Mass Dependence of Vacuum Energy

TL;DR

This paper derives a differential equation relating vacuum energy to mass and cutoff parameters, providing integral solutions and recursive formulas for asymptotic coefficients, enhancing understanding of quantum field vacuum energies.

Contribution

It introduces a differential equation approach to analyze mass dependence of vacuum energy and offers new integral and recursive solutions for regularized energy expressions.

Findings

Derived a PDE for vacuum energy dependence on mass and cutoff

Provided integral formulas for positive mass vacuum energy

Established recursive relations for asymptotic coefficients

Abstract

The regularized vacuum energy (or energy density) of a quantum field subjected to static external conditions is shown to satisfy a certain partial differential equation with respect to two variables, the mass and the "time" (ultraviolet cutoff parameter). The equation is solved to provide integral expressions for the regularized energy (more precisely, the cylinder kernel) at positive mass in terms of that for zero mass. Alternatively, for fixed positive mass all coefficients in the short-time asymptotics of the regularized energy can be obtained recursively from the first nontrivial coefficient, which is the renormalized vacuum energy.