Motivated by the $2.8\sigma$ discrepancy observed between the BABAR measurement and the standard model prediction of the $CP$ asymmetry in $\tau \to K_S\pi {\nu }_{\tau }$ decays, as well as the prospects of future measurements at Belle II, we revisit this observable in this paper. Firstly, we reproduce the known $CP$ asymmetry due to $K^0-{\overline{K}}^0$ mixing by means of the reciprocal basis, which is convenient when a $K_{S(L)}$ is involved in the final state. As the $K\pi$ tensor form factor plays a crucial role in generating a nonzero direct $CP$ asymmetry that can arise only from the interference of vector and tensor operators, we then present a dispersive representation of this form factor, with its phase obtained in the context of chiral theory with resonances, which fulfills the requirements of unitarity and analyticity. Finally, the $\tau \to K_S\pi {\nu }_{\tau }$ decays are analyzed both within a model-independent low-energy effective theory framework and in a scalar leptoquark scenario. It is observed that the $CP$ anomaly can be accommodated in the model-independent framework, even at the $1\sigma$ level, together with the constraint from the branching ratio of ${\tau }^{-}\to K_S{\pi }^{-}{\nu }_{\tau }$ decay; it can be, however, marginally reconciled only at the $2\sigma$ level, due to the specific relation between the scalar and tensor operators in the scalar leptoquark scenario. Once the combined constraints from the branching ratio and the decay spectrum of this decay are taken into account, these possibilities are, however, both excluded, even without exploiting further the stronger bounds from the (semi)leptonic kaon decays under the assumption of lepton-flavor universality, as well as from the neutron electric dipole moment and $D-\overline{D}$ mixing under the assumption of $SU(2)$ invariance of the weak interactions.

• Received 28 October 2019

DOI:https://doi.org/10.1103/PhysRevD.100.113006

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Published by the American Physical Society