Fingerprinting CP-violating New Physics with $B \to K\mu^+\mu^-$

The $B \to K \ell^+ \ell^-$ system is particularly interesting to explore physics beyond the Standard Model. Such effects could provide new sources of CP violation, thereby giving rise to Wilson coefficients which could not only be real, as assumed in most analyses, but also be complex. We propose a new strategy to extract the complex $C_{9 \mu}$, $C_{10 \mu}$ from the data, utilising the complementarity of direct CP violation in $B^+\to K^+ \mu^+\mu^-$ and mixing-induced CP violation arising - in addition to direct CP violation - in $B^0_d \to K_{S} \mu^+\mu^-$, as well as the differential rate in appropriate $q^2$ bins. The long-distance effects, which play a key role to generate the direct CP asymmetries, cannot be reliably calculated due to their non-perturbative nature. In order to describe them, we apply a model by the LHCb Collaboration employing experimental data. Assuming specific scenarios, we demonstrate the fingerprinting of CP-violating New Physics and the distinction between ambiguities in the model of the long-distance contributions. Finally, New Physics could couple differently to muons and electrons, as probed through the lepton flavour universality ratio $R_K$. We discuss these effects in the presence of CP violation and present a new way to measure the direct CP asymmetries of the $B \to K e^+ e^-$ channels.