Chiral Dynamics and S-wave contributions in Semileptonic $D_s\mathrm{/}{B}_s$ decays into ${\pi }^{+}{\pi }^{-}$

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 $B_s\to J\mathrm{/}\psi {\pi }^{+}{\pi }^{-}$. 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 future will be helpful to examine our formalism and constrain the input parameters more precisely.