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 for the ${{\mathit D}^{+}}$. $\bar C_{T}{}\equiv$ $\vec {{\mathit p}}_{{{\mathit K}^{-}}}\cdot{}$ ($\vec {{\mathit p}}_{{{\mathit \pi}^{-}}}{\times }\vec {{\mathit p}}_{{{\mathit \pi}^{+}}}$) is the corresponding quantity for the ${{\mathit D}^{-}}$. 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 ${{\mathit K}_S^0}$ ${{\mathit K}^{\pm}}$ ${{\mathit \pi}^{+}}$ ${{\mathit \pi}^{-}}$ $\rightarrow$ ${{\mathit D}^{\pm}}$ is not accessible, while the $\mathit P$-conjugate process is.

VALUE ($ 10^{-3} $) EVTS DOCUMENT ID TECN  COMMENT $\bf{ -3 \pm8}$ OUR AVERAGE  Error includes scale factor of 1.1. $3.4$ $\pm8.7$ $\pm3.2$ 19k MOON 2023 BELL 980 fb${}^{-1}$ at $\sim{}{{\mathit \Upsilon}{(4S)}}$ $-12.0$ $\pm10.0$ $\pm4.6$ 21k LEES 2011 E BABR ${{\mathit e}^{+}}{{\mathit e}^{-}}$ $\approx{}{{\mathit \Upsilon}{(4S)}}$ • • We do not use the following data for averages, fits, limits, etc. • • $23$ $\pm62$ $\pm22$ 523 LINK 2005 E FOCS ${{\mathit \gamma}}$ A, ${{\overline{\mathit E}}}_{\gamma }{}\approx{}$180 GeV

Conservation Laws:

TIME REVERSAL ($\mathit T$) INVARIANCE

References