Complementarity of the constraints on New Physics from $B_s\to \mu^+\mu^-$ and from $B\to K \ell^+\ell^-$ decays

28 pp, 12 figures, 2 tablesWe discuss the advantages of combining the experimental bound on ${\rm Br}(B_s\to \mu^+\mu^-)$ and the measured ${\rm Br}(B\to K\ell^+\ell^-)$ to get the model independent constraints on physics beyond the Standard Model. Since the two decays give complementary information, one can study not only the absolute values of the Wilson coefficients that are zero in the Standard Model, but also their phases. To identify the sector in which the new physics might appear, information about the shapes of the transverse asymmetries in $B\to K^\ast \ell^+\ell^-$ at low $q^2$'s can be particularly useful. We also emphasize the importance of measuring the forward-backward asymmetry in $B\to K\ell^+\ell^-$ decay at large $q^2$'s