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Rare $B\to πl\bar l$ and $B\toρl \bar l$ decays in the relativistic quark model

The branching fractions of the rare weak $B\to πl^+l^-(ν\barν)$ and $B\toρl^+l^-(ν\barν)$ decays are calculated in the framework of the relativistic quark model based on the quasipotential approach. The form factors parametrizing weak decay matrix elements are explicitly determined in the whole kinematical $q^2$ range without additional assumptions and extrapolations. Relativistic effects are systematically taken into account including recoil effects in meson wave functions and contributions of the intermediate negative-energy states. New experimental data on the differential distributions in the semileptonic heavy-to-light $B\toπlν_l$ and $B\toρlν_l$ decays are analyzed in detail. Good agreement of the predictions and data is found. The obtained results for the branching fractions of the rare semileptonic decays are found to be in agreement with other theoretical estimates and recent experimental data available for the $B^+\toπ^+ μ^+μ^-$ decay.