Paper detail

$S$-wave resonance contributions to the $B^0_{(s)}\to J/ψπ^+π^-$ and $B_s\toπ^+π^-μ^+μ^-$ decays

We study $S$-wave resonance contributions to the $B^0_{(s)}\to J/ψπ^+π^-$ and $B_s\toπ^+π^-\ell^+\ell^-$ decays in the perturbative QCD (PQCD) framework by introducing two-hadron distribution amplitudes for final states. The Breit-Wigner formula for the $f_0(500)$, $f_0(1500)$ and $f_0(1790)$ resonances and the Flatté model for the $f_0(980)$ resonance are adopted to parameterize the time-like scalar form factors in the two-pion distribution amplitudes, which include both resonant and nonresonant contributions. The resultant branching fraction and differential branching fraction in the pion-pair invariant mass for each resonance channel are consistent with experimental data. The determined $S$-wave two-pion distribution amplitudes, containing the information of both resonant and nonresonant rescattering phases, can be employed to predict direct CP asymmetries of other three-body hadronic $B$ meson decays in various localized regions of two-pion phase space.