On Thermodynamic and Ultraviolet Stability of Yang-Mills

Abstract

We prove ultraviolet stable stability bounds for the pure Yang-Mills relativistic quantum theory in an imaginary-time, functional integral formulation. We consider the gauge groups $\mathcal G={\rm U}(N)$, ${\rm SU}(N)$ and let $d(N)$ denote their Lie algebra dimensions. We start with a finite hypercubic lattice $\Lambda\subset a\mathbb Z^d$, $d=2,3,4$, $a\in(0,1]$, $L$ sites on a side, and with free boundary conditions. The Wilson partition function $Z_{\Lambda,a}\equiv Z_{\Lambda,a,g^2,d}$ is used, where the action is a sum over gauge-invariant plaquette actions with a pre-factor $[a^{d-4}/g^2]$, where $g^2\in(0,g_0^2]$, $0<g_0<\infty$, defines the gauge coupling. By a judicious choice of gauge fixing, which involves gauging away the bond variables belonging to a maximal tree in $\Lambda$, and which does not alter the value of $Z_{\Lambda,a}$, we retain only $\Lambda_r$ bond variables, which is of order $[(d-1)L^d]$, for large $L$. We prove that the normalized partition function $Z^n_{\Lambda,a}=(a^{(d-4)}/g^2)^{d(N)\Lambda_r/2}Z_{\Lambda,a}$ satisfies the stability bounds $e^{c_\ell d(N)\Lambda_r}\leq Z^n_{\Lambda,a}\leq e^{c_ud(N)\Lambda_r}$, with finite $c_\ell,\,c_u\in\mathbb R$ independent of $L$, the lattice spacing $a$ and $g^2$. In other words, we have extracted the {\em exact} singular behavior of the finite lattice free-energy. For the normalized free-energy $f^n=[d(N)\,\Lambda_r]^{-1}\,\ln Z^n_{\Lambda,a}$, our stability bounds imply, at least in the sense of subsequences, that a finite thermodynamic limit $\Lambda\nearrow a\mathbb Z^d$ exists. Subsequently, the continuum $a\searrow 0$ limit also exists.