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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 $\beta=g^{-2}>0$ 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 $\beta$, 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.