On vacuum integration

TL;DR

This paper demonstrates that massive vacuum loop integrations can be reformulated using massless loop integrations with specific prefactors, facilitating applications in conformal symmetry analysis.

Contribution

It introduces a method to express massive vacuum diagrams via massless integrals, addressing the $ ext{delta}(0)$-singularity issue, thus linking vacuum diagrams to conformal symmetry tools.

Findings

Massive vacuum integrals can be rewritten as massless integrals with prefactors.

A special treatment of $ ext{delta}(0)$-singularity is required.

The approach aids in applying conformal symmetry techniques.

Abstract

The effective potential is known to be given by the vacuum diagrams. In this paper we show that the massive loop integrations corresponding to the vacuum diagrams can be expressed through the massless loop integrations with the corresponding massive prefactor provided the special treatment of $\delta(0)$-singularity. In its turn, the massless loop integrations provide the useful instrument for the conformal symmetry application.