Overview of $B \to K^{(*)}\ell \ell$ Theoretical Calculations and Uncertainties

Abstract: The search for New Physics (NP) beyond the Standard Model (SM) has been a central focus of particle physics, including in the context of $B$-meson decays involving $b \to s \ell \ell$ transitions. These transitions, mediated by flavour-changing neutral currents, are highly sensitive to small NP effects due to their suppression in the SM. While direct searches at colliders have not yet led to NP discoveries, indirect probes through semi-leptonic decays have revealed anomalies in observables such as the branching fraction $\mathcal{B}(B \to K \mu \mu)$ and the angular observable $P_5'(B \to K^* \mu \mu)$. In order to assess the observed tensions, it is essential to ensure an accurate SM prediction. In this review, we examine the theoretical basis of the $B \to K^{(*)} \ell \ell$ decays, addressing in particular key uncertainties arising from local and non-local form factors. We also discuss the impact of QED corrections to the Wilson coefficients, as well as the effect of CKM matrix elements on the predictions and the tension with the experimental measurements. We discuss the most recent results, highlighting ongoing efforts to refine predictions and to constrain potential signs of NP in these critical decay processes.