  We point out that perturbation theory in conjunction with the renormalization
group (RG) puts a severe constraint on the structure of the large-N
non-perturbative glueball propagators in SU(N) pure YM, in QCD and in n=1 SUSY
QCD with massless quarks, or in any confining asymptotically-free gauge theory
massless in perturbation theory. For the scalar and pseudoscalar glueball
propagators in pure YM and QCD with massless quarks we check in detail the
RG-improved estimate to the order of the leading and next-to-leading logarithms
by means of a remarkable three-loop computation by Chetyrkin et al. We
investigate as to whether the aforementioned constraint is satisfied by any of
the scalar or pseudoscalar glueball propagators computed in the framework of
the AdS String/ large-N Gauge Theory correspondence and of a recent proposal
based on a Topological Field Theory underlying the large-N limit of YM. We find
that none of the proposals for the scalar or the pseudoscalar glueball
propagators based on the AdS String/ large-N Gauge Theory correspondence
satisfies the constraint, actually as expected, since the gravity side of the
correspondence is in fact strongly coupled in the ultraviolet. On the contrary,
the Topological Field Theory satisfies the constraint that follows by the
asymptotic freedom.
