Study of the $B_{s}$ ${\to}$ ${\phi}f_{0}(980)$ ${\to}$ ${\phi}\,{\pi}^{+}{\pi}^{-}$ decay with perturbative QCD approach

TL;DR

This paper calculates the branching ratio of the rare decay $B_s o \, \phi f_0(980) \to \, \phi \pi^+ \pi^-$ using perturbative QCD, incorporating nonperturbative effects, and finds results consistent with recent experimental measurements.

Contribution

The study introduces a detailed perturbative QCD approach with nonperturbative two-pion distribution amplitudes and Flatté parameterization for the $f_0(980)$ resonance, providing a precise theoretical prediction.

Findings

Predicted branching ratio: approximately 1.31 x 10^{-6}.

Results agree with recent LHCb measurements within uncertainties.

Uncertainty analysis includes parameters $a_{\pi\pi}$, $m_b$, and CKM factors.

Abstract

The rare cascade $B_{s}$ ${\to}$ ${\phi}f_{0}(980)$ ${\to}$ ${\phi}\,{\pi}^{+}{\pi}^{-}$ decay is studied with the perturbative QCD approach based on the formula for the quasi two-body decay, where the two-pion pair originates from the $S$-wave resonant $f_{0}(980)$ state. It is found that with the introduction of the nonperturbative two-pion distribution amplitudes and the Flatt\'{e} parameterization of the scalar form factor for the $f_{0}(980)$ resonance, the branching ratio in the mass range 400 MeV $<$ $m({\pi}^{+}{\pi}^{-})$ $<$ 1600 MeV is ${\cal B}_{\rm theo}(B_{s}^{0}\, {\to}\,{\phi}\,f_{0}(980)\,{\to}\,{\phi}\,{\pi}^{+}\,{\pi}^{-})$ $=$ $[1.31_{-0.31}^{+0.40}(a_{{\pi}{\pi}}){}_{-0.16}^{+0.19}(m_{b}) {}_{-0.09}^{+0.10}(\text{CKM})]{\times}10^{-6}$, where the uncertainties come from the parameter $a_{{\pi}{\pi}}$ of the two-pion distribution amplitudes, the $b$ quark mass $m_{b}$, the Cabibbo-Kobayashi-Maskawa (CKM) factors, respectively. This result agrees with the recent LHCb measurement within uncertainties, ${\cal B}_{\rm exp}(B_{s}^{0}\,{\to}\,{\phi}\, f_{0}(980)\,{\to}\,{\phi}\,{\pi}^{+}\,{\pi}^{-})$ $=$ $(1.12{\pm}0.16^{+0.09}_{-0.08}{\pm}0.11){\times}10^{-6}$, where the errors are statistical, systematic and from the normalization, respectively.