Extracting $\gamma$ from ${\mathrm{B_{s(d)}}}\to {\mathrm J}/\psi {\mathrm{K_S}}$ and ${\mathrm{B_{d(s)}}}\to {\mathrm{D^{+}_{d(s)}}} {\mathrm{D^{-}_{d(s)}}}$

R. Fleischer

doi:doi:10.1007/s100529900099

2021 Impact factor 4.991

Particles and Fields

Eur. Phys. J. C 10, 299-306
DOI 10.1007/s100529900099

Extracting $\gamma$ from ${\mathrm{B_{s(d)}}}\to {\mathrm J}/\psi {\mathrm{K_S}}$ and ${\mathrm{B_{d(s)}}}\to {\mathrm{D^{+}_{d(s)}}} {\mathrm{D^{-}_{d(s)}}}$

R. Fleischer

Theory Division, CERN, CH-1211 Geneva 23, Switzerland

Received: 24 March 1999 / Published online: 15 July 1999

Abstract
A completely general parametrization of the time-dependent decay rates of the modes ${\mathrm{B_s}}\to {\mathrm J}/\psi {\mathrm{K_S}}$ and ${\mathrm{B_d}}\to {\mathrm J}/\psi {\mathrm{K_S}}$ is given, which are related to each other through the U-spin flavour symmetry of strong interactions. Owing to the interference of current-current and penguin processes, the ${\mathrm{B_s}}\to {\mathrm J}/\psi {\mathrm{K_S}}$ observables probe the angle $\gamma$ of the unitarity triangle. Using the U-spin symmetry, the overall normalization of the ${\mathrm{B_s}}\to {\mathrm J}/\psi {\mathrm{K_S}}$ rate can be fixed with the help of the CP-averaged ${\mathrm{B_d}}\to {\mathrm J}/\psi {\mathrm{K_S}}$ rate, providing a new strategy to determine $\gamma$ . This extraction of $\gamma$ is not affected by any final-state-interaction effects, and its theoretical accuracy is only limited by U-spin-breaking corrections. As a by-product, this strategy allows us to also take into account the penguin effects in the determination of $\beta$ from ${\mathrm{B_d}}\to {\mathrm J}/\psi {\mathrm{K_S}}$ ,which are presumably very small, and to predict the direct CP asymmetry arising in this mode. An analogous strategy is provided by the time-dependent ${\mathrm{B_d}}\to {\mathrm D}^+{\mathrm D}^-$ rate, if its overall normalization is fixed through the CP-averaged ${\mathrm{B_s}}\to {\mathrm D}^+_{\mathrm s}{\mathrm D}^-_{\mathrm s}$ rate.