CP Violation in $\tau \to 3\pi {\nu }_{\tau }$

We consider CP violating effects in the decays $\tau\rightarrow (3\pi)\nu_\tau$whereboththe${\rm J}^{\rm P}=1^+$resonance,$a_1$,and${\rm J}^{\rm P}=0^-$resonance,$\pi^\prime$, can contribute. The interference between the $a_1$ and ${\pi }^{\mathrm{\prime }}$ resonances can lead to enhanced CP-violating asymmetries whose magnitudes depend crucially on the ${\pi }^{\mathrm{\prime }}$ decay constant, $f_{{\pi }^{\mathrm{\prime }}}$. We make an estimate of $f_{{\pi }^{\mathrm{\prime }}}$ with a simplified chiral Lagrangian coupled to a massive pseudoscalar field, and we compare the estimates from the non-relativistic quark model and from the QCD sum rule with the estimate from the `mock’ meson model. We then estimate quantitatively the size of CP-violating effects in a multi-Higgs-doublet model and scalar-leptoquark models. We find that, while CP-violating effects in the scalar-leptoquark models may require more than ${10}^{10}$ $\tau$ leptons, CP-violating effects from the multi-Higgs-doublet model can be seen at the $2\sigma$ level with about ${10}^7$ $\tau$ leptons using the chiral Lagrangian estimate of $f_{{\pi }^{\mathrm{\prime }}}=(1\sim 5)\times {10}^{-3}$ GeV.