The Time Dependence of B 0 → f Decays

We calculate the time dependence of B 0 → f decays (and analogously D 0 → f decays) and use this result to obtain expressions for the time dependence of: an untagged sample (B 0 → f and B 0 → f decays combined), a sample containing charge-conjugate final states (B 0 → f and B 0 → ¯ f decays combined), and a sample containing all four decay modes together. For simplicity we assume CP violating effects are negligible. The time dependence of B 0 → f decays, in which f is not a CP eigenstate, is not purely exponential due to the presence of B 0-B 0 mixing. This mixing arises due to either a mass difference ∆m or a decay-width difference ∆γ between the mass eigenstates of the B 0-B 0 system. This note calculates an expression for the time dependence in the presence of such mixing. It is assumed that CP violation, if present, occurs at a negligible level. This assumption is well-motivated in the D 0 system, for which our results can be applied. In particular, our results can be applied to the decay D 0 → K − π + [1]. There are two strong eigenstates, B 0 and B 0 , which are not eigenstates of the full Hamiltonian due to the weak interaction. Thus, they most generally evolve according to the Schrödinger equation: