The cross section of coherent neutrino-nucleus scattering is formulated in this paper, where the vector nuclear form factor F_V(Q2) is assumed to be the same as axial nuclear form factor F_A(Q2) without further explanation. Is it obvious to be so? Is it related to neutrino energy? I searched in Google, and found some paper talking about measuring axial nucleon form factor using high energy neutrino beams.
Article Probing new physics with coherent neutrino scattering off nuclei
It is not obvious that the axial and vector form factors are the same. The dependence of them as functions of the momentum transfer (Q^2) is typically assumed to be of the dipole form, i.e. F=F(0)/(1+Q^2/M^2)^2 where M is the cutoff mass. If, F_V(Q^2) and F_A(Q^2) are normalized one they will have similar behavior at low momentum transfer (meaning low neutrino energy). However, the so-called cut-off masses M_A and M_V are different. Consequently, the behavior at large Q^2 (i.e. for the neutrino energies of a few hundreds of MeVs) should be quite different. Typically, one is using M_A=1.014 GeV and M_V^2=0.71 GeV^2. For the vector form factor it exists also expansions beyond the dipole approximation which I use in my codes.
The lectures of S. Coleman, ``Soft pions'', at the 1966 Erice summer school (reprinted in ``Aspects of Symmetry'') are recommended reading for this question.
I remember that talk of Sidney Coleman, when he was balancing a cigarett between two fingers of his right hand , and a piece of chalk between two other.
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