On Yang-Mills Stability Bounds and Plaquette Field Generating Function

TL;DR

This paper establishes stability bounds and analyzes the ultraviolet behavior of lattice Yang-Mills theory with gauge group U(N), providing new bounds on correlations and insights into asymptotic freedom.

Contribution

It introduces novel stability bounds and a generating function bound for lattice Yang-Mills fields, advancing understanding of UV limits and correlation behaviors.

Findings

Thermodynamic and stability bounds are independent of lattice size and coupling.

The generating function for scaled plaquette correlations is absolutely bounded.

The physical two-plaquette correlation diverges as the lattice spacing approaches zero, indicating UV asymptotic freedom.

Abstract

We consider the Yang-Mills (YM) QFT with group $U(N)$. We take a finite lattice regularization $\Lambda\subset a\mathbb Z^d$, $d = 2,3,4$, 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.