Study of quasi-two-body $$B_{(s)}\rightarrow \phi (f_0(980)/f_2(1270)\rightarrow )\pi \pi $$ B ( s ) → ϕ ( f 0 ( 980 ) / f 2 ( 1270 ) → ) π π decays in perturbative QCD approach

In 2017, LHCb collaboration reported their first observation of the rare decays $$B_s \rightarrow \phi (f_0(980)$$ $$/f_2(1270) \rightarrow ) \pi ^+\pi ^-$$ and the evidence of $$B^0 \rightarrow \phi (f_0(980)/f_2(1270)\rightarrow )\pi ^+\pi ^-$$ . Motivated by this, we study these quasi-two-body decays in the perturbative QCD approach. The branching fractions, CP asymmetries and the polarization fractions are calculated. We find that within the appropriate two-meson wave functions, the calculated branching fractions are in agreement with the measurements of LHCb. Based on the narrow-width approximation, We also calculate the branching fractions of the quasi-two-body $$B_{d,s}\rightarrow \phi (f_0(980)/f_2(1270)\rightarrow ) \pi ^0\pi ^0$$ and $$B_{d,s}\rightarrow \phi (f_2(1270)\rightarrow ) K^+K^-$$ , and hope the predictions to be tested in the ongoing LHCb and Belle II experiments. Moreover, the processes $$B_{d,s}\rightarrow \phi f_2(1270)$$ are also analyzed under the approximation. We note that the CP asymmetries of these decays are very small, because these decays are either penguin dominant or pure penguin processes.