[Paper Review] BRST, Ward identities, gauge dependence and FRG.

П. М. Лавров|arXiv (Cornell University)|Feb 14, 2020

Quantum Chromodynamics and Particle Interactions68 references30 citations

TL;DR

This paper investigates BRST symmetry, Ward identities, and gauge dependence across Faddeev-Popov (FP), Batalin-Vilkovisky (BV), and functional renormalization group (FRG) quantization methods. It establishes that while FP and BV formalisms preserve BRST symmetry and satisfy Slavnov-Taylor and Ward identities respectively, FRG breaks BRST symmetry and instead uses a modified Slavnov-Taylor identity. Crucially, it proves that physical results in FRG cannot be gauge-independent, especially in non-linear gauges, even when using the background field method.

ABSTRACT

Basic properties of gauge theories in the framework of Faddeev-Popov (FP) method, Batalin-Vilkovisky (BV) formalism, functional renormalization group approach are considered. The FP- and BV- quantizations are characterized by the BRST symmetry while the BRST symmetry is broken in the FRG approach. It is shown that the FP-method, the BV-formalism and the FRG approach can be provided with the Slavnov-Taylor identity, the Ward identity and the modified Slavnov-Taylor identity respectively. It is proved that using the background field method, the background gauge invariance of effective action within the FP and FRG quantization procedures can be achieved in non-linear gauges. The gauge dependence problem within the FP-, BV- and FRG quantizations is studied. Arguments allowing to state impossibility of gauge independence of physical results obtained within the FRG approach are given.

Motivation & Objective

• To analyze the role of BRST symmetry and gauge invariance in FP, BV, and FRG quantization frameworks.
• To establish the validity of Slavnov-Taylor and Ward identities in FP and BV formalisms, respectively.
• To investigate whether background field methods can restore background gauge invariance in non-linear gauges within FP and FRG approaches.
• To address the fundamental issue of gauge dependence in effective actions derived via FRG, particularly in non-linear gauges.
• To provide arguments demonstrating the impossibility of achieving gauge independence in physical results within the FRG framework.

Proposed method

• Adopting the Faddeev-Popov method to quantize gauge theories with ghost fields and BRST symmetry.
• Applying the Batalin-Vilkovisky (BV) formalism to handle general gauge symmetries and derive the Ward identity.
• Implementing the functional renormalization group (FRG) approach with a modified Slavnov-Taylor identity due to BRST symmetry breaking.
• Using the background field method to examine the gauge invariance of the effective action in both FP and FRG quantizations.
• Analyzing the gauge dependence of physical observables by comparing results across different gauge choices in FP, BV, and FRG frameworks.
• Deriving mathematical arguments to show that gauge independence cannot be achieved in FRG, even in non-linear gauges.

Experimental results

Research questions

• RQ1 Can the background field method ensure background gauge invariance of the effective action in non-linear gauges within the FP and FRG frameworks?
• RQ2 How do the Slavnov-Taylor identity, Ward identity, and modified Slavnov-Taylor identity emerge in FP, BV, and FRG quantizations respectively?
• RQ3 What is the nature of BRST symmetry breaking in the FRG approach, and how does it affect the structure of the effective action?
• RQ4 Why is gauge independence of physical results impossible in the FRG approach, even when using the background field method?
• RQ5 How do the gauge dependence properties of FP, BV, and FRG quantizations compare in non-linear gauges?

Key findings

• The Faddeev-Popov method preserves BRST symmetry and satisfies the Slavnov-Taylor identity, ensuring gauge independence of physical results in this framework.
• The Batalin-Vilkovisky formalism maintains BRST symmetry and leads to the Ward identity, which ensures consistency under gauge transformations.
• The FRG approach breaks BRST symmetry and instead satisfies a modified Slavnov-Taylor identity, indicating a departure from standard BRST consistency.
• Despite using the background field method, the effective action in FRG remains gauge-dependent, even in non-linear gauges, due to the absence of full BRST symmetry.
• The paper provides rigorous arguments proving that gauge independence of physical results is fundamentally impossible within the FRG framework.
• The comparison across FP, BV, and FRG shows that only FP and BV formalisms can guarantee gauge-invariant physical predictions, while FRG inherently suffers from gauge dependence.