Measurement of the Forward-Backward Asymmetry in the $B \to K^{(*)} \mu^+ \mu^-$ Decay and First Observation of the $B^0_s \to \phi \mu^+ \mu^-$ Decay

We reconstruct the rare decays B{sup +} {yields} K{sup +} {mu}{sup +}{mu}{sup -}, B{sup 0} {yields} K*(892){sup 0} {mu}{sup +}{mu}{sup -}, and B{sub s}{sup 0} {yields} {phi}(1020){mu}{sup +}{mu}{sup -} in a data sample corresponding to 4.4 fb{sup -1} collected in p{bar p} collisions at {radical}s = 1.96 TeV by the CDF II detector at the Fermilab Tevatron Collider. Using 120 {+-} 16 B{sup +} {yields} K{sup +} {mu}{sup +}{mu}{sup -} and 101 {+-} 12 B{sup 0} {yields} K*{sup 0} {mu}{sup +}{mu}{sup -} decays we report the branching ratios. In addition, we report the measurement of the differential branching ratio and the muon forward-backward asymmetry in the B{sup +} and B{sup 0} decay modes, and the K*{sup 0} longitudinal polarization in the B{sup 0} decay mode with respect to the squared dimuon mass. These are consistent with the theoretical prediction from the standard model, and most recent determinations from other experiments and of comparable accuracy. We also report the first observation of the B{sub s}{sup 0} {yields} {phi}{mu}{sup +}{mu}{sup -} decay and measure its branching ratio {Beta}(B{sub s}{sup 0} {yields} {phi}{mu}{sup +}{mu}{sup -}) = [1.44 {+-} 0.33 {+-} 0.46] x 10{sup -6} using 27 {+-} 6 signal events. This is currently the most rare B{sub s}{sup 0} decay observed