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We study $S$-wave resonance contributions to the $B^0_{(s)}\toJ/\psi\pi^+\pi^-$ and $B_s\to\pi^+\pi^-\ell^+\ell^-$ decays in the perturbativeQCD (PQCD) framework by introducing two-hadron distribution amplitudes forfinal states. The Breit-Wigner formula for the $f_0(500)$, $f_0(1500)$ and$f_0(1790)$ resonances and the Flatt\'e model for the $f_0(980)$ resonance areadopted to parameterize the time-like scalar form factors in the two-piondistribution amplitudes, which include both resonant and nonresonantcontributions. The resultant branching fraction and differential branchingfraction in the pion-pair invariant mass for each resonance channel areconsistent with experimental data. The determined $S$-wave two-piondistribution amplitudes, containing the information of both resonant andnonresonant rescattering phases, can be employed to predict direct CPasymmetries of other three-body hadronic $B$ meson decays in various localizedregions of two-pion phase space.

$S$-wave resonance contributions to the $B^0_{(s)}\to J/\psi\pi^+\pi^-$ and $B_s\to\pi^+\pi^-\mu^+\mu^-$ decays