Entanglement and Effective Field Theories

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.