Constraining $\lambda_{B_s}$ by $B_s\to \gamma^*$ and $B_s\to \phi$ form factors

We calculate the form factors $F_{V}(q^2,q'^2)$ and $F_{TV}(q^2,q'^2)$ describing the $B_s\to \gamma^*$ transition induced by the vector and tensor weak currents in a broad range of values of $q^2$ and $q'^2$ far below the quark thresholds in both $q^2$ and $q'^2$ channels. These form factors are calculated via the distribution amplitudes of the $B_s$-meson. We then interpolate the obtained results by a formula that contains pole at $q'^2=M_\phi^2$ and extract the residue which gives the $B_s\to \phi$ transition form factors $V(q^2)$ and $T_1(q^2)$. In this way we obtain theoretical predictions for these form factors without invoking quark-hadron duality and QCD sum rules. Furthermore, we calculate the relationship between $V(0)$ and $T_1(0)$ and the parameter $\lambda_{B_s}(\mu)$, the inverse moment of the $B_s$-meson distribution amplitude. Using the available predictions for $V(0)$ and $T_1(0)$ coming from approaches not referring to the $B_s$-meson distribution amplitudes, we obtain the estimate $\lambda_{B_s}(\mu\simeq m_b)=(0.62\pm 0.10)$ GeV.