Measurement of the Ratios of Branching Fractions B(B_s⁰ -> D_s^-π+π+π−)/B(B⁰ -> D−π+π+π−) and B(B_s⁰ -> D−sπ+)/B(B⁰ -> D−π+)

Using 355 pb−1 of data collected by the CDF II detector in pp¯ collisions at s√=1.96 TeV at the Fermilab Tevatron, we study the fully reconstructed hadronic decays B0(s)→D−(s)π+ and B0(s)→D−(s)π+π+π−. We present the first measurement of the ratio of branching fractions B(B0s→D−sπ+π+π−)/B(B0→D−π+π+π−)=1.05±0.10(stat)±0.22(syst). We also update our measurement of B(B0s→D−sπ+)/B(B0→D−π+) to 1.13±0.08(stat)±0.23(syst), improving the statistical uncertainty by more than a factor of 2. We find B(B0s→D−sπ+)=[3.8±0.3(stat)±1.3(syst)]×10−3 and B(B0s→D−sπ+π+π−)=[8.4±0.8(stat)±3.2(syst)]×10−3