Precise Discussion of Time-Reversal Asymmetries in B-meson decays

BaBar collaboration announced that they observed time reversal (T) asymmetry through $B$ meson system. In the experiment, time dependencies of two distinctive processes, $B_- \rightarrow \bar{B^0}$ and $\bar{B^0}\rightarrow B_-$($-$ expresses CP value) are compared with each other. In our study, we examine event number difference of these two processes. In contrast to the BaBar asymmetry, the asymmetry of events number includes the overall normalization difference for rates. Time dependence of the asymmetry is more general and it includes terms absent in one used by BaBar collaboration. Both of the BaBar asymmetry and ours are naively thought to be T-odd since two processes compared are related with flipping time direction. We investigate the time reversal transformation property of our asymmetry. Using our notation, one can see that the asymmetry is not precisely a T-odd quantity, taking into account indirect CP and CPT violation of K meson systems. The effect of $\epsilon_K$ is extracted and gives rise to $\mathcal{O}(10^{-3})$ contribution. The introduced parameters are invariant under rephasing of quarks so that the coefficients of our asymmetry are expressed as phase convention independent quantities. Some combinations of the asymmetry enable us to extract parameters for wrong sign decays of $B_d$ meson, CPT violation, etc. We also study the reason why the T-even terms are allowed to contribute to the asymmetry, and find that several conditions are needed for the asymmetry to be a T-odd quantity.