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$CP$ asymmetry in the angular distributions of $τ\to K_Sπν_τ$ decays -- II: general effective field theory analysis

We proceed to study the $CP$ asymmetry in the angular distributions of $τ\to K_Sπν_τ$ decays within a general effective field theory framework including four-fermion operators up to dimension-six. It is found that, besides the commonly considered scalar-vector interference, the tensor-scalar interference can also produce a non-zero $CP$ asymmetry in the angular distributions. Bounds on the effective couplings of the non-standard scalar and tensor interactions are obtained under the combined constraints from the measured $CP$ asymmetries and the branching ratio of $τ^-\to K_Sπ^-ν_τ$ decay, with $\mathrm{Im}[\hatε_S]=-0.008\pm0.027$ and $\mathrm{Im}[\hatε_T]=0.03\pm0.12$, at the scale $μ_τ=2~\mathrm{GeV}$ in the $\mathrm{\overline{MS}}$ scheme. Using the best-fit values, we also find that the distributions of the $CP$ asymmetries can deviate significantly from the SM expectation in almost the whole $Kπ$ invariant-mass region. Nevertheless, the current bounds are still plagued by large experimental uncertainties, but will be improved with more precise measurements from Belle II as well as the proposed Tera-Z and STCF facilities. Assuming further that the non-standard scalar and tensor interactions originate from a weakly-coupled heavy new physics well above the electroweak scale, the $SU(2)_L$ invariance of the resulting SMEFT Lagrangian would indicate that very strong limits on $\mathrm{Im}[\hatε_S]$ and $\mathrm{Im}[\hatε_T]$ could also be obtained from the neutron electric dipole moment and the $D^0-\bar{D}^0$ mixing. With the bounds from these processes taken into account, it is then found that, unless there exist extraordinary cancellations between the new physics contributions, neither the scalar nor the tensor interaction can produce any significant effects on the $CP$ asymmetries in the processes considered.