Abstract
We consider the free energy
W[J]=
W
k
(H)
of QCD coupled to an external source
J
b
μ
(x)=
H
b
μ
cos(k⋅x)
, where
H
b
μ
is, by analogy with spin models, an external "magnetic" field with a color index that is modulated by a plane wave. We report an optimal bound on
W
k
(H)
and an exact asymptotic expression for
W
k
(H)
at large
H
. They imply confinement of color in the sense that the free energy per unit volume
W
k
(H)/V
and the average magnetization $m(k, H) ={1 \over V} {\p W_k(H) \over \p H}$ vanish in the limit of constant external field
k→0
. Recent lattice data indicate a gluon propagator
D(k)
which is non-zero,
D(0)≠0
, at
k=0
. This would imply a non-analyticity in
W
k
(H)
at
k=0
. We also give some general properties of the free energy
W(J)
for arbitrary
J(x)
. Finally we present a model that is consistent with the new results and exhibits (non)-analytic behavior. Direct numerical tests of the bounds are proposed.