Gauging Open EFTs from the top down

TL;DR

This paper develops a comprehensive top-down approach to calculating open effective field theories for gauged systems, emphasizing gauge fixing, BRST symmetry, and gauge invariance in various phases and models.

Contribution

It introduces a systematic method for deriving influence functionals in gauge theories, incorporating gauge fixing and BRST formalism, with explicit examples across different models and phases.

Findings

Influence functional remains gauge invariant under two copies of gauge symmetries.

BRST symmetry reduces to a single diagonal copy due to in-in boundary conditions.

Demonstrated gauge invariance in models including scalar QED and Higgs-Kibble.

Abstract

We present explicit top-down calculations of Open EFTs for gauged degrees of freedom with a focus on the effects of gauge fixing. Starting from the in-in contour with two copies of the action, we integrate out the charged matter in various $U(1)$ gauge theories to obtain the Feynman-Vernon influence functional for the photon, or, in the case of symmetry breaking, for the photon and St\"uckelberg fields. The influence functional is defined through a quantum path integral, which -- as is always the case when quantizing gauge degrees of freedom -- contains redundancies that must be eliminated via a gauge-fixing procedure. We implement the BRST formalism in this setting. The in-in boundary conditions break the two copies of BRST symmetry down to a single diagonal copy. Nevertheless the single diagonal BRST is sufficient to ensure that the influence functional is itself gauge invariant under two copies of gauge symmetries, retarded and advanced, regardless of the choice of state or symmetry-breaking phase. We clarify how this is consistent with the decoupling limit where the global advanced symmetry is generically broken by the state. We illustrate our results with several examples: a gauge field theory analogue of the Caldeira-Leggett model, spinor QED with fermions integrated out, scalar QED in a thermal state, the Abelian Higgs-Kibble model in the spontaneously broken state with the Higgs integrated out, and Abelian Higgs-Kibble model coupled to a charged bath in a symmetry-broken phase. The latter serves as an example of an open system for St\"uckelberg/Goldstone fields.