Working towards a gauge-invariant description of the Higgs model: from local composite operators to spectral density functions

TL;DR

This paper develops a gauge-invariant framework for analyzing the Higgs model and Yang-Mills theories, focusing on spectral functions of composite operators to distinguish physical states and improve understanding of gauge dependence.

Contribution

It introduces a gauge-invariant description of the spectrum using local composite operators and analyzes their spectral functions at one-loop, surpassing issues with gauge dependence of elementary fields.

Findings

Spectral functions of elementary fields depend strongly on gauge parameters.

Gauge-invariant composite operators have positive, well-defined spectral densities.

The approach clarifies physical states in Higgs and Yang-Mills models.

Abstract

We analyze different BRST invariant solutions for the introduction of a mass term in Yang-Mills (YM) theories. First, we analyze the non-local composite gauge-invariant field $A^h_{\mu}(x)$, which can be localized by the Stueckelberg-like field $\xi^a(x)$. This enables us to introduce a mass term in the $SU(N)$ YM model, a feature that has been indicated at a non-perturbative level by both analytical and numerical studies. We also consider the unitary Abelian Higgs model and investigate its spectral functions at one-loop order. This analysis allows to disentangle what is physical and what is not at the level of the elementary particle propagators, in conjunction with the Nielsen identities. We highlight the role of the tadpole graphs and the gauge choices to get sensible results. We also introduce an Abelian Curci-Ferrari action coupled to a scalar field to model a massive photon which, like the non-Abelian Curci-Ferarri model, is left invariant by a modified non-nilpotent BRST symmetry. Finally, the spectral properties of a set of local gauge-invariant composite operators are investigated in the $U(1)$ and $SU(2)$ Higgs model quantized in the 't Hooft $R_{\xi}$ gauge. These operators enable us to give a gauge-invariant description of the spectrum of the theory, thereby surpassing certain incommodities when using the standard elementary fields. The corresponding two-point correlation functions are evaluated at one-loop order and their spectral functions are obtained explicitly. It is shown that the spectral functions of the elementary fields suffer from a strong unphysical dependence from the gauge parameter $\xi$, and can even exhibit positivity violating behaviour. In contrast, the BRST invariant local operators exhibit a well defined positive spectral density.