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The ambiguities associated with the lack of gauge invariance in the non-perturbative truncations of Schwinger-Dyson equations (SDEs) are a challenging problem which has not yet been resolved in a deci- sive fashion. Pursuing this aim, we study dynamical chiral symmetry breaking in quantum electrodynamics in three space-time dimensions (QED3). We investigate the gauge dependence of the chiral condensate both in the quenched and the unquenched versions of the theory and emphasize the importance of taking into ac- count the gauge covariance properties of the fermion propagator as dictated by its Landau-Khalatnikov-Fradkin transformation (LKFT). We present numerical solutions of the SDE of the fermion propagator which respect Ward-Green-Takahashi identities (WGTI) and LKFT simultaneously. As a striking consequence, we obtain a practically gauge independent chiral condensate. Gauge theories of fundamental interactions have been highly successful in collating experimental results in the per- turbative regime. However, not all interesting phenomena can be understood in this approximation scheme. Confinement of quarks and gluons and the origin of hadronic masses are two examples. It is well known that on the distance scale of the order of hadron size, quarks behave as though their mass were 300 MeV, much larger than their vanishingly small cur- rent mass. In the context of covariant gauges, this effect may be attributed to an interplay of the behaviour of gluon and ghost propagators in the infrared, (1). These studies through Schwinger-Dyson equations (SDEs) have only been carried out in the Landau gauge. It is well known that if care is not taken in their truncation, solutions can be gauge dependent. Let alone the complicated battle ground of non-Abelian the- ories, the ambiguities associated with the lack of gauge invari- ance in the non-perturbative truncations of SDEs in Abelian theories such as QED are also a challenging problem. Among others isuues, it has ultraviolet divergences. Consequently, a neater testing ground for the validity of various truncations is provided by QED3, which is super-renormalizable. However, a recent study, (2), has revealed that the problem of gauge in- variance persists even in this seemingly simple scenario. In addition to being a toy model for SDEs studies, QED3 is an attractive theory in its own right. In condensed matter physics, for example, it is used in the context of high-Tc su- perconductors (3), studies which have received a boost from new experiments (4), and in the recently explored unconven- tional Quantum Hall Effect in Graphene (5). Additionally, in the realm of dynamical generation of fundamental fermion masses, numerical findings on the lattice, results obtained by employing SDEs, (6), and alternative methods, (7), have yet to arrive at a final consensus, and continue to provide a popular battle ground. The problem of gauge invariance in QED3 can be traced back to not employing (or doing so incorrectly) the gauge identities such as the Ward-Green-Takahashi identi- ties (WGTI) (8), the Nielsen identities (9) and the Landau- Khalatnikov-Fradkin transformations (LKFT) (10). Here we address this issue in the light of the LKFT, following and ex- tending the analysis of Ref. (11). These transformations de- scribe the specific manner in which Green functions transform under a variation of gauge. These are non-perturbative in na- ture, and thus are expected to be very helpful in studies of phe- nomena realized only non perturbatively, such as dynamical chiral symmetry breaking. Initial steps were taken in (12) to apply LKFT directly to the dynamically generated mass func- tion, whereas the complete numerical implementation of LK- FTs on solutions to SDEs in various truncation schemes was performed in (11) to study the gauge dependence of the chiral condensate in QED3. In this work we extend the analysis to a wider range of values of the gauge parameter.

Gauge Covariance and the Chiral Condenate in QED3