Search for $B_c^{+}\to {\pi }^{+}{\mu }^{+}{\mu }^{-}$ decays and measurement of the branching fraction ratio $\mathcal{B}(B_c^{+}\to \psi (2S){\pi }^{+})\mathrm{/}\mathcal{B}(B_c^{+}\to J\mathrm{/}\psi {\pi }^{+})$

The first search for nonresonant $B_c^{+}\to {\pi }^{+}{\mu }^{+}{\mu }^{-}$ decays is reported. The analysis uses proton-proton collision data collected with the LHCb detector between 2011 and 2018, corresponding to an integrated luminosity of 9 fb${}^{-1}$. No evidence for an excess of signal events over background is observed and an upper limit is set on the branching fraction ratio $\mathcal{B}(B_c^{+}\to {\pi }^{+}{\mu }^{+}{\mu }^{-})\mathrm{/}\mathcal{B}(B_c^{+}\to J\mathrm{/}\psi {\pi }^{+})<2.1\times {10}^{-4}$ at $90\mathrm{\%}$ confidence level. Additionally, an updated measurement of the ratio of the $B_c^{+}\to \psi (2S){\pi }^{+}$ and $B_c^{+}\to J\mathrm{/}\psi {\pi }^{+}$ branching fractions is reported. The ratio $\mathcal{B}(B_c^{+}\to \psi (2S){\pi }^{+})\mathrm{/}\mathcal{B}(B_c^{+}\to J\mathrm{/}\psi {\pi }^{+})$ is measured to be $0.254\pm 0.018\pm 0.003\pm 0.005$, where the first uncertainty is statistical, the second systematic, and the third is due to the uncertainties on the branching fractions of the leptonic $J\mathrm{/}\psi$ and $\psi (2S)$ decays. This measurement is the most precise to date and is consistent with previous LHCb results.