Non-Perturbative Mass Renormalization in Quenched QED from the Worldline Variational Approach

TL;DR

This paper develops a non-perturbative variational approach using the worldline formulation to compute the anomalous mass dimension in quenched QED, revealing new insights into its behavior at strong coupling.

Contribution

It introduces a novel variational method for quenched QED employing Grassmann trajectories and supersymmetry, providing an analytic solution for the anomalous mass dimension.

Findings

Derived a simple analytic expression for gamma_m(alpha) in the MS scheme.

Found gamma_m(alpha) scales as alpha^(1/2) at large couplings.

Identified divergence of perturbative expansion at alpha > 0.7934.

Abstract

Following Feynman's successful treatment of the polaron problem we apply the same variational principle to quenched QED in the worldline formulation. New features arise from the description of fermions by Grassmann trajectories, the supersymmetry between bosonic and fermionic variables and the much more singular structure of a renormalizable gauge theory like QED in 3+1 dimensions. We take as trial action a general retarded quadratic action both for the bosonic and fermionic degrees of freedom and derive the variational equations for the corresponding retardation functions. We find a simple analytic, non-perturbative, solution for the anomalous mass dimension gamma_m(alpha) in the MS scheme. For small couplings we compare our result with recent four-loop perturbative calculations while at large couplings we find that gamma_m(alpha) becomes proportional to (alpha)^(1/2). The anomalous mass dimension shows no obvious sign of the chiral symmetry breaking observed in calculations based on the use of Dyson-Schwinger equations, however we find that a perturbative expansion of gamma_m(alpha) diverges for alpha > 0.7934. Finally, we investigate the behaviour of gamma_m(alpha) at large orders in perturbation theory.