Improved variational description of the Wick-Cutkosky model with the most general quadratic trial action

TL;DR

This paper enhances the worldline variational approach for the Wick-Cutkosky model by incorporating anisotropic trial actions, leading to improved results, lower critical coupling, and new insights into the dressed particle's quantum motion.

Contribution

It introduces a generalized anisotropic trial action in the variational approach, improving accuracy and understanding of the Wick-Cutkosky model's behavior.

Findings

Lowered critical coupling for instability

Dressed particle distorted along its momentum at large couplings

Established relation between renormalized coupling and propagator

Abstract

We generalize the worldline variational approach to field theory by introducing a trial action which allows for anisotropic terms to be induced by external 4-momenta of Green's functions. By solving the ensuing variational equations numerically we demonstrate that within the (quenched) scalar Wick-Cutkosky model considerable improvement can be achieved over results obtained previously with isotropic actions. In particular, the critical coupling associated with the instability of the model is lowered, in accordance with expectations from Baym's proof of the instability in the unquenched theory. The physical picture associated with a different quantum mechanical motion of the dressed particle along and perpendicular to its classical momentum is discussed. Indeed, we find that for large couplings the dressed particle is strongly distorted in the direction of its four-momentum. In addition, we obtain an exact relation between the renormalized coupling of the theory and the propagator. Along the way we introduce new and efficient methods to evaluate the averages needed in the variational approach and apply them to the calculation of the 2-point function.