Forward backward CP asymmetry in $\tau^- \to K \pi \nu_{\tau}$ in the Left-Right Inverse seesaw model

Recent measurements of the integrated CP asymmetry in $\tau \to K\pi\nu_\tau$ decays by the BaBar collaboration exhibit a $2.8\sigma$ deviation from the Standard Model (SM) prediction. In this work, we investigate CP-violating effects in $\tau \to K\pi\nu_\tau$ within the model of the Left--Right Inverse Seesaw (LRIS) model. We show that, although the integrated asymmetry remains too small to account for the BaBar result, the model nevertheless predicts a pronounced signal in the \emph{differential} forward--backward CP asymmetry, $A_{\rm CP}^{\rm FB}(s)$. We derive the effective $|\Delta S| = 1$ Hamiltonian relevant for these decays and identify a dominant non-decoupling scalar operator, $g_S$, generated by a top-quark flavor-changing neutral current box diagram involving heavy neutrinos and scalar exchange. Our numerical analysis demonstrates that, while this contribution largely cancels in the integrated $A_{\rm CP}$, it significantly enhances $A_{\rm CP}^{\rm FB}(s)$ through interference with the SM vector current, leading to distinctive kinematic features near the $K^*(892)$ and $K_0^*(1430)$ resonances. These angular and differential observables provide a sensitive probe of the LRIS scalar sector at current and future flavor experiments, in particular Belle~II.