Demonstration of how the zeta function method for effective potential removes the divergences

TL;DR

This paper demonstrates how the zeta function method effectively removes divergences in calculating the effective potential's minimum, providing finite results without counter-terms by exploiting the regularity of the zeta function at s=0.

Contribution

It explicitly explains the divergence cancellation mechanism of the zeta function method and links it to established procedures for handling functions with simple poles.

Findings

Zeta function method yields finite effective potential without counter-terms.

Divergence cancellation occurs through the regularity of the zeta function at s=0.

The method aligns with known procedures for functions with simple poles.

Abstract

The calculation of the minimum of the effective potential using the zeta function method is extremely advantagous, because the zeta function is regular at $s=0$ and we gain immediately a finite result for the effective potential without the necessity of subtratction of any pole or the addition of infinite counter-terms. The purpose of this paper is to explicitly point out how the cancellation of the divergences occurs and that the zeta function method implicitly uses the same procedure used by Bollini-Giambiagi and Salam-Strathdee in order to gain finite part of functions with a simple pole.