Entanglement and Effective Field Theories (2502.11819v2)

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.

• The paper demonstrates that the Berry phase acts as a local chiral transformation linking the vacuum structure to the chiral anomaly in gauge theories.
• The paper introduces a dynamic treatment of the vacuum angle in four-dimensional QED, revealing geometric corrections to conventional chiral symmetry.
• The paper uncovers a novel connection between entanglement entropy and the Berry phase, offering insights into topological effects and infrared divergence regularization.

Analyzing Geometric Phases and Chiral Transformations in Gauge Theories

The paper investigates the emergence of geometric phases in chiral transformations within gauge theories, with a focus on understanding how these phases affect the vacuum structure and anomalies. By exploring interactions between anomalies and vacuum configurations at lower energy levels, the research seeks to establish a framework where geometric phases in chiral transformations introduce nontrivial corrections.

The analysis commences with the exploration of the Schwinger model in (1+1) dimensions, where chiral symmetry is explicitly impacted by the dynamical mass generation of photons. This model serves as a platform to paper the connection between vacuum structures and anomalies. Extending the investigation to four-dimensional Quantum Electrodynamics (QED), the paper sets the vacuum angle $\theta$ as a dynamic field $\theta(x)$. This novel approach allows a deeper examination of how axial anomalies and nontrivial vacuum structures modify conventional chiral symmetry.

Key Findings

1. Schwinger Model Insights: The research illustrates that the Berry phase and chiral charge in the system are intrinsically linked, manifesting through the chiral anomaly. Within this framework, the generated photon mass signifies topological structures captured by the Berry phase. The paper demonstrates that in the Schwinger model, the Berry phase acts as a local chiral transformation, thereby embodying a geometric connection in the model's parameter space. This insight reveals how the Berry phase serves as a chiral gauge transformation intrinsically related to the chiral anomaly.
2. QED in Four Dimensions: Transitioning to QED, the paper adopts a strategy where $\theta$ becomes a dynamical field $\theta(x)$, thus facilitating the computation of the fermionic determinant under the adiabatic approximation. The introduction of the Berry phase modifies the conservation of the chiral current in this low-energy domain. Here, $\theta(x)$ introduces a geometric correction to chiral transformations, illustrating that an effective description incorporates the Berry connection in parameter space. Consequently, the paper suggests that chiral symmetry acquires a geometrical dependence, which could yield topological effects by reinterpreting the anomaly as related to the curvature in parameter space geometry.
3. Interconnection with Quantum Entanglement: The authors posit a fundamental connection between entanglement entropy and the Berry phase through the chiral anomaly, a relation not previously illuminated. The defined entanglement entropy in the Schwinger model is shown to acquire corrections from the Berry phase, revealing deeper entanglements with geometric aspects of the theory. In four dimensions, the entanglement entropy equation manifests a dependence on the Berry phase, further underscoring the novel interplay between these factors.

Implications and Future Directions

The research establishes an essential bridge linking chiral symmetries, anomalies, and geometric phases in gauge theories. This connection allows a unified approach to account for topological effects, vacuum structure alterations, and infrared modifications within these theories. One of the proposed implications of the Berry phase is a potential mechanism for regularizing infrared divergences in theories with axial anomalies, offering a geometrical and topological perspective on longstanding problems, such as those related to soft photons in QED.

As a speculative aspect, further developments could probe the complete resolution of infrared divergence issues through Berry phases or discover alternative solutions within gauge field theories. Moreover, the axial anomaly's reinterpretation in terms of parameter space geometry invites broader research into effective field theories (EFTs), opening avenues to refine conceptual models in high-energy physics. As a conclusion, this paper contributes significantly to advanced theoretical paradigms, leveraging geometric phases to deepen the understanding of anomalies and topology in gauge theories.