PCAC and Shadowing of Low Energy Neutrinos
  The Adler relation between reactions initiated by neutrinos and pions is easy to misinterpret as a manifestation of the pion pole dominance. An axial current, however, cannot fluctuate into a pion, but only to heavy axial-vector states, since the lepton current is transverse. This is the miracle of the PCAC hypothesis which dictates a specific conspiracy between the heavy fluctuations, so that all together they mock the pion pole. Indeed, the observed Q^2 dependence of the axial form factor is controlled by the effective mass m_A \sim 1 GeV, rather than the pion mass. On the contrary, the onset of nuclear shadowing is governed by the small pion mass, rather than by the large axial mass scale. This is in variance with the conventional wisdom which equates the fluctuation lifetime and the coherence time. For the case of axial current they are different by almost two orders of magnitude. As a result, neutrino interactions are shadowed at very low energies of few hundred MeV, while energy of about 10 GeV is needed to access nuclear shadowing for the vector current. On the contrary to naive expectations, nuclear absorption enhances, rather than suppresses the cross section of coherent neutrino-production of pions which is the strongest channel (half of the total cross section) in the black disc limit. 
B.Z. Kopeliovich