Probing New Physics Through CP Violation in $B_{(s)}\to V\mu^+\mu^-$ Decays

Rare decays of the kind $B\to K^*\mu^+\mu^-$ and $B_s\to \phi\mu^+\mu^-$ are key players for testing the Standard Model. The current experimental data for their decay rates and angular observables show tensions with the theoretical predictions that may be indications of New Physics. We present a strategy to extract the relevant short-distance coefficients in the presence of new sources of CP violation, utilizing a synergy with $B\to K\mu^+\mu^-$ decays. Using the current data as a guideline, we illustrate the new method to determine the complex coefficients $C_9^{(\prime)}$ and $C_{10}^{(\prime)}$ using only four angular observables. Interestingly, the current experimental picture leaves significant room for CP-violating New Physics. We discuss also the link to leptonic $B^0_s\to\mu^+\mu^-$ decays. We are looking forward to the implementation of these strategies at the future high-precision frontier of flavour physics.