$B \rightarrow D^*$ vector, axial-vector and tensor form factors for the full $q^2$ range from lattice QCD

We compute the complete set of SM and tensor $B_{(s)}\to D_{(s)}^*\ell\bar{\nu}$ semileptonic form factors across the full kinematic range of the decay using second generation MILC $n_f=2+1+1$ HISQ gluon field configurations and HISQ valence quarks, with the heavy-HISQ method. Lattice spacings range from $0.09\mathrm{fm}$ to $0.044\mathrm{fm}$ with pion masses from $\approx 300\mathrm{MeV}$ down to the physical value and heavy quark masses ranging between $\approx 1.5 m_c$ and $4.1 m_c \approx 0.9 m_b$; currents are normalised nonperturbatively. Using the recent $B_{(s)}\to D^*_{(s)}\ell\bar{\nu}_\ell$ data from Belle and LHCb together with our form factors we determine a model independent value of $V_{cb}=39.03(56)_\mathrm{exp}(67)_\mathrm{latt}\times 10^{-3}$, in agreement with previous exclusive determinations and in tension with the inclusive result at the level of $3.6\sigma$. We observe a $\approx 1\sigma$ tension between the shape of the differential decay rates computed using our form factors and those measured by Belle. We compute a lattice-only SM value for the ratio of semitauonic and semimuonic decay rates, $R(D^*)=0.273(15)$, which we find to be closer to the recent Belle measurement and HFLAV average than theory predictions using fits to experimental differential rate data for $B\to D^*\ell\bar{\nu}_\ell$. Determining $V_{cb}$ using the total rate for $B\to D^*\ell\nu$ gives a value in agreement with inclusive results. We compute the longitudinal polarisation fraction for the semitauonic mode, $F_L^{D^*}=0.395(24)$, which is in tension at the level of $2.2\sigma$ with the recent Belle measurement. Our calculation combines $B\to D^*$ and $B_s\to D_s^*$ lattice results, and we provide an update which supersedes our previous lattice computation of the $B_s\to D_s^*$ form factors. We also give the chiral perturbation theory needed to analyse the tensor form factors.