Transverse Takahashi Identities and Their Implications for Gauge Independent Dynamical Chiral Symmetry Breaking

TL;DR

This paper uses transverse Takahashi identities to impose non-perturbative constraints on the fermion-photon vertex, ensuring gauge invariance and multiplicative renormalizability, and analyzes dynamical chiral symmetry breaking in QED.

Contribution

It introduces a framework incorporating transverse Takahashi identities to improve the understanding of gauge invariance and chiral symmetry breaking in quantum electrodynamics.

Findings

Excellent fit to Miransky scaling law.

Natural emergence of anomalous dimension γ_m=1.

Numerical confirmation of gauge invariance of the critical coupling.

Abstract

In this article, we employ transverse Takahashi identities to impose valuable non-perturbative constraints on the transverse part of the fermion-photon vertex in terms of new form factors, the so called $Y_i$ functions. We show that the implementation of these identities is crucial in ensuring the correct local gauge transformation of the fermion propagator and its multiplicative renormalizability. Our construction incorporates the correct symmetry properties of the $Y_i$ under charge conjugation operation as well as their well-known one-loop expansion in the asymptotic configuration of incoming and outgoing momenta. Furthermore, we make an explicit analysis of various existing constructions of this vertex against the demands of transverse Takahashi identities and the previously established key features of quantum electrodynamics, such as gauge invariance of the critical coupling above which chiral symmetry is dynamically broken. We construct a simple example in its quenched version and compute the mass function as we vary the coupling strength and also calculate the corresponding anomalous dimensions $\gamma_m$. There is an excellent fit to the Miransky scalling law and we find $\gamma_m=1$ rather naturally in accordance with some earlier results in literature, using arguments based on Cornwall-Jackiw-Tomboulis effective potential technique. Moreover, we numerically confirm the gauge invariance of this critical coupling.

Figures

Tables

Table 1. Table 1: Parameters for different gauges

$\xi$	${\alpha }_c=\pi /3$	$\kappa$	$\varphi$
$1$	$0.69\%$	$0.695$	$0.481$
$0$	$0.82\%$	$0.970$	$1.383$
$-1$	$0.51\%$	$1.112$	$1.604$
$-2$	$0.51\%$	$1.199$	$1.679$
$-3$	$2.81\%$	$1.283$	$1.829$

Equations

$$i{q}_{\mu }{\Gamma }_{\mu }(k,p)=S^{-1}(k)-S^{-1}(p),$$

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

$${\Gamma }_{\mu }^L(k,p)=a(k^2,p^2){\gamma }_{\mu }+\frac{b(k^2,p^2)}{2}{t}_{\mu }\gamma \cdot t-ic(k^2,p^2)t_{\mu }$$

$$a(k^2,p^2)$$

$$b(k^2,p^2)$$

$$c(k^2,p^2)$$

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

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

$${\Gamma }_{\mu }^T(k,p)=\sum _{i=1}^8{\tau }_i(k,p)T_{\mu }^i(k,p).$$

$$T_{\mu }^1(k,p)$$

$$T_{\mu }^2(k,p)$$

$$T_{\mu }^3(k,p)$$

$$T_{\mu }^4(k,p)$$

$$T_{\mu }^5(k,p)$$

$$T_{\mu }^6(k,p)$$

$$T_{\mu }^7(k,p)$$

$$T_{\mu }^8(k,p)$$

$${\sigma }_{\nu \rho }=\frac{i}{2}\left[{\gamma }_{\nu },{\gamma }_{\rho }\right].$$

$${\tau }_i(k,p)={\tau }_i(p,k),i=1,2,3,5,7,8,$$

$${\tau }_i(k,p)=-{\tau }_i(p,k),i=4,6.$$

$$q_{\mu }{\Gamma }_{\nu }(k,p)-q_{\nu }{\Gamma }_{\mu }(k,p)$$

$$q_{\mu }{\Gamma }_{\nu }^A(k,p)-q_{\nu }{\Gamma }_{\mu }^A(k,p)$$

$$T_{\mu \nu }^1$$

$$T_{\mu \nu }^2$$

$$q\cdot tt\cdot \Gamma (k,p)$$

$$q\cdot t\gamma \cdot \Gamma (k,p)$$

$$i{T}_{\mu \nu }^1{V}_{\mu \nu }^A$$

$$i{T}_{\mu \nu }^2{V}_{\mu \nu }^A$$

$${\tau }_1(k,p)$$

$${\tau }_2(k,p)$$

$${\tau }_3(k,p)$$

$${\tau }_4(k,p)$$

$${\tau }_5(k,p)$$

$${\tau }_6(k,p)$$

$${\tau }_7(k,p)$$

$${\tau }_8(k,p)$$

$$\nabla (k,p)=k^2{p}^2-(k\cdot p{)}^2.$$

$$Y_i(k,p)=Y_i(p,k),i=2,6^S,7^S,8^S,$$

$$Y_i(k,p)=-Y_i(p,k),i=1,3,4,5,6^A,7^A,8^A,$$

$$Y_i(k,p)=Y_i^S(k,p)+Y_i^A(k,p),$$

$$Y_6^S(k,p)$$

Full text

Schwinger-Dyson equations, QED, three-point fermion-photon vertex, transverse Takahashi indentity, multiplicative renormalizability

I Introduction

A quantum field theory (QFT) can be considered completely solved if we are able to compute its full set of $n$-point Green functions. However, these Green functions are infinite in number. Moreover, these are all intertwined through highly non-linear coupled integral equations, known as the Schwinger-Dyson equations (SDEs). A brute force method to compute them is a wild goose chase.

We believe that a satisfactory determination of relevant physical observables through a systematic truncation scheme for this infinite tower of equations is achievable if we preserve the key features and symmetries of the underlying theory. Perturbation theory (PT) provides an excellent example of such an approximation scheme. However, when the interaction strength grows and can no longer be used as a perturbative expansion parameter, one resorts to truncations which need to be carefully constructed in order to retain the essential features of the original theory, while maintaining contact with experimental data at the same time. Quantum chromodynamics (QCD) is a realization of this scenario in its infrared domain. Considerable progress has been made in the last decades to study its first few Green functions, e.g., the gluon propagator Boucaud et al. (2008); Aguilar et al. (2008); Pennington and Wilson (2011); Ayala et al. (2012); Aguilar et al. (2012); Strauss et al. (2012); Bashir et al. (2013); Huber (2015) and the quark-gluon vertex Alkofer et al. (2009); Chang and Roberts (2009); Kizilersu and Pennington (2009); Bashir et al. (2012a); Hopfer et al. (2012); Rojas et al. (2013); Aguilar et al. (2014); Peláez et al. (2015); Williams et al. (2016); Aguilar et al. (2018) whose knowledge consequently provides predictions for QCD and hadron physics, e.g., Alkofer et al. (2008); Chang et al. (2013); Williams (2015); Raya et al. (2015); Gómez-Rocha et al. (2016); also see reviews Bashir and Raya (2004); Bashir et al. (2012b); Aznauryan et al. (2013); Eichmann et al. (2016); Huber (2018) and references therein.

In several hadronic physics studies, such as electromagnetic and transition form factors Chang et al. (2013); Raya et al. (2015, 2017); Weil et al. (2017); Eichmann et al. (2017); Ding et al. (2018), probes are generally electromagnetic in nature and many SDE calculations crucially rely on how photons interact with quarks. Thus, quantum electrodynamics (QED) serves as a useful platform to study SDE truncations and provide improvements to preserve its key features, such as its gauge invariance, renormalizability and the recuperation of the well-known S-matrix perturbative expansion for its Green functions in the weak coupling regime, which it maintains at all accessible energies. In particular, study of the fermion propagator in QED generally amounts to requiring a physically meaningful and reliable Ansatz for the three-point fermion-photon vertex. Gauge invariance provides an essential ingredient in this connection. The gauge technique of Salam, Delbourgo and collaborators was developed to solve the constraints of the well-known Ward-Fradkin-Green-Takahashi identity (WFGTI) Ward (1950); Fradkin (1955); Green (1953); Takahashi (1957), writing the Green functions in terms of spectral representations, Salam and Delbourgo (1964); Delbourgo and West (1977); Delbourgo (1979). However, such approach, despite its elegant and formal results, Delbourgo et al. (1981); Delbourgo and Zhang (1984), is not amicable to straightforward computations, Rembiesa (1986). The WFGTI allows us to expand out the vertex in terms of a well-constrained longitudinal part Ball and Chiu (1980) and an undetermined transverse part. Several efforts alternative to the gauge technique start from making an Ansatz for this latter part and proceed from thereon.

A natural question to ask is if the transverse part can be constrained through any other symmetry principle?

Whereas the usual WFGTI relates the divergence of the three-point fermion-photon vertex to the inverse fermion propagator, there exist transverse Takahashi identities (TTI) which play a similar role for the curl of the fermion-photon vertex Takahashi (1985); Kondo (1997); He et al. (2000); He (2006, 2007). However, in addition to the inverse fermion propagator and the vector vertex, these identities also bring into play a non-local axial-vector vertex as well as new inhomogeneous tensor and axial-tensor vertices. Consequently, TTI are richer and more complicated in their structure. In past, they have been verified to one-loop order, Pennington and Williams (2006); He and Khanna (2006). More recently, practical implications of TTI have been investigated in Qin et al. (2013, 2014) to get insight into the non-perturbative forms of vector and axial-vector vertices.

In this article, we intend to study constraints of TTI on the transverse part of the fermion-photon vertex. Note that TTI do not modify the usual WFGTI in any way. However, we realize that they are crucially connected to another consequence of local gauge covariance, namely, Landau-Khalatnikov-Fradkin transformations (LKFT), derived in Landau and Khalatnikov (1956); Fradkin (1955); Johnson and Zumino (1959); Zumino (1960). LKFT are a well defined set of transformations which describe the response of the Green functions to an arbitrary gauge transformation. These transformations leave the SDEs and the WFGTI form-invariant. LKFT potentially play an important role in imposing valuable constraints on the fermion-photon vertex and obtaining gauge invariant chiral symmetry breaking, see for example Refs. Burden and Roberts (1993); Bashir et al. (1999); Bashir (2000); Bashir and Raya (2002); Bashir and Delbourgo (2004); Fischer et al. (2004); Bashir and Raya (2007); Bashir et al. (2006, 2007, 2008, 2009); Ahmadiniaz et al. (2016); Ahmad et al. (2016); Jia and Pennington (2016, 2017a, 2017b). More recently, these transformations have also been derived for QCD De Meerleer et al. (2018); Aslam et al. (2015).

Both the TTI and the LKFT (through the multiplicative renormalizability (MR) of the fermion propagator) constrain the transverse fermion-photon vertex. Therefore, it is reasonable to seek a combined constraint which would help us converge on pinning down this elusive part of the vertex. The fact that MR constrains the transverse vertex has already been known for some time  Parker (1984); Delbourgo and Zhang (1984); Cornwall (1982); King (1983). Later works in the literature involving similar considerations in constructing a refined fermion-photon vertex can be found in Curtis and Pennington (1990, 1991, 1993); Dong et al. (1994); Bashir and Pennington (1994, 1996); Bashir et al. (1998); Kizilersu and Pennington (2009); Fernandez-Rangel et al. (2016).

An important issue relevant to our current work concerns the usage of the TTI-constrained vertex to study dynamical chiral symmetry breaking (DCSB) or dynamical fermions mass generation as a consequence of enhanced interaction strength. This is a strictly non-perturbative phenomenon and a transcendental topic in QCD, where it induces measurable effects in numerous hadron observables. Therefore, physically meaningful truncations of QCD’s SDEs demand incorporation of DCSB through the relevant Green functions, in particular the quark propagator and the quark-gluon vertex. Regarding the latter, valuable progress has been made both in lattice, Skullerud et al. (2003, 2005); Kizilersu et al. (2007); Oliveira et al. (2016); Sternbeck et al. (2017); Oliveira et al. (2018a, b) and continuum studies. However, due to the non-abelian nature of QCD, investigating the impact of DCSB on the quark-gluon vertex, and vice versa, from the first principles, is still a theoretical challenge. A thorough investigation of the fermion-photon vertex and chiral symmetry breaking in QED is likely to provide a bench mark for the corresponding studies in QCD.

Although QED manifests a perturbative behavior at all observable scales, an intense background electromagnetic field can trigger a transition from perturbative to non-perturbative dynamics, the well-known magnetic catalysis, see for example Gusynin et al. (1995a, b); Lee et al. (1997); Hong (1998); Ferrer and de la Incera (2000); Ayala et al. (2006); Rojas et al. (2008); Watson and Reinhardt (2014); Mueller and Pawlowski (2015). Even a toy QED with an artificially scaled up coupling exhibits this phenomenon. Such a phase transition has long been studied. It is characterized by a critical coupling, $\alpha_{c}$, above which DCSB takes place, see Curtis and Pennington (1990); Bashir et al. (2012a); Kizilersu et al. (2015) and references therein. Since this critical coupling corresponds to a recognizable phase transition, it is considered to be a physical observable, and hence a gauge invariant parameter. This independence of $\alpha_{c}$ on the gauge parameter has long been used as a further requirement to constrain the transverse vertex Bashir et al. (2012a), and we follow this argument in the present article.

The TTI connect the transverse structure of the fermion-photon vertex to a set of unknown scalar functions $Y_{i}$ related to a non-local axial-tensor mentioned before. MR of the electron propagator implies that these functions cannot be ignored. Instead, MR constrains their form. This procedure involves an unknown function $W(x)$ of a dimensionless ratio $x$ of the incoming and outgoing fermion momenta. It satisfies an integral constraint which guarantees MR of the electron propagator in the leading logarithm approximation (LLA). Implementing charge conjugation symmetry on the integration kernels involved in the fermion propagator SDE, it is possible to parameterize $Y_{i}$ in terms of one single scalar, a priori unknown function ${\cal{T}}(k^{2},p^{2})$, which encodes the effect of the fully-dressed fermion-photon vertex on the fermion propagator. This general procedure fixes three of the $Y$-functions. An additional constraint comes from demanding gauge independent chiral symmetry breaking. For the so called quenched approximation, it yields a self-consistent solution for ${\cal{T}}(k^{2},p^{2})$.

In this article, we work in Euclidean space. Thus, for $\gamma$-matrices we have: $\left\{\gamma_{\mu},\gamma_{\nu}\right\}=2\delta_{\mu\nu}$ and $\gamma^{\dagger}_{\mu}=\gamma_{\mu}$, where $\delta_{\mu\nu}$ is the Euclidean metric. Furthermore, we define $\gamma_{5}=\gamma_{4}\gamma_{1}\gamma_{2}\gamma_{3}$, with $\hbox{Tr}\left[\gamma_{5}\gamma_{\mu}\gamma_{\nu}\gamma_{\alpha}\gamma_{\beta}\right]=-4\epsilon_{\mu\nu\alpha\beta}$.

This paper is organized as follows: in Section II, we review the WFGTI for the three-point vertex in QED, define the longitudinal vertex, write down the transverse part in a general basis, following Ball and Chiu (1980), and highlight the symmetry properties of the transverse form factors under charge conjugation operation. In Section III, we introduce abelian TTI for the vertex and expand out the transverse form factors in terms of the $Y$-functions. We then invert these relations and impose a perturbative constraint on $Y_{i}$ in the asymptotic limit of $k^{2}\gg p^{2}$. In Section IV, SDE for the fermion propagator is presented. We write it in terms of the $Y_{i}$ functions and discuss the quenched approximation. In Section V, We study the requirement of MR and the power law solution for the wave function renormalization of the fermion propagator within the LLA. We show how the requirement of MR imposes an integral constraint on the form factors of the transverse vertex, in terms of the function $W(x)$. In Section VI, we discuss a few examples illustrating the need and importance of the $Y$-functions. In Section VII, we construct simple examples to study DCSB and see how it naturally incorporates gauge independence of the critical coupling $\alpha_{c}$. In Section VIII, we present our conclusions and discuss prospects for future research.

II Vertex decomposition

In its general decomposition, the three-point fermion-boson vertex can be written in terms of 12 independent spin structures. For the kinematical configuration of Fig. 1, the WFGTI associated with this vertex takes the form

[TABLE]

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 and Chiu (1980):

[TABLE]

The longitudinal part $\Gamma^{L}_{\mu}(k,p)$ alone satisfies the WFGTI (1), and consumes four of the twelve independent spin structures (one of them is zero in QED), so that, Ball and Chiu (1980):

[TABLE]

with $t=k+p$, and

[TABLE]

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

[TABLE]

At the tree level, $F(k^{2},\Lambda^{2})=1$ and ${\cal{M}}(k^{2},\Lambda^{2})=m_{0}$, where $m_{0}$ is the bare mass of the fermion.

The transverse part $\Gamma^{T}_{\mu}(k,p)$ of the vertex decomposition (2), which remains undetermined by the WFGTI, is naturally constrained by

[TABLE]

In general, the ultraviolet finite transverse vertex can be expanded out in terms of 8 vector structures, and their corresponding scalar form factors $\tau_{i}(k,p)$ Ball and Chiu (1980):

[TABLE]

Moreover, for the kinematical configuration of Fig. 1, we define

[TABLE]

with

[TABLE]

This basis is not exactly the one adopted in Ball and Chiu (1980). We choose to work with a modification of this initial basis which was put forward in Kizilersu et al. (1995) and later employed in Davydychev et al. (2001) as well. This latter choice 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 Section 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 (7) to be symmetric under the interchange $k\leftrightarrow p$, except $\tau_{4}$ and $\tau_{6}$, which are odd:

[TABLE]

From Eq. (4), it is obvious that $a(k^{2},p^{2})$, $b(k^{2},p^{2})$ and $c(k^{2},p^{2})$ 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.

Although the longitudinal scalar functions (4) are fixed by the WFGTI (1), the transverse scalar functions in decomposition (7) remain unknown. In the next section, we introduce the TTIs for the three-point vertex in QED, which provide a powerful tool in constructing these non-perturbative transverse functions.

III Transverse Takahashi identities

The TTIs for vector ($\Gamma_{\mu}$) and axial-vector ($\Gamma_{\mu}^{A}$) vertices in QED, related to a fermion with bare mass $m_{0}$, read Qin et al. (2013):

[TABLE]

where $\sigma^{5}_{\mu\nu}=\gamma_{5}\sigma_{\mu\nu}$, and $\Gamma_{\mu\nu}(k,p)$ is an inhomogeneous tensor vertex. The last two tensor structures in Eqs. (12,13), $A^{V}_{\mu\nu}$ and $V^{A}_{\mu\nu}$, are related to the momentum space expressions for non-local axial-vector and vector vertices, whose definitions involve a gauge-field-dependent line integral. These non-perturbative identities are valid for any covariant gauge, and they do not have explicit dependence on the covariant gauge parameter.

The vector and axial-vector TTIs are intricately coupled to each other via the non-local terms $A^{V}_{\mu\nu}(k,p)$ and $V^{A}_{\mu\nu}(k,p)$, which are complicated even at one-loop order, Pennington and Williams (2006); He and Khanna (2006). Following the procedure described in Ref. Qin et al. (2013), useful progress has been made to disentangle this interdependence. In order to project out transverse form factors from the TTIs, Eqs. (12,13), it is convenient to introduce the following tensors

[TABLE]

By contracting the axial-vector identity (13) with tensors (14) and (15), the left-hand sides of the resulting equations reduce to zero, while the right-hand sides yield the following result:

[TABLE]

These expressions only involve the vector vertex $\Gamma_{\mu}(k,p)$, and do not contain explicit dependence on the fermion mass $m_{0}$. Information about the axial-vector vertex $\Gamma^{A}_{\mu}(k,p)$ can be obtained through analogous procedure involving the axial-vector TTI, Eq. (12). Although the terms $T^{1}_{\mu\nu}V_{\mu\nu}^{A}$ and $T^{2}_{\mu\nu}V_{\mu\nu}^{A}$ are still equally unknown, they are Lorentz scalar objects and can thus be conveniently expressed as follows:

[TABLE]

where $Y_{i}(k,p)$ are hitherto unconstrained scalar functions, and $\mathbf{I}_{D}$ is the identity matrix. Projections of Eqs. (16,17) lead to a set of eight linearly independent, coupled linear equations that fix the eight transverse scalar functions $\tau_{i}$ in terms of the $Y$-functions defined via Eqs. (18,19).

From Eqs. (7,8,16-19), it is possible to project out the scalar form factors $\tau_{i}$:

[TABLE]

where we have employed the obvious simplifying notation $Y_{i}\equiv Y_{i}(k,p)$. Moreover, we have introduced the Gram determinant

[TABLE]

In addition, the vertex transformation properties under charge conjugation determine the symmetry properties of the $Y$-functions:

[TABLE]

where we conveniently introduce the decomposition

[TABLE]

for $i=6,7,8$, where the superscripts $S$ and $A$ stand for the symmetric and antisymmetric parts of the corresponding $Y_{i}$s, under $k\leftrightarrow p$. Note that in Eqs. (20-27), there is no contribution of $Y_{6}^{S}$, $Y_{7}^{A}$ and $Y_{8}^{S}$. This is a consequence of the properties (10) and (11), which entail

[TABLE]

It is also worth noting that the trivial choice $Y_{i}(k,p)=0$ for all $Y$-functions completely fixes the transverse vertex, defined through Eqs. (7,8,21,27), in terms of the fermion wave function renormalization, as reported in ref. Qin et al. (2013). However, we shall show that MR of the electron propagator implies that these $Y$-functions cannot all be zero simultaneously.

We can invert relations (20-27) to write out the $Y$-functions in terms of $\tau_{i}$:

[TABLE]

Here, we have conveniently defined:

[TABLE]

We expect the study in terms of $Y_{i}(k,p)$ to be numerically amicable as the additional factor of $(k^{2}-p^{2})$ in the numerator eases out any kinematical singularities in the limit $k^{2}\rightarrow p^{2}$.

So far, we have shown that the TTIs relate the transverse vertex form factors to the fermion propagator and a non-local tensor vertex, but nevertheless this is not enough to elucidate the analytical behavior of the $Y$-functions. It is insightful to analyze the asymptotic behavior of the vertex. It has been shown that 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 reads (in our kinematical configuration) as Curtis and Pennington (1991):

[TABLE]

where $\xi$ is the gauge-fixing parameter, and

[TABLE]

On the other hand, from Eqs. (7,8,20-27), it is straightforward to see that the leading structure of the transverse vertex in the asymptotic limit acquires the following form:

[TABLE]

where we have used the fact that the one-loop expansion of the wave function renormalization yields $F(k^{2})=1+\beta\log(k^{2}/\Lambda^{2})$, where $\beta$ is a constant of order ${\cal{O}}(\alpha)$: we shall show in the next section that $\beta=\alpha\xi/(4\pi)$. Hence, the leading logarithmic expansion for the asymptotic limit of the vertex, Eq. (43), demands

[TABLE]

which must be fulfilled at least to second order of its perturbative expansion in powers of $k^{2}/p^{2}$, in order to ensure the correct asymptotic limit of the transverse vertex.

Although the TTIs, and in particular the identities (16) and (17), are potentially able to fix the transverse vertex, the construction of an Ansatz for this vertex is far from being complete since the $Y$-functions remain unknown. Additional requirements need to be implemented in order to compute them. In this spirit, we shall use the argument of MR for the fermion propagator, in the chirally symmetric limit, in order to derive an integral constraint for these $Y$-functions. We shall also restrict the structure of the vertex by implementing symmetry arguments and demanding a gauge independent breaking of chiral symmetry. To this end, we introduce the SDE for the fermion propagator in the next section.

IV Gap equation

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

Mathematically, the gap equation is written as:

[TABLE]

where the subscript $E$ indicates that the integral is performed in the Euclidean space, $\alpha=e^{2}/4\pi$ is the electromagnetic coupling, and $\Delta_{\mu\nu}(q)$ is the fully-dressed photon propagator. For an arbitrary gauge, it is defined as

[TABLE]

where $\Delta(q^{2})$ is the photon propagator dressing function. The subscript “0” in the first term of the right-hand side of Eq. (LABEL:gap_equation) denotes the tree level fermion propagator.

Recall from Eq. (5) that the fermion propagator is defined by the wave function renormalization and the mass function, so the gap equation, Eq. (LABEL:gap_equation), can be decomposed into two coupled, integral equations for ${\cal{M}}$ and $F$, which, in an arbitrary gauge, are respectively written as:

[TABLE]

where we have adopted the notation $F(k^{2})\equiv F(k^{2},\Lambda^{2})$ and conveniently defined

[TABLE]

The functions $G_{F}$ and $G_{{\cal{M}}}$ in Eqs. (LABEL:ProyM,LABEL:ProyF) encode the effective contribution of the fully-dressed fermion-photon vertex to the corresponding equations, and they are defined as

[TABLE]

where

[TABLE]

Moreover, we have defined massive ($\Lambda_{M}$) and non-massive ($\Lambda_{NM}$) functions as

[TABLE]

The reason for referring to $\Lambda_{M}$ and $\Lambda_{NM}$ as massive and non-massive functions, respectively, is the following: from Eqs. (LABEL:ProyM,52) it is straightforward to see that in the chiral limit, where $m_{0}=0$, a massless solution (${\cal{M}}=0$) is trivially achieved if $\Lambda_{M}$ has the mass function as a global factor, i.e. $\Lambda_{M}\sim{\cal{M}}$. On the other hand, the function $\Lambda_{NM}$ does not follow this argument, and therefore it does not have a dependence on ${\cal{M}}$ as a global factor. In the same spirit, we shall refer to $Y_{1}$, $Y_{6}^{A}$ and $Y_{7}^{S}$ as massive functions and $Y_{2}$, $Y_{3}$, $Y_{5}$ and $Y_{8}^{A}$ as massless functions.

Note that the contribution of $Y_{4}$ in Eqs. (LABEL:ProyM,LABEL:ProyF) cancels out. This is an indication that the vertex cannot be completely extracted solely from the fermion propagator SDE. Nonetheless, $Y_{4}$ can be modelled by relying on additional information, e.g., the expected anomalous electromagnetic moment for the corresponding fermion.

IV.1 Quenched QED

The system of Eqs. (LABEL:ProyM,LABEL:ProyF) has long been studied using different models for the photon propagator, see Ref. Kizilersu et al. (2015) and references therein. For the sake of simplicity, we limit this work to the well-known quenched approximation (qQED), where fermion loop contributions to the photon SDE are neglected and the coupling does not run, which in turn yields

[TABLE]

As we mentioned before, one of the goals of the present article is to study the impact of the transverse vertex on the DCSB and vice versa. In particular, we shall investigate the constraints imposed by demanding a gauge independent DCSB in Section VII. For this purpose, from now on, we focus our attention on the chiral limit ($m_{0}=0$) since this is the most insightful scenario to elucidate how QED undergoes a phase transition from perturbative to non-perturbative dynamics as we increase the electromagnetic coupling ($\alpha$) up to the critical value ($\alpha_{c}$) where DCSB is triggered. For $\alpha<\alpha_{c}$ the only possible solution to Eq. (LABEL:ProyM) in the chiral limit is ${\cal{M}}(k^{2})=0$, but as $\alpha\rightarrow\alpha_{c}$ a second non-zero solution bifurcates away from the trivial one. The theoretical prediction for the critical coupling above which DCSB takes place can be extracted from Eq. (LABEL:ProyM) through implementing bifurcation analysis.

In the vicinity of the critical coupling $\alpha\sim\alpha_{c}$, the dynamically generated fermion mass is rather small in comparison with any other mass scale. Therefore, quadratic and higher terms in the mass function can formally be neglected. In this case, Eq. (LABEL:ProyF) for $F$ and consequently its solution, reduce to that of a massless theory. Thus, the survey of the renormalization properties of the fermion propagator in massless QED, and the corresponding implications on the fermion-photon vertex, is mandatory.

In the next section we show that for a massless fermion in quenched QED, the wave function renormalization possesses a power law behavior, which is multiplicatively renormalizable. We also derive a non-perturbative constraint on the non-massive $Y$-functions that ensures a MR solution for $F$.

V MR constraints

It is well-known that in QED the gap equation (LABEL:gap_equation) leads to a fermion propagator that is logarithmically divergent. However, we can define renormalized propagators by absorbing these divergences into the renormailzation constants ${\cal Z}_{i}$. For massless QED, this multiplicative renormalization is accomplished by introducing renormalized fields, fermion field $\psi_{R}=\mathcal{Z}_{2}^{-1/2}\psi$, photon field $A_{\mu}^{R}=\mathcal{Z}_{3}^{-1/2}A_{\mu}$, and also renormalized coupling $e_{R}=\mathcal{Z}_{2}\mathcal{Z}_{3}^{1/2}e/\mathcal{Z}_{1}$. Thus, the MR of the fermion propagator requires renormalized $F_{R}$ to be related to unrenormalized $F$ through

[TABLE]

where $\mu$ plays the role of an arbitrary renormalization scale. In order to solve Eq. (58), the functions involved are expanded as perturbative series containing terms of the form $\alpha^{n}\ln^{n}$ (called leading logarithmic terms). This is known as the leading log approximation (LLA). In the LLA, we then have

[TABLE]

where $A_{n}$, $B_{n}$ and $C_{n}$ are unknown coefficients but can be calculated in perturbation theory to any desired order. However, MR condition (58) restricts the coefficients to be interrelated as follows:

[TABLE]

so that the functions $F$, $F_{R}$ and $\mathcal{Z}_{2}$ obey a power law behavior. Then, the infinite order solution of (59) for $F$ can be summed up as follows:

[TABLE]

where we define $\beta=\alpha A_{1}$. This is the LLA. Beyond it, $\beta$ would have terms of ${\cal O}(\alpha^{2})$. Naturally, PT allows us to evaluate the anomalous dimension $\beta$ at different orders of approximation.

The one-loop contribution to the fermion propagator can be evaluated by taking the tree level expressions for $S(p)$, $\Gamma_{\mu}(k,p)$ and $\Delta_{\mu\nu}(q)$ on the right-hand side of Eq. (LABEL:gap_equation). In the massless limit, ${\cal{M}}=0$, and the resulting expression for $F$ is

[TABLE]

Angular integration of Eq. (64) leads to

[TABLE]

Carrying out radial integration in Eq. (65) yields

[TABLE]

Comparing expression (66) with the perturbative expansion (59) to one-loop order, we see that $A_{1}=\xi/4\pi$. Therefore, PT fixes the ${\cal O}(\alpha)$ anomalous dimension in Eq. (63) to be

[TABLE]

The power law behavior of $F$ in Eq. (63), with $\beta$ given in Eq. (67), is the solution of

[TABLE]

Note that Eq. (68) is non-perturbative in nature and serves as a requirement of MR for the wave function renormalization $F$: any Ansatz for the three-point vertex must guarantee that the wave function renormalization $F$ in Eq. (LABEL:gap_equation) satisfies Eq. (68). We shall now proceed to show how the requirement of MR for the fermion propagator, embodied in Eq. (68), constrains the massless $Y$-functions.

In the massless limit, Eq. (LABEL:ProyF) reduces to

[TABLE]

Angular integration of the last term on the right-hand side of the above Eq. (69) vanishes in qQED, since

[TABLE]

where $\varphi$ is the angle between $k$ and $p$. Bearing in mind the latter result, Eq. (70), it is straightforward to see from Eq. (69) that if we set, quite generally,

[TABLE]

${\cal{T}}$ being an a priori arbitrary, dimensionless function of $k^{2}$ and $p^{2}$ alone, then Eq. (68) is trivially fulfilled in qQED, i.e., a multiplicatively renormalizable solution for $F(k^{2})$ is ensured.

Symmetry properties of the $Y$-functions, Eqs. (29,30), restrict ${\cal{T}}(k^{2},p^{2})$ in Eq. (71) to be fully symmetric under $k^{2}\leftrightarrow p^{2}$. Furthermore, in order to ensure that in PT the transverse form factors start at ${\cal{O}}(\alpha)$, the perturbative expansion for ${\cal{T}}$ is required to begin at the same order.

The above expression for $\Lambda_{NM}(k,p)$, Eq. (71), provides a non-perturbative Ansatz for the corresponding linear combination of the massless $Y$-functions, see Eq. (56). Although Eq. (71) does not fix the $Y$-functions individually, we shall show in Section VII that it suffices (along with additional constraints on the remaining, relevant $Y$-functions) to investigate DCSB in the fermion propagator. Moreover, in Eq. (71), we assume that the $q^{2}$-dependence of the $Y$-functions involved is effectively incorporated by means of the function $u(k,p)$. However, more realistic Ansätze are expected to possess a more complex $q^{2}$-dependence.

For an arbitrary $q^{2}$-dependence of $\Lambda_{NM}(k,p)$, it seems impossible to proceed any further in integrating Eq. (69) because of the unknown dependence of the $Y$-functions on the angle $\varphi$. To circumvent this problem, we shall work with “effective” functions, denoted as $Y_{i}(k^{2},p^{2})$, whose relation with the “real” ones, $Y_{i}(k,p)$, is defined exactly in analogy with Bashir et al. (2011, 1998) as follows:

[TABLE]

where we have defined

[TABLE]

where $\theta$ is the usual step function:

[TABLE]

Using aforementioned effective functions, angular integration of Eq. (69) in qQED leads to

[TABLE]

In order to ensure MR of the fermion propagator, we demand $F(k^{2})$ on the left-hand side of Eq. (LABEL:Gap_equation_Angular_integrated) to satisfy Eq. (68); this imposes the following restriction:

[TABLE]

This requirement encodes the fact that all divergences have already been absorbed in the MR solution for the wave function renormalization $F$. As a consequence, there is no necessity of regularizing Eq. (82) and we can take $\Lambda^{2}\rightarrow\infty$ in the integration limit. It is convenient to introduce a dimensionless variable $x$, defined as

[TABLE]

so that Eq. (82) is now expressed as

[TABLE]

with

[TABLE]

where we have defined the function

[TABLE]

The presence of the anomalous dimension $\beta$ as an exponent in Eqs. (86,87), is related to the terms $F(k^{2})/F(p^{2})$ in (82), which can be expressed as $(k^{2}/p^{2})^{\beta}$ in the light of Eq. (63). Furthermore, in Eq. (86), we have defined

[TABLE]

These are dimensionless functions, satisfying the properties

[TABLE]

Moreover, in Eqs. (88,89), we have conveniently defined scalar functions

[TABLE]

Employing $x=p^{2}/k^{2}$ in Eq. (86), and using definitions (88,89,92,93), we have

[TABLE]

where we have defined

[TABLE]

which enters the definition of $r(p^{2}/k^{2})$ through

[TABLE]

Eqs. (85-96) constitute non-perturbative constraints on the fermion-photon vertex: for every Ansatz for the $Y$-functions, the resulting function $W$ is restricted to guarantee the integral constraint (85), so that the MR of the fermion propagator is ensured. To bring out the applicability and scope of the integral constraint on the massless $Y$-functions, Eq. (85), we now proceed to analyze an existing, rather general transverse vertex Ansatz, which was constructed in qQED to implement the requirement of MR for massless fermion propagator in addition to all other key features of QED mentioned before, Bashir et al. (2012b). Different choices of the free parameters defining this Ansatz correspond to numerous vertices constructed in the past. We will make reference to all these constructions along the way.

VI Examples

The Bashir-Bermudez-Chang-Roberts (BBCR) vertex, ref. Bashir et al. (2012a), is an Ansatz for the dressed fermion-photon vertex in QED, whose construction is constrained primarily by two requirements: to provide MR of the fermion propagator and to produce gauge independent critical coupling for DCSB. As it involves projecting the vertex onto the gap equation, it is natural that it is expressed only in terms of the functions which appear in the full fermion propagator, namely $F(k^{2})$ and ${\cal{M}}(k^{2})$. Moreover, its simplicity lies in the fact that its functional dependence on these entities is solely through the forms which enter the longitudinal vertex, namely, $b(k^{2},p^{2})$ and $c(k^{2},p^{2})$. In our kinematical configuration and notation, the transverse form factors for the BBCR vertex read as

[TABLE]

where the coefficients $a_{i}$ are constants. We will consider this example in detail because different choices of $a_{i}$ correspond to several vertices proposed in the literature, see Ref. Aslam et al. (2015), e.g., the Ball-Chiu vertex, Ball and Chiu (1980), the Curtis-Pennington vertex Curtis and Pennington (1990) and the Qin-Chang vertex Qin et al. (2013).

From Eqs. (35-41,97-104), we see that the corresponding $Y$-functions for the BBCR vertex read as

[TABLE]

It is mathematically straightforward to show that the asymptotic expansion of Eqs. (106,107,109) in powers of $k^{2}/p^{2}$, for $p^{2}>>k^{2}>>m_{0}^{2}$, fulfills the PT requirement (46) up to second order if Bashir et al. (2012a)

[TABLE]

In order to verify that if, for the massless case, the BBCR vertex satisfies the integral constraint for $W$, Eq. (85), it is necessary to compute the corresponding massless effective $Y$-functions by means of Eqs. (72-LABEL:Y8_Antisym_effective_in_terms_of_the_real), which in turn yield

[TABLE]

Using Eqs. (114-117) in the expression for $W(p^{2}/k^{2})$, Eq. (94), we find

[TABLE]

If we insert the above expression for $W(x)$, Eq. (LABEL:W(x)_for_Rocio_Vertex), in its corresponding integral restriction, Eq. (85), we find that the integral vanishes if and only if the following two conditions are met:

[TABLE]

which are the constraints reported in Bashir et al. (2012a) for the coefficients $a_{i}$ in order to ensure the MR of the fermion propagator.

As a counterexample, we could take all $Y$-functions equal to zero. If we do so, the resulting function $W(x)$ reads as

[TABLE]

which does not satisfy the integral constraint (85). Therefore, setting all $Y$-functions equal to zero does not ensure the MR of the fermion propagator.

Throughout Sections V and VI, we have investigated MR solution for massless fermion propagator in qQED, within the LLA, and derived a consequent non-perturbative, integral constraint for the massless $Y$-functions. In the next section, we study DCSB and implement the argument of a gauge independent critical coupling to impose further constraints on the transverse vertex.

VII Gauge independent DCSB

In order to study DCSB through the gap equation by employing fully-dressed fermion-photon vertex, Eqs. (2-4,7,8,20-27), we propose an Ansatz for the functions $Y_{i}$ appearing in Eqs. (LABEL:ProyM-56). Naturally, we look for the simplest construction which incorporates all the key constraints we have enlisted and studied so far. This can be achieved by requiring the following :

• •

1) The massive $Y$-functions in the gap equation are expressed solely in terms of the fermion dressing functions, $F$ and ${\cal{M}}$.

• •

2) The antisymmetric contribution of $G_{\cal{M}}(k,p)$ and $G_{F}(k,p)$ vanishes under $k\leftrightarrow p$.

• •

3) The functions $G_{\cal{M}}$ and $G_{F}$ are the same (up to a constant factor).

These simplifying requirements do not jeopardize the MR of the fermion propagator which can still be ensured in massless qQED.

Assumptions 1) and 2) are fulfilled if we choose

[TABLE]

Moreover, we can implement the simplifying requirement 3) by demanding the MR condition on the massless functions, Eq. (71), and fixing $Y_{5}$ as follows:

[TABLE]

In fact, the Ansatz for the $Y$-functions, constituted through Eqs. (71,122-125), yields

[TABLE]

which fulfills assumption 2) in addition to 3). Moreover, for the massless limit in qQED, it simplifies Eq. (LABEL:ProyF) as

[TABLE]

After angular integration, $F$ satisfies Eq. (68) as expected, i.e., it has the power law behavior of Eq. (63) as constrained by MR, with the anomalous dimension $\beta$ given in Eq. (67).

Let us summarize below the important characteristics of the function ${\cal{T}}$:

• •

i) It must be a dimensionless function of $k^{2}$ and $p^{2}$. We assume it to be $q^{2}$-independent in order to ensure the MR of $F(k^{2})$.

• •

ii) It must be fully symmetric under $k^{2}\leftrightarrow p^{2}$.

• •

iii) Its perturbative expansion must start at ${\cal O}(\alpha)$.

• •

iv) It must vanish in the Landau gauge, $\xi=0$.

Recall from Section V that condition i) is required to ensure a MR solution for the massless fermion propagator in qQED, while the conditions ii) and iii) follow from the symmetry properties of the vertex and its perturbative expansion, respectively. The additional condition iv) is imposed in order to facilitate the extraction of an $\xi$-independent critical coupling, and an anomalous dimension for the mass function which, at criticality, is independent of the choice of the vertex, as we shall discuss now.

In the vicinity of the critical coupling, $\alpha_{c}$, above which chiral symmetry is broken dynamically, the generated fermion mass is negligible in comparison with any other mass scale. Hence, for $\alpha\sim\alpha_{c}$, we can formally neglect quadratic and higher powers (if any) of the mass function in Eq. (LABEL:ProyM). In the limit $m_{0}=0$, it reduces to

[TABLE]

where our Ansatz for the $Y$-functions, Eqs. (71,122-125), has been implicitly embedded through the function $G(k,p)$, defined in Eq. (126). Moreover, neglecting terms quadratic in ${\cal{M}}$, the equation for $F$, Eq. (LABEL:ProyF), reduces to that of a massless theory and decouples from that of the mass function ${\cal{M}}$. In the quenched approximation, it yields Eq. (127). Therefore, from Eqs. (127,128) we see that in the vicinity of the critical coupling, in qQED, the mass function satisfies the following equation:

[TABLE]

In the neighborhood of $\alpha_{c}$, MR forces a power law behavior for the mass function which must hold at all momenta:

[TABLE]

where $B_{\Lambda}$ is a constant (that depends on $\Lambda$), and the exponent $s=1-\gamma_{m}/2$ is defined in terms of the anomalous dimension of the mass function, $\gamma_{m}$. We assume $0<s\leq 1$ to comply with perturbation theory.

Bardeen et al. demonstrated that, at $\alpha=\alpha_{c}$, the mass anomalous dimension is $\gamma_{m}=1$ Bardeen et al. (1986); Leung et al. (1986). Some further analyses, based on Cornwall-Jackiw-Tomboulis effective potential technique, tend to argue that, at criticality, this value holds true regardless of the choice of the vertex Akram et al. (2013); Holdom (1988, 1989). In Ref. Bashir and Pennington (1994), this values is quite close to unity, though not exactly equal to it. Setting $\gamma_{m}=1$ results in the four-fermion interaction operator $(\overline{\psi}\psi)^{2}$ acquiring dynamical dimension $d=2(3-\gamma_{m})=4$ in contrast to its canonical dimension $d=6$. Therefore, four-fermion interaction becomes marginal. It must then be included in order to render non-perturbative QED a self-consistent, closed theory Akram et al. (2013); Bardeen et al. (1990). Depending upon the non-perturbative details of the fermion-photon interaction, it is plausible to have $\gamma_{m}>1$, implying $d<4$, which would modify the status of the four-point operators from marginal to relevant; see, e.g., the review article Bashir and Raya (2004) and references therein.

At $\alpha=\alpha_{c}$, the anomalous dimension $\gamma_{m}=1$ and its corresponding critical value $s_{c}=1/2$ can be obtained by constraining the fermion-photon vertex, a line of action which is followed in Bashir and Pennington (1996); Akram et al. (2013). In our analysis, this critical value is readily derived from Eq. (128) in Landau gauge, $\xi=0$, if we demand ${\cal{T}}(k^{2},p^{2})$ to fulfill condition iv): from Eq. (127) we see that in the Landau gauge $F(k^{2})=1$, and therefore Eq. (129) reduces to

[TABLE]

For the MR solution of the mass function, Eq. (130), the above Eq. (131) results in

[TABLE]

where $\alpha_{c}$ stands for the critical coupling, which signals the point where the two possible solution for $s$ match each other, and a non-trivial solution for the mass function111In addition to positive-definite solutions for the mass function (and their corresponding mirror), an arbitrary vertex may produce spurious oscillatory solutions for ${\cal{M}}$. However, it has been argued that a realistic vertex might only produce monotonically decreasing and increasing non-trivial solutions Raya et al. (2013). bifurcates away from the perturbative one (${\cal{M}}=0$): for $\alpha>\alpha_{c}$, the solution for the mass function enters the complex plane indicating that DCSB has taken place. In this case, the critical coupling is:

[TABLE]

thus revealing a Miransky scaling law for the interaction strength $\alpha$ Fomin and Miransky (1976); Fomin et al. (1978); Maskawa and Nakajima (1974), which has been derived using a bare vertex Miransky (1985); Fomin et al. (1983). For $\alpha=\alpha_{c}=\pi/3$ in Eq. (132), the expected critical value for the anomalous mass dimension is obtained, i.e.

[TABLE]

Since the critical coupling pinpoints a phase transition from perturbative to non-perturbative dynamics, it is potentially a physical observable, and hence it is expected to be independent of the gauge parameter. Thus, Eq. (131), and its corresponding MR solution, Eq. (130), must hold in all gauges. Therefore, from Eqs. (129,131), we see that in order for $\alpha_{c}$ and $s_{c}$ to be $\xi$-independent, the mass function and the vertex must satisfy the following equation in qQED:

[TABLE]

After performing angular integration, and introducing the dimensionless variables defined in Eqs. (83,84), the above Eq. (135) can be cast in the following form:

[TABLE]

with

[TABLE]

where $s$ and $\beta$, defined in Eqs. (67,134), appear in the light of the MR solutions for $F$ and ${\cal{M}}$, Eqs. (63,130), respectively. Furthermore, in Eq. (137) we have defined

[TABLE]

which is independent of $k^{2}$, and satisfies the following property:

[TABLE]

It is important to stress the fact that Eq. (136) stands for a non-perturbative constraint on the vertex: any Ansatz for ${\cal{T}}(k^{2},p^{2})$ must provide a function $V$ that should satisfy Eq. (136). Conversely, from a particular solution for $V$ in the latter equation, one can derive the corresponding function ${\cal{T}}$ by means of Eq. (137). However, there exist an infinite number of solutions for $V(x)$ satisfying Eq. (136). In addition, such a solution for $V$ must also satisfy (cf. Eq. (137))

[TABLE]

A simple choice satisfying Eqs. (136,140) at criticality reads:

[TABLE]

valid for $-\nicefrac{{1}}{{2}}\leq\beta\leq\nicefrac{{1}}{{2}}$ but $\beta\neq 0$. In the Landau gauge, Eqs. (136,140) are satisfied with the trivial solution

[TABLE]

It is worth reminding that Eq. (136), and consequently Eqs. (137,140), are rigourously valid only at criticality. Therefore, for $\alpha_{c}=\pi/3$ and $s_{c}=1/2$, the resulting function ${\cal{T}}(k^{2},p^{2})$, derived form Eqs. (137,141), reads (for $x=p^{2}/k^{2}$ and $-6\leq\xi\leq 6$, but $\xi\neq 0$) as

[TABLE]

whereas Eqs. (137,142) yield (for Landau gauge)

[TABLE]

The above expressions for ${\cal{T}}$ fulfill conditions i)-iv), as expected, but a few observations must be made: a) MR of the wave function renormalization entails $\nicefrac{{F(p^{2},\Lambda^{2})}}{{F(k^{2},\Lambda^{2})}}=\nicefrac{{F(p^{2},\mu^{2})}}{{F(k^{2},\mu^{2})}}$ for some renormalization scale, $\mu^{2}$, ensuring the $\Lambda^{2}$-independence of the second term on the RHS of Eq. (LABEL:ansatz_for_T(k2,p2)); and b) although $F=1$ for $\xi=0$ (in the LLA of qQED), leading to ${\cal{T}}_{\xi=0}=0$, the function ${\cal{T}}_{\xi=0}(k^{2},p^{2})$ defined in Eq. (144) does not necessarily vanish in the Landau gauge beyond the LLA and the quenched approximation.

Numerical evaluation of the ratio between the euclidean mass $m_{E}$, defined as $m_{E}={\cal{M}}(m_{E}^{2})$, and the ultraviolet cut-off $\Lambda$ is shown in Fig. 3 for different gauges implementing the Ansätze introduced in Eqs. (LABEL:ansatz_for_T(k2,p2),144). In the plot, points with $\alpha<1.25$ (highlighted as filled markers) are fitted to the Miransky scaling law Fomin and Miransky (1976); Fomin et al. (1978); Maskawa and Nakajima (1974); Miransky (1985); Fomin et al. (1983)

[TABLE]

Each fit yields the critical coupling $\alpha_{c}=\pi/3$ with a numerical error of less than 1% for $\xi=1,0,-1,-2$ (and $\sim$ 2.8% for $\xi=-3$) as indicated in the second column of the Table 1. It is worth reminding that our Ansatz reveals $\alpha_{c}=\pi/3$, just like the bare vertex in the Landau gauge. However, the bare vertex leads to a highly gauge dependent $\alpha_{c}$, including no chiral symmetry breaking for $\xi=-3$. In our case, chiral symmetry is broken in every gauge with the same critical coupling. The result for $\xi=-3$ particularly emphasizes this point.

It is well-known that DCSB manifests itself in the three-point vertex through its massive form factors Braun et al. (2016); Mitter et al. (2015), and thus a physically meaningful Ansatz for ${\cal{T}}(k^{2},p^{2})$ should incorporate the mass function. In the present work, we propose a simple, numerically tractable Ansatz for $\xi=0$,

[TABLE]

which is an extension of Eq. (144), with a mass term weighted by a real constant $\rho$. The fact that the last term of Eq. (146) contains quadratic powers in ${\cal{M}}$ ensures that this contribution can be neglected at criticality, which in turn yields a gauge independent critical coupling. In addition, characteristics i)-iv) remain preserved.

Numerical evaluation of $m_{E}/\Lambda$ using Eq. (146) is shown in Fig. 4 for different values of $\rho$. Again, points with $\alpha<1.25$ are fitted to Eq. (145), indicating a critical coupling independent of the $\rho$ parameter and equal to $\pi/3$, within a margin of error smaller than $1\%$.

Numerical results for the Euclidean mass plotted in Figs. 3 and 4 support the argument that any function ${\cal{T}}(k^{2},p^{2})$ preserving characteristics i)-iv) and satisfying conditions defined through Eqs. (136,140) will ensure a gauge independent critical coupling $\alpha_{c}=\pi/3$ in qQED. Eqs. (LABEL:ansatz_for_T(k2,p2),144) define simple, numerically friendly Ansätze for the transverse vertex contribution ${\cal{T}}$ to the gap equation. In Landau gauge, an extension of the Ansatz defined through Eq. (146) explicitly incorporates DCSB and still ensures a critical coupling independent of the $\rho$ parameter.

VIII Conclusions

In this article, we have investigated combined constraints of TTI, LKFT, MR of the massless fermion propagator, gauge-independence of the critical coupling $\alpha_{c}$ in quenched QED and one-loop perturbation theory in the asymptotic limit to construct a general fermion-photon vertex. We work explicitly with $Y_{i}$ functions, which arise naturally on the implementation of the TTI, providing, along the way, their symmetry properties under the charge conjugation operation. Through an exact relation, we define effective $Y_{i}$ for which the angular dependence on the variable $q^{2}$ has been integrated out to make their implementation in the gap equation more efficient. As a simplifying consequence of working with $Y_{i}$, we observe that the kernel dependence on the Gram determinant $\nabla(k,p)$ for the mass function disappears altogether. Moreover, our study reveals that we cannot force all $Y_{i}$ to be simultaneously equal to zero. It will violate the LKFT transformation law and the MR of the massless fermion propagator. We work with quite a general vertex construction Bashir et al. (2012a), formulated in terms of $Y_{i}$. We also provide simple examples of this fermion-photon vertex and carry out its numerical study to compute the mass function and its variation as a function of the coupling strength. The results clearly follow Miransky scaling law and provide $\alpha_{c}=\pi/3$. Moreover, anomalous mass dimension $\gamma_{m}=1$, as has been advocated in several previous works Holdom (1988, 1989); Bashir and Pennington (1996); Akram et al. (2013). Also, this critical coupling is gauge independent. As mentioned before, fermion-photon vertex enters the SDE study of several hadronic obseravbles, such as form factor calculations, where photons interact with quarks. Therefore, an improved understanding of this vertex, such as the one detailed in this article, is very important. Moreover, a natural extension of our work for the quark-gluon vertex in QCD is currently underway.

Acknowledgements: We dedicate this article to the fond memory of Prof. M.R. Pennington with whom the hunt for a gauge independent DCSB began about three decades ago. We are grateful to C. Schubert and R. Williams for helpful comments on the draft version of this article. This research was partly supported by Coordinación de la Investigación Científica (CIC) of the University of Michoacan and Consejo Nacional de Ciencia y Tecnología (CONACyT), Mexico, Grant nos. 4.10 and CB2014-22117, as well as grants CNPq no. 307485/2017-0 and FAPESP no. 2016/03154-7.