Nambu-Goldstone Mechanism in Real-Time Thermal Field Theory

TL;DR

This paper proves that in a one-generation fermion condensate model at finite temperature, spontaneous symmetry breaking can produce a Higgs boson and Nambu-Goldstone bosons, with the Goldstone Theorem holding exactly for degenerate fermions.

Contribution

It rigorously demonstrates the validity of the Goldstone Theorem in a real-time thermal field theory framework for electroweak symmetry breaking with degenerate fermions.

Findings

Massless Nambu-Goldstone bosons emerge at finite T for degenerate fermions.

Goldstone Theorem is approximately valid for non-degenerate fermions at low energy.

Pinch singularities cancel in the real-time thermal field theory.

Abstract

In a one-generation fermion condensate scheme of electroweak symmetry breaking, it is proven based on Schwinger-Dyson equation in the real-time thermal field theory in the fermion bubble diagram approximation that, at finite temperature $T$ below the symmetry restoration temperature $T_c$, a massive Higgs boson and three massless Nambu-Goldstone bosons could emerge from the spontaneous breaking of electroweak group $SU_L(2)\times U_Y(1) \to U_Q(1)$ if the two fermion flavors in the one generation are mass-degenerate, thus Goldstone Theorem is rigorously valid in this case. However, if the two fermion flavors have unequal masses, owing to "thermal flactuation", the Goldstone Theorem will be true only approximately for a very large momentum cut-off $\Lambda$ in zero temperature fermion loop or for low energy scales. All possible pinch singularities are proven to cancel each other, as is expected in a real-time thermal field theory.