[Paper Review] Towards a manifestly gauge invariant and universal calculus for Yang-Mills theory

Stefano Arnone, Antonio Gatti+1 more|ArXiv.org|Sep 16, 2002

Particle physics theoretical and experimental studies1 references3 citations

TL;DR

This paper proposes a manifestly gauge-invariant exact renormalization group (ERG) framework for pure SU(N) Yang-Mills theory, using a spontaneously broken SU(N|N) super-gauge theory to implement a gauge-invariant effective cutoff. The method computes the one-loop beta function at finite N without gauge fixing or ghosts, yielding the universal perturbative result, confirming the formalism's consistency and universality while preserving gauge invariance at every step.

ABSTRACT

A manifestly gauge invariant exact renormalization group for pure SU(N) Yang-Mills theory is proposed, along with the necessary gauge invariant regularisation which implements the effective cutoff. The latter is naturally incorporated by embedding the theory into a spontaneously broken SU(N|N) super-gauge theory, which guarantees finiteness to all orders in perturbation theory. The effective action, from which one extracts the physics, can be computed whilst manifestly preserving gauge invariance at each and every step. As an example, we give an elegant computation of the one-loop SU(N) Yang-Mills beta function, for the first time at finite N without any gauge fixing or ghosts. It is also completely independent of the details put in by hand, e.g. the choice of covariantisation and the cutoff profile, and, therefore, guides us to a procedure for streamlined calculations.

Motivation & Objective

• To develop a manifestly gauge-invariant exact renormalization group (ERG) framework for pure SU(N) Yang-Mills theory.
• To resolve the conflict between momentum-shell integration and gauge invariance in ERG by embedding the theory in a spontaneously broken SU(N|N) super-gauge theory.
• To eliminate the need for gauge fixing and ghosts, thereby avoiding the Gribov problem and ensuring that all computations preserve exact gauge symmetry.
• To demonstrate the universality and independence of the formalism by computing the one-loop beta function at finite N, confirming the expected perturbative result without dependence on arbitrary choices like cutoff profile or covariantisation.

Proposed method

• Embedding the Yang-Mills theory in a spontaneously broken SU(N|N) super-gauge theory to realize a non-perturbative, gauge-invariant effective cutoff.
• Using a superfield formalism to write the ERG flow equation in a manifestly gauge-invariant way, avoiding explicit gauge fixing.
• Implementing a regularisation scheme via higher-derivative terms in the action, covariantised using the covariant derivative and a smooth cutoff profile c(p²/Λ²), ensuring gauge invariance.
• Applying the ERG flow equation to compute the effective action, with all vertices and kernels derived from the flow equations, ensuring gauge symmetry is preserved throughout.
• Using dimensional regularisation as a preregulator to handle divergences without breaking SU(N|N) symmetry, enabling clean cancellation of divergent terms.
• Employing a diagrammatic technique to track contributions from vertices and kernels, focusing on universal terms that depend only on two-point functions and integrated kernels.

Experimental results

Research questions

• RQ1 Can a manifestly gauge-invariant exact renormalization group framework be constructed for Yang-Mills theory without gauge fixing or ghosts?
• RQ2 How can a non-perturbative, gauge-invariant effective cutoff be implemented in a way that preserves all symmetries and ensures finiteness?
• RQ3 Does the proposed formalism yield the correct one-loop beta function for SU(N) Yang-Mills theory at finite N, independent of arbitrary choices like cutoff profile or covariantisation?
• RQ4 To what extent can gauge invariance alone guide the computation of physical quantities, eliminating the need for auxiliary fields like ghosts or boundary conditions?
• RQ5 Can the formalism be generalised to non-perturbative approximations and extended to include matter fields or supersymmetry?

Key findings

• The one-loop beta function for SU(N) Yang-Mills theory is computed correctly and universally, matching the standard perturbative result without any gauge fixing or ghost fields.
• The calculation is completely independent of the choice of cutoff profile and covariantisation scheme, confirming the universality of the result.
• Gauge invariance is preserved manifestly at every step of the computation, including the flow equations and vertex contractions.
• Potentially universal contributions arise only from terms involving two-point functions and integrated kernels, simplifying the structure of the final result.
• The method successfully avoids the Gribov problem by eliminating the need for gauge fixing and boundary conditions.
• The formalism is general enough to be extended to non-perturbative approximations, matter fields, and space-time supersymmetry, with potential applications to QCD and AdS/CFT.