The $S$-wave resonance contributions to the three-body decays $B_{(s)}^0\to {\eta }_c{f}_0(X)\to {\eta }_c{\pi }^{+}{\pi }^{-}$ in perturbative QCD approach

Abstract

In this paper, we study the three-body decays $B^0/B_s^0\to {\eta }_c{f}_0(X)\to {\eta }_c{\pi }^{+}{\pi }^{-}$ by employing the perturbative QCD (PQCD) factorization approach. We evaluate the $S$-wave resonance contributions by using the two-pion distribution amplitude ${\Phi }_{\pi \pi }^{\mathrm{S}}$. The Breit-Wigner formula for the $f_0(500)$, $f_0(1500)$, and $f_0(1790)$ resonances and the Flatté model for the $f_0(980)$ resonance are adopted to parameterize the time-like scalar form factors $F_s({\omega }^2)$. We also use the D.~V.~Bugg model to parameterize the $f_0(500)$ and compare the relevant theoretical predictions from different models. We found the following results: (a) the PQCD predictions for the branching ratios are $\mathcal{B}(B^0\to {\eta }_c{f}_0(500)[{\pi }^{+}{\pi }^{-}])=\left(1.53_{-0.35}^{+0.76}\right)\times 10^{-6}$ for Breit-Wigner model and $\mathcal{B}(B^0\to {\eta }_c{f}_0(500)[{\pi }^{+}{\pi }^{-}])=\left(2.31_{-0.48}^{+0.96}\right)\times 10^{-6}$ for D.~V.~Bugg model; (b) $\mathcal{B}(B_s\to {\eta }_c{f}_0(X)[{\pi }^{+}{\pi }^{-}])=\left(5.02_{-1.08}^{+1.49}\right)\times 10^{-5}$ when the contributions from $f_0(X)=(f_0(980),f_0(1500),f_0(1790))$ are all taken into account; and (c) The considered decays could be measured at the ongoing LHCb experiment, consequently, the formalism of two-hadron distribution amplitudes could also be tested by such experiments.

Comments: 13 pages, 2 figures, Some modifications. Final version to be published in EPJC