Entanglement and Effective Field Theories

TL;DR

This paper explores how geometric phases, specifically Berry phases, emerge in gauge theories with fermions, revealing their role in vacuum structure, anomalies, and infrared behavior, especially in QED.

Contribution

It introduces a novel framework linking chiral symmetries, anomalies, and geometric phases, extending insights from the Schwinger model to four-dimensional QED with a dynamical vacuum angle.

Findings

Chiral transformations acquire a Berry phase dependent on vacuum topology.

The framework connects anomalies with geometric phases, affecting low-energy physics.

Berry phases may help regularize infrared divergences in gauge theories.

Abstract

We investigate the emergence of geometric phases in chiral transformations within gauge theories coupled to fermions. We begin by analyzing the Schwinger model in (1+1) dimensions, where chiral symmetry is explicitly modified due to the dynamical generation of a photon mass. This model provides a controlled setting to study the interplay between anomalies and vacuum structure. Building on these insights, we extend our analysis to four-dimensional QED by promoting the vacuum angle $\theta$ to a dynamical field $\theta(x)$. This generalization allows us to explore how the axial anomaly and the presence of a nontrivial vacuum structure modify the conventional chiral symmetry. Using the adiabatic approximation, we demonstrate that chiral transformations are modified by the emergence of a nontrivial Berry phase, which introduces a geometric correction that depends on the topological properties of the vacuum. This result suggests that chiral transformations acquire an effective gauge structure in parameter space, in the presence of a dynamical $\theta(x)$ field, leading to new physical consequences at low energies. This framework establishes a novel connection between chiral symmetries, anomalies, and geometric phases, offering a unified approach to describing topological effects, vacuum structure, and infrared modifications in gauge theories with fermions. Moreover, our results suggest that Berry phases play a crucial role in the infrared structure of QED, potentially providing a mechanism for regularizing infrared divergences in theories with axial anomalies.