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

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.