Measurement of the relative yields of the decay modes $B^0\to D^- \pi^+$, $B^0\to D^- K^+$, $B^0_s \to D^-_s \pi^+$, and determination of $f_s/f_d$ for 7~TeV $pp$ collisions

The relative abundances of the three decay modes $B^0\to D^- K^+$, $B^0\to D^- \pi^+$, and $B^0_s \to D^-_s \pi^+$ produced in 7~TeV $pp$ collisions at the LHC are determined from data taken during the 2010 running period, corresponding to an integrated luminosity of $35$~pb$^{-1}$. From these, the relative branching ratio of $B^0\to D^- K^+$ with respect to $B^0\to D^- \pi^+$ is determined, resulting in $B\left(B^0\to D^- K^+\right) = (2.02 \pm 0.17^{\textrm{stat}} \pm 0.12^{\textrm{syst}})\times 10^{-4}$. The ratio of fragmentation fractions $f_s/f_d$~is determined through the relative abundances of $B^0\to D^- K^+$ and $B^0\to D^- \pi^+$ with respect to $B^0_s \to D^-_s \pi^+$, leading to $f_s/f_d = 0.245 \pm 0.017^{\textrm{stat}} \pm 0.018^{\textrm{syst}} \pm 0.018^{\textrm{theor}}$.