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 $|{ϵ}_K|$, $\Delta m_{B_{d,s}}$ and $|V_{ub}/V_{cb}|$. We also include the recent information on $\sin 2\beta$ 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 $|{ϵ}_K|$, $\Delta m_{B_{d,s}}$, $|V_{ub}/V_{cb}|$, $\sin 2\beta$ 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.

Comments: Revised verion with the new experimental data from Belle and Babar included in the analysis to obtain the global fit for the CP violating phase gamma. RevTex, 32 pages and 8 figures