∎

11institutetext: Orlando Oliveira 22institutetext: CFisUC, Departament of
Physics, University of Coimbra, 3004-516 Coimbra, Portugal
22email<EMAIL_ADDRESS>33institutetext: Helena Lessa Macedo 44institutetext:
CFisUC, Departament of Physics, University of Coimbra, 3004-516 Coimbra,
Portugal
44email<EMAIL_ADDRESS>55institutetext: Rodrigo Carmo Terin
66institutetext: Instituto Tecnológico de Aeronáutica, DCTA, 12.228-900 São
José dos Campos, Brazil
66email<EMAIL_ADDRESS>
# Looking at QED with Dyson-Schwinger equations: basic equations, Ward-
Takahashi identities and the two-photon-two-fermion irreducible vertex

Orlando Oliveira  Helena Lessa Macedo  Rodrigo Carmo Terin

###### Abstract

A minimal truncated set of the integral Dyson-Schwinger equations, in
Minkowski spacetime, that allows to explore QED beyond its perturbative
solution is derived for general linear covariant gauges. The minimal set
includes the equations for the fermion and photon propagators, the photon-
fermion vertex, and the two-photon-two-fermion one-particle-irreducible
diagram. If the first three equations are exact, to build a closed set of
equations, the two-photon-two-fermion equation is truncated ignoring the
contribution of Green functions with large number of external legs. It is
shown that the truncated equation for the two-photon-two-fermion vertex
reproduces the lowest-order perturbative result in the limit of the small
coupling constant. Furthermore, this equation allows to define an iterative
procedure to compute higher order corrections in the coupling constant. The
Ward-Takahashi identity for the two-photon-two-fermion irreducible vertex is
derived and solved in the soft photon limit, where one of the photon momenta
vanish, in the low photon momenta limit and for general kinematics. The
solution of the Ward-Takahashi identity determines the longitudinal component
of the two-photon-two-fermion irreducible vertex, while it is proposed to use
the Dyson-Schwinger equation to determine the transverse part of this
irreducible diagram. The two-photon-two-fermion DSE is solved in heavy fermion
limit, considering a simplified version of the QED vertices. The contribution
of this irreducible vertex to a low-energy effective photon-fermion vertex is
discussed and the fermionic operators that are generated are computed in terms
of the fermion propagator functions.

###### Contents

  1. 1 Introduction and Motivation
  2. 2 Quantum Electrodynamics - definitions and generating functionals
  3. 3 The Dyson-Schwinger Equations
    1. 3.1 The fermion gap equation
    2. 3.2 The Dyson-Schwinger equation for the photon propagator
    3. 3.3 A Dyson-Schwinger equation for the photon-fermion vertex
  4. 4 Ward-Takahashi Identities
  5. 5 Solving the Ward-Takahashi identities
    1. 5.1 The Ward-Takahashi identity for the photon-fermion vertex
    2. 5.2 The Ward-Takahashi identity for the two-photon-two-fermion vertex
  6. 6 The Dyson-Schwinger equation for the two-photon-two-fermion irreducible vertex
    1. 6.1 Perturbative analysis of the DSE for the two-photon-two-fermion vertex
    2. 6.2 Infrared divergences and linear covariant gauges
    3. 6.3 On the two-photon-two-fermion tensor basis
    4. 6.4 An approximate solution of the DSE for heavy fermions
  7. 7 Two-photon-two-fermion vertex and its low energy contribution to the effective photon-fermion vertex
  8. 8 Renormalization
  9. 9 Summary, Dicussions and Conclusions
  10. A Decomposing the connected Green’s functions
  11. B Ward-Takahashi identities in momentum space

##  1 Introduction and Motivation

Quantum Electrodynamics (QED) is one of the first quantum field theories
encountered by a newcomer to the subject. The standard textbook approach to
solve QED is perturbative theory Peskin:1995ev . However, QED can be solved
with different techniques as the Dyson-Schwinger equations (DSE) Dyson:1949ha
; Schwinger:1951ex ; Schwinger:1951hq , that define an infinite tower of
integral equations, its lattice regularized formulation Wilson:1974sk  via
Monte Carlo simulations, see e.g. Loveridge:2021qzs ; Loveridge:2021wav ;
Loveridge:2022sih  and references, or using the functional renormalization
group approach, see e.g. Gies:2006wv  for introduction and references. In all
cases, perturbation theory reappears as a special case when the solution is
written as a power series in the coupling constant. Still, there are several
features of the dynamics of a quantum field theory (QFT) that cannot be
explained within a perturbative solution. This is the case, for example, of
the dynamical generation of masses in QED Schwinger:1962tn ; Polyakov:1976fu ;
Atkinson:1989fp ; Rojas:2008sg ; Fischer:2004nq ; Kizilersu:2014ela ;
Loveridge:2021qzs , that also occurs in QCD Pagels:1978ba ; Oliveira:2018lln ,
together with confinement Alkofer:2010ue ; Loveridge:2022sih , or the
formation of bound states.

The Dyson-Schwinger equations are a set of integral equations relating the
full set of the QED Green functions Roberts:1994dr . From a practical point of
view, it is impossible to handle simultaneously all the Green functions and
only a subset can be investigated at a time. In principle, by including as
many as possible Green functions in the analysis of the DSE, one becomes
closer and closer to the exact solution for QED Bender:2022eze . Of the DSE
for QED it is the equation for the fermion propagator, also known as the gap
equation, together with the equation for the photon propagator that have been
most investigated in the literature, However, to build a solution for these
equations the photon-fermion vertex needs to be provided Roberts:1994dr ;
Marciano:1977su ; Alkofer:2000wg . For example, for a tree level like photon-
fermion vertex, the fermionic gap equation allows for the dynamical generation
of a fermion mass if the coupling constant is sufficiently large
Atkinson:1989fp ; Rojas:2008sg ; Kizilersu:2014ela . The corresponding
equation for the photon propagator has a massive solution if the
aforementioned vertex is singular at zero momentum Schwinger:1962tn ;
Cornwall:1981zr ; Aguilar:2010cn . In a way, the characteristics of the
solutions for the propagators are a function of the dressing of the photon-
fermion vertex used Curtis:1990zs ; Bashir:1997qt . Typically, in QED the
photon-fermion vertex is built by solving the vertex Ward-Takahashi identity
(WTI) and adding perturbative corrections. However, the WTI does not provides
a complete description of the photon-fermion vertex as it only its
longitudinal component relative to the photon momentum Guzman:2023hzq .

The DSE for the photon-fermion vertex requires the two-photon-two-fermion
vertex, whose DSE calls for further Green functions with larger number of
external legs. From the point of view of building a solution of QED that is
written as a power series in $\alpha$, these higher-order Green functions
appear as corrections that are associated with higher powers of the coupling
constant $\alpha$ and, for a small enough coupling constant, they are
suppressed. However, by relying on an enlarged set of DSE, at least a finite
set of these would-be corrections can be considered in the solution of the
theory. Our main goal is to solve QED via a minimal set of DSE that go beyond
the propagator equations. Eventually, this will allow to improve the kernel
that is used to describe the phenomenology associated with QED that uses the
Bethe-Salpeter or the Faddeev equations to tackle two and three body problems,
respectively.

In order to address QED through the Dyson-Schwinger equations and to consider
a minimal set of Green functions to solve this QFT, we will look only at those
Green functions mentioned, i.e. the propagators, the photon-fermion vertex and
the two-photon-two-fermion irreducible vertex $\Gamma^{\mu\nu}$. No further
QED Green functions will be considered in the present work. In particular, for
the two-photon-two-fermion irreducible vertex its DSE will be derived. This
integral equation is complex and requires a five point Green functions and, to
arrive at a manageable set of equations, an approximated equation for
$\Gamma^{\mu\nu}$ will be built.

The relevance of the two-photon-two-fermion irreducible vertex goes beyond its
contribution to the solution of QED. Indeed, $\Gamma^{\mu\nu}$ has been
considered long ago in the study of non-linear interactions between
electromagnetic fields that include light-by-light scattering, two-quantum
pair creation, and the scattering of light by an external electromagnetic
field Karplus:1950zza ; Karplus:1950zz . It is also responsible for the main
contribution of the two-photon exchange corrections to the electron-proton and
electron-hadron processes Akhmadaliev:1998zz ; Arrington:2011dn ;
Eichmann:2012mp ; Afanasev:2017gsk ; Qweak:2021ijt  that determine the nucleon
and nuclear form factors, it impacts the electro-production cross section of
resonances and pions, the $\gamma Z$ interference in parity-violating electron
scattering, etc. The two-photon-two-fermion vertex is also relevant in the
computation of the muonic-hydrogen Lamb shift and hyperfine splitting of
muonic atoms Carlson:2011zd ; Antognini:2013rsa ; Gorchtein:2014hla ;
Krauth:2021foz ; Fu:2022fgh , in the calculation of the muon magnetic anomaly,
see Keshavarzi:2021eqa  and references therein, and for the extraction of the
proton charge radius, see Gao:2021sml  for a recent review and references
therein. This irreducible vertex is also relevant to virtual Compton
scattering processes Scherer:1996ux ; CLAS:2001wjj ; Belitsky:2001ns  and the
computation and extraction of generalized Parton distributions Belitsky:2005qn
; Shiells:2021xqo ; Chavez:2021koz . Non-linear electrodynamics effects
observed in intense laser beams Karbstein:2019oej ; Blackburn:2019rfv  have
also to consider multiple photon signals and, therefore, must take into
account $\Gamma^{\mu\nu}$ and other irreducible diagrams. Note that,
presently, there are ongoing experimental programs to investigate two-photon
processes Rachek:2015ymm ; OLYMPUS:2016gso ; Abramowicz:2019gvx ; CLAS:2021ovm
among other effects. These experimental programs are an extra motivation to
have a closer look at the two-photon-two-fermion irreducible diagrams.

As stated previously, herein, we aim to define a theoretical setup, based on
the Dyson-Schwinger equations, to investigate, from first principles, QED and
to go beyond the perturbative approach to Quantum Field Theories looking, in
particular, also to the two-photon-two-fermion vertex. Then, to have a
consistent notation for the approach that we are aiming for, besides the DSE
for the two-photon-two-fermion irreducible vertex, the gap equation for the
fermion and for the photon propagators, together with the photon-fermion
vertex DSE are re-derived. For the two-photon-two-fermion Green function, the
corresponding DSE is analysed in an approximated (truncated) version that, at
the lowest order in the coupling constant, reproduces the perturbative result.
As discussed below, in a perturbative-like solution of this DSE, the truncated
integral equation reproduces the lowest order perturbative result, and it
allows the building of an iterative procedure to estimate higher order
corrections. Moreover, we are able to solve, within a particular approximation
and in the heavy quark limit, the DSE for the two-photon-two-fermion vertex.

Besides the derivation of the DSE for the Green functions, the Ward-Takahashi
identities for the two-photon-two-fermion irreducible vertex and for the
photon-fermion vertex are derived and solved. For the photon-fermion vertex,
the solution of the Ward-Takahashi identity results in the Ball-Chiu photon-
fermion vertex Ball:1980ay  that takes into account only longitudinal tensors,
relative to the photon momentum, and describes the vertex in terms of the
fermion propagator functions. Similarly as for the photon-fermion vertex, the
solution of the WTI for the two-photon-two-fermion irreducible vertex that is
built is free of kinematical singularities, it provides only a longitudinal
component of this vertex, relative to the photon momenta, and writes this
component in terms of the fermion propagator functions and of the full photon-
fermion vertex. This, eventually, will allow in the modelling
$\Gamma^{\mu\nu}$ in connection with the modelling of the photon-fermion
vertex. Moreover, exact solutions of the WTI identity in the low energy limit
are also derived, namely for the case where one of the photon momenta is set
to zero (soft photon limit) or by taking into account only the linear terms
the photon momenta.

Along the way to achieve the goals described above, i.e., to arrive at the
various DSE equations, one needs to investigate the decomposition of the QED
Green functions in terms of connected functions and in terms of one-particle
irreducible diagrams. The decomposition follows the usual rules of QFT, and
its discussion helps to set the notation. In the current manuscript, the
reader can find details on how to perform such decompositions and, using our
notation, can find expressions for the various cases that were considered. The
motivation to include such information is twofold. To help possible newcomers
to the field and to provide details on the decomposition of the Green
functions that, in some cases, are valid for theories other than QED with
minimal modifications. In the current work, given the complexity of the
underlying mathematical problem, no attempt is made to solve any of the
integral equations. Indeed, the analysis and computation of the solutions for
the DSE is a complex task, that involves various subtleties, and will be the
subject of future publications. All the computations reported are performed in
Minkowski spacetime.

This manuscript is organized as follows. In Sec. 2 we provide the definitions
of the various functional generators and some useful relations used throughout
the paper. The Dyson-Schwinger equations for the fermion propagator, for the
photon propagator, and for the photon-fermion vertex are derived in Sec. 3,
while the Ward-Takahashi identities are computed in Sec. 4. The solution of
the WTI for the photon-fermion vertex, that fixes its longitudinal part, is
revisited in Sec . 5.1. The solution of the Ward-Takahashi identity for the
two-photon-two-fermion vertex is discussed in Sec. 5.2. We give solutions to
this WTI in the soft limit, in the linear approximation in the photon momenta
(low energy limit) and for a general kinematics. The DSE for the two-photon-
two-fermion irreducible vertex is derived in Sec. 6, together with its
approximate version. Furthermore, we show that the approximate DSE reproduces
the results of perturbation theory in the appropriate limit and discuss the
implementation of an iterative procedure to improve the lowest-order
expression. The problem of building a minimal tensor basis for
$\Gamma^{\mu\nu}$ is touched in Sec. 6.3 and an approximate solution to this
four-point Green function is built in Sec 6.4. In Sec. 7 we discuss the
contribution of the two-photon-two-fermion irreducible diagram to an effective
low-energy photon-fermion vertex and, in particular, what type of effective
operators emerge. The renormalization of QED Dyson-Schwinger equations is
discussed in Sec. 8. Finally, Sec. 9 summarizes and concludes. In the
appendices, we collect several auxiliary results that include the
decomposition of the Green functions in terms of one particle irreducible
diagrams and the definitions in momentum space (App.A) and the writing of the
Ward-Takahashi identities in momentum space (App. B). Our manuscript includes
many computational details that can be skipped in a first reading.

##  2 Quantum Electrodynamics - definitions and generating functionals

The classical theory of electromagnetism considers the interaction of the
four-potential field $A^{\mu}(x)=(V(x),\,\vec{A}(x))$, where $V$ is the scalar
potential and $\vec{A}$ the vector potential, with a Dirac spinorial field
$\psi$ that is described by the Lagrangian density

$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\bar{\psi}(x)\Big{(}i\gamma^{\mu}D_{\mu}-m\Big{)}\psi(x)\
,$ (1)

where the covariant derivative and the Abelian field strength are given by

$\displaystyle D_{\mu}(x)=\partial_{\mu}+i\,g\,A_{\mu}(x)\qquad\mbox{ and
}\qquad F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\ ,$ (2)

while $g$ refers to the electric charge. The Lagragian density $\mathcal{L}$
is invariant under the gauge transformation

$\displaystyle\psi(x)\rightarrow e^{i\,\theta(x)}\,\psi(x)\
,\qquad\bar{\psi}(x)\rightarrow\bar{\psi}(x)\,e^{-\,i\,\theta(x)}$
$\displaystyle\qquad\mbox{and}\qquad A_{\mu}(x)\rightarrow
A_{\mu}(x)-\frac{1}{g}\partial_{\mu}\theta(x)\ .$ (3)

The set of transformations associated with the fermion field defines the
$U(1)$ gauge symmetry group of electromagnetism. The definition of the
corresponding quantum field theory requires the introduction of a gauge fixing
term to build the generating functional for the Green functions $Z[J]$. The
Faddeev-Popov construction results in

$\displaystyle
Z[J,\bar{\eta},\eta]=e^{i\,W[J,\bar{\eta},\eta]}=\int\mathcal{D}A\,\mathcal{D}\bar{\psi}\,\mathcal{D}\psi\,$
$\displaystyle\exp\left\\{i\int
d^{4}x\Big{[}\mathcal{L}_{QED}(A,\bar{\psi},\psi)+J^{\mu}(x)A_{\mu}(x)+\bar{\eta}(x)\,\psi(x)+\bar{\psi}(x)\,\eta(x)\Big{]}\right\\}$
(4)

where

$\displaystyle\mathcal{L}_{QED}$ $\displaystyle=$
$\displaystyle\mathcal{L}-\frac{1}{2\xi}\left(\partial A\right)^{2}$ (5)

and the fermion fields $\psi$, $\bar{\psi}$ together with the fermionic
sources $\eta$, $\bar{\eta}$ are Grassmann fields. The real parameter $\xi$ is
the gauge fixing parameter that defines the linear covariant gauge. The Landau
gauge is defined by taking $\xi=0$. In addition, $W[J,\bar{\eta},\eta]$ is the
generating functional of the connected Green functions.

In the functional integration over the fundamental fields $A_{\mu}$, $\psi$ or
$\bar{\psi}$, these can be replaced by functional derivatives111For the
differentiation with respect to the Grassmann numbers we will consider left
differentiation. with respect to the sources following the rules

$\psi(x)\leftrightarrow\frac{\delta}{i\,\delta\bar{\eta}(x)}\
,\qquad\bar{\psi}(x)\leftrightarrow\frac{\delta}{-i\,\delta{\eta(x)}}\quad\mbox{and}\quad
A^{\mu}(x)\leftrightarrow\frac{\delta}{i\,\delta J_{\mu}(x)}\ .$ (6)

By introducing the classical fields

$A_{cl,\mu}(x)=\frac{\delta W}{\delta J^{\mu}(x)}\
,\qquad\quad\bar{\psi}_{cl}(x)=-\,\frac{\delta W}{\delta\eta(x)}\
,\qquad\quad\psi_{cl}(x)=\frac{\delta W}{\delta\bar{\eta}(x)}$ (7)

the generating functional for the one-particle irreducible diagrams is defined
via the Legendre transformation

$\Gamma[A_{cl},\,\bar{\psi}_{cl},\,\psi_{cl}]=W[J,\eta,\bar{\eta}]-(J,A_{cl})-(\bar{\eta},\psi_{cl})-(\bar{\psi}_{cl},\eta)$
(8)

where

$\displaystyle(J,A_{cl})=\int d^{4}x~{}J^{\mu}(x)\,A_{cl,\,\mu}(x)\
,\quad(\bar{\eta},\psi_{cl})=\int
d^{4}x~{}\bar{\eta}_{\alpha}(x)\,\psi_{cl,\,\alpha}(x)$
$\displaystyle\qquad\mbox{and}\qquad(\bar{\psi}_{cl},\eta)=\int
d^{4}x~{}\bar{\psi}_{cl,\,\alpha}(x)\,\eta_{\alpha}(x)\ .$ (9)

It follows from the definitions given in Eqs (7) and (8) and the rules of
functional derivation that

$\frac{\delta\,\Gamma}{\delta
A_{cl,\mu}(x)}=-\,J^{\mu}(x),\qquad\quad\frac{\delta\,\Gamma}{\delta\psi_{cl,\alpha}(x)}=\bar{\eta}_{\alpha}(x),\qquad\quad\frac{\delta\,\Gamma}{\delta\bar{\psi}_{cl,\alpha}(x)}=-\,\eta_{\alpha}(x).$
(10)

Furthermore, the classical fields are independent fields which are translated
into the relations

$\displaystyle\int\,d^{4}z~{}\frac{\delta^{2}W}{\delta J^{\mu}(x)~{}\delta
J^{\zeta}(z)}~{}\frac{\delta^{2}\Gamma}{\delta A_{cl,\,\zeta}(z)~{}\delta
A_{cl}^{\,\nu}(y)}$ $\displaystyle=$
$\displaystyle-~{}g_{\mu\nu}\,\delta(x-y)\ ,$ (11)
$\displaystyle\int\,d^{4}z~{}\frac{\delta^{2}W}{\delta\eta_{\iota}(z)~{}\delta\bar{\eta}_{\alpha}(x)}~{}\frac{\delta^{2}\Gamma}{\delta\psi_{cl,\,\beta}(y)~{}\delta\bar{\psi}_{cl,\,\iota}(z)}$
$\displaystyle=$ $\displaystyle-~{}\delta_{\alpha\beta}\,\delta(x-y)\ ,$ (12)
$\displaystyle\int\,d^{4}z~{}\frac{\delta^{2}W}{\delta\bar{\eta}_{\iota}(z)~{}\delta\eta_{\alpha}(x)}~{}\frac{\delta^{2}\Gamma}{\delta\bar{\psi}_{cl,\,\beta}(y)~{}\delta\psi_{cl,\,\iota}(z)}$
$\displaystyle=$ $\displaystyle-~{}\delta_{\alpha\beta}\,\delta(x-y)\ ,$ (13)

that hold in the limit where all the sources vanish. The results in Eqs (11) -
(13) also connect the second derivatives of the functional generator $\Gamma$
with the inverse of the propagators for QED. Indeed, in perturbation theory
and at the lowest order in the coupling constant,

$W[J,\bar{\eta},\eta]=-\,\frac{1}{2}
\,\left(J,\,D^{(0)}J\right)-\left(\bar{\eta},\,S^{(0)}\eta\right)+\mathcal{O}(g)\
,$ (14)

where $D^{(0)}$ and $S^{(0)}$ are the tree level photon and fermion
propagators, respectively, i.e.

$\displaystyle\left[D^{(0)}(x-y)\right]^{\mu\nu}=\int\frac{d^{4}k}{(2\,\pi)^{4}}~{}e^{-\,i\,k\,(x-y)}~{}\frac{1}{k^{2}+i\,\epsilon}\left[-g^{\mu\nu}+\left(1-\xi\right)\frac{k^{\mu}k^{\nu}}{k^{2}}\right]\
,$ (15) $\displaystyle
S^{(0)}(x-y)=\int~{}\frac{d^{4}p}{(2\,\pi)^{4}}~{}e^{-\,i\,p\,(x-y)}~{}\frac{\not{p}+m}{p^{2}-m^{2}+i\,\epsilon}$
(16)

for a fermion of mass $m$. Then, it follows from Eq. (14) that

$\displaystyle\left.\frac{\delta^{2}W}{\delta J_{\mu}(x)~{}\delta
J_{\nu}(y)}\right|_{J=\bar{\eta}=\eta=0}$ $\displaystyle=$
$\displaystyle-\left(D^{(0)}\right)^{\mu\nu}(x-y)+\cdots$ (17)
$\displaystyle\left.\frac{\delta^{2}W}{\delta\eta_{\beta}(y)~{}\delta\bar{\eta}_{\alpha}(x)}\right|_{J=\bar{\eta}=\eta=0}$
$\displaystyle=$
$\displaystyle-\,\left.\frac{\delta^{2}W}{\delta\bar{\eta}_{\alpha}(x)~{}\delta\eta_{\beta}(y)}\right|_{J=\bar{\eta}=\eta=0}$
(18) $\displaystyle=$
$\displaystyle-\left(S^{(0)}\right)_{\alpha\beta}(x-y)+\cdots$

and from Eqs (11) to (13) that the second derivatives of $\Gamma$ are the
inverse of the propagators.

Finally and before proceeding further let us comment on the definition of the
Landau gauge, that we take as the limit $\xi\rightarrow 0^{+}$ of the linear
covariant gauges. From the formal point of view, the Landau gauge within this
formalism cannot be treated since there is no inverse for the photon
propagator and, therefore, some of the formal manipulations are meaningless if
one sets $\xi=0$ from the very beginning. In the following, the expressions
for the Landau gauge are therefore derived from those of the linear covariant
gauges after setting the gauge fixing parameter to zero.

##  3 The Dyson-Schwinger Equations

The Dyson-Schwinger equations are the Green functions quantum equations of
motion. They can be used to define the theory beyond perturbation theory, as
they are valid for all values of the coupling constants, and are an infinite
tower of integral equations relating all QED Green functions. In the
following, to set the notation and to prepare the discussion of the DSE for
the two-photon-two-fermion one-particle irreducible diagram, the relevant
equations will be derived.

###  3.1 The fermion gap equation

Figure 1: The photon-fermion one-particle irreducible vertex in momentum
space: see Eqs (26) and (30) for definitions. All Feynman diagrams were built
using Axodraw Collins:2016aya .

The derivation of the DSE relies on the vanishing of the functional
integration of the derivatives with respect to the fields. In particular, for
the fermion propagator one starts from the identity

$\displaystyle
0=\int\mathcal{D}A\,\mathcal{D}\bar{\psi}\,\mathcal{D}\psi~{}\frac{\delta}{
\delta\bar{\psi}_{\alpha}(x)}~{}$ $\displaystyle\exp\left\\{i\int
d^{4}x\Big{[}\mathcal{L}_{QED}(A,\bar{\psi},\psi)+J^{\mu}(x)A_{\mu}(x)+\bar{\eta}(x)\,\psi(x)+\bar{\psi}(x)\,\eta(x)\Big{]}\right\\}$
(19)

and given that

$\displaystyle\frac{\delta S}{\delta\bar{\psi}_{\alpha}(x)}$ $\displaystyle=$
$\displaystyle\frac{\delta}{\delta\bar{\psi}_{\alpha}(x)}\int\,d^{4}z~{}\mathcal{L}_{QED}(z)$
(20) $\displaystyle=$
$\displaystyle\left(i\not{\partial}-m\right)_{\alpha\beta}\,\psi_{\beta}(x)-g\,\left(\gamma^{\mu}\right)_{\alpha\beta}\,A_{\mu}(x)\,\psi_{\beta}(x)\
,$

see Eq. (5), it allows to rewrite Eq. (19) as

$\displaystyle\Bigg{\\{}\bigg{[}i\not{\partial}-m\bigg{]}_{\alpha\beta}\left(\frac{\delta}{i\,\delta\bar{\eta}_{\beta}(x)}\right)~{}$
$\displaystyle\qquad-~{}g\,\left(\gamma^{\mu}\right)_{\alpha\beta}\left(\frac{\delta}{i\,\delta\bar{\eta}_{\beta}(x)}\right)\left(\frac{\delta}{i\,\delta
J^{\mu}(x)}\right)~{}+~{}\eta_{\alpha}(x)\Bigg{\\}}~{}Z[J,\bar{\eta},\eta]=0\
.$ (21)

The gap equation is obtained from this equation after taking its derivative
with respect to $i\,\delta/\delta\eta_{\theta}(y)$ resulting in

$\displaystyle\Bigg{\\{}i\,\bigg{[}i\not{\partial}_{x}-m\bigg{]}_{\alpha\beta}\left(\frac{\delta^{2}W[J,\bar{\eta},\eta]}{\delta\eta_{\theta}(y)\,\delta\bar{\eta}_{\beta}(x)}\right)~{}-~{}g\,\left(\gamma^{\mu}\right)_{\alpha\beta}\left(\frac{\delta^{3}W[J,\bar{\eta},\eta]}{\delta\eta_{\theta}(y)\,\delta\bar{\eta}_{\beta}(x)\,\delta
J^{\mu}(x)}\right)$
$\displaystyle\qquad\qquad~{}+~{}i\,\delta_{\alpha\theta}\,\delta(x-y)+\cdots\Bigg{\\}}~{}Z[J,\bar{\eta},\eta]=0\
,$ (22)

where $\cdots$ represent terms that vanish when the sources are set to zero.
Inserting the fermion propagator

$S_{\alpha\beta}(x-y)=\left.\frac{\delta^{2}W[J,\bar{\eta},\eta]}{\delta\bar{\eta}_{\alpha}(x)~{}\delta\eta_{\beta}(y)\,}\right|_{J=\bar{\eta}=\eta=0}$
(23)

and the photon propagator

$\left.\frac{\delta^{2}W[J,\bar{\eta},\eta]}{\delta J^{\mu}(x)~{}\delta
J^{\nu}(y)}\right|_{J=\eta=\bar{\eta}=0}~{}=~{}-~{}D_{\mu\nu}(x-y)$ (24)

in Eq. (22) it becomes, after some algebra,

$\displaystyle\left[S^{-1}(x-z)\right]_{\alpha\beta}=\left[i\not{\partial}_{x}-m\right]_{\alpha\beta}\delta(x-z)$
$\displaystyle\qquad-\,i\,g\,\left(\gamma^{\mu}\right)_{\alpha\alpha^{\prime}}\,\int\,d^{4}y~{}\frac{\delta^{3}W}{\delta
J^{\mu}(x)~{}\delta\eta_{\beta^{\prime}}(y)~{}\delta\bar{\eta}_{\alpha^{\prime}}(x)\,}\left[S^{-1}(y-z)\right]_{\beta^{\prime}\beta}\
.$ (25)

The gap equation can be rewritten using the decomposition of the connected
three-point Green function in terms of one-particle irreducible functions, see
App. A and in particular Eqs (127) and (128) , as

$\displaystyle S^{-1}(x-y)=\left[i\not{\partial}_{x}-m\right]\delta(x-y)~{}$
$\displaystyle\qquad-~{}i\,g^{2}\,\int
d^{4}z_{1}\,d^{4}z_{2}~{}D_{\mu\nu}(x-z_{1})~{}\Big{[}\gamma^{\mu}~{}S(x-z_{2})~{}{\Gamma}^{\nu}(z_{2},y;z_{1})\Big{]}$
(26)

where the photon-fermion vertex is defined as

$\left.\frac{\delta^{3}\Gamma}{\delta
A_{c,\,\mu}(z)\,\delta\psi_{c,\,\beta}(y)\,\delta\bar{\psi}_{c,\,\alpha}(x)}\right|_{J=\bar{\eta}=\eta=0}=-\,g\,\Big{(}{\Gamma}^{\mu}\Big{)}_{\alpha\beta}(x,y;z)$
(27)

that is represented in Fig. 1 in momentum space - see also Eq. (30). The
translation of the gap equation into momentum space uses the definitions

$\displaystyle
S(x-y)~{}=~{}\int\frac{d^{4}p}{(2\,\pi)^{4}}~{}e^{-i\,p(x-y)}~{}S(p)\ ,$ (28)
$\displaystyle
D_{\mu\nu}(x-y)~{}=~{}\int\frac{d^{4}k}{(2\,\pi)^{4}}~{}e^{-i\,k(x-y)}~{}D_{\mu\nu}(k)\
,$ (29)
$\displaystyle{\Gamma}^{\mu}(x,y;z)~{}=~{}\int\frac{d^{4}p^{\prime}}{(2\,\pi)^{4}}\,\frac{d^{4}p}{(2\,\pi)^{4}}\,\frac{d^{4}k}{(2\,\pi)^{4}}~{}e^{-i\,\left(p^{\prime}x+py+kz\right)}$
$\displaystyle\hskip
142.26378pt~{}(2\,\pi)^{4}\,\delta(p^{\prime}+p+k)~{}{\Gamma}^{\mu}(p^{\prime},p;k)$
(30)

and, after some algebra, Eq. (26) becomes

$S^{-1}(p)=\left(\not{p}-m\right)-\,i\,g^{2}\,\int\frac{d^{4}k}{(2\,\pi)^{4}}~{}D_{\mu\nu}(k)~{}\Big{[}\gamma^{\mu}~{}S(p-k)~{}{\Gamma}^{\nu}(p-k,-p;k)\Big{]}\
.$ (31)

The equation above is represented diagramatically in Fig. 2, where the filled
blob stands for full vertices or propagators, while the tree level photon-
fermion vertex is represented by a dot. The Green functions appearing so far
are bare quantities. The renormalization program for QED will be discussed
later in Sec. 8.

Figure 2: The fermion gap equation. For the vertex the filled blob represents
the corresponding one-particule irreducible vertex, while for the propagators
the filled blobs represent full propagators. The vertex with a dot refers to
tree level photon-fermion vertex. Similar notation is used in all the Figs. in
the manuscript.

###  3.2 The Dyson-Schwinger equation for the photon propagator

Similarly as in the derivation of the fermionic gap equation, the starting
point to derive the DSE for the photon propagator is the identity

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int\mathcal{D}A\,\mathcal{D}\bar{\psi}\,\mathcal{D}\psi~{}\frac{\delta}{
\delta A_{\mu}(x)}~{}$ (32) $\displaystyle\quad\exp\left\\{i\int
d^{4}x\Big{[}\mathcal{L}_{QED}(A,\bar{\psi},\psi)+J^{\mu}(x)A_{\mu}(x)+\bar{\eta}(x)\,\psi(x)+\bar{\psi}(x)\,\eta(x)\Big{]}\right\\}$
$\displaystyle=$
$\displaystyle\Bigg{\\{}\left[\square_{x}\,g^{\mu\nu}-\left(1-\frac{1}{\xi}\right)\partial^{\mu}_{x}\partial^{\nu}_{x}\right]\left(\frac{\delta}{i\,\delta
J^{\nu}(x)}\right)$ $\displaystyle\hskip
42.67912pt-g\,\left(\gamma^{\mu}\right)_{\alpha\beta}\left(\frac{\delta}{-i\,\delta\eta_{\alpha}(x)}\right)\left(\frac{\delta}{i\,\delta\bar{\eta}_{\beta}(x)}\right)+J^{\mu}(x)\Bigg{\\}}\,Z[J,\,\eta,\,\bar{\eta}]\
.$

Performing the functional derivative with respect to $\delta/\delta
J^{\nu}(y)$ of this equation and after setting the sources to zero one gets

$\displaystyle\left[\square_{x}\,g^{\mu\mu^{\prime}}-\left(1-\frac{1}{\xi}\right)\,\partial^{\mu}_{x}\,\partial^{\mu^{\prime}}_{x}\right]\,D_{\mu^{\prime}\nu}(x-y)$
$\displaystyle\qquad\qquad+~{}i\,g\,\left(\gamma^{\mu}\right)_{\alpha\beta}\,\frac{\delta^{3}W}{\delta
J^{\nu}(y)\,\delta\eta_{\alpha}(x)\,\delta\bar{\eta}_{\beta}(x)}~{}-~{}g^{\mu}_{~{}~{}\nu}\,\delta(x-y)=0\
.$ (33)

In Minkowski spacetime, the photon propagator is given by

$D_{\mu\nu}(x-y)=\int\frac{d^{4}k}{(2\,\pi)^{4}}~{}e^{-\,i\,k\,(x-y)}\underbrace{\bigg{(}-\,P^{\perp}_{\mu\nu}(k)\,D(k^{2})-\frac{\xi}{k^{2}}\,P^{L}_{\mu\nu}(k)\bigg{)}}_{D_{\mu\nu}(k)}\
,$ (34)

where

$P^{\perp}_{\mu\nu}(k)=g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\qquad\mbox{ and
}\qquad P^{L}_{\mu\nu}(k)=\frac{k_{\mu}k_{\nu}}{k^{2}}$ (35)

are the photon transverse and longitudinal projectors operators, respectively.
For the linear covariant gauges where $\xi\neq 0$, Eq. (33) can be multiplied
by the inverse of the photon propagator and, after taking into account the
decomposition of the three-point connected Green function (127), one arrives
at

$\displaystyle\left(D^{-1}\right)^{\mu\nu}(x-y)=\left[\square_{x}\,g^{\mu\nu}-\left(1-\frac{1}{\xi}\right)\,\partial^{\mu}_{x}\,\partial^{\nu}_{x}\right]\,\delta(x-y)$
$\displaystyle\hskip 28.45274pt+~{}i~{}g^{2}\,\int
d^{4}x^{\prime}\,d^{4}x^{\prime\prime}~{}\textasciiacute\text{Tr}\Big{[}\gamma^{\mu}~{}S(x-x^{\prime})~{}{\Gamma}^{\nu}(x^{\prime},x^{\prime\prime};y)~{}S(x^{\prime\prime}-x)\Big{]}$
(36)

that in momentum space reads

$\displaystyle\frac{P^{\perp}_{\mu\nu}(k)}{D(k^{2})}$ $\displaystyle=$
$\displaystyle k^{2}\,P^{\perp}_{\mu\nu}(k)$ (37)
$\displaystyle\quad-i\,g^{2}\int\frac{d^{4}p}{(2\,\pi)^{4}}~{}\mbox{Tr}\Big{[}\gamma_{\mu}\,S(p)\,{\Gamma}_{\nu}(p,-p+k;-k)\,S(p-k)\Big{]}$

and performing the contraction of the Lorentz indices it reduces to

$\displaystyle\frac{1}{D(k^{2})}=k^{2}-i\,\frac{g^{2}}{3}\,\int\frac{d^{4}p}{(2\,\pi)^{4}}~{}\text{Tr}\Big{[}\gamma_{\mu}\,S(p)\,{\Gamma}^{\mu}(p,-p+k;-k)\,S(p-k)\Big{]}\
.$ (38)

This equation is represented diagramatically in Fig. 3. In the derivation of
Eq. (38) the photon propagator must be invertible, a condition that is not
fulfilled in the Landau gauge.

Figure 3: The photon gap equation

In QED the knowledge of the fermion-photon vertex determines univocally the
fermion and the photon propagator by solving Eqs. (31) and (38), respectively.
Apparently, these equations do not depend on the gauge but the vertex itself
has a non-trivial dependence on the gauge fixing parameter that feeds the
propagator equations and, in this way, they become functions of $\xi$
themselves.

###  3.3 A Dyson-Schwinger equation for the photon-fermion vertex

As described previously, starting from Eq. (21) and taking an additional
functional derivative one arrives at the gap equation (22). A Dyson-Schwinger
equation for the vertex can be derived from this last equation by evaluating
its functional derivative with respect to $\delta/\,i\,\delta J^{\nu}(w)$ and
taking into account those terms not represented explicitly in Eq. (22). Then,
setting the sources to zero, the vertex equation reads

$\displaystyle\left[i\not{\partial}_{x}-m\right]_{\alpha\beta^{\prime}}\,\frac{\delta^{3}W}{\delta
J^{\nu}(w)\,\delta\eta_{\beta}(y)\,\delta\bar{\eta}_{\beta^{\prime}}(x)}$
$\displaystyle\qquad\qquad+i\,g\,\left(\gamma^{\mu}\right)_{\alpha\beta^{\prime}}\,\frac{\delta^{4}W}{\delta
J^{\nu}(w)\,\delta
J^{\mu}(x)\,\delta\eta_{\beta}(y)\,\delta\bar{\eta}_{\beta^{\prime}}(x)}$
$\displaystyle\qquad\qquad\qquad\qquad-~{}g\,\left(\gamma^{\mu}\right)_{\alpha\beta^{\prime}}~{}S_{\beta^{\prime}\beta}(x-y)~{}D_{\mu\nu}(x-w)=0\
$ (39)

with the fermion propagator defined in Eq. (23) and the photon propagator
settled in Eq. (24). Note that besides the propagators, the equation requires
also the knowledge of the two-photon-two-fermion connected Green function. The
decomposition of the Green functions that appear in (39) in terms of one-
particle irreducible functions is discussed in App. A; see, in particular, Eqs
(126), (127), (129) and the definition (128) for the decomposition of the
three-point function and Eqs (130), (134) and the definitions (132), (133) for
the decomposition of the four-point function. In momentum space the Dyson-
Schwinger equation for the vertex is given by

$\displaystyle{\Gamma}^{\mu}(p,\,-p-k;\,k)=\gamma^{\mu}~{}+~{}i\,g^{2}\,\int\frac{d^{4}q}{(2\,\pi)^{4}}~{}D_{\zeta\zeta^{\prime}}(q)$
$\displaystyle\Bigg{\\{}\gamma^{\zeta}\,S(p-q)\,{\Gamma}^{\mu}(p-q,\,-p-k+q;\,k)\,S(p+k-q)\,{\Gamma}^{\zeta^{\prime}}(p+k-q,\,-p-k;q)$
$\displaystyle\qquad\qquad+~{}\gamma^{\zeta}\,S(p-q)\,{\Gamma}^{\zeta^{\prime}\mu}(p-q,\,-p-k;\,q,\,k)~{}\Bigg{\\}}\
.$ (40)

In the derivation of the last expression the gap equation (31) was used to
simplify the final result. Furthermore, the original Dyson-Schwinger equation
includes a boson propagator that needs to be canceled out to arrive at Eq.
(40). We recall that there is no inverse for the photon propagator in the
Landau gauge and, therefore, the cancelation of the propagator is possible
only for the linear covariant gauges. The diagramatic representation of Eq.
(40) is given in Fig. 4.

Figure 4: The Dyson-Schwinger equation for the photon-fermion vertex.

##  4 Ward-Takahashi Identities

For QED the Ward-Takahashi identities can be derived by taking the variation
of the generating functional (4) with respect to an infinitesimal gauge
transformation (3) that results in

$\displaystyle\frac{1}{g\,\xi}\,\square_{x}\,\partial^{\mu}_{x}\,\left(\frac{\delta
Z[J,\bar{\eta},\eta]}{i\,\delta
J^{\mu}(x)}\right)+\frac{1}{g}\,\left(\partial^{\mu}_{x}\,J_{\mu}(x)\right)\,Z[J,\bar{\eta},\eta]$
$\displaystyle\qquad\qquad+~{}\bar{\eta}_{\alpha}(x)\,\left(\frac{\delta
Z[J,\bar{\eta},\eta]}{\delta\bar{\eta}_{\alpha}(x)}\right)+\left(\frac{\delta
Z[J,\bar{\eta},\eta]}{\delta\eta_{\alpha}(x)}\right)\,\eta_{\alpha}(x)=0\ .$
(41)

By differentiating this equation and after setting all the sources to zero,
one can arrive at the usual Ward-Takahashi identity for the fermion-photon
vertex

$\displaystyle\frac{1}{g\,\xi}\,\square_{x}\,\partial^{\mu}_{x}\,\left(\frac{\delta^{3}Z}{i\,\delta
J^{\mu}(x)\,\delta\eta_{\beta}(z)\,\delta\bar{\eta}_{\zeta}(y)}\right)$
$\displaystyle\qquad+\delta(x-y)\left(\frac{\delta^{2}Z}{\delta\eta_{\beta}(z)\,\delta\bar{\eta}_{\zeta}(x)}\right)-\delta(x-z)\left(\frac{\delta^{2}Z}{\delta\eta_{\beta}(x)\,\delta\bar{\eta}_{\zeta}(y)}\right)=0$
(42)

and, after further differentiation, to the Ward-Takahashi identity for the
two-photon-two-fermion vertex

$\displaystyle\frac{1}{g\,\xi}\,\square_{x}\,\partial^{\mu}_{x}\,\left(\frac{\delta^{4}Z}{\delta
J^{\nu}(w)\,i\,\delta
J^{\mu}(x)\,\delta\eta_{\beta}(z)\,\delta\bar{\eta}_{\zeta}(y)}\right)$
$\displaystyle\qquad\quad+\frac{1}{g}~{}\bigg{(}\partial^{x}_{\nu}\,\delta(x-w)\bigg{)}~{}\left(\frac{\delta^{2}Z}{\delta\eta_{\beta}(z)\,\delta\bar{\eta}_{\zeta}(y)}\right)$
$\displaystyle\qquad\qquad\qquad\quad+~{}\delta(x-y)\left(\frac{\delta^{3}Z}{\delta
J^{\nu}(w)\,\delta\eta_{\beta}(z)\,\delta\bar{\eta}_{\zeta}(x)}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad-\delta(x-z)\left(\frac{\delta^{3}Z}{\delta
J^{\nu}(w)\,\delta\eta_{\beta}(x)\,\delta\bar{\eta}_{\zeta}(y)}\right)=0\ .$
(43)

The identities given in Eqs (42) and (43) can be written in terms of one-
particle irreducible Green functions and in momentum space following the usual
rules, see (28) - (30) for example, and taking into account the results and
definitions given in App. A. Indeed, after some straightforward algebra Eq.
(42) becomes

$k_{\mu}\,{\Gamma}^{\mu}(p,\,-p-k;\,k)=S^{-1}(p+k)\,-\,S^{-1}(p)\ ,$ (44)

while Eq. (43) results in

$k_{\mu}\,\Gamma^{\mu\nu}(p,\,-p-k-q;\,k,\,q)~{}=~{}\Gamma^{\nu}(p,\,-p-q;\,q)~{}-~{}\Gamma^{\nu}(p+k,\,-p-k-q;\,q)\
.$ (45)

Details on the derivation of the WTI in momentum space can be found in App. B.
In Scherer:1996ux ; Eichmann:2012mp  the Ward-Takahashi identity for the two-
photon-two-fermion vertex is also given. Furthermore, by the contraction of
Eq. (45) with momentum $q$ one arrives at the “scalar” WTI

$\displaystyle k_{\mu}\,q_{\nu}\,\Gamma^{\mu\nu}(p,\,-p-k-q;\,k,\,q)$
$\displaystyle=$ $\displaystyle S^{-1}(p+k)~{}+~{}S^{-1}(p+q)$ (46)
$\displaystyle\qquad~{}-~{}S^{-1}(p+k+q)~{}-~{}S^{-1}(p)\,.$

that is symmetric under interchange of $k\leftrightarrow q$ and the r.h.s.
requires only the knowledge of the fermion propagator.

For massless QED in 1+1 spacetime dimensions, also named the Schwinger model,
the Ward-Takahashi identity for the two-photon-two-fermion vertex was deduced
and explored in Radozycki:1998pf . The WTI’s for QED, see Eq. (45), and for
the Schwinger model are related to their Lagrangian densities that differ
mainly in their dimensions. Moreover, the derivation of the WTIs are
independent of the fermion mass and the Ward-Takahashi identity for QED and
for the Schwinger model are identical. We call the reader attention that in
Radozycki:1998pf  the “vertex function” $\Gamma^{\mu\nu}$ does not have the
same meaning as in the present manuscript. Indeed, in Radozycki:1998pf  the
function $\Gamma^{\mu\nu}$ represents the four point connect Green function,
see their App., while here it represent the two-photon-two-fermion one-
particle irreducible Green function. Once this difference is taken into
account, the WTI become identical.

##  5 Solving the Ward-Takahashi identities

The Ward-Takahashi identities translate into the Green functions of QED the
constraints due the gauge symmetry. Then, any realistic description of QED
should comply with the WTI and, therefore, they play a major role in modelling
the vertices. The constraints due to the WTI do not apply to the tensor
structures that describe the full vertices but constraint only its
longitudinal components, relative to the photon momenta.

###  5.1 The Ward-Takahashi identity for the photon-fermion vertex

The WTI for the photon-fermion vertex given in Eq. (44) has been studied long
ago Ball:1980ay  and is commonly used to fix the longitudinal part of the
vertex. In the following lines, we review the Ball-Chiu construction for the
longitudinal part of the photon-fermion vertex also to fix the notation.

The photon-fermion vertex Ball:1980ay ; Oliveira:2018ukh  can be written in
terms of a longitudinal $\Gamma_{L}$ and transverse $\Gamma_{T}$ components,
relative to the photon momentum, as

$\Gamma^{\mu}(p_{2},\,p_{1};\,p_{3})=\Gamma^{\mu}_{L}(p_{2},\,p_{1};\,p_{3})+\Gamma^{\mu}_{T}(p_{2},\,p_{1};\,p_{3})\
,$ (47)

where $p_{2}$ is the incoming fermion momentum, $-p_{1}$ is the outgoing
fermion momentum and $p_{3}$ the incoming photon momenta. The momentum are
such that $p_{1}+p_{2}+p_{3}=0$. The Feynman diagram representing the vertex
is given in Fig. 1. It follows from the definition that

$p_{3\,_{\mu}}\,\Gamma^{\mu}_{T}(p_{2},\,p_{1};\,p_{3})=0$ (48)

and the WTI (44) reads

$p_{3\,_{\mu}}\,{\Gamma}^{\mu}(p_{2},p_{1};\,p_{3})=p_{3\,_{\mu}}\,{\Gamma}^{\mu}_{L}(p_{2},p_{1};\,p_{3})=S^{-1}(-p_{1})\,-\,S^{-1}(p_{2})\
.$ (49)

By writing the longitudinal vertex using the Ball-Chiu basis of operators
Ball:1980ay , i.e. setting

$\displaystyle\Gamma^{\mu}_{L}(p_{2},\,p_{1};\,p_{3})=\lambda_{1}(p^{2}_{1},\,p^{2}_{2},\,p^{2}_{3})\,\gamma^{\mu}~{}+~{}\lambda_{2}(p^{2}_{1},\,p^{2}_{2},\,p^{2}_{3})\,\Big{(}\not{p}_{1}-\not{p}_{2}\Big{)}\Big{(}p_{1}-p_{2}\Big{)}^{\mu}$
$\displaystyle\qquad\quad+~{}\lambda_{3}(p^{2}_{1},\,p^{2}_{2},\,p^{2}_{3})~{}\Big{(}p_{1}-p_{2}\Big{)}^{\mu}~{}+~{}\lambda_{4}(p^{2}_{1},\,p^{2}_{2},\,p^{2}_{3})~{}\sigma^{\mu\nu}\,\Big{(}p_{1}-p_{2}\Big{)}_{\nu}\
,$ (50)

where $\lambda_{i}$ are Lorentz scalar form factors,
$\sigma_{\mu\nu}=[\gamma_{\mu}\,,\,\gamma_{\nu}]/2$, then, writing the inverse
of the fermion propagator as

$S^{-1}(p)=A(p^{2})\,\not{p}-B(p^{2})\ ,$ (51)

one can solve the WTI and compute the various form factors in terms of $A$ and
$B$ that are given by

$\displaystyle\lambda_{1}(p^{2}_{1},\,p^{2}_{2},\,p^{2}_{3})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{(}A\big{(}p^{2}_{1}\big{)}+A\big{(}p^{2}_{2}\big{)}\bigg{)}\
,$ (52) $\displaystyle\lambda_{2}(p^{2}_{1},\,p^{2}_{2},\,p^{2}_{3})$
$\displaystyle=$
$\displaystyle\frac{1}{2\,\left(p^{2}_{1}-p^{2}_{2}\right)}\bigg{(}A\big{(}p^{2}_{1}\big{)}-A\big{(}p^{2}_{2}\big{)}\bigg{)}\
,$ (53) $\displaystyle\lambda_{3}(p^{2}_{1},\,p^{2}_{2},\,p^{2}_{3})$
$\displaystyle=$
$\displaystyle\frac{1}{p^{2}_{1}-p^{2}_{2}}\bigg{(}B\big{(}p^{2}_{1}\big{)}-B\big{(}p^{2}_{2}\big{)}\bigg{)}\
,$ (54) $\displaystyle\lambda_{4}(p^{2}_{1},\,p^{2}_{2},\,p^{2}_{3})$
$\displaystyle=$ $\displaystyle 0\ .$ (55)

Such as parameterisation of the vertex is known in the literature as the Ball-
Chiu vertex and will be used to describe the longitudinal part of the photon-
fermioin vertex here. The non-vanishing form factors of the Ball-Chiu vertex,
i.e. $\lambda_{i}$ with $i=1,2,3$, are symmetric functions under exchange of
the first two arguments as required by charge conjugation, see e.g.
Davydychev:2000rt . Moreover, the form factors that solve the WTI have no
kinematic singularities even in the limit where $p_{1}\rightarrow p_{2}$.
Indeed, setting $p_{1}=p_{2}+\delta$, assuming that $A$ and $B$ are smooth
functions of the momentum, then after expanding all functions in powers of
$\delta$ and taking the limit $\delta\rightarrow 0$ it follows that
$\lambda_{2}$ and $\lambda_{3}$ are proportional to derivatives of $A$ and $B$
and, therefore, these form factors, being smooth functions of momentum, are
not singular for $p_{1}=p_{2}$.

###  5.2 The Ward-Takahashi identity for the two-photon-two-fermion vertex

Before discussing the solution of the WTI for the two-photon-two-fermion
irreducible Green function, let us complete the description of photon-fermion
vertex, whose knowledge is required by WTI itself. The longitudinal component
of the photon fermion vertex was described in (50) using the Ball-Chiu basis
and the $\lambda_{i}$ are taken from the WTI. Similarly as for the
longitudinal part of the vertex, let as introduce a basis of operators to
describe $\Gamma_{T}$. The transverse component of the vertex requires eight
form factors, i.e.

$\Gamma_{T\,\mu}(p_{2},\,p_{1};p_{3})=\sum^{8}_{i=1}\tau_{i}(p^{2}_{1},\,p^{2}_{2},\,p^{2}_{3})\,T^{(i)}_{\mu}(p_{1},\,p_{2},\,p_{3})\
.$ (56)

Here, we take the Kızılersu-Reenders-Pennington Kizilersu:1995iz  basis of
operators

$\displaystyle T^{(1)}_{\mu}(p_{1},\,p_{2},\,p_{3})$ $\displaystyle=$
$\displaystyle p_{1\,_{\mu}}\big{(}p_{2}\cdot
p_{3}\big{)}-p_{2\,_{\mu}}\big{(}p_{1}\cdot p_{3}\big{)}\ ,$ (57)
$\displaystyle T^{(2)}_{\mu}(p_{1},\,p_{2},\,p_{3})$ $\displaystyle=$
$\displaystyle-\,T^{(1)}_{\mu}(p_{1},\,p_{2},\,p_{3})~{}\big{(}\not{p}_{1}-\not{p}_{2}\big{)}\
,$ (58) $\displaystyle T^{(3)}_{\mu}(p_{1},\,p_{2},\,p_{3})$ $\displaystyle=$
$\displaystyle p^{2}_{3}\,\gamma_{\mu}-p_{3\,_{\mu}}\,\not{p}_{3}\ ,$ (59)
$\displaystyle T^{(4)}_{\mu}(p_{1},\,p_{2},\,p_{3})$ $\displaystyle=$
$\displaystyle
T^{(1)}_{\mu}(p_{1},\,p_{2},\,p_{3})~{}\sigma_{\alpha\beta}\,p^{\alpha}_{1}\,p^{\beta}_{2}\
,$ (60) $\displaystyle T^{(5)}_{\mu}(p_{1},\,p_{2},\,p_{3})$ $\displaystyle=$
$\displaystyle\sigma_{\mu\nu}\,p^{\nu}_{3}\ ,$ (61) $\displaystyle
T^{(6)}_{\mu}(p_{1},\,p_{2},\,p_{3})$ $\displaystyle=$
$\displaystyle\gamma_{\mu}\big{(}p^{2}_{1}-p^{2}_{2}\big{)}+\big{(}p_{1}-p_{2}\big{)}_{\mu}\,\not{p}_{3}\
,$ (62) $\displaystyle T^{(7)}_{\mu}(p_{1},\,p_{2},\,p_{3})$ $\displaystyle=$
$\displaystyle-\,\frac{1}{2}\,\big{(}p^{2}_{1}-p^{2}_{2}\big{)}\,\big{[}\gamma_{\mu}\,\big{(}\not{p}_{1}-\not{p}_{2}\big{)}-\big{(}p_{1}-p_{2}\big{)}_{\mu}\big{]}$
(63)
$\displaystyle\qquad\qquad-\big{(}p_{1}-p_{2}\big{)}_{\mu}~{}\sigma_{\alpha\beta}\,p^{\alpha}_{1}\,p^{\beta}_{2}\
,$ $\displaystyle T^{(8)}_{\mu}(p_{1},\,p_{2},\,p_{3})$ $\displaystyle=$
$\displaystyle-\,\gamma_{\mu}\,\sigma_{\alpha\beta}\,p^{\alpha}_{1}\,p^{\beta}_{2}\,+\,p_{1\,_{\mu}}\not{p}_{2}\,-\,p_{2\,_{\mu}}\not{p}_{1}\
,$ (64)

that verify

$p^{\mu}_{3}\,T^{(i)}_{\mu}(p_{1},\,p_{2},\,p_{3})=0\ .$ (65)

The transverse form factors are free of kinematical singularities. Having
chosen a basis of operators, computing the vertex is equivalent to determine
the form factors $\tau_{i}$. The computation of the $\tau_{i}$’s can be done
in several ways, for example in perturbation theory Kizilersu:1995iz  or
solving the DSE (40). To proceed with the computation of a solution of the
two-photon-two-fermion WTI it will be assumed that the full photon-fermion
vertex is known.

Let us turn our attention to the Ward-Takahashi idendity for the two-photon-
two-fermion vertex, see Eq. (45), and discuss its solutions starting by
studying the soft photon limit where the photon momenta $q$ vanishes. For this
kinematics the r.h.s. of the WTI is determined by photon-fermion vertices with
a vanishing photon momentum $q=0$ that requires only its longitudinal
component. Inserting the longitudinal components, see Eqs (52) to (55), on the
r.h.s. of the WTI, it is straightforward to show that

$\displaystyle\Gamma^{\mu\nu}_{L}(p,\,-p-k;\,k,\,0)=g^{\mu\nu}~{}\bigg{[}2\,\lambda_{3}\left((p+k)^{2},\,(p+k)^{2},\,0\right)\bigg{]}$
$\displaystyle\quad+~{}\frac{k^{\mu}\,p^{\nu}}{k^{2}}~{}\bigg{[}2\,\left(\lambda_{3}\left((p+k)^{2},\,(p+k)^{2},\,0\right)-\lambda_{3}\left(p^{2},\,p^{2},\,0\right)\right)\bigg{]}$
$\displaystyle\quad+~{}\frac{k^{\mu}\,\gamma^{\nu}}{k^{2}}~{}\bigg{[}\lambda_{1}\left(p^{2},\,p^{2},\,0\right)-\lambda_{1}\left((p+k)^{2},\,(p+k)^{2},\,0\right)\bigg{]}$
$\displaystyle\quad+~{}\not{k}~{}\bigg{(}~{}g^{\mu\nu}~{}+~{}\frac{k^{\mu}\,p^{\nu}}{k^{2}}\bigg{)}~{}\bigg{[}-4\,\lambda_{2}\left((p+k)^{2},\,(p+k)^{2},\,0\right)\bigg{]}$
$\displaystyle\quad+~{}\not{p}~{}\Bigg{\\{}~{}g^{\mu\nu}\bigg{[}-4\,\lambda_{2}\left((p+k)^{2},\,(p+k)^{2},\,0\right)\bigg{]}$
$\displaystyle\qquad\quad~{}+~{}\frac{k^{\mu}\,p^{\nu}}{k^{2}}\bigg{[}4\,\left(\lambda_{2}\left(p^{2},\,p^{2},\,0\right)-\lambda_{2}\left((p+k)^{2},\,(p+k)^{2},\,0\right)\right)\bigg{]}\Bigg{\\}}\
.$ (66)

Note that we wrote $\Gamma^{\mu\nu}_{L}$ instead of $\Gamma^{\mu\nu}$ as the
WTI is blind to any component orthogonal to the photon momenta $k$. For a
vanishing photon momenta $\lambda_{2}$ and $\lambda_{3}$ are proportional to
derivatives of $A$ and $B$, respectively, and if these functions are
essentially constant then the above solution of the WTI can be approximated by

$\Gamma^{\mu\nu}_{L}(p,\,-p-k;\,k,\,0)=\frac{k^{\mu}\,\gamma^{\nu}}{k^{2}}~{}\bigg{[}A(p^{2})\,-\,A((p+k)^{2})\bigg{]}\
.$ (67)

As for the photon-fermion vertex, the solutions of the WTI in the soft photon
are determined by the inverse fermion propagator form factors $A$ and $B$, and
its derivatives.

Let’s proceed and discuss the low momenta approximation to (45). Our
understanding of the low momenta approximation is a linear approximation in
$q$ and $k$ to the r.h.s. of the WTI, disregarding all higher order
contributions in any of the photon momenta. Then, the longitudinal components
of the photon-fermion vertex to the l.h.s. of the WTI read

$\displaystyle
k_{\mu}\Bigg{\\{}2\,B^{\prime}(p^{2})\,g^{\mu\nu}~{}+~{}4\,B^{\prime\prime}(p^{2})\,p^{\mu}\,p^{\nu}~{}-~{}2\,A^{\prime}(p^{2})\,\left(p^{\mu}\,\gamma^{\nu}\,+\,p^{\nu}\,\gamma^{\mu}\right)$
$\displaystyle\qquad\qquad\qquad~{}-~{}2\,A^{\prime}(p^{2})\,\not{p}\,g^{\mu\nu}~{}-~{}4\,A^{\prime\prime}(p^{2})\,\not{p}\,p^{\mu}\,p^{\nu}\Bigg{\\}}$
(68)

and, therefore, it follows that

$\displaystyle\Gamma^{\mu\nu}_{L}(p,\,-p-k-q;\,k,\,q)=2\,B^{\prime}(p^{2})\,g^{\mu\nu}~{}+~{}4\,B^{\prime\prime}(p^{2})\,p^{\mu}\,p^{\nu}$
$\displaystyle~{}-~{}2\,A^{\prime}(p^{2})\,\left(p^{\mu}\,\gamma^{\nu}\,+\,p^{\nu}\,\gamma^{\mu}\right)~{}-~{}2\,A^{\prime}(p^{2})\,\not{p}\,g^{\mu\nu}~{}-~{}4\,A^{\prime\prime}(p^{2})\,\not{p}\,p^{\mu}\,p^{\nu}\
,$ (69)

modulo possible contributions coming from the transverse part of the photon-
fermion vertex. $A^{\prime}$ and $A^{\prime\prime}$ stand for the first and
second derivatives of $A$ with respect to its argument, with $B^{\prime}$ and
$B^{\prime\prime}$ having a similar meaning for the function $B$. Now, let us
discuss the contribution of the transverse operators $T^{(i)}_{\mu}$ given in
Eqs (57) to (64). The operators $T^{(i)}_{\mu}$ are either linear of quadratic
in $q$ and $k$ and, therefore, they are associated with higher order
contributions in the photon momenta to the r.h.s. of the WTI. Since the linear
contribution to the r.h.s of the WTI vanishes, then the expression in Eq. (69)
solves the WTI within the linear approximation. This solution is symmetric
under exchange of $\mu\leftrightarrow\nu$, as required by Bose symmetry, and
calls for the first and second derivatives of the fermion propagator form
factors, evaluated at the incoming fermion momentum. At large $p^{2}$ one
expects to recover the perturbative result to be a valid approximation. At
large momenta, $A$ and $B$ are essentially constants and, therefore, give sub-
leading contributions to $\Gamma^{\mu\nu}_{L}$, i.e. in the ultraviolet regime
$\Gamma^{\mu\nu}_{L}\sim 0$.

The analysis of the linear approximation to the WTI identifies an hierarchy of
tensors that are required to described the longitudinal part of the two-
photon-two-fermion vertex that are associated with the longitudinal photon-
fermion vertex. This suggest a possible construction for a partial tensor
basis to write $\Gamma^{\mu\nu}_{L}$. Indeed, by looking at the contribution
of the longitudinal photon-fermion vertex to the WTI and including only the
linear, in the photon momenta, transverse operators (after imposing Bose
symmetry), one can arrives at

$\displaystyle\Gamma^{\mu\nu}(p,\,-p-k-q;\,k,\,q)_{L}=\Gamma_{0}\,g^{\mu\nu}~{}+~{}\Gamma_{1}\,p^{\mu}\,p^{\nu}~{}+~{}\Gamma_{2}\,\left(p^{\mu}\,\gamma^{\nu}\,+\,p^{\nu}\,\gamma^{\mu}\right)$
$\displaystyle~{}+~{}\Gamma_{3}\,\not{p}\,g^{\mu\nu}~{}+~{}\Gamma_{4}\,\not{p}\,p^{\mu}\,p^{\nu}~{}+~{}\Gamma_{5}\,\left(\frac{k^{\mu}}{k^{2}}\sigma^{\nu\alpha}q_{\alpha}\,+\,\frac{q^{\nu}}{q^{2}}\sigma^{\mu\alpha}k_{\alpha}\right)$
$\displaystyle\quad+~{}\Gamma_{6}\,\Bigg{(}\frac{k^{\mu}}{k^{2}}\Big{(}\,(pq)\gamma^{\nu}\,-\,p^{\nu}\,\not{q}\,\Big{)}\,+\,\frac{q^{\nu}}{q^{2}}\Big{(}\,(pk)\gamma^{\mu}\,-\,p^{\mu}\,\not{k}\,\Big{)}\Bigg{)}$
$\displaystyle\quad+~{}\Gamma_{7}\,\Bigg{(}\frac{k^{\mu}}{k^{2}}\Big{[}\,(pq)\Big{(}\gamma^{\nu}\,\not{p}-p^{\nu}\Big{)}+p^{\nu}\,\sigma_{\alpha\beta}\,p^{\alpha}q^{\beta}\Big{]}$
$\displaystyle\hskip
85.35826pt\,+\,\frac{q^{\nu}}{q^{2}}\Big{[}\,(pk)\Big{(}\gamma^{\mu}\,\not{p}-p^{\mu}\Big{)}+p^{\mu}\,\sigma_{\alpha\beta}\,p^{\alpha}k^{\beta}\Big{]}\Bigg{)}$
$\displaystyle\quad+~{}\Gamma_{8}\,\Bigg{(}\frac{k^{\mu}}{k^{2}}\Big{[}\gamma^{\nu}\sigma_{\alpha\beta}\,q^{\alpha}p^{\beta}+p^{\nu}\,\not{q}-q^{\nu}\,\not{p}\Big{]}$
$\displaystyle\hskip
85.35826pt\,+\,\frac{q^{\nu}}{q^{2}}\Big{[}\gamma^{\mu}\sigma_{\alpha\beta}\,k^{\alpha}p^{\beta}+p^{\mu}\,\not{k}-k^{\nu}\,\not{p}\Big{]}\Bigg{)}\
.$ (70)

The discussion of this particular basis and its relation with other solutions
will be postponed until Sec. 6.3.

Let us start to discuss the solution of the WTI (45) by looking at its
contraction with $q$, that results in the scalar version of the WTI given in
(46). The scalar WTI is symmetric under interchange of photon momenta and
suggests the following “solution”

$\displaystyle\Gamma^{\mu\nu}_{L}(p,\,-p-k-q;\,k,\,q)$
$\displaystyle=\frac{k^{\mu}\,q^{\nu}}{k^{2}q^{2}}\bigg{(}k_{\mu}\Gamma^{\mu}(p,\,-p-k;k)-k_{\mu}\Gamma^{\mu}(p+q,\,-p-k-q;k)\bigg{)}$
$\displaystyle=\frac{k^{\mu}\,q^{\nu}}{k^{2}q^{2}}\bigg{(}S^{-1}(p+k)~{}+~{}S^{-1}(p+q)~{}-~{}S^{-1}(p+k+q)~{}-~{}S^{-1}(p)\bigg{)}\\!.$
(71)

The term in parenthesis on r.h.s. vanish when $k$ and/or $q$ approach zero and
it opens the possibility of having a “solution” with no kinematical
singularities at zero momenta. The above expression does not solve the WTI
exactly. Going back to the WTI (45) and consider the ansatz

$\displaystyle\Gamma^{\mu\nu}_{L}(p,\,-p-k-q;\,k,\,q)=\frac{k^{\mu}}{k^{2}}\bigg{(}\Gamma^{\nu}(p,\,-p-q;\,q)~{}-~{}\Gamma^{\nu}(p+k,\,-p-k-q;\,q)\bigg{)}$
$\displaystyle\qquad\qquad+~{}\frac{q^{\nu}}{q^{2}}\bigg{(}\Gamma^{\mu}(p,\,-p-k;\,k)~{}-~{}\Gamma^{\mu}(p+q,\,-p-k-q;\,k)\bigg{)}$
$\displaystyle\qquad\qquad+~{}P^{\perp\,\mu\nu}(k)~{}\widetilde{A}(k,q)+~{}P^{L\,\mu\nu}(k)~{}\widetilde{B}(k,q)$
$\displaystyle\qquad\qquad+~{}P^{\perp\,\mu\nu}(q)~{}\widetilde{A}(q,k)+~{}P^{L\,\mu\nu}(q)~{}\widetilde{B}(q,k)\
,$ (72)

where $\widetilde{A}$ and $\widetilde{B}$ are functions of the photon momenta.
This ansatz solves the two-photon-two-fermion WTI if, for all $k$ and $q$,

$\displaystyle\frac{q^{\nu}}{q^{2}}\bigg{(}S^{-1}(p+k)~{}+~{}S^{-1}(p+q)~{}-~{}S^{-1}(p)~{}-~{}S^{-1}(p+k+q)\bigg{)}$
$\displaystyle~{}+\bigg{(}k^{\nu}-\frac{(k\cdot
q)}{q^{2}}q^{\nu}\bigg{)}\,\widetilde{A}(q,k)~{}+~{}k^{\nu}\,\widetilde{B}(k,q)~{}+~{}\frac{(k\cdot
q)}{q^{2}}\,q^{\nu}\,\widetilde{B}(q,k)=0.$ (73)

To arrive at this later expression, the photon-fermion WTI (44) was used to
rewrite the equation. A general solution of this equation can be found if the
functions $\widetilde{A}$ and $\widetilde{B}$ are symmetric under interchange
of $k$ and $q$. In this case, it follows that

$\displaystyle\widetilde{B}(k,q)=-\,\widetilde{A}(q,k)\ ,$ (75)
$\displaystyle\widetilde{A}(q,k)=$ $\displaystyle~{}\frac{1}{2\,(k\cdot
q)}\bigg{(}S^{-1}(p+k)~{}+~{}S^{-1}(p+q)~{}-~{}S^{-1}(p)~{}-~{}S^{-1}(p+k+q)\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{1}{2\,(kq)}\Bigg{\\{}\bigg{[}A\left((p+k)^{2}\right)+A\left((p+q)^{2}\right)-A\left((p+k+q)^{2}\right)-A\left(p^{2}\right)\bigg{]}~{}\not{p}$
$\displaystyle\quad\quad+~{}\bigg{[}A\left((p+k)^{2}\right)-A\left((p+k+q)^{2}\right)\bigg{]}~{}\not{k}$
$\displaystyle\quad\quad+~{}\bigg{[}A\left((p+q)^{2}\right)-A\left((p+k+q)^{2}\right)\bigg{]}~{}\not{q}$
$\displaystyle\quad\quad+~{}B\left((p+k+q)^{2}\right)+B\left(p^{2}\right)-B\left((p+k)^{2}\right)-B\left((p+q)^{2}\right)\Bigg{\\}}.$

The solution of the WTI summarized in Eqs (72), (75) and (75) writes
$\Gamma^{\mu\nu}_{L}$ in terms of the photon-fermion vertex and the form
factors $A$ and $B$ that describe the fermion propagator. Furthermore, the
expression in Eq. (75) is free of kinematical singularities and, therefore,
$\Gamma^{\mu\nu}_{L}$ is also free of kinematical singularities. Moreover, for
the perturbative tree level solution of the theory, where $A=1$, $B=m$ and the
photon-fermion vertex reads $\Gamma^{\mu}=\gamma^{\mu}$, the longitudinal
component of the two-photon-two-fermion irreducible vertex vanishes as
expected, and a non-vanishing $\Gamma^{\mu\nu}_{L}$ within the perturbative
solution of QED can only be associated with the loop corrections.

##  6 The Dyson-Schwinger equation for the two-photon-two-fermion irreducible
vertex

Figure 5: The two-photon-two-fermion one-particle irreducible vertex: see Eq.
(76) for defintions.

As discussed in the previous section, the longitudinal part of the two-photon-
two-fermion vertex is determined by the corresponding WTI. However, the full
$\Gamma^{\mu\nu}$ certainly goes beyond $\Gamma^{\mu\nu}_{L}$ and one needs to
be able to compute it. Here, we consider the DySon-Schwinger equation for this
irreducible vertex that, in principle, can allow to access the remaining
components.

The one-particle irreducible two-photon-two-fermion vertex represented in Fig.
5 and defined in Eq. (132),

$\displaystyle g^{2}\,{\Gamma}^{\mu\nu}_{\alpha\beta}(x,y;z,w)$
$\displaystyle=$ $\displaystyle-~{}\frac{\delta^{4}\Gamma}{\delta
A_{cl,\,\mu}(z)~{}\delta
A_{cl,\,\nu}(w)~{}\delta\psi_{cl,\,\beta}(y)~{}\delta\bar{\psi}_{cl,\,\alpha}(x)}$
(76) $\displaystyle=$ $\displaystyle
g^{2}\int\frac{d^{4}p^{\prime}}{(2\,\pi)^{4}}\frac{d^{4}p}{(2\,\pi)^{4}}\frac{d^{4}k}{(2\,\pi)^{4}}\frac{d^{4}q}{(2\,\pi)^{4}}~{}e^{-i\,\left(p^{\prime}x+py+kz+qw\right)}$
$\displaystyle\hskip
36.98866pt\,(2\,\pi)^{4}~{}\delta(p^{\prime}+p+k+q)~{}{\Gamma}^{\mu\nu}_{\alpha\beta}(p^{\prime},\,p;\,k,\,q)\
,$

is symmetric under interchange of the photon indices. As the photon-fermion
irreducible vertex, ${\Gamma}^{\mu\nu}$ can be decomposed into a longitudinal
${\Gamma}^{\mu\nu}_{L}$ and an orthogonal ${\Gamma}^{\mu\nu}_{T}$ component,
relative to the photon momenta, i.e. in momentum space

$\displaystyle{\Gamma}^{\mu\nu}={\Gamma}^{\mu\nu}_{L}+{\Gamma}^{\mu\nu}_{T}$
$\displaystyle\qquad\mbox{ with }\qquad
k_{\mu}\,{\Gamma}^{\mu\nu}_{T}(p^{\prime},\,p;\,k,\,q)~{}=~{}q_{\nu}\,{\Gamma}^{\mu\nu}_{T}(p^{\prime},\,p;\,k,\,q)~{}=~{}0.$
(77)

The Ward-Takahashi identity (45) constrains only the longitudinal part of
$\Gamma^{\mu\nu}$. The solution of the WTI discussed in Sec. LABEL:Sec:SolWTI,
see Eqs (72) and (75), writes $\Gamma^{\mu\nu}_{L}$ in terms of the photon-
fermion vertex and the fermion propagator functions $A$ and $B$.

The Dyson-Schwinger equation for the two-photon-two-fermion Green function
$\Gamma^{\mu\nu}$ can be obtained in the usual way after taking various
functional derivatives of Eq. (21), and after setting the sources to zero it
reads

$\displaystyle
0=\left(i\,\not{\partial}_{x}-m\right)_{\alpha\beta}\frac{\delta^{4}W}{\delta
J^{\rho}(z)\,\delta
J^{\nu}(w)\,\delta\eta_{\gamma}(y)\,\delta\bar{\eta}_{\beta}(x)}$
$\displaystyle\hskip
28.45274pt+~{}i~{}g\,\left(\gamma^{\mu}\right)_{\alpha\beta}\,\frac{\delta^{5}W}{\delta
J^{\rho}(z)\,\delta J^{\nu}(w)\,\delta
J^{\mu}(x)\,\delta\eta_{\gamma}(y)\,\delta\bar{\eta}_{\beta}(x)}$
$\displaystyle\hskip
42.67912pt-~{}g\,\left(\gamma^{\mu}\right)_{\alpha\beta}\,\frac{\delta^{3}W}{\delta
J^{\nu}(w)\,\delta\eta_{\gamma}(y)\,\delta\bar{\eta}_{\beta}(x)}\,\frac{\delta^{2}W}{\delta
J^{\rho}(z)\,\delta J^{\mu}(x)}$ $\displaystyle\hskip
56.9055pt-~{}g\,\left(\gamma^{\mu}\right)_{\alpha\beta}\,\frac{\delta^{3}W}{\delta
J^{\rho}(z)\,\delta\eta_{\gamma}(y)\,\delta\bar{\eta}_{\beta}(x)}\,\frac{\delta^{2}W}{\delta
J^{\nu}(w)\,\delta J^{\mu}(x)}$ $\displaystyle\hskip
71.13188pt-~{}g\,\left(\gamma^{\mu}\right)_{\alpha\beta}\,\frac{\delta^{2}W}{\delta\eta_{\gamma}(y)\,\delta\bar{\eta}_{\beta}(x)}\,\frac{\delta^{3}W}{\delta
J^{\rho}(z)\,\delta J^{\nu}(w)\,\delta J^{\mu}(x)}$
$\displaystyle+~{}\Bigg{(}i\,\left(i\,\not{\partial}_{x}-m\right)_{\alpha\beta}\,\frac{\delta^{2}W}{\delta\eta_{\gamma}(y)\,\delta\bar{\eta}_{\beta}(x)}~{}-~{}g\,\left(\gamma^{\mu}\right)_{\alpha\beta}\,\frac{\delta^{3}W}{\delta
J^{\mu}(x)\,\delta\eta_{\gamma}(y)\,\delta\bar{\eta}_{\beta}(x)}$
$\displaystyle\hskip
133.72786pt~{}+~{}i\,\delta_{\alpha\gamma}\,\delta(x-y)\Bigg{)}\,\frac{\delta^{2}W}{\delta
J^{\rho}(z)\,\delta J^{\nu}(w)}\ .$ (78)

The term in parenthesis, see the last line, is the fermion gap equation (22)
and, therefore, it vanishes. The last term in Eq. (78) before the one
proportional to the gap equation is proportional to the three-photon one-
particle irreducible vertex and it vanishes for QED (Furry’s theorem). It
turns out that the DSE for the two-photon-two-fermion reads

$\displaystyle\left(i\,\not{\partial}_{x}-m\right)_{\alpha\beta}\frac{\delta^{4}W}{\delta
J^{\rho}(z)\,\delta
J^{\nu}(w)\,\delta\eta_{\gamma}(y)\,\delta\bar{\eta}_{\beta}(x)}$
$\displaystyle\hskip
17.07182pt+~{}i~{}g\,\left(\gamma^{\mu}\right)_{\alpha\beta}\,\frac{\delta^{5}W}{\delta
J^{\rho}(z)\,\delta J^{\nu}(w)\,\delta
J^{\mu}(x)\,\delta\eta_{\gamma}(y)\,\delta\bar{\eta}_{\beta}(x)}$
$\displaystyle\hskip
34.14322pt-~{}g\,\left(\gamma^{\mu}\right)_{\alpha\beta}\,\frac{\delta^{3}W}{\delta
J^{\nu}(w)\,\delta\eta_{\gamma}(y)\,\delta\bar{\eta}_{\beta}(x)}\,\frac{\delta^{2}W}{\delta
J^{\rho}(z)\,\delta J^{\mu}(x)}$ $\displaystyle\hskip
51.21504pt-~{}g\,\left(\gamma^{\mu}\right)_{\alpha\beta}\,\frac{\delta^{3}W}{\delta
J^{\rho}(z)\,\delta\eta_{\gamma}(y)\,\delta\bar{\eta}_{\beta}(x)}\,\frac{\delta^{2}W}{\delta
J^{\nu}(w)\,\delta J^{\mu}(x)}=0\ .$ (79)

To arrive at a solution of Eq. (79) for the two-photon-two-fermion vertex, all
the connected Green functions have to be written in terms of one-particle
irreducible vertices. The second term of this equation requires a five-point
connected Green function, whose decomposition in terms of one-particle
irreducible vertices can be achieved following the type of calculations
discussed in App A. For the five-point function in (79), after performing the
necessary functional derivatives of the generating functionals, after some
algebra and after setting the sources to zero one arrives at

$\displaystyle\frac{\delta^{5}W}{\delta J_{\rho}(s)\,\delta J_{\nu}(w)\,\delta
J_{\mu}(z)\,\delta\eta_{\beta}(y)\,\delta\bar{\eta}_{\alpha}(x)\,}~{}=~{}$
$\displaystyle=~{}\int
d^{4}v_{1}\,d^{4}v_{2}\,d^{4}v_{3}\,d^{4}v_{4}\,d^{4}v_{5}~{}\frac{\delta^{2}W}{\delta
J_{\mu}(z)~{}\delta J^{\mu^{\prime}}(v_{1})}~{}\frac{\delta^{2}W}{\delta
J_{\nu}(w)~{}\delta J^{\nu^{\prime}}(v_{2})}~{}\frac{\delta^{2}W}{\delta
J_{\rho}(s)~{}\delta J^{\rho^{\prime}}(v_{3})}$ $\displaystyle\hskip
85.35826pt\frac{\delta^{2}W}{\delta\eta_{\alpha^{\prime}}(v_{4})~{}\delta\bar{\eta}_{\alpha}(x)}~{}\frac{\delta^{2}W}{\delta\eta_{\beta}(y)~{}\delta\bar{\eta}_{\beta^{\prime}}(v_{5})}$
$\displaystyle\hskip 85.35826pt\frac{\delta^{5}\Gamma}{\delta
A_{c,\,\rho^{\prime}}(v_{3})~{}\delta A_{c,\,\nu^{\prime}}(v_{2})~{}\delta
A_{c,\,\mu^{\prime}}(v_{1})~{}\delta\psi_{c,\,\beta^{\prime}}(v_{5})~{}\delta\bar{\psi}_{c,\,\alpha^{\prime}}(v_{4})}$
$\displaystyle+~{}\int
d^{4}v_{1}\,d^{4}v_{2}\,d^{4}v_{3}~{}\frac{\delta^{2}W}{\delta
J_{\rho}(s)~{}\delta
J^{\rho^{\prime}}(v_{1})}~{}\frac{\delta^{2}W}{\delta\eta_{\alpha^{\prime}}(v_{2})~{}\delta\bar{\eta}_{\alpha}(x)}~{}\frac{\delta^{4}W}{\delta
J_{\nu}(w)~{}\delta
J_{\mu}(z)~{}\delta\eta_{\beta}(y)~{}\delta\bar{\eta}_{\beta^{\prime}}(v_{3})}$
$\displaystyle\hskip 85.35826pt\frac{\delta^{3}\Gamma}{\delta
A_{c,\,\rho^{\prime}}(v_{1})~{}\delta\psi_{c,\,\beta^{\prime}}(v_{3})~{}\delta\bar{\psi}_{c,\,\alpha^{\prime}}(v_{2})}$
$\displaystyle+~{}\int
d^{4}v_{1}\,d^{4}v_{2}\,d^{4}v_{3}~{}\frac{\delta^{2}W}{\delta
J_{\nu}(w)~{}\delta
J^{\nu^{\prime}}(v_{1})}~{}\frac{\delta^{2}W}{\delta\eta_{\alpha^{\prime}}(v_{2})~{}\delta\bar{\eta}_{\alpha}(x)}~{}\frac{\delta^{4}W}{\delta
J_{\rho}(s)~{}\delta
J_{\mu}(z)~{}\delta\eta_{\beta}(y)~{}\delta\bar{\eta}_{\beta^{\prime}}(v_{3})}$
$\displaystyle\hskip 85.35826pt\frac{\delta^{3}\Gamma}{\delta
A_{c,\,\nu^{\prime}}(v_{1})~{}\delta\psi_{c,\,\beta^{\prime}}(v_{3})~{}\delta\bar{\psi}_{c,\,\alpha^{\prime}}(v_{2})}$
$\displaystyle+~{}\int
d^{4}v_{1}\,d^{4}v_{2}\,d^{4}v_{3}~{}\frac{\delta^{2}W}{\delta
J_{\mu}(z)~{}\delta
J^{\mu^{\prime}}(v_{1})}~{}\frac{\delta^{2}W}{\delta\eta_{\alpha^{\prime}}(v_{2})~{}\delta\bar{\eta}_{\alpha}(x)}~{}\frac{\delta^{4}W}{\delta
J_{\rho}(s)~{}\delta
J_{\nu}(w)~{}\delta\eta_{\beta}(y)~{}\delta\bar{\eta}_{\beta^{\prime}}(v_{3})}$
$\displaystyle\hskip 85.35826pt\frac{\delta^{3}\Gamma}{\delta
A_{c,\,\mu^{\prime}}(v_{1})~{}\delta\psi_{c,\,\beta^{\prime}}(v_{3})~{}\delta\bar{\psi}_{c,\,\alpha^{\prime}}(v_{2})}$
$\displaystyle+~{}\int
d^{4}v_{1}\,d^{4}v_{2}\,d^{4}v_{3}~{}\frac{\delta^{2}W}{\delta\eta_{\alpha^{\prime}}(v_{1})~{}\delta\bar{\eta}_{\alpha}(x)}~{}\frac{\delta^{2}W}{\delta\eta_{\beta}(y)~{}\delta\bar{\eta}_{\beta^{\prime}}(v_{2})}~{}\frac{\delta^{4}W}{\delta
J_{\rho}(s)~{}\delta J_{\nu}(w)~{}\delta J_{\mu}(z)~{}\delta
J^{\zeta^{\prime}}(v_{3})}$ $\displaystyle\hskip
85.35826pt\frac{\delta^{3}\Gamma}{\delta
A_{c,\,\zeta^{\prime}}(v_{3})~{}\delta\psi_{c,\,\beta^{\prime}}(v_{2})~{}\delta\bar{\psi}_{c,\,\alpha^{\prime}}(v_{1})}$
$\displaystyle+~{}\int
d^{4}v_{1}\,d^{4}v_{2}\,d^{4}v_{3}\,d^{4}v_{4}~{}\frac{\delta^{2}W}{\delta
J_{\rho}(s)~{}\delta J^{\rho^{\prime}}(v_{1})}~{}\frac{\delta^{2}W}{\delta
J_{\nu}(w)~{}\delta
J^{\nu^{\prime}}(v_{2})}~{}\frac{\delta^{2}W}{\delta\eta_{\alpha^{\prime}}(v_{3})~{}\delta\bar{\eta}_{\alpha}(x)}$
$\displaystyle\hskip 39.83368pt\frac{\delta^{3}W}{\delta
J_{\mu}(z)~{}\delta\eta_{\beta}(y)~{}\delta\bar{\eta}_{\beta^{\prime}}(v_{4})}~{}\frac{\delta^{4}\Gamma}{\delta
A_{c,\,\rho^{\prime}}(v_{1})~{}\delta
A_{c,\,\nu^{\prime}}(v_{2})~{}\delta\psi_{c,\,\beta^{\prime}}(v_{4})~{}\delta\bar{\psi}_{c,\,\alpha^{\prime}}(v_{3})}$
$\displaystyle+~{}\int
d^{4}v_{1}\,d^{4}v_{2}\,d^{4}v_{3}\,d^{4}v_{4}~{}\frac{\delta^{2}W}{\delta
J_{\rho}(s)~{}\delta J^{\rho^{\prime}}(v_{1})}~{}\frac{\delta^{2}W}{\delta
J_{\mu}(z)~{}\delta
J^{\mu^{\prime}}(v_{2})}~{}\frac{\delta^{2}W}{\delta\eta_{\alpha^{\prime}}(v_{3})~{}\delta\bar{\eta}_{\alpha}(x)}$
$\displaystyle\hskip 39.83368pt\frac{\delta^{3}W}{\delta
J_{\nu}(w)~{}\delta\eta_{\beta}(y)~{}\delta\bar{\eta}_{\beta^{\prime}}(v_{4})}~{}\frac{\delta^{4}\Gamma}{\delta
A_{c,\,\rho^{\prime}}(v_{1})~{}\delta
A_{c,\,\mu^{\prime}}(v_{2})~{}\delta\psi_{c,\,\beta^{\prime}}(v_{4})~{}\delta\bar{\psi}_{c,\,\alpha^{\prime}}(v_{3})}$
$\displaystyle+~{}\int
d^{4}v_{1}\,d^{4}v_{2}\,d^{4}v_{3}\,d^{4}v_{4}~{}\frac{\delta^{2}W}{\delta
J_{\nu}(w)~{}\delta J^{\nu^{\prime}}(v_{1})}~{}\frac{\delta^{2}W}{\delta
J_{\mu}(z)~{}\delta
J^{\mu^{\prime}}(v_{2})}~{}\frac{\delta^{2}W}{\delta\eta_{\alpha^{\prime}}(v_{3})~{}\delta\bar{\eta}_{\alpha}(x)}$
$\displaystyle\hskip 39.83368pt\frac{\delta^{3}W}{\delta
J_{\rho}(s)~{}\delta\eta_{\beta}(y)~{}\delta\bar{\eta}_{\beta^{\prime}}(v_{4})}~{}\frac{\delta^{4}\Gamma}{\delta
A_{c,\,\nu^{\prime}}(v_{1})~{}\delta
A_{c,\,\mu^{\prime}}(v_{2})~{}\delta\psi_{c,\,\beta^{\prime}}(v_{4})~{}\delta\bar{\psi}_{c,\,\alpha^{\prime}}(v_{3})}\
.$ (80)

The decomposition given in Eq. (80) for the five-point Green function is
symmetric under the interchange of any pair of the bosonic degrees of freedom
as required by Bose symmetry. The decomposition of the five-point Green
function requires further Green functions not considered so far. In order to
arrive at a manageable equation and to avoid the introduction of new vertices,
we consider a truncation of Eq. (80) and ignore the terms that are
proportional to the one-particle irreducible five-point Green function (first
term on r.h.s.) and the four-photon irreducible diagrams that appear in the
decomposition of the four-photon connected Green function. By disregarding the
four-photon irreducible vertices it turns out that the contribution of the
connected four-photon Green function, that is a sum of a term containing the
irreducible four-photon vertex and terms proportional to the three-photon
irreducible vertices, that vanish in QED (Furry’s theorem), is not taken into
consideration within the truncation considered. By taking into account the
definition of the photon and fermion propagators, the five-point Green
function given in Eq. (80) can be simplified using the definitions given in
Eqs (128) and (132) for the one-particle irreducible Green functions. In
momentum space the above equation becomes

$\displaystyle\frac{\delta^{5}W}{\delta J^{\rho}(s)\,\delta J^{\nu}(w)\,\delta
J^{\mu}(z)\,\delta\eta_{\beta}(y)\,\delta\bar{\eta}_{\alpha}(x)\,}~{}=~{}$
$\displaystyle=~{}g^{3}~{}\int\frac{d^{4}k_{1}}{(2\,\pi)^{4}}\,\frac{d^{4}k_{2}}{(2\,\pi)^{4}}\,\frac{d^{4}k_{3}}{(2\,\pi)^{4}}\,\frac{d^{4}p}{(2\,\pi)^{4}}~{}e^{-\,i\,k_{1}z-\,i\,k_{2}w-\,i\,k_{3}s-\,i\,p\,x+\,i\,(p+k_{1}+k_{2}+k_{3})y}$
$\displaystyle\qquad\qquad
D_{\rho\rho^{\prime}}(k_{3})~{}D_{\nu\nu^{\prime}}(k_{2})~{}D_{\mu\mu^{\prime}}(k_{1})~{}\Bigg{\\{}~{}S(p)~{}\bigg{[}$
$\displaystyle~{}\Gamma^{\rho^{\prime}}(p,\,-p-k_{3};\,k_{3})\,S(p+k_{3})\,\Gamma^{\nu^{\prime}}(p+k_{3},\,-p-k_{2}-k_{3};k_{2})\,S(p+k_{2}+k_{3})\,$
$\displaystyle\hskip
113.81102pt\Gamma^{\mu^{\prime}}(p+k_{2}+k_{3},\,p-k_{1}-k_{2}-k_{3};k_{1})$
$\displaystyle+~{}\Gamma^{\rho^{\prime}}(p,\,-p-k_{3};\,k_{3})\,S(p+k_{3})\,\Gamma^{\mu^{\prime}}(p+k_{3},\,-p-k_{1}-k_{3};k_{1})\,S(p+k_{1}+k_{3})\,$
$\displaystyle\hskip
113.81102pt\Gamma^{\nu^{\prime}}(p+k_{1}+k_{3},\,-p-k_{1}-k_{2}-k_{3};k_{2})$
$\displaystyle+~{}\Gamma^{\nu^{\prime}}(p,\,-p-k_{2};\,k_{2})\,S(p+k_{2})\,\Gamma^{\rho^{\prime}}(p+k_{2},\,-p-k_{2}-k_{3};k_{3})\,S(p+k_{2}+k_{3})\,$
$\displaystyle\hskip
113.81102pt\Gamma^{\mu^{\prime}}(p+k_{2}+k_{3},\,-p-k_{1}-k_{2}-k_{3};k_{1})$
$\displaystyle+~{}\Gamma^{\nu^{\prime}}(p,\,-p-k_{2};\,k_{2})\,S(p+k_{2})\,\Gamma^{\mu^{\prime}}(p+k_{2},\,-p-k_{1}-k_{2};k_{1})\,S(p+k_{1}+k_{2})\,$
$\displaystyle\hskip
113.81102pt\Gamma^{\rho^{\prime}}(p+k_{1}+k_{2},\,-p-k_{1}-k_{2}-k_{3};k_{3})$
$\displaystyle+~{}\Gamma^{\mu^{\prime}}(p,\,-p-k_{1};\,k_{1})\,S(p+k_{1})\,\Gamma^{\rho^{\prime}}(p+k_{1},\,-p-k_{1}-k_{3};k_{3})\,S(p+k_{1}+k_{3})\,$
$\displaystyle\hskip
113.81102pt\Gamma^{\nu^{\prime}}(p+k_{1}+k_{3},\,-p-k_{1}-k_{2}-k_{3};k_{2})$
$\displaystyle+~{}\Gamma^{\mu^{\prime}}(p,\,-p-k_{1};\,k_{1})\,S(p+k_{1})\,\Gamma^{\nu^{\prime}}(p+k_{1},\,-p-k_{1}-k_{2};k_{2})\,S(p+k_{1}+k_{2})\,$
$\displaystyle\hskip
113.81102pt\Gamma^{\rho^{\prime}}(p+k_{1}+k_{2},\,-p-k_{1}-k_{2}-k_{3};k_{3})$
$\displaystyle+~{}\Gamma^{\rho^{\prime}}(p,\,-p-k_{3};\,k_{3})\,S(p+k_{3})\,\Gamma^{\nu^{\prime}\mu^{\prime}}(p+k_{3},\,-p-k_{1}-k_{2}-k_{3};\,k_{2},\,k_{1})$
$\displaystyle+~{}\Gamma^{\nu^{\prime}}(p,\,-p-k_{2};\,k_{2})\,S(p+k_{2})\,\Gamma^{\rho^{\prime}\mu^{\prime}}(p+k_{2},\,-p-k_{1}-k_{2}-k_{3};\,k_{3},\,k_{1})$
$\displaystyle+~{}\Gamma^{\mu^{\prime}}(p,\,-p-k_{1};\,k_{1})\,S(p+k_{1})\,\Gamma^{\rho^{\prime}\nu^{\prime}}(p+k_{1},\,-p-k_{1}-k_{2}-k_{3};\,k_{3},\,k_{2})$
$\displaystyle+~{}\Gamma^{\rho^{\prime}\nu^{\prime}}(p,\,-p-k_{2}-k_{3};\,k_{3},\,k_{2})\,S(p+k_{2}+k_{3})\,$
$\displaystyle\hskip
113.81102pt\Gamma^{\mu^{\prime}}(p+k_{2}+k_{3},\,-p-k_{1}-k_{2}-k_{3};\,k_{1})$
$\displaystyle+~{}\Gamma^{\rho^{\prime}\mu^{\prime}}(p,\,-p-k_{1}-k_{3};\,k_{3},\,k_{1})\,S(p+k_{1}+k_{3})\,$
$\displaystyle\hskip
113.81102pt\Gamma^{\nu^{\prime}}(p+k_{1}+k_{3},\,-p-k_{1}-k_{2}-k_{3};\,k_{2})$
$\displaystyle+~{}\Gamma^{\nu^{\prime}\mu^{\prime}}(p,\,-p-k_{1}-k_{2};\,k_{2},\,k_{1})\,S(p+k_{1}+k_{2})\,$
$\displaystyle\hskip
113.81102pt\Gamma^{\rho^{\prime}}(p+k_{1}+k_{2},\,-p-k_{1}-k_{2}-k_{3};\,k_{3})~{}\bigg{]}$
$\displaystyle~{}S(p+k_{1}+k_{2}+k_{3})~{}\Bigg{\\}}_{\alpha\beta}\ .$ (81)

Proceeding, as usual, starting from Eq. (79), doing the decomposition of the
Green functions in terms of one-particle irreducible vertices and propagators,
performing the necessary Fourier transformations and doing the required
inversions, one arrives at the truncated Dyson-Schwinger equation for two-
photon-two-fermion one-particle irreducible vertex in momentum space

$\displaystyle\Gamma^{\mu\nu}(p,\,-p-k-q;\,k,\,q)~{}=$
$\displaystyle\quad=~{}Z^{\mu}(p,k)~{}\bigg{[}S(p+k)~{}\Gamma^{\nu}(p+k,\,-p-k-q;\,q)\bigg{]}~{}$
$\displaystyle\qquad+~{}Z^{\nu}(p,q)~{}\bigg{[}S(p+q)~{}\Gamma^{\mu}(p+q,\,-p-k-q;\,k)\bigg{]}$
$\displaystyle\qquad-~{}\Gamma^{\mu}(p,\,-p-k;\,k)~{}S(p+k)~{}\Gamma^{\nu}(p+k,\,-p-k-q;\,q)$
$\displaystyle\qquad-~{}\Gamma^{\nu}(p,\,-p-q;\,q)~{}S(p+q)~{}\Gamma^{\mu}(p+q,\,-p-k-q;\,k)$
$\displaystyle\quad\quad+\,i\,g^{2}~{}\int\frac{d^{4}w}{(2\,\pi)^{4}}~{}D_{\zeta\zeta^{\prime}}(w)~{}\gamma^{\zeta}~{}S(p-w)$
$\displaystyle\bigg{\\{}~{}\Gamma^{\mu}(p-w,\,-p-k+w;\,k)~{}S(p+k-w)~{}$
$\displaystyle\hskip 56.9055pt\Gamma^{\nu}(p+k-w,\,-p-k-q+w;\,q)$
$\displaystyle\hskip
56.9055ptS(p+k+q-w)~{}\Gamma^{\zeta^{\prime}}(p+k+q-w,\,-p-k-q;\,w)$
$\displaystyle~{}+~{}~{}\Gamma^{\nu}(p-w,\,-p-q+w;\,q)~{}S(p+q-w)~{}$
$\displaystyle\hskip 56.9055pt\Gamma^{\mu}(p+q-w,\,-p-k-q+w;\,k)$
$\displaystyle\hskip
56.9055ptS(p+k+q-w)~{}\Gamma^{\zeta^{\prime}}(p+k+q-w,\,-p-k-q;\,w)$
$\displaystyle~{}+~{}\Gamma^{\mu}(p-w,\,-p-k+w;\,k)~{}S(p+k-w)~{}$
$\displaystyle\hskip 56.9055pt\Gamma^{\zeta^{\prime}}(p+k-w,\,-p-k;\,w)$
$\displaystyle\hskip 56.9055ptS(p+k)~{}\Gamma^{\nu}(p+k,\,-p-k-q;\,q)$
$\displaystyle~{}+~{}~{}\Gamma^{\nu}(p-w,\,-p-q+w;\,q)~{}S(p+q-w)~{}$
$\displaystyle\hskip 56.9055pt\Gamma^{\zeta^{\prime}}(p+q-w,\,-p-q;\,w)$
$\displaystyle\hskip 56.9055ptS(p+q)~{}\Gamma^{\mu}(p+q,\,-p-k-q;\,k)$
$\displaystyle~{}+~{}~{}\Gamma^{\mu}(p-w,\,-p-k+w;\,k)~{}S(p+k-w)~{}$
$\displaystyle\hskip
56.9055pt\Gamma^{\nu\zeta^{\prime}}(p+k-w,\,-p-k-q;\,q,\,w)$
$\displaystyle~{}+~{}~{}\Gamma^{\nu}(p-w,\,-p-q+w;\,q)~{}S(p+q-w)~{}$
$\displaystyle\hskip
56.9055pt\Gamma^{\mu\zeta^{\prime}}(p+q-w,\,-p-k-q;\,k,\,w)$
$\displaystyle~{}+~{}~{}\Gamma^{\mu\nu}(p-w,\,-p-k-q+w;\,k,\,q)~{}S(p+k+q-w)~{}$
$\displaystyle\hskip
56.9055pt\Gamma^{\zeta^{\prime}}(p+k+q-w,\,-p-k-q;\,w)~{}\bigg{\\}}$ (82)

where

$\displaystyle Z^{\mu}(p,k)=\gamma^{\mu}~{}+$
$\displaystyle~{}i\,g^{2}\,\int\frac{d^{4}w}{(2\,\pi)^{4}}~{}D_{\zeta\zeta^{\prime}}(w)~{}\bigg{[}\gamma^{\zeta}~{}S(p-w)~{}\Gamma^{\mu\zeta^{\prime}}(p-w,\,-p-k;\,k,\,w)\bigg{]}\
.$ (83)

The definition of $Z^{\mu}$ replicates (up to a sign) the correction of the
fermion self-energy after replacing $\Gamma^{\mu}$ by the one-particle
irreducible vertex $\Gamma^{\mu\nu}$. Moreover, equation (82) is symmetric
under the interchange of any pair of bosonic indices as required by Bose
statistics. The above equation can be further simplified with the help of the
Dyson-Schwinger equation for the vertex (40). Indeed, replacing in the third
and fourth lines the vertex that appears on the left by Eq. (40), it turns out
that it cancels exactly the first and the second lines of Eq. (82) and also
the third and fourth lines under the integral. Then, it follows that the
truncated equation determining the two-photon-two-fermion one-particle
irreducible vertex is

$\displaystyle\Gamma^{\mu\nu}(p,\,-p-k-q;\,k,\,q)~{}=$
$\displaystyle=i\,g^{2}~{}\int\frac{d^{4}w}{(2\,\pi)^{4}}~{}D_{\zeta\zeta^{\prime}}(w)~{}\gamma^{\zeta}~{}S(p-w)$
$\displaystyle\hskip
14.22636pt\bigg{\\{}~{}\Gamma^{\mu}(p-w,\,-p-k+w;\,k)~{}S(p+k-w)~{}$
$\displaystyle\hskip 85.35826pt\Gamma^{\nu}(p+k-w,\,-p-k-q+w;\,q)$
$\displaystyle\hskip
85.35826ptS(p+k+q-w)~{}\Gamma^{\zeta^{\prime}}(p+k+q-w,\,-p-k-q;\,w)$
$\displaystyle\hskip
14.22636pt~{}+~{}~{}\Gamma^{\nu}(p-w,\,-p-q+w;\,q)~{}S(p+q-w)~{}$
$\displaystyle\hskip 85.35826pt\Gamma^{\mu}(p+q-w,\,-p-k-q+w;\,k)$
$\displaystyle\hskip
85.35826ptS(p+k+q-w)~{}\Gamma^{\zeta^{\prime}}(p+k+q-w,\,-p-k-q;\,w)$
$\displaystyle\hskip
14.22636pt~{}+~{}~{}\Gamma^{\mu}(p-w,\,-p-k+w;\,k)~{}S(p+k-w)~{}$
$\displaystyle\hskip
85.35826pt\Gamma^{\nu\zeta^{\prime}}(p+k-w,\,-p-k-q;\,q,\,w)$
$\displaystyle\hskip
14.22636pt~{}+~{}~{}\Gamma^{\nu}(p-w,\,-p-q+w;\,q)~{}S(p+q-w)~{}$
$\displaystyle\hskip
85.35826pt\Gamma^{\mu\zeta^{\prime}}(p+q-w,\,-p-k-q;\,k,\,w)$
$\displaystyle\hskip
14.22636pt~{}+~{}~{}\Gamma^{\mu\nu}(p-w,\,-p-k-q+w;\,k,\,q)~{}S(p+k+q-w)~{}$
$\displaystyle\hskip
85.35826pt\Gamma^{\zeta^{\prime}}(p+k+q-w,\,-p-k-q;\,w)~{}\bigg{\\}}$ (84)

whose representation in terms of Feynman diagrams is given in Fig. 6. This
equation can, in principle, be solved self-consistently as the computation of
the r.h.s. requires the knowledge of $\Gamma^{\mu\nu}$ itself. We end this
section by calling the reader attention that the derivation of Eq. (84) calls
for the inverse of the photon propagator and of the fermion propagator that
are not well defined in the Landau gauge.

Figure 6: The truncated Dyson-Schwinger equation for the two-photon-two-
photon-fermion one-particle irreducible vertex as a function of the one-
particle irreducible vertices and full propagators. Recall that only vertices
with a number of external legs smaller than four were considered.

###  6.1 Perturbative analysis of the DSE for the two-photon-two-fermion
vertex

The Dyson-Schwinger Eq. (84) is an approximation to the bare DSE for the two-
photon-two-fermion irreducible vertex. This equation was built ignoring
contributions from higher order Green functions and, therefore, the truncation
can introduce unwanted terms that have to be removed. However, the diagramatic
representation of the equation, see Fig. 6, and the perturbative analysis of
Eq. (84) shows that, for the truncation considered, this is not the case.

In perturbation theory all quantities are written as series in the coupling
constant. To proceed let us introduce the notation for the photon-fermion
vertex

$\Gamma^{\mu}(p_{2},\,p_{1};\,p_{3})=\gamma^{\mu}+\Delta\Gamma^{\mu}(p_{2},\,p_{1};\,p_{3})\
,$ (85)

where $\Delta\Gamma^{\mu}$ represents the corrections to the tree-level
vertex. In the perturbative solution of QED, $\Delta\Gamma^{\mu}$ starts at
order $g^{2}$. Indeed, in the lowest order in the coupling constant, see Eq.
(40), it is given by

$\displaystyle\Delta{\Gamma}^{\mu}(p,\,-p-k;\,k)~{}=~{}i\,g^{2}\,\int\frac{d^{4}w}{(2\,\pi)^{4}}~{}D_{\zeta\zeta^{\prime}}(w)~{}$
$\displaystyle\hskip
133.72786pt\bigg{[}\gamma^{\zeta}\,S(p-w)\,\gamma^{\mu}~{}S(p+k-w)~{}\gamma^{\zeta^{\prime}}\bigg{]}\
.$ (86)

Inserting the definition (85) into Eq. (84), it follows that the lowest order
contribution to $\Gamma^{\mu\nu}$ is of order $g^{2}$. Thus, in lowest order
in the coupling constant one has

$\displaystyle\Gamma^{\mu\nu}(p,\,-p-k-q;\,k,\,q)=i\,g^{2}\,\int\frac{d^{4}w}{(2\,\pi)}~{}D_{\zeta\zeta^{\prime}}(w)$
$\displaystyle\hskip 56.9055pt\gamma^{\zeta}~{}S(p-w)\times$
$\displaystyle\hskip
71.13188pt\times\bigg{\\{}~{}\gamma^{\nu}~{}S(p+q-w)~{}\gamma^{\mu}~{}+~{}\gamma^{\mu}~{}S(p+k-w)~{}\gamma^{\nu}\bigg{\\}}\times$
$\displaystyle\hskip 85.35826pt\times~{}S(p+k+q-w)~{}\gamma^{\zeta^{\prime}}$
(87)

that represents the box diagrams that are the lowest order in $g$ perturbative
contribution to $\Gamma^{\mu\nu}$ and, therefore, the approximate Dyson-
Schwinger equation reproduces the lowest order results of perturbation theory
for the two-photon-two-fermion irreducible vertex. Then, considering the
definitions in Eqs (85) and (86) and the lowest order (87) for the two-photon-
two-fermion irreducible vertex one can use these results in Eq. (84) to
generate the $g^{4}$ correction to the one-particle irreducible vertex under
investigation. By iterating this procedure further higher order corrections
can be computed. This interactive procedure requires that renormalization
should be performed in an order-by-order approach. At the lowest order in the
coupling constant, a naive power count shows that the two-photon-two-fermion
irreducible vertex is UV and, for massive fermions, IR finite. The
renormalization of the DSE is discussed in Sec. 8.

###  6.2 Infrared divergences and linear covariant gauges

A closer look at the loop integrals shows that, in general, $\Gamma^{\mu\nu}$
is combined with a photon propagator as follows

$\displaystyle
D_{\zeta\zeta^{\prime}}(w)\,\Gamma^{\mu\zeta^{\prime}}(p-w,\,-p-k;\,k,\,w)=$
$\displaystyle\hskip
28.45274pt=~{}-\,\left(P^{\perp}\right)_{\zeta\zeta^{\prime}}(w)\,D(w^{2})\,\Gamma^{\mu\zeta^{\prime}}(p-w,\,-p-k;\,k,\,w)$
$\displaystyle\hskip
71.13188pt-\,\frac{\xi}{w^{2}}\,\left(P^{L}\right)_{\zeta\zeta^{\prime}}(w)\,\Gamma^{\mu\zeta^{\prime}}(p-w,\,-p-k;\,k,\,w)$
(88)

where

$\left(P^{\perp}\right)_{\zeta\zeta^{\prime}}(w)=g_{\zeta\zeta^{\prime}}-\frac{w_{\zeta}w_{\zeta}^{\prime}}{w^{2}}\qquad\mbox{
and
}\qquad\left(P^{L}\right)_{\zeta\zeta^{\prime}}(w)=\frac{w_{\zeta}w_{\zeta}^{\prime}}{w^{2}}$
(89)

are the transverse and longitudinal momentum projection operators,
respectively. The exception to this rule is the last term appearing in Eq.
(84) where $\Gamma^{\mu\zeta^{\prime}}$ should be replaced by
$\Gamma^{\zeta^{\prime}}$. The gauge-dependent part can be simplified with the
help of the Ward-Takahashi identity (45), and for the last term one should use
instead Eq. (44). The contribution of the two-photon-two-fermion to the loop
integrals can be replaced by the terms that are proportional to the photon-
fermion vertex as follows

$\displaystyle
D_{\zeta\zeta^{\prime}}(w)\,\Gamma^{\mu\zeta^{\prime}}(p-w,\,-p-k;\,k,\,w)=$
$\displaystyle\hskip
19.91684pt=~{}-\,\left(P^{\perp}\right)_{\zeta\zeta^{\prime}}(w)\,D(w^{2})\,\Gamma^{\mu\zeta^{\prime}}(p-w,\,-p-k;\,k,\,w)$
$\displaystyle\hskip
34.14322pt-\,\frac{\xi}{w^{4}}\,w_{\zeta}\,\bigg{(}\Gamma^{\mu}(p-w,\,-p-k+w;\,k)-\Gamma^{\mu}(p,\,-p-k;\,k)\bigg{)}\
.$ (90)

This expression suggests that the problem of the infrared divergences in the
linear covariant gauges becomes more difficult when $\xi$ moves away from
zero, i.e. when one departure from the Landau gauge. However, the gauge-
dependent part of the last equation can also be viewed as constraining the
vertex for small $w$ where one can do the approximation

$\displaystyle-\,\frac{\xi}{w^{4}}\,w_{\zeta}\,\bigg{(}\Gamma^{\mu}(p-w,\,-p-k+w;\,k)-\Gamma^{\mu}(p,\,-p-k;\,k)\bigg{)}\approx$
$\displaystyle\hskip
45.52458pt\approx\frac{\xi}{w^{4}}\,w_{\zeta}w_{\iota}\,\left(\frac{\partial\Gamma^{\mu}(p,\,-p-k;\,k)}{\partial{p_{1\,\iota}}}-\frac{\partial\Gamma^{\mu}(p,\,-p-k;\,k)}{\partial{p_{2\,\iota}}}\right),$
(91)

where $\partial\Gamma^{\mu}/\partial p_{j}$ refers to differentiation with
respect to argument $j$, up to higher order terms in $w$. A naive power count
in $w$ shows that the gauge dependent term is $1/w^{2}$, reproducing the
perturbative like behaviour of the photon propagator. This power like behavior
seems to solve possible infrared divergences that are associated with the
gauge-dependent terms in the loop integrals. For the type of contributions
under discussion, the problem of the infrared divergences in the loop
integrals for the gauge dependent part is solved either if for small momenta
the photon-fermion vertex does not depend on the fermion momenta, as is
sometimes assumed, or if the derivative dependence on both fermion momenta is
the same. In both cases the r.h.s. of Eq. (91) vanishes.

Repeating the same argument for the last term in the loop integral of Eq.
(84), then

$\displaystyle-\,\frac{\xi}{w^{4}}\,w_{\zeta}\,w_{\zeta^{\prime}}\,\Gamma^{\zeta^{\prime}}(p+k+q-w,\,-p-k-q;\,w)=$
$\displaystyle\hskip
22.76228pt=-\,\frac{\xi}{w^{4}}\,w_{\zeta}\,\bigg{(}S^{-1}(p+k+q)-S^{-1}(p+k+q-w)\bigg{)}$
$\displaystyle\hskip
22.76228pt\approx\frac{\xi}{w^{4}}\,w_{\zeta}\,w_{\iota}\,\frac{\partial
S^{-1}(p+k+q)}{\partial p_{\iota}}+\cdots$ (92)

and the infrared behaviour of the gauge-dependent part reproduces the same
type of behaviour as before. Inspired on perturbation theory, one can conclude
that the infrared divergences for any linear covariant gauges should be of the
same nature as the possible infrared divergences that appear in the Landau
gauge defined by $\xi=0$.

We conclude this section noting that the combination of Eq. (84), after
contraction with $k_{\mu}\,q_{\nu}$, with the “scalar” Ward-Takahashi given in
Eq. (46) gives a lengthy sum rule that combines the fermion-photon vertex and
the fermion propagator.

###  6.3 On the two-photon-two-fermion tensor basis

The discussion of the solution for the two-photon-two-fermion Ward-Takahashi
identity in Sec. LABEL:Sec:SolWTI suggested a minimal basis of operators to
describe, in momentum space, $\Gamma^{\mu\nu}_{L}$. The minimal basis involves
eight different tensors, see Eq. (70), and was built taking into consideration
the solution of the WTI and including the set of linear operators, in the
photon momenta, that are associated with the transverse components of the
photon-fermion vertex appearing in the two-photon-two-fermion WTI. However, as
discussed in Tarrach:1975tu ; Perrottet:1973qw , see also Scherer:1996ux ;
Eichmann:2012mp  for particular cases, the most general basis of operators to
describe $\Gamma^{\mu\nu}$ requires a larger basis. For example, in
Tarrach:1975tu  a minimal basis of gauge-invariant tensors with no kinematical
singularities was constructed explicitly and it uses eighteen different
operators, that require eighteen different Lorentz scalar form factors. A
comparison of Eq. (70) with this explicit construction shows that all the
operators reported in Eq. (70) also appear in Tarrach:1975tu , either
explicitly or as linear combinations of what the author calls as $\tau_{i}$.
The minimal basis reported earlier can be extended to include some transverse
components following the procedure devised in Bardeen:1968ebo  and worked out
in Tarrach:1975tu  that requires the replacement

$\mathcal{O}^{\mu\nu}(k,q)~{}\longmapsto~{}\left(g^{\mu\mu^{\prime}}-\frac{k^{\mu}k^{\mu^{\prime}}}{k^{2}}\right)\left(g^{\nu\nu^{\prime}}-\frac{q^{\nu}q^{\nu^{\prime}}}{q^{2}}\right)~{}\mathcal{O}_{\mu^{\prime}\nu^{\prime}}(k,q)\
,$ (93)

where $\mathcal{O}^{\mu\nu}(k,q)$ stands for a generic operator that complies
with Bose symmetry, and, when necessary, multiplied by Lorentz invariant
kinematical factors to ensure that the tensor basis is free of kinematical
singularities. Even after performing such a construction, the total number of
operators is under eighteen.

###  6.4 An approximate solution of the DSE for heavy fermions

In this section, we would like to discuss an approximate solution to Eq. (84).
In order to do so, we try to identify the dominant form factors for
$\Gamma_{\mu\nu}$. This is difficult to achieve in a completely satisfactory
way, but we turn our attention to the low momentum solutions given in Eqs (67)
and (69). With the aim of simplifying the DSE (84) we take the heavy fermion
limit and consider that

$S(p)\approx\frac{1}{m}\ .$ (94)

Then, in the heavy fermion limit the DSE becomes

$\displaystyle\Gamma^{\mu\nu}(p,\,-p-k-q;\,k,\,q)~{}\approx$
$\displaystyle\quad\approx
i\,\frac{g^{2}}{m}~{}\int\frac{d^{4}w}{(2\,\pi)^{4}}~{}D_{\zeta\zeta^{\prime}}(w)~{}\gamma^{\zeta}$
$\displaystyle\hskip
19.91684pt\bigg{\\{}~{}\frac{1}{m^{2}}~{}\Gamma^{\mu}(p-w,\,-p-k+w;\,k)~{}$
$\displaystyle\hskip 85.35826pt\Gamma^{\nu}(p+k-w,\,-p-k-q+w;\,q)$
$\displaystyle\hskip
113.81102pt~{}\Gamma^{\zeta^{\prime}}(p+k+q-w,\,-p-k-q;\,w)$
$\displaystyle\hskip
19.91684pt~{}+~{}\frac{1}{m^{2}}~{}~{}\Gamma^{\nu}(p-w,\,-p-q+w;\,q)~{}$
$\displaystyle\hskip 85.35826pt\Gamma^{\mu}(p+q-w,\,-p-k-q+w;\,k)$
$\displaystyle\hskip
113.81102pt~{}\Gamma^{\zeta^{\prime}}(p+k+q-w,\,-p-k-q;\,w)$
$\displaystyle\hskip
19.91684pt~{}+~{}\frac{1}{m}~{}~{}\Gamma^{\mu}(p-w,\,-p-k+w;\,k)~{}$
$\displaystyle\hskip
85.35826pt~{}\Gamma^{\nu\zeta^{\prime}}(p+k-w,\,-p-k-q;\,q,\,w)$
$\displaystyle\hskip
19.91684pt~{}+~{}\frac{1}{m}~{}~{}\Gamma^{\nu}(p-w,\,-p-q+w;\,q)~{}$
$\displaystyle\hskip
85.35826pt~{}\Gamma^{\mu\zeta^{\prime}}(p+q-w,\,-p-k-q;\,k,\,w)$
$\displaystyle\hskip
19.91684pt~{}+~{}\frac{1}{m}~{}~{}\Gamma^{\mu\nu}(p-w,\,-p-k-q+w;\,k,\,q)~{}$
$\displaystyle\hskip
85.35826pt~{}\Gamma^{\zeta^{\prime}}(p+k+q-w,\,-p-k-q;\,w)~{}\bigg{\\}}$
$\displaystyle\quad\approx
i\,\frac{g^{2}}{m^{2}}~{}\int\frac{d^{4}w}{(2\,\pi)^{4}}~{}D_{\zeta\zeta^{\prime}}(w)~{}\gamma^{\zeta}~{}\bigg{\\{}$
$\displaystyle\hskip
45.52458pt~{}\gamma^{\mu}~{}\Gamma^{\nu\zeta^{\prime}}(p+k-w,\,-p-k-q;\,q,\,w)$
$\displaystyle\hskip
73.97733pt~{}+~{}\gamma^{\nu}~{}\Gamma^{\mu\zeta^{\prime}}(p+q-w,\,-p-k-q;\,k,\,w)$
$\displaystyle\hskip
102.43008pt~{}+~{}~{}\Gamma^{\mu\nu}(p-w,\,-p-k-q+w;\,k,\,q)~{}\gamma^{\zeta^{\prime}}\bigg{\\}}$
(95)

where in the last line we took only the leading term in $1/m$ and set the
photon-fermion vertex to its tree level expression. For the approximation
considered here, the terms proportional to the derivatives of $B$ in Eq. (69)
can be ignored and, in this case, the solution of the WTI requires a single
tensor operator

$\Gamma^{\mu\nu}_{L}(p,\,-p-k-q;\,k,\,q)~{}~{}=~{}~{}-\,\,2\,\,A^{\prime}(p^{2})\Big{(}\not{p}\,g^{\mu\nu}+p^{\mu}\,\gamma^{\nu}\,+\,p^{\nu}\,\gamma^{\mu}\Big{)}$
(96)

and suggests to write in the r.h.s. of the DSE

$\displaystyle\Gamma^{\mu\nu}(p,\,-p-k-q;\,k,\,q)=F_{0}\,\Bigg{(}\not{p}\,g^{\mu\nu}+p^{\mu}\,\gamma^{\nu}\,+\,p^{\nu}\,\gamma^{\mu}\Bigg{)}$
$\displaystyle+~{}F_{1}\,\left(g^{\mu\mu^{\prime}}-\frac{k^{\mu}k^{\mu^{\prime}}}{k^{2}}\right)\left(g^{\nu\nu^{\prime}}-\frac{q^{\nu}q^{\nu^{\prime}}}{q^{2}}\right)\,\Bigg{(}\not{p}\,g_{\mu^{\prime}\nu^{\prime}}+p_{\mu^{\prime}}\,\gamma_{\nu^{\prime}}\,+\,p_{\nu^{\prime}}\,\gamma_{\mu^{\prime}}\Bigg{)}$
(97)

where $F_{0}$ and $F_{1}$ are Lorentz scalar functions of $p^{2}$,
$(p+k+q)^{2}$, $k^{2}$ and $q^{2}$. To proceed, let us ignore in Eq. (97) the
transverse part of $\Gamma^{\mu\nu}$, set $F_{0}=-2A^{\prime}(p^{2})$, and
take the limit $k=q=0$ of Eq. (95). Then, this last equation can written as

$\displaystyle\Gamma^{\mu\nu}(p,\,-p;\,0,\,0)~{}\approx\,2\,i\,\frac{g^{2}}{m^{2}}~{}\int\frac{d^{4}w}{(2\,\pi)^{4}}~{}D(w^{2})~{}A^{\prime}((p-w)^{2})$
$\displaystyle\bigg{\\{}3\,\,{g}^{\mu\nu}\,\,\not{p}~{}+~{}{g}^{\mu\nu}\,\,\not{w}~{}-~{}2\,\,{\gamma}^{\nu}\,{p}^{\mu}~{}-~{}2\,\,{\gamma}^{\mu}\,{p}^{\nu}~{}+~{}{\gamma}^{\mu}\,{w}^{\nu}~{}+~{}{\gamma}^{\nu}\,{w}^{\mu}$
$\displaystyle\qquad-4~{}\frac{({p}{w})}{\,{w}^{2}}\,\,\not{w}\,\,{g}^{\mu\nu}~{}-~{}5\,\,\not{w}\,\,\frac{{p}^{\mu}\,{w}^{\nu}}{{w}^{2}}~{}-~{}5\,\,\not{w}\,\,\frac{{p}^{\nu}\,{w}^{\mu}}{{w}^{2}}$
$\displaystyle\quad~{}-~{}2\,\,\not{p}\,\,\frac{{w}^{\mu}\,{w}^{\nu}}{{w}^{2}}~{}+~{}12\,\,\not{w}\,\,\frac{{w}^{\mu}\,{w}^{\nu}}{{w}^{2}}~{}~{}+~{}\frac{({p}{w})}{\,{w}^{2}}\,\,{\gamma}^{\mu}\,{w}^{\nu}~{}+~{}\frac{({p}{w})}{{w}^{2}}\,\,{\gamma}^{\nu}\,{w}^{\mu}$
$\displaystyle\quad~{}-~{}i\,\,\frac{\,{w}^{\nu}\,}{\,{w}^{2}}\,\,{\epsilon}^{\eta\mu\alpha\beta}\,{p}_{\alpha}\,{w}_{\beta}\,\,{\gamma}_{\eta}\,{\gamma}_{5}~{}-~{}i\,\,\frac{\,{w}^{\mu}\,}{\,{w}^{2}}\,\,{\epsilon}^{\eta\nu\alpha\beta}\,{p}_{\alpha}\,{w}_{\beta}\,\,{\gamma}_{\eta}\,{\gamma}_{5}\bigg{\\}}$
(98)

and the computation of the two-photon-two-fermion irreducible Green function
requires the knowledge of the fermion propagator, i.e. of $A$, that calls for
a solution of the coupled equations considered in the present work or a model
of the fermion propagator. However, the analysis of Eq. (98) shows that, in
the approximation considered, the tensor structure of the Green function is a
linear combination of the operators

$\not{p}\,\,g^{\mu\nu},\qquad
p^{\mu}\gamma^{\nu}+p^{\nu}\gamma^{\mu},\qquad\not{p}\,\,p^{\mu}p^{\nu}\qquad\mbox{
and }\qquad
i\,\,\epsilon^{\eta\mu\nu\alpha}\,\,p_{\alpha}\,\,\gamma_{\eta}\,\gamma_{5}$
(99)

with different coefficients that are functions of $p^{2}$. The description of
$\Gamma^{\mu\nu}$ is certainly more elaborated than the ansatz (96) used to
feed the r.h.s. of the truncated DSE.

For the Schwinger model, the authors of Radozycki:1998pf  also derived a
Dyson-Schwinger equation for the two-photon-two-fermion one particle
irreducible Green function and built a self-consistence solution of the
corresponding equation in coordinate space. We postpone the solution of the
above approximate Dyson-Schwinger equation and, therefore, we are unable to
comment on similarities and differences.

##  7 Two-photon-two-fermion vertex and its low energy contribution to the
effective photon-fermion vertex

The Dyson-Schwinger equation for the photon-fermion vertex, see Eq. (40),
shows that this vertex gets a contribution coming from the two-photon-two-
fermion irreducible diagram that reads

$\displaystyle\widetilde{\widetilde{\Gamma}}\,^{\mu}(p,\,-p-k;\,k)=$
$\displaystyle=i\,g^{2}\,\int\frac{d^{4}q}{(2\,\pi)^{4}}~{}D_{\zeta\zeta^{\prime}}(q)~{}\Bigg{\\{}\gamma^{\zeta}\,S(p-q)\,{\Gamma}^{\zeta^{\prime}\mu}(p-q,\,-p-k;\,q,\,k)\Bigg{\\}}$
$\displaystyle\approx~{}i\,g^{2}\,\int\frac{d^{4}q}{(2\,\pi)^{4}}~{}D_{\zeta\zeta^{\prime}}(q)~{}\Bigg{\\{}\gamma^{\zeta}\,S(p-q)\,{\Gamma}^{\zeta^{\prime}\mu}(p-q,\,-p;\,q,\,0)\Bigg{\\}}$
$\displaystyle\qquad+~{}\mathcal{O}(k)\ .$ (100)

Its low energy limit, defined as the soft photon limit given by taking $k=0$,
requires the knowledge of the two-photon-two-fermion vertex in the soft photon
limit, whose longitudinal part can be found in Eq. (66). For the Landau gauge,
where the photon propagator is transverse, the only terms in Eq. (66) that
contribute to the vertex are

$\displaystyle\Gamma^{\mu\nu}(p,\,-p-k;\,k,\,0)=$
$\displaystyle=2~{}g^{\mu\nu}~{}\Bigg{\\{}\,\lambda_{3}\left(p^{2},\,p^{2},\,0\right)~{}-~{}2\,\not{q}\,\lambda_{2}\left(p^{2},\,p^{2},\,0\right)~{}-~{}2\,\not{p}\,\lambda_{2}\left(p^{2},\,p^{2},\,0\right)\bigg{\\}}$
$\displaystyle=2~{}g^{\mu\nu}~{}\Bigg{\\{}\,B^{\prime}\left(p^{2}\right)~{}-~{}\,\not{q}\,A^{\prime}\left(p^{2}\right)~{}-~{}\,\not{p}\,A^{\prime}\left(p^{2}\right)\bigg{\\}}$
(101)

where to write the last line we used Eqs (55), $A^{\prime}(x)=dA(x)/dx$,
$B^{\prime}(x)=dB(x)/dx$ and assumed that the fermion functions $A$ and $B$
are differentiable. Then, inserting the photon propagator decomposition, see
Eq. (34), in Eq. (100) it follows that

$\displaystyle\widetilde{\widetilde{\Gamma}}\,^{\mu}(p,\,-p-k;\,k)=-\,2\,i\,g^{2}\,\int\frac{d^{4}q}{(2\,\pi)^{4}}~{}D(q^{2})\left(g_{\zeta}^{\mu}-\frac{q_{\zeta}q^{\mu}}{q^{2}}\right)~{}$
$\displaystyle\qquad\qquad\qquad\gamma^{\zeta}\,S(p-q)\,\bigg{(}B^{\prime}\left(p^{2}\right)-\,\left(\not{q}+\not{p}\right)\,A^{\prime}\left(p^{2}\right)\bigg{)}+\mathcal{O}(k)\
.$ (102)

and the contribution of the two-photon-two-fermion vertex to the photon-
fermion vertex can be expressed in terms of the fermion propagator functions.
Significant contributions to the photon-fermion vertex can occur when the
derivatives of the $A$ and $B$ become large. Moreover, the decomposition of
Eq. (102) into the Dirac basis shows that the two-photon-two-fermion
contribution to the photon-fermion vertex results in electric and magnetic
dipole-like operators that are proportional to the derivatives of the fermion
propagator functions. Indeed, performing the Dirac algebra, in the limit of
$k\rightarrow 0$, Eq. (102) results in terms involving the operators

$\gamma^{\mu},\qquad p^{\mu},\qquad\sigma^{\mu\alpha}p_{\alpha}\quad\mbox{ and
}\quad p^{\mu}\,\not{p}\ .$ (103)

Corrections in $k$ allow for new classes of operators that are not listed and
that require either $A^{\prime}$, $B^{\prime}$ or either order derivatives of
these functions.

The analysis of the first term in Eq. (40) shows that the photon-fermion
vertex in soft photon limit is not determined only by the fermion propagator
functions $A$ and $B$ and its derivatives. Indeed, the first line in Eq. (40)
includes a contribution of the vertex himself that requires only its soft
photon limit and a second term that calls for the full vertex. It is only the
contribution of the two-photon-two-fermion irreducible diagram that, in the
soft photon limit, can be written in terms of the fermion propagator
functions.

##  8 Renormalization

The equations derived so far involve only bare quantities and, therefore, it
remains to write all the Eqs in term of physical quantities. In order to do
so, one has to go through the renormalization procedure. For completeness,
this is worked out in this section. Herein, the use of the index “phys” refers
to physical quantities, while the bare quantities have no index.

For QED the renormalization constants, generically named $Z$, are

$\displaystyle
A_{\mu}=Z^{\frac{1}{2}}_{3}\,A^{(phys)}_{\mu},\quad\psi=Z^{\frac{1}{2}}_{2}\,\psi^{(phys)},\quad
g=\frac{Z_{1}}{Z_{2}\,Z^{\frac{1}{2}}_{3}}g^{(phys)},$ $\displaystyle
m=\frac{Z_{0}}{Z_{2}}\,m^{(phys)}\quad\mbox{and}\quad\xi=Z_{3}\,\xi^{(phys)}.$
(104)

These definitions imply in the following relations between bare and
renormalized propagators

$D_{\mu\nu}(k^{2})=Z_{3}\,D^{(phys)}_{\mu\nu}(k^{2})\qquad\mbox{ and }\qquad
S(p)=Z_{2}\,S^{(phys)}(p)$ (105)

and, from the definitions Eqs (128) and (132), one has

$\Gamma_{\mu}=\,\frac{\Gamma^{(phys)}_{\mu}}{Z_{1}}\qquad\mbox{ and
}\qquad\Gamma_{\mu\nu}=\,\frac{Z_{2}}{Z^{2}_{1}}\,\Gamma^{(phys)}_{\mu\nu}\ .$
(106)

On the other hand, the Ward identities (44) and (45) imply that

$\Gamma_{\mu}=\frac{\Gamma^{(phys)}_{\mu}}{Z_{2}}\qquad\mbox{ and
}\qquad\Gamma_{\mu\nu}=\frac{\Gamma^{(phys)}_{\mu\nu}}{Z_{2}}$ (107)

and, therefore,

$Z_{1}=Z_{2}\ .$ (108)

The renormalization program for QED requires three renormalization constants,
from three renormalization conditions computed from the fermion gap equation
(31), that provide $Z_{1}(=Z_{2})$ and $Z_{0}$, and the equivalent photon gap
equation (38), which determines $Z_{3}$. Note that with the above definitions,
the coupling constant is renormalized by $\sqrt{Z_{3}}$ and the combination
$g^{2}\,D(k^{2})$ is independent of the renormalization scale.

Before proceeding any further, let us write the Dyson-Schwinger equations in
their renormalized form where all quantities are finite. To simplify the
notation, from now on, in the renormalized equations we will omit the suffix
$(phys)$. The renormalized fermion gap equation (31) reads

$\displaystyle S^{-1}(p)$ $\displaystyle=$ $\displaystyle
Z_{2}\,\not{p}-Z_{0}\,m$ (109) $\displaystyle\hskip
17.07182pt-\,i\,g^{2}\,Z_{2}\,\int\frac{d^{4}k}{(2\,\pi)^{4}}~{}D_{\mu\nu}(k)~{}\Big{[}\gamma^{\mu}~{}S(p-k)~{}{\Gamma}^{\nu}(p-k,-p;k)\Big{]}$
$\displaystyle=$ $\displaystyle Z_{2}\,\not{p}-Z_{0}\,m-\,i\,g^{2}\,\Sigma(p)\
,$

while the renormalized photon gap equation (38) is given by

$\displaystyle\frac{1}{D(k^{2})}$ $\displaystyle=$ $\displaystyle
Z_{3}\,k^{2}-i\,\frac{g^{2}}{3}\,Z_{2}\,\int\frac{d^{4}p}{(2\,\pi)^{4}}~{}\text{Tr}\Big{[}\gamma_{\mu}\,S(p)\,{\Gamma}^{\mu}(p,-p+k;-k)\,S(p-k)\Big{]}$
(110) $\displaystyle=$ $\displaystyle
Z_{3}\,k^{2}-i\,\frac{g^{2}}{3}\,k^{2}\,\Pi(k^{2})$

where $\Sigma(p)$ and $\Pi(k^{2})$ are the fermion and the photon self-
energies, respectively. Then, the renormalized Dyson-Schwinger equation for
the vertex (40) is

$\displaystyle{\Gamma}^{\mu}(p,\,-p-k;\,k)=Z_{2}\,\gamma^{\mu}~{}+~{}i\,g^{2}\,Z_{2}\,\int\frac{d^{4}q}{(2\,\pi)^{4}}~{}D_{\zeta\zeta^{\prime}}(q)$
$\displaystyle\Bigg{\\{}\gamma^{\zeta}\,S(p-q)\,{\Gamma}^{\mu}(p-q,\,-p-k+q;\,k)\,S(p+k-q)\,{\Gamma}^{\zeta^{\prime}}(p+k-q,\,-p-k;q)$
$\displaystyle\qquad\qquad~{}+~{}\gamma^{\zeta}\,S(p-q)\,{\Gamma}^{\zeta^{\prime}\mu}(p-q,\,-p-k;\,q,\,k)\Bigg{\\}}$
$\displaystyle=~{}Z_{2}\,\gamma^{\mu}~{}+~{}i\,g^{2}\,\Delta\Gamma^{\mu}(p,k)\
.$ (111)

It is convenient to introduce some extra notation and write the inverse of the
renormalized fermion propagator as

$S^{-1}(p)=A(p^{2})\,\not{p}-B(p^{2})$ (112)

and, therefore,

$S(p)=Z(p^{2})\,\frac{\not{p}+M(p^{2})}{p^{2}-M^{2}(p^{2})+i\,\epsilon}$ (113)

with $Z(p^{2})=1/A(p^{2})$ and $M(p^{2})=B(p^{2})/A(p^{2})$. It is also common
practice to write the fermion self-energy as

$\Sigma(p)=\Sigma_{v}(p^{2})\,\not{p}+\Sigma_{s}(p^{2})\ ,$ (114)

that allows to split the fermion gap equation into a vectorial and a scalar
part as follows

$\displaystyle
A(p^{2})=Z_{2}\,-\,i\,\frac{g^{2}}{p^{2}}\,Z_{2}\,\int\frac{d^{4}k}{(2\,\pi)^{4}}~{}D_{\mu\nu}(k)~{}\text{Tr}\Big{[}\,\not{p}~{}\gamma^{\mu}~{}S(p-k)~{}{\Gamma}^{\nu}(p-k,-p;k)\,\Big{]}$
$\displaystyle\hskip 28.45274pt=Z_{2}-\,i\,g^{2}\,\Sigma_{v}(p^{2})\ ,$ (115)
$\displaystyle
B(p^{2})=Z_{0}\,m+\,i\,g^{2}\,Z_{2}\,\int\frac{d^{4}k}{(2\,\pi)^{4}}~{}D_{\mu\nu}(k)~{}\text{Tr}\Big{[}\,\gamma^{\mu}~{}S(p-k)~{}{\Gamma}^{\nu}(p-k,-p;k)\,\Big{]}$
$\displaystyle\hskip 28.45274pt=Z_{0}\,m+i\,g^{2}\,\Sigma_{s}(p^{2})\ .$ (116)

One way to define the renormalization constants is choosing renormalization
conditions implemented using Eqs (110), (115) and (116). Thus, demanding that

$A(\mu^{2})=1,\qquad B(\mu^{2})=m\qquad\mbox{ and }\qquad
D(\mu^{2})=\frac{1}{\mu^{2}}$ (117)

it comes that

$\displaystyle Z_{2}$ $\displaystyle=$ $\displaystyle
1\,+\,i\,g^{2}\,\Sigma_{v}(\mu^{2})\ ,$ (118) $\displaystyle Z_{0}$
$\displaystyle=$ $\displaystyle 1\,-i\,g^{2}\,\Sigma_{s}(\mu^{2})/m\ ,$ (119)
$\displaystyle Z_{3}$ $\displaystyle=$ $\displaystyle
1\,+\,\frac{i}{3}\,g^{2}\,\Pi(\mu^{2}),$ (120)

recall that $\Sigma_{v}$, $\Sigma_{s}$, $\Pi$ include a $Z_{2}$ factor in its
definition, and the renormalized equations for the propagators are

$\displaystyle
A(p^{2})=1-\,i\,g^{2}\,\bigg{(}\Sigma_{v}(p^{2})-\Sigma_{v}(\mu^{2})\bigg{)}\
,$ (121) $\displaystyle
B(p^{2})=\,m+i\,g^{2}\,\bigg{(}\Sigma_{s}(p^{2})-\Sigma_{s}(\mu^{2})\bigg{)}\
,$ (122)
$\displaystyle\frac{1}{D(k^{2})}=k^{2}\bigg{(}1-\frac{i}{3}\,g^{2}\,\Big{(}\Pi(k^{2})-\Pi(\mu^{2})\Big{)}\bigg{)}\
.$ (123)

Here, we have used the same renormalization scale for the fermion and the
photon, but one could use instead different mass scales for the
renormalization of fermionic and bosonic fields.

##  9 Summary, Dicussions and Conclusions

In the current work, the Dyson-Schwinger equations in Minkowski spacetime are
considered to address QED in a general linear covariant gauge. The DSE
equations are an infinite tower of integral equations and, therefore, only
approximate solutions can be built by dealing with truncated versions of the
full set of equations. A minimal set of integral equations to tackle both the
propagators and the photon-fermion vertex in QED is provided. These equations
for the two-point and three-point Green functions are exact but, to build a
closed set of equations, one also consider a truncated Dyson-Schwinger
equation for the two-photon-two-fermion one-particle irreducible vertex.

The Dyson-Schwinger equations can, in principle, be solved for any regime of
the theory that includes the small and strong coupling regimes and, therefore,
allow to solve QED beyond perturbation theory. The analysis of the truncated
two-photon-two-fermion DSE in a perturbative-type of solution, shows that it
reproduces the lowest order perturbative solution, as expected. Moreover, by
combining the various integral equations derived, it is possible to build an
iterative procedure that allows to compute higher order corrections in the
coupling constant, mimicking the perturbative expansion. More, we were able to
solve the truncated integral equation for $\Gamma^{\mu\nu}$ for a particular
kinematical limit, in the heavy quark model, feeding the equation with a
simplified model of this vertex. The closed set of equations considered allow
to investigate the solutions of QED for the propagators, for the photon-
fermion vertex and for the two-photon-two-fermion vertex and to study their
gauge dependence. In particular, the study of the gauge dependence can be
confronted with the results of the Landau-Khalatnikov-Fradkin transformations
Landau:1955zz ; Fradkin:1955jr ; DeMeerleer:2018txc ; DeMeerleer:2019kmh ;
DallOlio:2021njq  as a test to the truncation itself.

Along with the derivation of the Dyson-Schwinger equations, the Ward-Takahashi
identities for the photon-fermion and the two-photon-two-fermion vertices are
also derived. As discussed, the WTI for the photon-fermion vertex fixes its
longitudinal component relative to the photon momentum, providing the so-
called Ball-Chiu vertex. In the Ball-Chiu vertex the form factors are
functions of the fermion propagators. The compatibility of the DSE for the
vertex (40) with the WTI (44) is, in practise, difficult to achieve in a
solution of the vertex equation. Given that the WTI results from the gauge
symmetry directly, set the longitudinal part of the vertex, a solution of (40)
that does not spoils gauge symmetry can be built by considering only its
transverse component, i.e. by taking (52) - (55) for $\Gamma^{\mu}_{L}$ and
solve the equation resulting from contracting (40) with the transverse
operator $P^{\perp}_{\mu\nu}(k)$. In this way, a dynamical equation for the
transverse part of the photon-fermion vertex is engineered keeping its
longitudinal part compatible with the WTI.

The Ward-Takahashi identity for the two-photon-two-fermion vertex calls for
the full photon-fermion vertex and, this vertex being more complex that the
fermion propagators, make its analysis rather involved. However, we are able
to solve this WTI exactly for the longitudinal component, in the low photon
momenta limit. For the soft photon limit, the solution is given in Eq. (66)
and writes $\Gamma^{\mu\nu}_{L}$ in terms of the photon-fermion vertex
longitudinal form factors, that are functions of the fermion propagator
functions $A$ and $B$ themselves. A solution of the WTI keeping only its
linear terms in the photon momenta can be found in (69), suggesting a minimal
basis to describe the two-photon-two-fermion vertex. Further, the WTI for any
kinematics is solved and its solution determines $\Gamma^{\mu\nu}_{L}$ for a
general configuration of momenta, see Eqs (72) and (75). In what concerns the
transverse part of $\Gamma^{\mu\nu}$, the proposal is to computed it by
solving the truncated DSE for the vertex, i.e. Eq. (84) and, indeed, an
approximate solution is built that include further operators than the original
ansatz used to build the solution.

The compatibility of the DSE for the two-photon-two-fermion vertex, i.e. Eq.
(84), with gauge symmetry, expressed through the WTI in Eq. (45), is similar
to the computation of the solution for the vertex equation (40) that is
compatible with the WTI (44). Once more this can be solved by setting
$\Gamma^{\mu\nu}_{L}$ as given by the solution of the two-photon-two-fermioin
WTI identity, and replacing Eq. (84) by its transverse part that results from
contractions with projection operators

$\Gamma_{T}^{\mu\nu}(p,-p-k-q;k,q)=P^{\perp}\,{}^{\mu}\,_{\mu^{\prime}}(k)~{}~{}P^{\perp}\,{}^{\nu}\,_{\nu^{\prime}}(q)~{}~{}\Gamma^{\mu^{\prime}\nu^{\prime}}(p,-p-k-q;k,q)$
(124)

The complexity of the closed set of integral equations prevent us from
attempting to solve the set of coupled equations for the various Green
functions. However, a perturbative treatment of the equations that were
derived recover, at least to lhe lowest order in the coupling constant, the
results of perturbation theory. Furthermore, we recall the reader that by
solving the Ward-Takahashi identities the longitudinal parts of the photon-
fermion and two-photon-two-fermion vertices are determined exactly. The
computation of solutions of the integral equations, together with its
phenomenological implications, explored via the corresponding Bethe-Salpeter
equation, Faddeev equations, etc., will be the subject of future publications,
.

## Acknowledgments

This work was partly supported by the FCT – Fundação para a Ciência e a
Tecnologia, I.P., under Projects Nos. UIDB/04564/2020 and UIDP/04564/2020. H.
L. Macedo acknowledges financial support via the Research Fellowship - LUGUS
781687 within the R&D Unit CFisUC - Centro de Física de Universidade de
Coimbra, reference: UIDB/04564/2020, financed by FCT - Fundação para a Ciência
e Tecnologia. R. C. Terin acknowledges the Technological Institute of
Aeronautics (ITA) for the hospitality during his Post-Doc financed by the
National Council for Scientific and Technological Development (CNPq) being
supervised at that period by Prof. Tobias Frederico (ITA).

##  Appendix A Decomposing the connected Green’s functions

In this appendix, we discuss the decomposition of the connected Green
functions that are required in the current work. Starting from the last
relation given in the set of Eqs (10) and taking its second derivative with
respect to $A_{c}$ after $\eta$, followed by some algebra, and after setting
the sources to zero, one arrives at

$\displaystyle\int
d^{4}w_{1}\,d^{4}w_{2}~{}\frac{\delta^{2}\Gamma}{\delta\psi_{cl,\,\iota}(w_{1})\,\delta\bar{\psi}_{cl,\,\alpha}(x)}\,\frac{\delta^{2}\Gamma}{\delta
A_{cl,\,\nu}(z)\,\delta
A_{cl,\,\nu^{\prime}}(w_{2})}\,\frac{\delta^{3}W}{\delta
J^{\nu^{\prime}}(w_{2})\,\delta\eta_{\beta}(y)\,\delta\bar{\eta}_{\iota}(w_{1})}=$
$\displaystyle\hskip 28.45274pt=\int d^{4}w~{}\frac{\delta^{3}\Gamma}{\delta
A_{cl,\,\nu}(z)\,\delta\psi_{cl,\,\iota}(w)\,\delta\bar{\psi}_{cl,\,\alpha}(x)}\,\frac{\delta^{2}W}{\delta\eta_{\beta}(y)\,\delta\bar{\eta}_{\iota}(w)}\
.$ (125)

This equation can be solved for the connected Green function using the
orthogonality relations (11) - (13) and gives

$\displaystyle\frac{\delta^{3}W}{\delta
J_{\mu}(z)~{}\delta\eta_{\beta}(y)~{}\delta\bar{\eta}_{\alpha}(x)}=\int
d^{4}w_{1}~{}d^{4}w_{2}~{}d^{4}w_{3}$
$\displaystyle\qquad\qquad\qquad\frac{\delta^{2}W}{\delta J_{\mu}(z)\,\delta
J^{\mu^{\prime}}(w_{1})}\,\frac{\delta^{2}W}{\delta\eta_{\alpha^{\prime}}(w_{2})\,\delta\bar{\eta}_{\alpha}(x)}\frac{\delta^{2}W}{\delta\eta_{\beta}(y)\,\delta\bar{\eta}_{\beta^{\prime}}(w_{3})}\,$
$\displaystyle\qquad\qquad\qquad\frac{\delta^{3}\Gamma}{\delta
A_{cl,\,\mu^{\prime}}(w_{1})~{}\delta\psi_{cl,\,\beta^{\prime}}(w_{3})~{}\delta\bar{\psi}_{cl,\,\alpha^{\prime}}(w_{2})}$
(126)

that in terms of the full fermion propagator (23) and of the full gauge boson
propagator (24), i.e. after setting the sources to zero, reads

$\displaystyle\frac{\delta^{3}W}{\delta
J^{\mu}(z)\,\delta\eta_{\beta}(y)\,\delta\bar{\eta}_{\alpha}(x)}=$
$\displaystyle=~{}g\,\int\,d^{4}w_{1}\,d^{4}w_{2}\,d^{4}w_{3}~{}D_{\mu\mu^{\prime}}(z-w_{1})~{}\bigg{[}S(x-w_{2})~{}{\Gamma}^{\mu^{\prime}}(w_{2},w_{3};w_{1})~{}S(w_{3}-y)\bigg{]}_{\alpha\beta}$
(127)

where the photon-fermion vertex was defined as

$\left.\frac{\delta^{3}\Gamma}{\delta
A_{c,\,\mu}(z)\,\delta\psi_{c,\,\beta}(y)\,\delta\bar{\psi}_{c,\,\alpha}(x)}\right|_{J=\bar{\eta}=\eta=0}=-\,g\,\Big{(}{\Gamma}^{\mu}\Big{)}_{\alpha\beta}(x,y;z)\
.$ (128)

In momentum space, given the definitions in Eqs (28), (29), and (30), the
decomposition of the photon-fermion irreducible vertex is written as

$\displaystyle\frac{\delta^{3}W}{\delta
J^{\mu}(z)\,\delta\eta_{\beta}(y)\,\delta\bar{\eta}_{\alpha}(x)}$
$\displaystyle=$ $\displaystyle
g\,\int\,\frac{d^{4}k}{(2\,\pi)^{4}}\,\frac{d^{4}p^{\prime}}{(2\,\pi)^{4}}\,\frac{d^{4}p}{(2\,\pi)^{4}}\quad
e^{-\,i\,k\,z~{}-\,i\,p^{\prime}\,x~{}+\,i\,p\,y}\quad(2\,\pi)^{4}\,\delta(p^{\prime}-p+k)~{}$
(129) $\displaystyle\hskip
56.9055ptD_{\mu\mu^{\prime}}(k)~{}\Big{[}S(p^{\prime})~{}{\Gamma}^{\mu^{\prime}}(p^{\prime},\,-\,p;\,k)~{}S(p)\Big{]}_{\alpha\beta}\
.$

The Dyson-Schwinger equation for the photon-fermion vertex (39) calls for the
three-point connected Green function, see Eq. (127), and the four-point
connected Green with two-fermion and two-gauge boson external lines. The
latter Green function can be decomposed in terms of irreducible functions
following the same procedure as for the three-point photon-fermion correlation
function. Starting from Eq. (126) and performing the required derivatives,
after some straightforward algebra and after setting all the sources to zero,
one arrives at the decomposition of the four-point correlation function in
terms of one-particle irreducible functions

$\displaystyle\frac{\delta^{4}W}{\delta J^{\mu}(z)\,\delta
J^{\nu}(w)\,\delta\eta_{\beta}(y)\,\delta\bar{\eta}_{\alpha}(x)}=$
$\displaystyle\qquad=~{}~{}-~{}g^{2}\,\int
d^{4}u_{1}\,d^{4}u_{2}\,d^{4}u_{3}\,d^{4}u_{4}~{}~{}D_{\mu\mu^{\prime}}(z-u_{1})\,D_{\nu\nu^{\prime}}(w-u_{2})$
$\displaystyle\hskip
85.35826pt\Big{[}S(x-u_{3})~{}{\Gamma}^{\mu^{\prime}\nu^{\prime}}(u_{3},u_{4};u_{1},u_{2})S(u_{4}-y)\Big{]}_{\alpha\beta}$
$\displaystyle\qquad~{}-~{}g^{2}\,\int
d^{4}u_{1}\,d^{4}u_{2}\,d^{4}u_{3}\,d^{4}u_{4}\,d^{4}u_{5}\,d^{4}u_{6}~{}~{}D_{\mu\mu^{\prime}}(z-u_{1})\,D_{\nu\nu^{\prime}}(w-u_{2})~{}D_{\bar{\nu}\bar{\mu}}(u_{4}-u_{3})$
$\displaystyle\hskip
85.35826pt\Big{[}S(x-u_{5})~{}\Gamma^{\bar{\nu}}(u_{5},u_{6};u_{4})~{}S(u_{6}-y)\Big{]}_{\alpha\beta}~{}~{}{\Gamma}^{\mu^{\prime}\nu^{\prime}\bar{\mu}}(u_{1},u_{2},u_{3})$
$\displaystyle\qquad-~{}g^{2}\,\int
d^{4}u_{1}\,d^{4}u_{2}\,d^{4}u_{3}\,d^{4}u_{4}\,d^{4}u_{5}\,d^{4}u_{6}~{}~{}D_{\mu\mu^{\prime}}(z-u_{1})\,D_{\nu\nu^{\prime}}(w-u_{2})$
$\displaystyle\hskip
56.9055pt\Big{[}S(x-u_{3})~{}{\Gamma}^{\nu^{\prime}}(u_{3},u_{5};u_{2})~{}S(u_{5}-u_{6})~{}{\Gamma}^{\mu^{\prime}}(u_{6},u_{4};u_{1})~{}S(u_{4}-y)\Big{]}_{\alpha\beta}$
$\displaystyle\qquad-~{}g^{2}\,\int
d^{4}u_{1}\,d^{4}u_{2}\,d^{4}u_{3}\,d^{4}u_{4}\,d^{4}u_{5}\,d^{4}u_{6}~{}~{}D_{\mu\mu^{\prime}}(z-u_{1})\,D_{\nu\nu^{\prime}}(w-u_{2})$
$\displaystyle\hskip
56.9055pt\Big{[}S(x-u_{3})~{}{\Gamma}^{\mu^{\prime}}(u_{3},u_{5};u_{1})~{}S(u_{5}-u_{6})~{}{\Gamma}^{\nu^{\prime}}(u_{6},u_{4};u_{2})~{}S(u_{4}-y)\Big{]}_{\alpha\beta}$
(130)

where we have used the relation

$\displaystyle\frac{\delta^{3}W}{\delta J^{\mu}(x)~{}\delta
J^{\nu}(y)~{}\delta J^{\zeta}(z)}=g~{}\int
d^{4}u_{1}\,d^{4}u_{2}\,d^{4}u_{3}~{}$ $\displaystyle\hskip
56.9055ptD_{\mu\mu^{\prime}}(x-u_{1})~{}D_{\nu\nu^{\prime}}(y-u_{2})~{}D_{\zeta\zeta^{\prime}}(z-u_{3})~{}\Gamma^{\mu^{\prime}\nu^{\prime}\zeta^{\prime}}(u_{1},u_{2},u_{3})$
(131)

that follows from taking functional derivatives of Eq. (11), using this
orthogonality relation and setting the sources to zero, and where the notation

$\displaystyle\frac{\delta^{4}\Gamma}{\delta A_{cl,\,\mu}(x)~{}\delta
A_{cl,\,\nu}(y)~{}\delta\psi_{cl,\,\beta}(z)~{}\delta\bar{\psi}_{cl,\,\alpha}(w)}$
$\displaystyle=$
$\displaystyle~{}-\,g^{2}\,\Big{(}{\Gamma}^{\mu\,\nu}\Big{)}_{\alpha\beta}(w,z;x,y)\
,$ (132) $\displaystyle\frac{\delta^{3}\Gamma}{\delta A_{cl,\,\mu}(x)~{}\delta
A_{cl,\,\nu}(y)~{}\delta A_{cl,\,\zeta}(z)}$ $\displaystyle=$
$\displaystyle~{}-\,g~{}{\Gamma}^{\mu\,\nu\,\zeta}(x,y,z)$ (133)

was used, together with the bose symmetry of the vertices, i..e, that the
vertices with multiple bosonic lines are symmetric under interchange of pair
of indices. In QED, Furry’s theorem tell us that the three photon vertex
vanishes and the contribution proportional to $\Gamma^{\mu\nu\zeta}$ can be
ignored. Following the same rules as those used to write Eq. (127) in momentum
space, see Eq. (129), the decomposition of the connected four-point Green
function (130) in momentum space reads

$\displaystyle\frac{\delta^{4}W}{\delta J^{\mu}(z)\,\delta
J^{\nu}(w)\,\delta\eta_{\beta}(y)\,\delta\bar{\eta}_{\alpha}(x)}~{}=$
$\displaystyle=~{}-~{}g^{2}\,\int\,\frac{d^{4}k}{(2\,\pi)^{4}}\,\frac{d^{4}q}{(2\,\pi)^{4}}\,\frac{d^{4}p^{\prime}}{(2\,\pi)^{4}}\,\frac{d^{4}p}{(2\,\pi)^{4}}~{}$
$\displaystyle\qquad
e^{-\,i\,k\,z~{}-i\,q\,w~{}-\,i\,p^{\prime}\,x~{}+\,i\,p\,y}~{}(2\,\pi)^{4}~{}\delta(p^{\prime}-p+q+k)~{}~{}D_{\mu\mu^{\prime}}(k)\,D_{\nu\nu^{\prime}}(q)$
$\displaystyle\qquad\Bigg{\\{}~{}\Big{[}S(p^{\prime})~{}{\Gamma}^{\nu^{\prime}\mu^{\prime}}(p^{\prime},-p;q,k)~{}S(p)\Big{]}_{\alpha\beta}$
$\displaystyle\hskip
36.98866pt+~{}\Big{[}S(p^{\prime})~{}{\Gamma}^{\nu^{\prime}}(p^{\prime},-p^{\prime}-q;q)~{}S(p^{\prime}+q)~{}{\Gamma}^{\mu^{\prime}}(p^{\prime}+q,-p;k)~{}S(p)\Big{]}_{\alpha\beta}$
$\displaystyle\hskip
36.98866pt+~{}\Big{[}S(p^{\prime})~{}{\Gamma}^{\mu^{\prime}}(p^{\prime},-p^{\prime}-k;k)~{}S(p^{\prime}+k)~{}{\Gamma}^{\nu^{\prime}}(p-q,-p;q)~{}S(p)\Big{]}_{\alpha\beta}~{}\Bigg{\\}}\
.$ (134)

Recall that Eqs (130) and (134) are valid only for
$J=\eta=\bar{\eta}=A_{c}=\psi_{c}=\bar{\psi}_{c}=0$.

##  Appendix B Ward-Takahashi identities in momentum space

The Ward-Takahashi identities (41) and (42) are given in coordinate space,
whose translation into momentum space (44) and (45) is discussed now. Starting
with the WTI for the photon-fermion vertex (41) that in terms of the connected
Green functions, i.e. with respect to derivatives of the generating functional
$W$, see Eq. (4) for definitions, reads

$\displaystyle\frac{1}{g\,\xi}\,\square_{x}\,\partial^{\mu}_{x}\,\left(\frac{\delta^{3}W}{\delta
J^{\mu}(x)\,\delta\eta_{\beta}(z)\,\delta\bar{\eta}_{\zeta}(y)}\right)$
$\displaystyle\hskip
56.9055pt-\,i\,\delta(x-y)~{}S_{\zeta\beta}(x-z)+\,i\,\delta(x-z)~{}S_{\zeta\beta}(y-x)=0\
,$ (135)

where we have used the definition of the fermion propagator (23), and
considered the decomposition of the three point function given in Eq. (129),
the photon propagator in momentum space (34) and the Fourier decomposition of
the fermion propagator (28) one arrives, after some algebra, at

$\displaystyle\int\frac{d^{4}k}{(2\,\pi)^{4}}\,\frac{d^{4}p^{\prime}}{(2\,\pi)^{4}}\,\frac{d^{4}p}{(2\,\pi)^{4}}~{}e^{-i\,kx\,-i\,p^{\prime}y\,-i\,pz}~{}(2\,\pi)^{4}\,\delta(p^{\prime}-p+k)~{}k_{\mu}\Big{[}S(p^{\prime})\,\Gamma^{\mu}(p^{\prime},\,-p;\,k)\,S(p)\bigg{]}$
$\displaystyle\hskip
28.45274pt+~{}\int\frac{d^{4}p^{\prime}}{(2\,\pi)^{4}}\,\frac{d^{4}p}{(2\,\pi)^{4}}~{}e^{-i\,p(x-y)\,-i\,p^{\prime}(y-z)}~{}S(p^{\prime})~{}$
$\displaystyle\hskip
56.9055pt-~{}\int\frac{d^{4}p^{\prime}}{(2\,\pi)^{4}}\,\frac{d^{4}p}{(2\,\pi)^{4}}~{}e^{-i\,p(x-z)\,-i\,p^{\prime}(y-x)}~{}S(p^{\prime})~{}=~{}0\
.$ (136)

Then, performing an inverse Fourier transformation and multiplying the
resulting equation by the inverse of fermion propagator one arrives at Eq.
(44).

The rewriting of the two-photon-two-fermion WTI in momentum space follows the
same steps as those performed for the photon-fermion identity. In terms of the
connected Green functions, Eq. (43) is given by

$\displaystyle\frac{1}{g\,\xi}\,\square_{x}\,\partial^{\mu}_{x}\,\left(\frac{\delta^{4}W}{\delta
J^{\nu}(w)\,\delta
J^{\mu}(x)\,\delta\eta_{\beta}(z)\,\delta\bar{\eta}_{\zeta}(y)}~{}+~{}i\,D_{\mu\nu}(x-w)\,S_{\zeta\beta}(y-z)\right)$
$\displaystyle\qquad-\frac{i}{g}~{}\partial^{\mu}_{x}\Big{(}g_{\mu\nu}\,\delta(x-w)\Big{)}\,S_{\zeta\beta}(y-z)$
$\displaystyle\qquad\qquad+~{}i\,\delta(x-y)\left(\frac{\delta^{3}W}{\delta
J^{\nu}(w)\,\delta\eta_{\beta}(z)\,\delta\bar{\eta}_{\zeta}(x)}\right)~{}$
$\displaystyle\qquad\qquad\qquad-~{}i\,\delta(x-z)\left(\frac{\delta^{3}W}{\delta
J^{\nu}(w)\,\delta\eta_{\beta}(x)\,\delta\bar{\eta}_{\zeta}(y)}\right)=0$
(137)

where we have used the definition of the fermion propagator, see Eq. (23).
From the Fourier decomposition of the photon propagator (29) it follows that

$\square_{x}\,\partial^{\mu}_{x}\,D_{\mu\nu}(x-w)=\xi\,\partial_{\mu}^{x}\bigg{(}\delta(x-w)\bigg{)}$
(138)

and, therefore, the second and third terms in Eq. (137) cancel exactly. Then,
the WTI simplifies into

$\displaystyle\frac{1}{g\,\xi}\,\square_{x}\,\partial^{\mu}_{x}\,\left(\frac{\delta^{4}W}{\delta
J^{\nu}(w)\,\delta
J^{\mu}(x)\,\delta\eta_{\beta}(z)\,\delta\bar{\eta}_{\zeta}(y)}\right)$
$\displaystyle\qquad\qquad+~{}i\,\delta(x-y)\left(\frac{\delta^{3}W}{\delta
J^{\nu}(w)\,\delta\eta_{\beta}(z)\,\delta\bar{\eta}_{\zeta}(x)}\right)~{}-~{}i\,\delta(x-z)\left(\frac{\delta^{3}W}{\delta
J^{\nu}(w)\,\delta\eta_{\beta}(x)\,\delta\bar{\eta}_{\zeta}(y)}\right)$
$\displaystyle\qquad\qquad\qquad\qquad=0\ .$ (139)

Inserting the decompositions of the four point vertex (134) and of the three
point vertex (129), after some straightforward algebra and after performing an
inverse Fourier transformation, this equation reduces to

$\displaystyle
D_{\nu\nu^{\prime}}(q)~{}k_{\mu}~{}S(p)~{}\Big{\\{}~{}\Gamma^{\mu}(p,\,-p-k;\,k)~{}S(p+k)~{}\Gamma^{\nu^{\prime}}(p+k,\,-p-k-q;\,q)$
$\displaystyle\hskip
85.35826pt+~{}\Gamma^{\nu^{\prime}}(p,\,-p-q;\,q)~{}S(p+q)~{}\Gamma^{\mu}(p+q,\,-p-k-q;\,k)$
$\displaystyle\hskip
85.35826pt+~{}\Gamma^{\mu\nu^{\prime}}(p,\,-p-k-q;\,k,\,q)~{}\Big{\\}}~{}S(p+k+q)$
$\displaystyle\qquad+~{}D_{\nu\nu^{\prime}}(q)~{}S(p+k)~{}\Gamma^{\nu^{\prime}}(p+k,\,-p-k-q;\,q)~{}S(p+k+q)$
$\displaystyle\qquad-~{}D_{\nu\nu^{\prime}}(q)~{}S(p)~{}\Gamma^{\nu^{\prime}}(p,\,-p-q;\,q)~{}S(p+q)~{}=~{}0\
.$ (140)

When rewriting Eq. (137) we considered Furry theorem and ignored the term
proportional to the three-photon irreducible vertex. Then, multiplying the
above expression by the inverse of the photon propagator, the inverse of the
fermion propagators, after using the WTI (44) to simplify the contractions of
$k_{\mu}$ with the photon-fermion vertex, one arrives at

$\Gamma^{\nu}(p+k,\,-p-k-q;\,q)~{}-~{}\Gamma^{\nu}(p,\,-p-q;\,q)~{}+~{}k_{\mu}\,\Gamma^{\mu\nu}(p,\,-p-k-q;\,k,\,q)~{}=~{}0\
.$ (141)

that is the momentum space WTI (45 )appearing in the main text. We call the
reader attention that the derivation of these expressions requires explicitly
the use of the inverse of the photon propagator that in the Landau gauge and
within this formalism is not defined.

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