  We present a determination of the neutral current weak axial charge
$G^Z_A(0)=-0.654(3)_{\rm stat}(5)_{\rm sys}$ using the strange quark axial
charge $G^s_A(0)$ calculated with lattice QCD. We then perform a
phenomenological analysis, where we combine the strange quark electromagnetic
form factor from lattice QCD with (anti)neutrino-nucleon scattering
differential cross section from MiniBooNE experiments in a momentum transfer
region $0.24\lesssim Q^2 \lesssim 0.71$ GeV$^2$ to determine the neutral
current weak axial form factor $G^Z_A(Q^2)$ in the range of $0\lesssim Q^2\leq
1$ GeV$^2$. This yields a phenomenological value of $G^Z_A(0)=-0.687(89)_{\rm
stat}(40)_{\rm sys}$. The value of $G^Z_A(0)$ constrained by the lattice QCD
calculation of $G^s_A(0)$, when compared to its phenomenological determination,
provides a significant improvement in precision and accuracy and can be used to
provide a constraint on the fit to $G^Z_A(Q^2)$ for $Q^2>0$. This constrained
fit leads to an unambiguous determination of (anti)neutrino-nucleon neutral
current elastic scattering differential cross section near $Q^2=0$ and can play
an important role in numerically isolating nuclear effects in this region. We
show a consistent description of $G^Z_A(Q^2)$ obtained from the
(anti)neutrino-nucleon scattering cross section data requires a nonzero
contribution of the strange quark electromagnetic form factor. We demonstrate
the robustness of our analysis by providing a post-diction of the BNL E734
experimental data.
