Nonperturbative mass renormalization in quenched QED from the worldline variational approach

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 such as QED in 3+1 dimensions. We take as a 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 γm(α) in the MS scheme. For small couplings we compare our result with recent four-loop perturbative calculations while at large couplings we find that γm(α) becomes proportional to √α. 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 γm(α) diverges for α>0.7934. Finally, we investigate the behavior of γm(α) at large orders in perturbation theory.