  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 future will
be helpful to examine our formalism and constrain the input parameters more
precisely.
