The $b$ quark fragmentation fractions at LHCb and meson decays with heavy quark spectators

We study the current estimates of $B_c\to B_s\pi$ to extract the fragmentation fraction $f_c\mathrm{/}{f}_s$ at the LHCb. A rather robust estimate of $Br(B_c\to B_s\pi )$ based on factorization and lattice results for the form factor gives $f_c\mathrm{/}{f}_s\sim 0.056$ with a $16\mathrm{\%}$ error. We also revisit the extraction of $f_s\mathrm{/}{f}_d$ using $B\to D\pi$ instead of the theoretical cleaner but more suppressed channel $B\to DK$. We also find a tension on the predictions of $Br(B_c\to J\mathrm{/}\psi \pi )$ and $Br(B_c\to B_s\pi )$ considering the measurements of these modes at LHCb, and find that, within a $23\mathrm{\%}$ uncertainty, only the lower end of the current prediction range $Br(B_c\to J\mathrm{/}\psi )\sim 0.4\mathrm{\%}-1.7\mathrm{\%}$ would be consistent with the LHCb measurements.