  We revisit a low-energy theorem (LET) of NSVZ type in SU($N$) QCD with $N_f$
massless quarks derived in [1] by implementing it in dimensional
regularization. The LET relates $n$-point correlators in the lhs to $n+1$-point
correlators with the extra insertion of Tr$F^2$ at zero momentum in the rhs.
First, we demonstrate that, for $2$-point correlators of an operator $O$ in the
lhs, the LET implies that, in general, the integrated $3$-point correlator in
the rhs needs in perturbation theory an infinite additive renormalization in
addition to the multiplicative one. Second, we relate the above counterterm --
that is completely fixed by the LET -- to a corresponding divergent contact
term in a certain coefficient of the OPE of Tr$F^2$ with $O$ in the momentum
representation, thus extending by means of the LET to any operator $O$ an
independent argument that first appeared for $O$=Tr$F^2$ in [2]. Third, we
verify by direct computation that the latter divergent contact term first
computed in [3] to order $g^4$ in perturbation theory and to all orders in [2]
actually agrees with the one implied by the LET. Fourth, we evaluate the
divergent contact terms for the above OPE coefficient both in the coordinate
and momentum representation and discuss their relation. Fifth, we demonstrate
that in the asymptotically free phase of QCD the aforementioned counterterm in
the LET -- though divergent order by order in perturbation theory -- is
actually finite nonperturbatively after resummation to all perturbative orders.
Finally, we briefly recall the implications of the LET in the gauge-invariant
framework of dimensional regularization for the perturbative and
nonperturbative renormalization in large-$N$ QCD. The implications of the LET
inside and above the conformal window of SU($N$) QCD with $N_f$ massless quarks
will appear in a forthcoming paper.
