Higgs Mechanism with Type-II Nambu-Goldstone Bosons at Finite Chemical Potential

TL;DR

This paper investigates how gauge bosons gain mass via the Higgs mechanism in systems with non-Lorentz invariant symmetry breaking, focusing on the role of type-II Nambu-Goldstone bosons at finite chemical potential.

Contribution

It demonstrates that the physical spectrum remains unchanged despite the presence of type-II NG bosons and scalar mixings induced by chemical potential in gauge theories.

Findings

Massless and massive scalar bosons are absorbed into gauge bosons.

Chemical potential induces scalar and gauge boson mixing but does not alter physical spectra.

Total physical modes count remains consistent even with NG boson and broken generator mismatch.

Abstract

When the spontaneous symmetry breaking occurs for systems without Lorentz covariance, there arises possible mismatch, $N_{\rm NG} < N_{\rm BG}$, between numbers of Nambu-Goldstone (NG) bosons ($N_{\rm NG}$) and the numbers of broken generators ($N_{\rm BG}$. In such a situation, so-called type-II NG bosons emerge. We study how the gauge bosons acquire masses through the Higgs mechanism under this mismatch by employing gauge theories with complex scalar field at finite chemical potential and by enforcing "charge" neutrality. To separate the physical spectra from unphysical ones, the $R_{\xi}$ gauge is adopted. Not only massless NG bosons but also massive scalar bosons generated by the chemical potential are absorbed into spatial components of the gauge bosons. Although the chemical potential induces a non-trivial mixings among the scalar bosons and temporal components of the gauge bosons, it does not affect the structure of the physical spectra, so that the total number of physical modes is not modified even for $N_{\rm NG} < N_{\rm BG}$.