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 G = U(N ). We use a lattice ultraviolet regularization, starting with the model defined on a finite hypercubic lattice Λ ⊂ aZ d , d = 2, 3, 4, with lattice spacing a ∈ (0, 1] and L ∈ 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 ∈ (0, g 2 0 ], 0 < g0 < ∞. 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 ∈ 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.

On Yang-Mills Stability Bounds and Plaquette Field Generating Function

“…Recently, in the unpublished papers [14,15], a simple proof of thermodynamic and ultraviolet stable (TUV) stability bounds is given by a direct analysis of the Wilson partition function with free boundary conditions (b.c.) in configuration space, starting with the model in a finite hypercubic lattice.…”

“…In this paper, we show detailed and much simplified proofs of the Theorems of Refs. [14,15]. Besides these simplifications, and in order to make clear how our results are obtained, we incorporate an analysis of the special case of the abelian gauge group U (1).…”