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

In this work, we study the semileptonic decay modes $B_s^0\to π^+π^-\ell^+\ell^-$ and $D_s^+\to π^+π^-\ell^+ ν$ in the kinematics region where the $π^+π^-$ system has a invariant mass in the range $0.5$-$1.3$ GeV. These processes are valuable towards the determination of S-wave $π^+π^-$ 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 $B_s\to J/ψπ^+π^-$. Then the $B_s\to π^+π^-$ and $D_s\to π^+π^-$ 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 $π^+π^-$. More accurate measurements at BEPC, LHC and KEKB in future will be helpful to examine our formalism and constrain the input parameters more precisely.