Generalized Yang-Mills theory: Interpolating between SDYM and YM

TL;DR

The paper constructs a generalized Yang-Mills theory with two real couplings $\epsilon$ and $g$ that interpolate between self-dual Yang-Mills and standard Yang-Mills, revealing an exact all-order relation $\beta_g/g^3=\beta_\epsilon/\epsilon^3$ and an RG-invariant combination that introduces a new dimensionless expansion parameter. It shows SDYM ( $\epsilon\to0$ ) hosts a finite density of self-dual defects but remains a non-unitary CFT with massless local correlators, while turning on $\epsilon$ induces a mass gap and confinement, connecting conformal and confining dynamics through a two-scale running structure with $\Lambda_g$ and $\Lambda_\epsilon$. The analysis uses a clear separation of kinetic and topological sectors, perturbative non-renormalization of the topological term, and semiclassical monopole-instanton dynamics, including a monopole–anti-monopole sector that yields an exponentially small mass gap $m_\epsilon^2 \sim L^{-2} \exp(-16\pi^2/(\epsilon^2 N))$ on $\mathbb{R}^3\times S^1$. Altogether, the work provides a controlled framework to study YM mass gap and confinement by perturbing a logarithmic CFT, with potential implications for nonperturbative YM physics in four dimensions and links to twistor and nonperturbative topological sectors.

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 $ε$ 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 $Λ_ε/ Λ_g$ 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.

Figures (1)

• Figure 1: The running of kinematic and topological couplings, and associated strong scales.