Relativistic description of exclusive heavy-to-light semileptonic decaysB→π(ρ)eν

The method of calculating electroweak decay matrix elements between heavy-heavy and heavy-light meson states is developed in the framework of a relativistic quark model based on the quasipotential approach in quantum field theory. This method is applied to the study of exclusive semileptonic $B\to \pi \left(\rho \right)$ decays. It is shown that the large value of the final $\pi \left(\rho \right)$ meson recoil momentum allows for the expansion in inverse powers of the $b$-quark mass of the decay form factors at $q^2=0\mathrm{}$, where $q$ is a momentum carried by the lepton pair. This $\frac{1}{m_b}$ expansion considerably simplifies the analysis of these decays and is carried out up to the second order. The $q^2$ dependence of the form factors is investigated. It is found that the $q^2$ behavior of the axial form factor $A_1$ is different from the other form factors. It is argued that the ratios $\frac{\Gamma (B\to \rho e\nu )}{\Gamma (B\to \pi e\nu )}$ and $\frac{{\Gamma }_L}{{\Gamma }_T}$ are sensitive probes of the $A_1{q}^2$ dependence, and, thus, their experimental measurement may discriminate between different approaches. We find $\Gamma (B\to \pi e\nu )=(3.0\mathrm{}\pm 0.6)\times {\left|V_{\mathrm{ub}}\right|}^2\times {10}^{12}{\mathrm{s}}^{-1\mathrm{}}$ and $\Gamma (B\to \rho e\nu )=(5.4\mathrm{}\pm 1.2)\times {\left|V_{\mathrm{ub}}\right|}^2\times {10}^{12}{\mathrm{s}}^{-1\mathrm{}}$. The relation between semileptonic and rare radiative $B$ decays is discussed.