Nambu-Goldstone mechanism at finite temperature in the imaginary-time and real-time formalism

TL;DR

This paper compares the imaginary-time and real-time formalisms in thermal field theory to analyze the Nambu-Goldstone mechanism of electroweak symmetry breaking, finding physical equivalence and conditions for the Goldstone theorem's validity.

Contribution

It demonstrates the equivalence of imaginary-time and real-time formalisms in analyzing the Nambu-Goldstone mechanism at finite temperature within a fermion condensate scheme.

Findings

Goldstone theorem holds exactly for mass-degenerate fermions.

Approximate validity of Goldstone theorem at low energies for non-degenerate masses.

Fluctuation effects on the Higgs are negligible with a large momentum cutoff.

Abstract

In the imaginay-time formalism of thermal field theory, and also in the real-time formalism but by means of some redefined physical propagators for scalar bound states by diagonalization of four-point function matrices, we reexamine the Nambu-Goldstone mechanism of electroweak symmetry breaking in a one-generation fermion condensate scheme, based on the Schwinger-Dyson equation in the fermion bubble diagram approximation, and compare the obtained results. We have reached the conclusion that in both the formalisms, the Goldstone theorem of spontaneous electroweak symmetry breaking is rigorously true for the case of mass-degenerate two flavors of fermions and only approximately valid at low energy scales for the mass-nondegenerate case, in spite of existence of some difference between the two formalisms in the imaginary parts of the denominators of the propagators for scalar bound states. When the two flavors of fermions have unequal nonzero masses, the induced possible fluctuation effect for the Higgs particle is negligible if the momentum cut-off in the zero temperature loops is large enough. All the results show physical equivalence of the two formalisms in the present discussed problems.