Investigating the potential of $R_{K^{(*)}}^{\tau \mu}$ to probe lepton flavor universality violation

In this work we study the potential of the lepton flavor ratios $R_{K}^{\tau \mu} \equiv \Gamma(B \to K \tau^+ \tau^-)/ \Gamma(B \to K \mu^+ \mu^-)$ and $ R_{K^{*}}^{\tau \mu} \equiv \Gamma(B \to K^* \tau^+ \tau^-)/ \Gamma(B \to K^* \mu^+ \mu^-)$ to probe lepton flavor universality (LFU) violation in $\tau-\mu$ sector. We show that these ratios can deviate from their SM values even if the new physics couplings are universal in nature, i.e., having equal couplings to $e$, $\mu$ and $\tau$ leptons. Therefore in order to utilize these observables to probe LFU violation, we need to compare the allowed range of $R_{K^{(*)}}^{\tau \mu}$ for class of solutions with only universal couplings to leptons and solutions having both universal and non-universal components. For the current $b \to s \ell \ell$ ($\ell=e,\,\mu$) data, we find that these two class of solutions can be discriminated provided the measured value of $ R_{K^{*}}^{\tau \mu} $ is greater than the SM prediction.