The CP violating phase $\gamma$ from global fit of rare charmless hadronic B decays

Abstract

We study constraints on the CP violating phase $\gamma$ in the Kobayashi-Maskawa model using available experimental data. We first follow the conventional method to up date the constraint on $\gamma$ by performing a $\chi^2$ analysis using data from $|\epsilon_K|$, $\Delta m_{B_{d,s}}$ and $|V_{ub}/V_{cb}|$. We also include the recent information on $\sin2\beta$ in the analysis. We obtain the best fit for $\gamma$ to be $66^\circ$ and the 95% C.L. allowed range to be $42^\circ \sim 87^\circ$. We then develop a method to carry out a $\chi^2$ analysis based on SU(3) symmetry using data from $B\to \pi \pi$ and $B\to K \pi$. We also discuss SU(3) breaking effects from model estimate. We find that present data on $B\to \pi\pi, K \pi$ can also give some constraint on $\gamma$ although weaker than the earlier method limited by the present experimental errors. Future improved data will provide more stringent constraint. Finally we perform a combined fit using data from $|\epsilon_K|$, $\Delta m_{B_{d,s}}$, $|V_{ub}/V_{cb}|$, $\sin2\beta$ and rare charmless hadronic B decays. The combined analysis gives $\gamma=67^\circ$ for the best fit value and $43^\circ \sim 87^\circ$ as the 95% C.L. allowed range. Several comments on other methods to determine $\gamma$ based on SU(3) symmetry are also provided.