$CP$ violation in $b \to s \ell \ell$: a model independent analysis

We perform a model-independent global fit to all germane and updated $b \to s \ell \ell$ ($\ell=e,\,\mu$) data assuming new physics couplings to be complex. Under the approximation that new physics universally affects muon and electron sectors and that either one or two related operators contribute at a time, we identify scenarios which provide a good fit to the data. It turns out that the favored scenarios remain the same as obtained for the real fit. Further, the magnitude of complex couplings can be as large as that of their real counterparts and these are reflected in the predictions of the direct $CP$ asymmetry, $A_{\rm CP}$, in $B \to (K,\, K^*) \mu^+ \mu^-$ along with a number of angular $CP$ asymmetries, $A_i$, in $B^0 \to K^{*0} \mu^+ \mu^-$ decay. The sensitivities of these observables to various solutions are different in the low and high-$q^2$ bins. We also determine observables which can serve as unique identifier for a particular new physics solution. Moreover, we examine correlations between $A_{\rm CP}$ and several $A_i$ observables. A precise measurement of $A_{\rm CP}$ and $A_i$ observables can not only confirm the existence of additional weak phases but can also enable unique determination of Lorentz structure of possible new physics in $b \to s \mu^+ \mu^-$ transition.