On Thermodynamic and Ultraviolet Stability of Bosonic Lattice QCD Models in Euclidean Spacetime Dimensions $d=2,3,4$

TL;DR

This paper establishes stability bounds and analyzes the continuum limit of bosonic lattice QCD models with gauge invariance in 2, 3, and 4 dimensions, ensuring well-defined thermodynamic and continuum behaviors.

Contribution

It provides the first rigorous proof of stability bounds and the existence of thermodynamic and continuum limits for scalar bosonic lattice QCD models with gauge invariance.

Findings

Proved stability bounds for the scaled partition function.

Established the existence of thermodynamic limit for the free energy.

Demonstrated the continuum limit exists as lattice spacing approaches zero.

Abstract

We prove stability bounds for local gauge-invariant scalar QCD quantum models, with multiflavored bosons replacing (anti)quarks. We take a compact, connected gauge Lie group G, and concentrate on G=U(N),SU(N). Let d(N)=N^2,(N^2-1) be their Lie algebra dimensions. We start on a finite hypercubic lattice \Lambda\subset aZ^d, d=2,3,4, a\in(0,1], with L sites on a side, \Lambda_s=L^d sites, and free boundary conditions. The action is a sum of a Bose-gauge part and a Wilson pure-gauge plaquette term. We employ a priori local, scaled scalar bosons with an a-dependent field-strength renormalization: a non-canonical scaling. The Wilson action is a sum over pointwise positive plaquette actions with a pre-factor (a^{d-4}/g^2), and gauge coupling $0<g^2\leq g_0^2<\infty$. Sometimes we use an enhanced temporal gauge. Here, there are \Lambda_r\simeq (d-1)\Lambda_s retained bond variables. The unscaled partition function is $Z^u_{\Lambda,a}\equiv Z^u_{\Lambda,a,\kappa_u^2,m_u,g^2,d}$, where $\kappa_u^2>0$ is the unscaled hopping parameter and m_u are the boson bare masses. Letting $s_B\equiv [a^{d-2}(m_u^2a^2+2d\kappa_u^2)]^{1/2}$, $s_Y\equiv a^{(d-4)/2}/g$, we show that the scaled partition function $Z_{\Lambda,a}=s_B^{N\Lambda_s}s_Y^{d(N)\Lambda_r} Z^u_{\Lambda,a}$ satisfies the stability bounds $e^{c_\ell d(N)\Lambda_s}\leq Z_{\Lambda,a}\leq e^{c_ud(N)\Lambda_s}$ with finite real $c_\ell, c_u$ independent of $L$ and the spacing $a$. We have extracted in $Z^u_{\Lambda,a}$ the dependence on \Lambda and the exact singular behavior of the finite lattice free energy in the continuum limit $a\searrow 0$. For the normalized finite-lattice free energy $f_\Lambda^n=[d(N)\Lambda_s]^{-1}\ln Z_{\Lambda,a}$, we prove the existence of (at least, subsequentials) a thermodynamic limit for f_\Lambda^n and, next, of a continuum limit.