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We present the first lattice QCD determination of the $B_c \rightarrowJ/\psi$ vector and axial-vector form factors. These will enable experimentalinformation on the rate for $B_c$ semileptonic decays to $J/\psi$ to beconverted into a value for $V_{cb}$. Our calculation covers the full physical$q^2$ range of the decay and uses non-perturbatively renormalised latticecurrents. We use the Highly Improved Staggered Quark (HISQ) action for allvalence quarks on the second generation MILC ensembles of gluon fieldconfigurations including $u$, $d$, $s$ and $c$ HISQ sea quarks. Our HISQ heavyquarks have masses ranging upwards from that of $c$; we are able to reach thatof the $b$ on our finest lattices. This enables us to map out the dependence onheavy quark mass and determine results in the continuum limit at the $b$. Weuse our form factors to construct the differential rates for $B_c^- \rightarrowJ/\psi \mu^- \bar{\nu}_\mu$ 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 of0.0150(11)(10)(3) ~with uncertainties from lattice QCD,$\eta_\mathrm{EW}V_{cb}$ and $\tau_{B_c}$ respectively.

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