Measurement of the ratios of branching fractions B(B_s⁰ → D_s^- π⁺)/B(B⁰ → D^-π⁺) and B(B⁺ → anti-D⁰π⁺)/B(B⁰ → D^-π⁺)

Using 355 pb-1 of data collected by the CDF II detector in p[overline p] collisions at sqrt(s)=1.96 TeV at the Fermilab Tevatron, we study the fully reconstructed hadronic decays B_(s)⁰-->D_(s)^-pi+ and B_(s)⁰-->D_(s)^-pi+pi+pi-. We present the first measurement of the ratio of branching fractions [script B](B_s⁰-->D_s^-pi+pi+pi-)/[script B](B0-->D-pi+pi+pi-)=1.05±0.10(stat)±0.22(syst). We also update our measurement of [script B](B_s⁰-->D_s^-pi+)/[script B](B0-->D-pi+) to 1.13±0.08(stat)±0.23(syst), improving the statistical uncertainty by more than a factor of 2. We find [script B](B_s⁰-->D_s^-pi+)=[3.8±0.3(stat)±1.3(syst)]×10-3 and [script B](B_s⁰-->D_s^-pi+pi+pi-)=[8.4±0.8(stat)±3.2(syst)]×10-3.