Electron-Photon Vertex and Dynamical Chiral Symmetry Breaking in Reduced QED An Advanced Study of Gauge Invariance L. Albino1A. Bashir1yA.J. Mizher2 3zand A. Raya1 3x

2025-04-29 0 0 836.85KB 11 页 10玖币

Electron-Photon Vertex and Dynamical Chiral Symmetry Breaking in Reduced QED:

An Advanced Study of Gauge Invariance

L. Albino,1, ∗A. Bashir,1, †A.J. Mizher,2, 3, ‡and A. Raya1, 3, §

1Instituto de F´ısica y Matem´aticas, Universidad Michoacana de San Nicol´as de Hidalgo, Morelia, Michoac´an 58040, M´exico.

2Instituto de F´ısica Te´orica, Universidade Estadual Paulista,

Rua Dr. Bento Teobaldo Ferraz, 271-Bloco II, 01140-070, S˜ao Paulo, SP, Brazil.

3Centro de Ciencias Exactas, Universidad del B´ıo-B´ıo. Avda. Andr´es Bello 720, Casilla 447, 3800708, Chill´an, Chile.

(Dated: October 5, 2022)

We study the eﬀect of a reﬁned electron-photon vertex on the dynamical breaking of chiral sym-

metry 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 opera-

tions. 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 gener-

ated 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 [1,2], is a physical

system with immense potential for technological appli-

cations. It has driven a lot of research in both the ap-

plied and theoretical physics, not only from the point of

view of condensed matter and materials sciences [3,4],

but also based on the quantum ﬁeld theoretic descrip-

tion within the domain of high energy physics [5] and

cosmology [6–8]. Its remarkable properties of high elec-

tric and thermal conductivity, stiﬀness, ﬂexibility and

transparency have opened the door to explore a growing

family of modern relativistic and relativistic-like mate-

rials in one, two and three spatial dimensions. The un-

derlying 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 oc-

currence is responsible for Klein tunneling [9] as well as

other exotic and novel phenomena [10] exhibited by rel-

ativistic systems only. That makes graphene an incar-

nation of quantum electrodynamics (QED) in condensed

matter realms. Along with quantum Hall systems [2,11–

13] and high-Tclayered cuprate superconductors [14–18],

graphene is also a system suitable for its description in

terms of relativistic quantum ﬁeld theoretical consider-

ations, developed and reﬁned in the domain of particle

physics [5]. This in turn allows for an exploration of

otherwise inaccessible particle physics phenomenology in

a more controlled and observable ambient of solid state

∗Electronic address: albino.fernandez@umich.mx

†Electronic address: adnan.bashir@umich.mx

‡Electronic address: ana.mizher@unesp.br

§Electronic address: alfredo.raya@umich.mx

physics. A representative example in this connection is

the so-called chiral magnetic eﬀect [19,20], which was

ﬁrst predicted to take place in relativistic heavy ion col-

lisions. 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 ef-

fect was proposed to probe the non-tivial vacuum struc-

ture of quantum chromodynamiocs (QCD). However, it

has not been observed in isobar collisions in the STAR

collaboration at RHIC [21]. 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[22], 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 [23–27].

Some theoretical ideas have also been developed to ob-

serve a similar eﬀect in 2D crystals such as graphene [28,

29]. More recently, it has been shown that in some 2D

materials, one may observe a novel quantum spin Hall

phenomenon [30]. A key ingredient for the realization

of these later phenomena is the description of electro-

magnetic and matter ﬁelds living in mixed dimensions.

Mixed dimensional theories emerge naturally in the de-

scription of 2D materials where experiments are carried

out with external electromagnetic ﬁelds which permeate

the whole space whereas the movement of the charge car-

riers remains restricted to a plane.

Two independent formulations have been proposed in

literature to describe QED of photons and electrons liv-

ing in diﬀerent space-dimensions. One vision exploits the

equivalence of a theory where electrons live in lower di-

mensions than photons and a Chern-Simons theory. It

has been dubbed as Pseudo-QED [31–34]. Alternatively,

a brane-world inspired scenario was developed in [35] to

arXiv:2210.01280v1 [hep-ph] 4 Oct 2022

explore the traits of dynamical chiral symmetry break-

ing (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 es-

tablished. It respects causality [33] and unitarity [34].

It exhibits a Coulomb static interaction in the case of

graphene [32,36]. It contains an infrared ﬁxed point of

the renormalization group as the Fermi velocity tends to

the speed of light [36–39]. Two-loop perturbative analy-

sis has been carried out in [40] and later improved with

renormalization group arguments [41]. Interestingly for

the immediate purpose of our manuscript, DCSB has

been explored within the Schwinger-Dyson [42] 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 pre-

serving ordinary QED3. In the latter theory, it is known

that there exists a critical number of fermion families

Ncabove 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 elec-

tromagnetic coupling αexceeds a critical value αc. 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 [43] in connection with the pres-

ence 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 chi-

ral symmetry. Finally, RQED has also been formulated

in curved spaces [44]. For a review discussing all this

properties and applications of RQED, see [45].

Studying DCSB and its gauge invariance in RQED

naturally requires its non-perturbative treatment. For-

tunately, extensive amount of analogous research in

QED4[46–50] and QED3[51–54] 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. Fo-

cusing on the quenched version of the theory, the sole

source of our starting ansatz is the electron-photon ver-

tex. We construct it by demanding all the key character-

istics of RQED to be faithfully respected:

•Ward-Fradkin-Green-Takahashi identity (WFGTI)

that connects the electron propagator with the lon-

gitudinal part of the electron-photon vertex is satis-

ﬁed non-perturbatively by construction, known as

the Ball-Chiu (BC) vertex [55].

•To expand the transverse part of the vertex, we em-

ploy the vector basis and its coeﬃcients in such a

manner as to ensure spurious kinematic singulari-

ties are absent from our construction [56–58].

•In the weak coupling regime, the vertex faithfully

reproduces its one-loop perturbative expansion for

the asymptotic limit of momenta k2p2, just

it has previously been done in QED4[46,47] and

QED3[59–61].

•The standard model of particle physics tells us of

the intimate connection between its renormalizabil-

ity and gauge invariance. In the same spirit, we re-

quire our vertex ansatz to guarantee the multiplica-

tive 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 con-

jugation.

Based upon the above-mentioned constraints, we are

able to achieve nearly gauge independent Euclidean mass

and critical coupling αcwhere the DCSB solution bi-

furcates away from the chirally symmetric one. We be-

lieve that obtaining gauge independent DCSB holds the

promise to study observable eﬀects in the 2Dmaterials

described by RQED in a reliable manner through contin-

uum SDEs.

The article has been organized as follows: Sect. II be-

gins with a brief introduction to the mathematical foun-

dations of RQED. In Sect. III, we provide necessary pre-

liminaries on the vertex decomposition and its general

features. In Sect. IV, we construct a family of Ans¨atze

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 propa-

gator in RQED and the appropriate use of the WFGTI in

order to ensure the MR of the massless electron propaga-

tor. 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 αcabove

which the massive solution bifurcates away from the per-

turbative 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 ini-

tially begins with the well-known QED Lagrangian:

LQED =¯

ψ(iγµ∂µ−m0)ψ+jµAµ

−1

4Fµν Fµν −1

2ξ(∂µAµ)2,(1)

where ψand Aµare electron and photon ﬁelds, re-

spectively, coupled to each other through the electro-

magnetic current jµ, where µ= 0,1,2,3. Further-

more, the 4-dimensional Dirac matrices γµsatisfy the

anti-commutation relation {γµ, γν}= 2gµν , with the

commonly used convention gµν = (+,−,−,−) for the

Minkowski space metric tensor. Additionally, m0is the

bare electron mass, ξis the covariant gauge parameter

and Fµν =∂µAν−∂νAµis the usual electromagnetic ﬁeld

tensor. The corresponding action SQED =Rd4xLQED

can be conveniently expressed as

SQED =S(3)

ψψ +1

2Zd4xhjµˆ

∆µν jν−Aµˆ

∆−1

µν Aνi,(2)

where the kinetic term for electrons, constrained to move

in a 3-dimensional space-time reads:

S(3)

ψψ =Zd3x¯

ψ(iγµ∂µ−m0)ψ , (3)

with µ= 0,1,2. Moreover, the diﬀerential operator ˆ

∆µν ,

namely the photon propagator in coordinate space, can

be cast in terms of its momentum space counterpart by

means of a Fourier transformation

∆µν =−Zd4q

(2π)4e−iq·x1

q2gµν −(1 −ξ)qµqν

q2,(4)

satisfying ˆ

∆−1

µα ˆ

∆αν =gν

µwith its corresponding inverse

propagator. Such a Green function accounts for a gauge

ﬁeld propagating through the whole 4-dimensional space-

time with µ, ν = 0,1,2,3.

To account for a mixed-dimension system described by

RQED with electrons restricted to move on a plane per-

pendicular to the x3-axis [31–35,62], the electromagnetic

current takes the form

jµ=−ie ¯

ψγµψδ(x3) for µ= 0,1,2,

0 for µ= 3 .(5)

Therefore, only the indices µ= 0,1,2 contribute to the

term jµˆ

∆µν jνin Eq. (2). The component µ= 3 can thus

be integrated out in Eq. (4), leading to [63]

∆µν =Zd3q

(2π)3e−iq·x1

2p−q2gµν −(1 −ξ)qµqν

2q2.(6)

It entails a non-local diﬀerential operator. This propaga-

tor can be obtained from the eﬀective action for RQED

(redeﬁning ξ= 2ζ−1)

SRQED =S(3)

ψψ +Zd3x jµAµ

Zd3x1

2Fµν

√−∂2Fµν +1

ζ∂µAµ1

√−∂2∂νAν.(7)

From now on, we work in the Euclidean space deﬁned

by the metric tensor δµν = (+,+,+) for µ, ν = 4,1,2.

In this space, the bare photon propagator takes the form

(c.f. Eq. (6))

∆(0)

µν (q) = 1

2qδµν −(1 −ξ)qµqν

2q2,(8)

where we have deﬁned q≡p−q2. Note that this prop-

agator has a softer infrared behavior than the photon

propagator in QED4and QED3. Several groups diﬀer in

conventions by the global factor of 1/2. When written in

terms of the variable ζ, this propagator can be separated

into a familiar longitudinal and a transverse component

(to qµ):

∆(0)

µν (q) = 1

2qδµν −qµqν

q2+ζqµqν

2q3,(9)

However, it must be emphasized that the gauge ﬁxing

parameter ζin RQED is diﬀerent from that in QED, ξ,

due to dimensional reduction. In the above expression

for the photon propagator, Eq. (9), the ζ-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 ξ-independent term in Eq. (8) does not de-

ﬁne a transverse propagator. Therefore, quantum correc-

tions aﬀect both ξ-dependent and ξ-independent compo-

nents. 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. (8,9) are equally suitable: we choose that of Eq. (8).

Note that the electron-photon vertex plays a vital role

in computing the non-perturbative solution of the elec-

tron propagator through its gap equation. Therefore, we

address this three-point Green function at length in the

following section.

III. THE VERTEX: GENERALITIES

q=k−p

Γµ

FIG. 1: Diagrammatic representation of the full electron-

photon vertex Γµ(k, p), with momentum ﬂow indicated.

In its general decomposition, the three-point electron-

photon vertex can be written in terms of twelve indepen-

dent spin structures. For the kinematical conﬁguration

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Electron-PhotonVertexandDynamicalChiralSymmetryBreakinginReducedQED:AnAdvancedStudyofGaugeInvarianceL.Albino,1,A.Bashir,1,yA.J.Mizher,2,3,zandA.Raya1,3,x1InstitutodeFsicayMatematicas,UniversidadMichoacanadeSanNicolasdeHidalgo,Morelia,Michoacan58040,Mexico.2InstitutodeFsicaTeorica,Universid...

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Electron-Photon Vertex and Dynamical Chiral Symmetry Breaking in Reduced QED An Advanced Study of Gauge Invariance L. Albino1A. Bashir1yA.J. Mizher2 3zand A. Raya1 3x

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