Motivated by the measurements of the branching fractions of the quasi-two-body decays $B^0\to K^{*0}(f_0(980)\to ){\pi }^{+}{\pi }^{-}$ and $B^0\to {\rho }^0(f_0(980)\to ){\pi }^{+}{\pi }^{-}$, we study the charmless $B\to V(f_0(980)\to ){\pi }^{+}{\pi }^{-}$ decays in the perturbative QCD approach. Supposing that $f_0(980)$ is a two-quark state and mixture of $n\overline{n}=(u\overline{u}+d\overline{d})/\sqrt{2}$ and $s\overline{s}$ 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 $[135°,155°]$, the calculated branching fractions of the $B^0\to K^{*0}{f}_0(980)\to K^{*0}{\pi }^{+}{\pi }^{-}$ and $B^0\to {\rho }^0{f}_0(980)\to {\rho }^0{\pi }^{+}{\pi }^{-}$ 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 $B\to V{f}_0(980)\to V{\pi }^0{\pi }^0$, which are half of the corresponding decays $B\to V{f}_0(980)\to V{\pi }^{+}{\pi }^{-}$. Moreover, the direct $CP$ asymmetries of these decays are also calculated, and some of them can be tested in the current experiments.

• Received 30 March 2021
• Accepted 24 May 2021

DOI:https://doi.org/10.1103/PhysRevD.103.113005

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Published by the American Physical Society