On Yang-Mills Stability Bounds and Plaquette Field Generating Function

Abstract

We consider the Yang-Mills (YM) QFT with group $U(N)$. We take a finite lattice regularization $Λ\subset a\mathbb Z^d$, $d = 2,3,4$, with $a\in (0,1]$ and $L$ (even) sites on a side. Each bond has a gauge variable $U\in U(N)$. The Wilson partition function is used and the action is a sum of gauge-invariant plaquette (minimal square) actions times $a^{d-4}/g^2$, $g^2\in(0,g_0^2]$, $0<g_0^2<\infty$. A plaquette action has the product of its four variables and the partition function is the integral of the Boltzmann factor with a product of $U(N)$ Haar measures. Formally, when $a\searrow 0$ our action gives the usual YM continuum action. For free and periodic b.c., we show thermodynamic and stability bounds for a normalized partition function of any YM model defined as before, with bound constants independent of $L,a,g$. The subsequential thermodynamic and ultraviolet limit of the free energy exist. To get our bounds, the Weyl integration formula is used and, to obtain the lower bound, a new quadratic global upper bound on the action is derived. We define gauge-invariant physical and scaled plaquette fields. Using periodic b.c. and the multi-reflection method, we bound the generating function of $r-$scaled plaquette correlations. A normalized generating function for the correlations of $r$ scaled fields is absolutely bounded, for any $L,a,g$, and location of the external fields. From the joint analyticity on the field sources, correlations are bounded. The bounds are new and we get $a^{-d}$ for the physical two-plaquette correlation at coincident points. Comparing with the $a\searrow 0$ singularity of the physical derivative massless scalar free field two-point correlation, this is a measure of ultraviolet asymptotic freedom in the context of a lattice QFT. Our methods are an alternative and complete the more traditional ones.