Conformal Symmetry and Effective Potential: I. Vacuum $V_{z,x}$-operation for the Green functions

TL;DR

This paper introduces a novel vacuum $V_{z,x}$-operation in conformal symmetry that transforms Green functions into vacuum integrals, aiding the study of multi-loop effective potential evolution in $^4$-theory.

Contribution

It presents a new $V_{z,x}$-operation for Green functions, providing an inverse approach to existing methods, and details the treatment of mass terms in effective potential calculations.

Findings

Defines the vacuum $V_{z,x}$-operation for Green functions.

Provides a method to generate effective potential via vacuum integrations.

Lays groundwork for multi-loop effective potential analysis in conformal symmetry.

Abstract

We begin a series of two papers that is devoted to the study of the multi-loop effective potential evolution in $\varphi^4$-theory using the conformal symmetry. In the first part, we introduce and describe in detail the vacuum $V_{z,x}$-operation ($"V"$ stems from "vacuum", $\{z,x\}$ imply the corresponding coordinates) that transforms the given Green functions to the corresponding vacuum integrations which generate the effective potential. Our operation can be considered as an inverse procedure compared to the Gorishni-Isaev method. To the final goal, it is necessary to introduce also the special treatment of the mass terms as sorts of "interaction" in an asymptotical expansion of the generating functional.