Tests for Coherence in Neutral B Meson Decays

A density-matrix method for the study of tagged states of neutral B mesons with arbitrary coherence properties is applied to several examples, including $e^+ e^-$ production both at and above the $\Upsilon(4S)$ resonance, and hadronic production. In the absence of coherence the only term modulating the exponential decay of a neutral B meson behaves as $\cos \Delta m t$, while a $\sin \Delta m t$ modulation is a signal of partial or full coherence. Decays to CP eigenstates are needed to fully specify the density matrix. We relate these results to more familiar expressions for the cases of the $\Upsilon(4S)$ and incoherent production.