Indirect $CP$ violation and implications for $D^0-{\overline{D}}^0$ and $B_s-{\overline{B}}_s$ mixing

The two kinds of indirect $CP$ violation in neutral meson systems are related, in the absence of new weak phases in decay. The result is a model-independent expression relating $CP$ violation in mixing, $CP$ violation in the interference of decays with and without mixing, and the meson mass and width differences. It relates the semileptonic and time-dependent $CP$ asymmetries, and pairs of time-dependent $CP$ asymmetries in $D^0$ decays to non-$CP$ eigenstates. $CP$ violation in the interference of decays with and without mixing is related to the mixing parameters of relevance to model building: the off-diagonal mixing matrix elements $|M_{12}|$, $|{\Gamma }_{12}|$, and ${\varphi }_{12}\equiv \arg (M_{12}/{\Gamma }_{12})$. Incorporating this relation into a fit to the $D^0-{\overline{D}}^0$ mixing data implies a level of sensitivity to $|{\varphi }_{12}^D|$ of 0.10 (rad) at $1\sigma$. The formalism is extended to include new weak phases in decay, and in ${\Gamma }_{12}$. The phases are highly constrained by direct $CP$ violation measurements. Consequently, the bounds on $|{\varphi }_{12}^D|$ are not significantly altered, and the effects of new weak phases in decay could be difficult to observe at a high luminosity flavor factory ($D^0$) or at the LHC ($B_s$) via violations of the above relations, unlike in direct $CP$ violation.