We reinvestigate a simple relation between the semileptonic $CP$ asymmetry $a_{sl}^s$, the decay rate difference $\Delta {\Gamma }_s$, the mass difference $\Delta M_s$ and $S_{\psi \varphi }$ extracted from the angular analysis of the decay $B_s\to \psi \varphi$, which is regularly used in the literature. We find that this relation is not suited to eliminate the theory prediction for ${\Gamma }_{12}$, it can, however be used to determine the size of the penguin contributions to the decay $B_s\to \psi \varphi$. Moreover we comment on the current precision of the theory prediction for ${\Gamma }_{12}$.

• Received 22 June 2011

DOI:https://doi.org/10.1103/PhysRevD.84.031501