In this work, we study the semileptonic decay modes $B_s^0\to {\pi }^{+}{\pi }^{-}{\ell }^{+}{\ell }^{-}$ and $D_s^{+}\to {\pi }^{+}{\pi }^{-}{\ell }^{+}\nu$ in the kinematics region where the ${\pi }^{+}{\pi }^{-}$ system has a invariant mass in the range 0.5–1.3 GeV. These processes are valuable towards the determination of S-wave ${\pi }^{+}{\pi }^{-}$ light-cone distribution amplitudes whose normalizations are scalar form factors. We compare the results for scalar form factors predicted in unitarized chiral perturbation theory and extracted from the data on the $B_s\to J/\psi {\pi }^{+}{\pi }^{-}$. Then the $B_s\to {\pi }^{+}{\pi }^{-}$ and $D_s\to {\pi }^{+}{\pi }^{-}$ form factors are calculated in light-cone sum rules, based on which predictions for differential decay widths are made. The results are in good agreement with the experimental data on the $B_s$ and $D_s$ decays into ${\pi }^{+}{\pi }^{-}$. More accurate measurements at BEPC, LHC and KEKB in the future will be helpful to examine our formalism and constrain the input parameters more precisely.

• Received 30 July 2015

DOI:https://doi.org/10.1103/PhysRevD.92.074038