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We present details of a lattice QCD calculation of the $B_s\to D_s^*$ axialform factor at zero recoil using the Highly Improved Staggered Quark (HISQ)formalism on the second generation MILC gluon ensembles that include up, down,strange and charm quarks in the sea. Using the HISQ action for all valencequarks means that the lattice axial vector current that couples to the $W$ canbe renormalized fully non-perturbatively, giving a result free of theperturbative matching errors that previous lattice QCD calculations have had.We calculate correlation functions at three values of the lattice spacing, andmultiple `$b$'-quark masses, for physical $c$ and $s$. The functionaldependence on the $b$-quark mass can be determined and compared to Heavy QuarkEffective Theory expectations, and a result for the form factor obtained at thephysical value of the $b$-quark mass. We find $\mathcal{F}^{B_s\to D_s^*}(1) =h^s_{A_1}(1) = 0.9020(96)_{\text{stat}}(90)_{\text{sys}}$. This is in agreementwith earlier lattice QCD results, which use NRQCD $b$ quarks, with a totaluncertainty reduced by more than a factor of two. We discuss implications ofthis result for the $B\to D^*$ axial form factor at zero recoil and fordeterminations of $V_{cb}$.

Lattice QCD form factor for $B_s\to D_s^* l\nu$ at zero recoil with non-perturbative current renormalisation