Vacuum Polarization Energy of a Proca Soliton

TL;DR

This paper calculates the quantum vacuum polarization energy of a soliton in an extended Proca model with scalar and vector fields in 1+1 dimensions, using spectral methods and analyzing the Jost function.

Contribution

It introduces a spectral method approach to compute the vacuum polarization energy in a Proca soliton model, addressing non-analytical components and normalization issues.

Findings

Numerical verification that non-analytical obstacles do not arise.

Real and imaginary momentum formulations of VPE agree.

The Born approximation is essential for proper renormalization.

Abstract

We study an extended Proca model with one scalar field and one massive vector field in one space and one time dimensions. We construct the soliton solution and subsequently compute the vacuum polarization energy (VPE) which is the leading quantum correction to the classical energy of the soliton. For this calculation we adopt the spectral methods approach which heavily relies on the analytic properties of the Jost function. This function is extracted from the interaction of the quantum fluctuations with a background potential generated by the soliton. Particularly we explore eventual non-analytical components that may be induced by mass gaps and the unconventional normalization for the longitudinal component of the vector field fluctuations. By numerical simulation we verify that these obstacles do actually not arise and that the real and imaginary momentum formulations of the VPE yield equal results. The Born approximation to the Jost function is crucial when implementing standard renormalization conditions. In this context we solve problems arising from the Born approximation being imaginary for real momenta associated with energies in the mass gap.