The three-body decays B(s) → ρf0(X)→ρπ−π+ in pQCD factorization approach

In this paper, we study three-body decays B(s)→ρf0(X)→ρπ−π+ in the perturbative QCD (pQCD) factorization approach. By using two-pion distribution amplitudes (DAs), these decays can proceed mainly via quasi-two-body channels. The Flatté model for the f0(980) resonance and the Breit-Wigner formula for the f0(500), f0(1500), and f0(1790) resonances are adopted to parameterize the time-like scalar form factors. We also use Bugg's model for the wide f0(500) as a comparison. We evaluate S-wave resonance contributions, and show the contributions to the B(s)→ρπ+π− decay spectrums with respect to the two-pion invariant mass M(π+π−). The pQCD prediction of B→ρf0(500) is B(B0→ρ0f0(500))=4.43×10−8, B(B+→ρ+f0(500))=3.84×10−8 for Breit-Wigner model, and B(B0→ρ0f0(500))=4.91×10−8, B(B+→ρ+f0(500))=4.07×10−8 for Bugg's model. We also predict other resonance contributions B(Bs→ρ0f0(980))=8.91×10−8, B(Bs→ρ0f0(1500))=1.43×10−8 and B(Bs→ρ0f0(1790))=2.78×10−9. This study in pQCD approach can provide a ready reference to the running and forthcoming experiments. Further improvements in theories and experiments are expected so that we can understand the inner structure of the scalar mesons f0.