}\delta(x\\!-\\!y_{1})\Gamma_{\\!Q_{\nu_{1}}^{e}}\\!\\!(x)\\!+\\!gf^{eda}Q_{\mu}^{d}(x)\Gamma_{\\!Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\mu}^{e}}\\!(y_{1},x)\\!+\\!\partial_{\mu}\Gamma_{\\!B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}}\\!(x,y_{1})\\!=\\!0\,.$ (80)

Setting the gluon field $Q=0$, and the one-point function
$\Gamma_{\\!Q_{\nu_{1}}^{e}}\\!\\!(x)=0$, we obtain

$\displaystyle\partial_{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}}(x,y_{1})=0\,,$ (81)

where $\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle 1}}}}(x,y_{1})$
is the inverse BQ propagator, with its $1/\xi_{Q}$ term removed. In momentum
space notation, Eq. (81) becomes

$q^{\mu}\left[q^{2}P_{\mu\nu}(q)+i{{\widetilde{\Pi}}_{\mu\nu}(q)}\right]=0\quad\Longrightarrow\quad
q^{\mu}{{\widetilde{\Pi}}_{\mu\nu}(q)}=0\,,$ (82)

expressing the exact transversality of the BQ self-energy.

Then, an additional differentiation of Eq. (80) with respect to
$Q_{\nu_{2}}^{b_{2}}(y_{2})$ yields

$\displaystyle\frac{\delta^{3}W}{\delta Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}(y_{1})\delta Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}(y_{1})\delta\vartheta^{a}(x)}=$ $\displaystyle
gf^{eb_{{\scriptscriptstyle
1}}a}\delta(x-y_{1})\Gamma_{Q_{\nu_{1}}^{e}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}}(x,y_{2})+gf^{eb_{{\scriptscriptstyle
2}}a}\delta(x-y_{2})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{e}}(y_{1},x)$ $\displaystyle+gf^{eda}Q_{\mu}^{d}(x)$
$\displaystyle\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\mu}^{e}}(y_{1},y_{2},x)+\partial_{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}}(x,y_{1},y_{2})=0\,.$ (83)

At this point, by setting all the fields to zero we obtain the Abelian STI for
the BQQ vertex in configuration space, namely

$\displaystyle\partial_{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}}(x,y_{1},y_{2})\\!=\\!-gf^{eab_{{\scriptscriptstyle
1}}}\delta(x\\!-\\!y_{1})\Gamma_{Q_{\nu_{1}}^{e}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}}(x,y_{2})\\!-\\!gf^{eab_{{\scriptscriptstyle
2}}}\delta(x\\!-\\!y_{2})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{e}}(y_{1},x)\,.$ (84)

Now, Fourier transforming the above equation to momentum space leads to

$\displaystyle q^{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}}(q,r,p)=gf^{ab_{{\scriptscriptstyle 1}}b_{{\scriptscriptstyle
2}}}\left[p^{2}P_{\mu\nu}(p)+{\Pi_{\mu\nu}(p)}\right]-gf^{ab_{{\scriptscriptstyle
1}}b_{{\scriptscriptstyle
2}}}\left[r^{2}P_{\mu\nu}(r)+{\Pi_{\mu\nu}(r)}\right]\,,$ (85)

which, with the definition of Eq. (3), becomes Eq. (100). The derivation of
the STI for the BBB vertex, given in Eq. (20), is completely analogous.

Next, we prove that the STI of the ${\rm BQ}^{n}$ vertex is given by

$\displaystyle-\partial_{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(x,y_{1},y_{2},\cdots,y_{n})=gf^{eab_{{\scriptscriptstyle
1}}}\delta(x-y_{1})\Gamma_{Q_{\nu_{1}}^{e}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(x,y_{2},y_{3},\cdots,y_{n})$ $\displaystyle\hskip
113.81102pt+gf^{eab_{{\scriptscriptstyle
2}}}\delta(x-y_{2})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{e}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(y_{1},x,y_{3},\cdots,y_{n})+\cdots$ $\displaystyle\hskip
113.81102pt+gf^{eab_{n}}\delta(x-y_{n})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n-1}}^{b_{n-1}}Q_{\nu_{n}}^{e}}(y_{1},y_{2},\cdots,y_{n-1},x)\,.$ (86)

To that end, we first differentiate Eq. (A) $n-2$ times. This procedure yields

$\displaystyle-\partial_{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(x,y_{1},y_{2},\cdots,y_{n})=gf^{eab_{{\scriptscriptstyle
1}}}\delta(x-y_{1})\Gamma_{Q_{\nu_{1}}^{e}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(x,y_{2},y_{3},\cdots,y_{n})$ $\displaystyle\hskip
85.35826pt+gf^{eab_{{\scriptscriptstyle
2}}}\delta(x-y_{2})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{e}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(y_{1},x,y_{3},\cdots,y_{n})$ $\displaystyle\hskip
85.35826pt+gf^{eax}\left\\{\frac{\delta^{n-2}}{\delta
Q_{\nu_{n}}^{b_{n}}(y_{n})\cdots\delta Q_{\nu_{3}}^{b_{{\scriptscriptstyle
3}}}(y_{3})}\left[Q_{\mu}^{x}(x)\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\mu}^{e}}(y_{1},y_{2},x)\right]\right\\}_{Q\rightarrow 0}\\!\\!\\!\,.$
(87)

Clearly, to demonstrate Eq. (A) we need to prove that

$\displaystyle gf^{eax}$ $\displaystyle\left\\{\frac{\delta^{n-2}}{\delta
Q_{\nu_{n}}^{b_{n}}(y_{n})\cdots\delta Q_{\nu_{3}}^{b_{{\scriptscriptstyle
3}}}(y_{3})}\left[Q_{\mu}^{x}(x)\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\mu}^{e}}(y_{1},y_{2},x)\right]\right\\}_{Q\rightarrow 0}=$
$\displaystyle\qquad\qquad\qquad\qquad\qquad gf^{eab_{{\scriptscriptstyle
3}}}\delta(x-y_{3})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\nu_{3}}^{e}Q_{\nu_{4}}^{b_{4}}\cdots
Q_{\nu_{n}}^{b_{n}}}(y_{1},y_{2},x,y_{4},\cdots,y_{n})+\cdots$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+gf^{eab_{n}}\delta(x-y_{n})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n-1}}^{b_{n-1}}Q_{\nu_{n}}^{e}}(y_{1},y_{2},\cdots,y_{n-1},x)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+gf^{eax}\left[Q_{\mu}^{x}(x)\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n}}^{b_{n}}Q_{\mu}^{e}}(y_{1},y_{2},\cdots,y_{n},x)\right]_{Q\rightarrow
0}\,.$ (88)

The proof proceeds by induction. First, it is clear that Eq. (A) holds for
$n=3$. Indeed, in this case one has to take a single derivative

$\displaystyle gf^{eax}\\!\left\\{\frac{\delta}{\delta
Q_{\nu_{{\scriptscriptstyle 3}}}^{b_{{\scriptscriptstyle
3}}}(y_{3})}\\!\left[Q_{\mu}^{x}(x)\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\mu}^{e}}(y_{1},y_{2},x)\right]\right\\}_{Q\rightarrow
0}\\!\\!\\!=gf^{eab_{{\scriptscriptstyle
3}}}\delta(x-y_{3})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}Q_{\nu_{3}}^{e}}(y_{1},y_{2},x)$
$\displaystyle\hskip
170.71652pt+gf^{eax}\left[Q_{\mu}^{x}(x)\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\nu_{3}}^{b_{3}}Q_{\mu}^{e}}(y_{1},y_{2},y_{3},x)\right]_{Q\rightarrow
0}\,,$ (89)

which is Eq. (A) for $n=3$.

Then, assume that Eq. (A) is true for $n=k$. Differentiating the result once
more with respect to $Q_{\nu_{k+1}}^{b_{k+1}}$ we obtain

$\displaystyle gf^{eax}\left\\{\frac{\delta^{k-1}}{\delta
Q_{\nu_{k+1}}^{b_{k+1}}(y_{k+1})\delta Q_{\nu_{k}}^{b_{k}}(y_{k})\cdots\delta
Q_{\nu_{3}}^{b_{{\scriptscriptstyle
3}}}(y_{3})}\left[Q_{\mu}^{x}(x)\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\mu}^{e}}(y_{1},y_{2},x)\right]\right\\}_{Q\rightarrow 0}=$
$\displaystyle\hskip 85.35826ptgf^{eab_{{\scriptscriptstyle
3}}}\delta(x-y_{3})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\nu_{3}}^{e}Q_{\nu_{4}}^{b_{4}}\cdots
Q_{\nu_{k}}^{b_{k}}Q_{\nu_{k+1}}^{b_{k+1}}}(y_{1},y_{2},x,y_{4},\cdots,y_{k},y_{k+1})+\cdots$
$\displaystyle\hskip
85.35826pt+gf^{eab_{k}}\delta(x-y_{k})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{k-1}}^{b_{k-1}}Q_{\nu_{k}}^{e}Q_{\nu_{k+1}}^{b_{k+1}}}(y_{1},y_{2},\cdots,y_{k-1},x,y_{k+1})$
$\displaystyle\hskip
85.35826pt+gf^{eab_{k+1}}\delta(x-y_{k+1})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{k}}^{b_{k}}Q_{\nu_{k+1}}^{e}}(y_{1},y_{2},\cdots,y_{k},x)\,.$
$\displaystyle\hskip
85.35826pt+gf^{eax}\left[Q_{\mu}^{x}(x)\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{k}}^{b_{k}}Q_{\nu_{k+1}}^{b_{k+1}}Q_{\mu}^{e}}(y_{1},y_{2},\cdots,y_{k},y_{k+1},x)\right]_{Q\rightarrow
0}\,,$ (90)

which is Eq. (A) for $n=k+1$. This completes the proof.

In momentum space, Eq. (A) is given by (suppressing a factor of $g$)

$\displaystyle iq_{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(q,p_{1},p_{2},\cdots,p_{n})=$ $\displaystyle
f^{eab_{{\scriptscriptstyle
1}}}\Gamma_{Q_{\nu_{1}}^{e}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(p_{1}+q,p_{2},p_{3},\cdots,p_{n})$ $\displaystyle+$
$\displaystyle f^{eab_{{\scriptscriptstyle
2}}}\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{e}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(p_{1},p_{2}+q,p_{3},\cdots,p_{n})+\cdots$
$\displaystyle+$ $\displaystyle
f^{eab_{n}}\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n-1}}^{b_{n-1}}Q_{\nu_{n}}^{e}}(p_{1},p_{2},p_{3},\cdots,p_{n}+q)\,.$
(91)

The corresponding WI is obtained by expanding Eq. (A) around $q=0$ and
collecting terms linear in $q$. Using $p_{n}=-\sum\limits_{i=1}^{n-1}p_{i}$,
we obtain

$\displaystyle i\Gamma_{\\!B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(0,p_{1},p_{2},\cdots,p_{n})\\!=\\!\left(\\!f^{eab_{{\scriptscriptstyle
1}}}\frac{\partial}{\partial
p_{1}^{\nu_{1}}}+\\!\cdots\\!+f^{eab_{n-1}}\frac{\partial}{\partial
p_{n-1}^{\nu_{n-1}}}\\!\right)\Gamma_{\\!Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}\cdots Q_{\nu_{n}}^{e}}(p_{1},\cdots,p_{n})\,.$ (92)

Note that the absence of a zeroth order term on the l.h.s. of Eq. (A) implies
the relation

$\displaystyle 0=$ $\displaystyle f^{eab_{{\scriptscriptstyle
1}}}\Gamma_{Q_{\nu_{1}}^{e}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(p_{1},p_{2},p_{3},\cdots,p_{n})+f^{eab_{{\scriptscriptstyle
2}}}\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{e}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(p_{1},p_{2},p_{3},\cdots,p_{n})+\cdots$
$\displaystyle\hskip
170.71652pt+f^{eab_{n}}\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n-1}}^{b_{n-1}}Q_{\nu_{n}}^{e}}(p_{1},p_{2},p_{3},\cdots,p_{n})\,,$
(93)

whose validity we have checked explicitly for $n=3,4$.

##  Appendix B Feynman rules for BFM vertices

In the Table 1 of this Appendix we list the Feynman rules for BFM vertices at
tree level.

Vertex | Feynman rule
---|---
|
$\displaystyle\widetilde{\Gamma}_{\alpha\mu\nu}^{(0)}(q,r,p)=(q-r)_{\nu}g_{\alpha\mu}$
$\displaystyle+$
$\displaystyle(r-p)_{\alpha}g_{\mu\nu}+(p-q)_{\mu}g_{\alpha\nu}$
$\displaystyle+$ $\displaystyle\xi_{{\scriptscriptstyle
Q}}^{-1}(g_{\alpha\nu}r_{\mu}-g_{\alpha\mu}p_{\nu})\,,$
|  $\displaystyle\widetilde{\Gamma}^{(0)}_{\mu}(r,p,q)=(r-p)_{\mu}\,,$ (95)
|
$\displaystyle\widetilde{\Gamma}^{(0)abcd}_{\alpha\beta\mu\nu}=f^{adx}f^{cbx}\left(g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\beta}g_{\mu\nu}\right)+f^{abx}f^{dcx}\left(g_{\alpha\nu}g_{\beta\mu}-g_{\alpha\mu}g_{\beta\nu}\right)$
(96)
$\displaystyle+f^{acx}f^{dbx}\left(g_{\alpha\nu}g_{\beta\mu}-g_{\alpha\beta}g_{\mu\nu}\right)\,,$
|  $\displaystyle\widehat{\Gamma}^{(0)abcd}_{\alpha\beta\mu\nu}$
$\displaystyle=\Gamma_{\alpha\beta\mu\nu}^{(0)abcd}+\xi_{{\scriptscriptstyle
Q}}^{-1}(f^{adx}f^{bcx}g_{\alpha\nu}g_{\beta\mu}-f^{acx}f^{dbx}g_{\alpha\mu}g_{\beta\nu})\,,$
(97)
|
$\displaystyle\widetilde{\Gamma}^{(0)abmn}_{\mu\nu}=f^{max}f^{xbn}g_{\mu\nu}\,,$
(98)
|
$\displaystyle\widehat{\Gamma}^{(0)abmn}_{\mu\nu}=g_{\mu\nu}\left(f^{max}f^{bnx}+f^{mbx}f^{anx}\right)\,.$
(99)
Table 1: The diagrammatic representations of the new vertices appearing in the
BFM and their respective Feynman rules at tree level Binosi and Papavassiliou
(2009). Notice that for the three-point functions we have factored out the
coupling $g$ and their respective color structure, following the definitions
of Eq. (3), while for the four-point functions, we have factored out only
$-ig^{2}$ as shown in Eq. (4).

##  Appendix C Abelian Slavnov-Taylor identities in the BFM

In the Table 2 we collect all the Abelian STI in the BFM necessary to
demonstrate the block-wise realization of the STI for the background three-
gluon vertex.

Vertex |  Abelian STI
---|---
BQQ |  $q^{\alpha}\,\widetilde{\Gamma}_{\alpha\mu\nu}(q,r,p)=\Delta^{-1}_{\mu\nu}(p)-\Delta^{-1}_{\mu\nu}(r)\,,\vspace{-0.9cm}$ (100)
${\rm B\bar{c}c}$ |  $q^{\mu}\,\widetilde{\Gamma}_{\mu}(r,p,q)=D^{-1}(p)-D^{-1}(r)\,,\vspace{-0.9cm}$ (101)
BQQQ |  $\displaystyle q^{\alpha}\widetilde{\Gamma}^{abcd}_{\alpha\beta\mu\nu}(q,r,p,t)=$ $\displaystyle f^{abx}f^{dcx}\Gamma_{\beta\mu\nu}(r+q,p,t)+f^{acx}f^{bdx}\Gamma_{\beta\mu\nu}(r,p+q,t)$ $\displaystyle+f^{adx}f^{cbx}\Gamma_{\beta\mu\nu}(r,p,t+q)\,,\vspace{-3cm}$
BBQQ |  $\displaystyle q^{\alpha}\widehat{\Gamma}^{abcd}_{\alpha\beta\mu\nu}(q,r,p,t)=$ $\displaystyle f^{abx}f^{dcx}\widetilde{\Gamma}_{\beta\mu\nu}(r+q,p,t)+f^{acx}f^{bdx}\widetilde{\Gamma}_{\beta\mu\nu}(r,p+q,t)$ $\displaystyle+f^{adx}f^{cbx}\widetilde{\Gamma}_{\beta\mu\nu}(r,p,t+q)\,,\vspace{-7cm}$
${\rm BQ\bar{c}c}$ |  $\displaystyle q^{\mu}\widetilde{\Gamma}^{abmn}_{\mu\nu}(q,r,p,t)=$ $\displaystyle f^{nax}f^{bmx}\Gamma_{\nu}(p,q+t,r)+f^{nbx}f^{max}\Gamma_{\nu}(q+p,t,r)$ $\displaystyle+f^{nmx}f^{abx}\Gamma_{\nu}(p,t,q+r)\,,\vspace{-5.5cm}$
${\rm BB\bar{c}c}$ |  $\displaystyle q^{\mu}\widehat{\Gamma}_{\mu\nu}^{abmn}(q,r,p,t)=$ $\displaystyle f^{abx}f^{mnx}\widetilde{\Gamma}_{\nu}(p,t,q+r)+f^{amx}f^{nbx}\widetilde{\Gamma}_{\nu}(q+p,t,r)$ $\displaystyle+f^{anx}f^{bmx}\widetilde{\Gamma}_{\nu}(p,q+t,r)\,,\vspace{-5.5cm}$
BBQQQ |  $\displaystyle q^{\alpha}\widehat{\Gamma}^{abcde}_{\alpha\beta\mu\nu\rho}(q,r,p,t,u)=f^{bax}\widetilde{\Gamma}_{\beta\mu\nu\rho}^{xcde}(r+q,p,t,u)+f^{cax}\widetilde{\Gamma}_{\beta\mu\nu\rho}^{bxde}(r,p+q,t,u)$ (106) $\displaystyle+f^{dax}\widetilde{\Gamma}_{\beta\mu\nu\rho}^{bcxe}(r,p,t+q,u)+f^{eax}\widetilde{\Gamma}_{\beta\mu\nu\rho}^{bcdx}(r,p,t,u+q)\,,$
${\rm BBQ\bar{c}c}$ |  $\displaystyle q^{\alpha}\widehat{\Gamma}^{abcmn}_{\alpha\mu\nu}(q,r,p,t,u)=f^{bax}\widetilde{\Gamma}_{\mu\nu}^{xcmn}(r+q,p,t,u)+f^{cax}\widetilde{\Gamma}_{\mu\nu}^{bxmn}(r,p+q,t,u)$ (107) $\displaystyle+f^{max}\widetilde{\Gamma}_{\mu\nu}^{bcxn}(r,p,t+q,u)+f^{nax}\widetilde{\Gamma}_{\mu\nu}^{bcmx}(r,p,t,u+q)\,.$
Table 2: The Abelian STIs satisfied by the BQQ, ${\rm B\bar{c}c}$, BQQQ, BBQQ,
${\rm BQ\bar{c}c}$, ${\rm BB\bar{c}c}$, BBQQQ and ${\rm BBQ\bar{c}c}$
vertices.

##  Appendix D Expressions for the two-loop ghost sector of the BBB SDE

The two-loop ghost sector of the SDE of the vertex BBB given by diagram $(e)$
in Fig. 2, whose expansion is given in Fig. 6, relate with the background
gluon self-energy by Eq. (57), where the expression for the diagrams
$(a_{{\scriptscriptstyle 7}})$, $(a_{{\scriptscriptstyle 8}})$,
$(a_{{\scriptscriptstyle 9}})$ and $(a_{{\scriptscriptstyle 10}})$, in Fig. 1
can be expressed as

$\displaystyle(a_{{\scriptscriptstyle 7}})^{ab}_{\alpha\mu}(q)$
$\displaystyle=ig^{4}h_{1}^{aedm}\\!\\!\int_{k}\int_{l}D(l)D(s)\Delta^{\beta}_{\alpha}(k)\widetilde{\Gamma}_{\mu\beta}^{bmde}(-q,k,l,s)\,,$
(108) $\displaystyle(a_{{\scriptscriptstyle 8}})^{ab}_{\alpha\mu}(q)$
$\displaystyle=\frac{1}{2}i\lambda^{2}\delta^{ab}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}R_{\sigma}(-l,l-k)\widetilde{R}_{\mu}^{\sigma\beta}(-k,q+k)\,,$
(109) $\displaystyle(a_{{\scriptscriptstyle 9}})^{ab}_{\alpha\mu}(q)$
$\displaystyle=-i\lambda^{2}\delta^{ab}\\!\\!\int_{k}\int_{l}D(q+l)\Delta_{\alpha}^{\beta}(k)R_{\beta}(k-l,l)\widetilde{\Gamma}_{\mu}(-l,q+l,-q)\,,$
(110) $\displaystyle(a_{{\scriptscriptstyle 10}})^{ab}_{\alpha\mu}(q)$
$\displaystyle=-\frac{1}{2}i\lambda^{2}\delta^{ab}\\!\\!\int_{k}\int_{l}D(q+l)\Delta_{\alpha}^{\beta}(k)R_{\beta}(l,k-l)\widetilde{\Gamma}_{\mu}(q+l,-l,-q)\,.$
(111)

The decomposition of diagram $(e)$, given in Eqs. (55) and (56), can be
separated in three groups: $(e_{1})^{abc}_{\alpha\mu\nu}$, with

$\displaystyle(e_{1,1})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{2}^{amdce}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}D(k)D(s)\widetilde{R}_{\nu}^{\beta\sigma}(l,-p-l)\widetilde{\Gamma}^{bedm}_{\mu\sigma}(r,l+p,s,k)\,,$
$\displaystyle(e_{1,2})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{2}^{cmade}\\!\\!\int_{k}\int_{l}D(s)\Delta_{\alpha}^{\beta}(l)\widetilde{R}_{\nu}(-p-k,k)\widetilde{\Gamma}^{bedm}_{\mu\beta}(r,l,s,k+p)\,,$
$\displaystyle(e_{1,3})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{2}^{amecd}\\!\\!\int_{k}\int_{l}D(k)\Delta_{\alpha}^{\beta}(l)\widetilde{R}_{\nu}(k,-k-p)\widetilde{\Gamma}^{bedm}_{\mu\beta}(r,l,k+p,s)\,,$
$\displaystyle(e_{1,4})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{1}^{amed}\\!\\!\int_{k}\int_{l}D(k)D(s)\Delta_{\alpha}^{\beta}(l)\widehat{\Gamma}_{\mu\nu\beta}^{bcedm}(r,p,l,s,k)\,,$
(112)

$(e_{2})^{abc}_{\alpha\mu\nu}$, with

$\displaystyle(e_{2,1})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{2}^{amebd}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}D(k)D(s)\widetilde{R}_{\mu}^{\beta\sigma}(l,-r-l)\widetilde{\Gamma}_{\nu\sigma}^{cdem}(p,r+l,s,k)\,,$
$\displaystyle(e_{2,2})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{i}{4}\lambda^{2}f^{abc}\\!\\!\int_{k}\int_{l}\Delta^{\beta}_{\alpha}(l)R_{\rho}(s,k)\widetilde{R}_{\mu}^{\sigma\rho}(p+l,q-l)\widetilde{\Gamma}_{\nu\beta\sigma}(p,l,-p-l)\,,$
$\displaystyle(e_{2,3})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{i}{4}\lambda^{2}f^{abc}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}\Delta^{\lambda\rho}(q-l)R_{\lambda}(s,k)\widetilde{R}_{\mu}^{\beta\sigma}(l,-r-l)\widetilde{\Gamma}_{\nu\rho\sigma}(p,q-l,l+r)\,,$
$\displaystyle(e_{2,4})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{i}{4}\lambda^{2}f^{abc}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}D(s)\widetilde{R}_{\nu}(-k-p,k)\widetilde{R}_{\mu}^{\beta\sigma}(l,-r-l)\Gamma_{\sigma}(s,k+p,l+r)\,,$
$\displaystyle(e_{2,5})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{1}^{xmbe}h_{2}^{amecx}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}D(s)\widetilde{R}_{\nu}(k,-k-p)\widetilde{R}_{\mu}^{\beta\sigma}(l,-r-l)\Gamma_{\sigma}(k+p,s,l+r)\,,$
$\displaystyle(e_{2,6})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{i}{2}g^{2}\lambda
f^{aem}\\!\\!\int_{k}\int_{l}\Delta_{\alpha}^{\beta}(l)\Delta^{\rho\sigma}(q-l)R_{\rho}(s,k)\widehat{\Gamma}_{\mu\nu\beta\sigma}^{bcem}(r,p,l,q-l)\,,$
(113)

and finally $(e_{3})^{abc}_{\alpha\mu\nu}$, with

$\displaystyle(e_{3,1})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{2}^{bmade}\\!\\!\int_{k}\int_{l}D(s)\Delta_{\alpha}^{\beta}(l)\widetilde{R}_{\mu}(-k-r,k)\widetilde{\Gamma}^{cedm}_{\nu\beta}(p,l,s,k+r)\,,$
$\displaystyle(e_{3,2})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{1}^{xmce}h_{2}^{amebx}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}D(k)\widetilde{R}_{\nu}^{\beta\sigma}(l,-l-p)R_{\sigma}(s,k+r)\widetilde{\Gamma}_{\mu}(-k-r,k,r)\,,$
$\displaystyle(e_{3,3})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{i}{4}\lambda^{2}f^{abc}\\!\\!\int_{k}\int_{l}\Delta_{\alpha}^{\beta}(l)\widetilde{R}_{\nu}(k,-k-p)R_{\beta}(k-q,s)\widetilde{\Gamma}_{\mu}(k+p,q-k,r)\,,$
$\displaystyle(e_{3,4})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{i}{4}\lambda^{2}f^{abc}\\!\\!\int_{k}\int_{l}\Delta_{\alpha}^{\beta}(l)\Gamma_{\beta}(s+p,k+r,l)\widetilde{R}_{\mu}(-k-r,k)\widetilde{R}_{\nu}(s,-s-p)\,,$
$\displaystyle(e_{3,5})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{i}{2}\lambda^{2}f^{abc}\\!\\!\int_{k}\int_{l}\Delta_{\alpha}^{\beta}(l)R_{\beta}(s,k-q)\widetilde{R}_{\mu}(-k-r,k)\widetilde{\Gamma}_{\nu}(q-k,k+r,p)\,,$
$\displaystyle(e_{3,6})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{i}{2}g^{2}\lambda
f^{ade}\\!\\!\int_{k}\int_{l}D(k)\Delta_{\alpha}^{\beta}(l)R_{\beta}(k-q,s)\widehat{\Gamma}^{cbde}_{\mu\nu}(r,p,k,q-k)\,,$
$\displaystyle(e_{3,7})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{2}^{amedb}\\!\\!\int_{k}\int_{l}D(s)\Delta_{\alpha}^{\beta}(l)\widetilde{R}_{\mu}(k,-k-r)\widetilde{\Gamma}^{cedm}_{\nu\beta}(p,l,k+r,s)\,,$
$\displaystyle(e_{3,8})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{1}^{xmce}h_{2}^{amebx}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}D(k)\widetilde{R}_{\nu}^{\beta\sigma}(l,-l-p)R_{\sigma}(k+r,s)\widetilde{\Gamma}_{\mu}(k,-k-r,r)\,,$
$\displaystyle(e_{3,9})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{i}{2}\lambda^{2}f^{abc}\\!\\!\int_{k}\int_{l}\Delta_{\alpha}^{\beta}(l)R_{\beta}(s,k-q)\widetilde{\Gamma}_{\mu}(q-k,k+p,r)\widetilde{R}_{\nu}(-p-k,k)\,,$
$\displaystyle(e_{3,10})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{i}{4}\lambda^{2}f^{abc}\\!\\!\int_{k}\int_{l}\Delta_{\alpha}^{\beta}(l)\Gamma_{\beta}(r-k,-t,l)\widetilde{R}_{\mu}(-k,k-r)\widetilde{R}_{\nu}(t,q+k+l)\,,$
$\displaystyle(e_{3,11})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{i}{2}\lambda^{2}f^{abc}\\!\\!\int_{k}\int_{l}\Delta_{\alpha}^{\beta}(l)R_{\beta}(k-q,s)\widetilde{R}_{\mu}(k,-k-r)\widetilde{\Gamma}_{\nu}(k+r,q-k,p)\,,$
$\displaystyle(e_{3,12})^{abc}_{\alpha\mu\nu}$ $\displaystyle=-ig^{2}\lambda
f^{ade}\\!\\!\int_{k}\int_{l}D(k)\Delta_{\alpha}^{\beta}(l)R_{\beta}(s,k-q)\widehat{\Gamma}^{bcde}_{\mu\nu}(r,p,q-k,k)\,.$
(114)

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