CP Violation in $τ\rightarrow 3πν_τ$

Abstract

We consider CP violating effects in the decays $τ\rightarrow (3π)ν_τ$ where both the ${\rm J}^{\rm P}=1^+$ resonance, $a_1$, and ${\rm J}^{\rm P}=0^-$ resonance, $π^\prime$, can contribute. The interference between the $a_1$ and $π^\prime$ resonances can lead to enhanced CP-violating asymmetries whose magnitudes depend crucially on the $π^\prime$ decay constant, $f_{π^\prime}$. We make an estimate of $f_{π^\prime}$ 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 $10^{10}$ $τ$ leptons, CP-violating effects from the multi-Higgs-doublet model can be seen at the $2σ$ level with about $10^7$ $τ$ leptons using the chiral Lagrangian estimate of $f_{π^\prime}=(1\sim 5)\times 10^{-3}$ GeV.