On Charge Conjugation, Correlations, Elitzur's Theorem and the Mass Gap Problem in Lattice $SU(N)$ Yang-Mills Models in $d=4$ Dimensions

TL;DR

This work provides a rigorous lattice formulation of four-dimensional SU(N) Yang–Mills theory using the Wilson action and the Osterwalder–Schrader–Seiler/Feynman–Kac framework, focusing on charge-conjugation symmetry and its impact on the physical spectrum. It proves an orthogonal decomposition of the physical Hilbert space and of correlation spaces into sectors of definite charge conjugation for $N\neq2$, while SU(2) remains in a single sector, and it establishes a version of Elitzur’s theorem that yields strong constraints on non-gauge-invariant observables and correlations. The analysis shows that the Wilson action contains no local mass term in the gluon-field expansion, but the Haar measure density exponent induces a positive local mass term, which, together with spectral representations, supports the emergence of a mass gap and the existence of two possibly distinct glueball states for $N\neq2$ at small $\beta$ (with SU(2) as a special case). The results hold in $d=3$ as well and illuminate the link between group-theoretic structure, symmetry constraints, and the Yang–Mills mass-gap problem, highlighting both the role of charge conjugation and the subtleties of gauge-measure contributions toward a continuum limit.

Abstract

We consider a four-dimensional Euclidean Wilson lattice Yang-Mills model with gauge group $SU(N)$, and the associated lattice Euclidean quantum field theory constructed by Osterwalder-Schrader-Seiler via a Feynman-Kac formula. In this model, to each lattice bond $b$ there is assigned a bond variable $U_b\in SU(N)$. Gluon fields are parameters in the Lie algebra of $SU(N)$. We define a charge conjugation operator $\mathcal C$ in the physical Hilbert space $H$ and prove that, for $N\not=2$, $H$ admits an orthogonal decomposition into two sectors with charge conjugation $\pm1$. There is only one sector for $N=2$. In the space of correlations, a charge conjugation operator $C_E$ is defined and a similar decomposition holds. Besides, a version of Elitzur's theorem is shown; applications are given. It is proven that the expectation averages of two distinct lattice vector potential correlators is zero. Surprisingly, the expectation of two distinct field strength tensors is also zero. In the gluon field parametrization it is known that Wilson action is bounded quadratically in the gluon fields. Towards solving the mass gap problem, in the gluon field expansion of the action, we show there are no local terms, such as a mass term. However, the action associated with the exponent of the exponentiated Haar measure density has a positive local mass term which is proportional to the square of the gauge coupling $β=g^{-2}>0$ and the $SU(N)$ quadratic Casimir operator eigenvalue. Our results also hold in dimension three. Of course, the orthogonal decomposition of the Hilbert space $H$ has consequences in the analysis of the truncated two-point plaquette field correlation and the Yang-Mills mass gap problem: for $N\not=2$ and at least for small $β$, a multiplicity-two one-particle glueball state (two mass gaps) is expected to be present in the energy-momentum spectrum of the Yang-Mills model.

Theorems & Definitions (3)

• Remark 1$\:$
• Lemma 1$\:$
• Theorem 1$\:$