  The nonlocal order parameters of the QCD chiral symmetry breaking, namely the
nonanomalous three-point Green functions of the quark bilinears belonging to
the odd intrinsic parity sector, are studied within the framework of the
Resonance chiral theory. The general form of these correlators is then matched
with various high energy constraints: the leading and sub-leading OPE for all
three momenta simultaneously large, the leading OPE for two momenta large and
the leading and sub-leading Brodsky-Lepage limit for the corresponding
transition form factor. In order to satisfy these constraints, the Resonance
chiral theory has to be enlarged with additional resonance multiplets and with
higher derivative operators as well. We discuss a minimal extension of this
kind in detail and present corresponding correlators in a closed form. The
remaining free parameters are then constrained from lattice data. As a
phenomenological application, we discuss the pion-pole contribution to the muon
$g-2$ and the decay $\pi^0\to e^+e^-$.
