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 Λ\subset aZ^d, d=2,3,4, a\in(0,1], with L sites on a side, Λ_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 $0<g^2\le g_0^2<\infty$. Sometimes we use an enhanced temporal gauge. Here, there are Λ_r\simeq (d-1)Λ_s retained bond variables. The unscaled partition function is $Z_{\Lambda ,a}^u\equiv Z_{\Lambda ,a,{\kappa }_u^2,m_u,g^2,d}^u$, where ${\kappa }_u^2>0$ is the unscaled hopping parameter and m_u are the boson bare masses. Letting $s_B\equiv [a^{d-2}(m_u^2{a}^2+2d{\kappa }_u^2){]}^{1/2}$, $s_Y\equiv a^{(d-4)/2}/g$, we show that the scaled partition function $Z_{\Lambda ,a}=s_B^{N{\Lambda }_s}{s}_Y^{d(N){\Lambda }_r}{Z}_{\Lambda ,a}^u$ satisfies the stability bounds $e^{c_{\ell }d(N){\Lambda }_s}\le Z_{\Lambda ,a}\le e^{c_{u}d(N){\Lambda }_s}$ with finite real $c_{\ell },c_u$ independent of $L$ and the spacing $a$. We have extracted in $Z_{\Lambda ,a}^u$ the dependence on Λand 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_Λ^n and, next, of a continuum limit.

References in corpus (2)

• A Perspective on Constructive Quantum Field Theory
• Revisiting SU(N) integrals