Modification of the effective potential of $\phi^4$ theory using the Boltzmann factor

TL;DR

This paper introduces a modification to the effective potential in $^4$ theory by incorporating a Boltzmann suppression factor for high-energy virtual particles, revealing a non-zero minimum of the potential.

Contribution

It applies a Boltzmann factor to the effective potential in $^4$ theory, showing the potential's minimum shifts away from zero, which is a novel approach.

Findings

Effective potential minimum occurs at non-zero 

Modified potential includes Bessel functions of the second kind

Suggests high-energy virtual particles are suppressed by Boltzmann factor

Abstract

The need for a cutoff in the Lamb shift calculation suggests that high-energy virtual photons do not interact with the real particles. In this study, we assume that the creation of high-energy virtual particles is suppressed by a Boltzmann factor $e^{-\beta E}$ and apply this Boltzmann factor to the effective potential of the $\phi^4$ theory. Consequently, the effective potential is transformed into $V(\phi) = 1/4!\phi^4 + \beta^{-4} \left \{ \phi^3 K_3(\phi) -\phi^2 K_2 (\phi) \right\}$. $K_n(x)$ is the modified Bessel function of the second kind of order $n$. We demonstrate that the minimum of the effective potential occurs for $\phi \ne 0$.