Model-Independent Determination of $B_c^+ \to η_c\, \ell^+\, ν$ Form Factors

Abstract

We derive model-independent bounds on the form factors for the decay $B_c^+ \to η_c\, \ell^+\, ν$ including full mass effects, i.e. $\ell = e,\, μ, \text{ and } τ$. The bounds are obtained by using the BGL parameterization for the form factors, and fitting to the preliminary lattice data of the HPQCD Collaboration. Our main result after bounding the form factors is the Standard Model (SM) prediction for the ratio of branching fractions $R(η_c) = \mathcal{B}(B_c^+ \to η_c\, τ^+\, ν_τ) / \mathcal{B}(B_c^+ \to η_c\, μ^+\, ν_μ)$. We find $R(η_c)|_{\text{SM}} = 0.31^{+0.04}_{-0.02}$, and argue that a measurement of $R(η_c)$ is within the reach of LHCb during the high luminosity run of the LHC. In addition, using the heavy-quark spin symmetry of the $B_c$ meson we relate our results for $B_c^+ \to η_c\, \ell^+\, ν$ to those for $B_c^+ \to J/ψ\, \ell^+\, ν$ yielding the estimate $R(J/ψ)|_{\text{SM}} = 0.26 \pm 0.02$ in good agreement with other determinations.