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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 kinematicsregion where the $\pi^+\pi^-$ system has a invariant mass in the range$0.5$-$1.3$ GeV. These processes are valuable towards the determination ofS-wave $\pi^+\pi^-$ light-cone distribution amplitudes whose normalizations arescalar form factors. We compare the results for scalar form factors predictedin 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 whichpredictions for differential decay widths are made. The results are in goodagreement 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 willbe helpful to examine our formalism and constrain the input parameters moreprecisely.

Chiral Dynamics and S-wave contributions in Semileptonic $D_s/B_s$ decays into $\pi^+\pi^-$