We consider the free energy W[J] = Wk(H) of QCD coupled to an external source Jbμ (x) = Hb μ cos(k · x), where Hb 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 Wk(H) and an exact asymptotic expression for Wk(H) at large H. They imply confinement of color in the sense that the free energy per unit volumeWk(H)/V and the average magnetization m(k,H) = 1 V ¶Wk(H) ¶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) 6=0, at k =0. This would imply a non-analyticity inWk(H) at k =0. We present a model that is consistent with the new results and exhibits (non)-analytic behavior. Direct numerical tests of the bounds are proposed.
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