$B_c \rightarrow J/\psi$ Form Factors for the full $q^2$ range from Lattice QCD

We present the first lattice QCD determination of the $B_c \rightarrow J/\psi$ vector and axial-vector form factors. These will enable experimental information on the rate for $B_c$ semileptonic decays to $J/\psi$ to be converted into a value for $V_{cb}$. Our calculation covers the full physical $q^2$ range of the decay and uses non-perturbatively renormalised lattice currents. We use the Highly Improved Staggered Quark (HISQ) action for all valence quarks on the second generation MILC ensembles of gluon field configurations including $u$, $d$, $s$ and $c$ HISQ sea quarks. Our HISQ heavy quarks have masses ranging upwards from that of $c$; we are able to reach that of the $b$ on our finest lattices. This enables us to map out the dependence on heavy quark mass and determine results in the continuum limit at the $b$. We use our form factors to construct the differential rates for $B_c^- \rightarrow J/\psi \mu^- \bar{\nu}_\mu$ and obtain a total rate with $7\%$ uncertainty: $\Gamma(B_c^-\rightarrow J/\psi \mu^-\bar{\nu}_{\mu})/|\eta_{\mathrm{EW}}V_{cb}|^2 = 1.73(12)\times 10^{13} ~\mathrm{s}^{-1}$. Including values for $V_{cb}$, $\eta_{\mathrm{EW}}$ and $\tau_{B_c}$ yields a branching fraction for this decay mode of 0.0150(11)(10)(3) ~with uncertainties from lattice QCD, $\eta_\mathrm{EW}V_{cb}$ and $\tau_{B_c}$ respectively.