Generalized Yang-Mills theory: Interpolating between SDYM and YM

TL;DR

This paper introduces a generalized Yang-Mills theory with two couplings that smoothly transitions between the self-dual and standard Yang-Mills theories, revealing new insights into their renormalization and vacuum structure.

Contribution

It constructs a two-coupling YM model, derives an exact relation between their beta functions, and explores the vacuum and confinement mechanisms across the interpolation.

Findings

Exact all-order relation between beta functions of the two couplings

Identification of a new RG-invariant dimensionless parameter

Demonstration of mass gap and confinement emergence as kinetic coupling increases

Abstract

We construct a generalized Yang-Mills (YM) theory with two real couplings, interpolating continuously between the Self-Dual Yang-Mills (SDYM) limit (also called Chalmers-Siegel theory) and physical Yang-Mills theory. The kinetic coupling $\epsilon$ controls local fluctuations and anti-instanton weight, while the topological coupling $g$ controls the instanton weight. Both couplings are asymptotically free. We derive an exact all-order relation between the beta functions of the two couplings, revealing a Renormalization Group invariant, a new dimensionless expansion parameter $\Lambda_\epsilon / \Lambda_g$ into the study of YM theory. In the SDYM limit, the vacuum is populated by a finite density of topological defects, yet local correlators decay algebraically, consistent with a non-unitary conformal field theory. We confirm this mechanism via compactification on arbitrary size $\mathbb{R}^3 \times S^1$, where the vacuum maps to a non-interacting ideal gas of monopole-instantons. As the kinetic coupling is turned on, a mass gap and confinement scale emerge.