Time-Reversal Invariance and the Phase of theρ−ωMixing

Abstract

We use the unitarity sum to rederive in a simple way an explicit formula for the $\rho -\omega$ mixing phase in terms of $\left|\epsilon \right|$, ${\Gamma }_{\omega }$, ${\Gamma }_{\rho }$, and $m_{\omega }^{}{}_{}{}^2-m_{\rho }^{}{}_{}{}^2$. We then present a systematic study of the condition imposed by $T$ (or $\mathrm{CPT}$) invariance and unitarity on the relative phase and strength of the mixing. It is shown that the departure from $T$ invariance in the $\rho -\omega$ system is, in principle, directly demonstrable through the measurement of the $\rho -\omega$ mixing phase in $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$. The connection between the phase discrepancy and the breakdown of microscopic reversibility is also discussed. Finally, two possible graphical representations for the $\rho -\omega$ mixing parameters which exhibit directly the condition imposed by unitarity and time-reversal invariance on the relative phases and strengths of the $\rho -\omega$ mixing are given.