On Yang-Mills Stability Bounds and Plaquette Field Generating Function

Abstract

We consider the local gauge-invariant Yang-Mills quantum field theory on the finite hyper-cubic lattice Λ ⊂ aℤ^d ⊂ ℝ^d, d = 2, 3, 4, a ∊ (0, 1], with L (even) sites on a side and with the gauge Lie groups $G$ = U(N), SU(N). To each Λ bond b, there is a unitary matrix gauge variable U_b from an irrep of $G$. The vector gauge potentials (gluon fields) are parameters in the Lie algebra of $G$. The Wilson finite lattice partition function Z_Λ (a) is used. The action A_Λ (a) is a sum of gauge-invariant plaquette actions times $\left[a^{d-4}/g^2\right]$, $g^2\in (0,g_0^2]$, $0<g_0^2<\infty$. Each plaquette action has the product of four bond variables; the partition function is the integral over the Boltzmann factor with a product over bonds of $G$ Haar measures. Formally, in the continuum, ultraviolet (UV) limit a &drarr; 0, the action gives the YM classical continuum action. For free and periodic boundary conditions (b.c.), and using scaled fields, defined with an a-dependent noncanonical scaling, we show thermodynamic and UV stable (TUV) stability bounds for a scaled partition function, with constants independent of L, a and g. Passing to scaled fields does not alter the model energy-momentum spectrum and can be interpreted as an a priori field strength renormalization, making the action more regular. With scaled fields, we can isolate the UV singularity of the finite lattice physical, unscaled free energy fΛ(a) = [ln ZΛ ]/Λ_s, where Λ_s = L^d is the total number of lattice sites. With this, we show the existence of, at least, the subsequential thermodynamic (Λ &nearr; dℤ^d) and UV limits of a scaled free energy. To obtain the TUV bounds, the Weyl integration formula is used in the gauge integral and the random matrix probability distributions of the CUE and GUE appear naturally. Using periodic b.c. and the multireflection method, the generating function of r scaled plaquette field correlations is bounded uniformly in L, a, g and the location/orientation of the r plaquette fields. Consequently, r-scaled plaquette field correlations are also bounded. We also show the physical two-plaquette field correlation at coincident points has an a^−d UV singular behaviour; the same as for the correlation of the derivative of free scalar unscaled fields at coincident points. Using the free scalar case as a reference, we then have a lattice characterization of UV asymptotic freedom.