Genuine lepton-flavor-universality-violating observables in the τ-μ sector of B to (K,\,K^*) ell ell decays

It was previously shown that unlike the ratios $R_K^{\mu e} \equiv R_K \equiv \Gamma(B \to K \mu^+ \mu^-)/\Gamma(B \to K e^+ e^-)$ and $R_{K^*}^{\mu e} \equiv R_{K^*} \equiv \Gamma(B \to K^* \mu^+ \mu^-)/\Gamma(B \to K^* e^+ e^-)$, the ratios $R_K^{\tau \mu}$ and $R_{K^*}^{\tau \mu}$ can deviate from their Standard Model (SM) predictions even with universal new physics couplings. This observation highlights the critical need to identify and establish genuine lepton flavor universality violating (LFUV) observables in the $\tau-\mu$ sector. This work embarks on establishing genuine LFUV ratio observables in $B \to K \ell \ell$ and $B \to K^* \ell \ell$ decays through comprehensive analysis of their angular distributions. We find that like $R_{K^*}^{\tau \mu}$, the ratios $R_{A_{FB}}^{\tau \mu}$ and $R_{f_L}^{\tau \mu}$ do not qualify as genuine LFUV observables, whereas the ratios of all optimized observables in $B \to K^* \ell \ell$ decays within the $\tau-\mu$ sector definitively do. In the case of $B \to K \ell \ell$ decays, similar to $R_K^{\tau \mu}$, the ratio $R_{F_H}$ is influenced by mass effects and therefore cannot be considered a genuine LFUV observable in the $\tau-\mu$ sector. However, the ratio $\Gamma_\tau(1-F_{H}^{\tau})/\Gamma_\mu(1-F_{H}^{\mu})$ stands as the sole genuine LFUV observable in $B \to K \ell \ell$ decays. Furthermore, by making use of new physics Lorentz structures which provide a better fit to the current $b \to s \ell \ell$ data as compared to the SM, we demonstrate how the non-genuine LFUV ratios $R_{A_{FB}}^{\tau \mu}$ and $R_{f_L}^{\tau \mu}$ can be employed to distinguish between framework with solely universal lepton couplings and those with both universal and non-universal couplings.