On charge conjugation, correlations, Elitzur’s theorem and the mass gap problem in lattice SU( N ) Yang–Mills models in d = 4 dimensions

Working in d = 4 Euclidean spacetime dimensions, we analyze a Wilson, Yang–Mills model on a hypercubic lattice, with gauge group SU(N), and constructed using the Osterwalder–Schrader–Seiler formalism via a Feynman–Kac formula. To each bond b there is a bond variable Ub ∈ SU(N) and gluon fields are parameters in the SU(N) Lie algebra. We define a charge conjugation operator C in the underlying physical quantum mechanical Hilbert space H and prove that, for N ≠ 2, H admits an orthogonal direct sum decomposition into two sectors with charge conjugation ±1. There is only one H sector for SU(2) due to the reality of the characters. In the space of correlations, a charge conjugation operator CE is defined and an analogous decomposition holds. Recall the mass gap property is related to the exponential distance decay of the truncated two gauge invariant plaquette field correlation. The exponential decay rate determines the glueball mass. Here, we first show that, if a mass gap occurs, a multiplicity-two glueball state (two mass gaps) is expected in the low-lying energy-momentum spectrum of the SU(N ≠ 2) Yang–Mills model, at least for small gauge coupling 0 β = g−2 ≪ 1. Besides, a version of Elitzur’s theorem is shown and applications are given. It is proven that the expectation averages of two distinct lattice vector potential correlators is zero. Surprisingly, the same holds for the expectation of two distinct field strength tensors. In the gluon field parametrization, the Wilson plaquette action has a quadratic upper bound in the gluon fields. Taking the gluon field expansion of the Wilson action, there is no local term, such as a mass term. However, the exponentiated Haar measure density has a positive local mass term, proportional to β and the quadratic Casimir eigenvalue. This is an indication that a mass gap may develop in Yang–Mills models. Our results also hold d = 3.