  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.
