On Yang-Mills Stability and Plaquette Field Generating Functional

Abstract

We consider the pure Yang-Mills relativistic quantum field theory in an imaginary time functional integral formulation. The gauge group is taken to be $\mathcal G = \mathrm U(N)$. We use a lattice ultraviolet regularization, starting with the model defined on a finite hypercubic lattice $\Lambda\subset a\mathbb Z^d$, $d = 2,3,4$, with lattice spacing $a\in (0,1]$ and $L\in\mathbb N$ sites on a side. The Wilson partition function is used where the action is a sum over four lattice bond variables of gauge-invariant plaquette (lattice minimal squares) actions with a prefactor $a^{d-4}/g^2$, where we take the gauge coupling $g\in(0,g_0^2]$, $0<g_0<\infty$. In a recent paper, for free boundary conditions, we proved that a normalized model partition function satisfies thermodynamic and ultraviolet stable stability bounds. Here, we extend the stability bounds to the Yang-Mills model with periodic boundary conditions, with constants which are also independent of $L$, $a$, $g$. Furthermore, we also consider a normalized generating functional for the correlations of $r\in\mathbb N$ gauge-invariant plaquette fields. Using periodic boundary conditions and the multireflection method, we then prove that this generating functional is bounded, with a bound that is independent of $L$, $a$, $g$ and the location and orientation of the $r$ plaquette fields. The bounds factorize and each factor is a single-bond variable, single-plaquette partition function. The number of factors is, up to boundary corrections, the number of non-temporal lattice bonds, such as $(d-1)L^d$. A new global quadratic upper bound in the gluon fields is proved for the Wilson plaquette action.