Exact solutions and zero modes in scalar field theory

TL;DR

This paper presents exact solutions involving Jacobi elliptic functions in scalar field theory with Z2 symmetry, revealing massive waves from massless equations and analyzing zero modes related to symmetry breaking and the Higgs mechanism.

Contribution

It introduces new exact solutions with elliptic functions in scalar field theory and explores the role of zero modes in symmetry breaking and the Higgs mechanism.

Findings

Exact solutions with Jacobi elliptic functions in scalar fields.

Zero modes are crucial for spontaneous symmetry breaking.

Zero modes may act like Goldstone bosons in the Higgs mechanism.

Abstract

We provide a set of exact solutions in field theory of scalar fields with $Z_2$ symmetry that involve Jacobi elliptic functions. These solutions have the interesting property to provide massive waves even if one starts from a massless equation. We analyze them classically providing also exact solutions to the corresponding equations for the Green functions needed to completely solve them in a strong coupling limit. This is accomplished using a functional expansion into powers of the current. It is shown that the spontaneous breaking of the $Z_2$ symmetry is due to the existence of a zero mode that persists also in the case of the Higgs mechanism. In this latter case, the zero mode seems to play a role similar to the Goldstone boson in the breaking of a continuous symmetry and so, it should be important at lower momenta as a long range excitation.