Search for $${{B} _{c} ^+} \!\rightarrow {{\pi } ^+} {\mu ^+\mu ^-} $$ Decays and Measurement of the Branching Fraction Ratio $${\mathcal {B}} ({{B} _{c} ^+} \!\rightarrow {\psi {(2S)}} {{\pi } ^+} )/{\mathcal {B}} ({{B} _{c} ^+} \!\rightarrow {{J \Hspace{-1.66656Pt}/\hspace{-1.111pt}\psi }} {{\pi } ^+} )$$

The first search for nonresonant $${{B} _{c} ^+} \!\rightarrow {{\pi } ^+} {\mu ^+\mu ^-} $$ B c + → π + μ + μ - 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 $$\,\text {fb} ^{-1}$$ 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 $${\mathcal {B}} ({{B} _{c} ^+} \!\rightarrow {{\pi } ^+} {\mu ^+\mu ^-} )/{\mathcal {B}} ({{B} _{c} ^+} \!\rightarrow {{J \hspace{-1.66656pt}/\hspace{-1.111pt}\psi }} {{\pi } ^+} ) < 2.1\times 10^{-4}$$ B ( B c + → π + μ + μ - ) / B ( B c + → J / ψ π + ) < 2.1 × 10 - 4 at $$90\%$$ 90 % confidence level. Additionally, an updated measurement of the ratio of the $${{B} _{c} ^+} \!\rightarrow {\psi {(2S)}} {{\pi } ^+} $$ B c + → ψ ( 2 S ) π + and $${{B} _{c} ^+} \!\rightarrow {{J \hspace{-1.66656pt}/\hspace{-1.111pt}\psi }} {{\pi } ^+} $$ B c + → J / ψ π + branching fractions is reported. The ratio $${\mathcal {B}} ({{B} _{c} ^+} \!\rightarrow {\psi {(2S)}} {{\pi } ^+} )/{\mathcal {B}} ({{B} _{c} ^+} \!\rightarrow {{J \hspace{-1.66656pt}/\hspace{-1.111pt}\psi }} {{\pi } ^+} )$$ B ( B c + → ψ ( 2 S ) π + ) / B ( B c + → J / ψ π + ) is measured to be $$0.254\pm 0.018 \pm 0.003 \pm 0.005$$ 0.254 ± 0.018 ± 0.003 ± 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 $${J \hspace{-1.66656pt}/\hspace{-1.111pt}\psi }$$ J / ψ and $$\psi {(2S)}$$ ψ ( 2 S ) decays. This measurement is the most precise to date and is consistent with previous LHCb results.