Physics-informed gauge theories (2503.22638v1)

Abstract: We use the physics-informed renormalisation group (PIRG) for the construction of gauge invariant renormalisation group flows. The respective effective action is a sum of a gauge invariant quantum part and the classical gauge fixing part which arranges for invertibility of the gauge field two-point function. Thus, the BRST transformations simply accommodate the gauge consistency of the gauge fixing sector, while the quantum part of the effective action is gauge and BRST invariant. We apply this physics-informed approach to Yang-Mills theory and gravity and show how the flowing gauge fields arrange for full gauge invariance. We also embed the background field approximation to the functional renormalisation group (fRG) in an exact gauge invariant PIRG flow. This allows us to discuss the dynamics of the correction terms, and the non-trivial ultraviolet or infrared relevant terms are elucidated within a one-loop approximation. The background field approximation of the latter is known for violating one-loop universality for specific regulators and we show how the present setup reinstates universality in a constructive way. Finally, we discuss the landscape of fRG flows in gauge theories through the lens of the novel PIRG approach as well as potential applications.

• The paper presents a novel physics-informed renormalization group (PIRG) method that integrates gauge invariance into RG flows while preserving quantum and BRST symmetry.
• It demonstrates the method's effectiveness in addressing one-loop universality violations and improving analysis in Yang-Mills theory and metric quantum gravity.
• The approach decomposes the effective action into a gauge-invariant quantum part and a classical gauge-fixing part, ensuring consistency with functional RG requirements.

Physics-informed Gauge Theories

Introduction

The paper "Physics-informed gauge theories" proposes a novel approach to constructing gauge-invariant renormalization group (RG) flows using physics-informed renormalization group (PIRG) techniques. This methodology seamlessly integrates gauge invariance into the renormalization process, maintaining the quantum and BRST symmetry automatically. The authors apply this framework to Yang-Mills theory and gravity and demonstrate its practical implications for functional renormalization groups (fRG).

Gauge Invariant PIRGs

The heart of the paper lies in developing a PIRG that ensures gauge invariance is preserved through its RG flows. The effective action is decomposed into a gauge-invariant quantum part and a gauge-fixing part treated classically. By embedding the background field approximation within an exact gauge-invariant PIRG flow, the approach elucidates the dynamics of correction terms related to the violation of one-loop universality in standard methods.

PIRGs and Properties

The method involves formulating the RG for the pair $\bigl( \Gamma_\phi[\phi], \dot\phi[\phi] \bigl)$, where $\Gamma_\phi[\phi]$ is the target action, and $\dot\phi[\phi]$ denotes the scale-dependent transformation of the flowing field. This framework allows tailoring the transformation to maintain gauge and BRST invariance, a critical requirement for consistent quantum field theory.

1. Trivial Nielsen Identity: The effective action depends only on the sum of background and flowing fields.
2. Background Gauge Invariance: The effective action remains invariant under transformations of the background field.
3. BRST Invariance: The master equation must hold, ensuring that the transformations respect quantum gauge invariance.

Flow Mechanisms and Applications

Implementing these concepts in practice involves solving the general flow and constraint equations for the flowing field and ensuring they satisfy the given symmetry properties at all scales. These flows can be applied directly to analyze Yang-Mills theory using gauge invariant PIRGs, allowing exploration of one-loop properties and comparing them with standard methods.

Specific Applications

• Yang-Mills Theory: Demonstrating how the PIRG can reconcile universal one-loop results that fail in the background field approximation when infrared-divergent regulators are applied.
• Metric Quantum Gravity: Addressing discrepancies and singularities observed in conventional background field approaches, suggesting improvements via PIRG methodologies.

Conclusion

The PIRG framework offers a robust, symmetry-consistent way to handle gauge theories in the context of the renormalization group. The approach enables a novel understanding of gauge theories' landscape through an exact RG framework, potentially reshaping future analytical and computational methods in quantum field theory. Its application to foundational theories like quantum gravity and Yang-Mills showcases its ability to address long-standing issues within these domains.