Quasi-two-body B(s) → Vππ decays with resonance f0 (980) in the PQCD approach

Motivated by the measurements of the branching fractions of the quasi-two-body decays B-0 -> K*(0) (f(0)(980) ->)pi(+)pi(-) and B-0 -> rho(0)(f(0) (980) ->)pi(+)pi(-), we study the charmless B -> V(f(0)(980) ->)pi(+)pi(-) decays in the perturbative QCD approach. Supposing that f(0) (980) is a two-quark state and mixture of n (n) over bar = (u (u) over bar + d (d) over bar)/root 2 and s (s) over bar with the mixing angle theta, we calculate the branching fractions of these decays with the new introduced s-wave pi(+)pi(-)-pair wave function. When the mixing angle theta lies in the range [135 degrees,155 degrees], the calculated branching fractions of the B-0 -> K*(0)f(0) (980) -> K*(0)pi(+)pi(-) and B-0 -> rho(0) f(0) (980) -> rho(0)pi(+)pi(-) decays are in agreement with the experimental data. The branching fractions of other decays could be measured in the current LHCb and Belle II experiments. Considering the isospin symmetry, we also estimate the branching fractions of the quasi-two-body decays B -> Vf(0) (980) -> V pi(0)pi(0), which are half of the corresponding decays B -> Vf(0)(980) -> V pi(+)pi(-). Moreover, the direct CP asymmetries of these decays are also calculated, and some of them can be tested in the current experiments.