One distinguishes between “true” $CP$-violating triple-product (TP) asymmetries which require no strong phases and “fake” asymmetries which are due to strong phases but require no $CP$ violation. So far a single true TP asymmetry has been measured in $K_L\to {\pi }^{+}{\pi }^{-}{e}^{+}{e}^{-}$. A general discussion is presented for $T$-odd TP asymmetries in four-body decays. It is shown that TP asymmetries vanish for two identical and kinematically indistinguishable particles in the final state. Two examples are $D^0\to K^{-}{\pi }^{+}{\pi }^{-}{\pi }^{+}$ and $D^{+}\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^0$. A nonzero TP asymmetry can be expected when nontrivial kinematic correlations exist, as in the decay $K_L\to e^{+}{e}^{-}{e}^{+}{e}^{-}$. Triple-product asymmetries measured in charmed particle decays indicate an interesting pattern of final-state interactions. We reiterate a discussion of TP asymmetries in $B$ meson decays to two vector mesons each decaying to a pseudoscalar pair, extending results to decays where one vector meson decays into a lepton pair. We derive expressions for time-dependent TP asymmetries for neutral $B$ decays to flavorless states in terms of the neutral $B$ mass difference $\Delta m$ and the width-difference $\Delta \Gamma$. Time-integrated true $CP$-violating asymmetries, measurable for untagged $B_s$ decays, are shown to be suppressed by neither ${\Gamma }_s/\Delta m_s$ nor $\Delta {\Gamma }_s/{\Gamma }_s$ if transversity amplitudes for $CP$-even and $CP$-odd states involve different weak phases. In contrast, fake asymmetries require flavor tagging and are suppressed by the former ratio when time integrated. We apply our results to $B\to K^{*}\varphi$ and $B_s\to \varphi \varphi$ data and suggest an application for $B_s\to J/\psi \varphi$.

• Received 11 July 2011

DOI:https://doi.org/10.1103/PhysRevD.84.096013