Bosons after Symmetry Breaking in Quantum Field Theory

TL;DR

This paper provides a unified analysis of spontaneous symmetry breaking in fermion field theories, clarifying the nature of bosons and the Goldstone theorem, with exact solutions in models like Thirring and QCD$_2$.

Contribution

It offers a new perspective on boson spectra post-symmetry breaking, showing the absence of Goldstone bosons in certain fermion models and analyzing the spectrum using Bethe ansatz.

Findings

No Goldstone boson in models like Nambu-Jona-Lasinio, Thirring, and QCD$_2$ after symmetry breaking.

Goldstone boson predicted by the theorem is virtual, representing a fermion-antifermion pair.

Bosons in QCD$_2$ have finite mass, contrary to some predictions.

Abstract

We present a unified description of the spontaneous symmetry breaking and its associated bosons in fermion field theory. There is no Goldstone boson in the fermion field theory models of Nambu-Jona-Lasinio, Thirring and QCD$_2$ after the chiral symmetry is spontaneously broken in the new vacuum. The defect of the Goldstone theorem is clarified, and the "massless boson" predicted by the theorem is virtual and corresponds to just a $free$ massless fermion and antifermion pair. Further, we discuss the exact spectrum of the Thirring model by the Bethe ansatz solutions, and the analytical expressions of all the physical observables enable us to understand the essence of the spontaneous symmetry breaking in depth. Also, we examine the boson spectrum in QCD$_2$, and show that bosons always have a finite mass for $SU(N_c)$ colors. The problem of the light cone prescription in QCD$_2$ is discussed, and it is shown that the trivial light cone vacuum is responsible for the wrong prediction of the boson mass.