Explicit proof of equivalence of two-point functions in the two formalisms of thermal field theory

TL;DR

This paper provides an explicit proof of the equivalence of two-point functions in two formalisms of thermal field theory at one-loop order, emphasizing the importance of separating the imaginary part of zero-temperature integrals.

Contribution

It presents a detailed proof of the equivalence of two-point functions in real-time formalism for thermal bbb3 theory, highlighting a key step involving imaginary part separation.

Findings

Confirmed equivalence at one-loop order in bbb3 theory.

Identified the critical role of separating imaginary parts of integrals.

Method applicable to other theories, including NJL model.

Abstract

We give an explicit proof of equivalence of the two-point function to one-loop order in the two formalisms of thermal $\lambda \phi^3$ theory based on the expressions in the real-time formalism. It is indicated that the key-point of completing the proof is to separate carefully the imaginary part of the zero-temperature loop integral from relevant expressions and this fact will certainly be very useful for examination of the equivalence problem of the two formalisms of thermal field theory in other theories, including the one of the propagators for scalar bound states in a NJL model.