On thermodynamic and ultraviolet stability bounds for bosonic lattice QCD models in Euclidean dimensions d = 2,3,4

Abstract

We prove thermodynamic and ultraviolet stable stability bounds for lattice scalar QCD quantum models, with multiflavor real or complex scalar Bose matter fields and a compact, connected gauge Lie group [Formula: see text], [Formula: see text] with Lie algebra dimension [Formula: see text]. Our models are defined on a finite hypercubic lattice [Formula: see text], [Formula: see text], [Formula: see text], with [Formula: see text], even, sites on a side, [Formula: see text] sites, and with free boundary conditions. The models action is a sum of a minimally coupled Bose-gauge part and a Wilson pure-gauge plaquette action. We use local, scaled scalar multiflavor Bose fields. The scaling is global, [Formula: see text]-dependent and noncanonical, and corresponds to an a priori renormalization. The Wilson action is a sum over positive plaquette actions times a factor [Formula: see text], with the gauge coupling [Formula: see text] in [Formula: see text], [Formula: see text]. By local gauge invariance, to eliminate the excess of gauge variables, sometimes we use an enhanced temporal gauge, leaving only [Formula: see text] for [Formula: see text], retained bonds. Fixing this gauge does not alter the value of the partition function. Considering the original physical, unscaled partition function [Formula: see text], where [Formula: see text] is the unscaled (bare) hopping parameter and [Formula: see text] are the boson fields bare masses, and letting [Formula: see text] and [Formula: see text], we show that the scaled partition function [Formula: see text] satisfies the thermodynamic and ultraviolet stable stability bounds [Formula: see text], with finite constants [Formula: see text], independent of the lattice size [Formula: see text] of [Formula: see text] and the spacing [Formula: see text]. For the normalized finite-lattice free energy [Formula: see text], a finite thermodynamic limit ([Formula: see text]) for [Formula: see text], and then the continuum limit [Formula: see text], both exist in the sense of subsequences. They give the model normalized free energies [Formula: see text]. The finiteness of [Formula: see text] is the only question addressed here! The use of the Weyl integration formula is essential in showing these bounds. It allows us to replace the gauge integral over [Formula: see text] gauge bond matrix elements by the integration over its [Formula: see text] eigenvalues. A new global upper bound on the Wilson plaquette action is obtained, which is quadratic in the gluon fields. Our method bypasses the use of diamagnetic inequality and can be extended to treat more general lattices and Lie gauge groups.