#### $\mathit CP$ VIOLATING ASYMMETRIES OF $\mathit P$-ODD ($\mathit T$-ODD) MOMENTS

The $\mathit CP$-sensitive $\mathit P$-odd ($\mathit T$-odd) correlation in ${{\mathit D}^{0}}$ , ${{\overline{\mathit D}}^{0}}$ decays. The ${{\mathit D}^{0}}$ and ${{\overline{\mathit D}}^{0}}$ are distinguished by the charge of the parent ${{\mathit D}^{*}}$ : ${{\mathit D}^{*+}}$ $\rightarrow$ ${{\mathit D}^{0}}{{\mathit \pi}^{+}}$ and ${{\mathit D}^{*-}}$ $\rightarrow$ ${{\overline{\mathit D}}^{0}}{{\mathit \pi}^{-}}$ .

#### $\mathit A_{\mathit Tviol}$( ${{\mathit K}^{+}}{{\mathit K}^{-}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ ) in ${{\mathit D}^{0}}$ , ${{\overline{\mathit D}}^{0}}$ $\rightarrow$ ${{\mathit K}^{+}}{{\mathit K}^{-}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$

C$_{T}{}\equiv$ $\vec {{\mathit p}}_{{{\mathit K}^{+}} }\cdot{}$ ($\vec {{\mathit p}}_{{{\mathit \pi}^{+}} }{\times }\vec {{\mathit p}}_{{{\mathit \pi}^{-}} }$) is a parity-odd correlation of the ${{\mathit K}^{+}}$ , ${{\mathit \pi}^{+}}$ , and ${{\mathit \pi}^{-}}$ momenta (evaluated in the ${{\mathit D}^{0}}$ rest frame) for the ${{\mathit D}^{0}}$ . $\bar C_{T}{}\equiv$ $\vec {{\mathit p}}_{{{\mathit K}^{-}} }\cdot{}$ ($\vec {{\mathit p}}_{{{\mathit \pi}^{-}} }{\times }\vec {{\mathit p}}_{{{\mathit \pi}^{+}} }$) is the corresponding quantity for the ${{\overline{\mathit D}}^{0}}$ . Then A$_{T}{}\equiv$ [$\Gamma (C_{T}>$ 0)$−$ $\Gamma (C_{T}<$ 0)] $/$ [$\Gamma (C_{T}>$ 0)$+$ $\Gamma (C_{T}<$ 0)], and $\bar A_{T}{}\equiv$ [$\Gamma (−\bar C_{T}>$ 0)$−$ $\Gamma (−\bar C_{T}<$ 0)] $/$ [$\Gamma (−\bar C_{T}>$ 0)$+$ $\Gamma (−\bar C_{T}<$ 0)], and A$_{Tviol}{}\equiv$ ${1\over 2}(A_{T}$ $−$ $\bar A_{T}$). C$_{T}$ and $\bar C_{T}$ are commonly referred to as $\mathit T$-odd moments, because they are odd under $\mathit T$ reversal. However, the $\mathit T$-conjugate process ${{\mathit K}^{+}}$ ${{\mathit K}^{-}}$ ${{\mathit \pi}^{+}}$ ${{\mathit \pi}^{-}}$ $\rightarrow$ ${{\mathit D}^{0}}$ is not accessible, while the $\mathit P$-conjugate process is.
VALUE ($10^{-3}$) EVTS DOCUMENT ID TECN  COMMENT
 $\bf{ 2.9 \pm2.2}$ OUR AVERAGE
$5.2$ $\pm3.7$ $\pm0.7$ 110k 1
 2019
BELL ${{\mathit e}^{+}}{{\mathit e}^{-}}$ at ${{\mathit \Upsilon}{(1S)}}$ $−$ ${{\mathit \Upsilon}{(6S)}}$
$1.8$ $\pm2.9$ $\pm0.4$ 171k
 2014 BC
LHCB ${{\mathit B}}$ $\rightarrow$ ${{\mathit D}^{0}}{{\mathit \mu}^{-}}{{\mathit X}}$
$1.0$ $\pm5.1$ $\pm4.4$ 47k
 2010
BABR ${{\mathit e}^{+}}{{\mathit e}^{-}}$ $\approx{}$ 10.6 GeV
• • We do not use the following data for averages, fits, limits, etc. • •
$10$ $\pm57$ $\pm37$ 0.8k
 2005 E
FOCS ${{\mathit \gamma}}$ A, ${{\overline{\mathit E}}}_{\gamma }{}\approx{}$180 GeV
 1 KIM 2019 also study $\mathit CP$-violating asymmetries in several other kinematic variables. No evidence for $\mathit CP$ violation is found in any of them.
Conservation Laws:
TIME REVERSAL ($\mathit T$) INVARIANCE
References:
 KIM 2019
PR D99 011104 Search for $CP$ violation with kinematic asymmetries in the $D^0 \to K^+ K^- \pi^+ \pi^-$ decay
 AAIJ 2014BC
JHEP 1410 005 Search for $\mathit CP$ Violation using T-odd Correlations in ${{\mathit D}^{0}}$ $\rightarrow$ ${{\mathit K}^{+}}{{\mathit K}^{-}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ Decays
 DEL-AMO-SANCHEZ 2010
PR D81 111103 Search for $\mathit CP$ Violation using $\mathit T$-odd Correlations in ${{\mathit D}^{0}}$ $\rightarrow$ ${{\mathit K}^{+}}{{\mathit K}^{-}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ Decays
PL B622 239 Search for $\mathit T$ Violation in Charm Meson Decays