Measurement of the ratio of the B${}_{\mathrm{c}}^{+}$ $\to$ J/$\psi$${\tau }^{+}{\nu }_{\tau }$ and B${}_{\mathrm{c}}^{+}$ $\to$ J/$\psi$ ${\mu }^{+}{\nu }_{\mu }$ branching fractions using three-prong $\tau$ lepton decays

The ratio between the B${}_{\mathrm{c}}^{+}$ $\to$ J/$\psi$${\tau }^{+}{\nu }_{\tau }$ and B${}_{\mathrm{c}}^{+}$ $\to$ J/$\psi$${\mu }^{+}{\nu }_{\mu }$ branching fractions is measured using a data sample of proton-proton collisions collected by CMS at a center-of-mass energy of 13 TeV in the years 2016$-$2018 and corresponding to an integrated luminosity of 138 fb${}^{-1}$. The J$\mathrm{/}\psi$ meson is identified through its J$\mathrm{/}\psi$ $\to$ ${\mu }^{+}{\mu }^{-}$ decay and the tau lepton is reconstructed in the hadronic three-prong final state. The measured ratio of branching fractions in this tau decay mode, ${\mathcal{R}}_{\mathrm{J}\mathrm{/}\psi }^{\text{had}}$ = 1.04${}_{-0.44}^{+0.50}$, is combined with the previous analysis based on the ${\tau }^{+}$ $\to$ ${\mu }^{+}{\nu }_{\mu }{\bar{\nu }}_{\tau }$ leptonic decay channel, leading to ${\mathcal{R}}_{\mathrm{J}\mathrm{/}\psi }$ = 0.49 $\pm$ 0.26. As this result is consistent with the standard model prediction of 0.258 $\pm$ 0.004, no evidence of lepton flavor universality violation is found.