# Electron-Photon Vertex and Dynamical Chiral Symmetry Breaking in Reduced
QED: An Advanced Study of Gauge Invariance

L. Albino<EMAIL_ADDRESS>Instituto de Física y Matemáticas,
Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán 58040,
México. A. Bashir<EMAIL_ADDRESS>Instituto de Física y Matemáticas,
Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán 58040,
México. A.J. Mizher<EMAIL_ADDRESS>Instituto de Física Teórica,
Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz, 271-Bloco II,
01140-070, São Paulo, SP, Brazil. Centro de Ciencias Exactas, Universidad del
Bío-Bío. Avda. Andrés Bello 720, Casilla 447, 3800708, Chillán, Chile. A. Raya
<EMAIL_ADDRESS>Instituto de Física y Matemáticas, Universidad
Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán 58040, México. Centro
de Ciencias Exactas, Universidad del Bío-Bío. Avda. Andrés Bello 720, Casilla
447, 3800708, Chillán, Chile.

###### Abstract

We study the effect of a refined electron-photon vertex on the dynamical
breaking of chiral symmetry in reduced quantum electrodynamics. We construct
an educated ansatz for this vertex which satisfies the required discrete
symmetries under parity, time reversal and charge conjugation operations.
Furthermore, it reproduces its asymptotic perturbative limit in the weak
coupling regime and ensures the massless electron propagator is
multiplicatively renormalizable in its leading logarithmic expansion.
Employing this vertex ansatz, we solve the gap equation to compute dynamically
generated electron mass whose dependence on the electromagnetic coupling is
found to satisfy Miransky scaling law. We also investigate the gauge
dependence of this dynamical mass as well as that of the critical coupling
above which chiral symmetry is dynamically broken. As a litmus test of our
vertex construction, both these quantities are rendered virtually gauge
independent within a certain interval of values considered for the covariant
gauge parameter.

##  I Introduction

Graphene, the wonder material Novoselov2005 ; Zhang2005 , is a physical system
with immense potential for technological applications. It has driven a lot of
research in both the applied and theoretical physics, not only from the point
of view of condensed matter and materials sciences CastroNeto:2007fxn ;
Geim2007 , but also based on the quantum field theoretic description within
the domain of high energy physics Gusynin:2007ix and cosmology Cortijo2007 ;
Vozmediano_2008 ; VOZMEDIANO2010109 . Its remarkable properties of high
electric and thermal conductivity, stiffness, flexibility and transparency
have opened the door to explore a growing family of modern relativistic and
relativistic-like materials in one, two and three spatial dimensions. The
underlying honeycomb array of tightly packed carbon atoms and its
crystallographic description in terms of two inter-imposed triangular sub-
lattices provide chiral and valley quantum labels to the charge carrier
electrons. This occurrence is responsible for Klein tunneling Katsnelson_2006
as well as other exotic and novel phenomena Katsnelson2006 exhibited by
relativistic systems only. That makes graphene an incarnation of quantum
electrodynamics (QED) in condensed matter realms. Along with quantum Hall
systems Zhang2005 ; Semenoff.PhysRevLett.53.2449 ; Haldane.PhysRevLett.61.2015
; Gusynin.PhysRevLett.95.146801 and high-$T_{c}$ layered cuprate
superconductors DOREY1992614 ; PhysRevLett.87.257003 ; PhysRevB.66.094504 ;
PhysRevB.66.054535 ; Farakos:1997qi , graphene is also a system suitable for
its description in terms of relativistic quantum field theoretical
considerations, developed and refined in the domain of particle physics
Gusynin:2007ix . This in turn allows for an exploration of otherwise
inaccessible particle physics phenomenology in a more controlled and
observable ambient of solid state physics. A representative example in this
connection is the so-called chiral magnetic effect Kharzeev:2007jp ;
Fukushima:2008xe , which was first predicted to take place in relativistic
heavy ion collisions. It involves chirality flip of quarks, caused by the
chiral anomaly. It is a necessary ingredient to produce a non dissipative
current as an observable effect. This effect was proposed to probe the non-
tivial vacuum structure of quantum chromodynamiocs (QCD). However, it has not
been observed in isobar collisions in the STAR collaboration at RHIC
STAR:2021mii . Nevertheless the same basic idea of a physical system in which
interactions drive a chirality flip of the fundamental degrees of freedom was
found in ZnTe5 Li:2014bha , where a neat non-dissipative current was observed
when this 3D crystal is subject to an array of adequately aligned electric and
magnetic fields. After this first observation, non-dissipative currents driven
by the chiral anomaly were also encountered in several other similar materials
Li2015 ; PhysRevX.5.031023 ; PhysRevB.93.121112 ; Arnold:2015vvs ; Zhang2017 .

Some theoretical ideas have also been developed to observe a similar effect in
2D crystals such as graphene Mizher:2018dtf ; PhysRevD.102.096023 . More
recently, it has been shown that in some 2D materials, one may observe a novel
quantum spin Hall phenomenon Dudal:2021ret . A key ingredient for the
realization of these later phenomena is the description of electromagnetic and
matter fields living in mixed dimensions. Mixed dimensional theories emerge
naturally in the description of 2D materials where experiments are carried out
with external electromagnetic fields which permeate the whole space whereas
the movement of the charge carriers remains restricted to a plane.

Two independent formulations have been proposed in literature to describe QED
of photons and electrons living in different space-dimensions. One vision
exploits the equivalence of a theory where electrons live in lower dimensions
than photons and a Chern-Simons theory. It has been dubbed as Pseudo-QED
MARINO1993551 ; marino_2017 ; Amaral_1992 ; PhysRevD.90.105003 .
Alternatively, a brane-world inspired scenario was developed in
PhysRevD.64.105028 to explore the traits of dynamical chiral symmetry breaking
(DCSB) in the so-called Reduced QED (RQED), a nomenclature we choose to adopt
in this article. The equivalence of these two visions has already been
established. It respects causality Amaral_1992 and unitarity
PhysRevD.90.105003 . It exhibits a Coulomb static interaction in the case of
graphene marino_2017 ; GONZALEZ1994595 . It contains an infrared fixed point
of the renormalization group as the Fermi velocity tends to the speed of light
GONZALEZ1994595 ; PhysRevLett.77.3589 ; PhysRevB.59.R2474 ; PhysRevB.63.134421
. Two-loop perturbative analysis has been carried out in PhysRevD.89.065038
and later improved with renormalization group arguments Dudal:2018pta .
Interestingly for the immediate purpose of our manuscript, DCSB has been
explored within the Schwinger-Dyson PhysRevD.87.125002 equations (SDEs) and
renormalization group frameworks exploiting the duality between the gap
equation in this theory and the corresponding $1/N$ leading truncation in
parity preserving ordinary QED3. In the latter theory, it is known that there
exists a critical number of fermion families $N_{c}$ above which DCSB is
restored. In comparison, it is observed that in the quenched version of RQED,
where electron-loop contributions to the photon propagator are neglected and
the photon dressing function reduces to its tree level expansion, DCSB occurs
provided the electromagnetic coupling $\alpha$ exceeds a critical value
$\alpha_{c}$. The particular values of these critical numbers depend on the
gauge parameter and provide a natural motivation for the work we present and
the solutions we provide in this article. For the sake of completeness, we
would like to mention that DCSB has also been considered in RQED at finite
temperature and in the presence of a Chern-Simons term. Parity violating
solutions to the gap equation have also been explored in
carrington.PhysRevB.99.115432 in connection with the presence of a Chern-
Simons term. This term plays the role of an effective dielectric constant,
hence having potential experimental realization in graphene related materials.
Effects of strain have also been considered, leading to a lower value of the
critical coupling required to break chiral symmetry. Finally, RQED has also
been formulated in curved spaces Caneda.PhysRevD.103.065010 . For a review
discussing all this properties and applications of RQED, see Olivares:2021svj
.

Studying DCSB and its gauge invariance in RQED naturally requires its non-
perturbative treatment. Fortunately, extensive amount of analogous research in
QED4 Curtis:1990zs ; Bashir:1994az ; Bashir:1995qr ; Bashir:2011dp ;
Albino:2018ncl and QED3 Bashir:2002dz ; Bashir:2004yt ; Bashir:2005wt ;
Bashir:2009fv provides necessary groundwork to carry out similar reliable
analysis in RQED. We report the results of this continuum study through state-
of-the-art truncation schemes in SDEs. Focusing on the quenched version of the
theory, the sole source of our starting ansatz is the electron-photon vertex.
We construct it by demanding all the key characteristics of RQED to be
faithfully respected:

  * •

Ward-Fradkin-Green-Takahashi identity (WFGTI) that connects the electron
propagator with the longitudinal part of the electron-photon vertex is
satisfied non-perturbatively by construction, known as the Ball-Chiu (BC)
vertex Ball:1980ay .

  * •

To expand the transverse part of the vertex, we employ the vector basis and
its coefficients in such a manner as to ensure spurious kinematic
singularities are absent from our construction Kizilersu:1995iz ;
Davydychev:2000rt ; Bermudez:2017bpx .

  * •

In the weak coupling regime, the vertex faithfully reproduces its one-loop
perturbative expansion for the asymptotic limit of momenta $k^{2}\gg p^{2}$,
just it has previously been done in QED4 Curtis:1990zs ; Bashir:1994az and
QED3 Bashir:1999bd ; Bashir:2000rv ; Bashir:2011vg .

  * •

The standard model of particle physics tells us of the intimate connection
between its renormalizability and gauge invariance. In the same spirit, we
require our vertex ansatz to guarantee the multiplicative renormalizability
(MR) of the massless electron propagator in its leading logarithmic expansion.

  * •

We also require our vertex to satisfy the discrete symmetries of parity, time
reversal and charge conjugation.

Based upon the above-mentioned constraints, we are able to achieve nearly
gauge independent Euclidean mass and critical coupling $\alpha_{c}$ where the
DCSB solution bifurcates away from the chirally symmetric one. We believe that
obtaining gauge independent DCSB holds the promise to study observable effects
in the $2D$ materials described by RQED in a reliable manner through continuum
SDEs.

The article has been organized as follows: Sect. II begins with a brief
introduction to the mathematical foundations of RQED. In Sect. III, we provide
necessary preliminaries on the vertex decomposition and its general features.
In Sect. IV, we construct a family of Ans$\ddot{a}$tze for the transverse
vertex in perhaps the most economical yet efficient manner by resorting to the
constraints of MR and its explicit form in the so-called asymptotic limit at
the one-loop level. In Sect. V, we set up the gap equation and engage in a
detailed discussion on the photon propagator in RQED and the appropriate use
of the WFGTI in order to ensure the MR of the massless electron propagator.
Sect. VI provides solution of the gap equation, first in the perturbative
realm and then the non-perturbative DCSB solution in terms of the critical
coupling $\alpha_{c}$ above which the massive solution bifurcates away from
the perturbative massless one. We primarily focus on obtaining gauge
independent DCSB. Sect. VII contains conclusions and offers perspectives for
future work.

##  II RQED foundations

In order to describe electrons, restricted to move in a dimensionally reduced
space-time, coupled to photons free to propagate through the bulk space-time,
one initially begins with the well-known QED Lagrangian:

$\displaystyle\mathcal{L}_{\text{QED}}$ $\displaystyle=$
$\displaystyle\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m_{0}\right)\psi+j_{\mu}A^{\mu}$
(1)
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2\xi}\left(\partial_{\mu}A^{\mu}\right)^{2}\,,$

where $\psi$ and $A^{\mu}$ are electron and photon fields, respectively,
coupled to each other through the electromagnetic current $j_{\mu}$, where
$\mu=0,1,2,3$. Furthermore, the 4-dimensional Dirac matrices $\gamma^{\mu}$
satisfy the anti-commutation relation
$\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}=2g^{\mu\nu}$, with the commonly
used convention $g^{\mu\nu}=(+,-,-,-)$ for the Minkowski space metric tensor.
Additionally, $m_{0}$ is the bare electron mass, $\xi$ is the covariant gauge
parameter and $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
usual electromagnetic field tensor. The corresponding action
$\mathcal{S}_{\text{QED}}=\int\mathrm{d}^{4}x\mathcal{L}_{\text{QED}}$ can be
conveniently expressed as

$\displaystyle\mathcal{S}_{\text{QED}}=\mathcal{S}_{\bar{\psi}\psi}^{(3)}+\frac{1}{2}\int\mathrm{d}^{4}x\left[j^{\mu}\hat{\Delta}_{\mu\nu}j^{\nu}-A^{\mu}\hat{\Delta}_{\mu\nu}^{-1}A^{\nu}\right]\,,$
(2)

where the kinetic term for electrons, constrained to move in a 3-dimensional
space-time reads:

$\displaystyle\mathcal{S}_{\bar{\psi}\psi}^{(3)}=\int\mathrm{d}^{3}x\,\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m_{0}\right)\psi\,,$
(3)

with $\mu=0,1,2$. Moreover, the differential operator $\hat{\Delta}_{\mu\nu}$,
namely the photon propagator in coordinate space, can be cast in terms of its
momentum space counterpart by means of a Fourier transformation

$\displaystyle\hat{\Delta}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\int\frac{\mathrm{d}^{4}q}{\left(2\pi\right)^{4}}e^{-iq\cdot
x}\frac{1}{q^{2}}\left[g_{\mu\nu}-\left(1-\xi\right)\frac{q_{\mu}q_{\nu}}{q^{2}}\right]\,,$
(4)

satisfying
$\hat{\Delta}_{\mu\alpha}^{-1}\hat{\Delta}^{\alpha\nu}=g_{\mu}^{\nu}$ with its
corresponding inverse propagator. Such a Green function accounts for a gauge
field propagating through the whole 4-dimensional space-time with
$\mu,\nu=0,1,2,3$.

To account for a mixed-dimension system described by RQED with electrons
restricted to move on a plane perpendicular to the $x_{3}$-axis MARINO1993551
; marino_2017 ; Amaral_1992 ; PhysRevD.90.105003 ; PhysRevD.64.105028 ;
PhysRevD.86.025005 , the electromagnetic current takes the form

$\displaystyle j^{\mu}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}-ie\bar{\psi}\gamma^{\mu}\psi\delta(x_{3})&~{}\text{for}~{}\mu=0,1,2\,,\\\
0&~{}\text{for}~{}\mu=3\,.\end{array}\right.$ (7)

Therefore, only the indices $\mu=0,1,2$ contribute to the term
$j^{\mu}\hat{\Delta}_{\mu\nu}j^{\nu}$ in Eq. (2). The component $\mu=3$ can
thus be integrated out in Eq. (4), leading to Dudal:2018mms

$\displaystyle\hat{\Delta}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}^{3}q}{\left(2\pi\right)^{3}}e^{-iq\cdot
x}\frac{1}{2\sqrt{-q^{2}}}\left[g_{\mu\nu}-\left(1-\xi\right)\frac{q_{\mu}q_{\nu}}{2q^{2}}\right]\,.$
(8)

It entails a non-local differential operator. This propagator can be obtained
from the effective action for RQED (redefining $\xi=2\zeta-1$)

$\displaystyle\mathcal{S}_{\text{RQED}}$ $\displaystyle=$
$\displaystyle\mathcal{S}_{\bar{\psi}\psi}^{(3)}+\int\mathrm{d}^{3}x\,j_{\mu}A^{\mu}$
(9)
$\displaystyle+\int\mathrm{d}^{3}x\left[\frac{1}{2}F_{\mu\nu}\frac{1}{\sqrt{-\partial^{2}}}F^{\mu\nu}+\frac{1}{\zeta}\partial_{\mu}A^{\mu}\frac{1}{\sqrt{-\partial^{2}}}\partial_{\nu}A^{\nu}\right].$

From now on, we work in the Euclidean space defined by the metric tensor
$\delta_{\mu\nu}=(+,+,+)$ for $\mu,\nu=4,1,2$. In this space, the bare photon
propagator takes the form (c.f. Eq. (8))

$\displaystyle\Delta_{\mu\nu}^{(0)}(q)=\frac{1}{2q}\left[\delta_{\mu\nu}-\left(1-\xi\right)\frac{q_{\mu}q_{\nu}}{2q^{2}}\right]\,,$
(10)

where we have defined $q\equiv\sqrt{-q^{2}}$. Note that this propagator has a
softer infrared behavior than the photon propagator in QED4 and QED3. Several
groups differ in conventions by the global factor of 1/2. When written in
terms of the variable $\zeta$, this propagator can be separated into a
familiar longitudinal and a transverse component (to $q_{\mu}$):

$\displaystyle\Delta_{\mu\nu}^{(0)}(q)=\frac{1}{2q}\left[\delta_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right]+\zeta\frac{q_{\mu}q_{\nu}}{2q^{3}}\,,$
(11)

However, it must be emphasized that the gauge fixing parameter $\zeta$ in RQED
is different from that in QED, $\xi$, due to dimensional reduction. In the
above expression for the photon propagator, Eq. (11), the $\zeta$-independent
term defines the transverse propagator which is the only component that gets
quantum corrections. This is the reason for using such a decomposition in
other works. In contrast, the $\xi$-independent term in Eq. (10) does not
define a transverse propagator. Therefore, quantum corrections affect both
$\xi$-dependent and $\xi$-independent components. In the present work, we
restrict ourselves to work in the quenched approximation. As there are no
fermion loops present, there are no quantum corrections to the bare photon
propagator. Therefore, both the expression, Eqs. (10,11) are equally suitable:
we choose that of Eq. (10).

Note that the electron-photon vertex plays a vital role in computing the non-
perturbative solution of the electron propagator through its gap equation.
Therefore, we address this three-point Green function at length in the
following section.

##  III The Vertex: Generalities

Figure 1: Diagrammatic representation of the full electron-photon vertex
$\Gamma_{\mu}(k,p)$, with momentum flow indicated.

In its general decomposition, the three-point electron-photon vertex can be
written in terms of twelve independent spin structures. For the kinematical
configuration of Fig. 1, the WFGTI associated with this vertex takes the form

$iq_{\mu}\Gamma_{\mu}(k,p)=S^{-1}(k)-S^{-1}(p)\,,$ (12)

where $q=k-p$. This identity allows us to split the vertex as a sum of
longitudinal and transverse components, as suggested by Ball and Chiu
Ball:1980ay :

$\Gamma_{\mu}(k,p)=\Gamma^{L}_{\mu}(k,p)+\Gamma^{T}_{\mu}(k,p)\,.$ (13)

The longitudinal part $\Gamma^{L}_{\mu}(k,p)$ alone satisfies the WFGTI (12),
and consumes four of the twelve independent spin structures (one of them is
zero in QED), so that, Ball:1980ay :

$\displaystyle\Gamma^{L}_{\mu}(k,p)=\lambda_{1}(k,p)\gamma_{\mu}+\lambda_{2}(k,p)t_{\mu}\gamma\cdot
t-i\lambda_{3}(k,p)t_{\mu}\,,$ (14)

with $t=k+p$, and

$\displaystyle\lambda_{1}(k,p)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{1}{F(k^{2},\Lambda^{2})}+\frac{1}{F(p^{2},\Lambda^{2})}\right]\,,$
$\displaystyle\lambda_{2}(k,p)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{1}{F(k^{2},\Lambda^{2})}-\frac{1}{F(p^{2},\Lambda^{2})}\right]\frac{1}{k^{2}-p^{2}}\,,$
$\displaystyle\lambda_{3}(k,p)$ $\displaystyle=$
$\displaystyle\left[\frac{{\cal{M}}(k^{2},\Lambda^{2})}{F(k^{2},\Lambda^{2})}-\frac{{\cal{M}}(p^{2},\Lambda^{2})}{F(p^{2},\Lambda^{2})}\right]\frac{1}{k^{2}-p^{2}}\,,$
(15)

where $\Lambda$ is an ultraviolet (UV) cut-off regulator. Note that
${\cal{M}}(k^{2},\Lambda^{2})$ and $F(k^{2},\Lambda^{2})$ are the mass
function and the wave function renormalization, respectively, related to the
electron propagator $S(k)$ through

$S(k)=\frac{F(k^{2},\Lambda^{2})}{i\gamma\cdot
k+{\cal{M}}(k^{2},\Lambda^{2})}\,,$ (16)

with $F(k^{2},\Lambda^{2})=1$ and ${\cal{M}}(k^{2},\Lambda^{2})=m_{0}$ at the
tree level.

The transverse part $\Gamma^{T}_{\mu}(k,p)$ of the vertex decomposition (13),
which remains undetermined by the WFGTI, is naturally constrained by

$q_{\mu}\Gamma^{T}_{\mu}(k,p)=0\,.$ (17)

In general, the ultraviolet finite transverse vertex can be expanded out in
terms of eight basis vector structures, and their corresponding scalar form
factors $\tau_{i}(k,p)$ Ball:1980ay :

$\Gamma^{T}_{\mu}(k,p)=\sum_{i=1}^{8}\tau_{i}(k,p)T^{i}_{\mu}(k,p)\,.$ (18)

For the kinematical configuration of Fig. 1, we define

$\displaystyle T^{1}_{\mu}(k,p)$ $\displaystyle=$ $\displaystyle
i\left[p_{\mu}(k\cdot q)-k_{\mu}(p\cdot q)\right]\,,$ $\displaystyle
T^{2}_{\mu}(k,p)$ $\displaystyle=$ $\displaystyle\left[p_{\mu}(k\cdot
q)-k_{\mu}(p\cdot q)\right]\left(\gamma\cdot t\right)\,,$ $\displaystyle
T^{3}_{\mu}(k,p)$ $\displaystyle=$ $\displaystyle
q^{2}\gamma_{\mu}-q_{\mu}\left(\gamma\cdot q\right)\,,$ $\displaystyle
T^{4}_{\mu}(k,p)$ $\displaystyle=$ $\displaystyle
iq^{2}\left[\gamma_{\mu}\left(\gamma\cdot
t\right)-t_{\mu}\right]+2q_{\mu}p_{\nu}k_{\rho}\sigma_{\nu\rho}\,,$
$\displaystyle T^{5}_{\mu}(k,p)$ $\displaystyle=$
$\displaystyle\sigma_{\mu\nu}q_{\nu}\,,$ $\displaystyle T^{6}_{\mu}(k,p)$
$\displaystyle=$
$\displaystyle-\gamma_{\mu}\left(k^{2}-p^{2}\right)+t_{\mu}\left(\gamma\cdot
q\right)\,,$ $\displaystyle T^{7}_{\mu}(k,p)$ $\displaystyle=$
$\displaystyle\frac{i}{2}(k^{2}-p^{2})\left[\gamma_{\mu}\left(\gamma\cdot
t\right)-t_{\mu}\right]+t_{\mu}p_{\nu}k_{\rho}\sigma_{\nu\rho}\,,$
$\displaystyle T^{8}_{\mu}(k,p)$ $\displaystyle=$
$\displaystyle-i\gamma_{\mu}p_{\nu}k_{\rho}\sigma_{\nu\rho}-p_{\mu}\left(\gamma\cdot
k\right)+k_{\mu}\left(\gamma\cdot p\right)\,,$ (19)

with

$\sigma_{\nu\rho}=\frac{i}{2}\left[\gamma_{\nu},\gamma_{\rho}\right]\,.$ (20)

This basis is not exactly the one adopted in Ball:1980ay . We choose to work
with a modification of this initial basis which was put forward in
Kizilersu:1995iz and later employed in Davydychev:2000rt as well. This
modified choice of the basis vectors ensures all transverse form factors of
the vertex are independent of any kinematic singularities in one-loop
perturbation theory in an arbitrary covariant gauge.

As stated earlier in Sect. I, any ansatz for the full vertex must have the
same transformation properties as the bare vertex under charge conjugation
operation. This requires all the $\tau_{i}$s in (18) to be symmetric under the
interchange $k\leftrightarrow p$, except $\tau_{4}$ and $\tau_{6}$, which are
odd:

$\displaystyle\tau_{i}(k,p)=\tau_{i}(p,k)\,,\hskip 17.78296pti=1,2,3,5,7,8,$
(21) $\displaystyle\tau_{i}(k,p)=-\tau_{i}(p,k)\,,\quad i=4,6.$ (22)

From Eq. (15), it is obvious that $\lambda_{1}(k,p)$, $\lambda_{2}(k,p)$ and
$\lambda_{3}(k,p)$ are symmetric under $k\leftrightarrow p$, as they should
be, in order to preserve the correct transformation properties under charge
conjugation operation for the full vertex.

In the following section, we construct the transverse vertex, putting forward
an ansatz which complies with the key requirements expected of it as detailed
in Sect. I. It is also effective, economical and yields practically gauge
invariant DCSB as we see in subsequent sections.

##  IV The transverse vertex

The transverse vertex is completely determined once the form factors in Eqs.
(18,19) are known. We now proceed to construct an ansatz for it.

###  IV.1 A general ansatz for the transverse vertex

We start out by recalling a fairly general ansatz which was first proposed in
Bashir:2011dp , generally referred to as the BB-vertex, and later successfully
employed in several related works, e.g., Bermudez:2017bpx ; Albino:2018ncl ;
Albino:2021rvj

$\displaystyle\tau_{1}(k,p)$ $\displaystyle=$
$\displaystyle\frac{a_{1}}{(k^{2}+p^{2})}\;\lambda_{3}(k,p)\,,$ (23)
$\displaystyle\tau_{2}(k,p)$ $\displaystyle=$
$\displaystyle\frac{2a_{2}}{(k^{2}+p^{2})}\;\lambda_{2}(k,p)\,,$ (24)
$\displaystyle\tau_{3}(k,p)$ $\displaystyle=$ $\displaystyle
2a_{3}\;\lambda_{2}(k,p)\,,$ (25) $\displaystyle\tau_{4}(k,p)$
$\displaystyle=$
$\displaystyle\frac{a_{4}(k^{2}-p^{2})}{4k^{2}p^{2}}\;\lambda_{3}(k,p)\,,$
(26) $\displaystyle\tau_{5}(k,p)$ $\displaystyle=$ $\displaystyle-
a_{5}\;\lambda_{3}(k,p)\,,$ (27) $\displaystyle\tau_{6}(k,p)$ $\displaystyle=$
$\displaystyle-\frac{2a_{6}(k^{2}+p^{2})}{(k^{2}-p^{2})}\;\lambda_{2}(k,p)\,,$
(28) $\displaystyle\tau_{7}(k,p)$ $\displaystyle=$
$\displaystyle-\left[\frac{a_{4}q^{2}}{2k^{2}p^{2}}+\frac{a_{7}}{k^{2}+p^{2}}\right]\;\lambda_{3}(k,p)\,,$
(29) $\displaystyle\tau_{8}(k,p)$ $\displaystyle=$ $\displaystyle
2a_{8}\;\lambda_{2}(k,p)\,.$ (30)

Before we proceed any further, we summarize the following important points:

  * •

All eight transverse form factors are taken into consideration which implies
the generality of this ansatz.

  * •

All form factors are proportional to the same structures which appear in the
longitudinal vertex, namely $\lambda_{2}(k,p)$ and $\lambda_{3}(k,p)$.

  * •

The form factor $\tau_{6}(k,p)$ has a kinematic singularity for $k^{2}=p^{2}$.
For the QED4 gap equation, we can employ this form factor with impunity
because it cancels with the corresponding term proportional to
$\lambda_{2}(k,p)$ coming from the longitudinal vertex. However, for RQED,
this cancellation does not take place and we need to modify this form factor
to avoid the kinematic singularity. We propose an economical and effective
modification:

$\displaystyle\tau_{6}(k,p)$ $\displaystyle=$ $\displaystyle
a_{6}\frac{(k^{2}-p^{2})}{(k^{2}+p^{2})}\;\lambda_{2}(k,p)\;.$ (31)

We now try to constrain the coefficients $a_{i}$ through the one-loop
expression for the transverse vertex. We may call this as the modified BB-
vertex for RQED.

###  IV.2 One-Loop electron-photon vertex

In perturbation theory, the one-loop order expansion of the electron-photon
vertex is

$\displaystyle\Gamma_{\mu}^{(1)}(k,p)=\gamma_{\mu}$
$\displaystyle-\frac{\alpha}{2\pi^{2}}\int{d^{3}\omega\gamma_{\alpha}S(p-\omega)\gamma_{\mu}S(k-\omega)\gamma_{\beta}\Delta_{\alpha\beta}(\omega)}\,.$
(32)

In analogy with Curtis:1991fb ; Bashir:1997qt ; Bashir:2011vg , we have
computed the one-loop corrections to the electron-photon vertex in the
asymptotic limit, defined as the perturbative expansion with $p^{2}\gg
k^{2}\gg m^{2}_{0}$. The leading logarithmic term of the transverse vertex at
this level of approximation reads:

$\displaystyle\Gamma_{\mu}^{T}(k,p)$ $\displaystyle\overset{p^{2}\gg
k^{2}}{=}$ $\displaystyle-\frac{\alpha}{8\pi
p^{2}}\left(\xi-\frac{1}{3}\right)\log\left(\frac{p^{2}}{k^{2}}\right)T_{\mu}^{asy}\,,$
(33)

with

$\displaystyle T_{\mu}^{asy}=p^{2}\gamma_{\mu}-p_{\mu}\gamma\cdot p\,.$ (34)

Hence, in this limit, the transverse vertex can be expressed as (cf. Eq. (60))

$\displaystyle\Gamma_{\mu}^{T}(k,p)$ $\displaystyle\overset{p^{2}\gg
k^{2}}{=}$
$\displaystyle\frac{1}{2}\left[\frac{1}{F(k^{2})}-\frac{1}{F(p^{2})}\right]\frac{T_{\mu}^{asy}}{k^{2}-p^{2}}\,.$
(35)

On the other hand, from Eqs. (18,19), it is straightforward to see that the
leading structure of the transverse vertex in this limit acquires the
following form:

$\displaystyle\Gamma_{\mu}^{T}(k,p)$ $\displaystyle\overset{p^{2}\gg
k^{2}}{=}$ $\displaystyle\left(\tau_{3}+\tau_{6}\right)T_{\mu}^{asy}\,,$ (36)

where $\tau_{3,6}\equiv\tau_{3,6}(k,p)$. Moreover, we have used the fact that,
in the asymptotic expansion, the dominant contributions to the transverse
vertex come from $T_{\mu}^{3}$ and $T_{\mu}^{6}$ which simplify to

$\displaystyle T_{\mu}^{3\,asy}=T_{\mu}^{6\,asy}\equiv T_{\mu}^{asy}\,.$ (37)

As the simplest construction of the transverse vertex, we can single out
$\tau_{3}(k,p)$ and $\tau_{6}(k,p)$ which correctly provide this desired
limit. Writing out these form factors explicitly,

$\displaystyle\tau_{3}(k,p)$ $\displaystyle=$ $\displaystyle
a_{3}\left[\frac{1}{F(k^{2})}-\frac{1}{F(p^{2})}\right]\frac{1}{k^{2}-p^{2}}\,,$
(38) $\displaystyle\tau_{6}(k,p)$ $\displaystyle=$ $\displaystyle
a_{6}\left[\frac{1}{F(k^{2})}-\frac{1}{F(p^{2})}\right]\frac{1}{k^{2}+p^{2}}\,.$
(39)

It is thus straightforward to see from Eqs. (35,36) that the one-loop
behaviour of the vertex in the asymptotic limit requires

$\displaystyle a_{3}-a_{6}=1/2\,.$ (40)

Our aim is to compute non-perturbative solutions of the gap equation using
various vertex ansätze. Our preferred choice is the electron-photon vertex
consonant with the WFGTI and perturbation theory. It consists of the
longitudinal BC-vertex, Eq. (14), and the transverse component constructed in
terms of the vector structures $T_{\mu}^{3}$ and $T_{\mu}^{6}$ alone. For the
sake of completeness, we also compute and depict some results obtained by
employing the bare vertex, the central BC-vertex and the full BC-vertex.

The starting point for such an endeavor is naturally the gap equation for the
electron propagator. This is what we proceed to set up and study in the next
section.

##  V Setting up the gap equation

The SDE for the electron propagator, also known as the electron gap equation,
is diagrammatically depicted in Fig. 2.

Figure 2: The gap equation for the electron propagator. The color-filled blobs
labelled with $S$, $\Delta_{\mu\nu}$ and $\Gamma_{\mu}$ stand for the fully-
dressed electron and photon propagators, and the three-point vertex,
respectively.

Mathematically, the gap equation is written as:

$\displaystyle S^{-1}(k)$ $\displaystyle=$ $\displaystyle
S_{0}^{-1}(k)+\frac{\alpha}{2\pi^{2}}\int_{E}{d^{3}p\,\gamma_{\nu}S(p)\Gamma_{\mu}(k,p)\Delta_{\mu\nu}(q)}\,,$

where the subscript $E$ indicates that the integral is performed in the
Euclidean space, the subscript “0” denotes the tree level of the corresponding
propagator and $\alpha=e^{2}/4\pi$ is the electromagnetic coupling. We have
already provided a detailed discussion on the full electron-photon vertex
$\Gamma_{\mu}(k,p)$. A similar analysis of the photon propagator
$\Delta_{\mu\nu}(q)$ is in place now.

###  V.1 The Photon Propagator

For the sake of an appropriate implementation of the WFGTI in the gap
equation, let us split the photon propagator of Eq. (10) in a longitudinal
piece

$\displaystyle\Delta_{\mu\nu}^{L}(q)$ $\displaystyle=$
$\displaystyle\varrho\,\kappa\frac{q_{\mu}q_{\nu}}{q^{3}}\,,$ (42)

and a remaining non-longitudinal component

$\displaystyle\Delta_{\mu\nu}^{NL}(q)$ $\displaystyle=$
$\displaystyle\,\frac{\varrho}{q}\left[\delta_{\mu\nu}-\chi\frac{q_{\mu}q_{\nu}}{q^{2}}\right]\,,$
(43)

expressed in Euclidean space such that

$\displaystyle\Delta_{\mu\nu}(q)$ $\displaystyle=$
$\displaystyle\Delta_{\mu\nu}^{L}(q)+\Delta_{\mu\nu}^{NL}(q)\,,$ (44)

where $\chi\equiv 1-\epsilon_{e}$, $\kappa\equiv\chi\xi$,
$\varrho=\Gamma\left[\chi\right]/\left(4\pi\right)^{\epsilon_{e}}$. It is
important to notice that RQED ($\epsilon_{e}=1/2$) yields $\chi=\varrho=1/2$
and $\kappa=\xi/2$. We have opted for this notation of arbitrary dimensions in
alignment with the one adopted in PhysRevD.86.025005 .

This separation of the photon propagator, Eqs. (42,43), plays an important
role in QED where it has been demonstrated that it is imperative to apply the
WFGTI to the divergence term $q_{\mu}\Gamma_{\mu}$ which arises from the
contraction of the vertex with the longitudinal (gauge dependent) contribution
of the photon propagator in the gap equation. This recipe ensures the absence
of spurious terms which are a characteristic of the UV cut-off regularization
scheme due to the fact that it breaks translational invariance and can violate
gauge invariance if not used carefully Dong:1994jr ; Bashir:1994az . Providing
a vertex that satisfies the WFGTI and employing this identity in the gap
equation only for the appropriate part of the photon propagator are crucial to
guarantee the local gauge covariance of the electron propagator. In the
quenched approximation, the electron wave function renormalization for QED4
has been shown to be explicitly renormalizable to all orders in perturbation
theory provided an ansatz for the vertex in close analogy with full QED4 is
constructed Curtis:1990zs . In the leading logarithmic approximation, the
electron wave function renormalization exhibits a power law behaviour:

$\displaystyle F(k^{2})$ $\displaystyle=$ $\displaystyle
1+\sum_{n=1}^{\infty}\frac{\beta^{n}}{n!}\log^{n}\left(\frac{k^{2}}{\Lambda^{2}}\right)=\left(\frac{k^{2}}{\Lambda^{2}}\right)^{\beta}\,,$
(45)

where $\beta=\beta(\alpha)$ is an unknown coefficient that can be computed
order by order in perturbation theory. In the leading logarithmic
approximation of QED4, it is known that $\beta=\alpha\xi/(4\pi)$. For quenched
RQED we demonstrate that $\beta=\alpha(\xi-1/3)/(4\pi)$.

As mentioned earlier, careless implementation of an ultraviolet cut-off
regularization breaks gauge invariance and it is manifest in the appearance of
spurious terms even in one-loop calculations. An appropriate use and
manipulation of the WFGTI in the gap equation is key to getting rid of such
spurious terms. For this purpose, it is worth noting that in quenched RQED we
can redefine the longitudinal and non-longitudinal components of the photon
propagator by shifting $\chi$ and $\kappa$ with an arbitrary factor $\eta$
such that the complete photon propagator, Eq. (44), remains the same:

$\displaystyle\chi$ $\displaystyle\rightarrow$ $\displaystyle\chi-\eta\,,$
(46) $\displaystyle\kappa$ $\displaystyle\rightarrow$
$\displaystyle\kappa-\eta\,.$ (47)

Bear in mind that the photon propagator and consequently the electron SDE, Eq.
(LABEL:gap_equation), remain invariant under the shifts prescribed by Eqs.
(46,47),

We apply the WFGTI, Eq. (12), on the longitudinal photon propagator term in
the gap equation without loss of generality, leading to

$\displaystyle\int_{E}{d^{3}p\,\gamma_{\nu}S(p)\Gamma_{\mu}(k,p)\Delta^{L}_{\mu\nu}(q)}$
$\displaystyle=$
$\displaystyle\varrho\,\kappa\int_{E}{\frac{d^{3}q}{q^{3}}\gamma\cdot q}$ (48)
$\displaystyle-i\varrho\,\kappa\int_{E}{\frac{d^{3}p}{q^{3}}\gamma\cdot
q\,S(p)S^{-1}(k)}\,,$

where we have shifted $d^{3}p\rightarrow d^{3}q$ in the first term on the
right-hand side of the above equation in order to show that such a term
vanishes in a translationally invariant theory as it is an odd integral.
Moreover, for the term $\Gamma_{\mu}(k,p)\Delta^{NL}_{\mu\nu}(q)$ in Eq.
(LABEL:gap_equation) we use the explicit form of the vertex defined through
Eqs. (13-15,18,19). After bringing out the subtle role played by the photon
propagator in RQED, we can now focus on the mathematical and technical details
of the gap equation itself.

###  V.2 The gap equation

We can project out two coupled, integral equations for ${\cal{M}}$ and $F$
from the matrix gap equation. In an arbitrary gauge, these equations can
respectively be written as:

$\displaystyle\frac{{\cal{M}}(k^{2})}{F(k^{2})}$ $\displaystyle=$
$\displaystyle
m_{0}+\frac{\alpha\varrho\kappa}{2\pi^{2}}\int_{E}\frac{d^{3}p}{q^{3}}\frac{F(p^{2})}{p^{2}+{\cal{M}}^{2}(p^{2})}\frac{1}{F(k^{2})}$
(49) $\displaystyle\hskip 56.9055pt\times\Big{\\{}{\cal{M}}(p^{2})\,q\cdot
k-{\cal{M}}(k^{2})\,q\cdot p\Big{\\}}$ $\displaystyle\hskip
8.53581pt+\frac{\alpha\varrho}{2\pi^{2}}\int_{E}d^{3}p\,\frac{F(p^{2})}{p^{2}+{\cal{M}}^{2}(p^{2})}\,G_{{\cal{M}}}(k,p)\,,$
$\displaystyle\frac{1}{F(k^{2})}$ $\displaystyle=$ $\displaystyle
1-\frac{\alpha\varrho\kappa}{2\pi^{2}}\int_{E}\frac{d^{3}p}{q^{3}}\frac{F(p^{2})}{p^{2}+{\cal{M}}^{2}(p^{2})}\frac{1}{F(k^{2})}$
(50) $\displaystyle\hskip 56.9055pt\times\Big{\\{}\,q\cdot
p+{\cal{M}}(k^{2}){\cal{M}}(p^{2})\,\frac{q\cdot k}{k^{2}}\Big{\\}}$
$\displaystyle\hskip
8.53581pt+\frac{\alpha\varrho}{2\pi^{2}}\int_{E}\frac{d^{3}p}{k^{2}}\frac{F(p^{2})}{p^{2}+{\cal{M}}^{2}(p^{2})}\,G_{F}(k,p)\,,$

where we have adopted the notation $F(k^{2})\equiv F(k^{2},\Lambda^{2})$ and
the same for ${\cal{M}}$. Moreover, the electron-photon vertex form factors
contribute to the gap equation via the scalar functions $G_{{\cal M}}$ and
$G_{F}$:

$\displaystyle q\,G_{{\cal M}}(k,p)$ $\displaystyle=$
$\displaystyle(3-\chi){\cal M}(p^{2})\lambda_{1}$ (51)
$\displaystyle+\left[t^{2}-\frac{(k^{2}-p^{2})^{2}}{q^{2}}\chi\right]{\cal
M}(p^{2})\lambda_{2}$ $\displaystyle-\left[t\cdot p-\frac{(q\cdot
p)(k^{2}-p^{2})}{q^{2}}\chi\right]\lambda_{3}$
$\displaystyle+\nabla(k,p)\tau_{1}+2\nabla(k,p){\cal M}(p^{2})\tau_{2}$
$\displaystyle+2q^{2}{\cal M}(p^{2})\tau_{3}$
$\displaystyle-2\left[(k^{2}-p^{2})(q\cdot p)+\nabla(k,p)\right]\tau_{4}$
$\displaystyle-2(q\cdot p)\,\tau_{5}-2(k^{2}-p^{2}){\cal M}(p^{2})\tau_{6}$
$\displaystyle-\left[(k^{2}-p^{2})(t\cdot p)-\nabla(k,p)\right]\tau_{7}\,,$

$\displaystyle q\,G_{F}(k,p)$ $\displaystyle=$
$\displaystyle\left[(1-3\chi)k\cdot p+2\chi\,u(k,p)\right]\lambda_{1}$ (52)
$\displaystyle-\left[(k\cdot p)t^{2}+2\nabla(k,p)-\frac{(k\cdot
p)(k^{2}-p^{2})^{2}}{q^{2}}\chi\right]\lambda_{2}$ $\displaystyle-\left[t\cdot
k-\frac{(q\cdot k)(k^{2}-p^{2})}{q^{2}}\chi\right]{\cal M}(p^{2})\lambda_{3}$
$\displaystyle+\nabla(k,p){\cal
M}(p^{2})\tau_{1}-(k^{2}+p^{2})\nabla(k,p)\tau_{2}$ $\displaystyle+2(q\cdot
k)(q\cdot p)\tau_{3}$ $\displaystyle+2\left[q^{2}(t\cdot
k)-\nabla(k,p)\right]{\cal M}(p^{2})\tau_{4}$ $\displaystyle+2(q\cdot k){\cal
M}(p^{2})\tau_{5}-2(k^{2}-p^{2})(k\cdot p)\,\tau_{6}$
$\displaystyle+\left[(k^{2}-p^{2})(t\cdot k)+\nabla(k,p)\right]{\cal
M}(p^{2})\tau_{7}$ $\displaystyle+\nabla(k,p)\tau_{8}\,,$

where we have used the simplifying notation
$\lambda_{i}\equiv\lambda_{i}(k,p)$ and $\tau_{i}\equiv\tau_{i}(k,p)$. We have
also defined

$\displaystyle\nabla(k,p)$ $\displaystyle=$ $\displaystyle k^{2}p^{2}-(k\cdot
p)^{2}\,,$ (53) $\displaystyle u(k,p)$ $\displaystyle=$ $\displaystyle
2\,k\cdot p-\frac{\nabla(k,p)}{q^{2}}\,.$ (54)

We are now in a position to go ahead and solve these coupled equations for any
given vertex ansatz. We take up this task in perturbative and non-perturbative
realms in the next section.

##  VI Solving the gap equation

In order to have a better intuitive grasp over the solutions of the gap
equation, we start from perturbation theory. It is only natural to demand that
any physically acceptable non-perturbative solution must reduce to its
perturbative counterpart in the weak coupling regime.

###  VI.1 Perturbative solution

Note that in the weak coupling regime, the gap equation is expected to
reproduce the perturbative result for the electron propagator. In the chiral
limit ($m_{0}=0$), the leading logarithmic expansion for the wave function
renormalization in quenched RQED at one-loop order reads:

$\displaystyle
F(k^{2})=1+\frac{\alpha\varrho}{\pi}\left(\kappa-\phi\right)\log\left(\frac{k^{2}}{\Lambda^{2}}\right)+\frac{2}{3}\frac{\alpha\varrho}{\pi}\phi\,,$
(55)

where the last term in the above Eq. (55), parameterized by the constant
$\phi=\chi-1/3$, spoils the power law behaviour of $F(k^{2})$, Eq. (45). This
term is spurious. It does not appear if we work in the dimensional
regularization scheme. Just like in QED4, its presence owes itself to the use
of the cut-off regulator which we cannot avoid for the non-perturbative
treatment presented in the next section. The good news is that just like in
QED4, we can cure the inadequacy of the cut-off regulator by an appropriate
use of the WFGTI which ensures this spurious term is formally removed. It is
achieved by redefining $\chi$ and $\kappa$ as prescribed through Eqs. (46,47).
To clarify how to achieve this, we focus on the survey of two special cases:

Case 1: Let us choose not to apply the WFGTI in any term of the kernel of the
gap equation. It is equivalent to taking $\eta=\kappa$ in Eqs. (46,47). It
implies making the following shift:

$\displaystyle\chi$ $\displaystyle\rightarrow$
$\displaystyle\chi^{\prime}=(1-\epsilon_{e})\left(1-\xi\right)\,,$ (56)
$\displaystyle\kappa$ $\displaystyle\rightarrow$
$\displaystyle\kappa^{\prime}=0\,,$ (57)

yielding $\phi=-(\xi-1/3)/2$.

Case 2: Like the case of QED4, we can be selective in applying the WFGTI on a
redefined longitudinal part of the photon propagator such that $\phi=0$.
According to Eqs. (46,47), it entails making the shift

$\displaystyle\chi$ $\displaystyle\rightarrow$
$\displaystyle\tilde{\chi}=1/3\,,$ (58) $\displaystyle\kappa$
$\displaystyle\rightarrow$
$\displaystyle\tilde{\kappa}=\epsilon_{e}-2/3+\xi(1-\epsilon_{e})\,.$ (59)

In this second case, the wave function renormalization exhibits the power law
behaviour of Eq. (45) in the leading logarithmic approximation with
$\beta=\alpha(\xi-1/3)/(4\pi)$ as stated before.

It is also worth noticing that for both the cases considered, we have

$\displaystyle\frac{1}{F(k^{2})}-\frac{1}{F(p^{2})}=-\frac{\alpha}{4\pi}\left(\xi-\frac{1}{3}\right)\log\left(\frac{k^{2}}{p^{2}}\right)\,,$
(60)

which proves to be a useful guide to express the one-loop corrections to the
electron-photon vertex in terms of the electron dressing functions, as we
study in section IV, thus helping us construct a non-perturbative ansatz for
the electron-photon vertex.

So far, we have shown that ensuring the MR for the massless electron
propagator, i.e., the power law behaviour of Eq. (45), at the one-loop level
only requires a specific shifting of $\chi$ and $\kappa$, cf. Eqs. (58,59).
However, two-loop and higher order perturbative expansions of the electron
propagator and its gauge covariance will be sensitive to the choice of the
vertex ansatz. For the sake of comparison, we compute the two-loop
contributions to the electron dressing functions for the bare vertex
($\Gamma_{\mu}=\gamma_{\mu}$) and the BC-vertex, Eqs. (14,15). For this
purpose, we first compute the one-loop corrections to $F$ and ${\cal{M}}$ from
Eqs. (49,50) for the case of a small mass (i.e., we neglect squared mass
terms). The result is used recursively in the gap equation in order to compute
two-loop contributions. Up to leading logarithmic terms, we arrive at the
following expressions:

$\displaystyle F(k^{2})$ $\displaystyle=$ $\displaystyle
1+\beta\log\left(\frac{k^{2}}{\Lambda^{2}}\right)+\frac{1}{2}\Phi_{F}\beta^{2}\log^{2}\left(\frac{k^{2}}{\Lambda^{2}}\right),$
(61) $\displaystyle\frac{{\cal{M}}(k^{2})}{m_{0}}$ $\displaystyle=$
$\displaystyle
1+\lambda\log\left(\frac{\Lambda^{2}}{k^{2}}\right)+\frac{1}{2}\Phi_{\cal{M}}\lambda^{2}\log^{2}\left(\frac{\Lambda^{2}}{k^{2}}\right),$
(62)

where $\beta=\alpha\varrho\left(\kappa-\chi+1/3\right)/\pi$ in consonance with
the one-loop expansion for $F$, Eq. (55), and $\lambda=8\alpha\varrho/(3\pi)$.
Moreover, for the bare vertex we have

$\displaystyle\Phi_{F}$ $\displaystyle=$
$\displaystyle\left(\kappa+\chi-1/3\right)/\left(\kappa-\chi+1/3\right)\,,$
$\displaystyle\Phi_{\cal{M}}$ $\displaystyle=$
$\displaystyle\left\\{23+6\left(\kappa+\chi\right)-3\left(\kappa-\chi\right)\left[8+3\left(\kappa-\chi\right)\right]\right\\}/32\,,$

and for the BC-vertex

$\displaystyle\Phi_{F}$ $\displaystyle=$
$\displaystyle\left(\kappa-\chi+4/3\right)/\left(\kappa-\chi+1/3\right)\,,$
$\displaystyle\Phi_{\cal{M}}$ $\displaystyle=$
$\displaystyle\left\\{40-3\left(\kappa-\chi\right)\left[1+3\left(\kappa-\chi\right)\right]\right\\}/32\,.$
(64)

It is important to notice from Eqs. (61,62) that a power law solution for $F$
and ${\cal{M}}$, cf. Eq. (55), demands $\Phi_{F}=\Phi_{\cal{M}}=1$. However,
there is no shift in $\chi$ and $\kappa$ that ensures this requirement.
Moreover, for both bare and BC vertices, the mass function picks up a gauge
dependence at second order in perturbation theory via $\kappa\sim\xi$ in
$\Phi_{\cal{M}}$. These results show that even the full longitudinal vertex
fails to ensure the local gauge covariance of the electron propagator and
highlights the necessity of incorporating a transverse component for the
electron-photon vertex.

In order to elucidate the analytical structure of the transverse vertex and
its impact on the gauge covariance of the electron propagator, it is worth
realizing that the leading logarithmic contributions to $F$ and ${\cal{M}}$ at
two-loops, Eqs. (61,62), arise from integrals of the form

$\displaystyle\int_{k}^{\Lambda}\frac{dp}{p}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(\frac{\Lambda^{2}}{k^{2}}\right)\,,$
$\displaystyle\int_{k}^{\Lambda}\frac{dp}{p}\log\left(\frac{p^{2}}{\Lambda^{2}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4}\log^{2}\left(\frac{\Lambda^{2}}{k^{2}}\right)\,,$

which in turn correspond to evaluating the integrals in the so-called
asymptotic regime where $p^{2}\gg k^{2}\gg m^{2}_{0}$. Since the two-loop
contributions to the electron propagator are determined by the one-loop
corrections of the electron-photon vertex, in particular its transverse
pieces, we reaffirm that not only a proper treatment of the photon propagator
and an adequate use of the WFGTI are essential to study the gap equation but
an inclusion of a refined transverse vertex is also required to incorporate
the constraints of gauge covariance and perturbation theory as advocated
before as well.

Figure 3: Dynamically generated dimensionless Euclidean mass $M_{E}/\Lambda$
as a function of the electromagnetic coupling $\alpha$ for the bare vertex
ansatz for different values of the covariant gauge parameter $\xi$. The gauge
dependence of $M_{E}/\Lambda$ and $\alpha_{c}$ is noticeably sizeable even in
a small interval of $\xi=$, i.e., (0,1).

###  VI.2 Non-perturbative solution and the DCSB

In order to compute non-perturbative solutions of the electron SDE or the gap
equation, Eq. (LABEL:gap_equation), it is necessary to choose an ansatz for
the electron-photon vertex defined through Eqs. (13,14,18,19). We compute
dynamically generated massive solutions for the electron propagator in the
chiral limit, i.e., when the bare electron mass $m_{0}=0$. In this case, the
electron mass generated is entirely dynamical in nature. In particular, we
compute $M_{E}={\cal M}(k^{2})$ which can be interpreted as the Euclidean
mass. Note that the only mass scale available in quenched RQED is the
ultraviolet regulator $\Lambda$. The dynamically generated mass $M_{E}$ is
naturally proportional to it. Therefore, we choose to plot $M_{E}/\Lambda$ as
a function of the electromagnetic coupling $\alpha$. At $\alpha=\alpha_{c}$, a
DCSB solution emerges, bifurcating away from the perturbative solution
$M_{E}=0$ which corresponds to the fact that no electron mass is generated at
any order in perturbation theory if we start from the bare mass $m_{0}=0$. As
$\alpha_{c}$ separates the phase of massless electrons from that of massive
electrons, we expect it to be gauge invariant just as the Euclidean mass
itself111It is the Minkowski pole mass which should strictly be gauge
invariant. However, as $M_{E}$ lies close to $M(0)$, we would expect it be at
least nearly gauge invariant.. Note that we only depict results after the
appropriate use of the WFGTI in the kernel of the gap equation. To start with,
we compare the results of the following two vertices:

Figure 4: Dynamically generated dimensionless Euclidean electron mass
$M_{E}/\Lambda$ as a function of the electromagnetic coupling $\alpha$ for our
proposed vertex construction for different values of the covariant gauge
parameter $\xi$. Both $M_{E}/\Lambda$ and $\alpha_{c}$ are now practically
gauge independent!

  * •

Bare, with $\lambda_{1}=1$ and $\lambda_{2,3}=\tau_{1,2,...,8}=0$.

  * •

Our ansatz, with $\lambda_{1,2,3}$ defined in Eq. (15) and all $\tau$’s set to
zero except $\tau_{6}$ defined in Eq. (39) with $a_{6}=-1/2$.

Fig. (3) clearly depicts the fact that the bare vertex results in considerable
amount of undesirable gauge dependence in the Euclidean mass $M_{E}$ and
$\alpha_{c}$. We now repeat the exercise for our proposed vertex ansatz which
not only agrees with the asymptotic one-loop result for the full vertex but
also guarantees the wave-function renormalization to be MR in the leading
logarithmic approximation, see Fig. (4).

The gauge independence is almost magically instated! We find that our choice
of the vertex yields $\alpha_{c}(\xi)\approx 0.212$. We confirm that the
$\alpha$-dependence of $M_{E}/\Lambda$ in the close vicinity of $\alpha_{c}$
satisfies Miransky scaling law, see Fig. (5):

$\displaystyle\frac{M_{E}}{\Lambda}={\rm
Exp}\left[-\frac{\pi\kappa_{1}}{\sqrt{\alpha/\alpha_{c}-1}}+\kappa_{2}\right]\,,$
(65)

where $\kappa_{1}=0.051$ and $\kappa_{2}=-12.857$.

Figure 5: Miransky Scaling using our vertex for $\xi=1/3$. Figure 6:
Dynamically generated dimensional Euclidean mass as a function of the
covariant gauge parameter $\xi$ for the bare vertex and our proposed vertex
ansatz. We choose $\alpha=0.35$ to draw the plot. Figure 7: The critical
coupling $\alpha_{c}$ as a a function of the covariant gauge parameter $\xi$
for four different choices of the vertex Ans$\ddot{a}$tze, mentioned in the
text.

At this point, it appears worth its while to plot the dynamical mass
$M_{E}/\Lambda$ as a function of the covariant gauge parameter $\xi$, Fig.
(6). We choose $\alpha=0.35$ merely as a representative value to bring out the
stark difference between the two truncation schemes as regards the gauge
dependence of this physical observable. It is remarkable to note the potency
of our proposal in this connection.

We also plot $\alpha_{c}(\xi)$ to have a more quantitative insight into its
gauge (in)dependence, Fig. (7). Moreover, one might consider it illustrative
to also compare the results with the following two vertices occasionally
adopted in literature:

  * •

Central Ball-Chiu (CBC), with $\lambda_{1}$ defined in Eq. (15) and
$\lambda_{2,3}=\tau_{1,2,...,8}=0$.

  * •

Ball-Chiu (BC), with $\lambda_{1,2,3}$ defined in Eq. (15) and
$\tau_{1,2,...,8}=0$.

Notably the bare vertex and the CBC-vertex perform almost equally badly! The
BC-vertex reduces the gauge dependence. However, our proposal renders
$\alpha_{c}$ practically gauge independent. These observations naturally lead
us to conclude the article on a positive note.

##  VII Conclusions and Perspectives

It is quite satisfactory to observe that a truncation of the infinite tower of
SDEs which respects key features of the underlying quantum field theory of
RQED, namely its discrete symmetries, matching with perturbation theory in the
domain of weak coupling and the MR of the massless electron propagator
expectedly leads to practically gauge invariant results, within the interval
$(0,1)$ of the gauge parameter $\xi$ for the observales under study, i.e., the
dynamically generated Euclidean mass and the critical coupling which marks the
onslaught of DCSB. What is remarkable is that only one transverse form factor
$\tau_{6}(k,p)$ suffices to construct our vertex ansatz and to achieve the
goal we set ourselves. Having constructed a reliable truncation of the
infinite tower of the SDEs, we believe we are now in a pole position to study
the physical properties of Graphene and other relevant materials of interest
within this formalism. This is for future.

## Acknowledgments

This work is partly funded by CONACyT through the post doctoral scholarship
for L. Albino. A. Bashir acknowledges the CIC(UMSNH) grant 4.10. A. J. Mizher
is grateful to the financial support from FAPESP under fellowship number
2016/12705-7 and A. Raya to the funding from the CONACYT Project FORDECYT-
PRONACES/61533/2020.

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