Transverse Takahashi identities and their implications for gauge independent dynamical chiral symmetry breaking

In this article, we employ transverse Takahashi identities to impose valuable nonperturbative constraints on the transverse part of the fermion-photon vertex in terms of new form factors, the so-called Y functions. We show that the implementation of these identities is crucial for 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 scaling law, and we find gamma(m) = 1 rather naturally in accordance with some earlier results in literature, using arguments based on the Comwall-Jackiw-Tomboulis effective potential technique. Moreover, we numerically confirm the gauge invariance of this critical coupling.