$\bar B^0 \to π^+ X$ in the Standard Model

Abstract

In this paper we investigate the possibility of studying $B\to π$ form factor using the semi-inclusive decays $\bar B^0 \to π^+ + X_q$. In general $B\to PX$ semi-inclusive decays involve several hadronic parameters. But for $\bar B^0 \to π^+ X_q$ decays we find that in the factorization approximation, the only unknown hadronic parameters are the form factors $F^{B\to π}_{0,1}$. Therefore these form factors can be studied in $\bar B^0 \to π^+ X_q$ decays. Using theoretical model calculations for the form factors the branching ratios for $\bar B^0 \to π^+ X_d(ΔS = 0) $ and $\bar B^0 \to π^+ X_s (ΔS = -1)$, with the cut $E_π > 2.1$ GeV, are estimated to be in the ranges of $(3.1\sim 4.9) \times 10^{-5}(F^{B\to π}_1(0)/0.33)^2$ and $(2.5\sim 4.2)\times 10^{-5}(F_1^{B\to π}(0)/0.33)^2$, respectively, depending on the value of $γ$. The combined branching ratio for $\bar B^0 \to π^+ (X_d+ X_s)$ is about $7.4\times 10^{-5} (F^{B\to π}_1(0)/0.33)^2$ and is insensitive to $γ$. We also discuss CP asymmetries in these decay modes.