$S$-wave resonance contributions to the $B^0_{(s)}\to \eta_c{(2S)}\pi^+\pi^-$ in the perturbative QCD factorization approach

TL;DR

This paper uses perturbative QCD to analyze $B^0_{(s)}$ decays involving $S$-wave resonances in the two-pion system, providing predictions for decay rates with different resonance models.

Contribution

It introduces a detailed PQCD analysis of $B^0_{(s)}$ decays with $S$-wave resonance contributions, comparing different scalar form factor models.

Findings

Predicted branching ratio for $B^0_s$ decay: approximately 2.67×10^{-5}.

Predicted branching ratio for $B^0$ decay with $f_0(500)$: around 1.4×10^{-6}.

Results vary depending on the resonance model used.

Abstract

By employing the perturbative QCD (PQCD) factorization approach, we study the quasi-two-body $B^0_{(s)}\to \eta_c{(2S)}\pi^+\pi^-$ decays, where the pion pair comes from the $S$-wave resonance $f_0(X)$. The Breit$-$Wigner formula for the $f_0(500)$ and $f_0(1500)$ resonances, and the Flatt\'e model for the $f_0(980)$ resonance are adopted to parameterize the time-like scalar form factors in the two-pion distribution amplitudes. As a comparison, Bugg's model is also used for the wide $f_0(500)$ in this work. For decay rates, we found the following PQCD predictions: (a) $ {\cal B}(B^0_s\to \eta_c(2S) f_0(X)[\pi^+\pi^-]_s )=\left ( 2.67^{+1.78}_{-1.08} \right )\times 10^{-5}$ when the contributions from $f_0(980)$ and $f_0(1500)$ are all taken into account; (b) ${\cal B}(B^0\to \eta_c(2S) f_0(500)[\pi^+\pi^-]_s)= \left ( 1.40 ^{+0.92}_{-0.56} \right ) \times 10^{-6}$ in the Breit-Wigner model and $ \left ( 1.53 ^{+0.97}_{-0.61} \right ) \times 10^{-6}$ in the Bugg's model.

Equations

$$\mathcal{A}={\varphi }_B\otimes H\otimes {\varphi }_{h_1{h}_2}\otimes {\varphi }_{h_3}.$$

$$p_B=\frac{m_B}{\sqrt{2}}(1,1,0_{\mathrm{T}}),p=p_1+p_2=\frac{m_B}{\sqrt{2}}(1-r^2,\eta ,0_{\mathrm{T}}),p_3=\frac{m_B}{\sqrt{2}}(r^2,1-\eta ,0_{\mathrm{T}})$$

$$k_B=\left(0,\frac{m_B}{\sqrt{2}}{x}_B,k_{BT}\right),k=\left(\frac{m_B}{\sqrt{2}}z(1-r^2),0,k_{\mathrm{T}}\right),k_3=\left(\frac{m_B}{\sqrt{2}}{r}^2{x}_3,\frac{m_B}{\sqrt{2}}(1-\eta )x_3,k_{3\mathrm{T}}\right),$$

$${\Phi }_{\pi \pi }^{S-wave}=\frac{1}{\sqrt{2N_c}}\left[p/{\Phi }_{v\nu =-}^{I=0}(z,\zeta ,{\omega }^2)+\omega {\Phi }_s^{I=0}(z,\zeta ,{\omega }^2)+\omega (n{/}_{+}n{/}_{-}-1){\Phi }_{t\nu =+}^{I=0}(z,\zeta ,{\omega }^2)\right],$$

$${\Phi }_{v\nu =-}^{I=0}$$

$$R_{f_0(500)}(s)=m_r{\Gamma }_1(s){\left[m_r^2-s-g_1^2\frac{s-s_A}{m_r^2-s_A}\left[j_1(s)-j_1(m_r^2)\right]-i{m}_r\sum _{i=1}^4{\Gamma }_i(s)\right]}^{-1},$$

$$m_r{\Gamma }_1(s)$$

$$g_1^2(s)$$

$$j_1(s)$$

$$m_r{\Gamma }_2(s)$$

$$m_r{\Gamma }_3(s)$$

$$m_r{\Gamma }_4(s)$$

$${\rho }_{4\pi }(s)$$

$${\Phi }_B=\frac{i}{\sqrt{2N_c}}({p/}_B+m_B){\gamma }_5{\varphi }_B({\mathbf{k}}_1).$$

$${\varphi }_B(x,b)$$

$$⟨{\eta }_c(2S)|\bar{c}(z{)}_{\alpha }c(0{)}_{\beta }|0⟩=-\frac{i}{\sqrt{2N_c}}{\int }_0^{1}dx{e}^{ix{p}_3\cdot z}\left[({\gamma }_{5}to0.0pt/\text{hss}{p}_3{)}_{\alpha \beta }{\psi }^v(x,b)+m({\gamma }_5{)}_{\alpha \beta }{\psi }^s(x,b)\right].$$

$${\Psi }^v(x,b)$$

$${\Psi }^s(x,b)$$

$$\mathcal{A}(B_{(s)}^0\to {\eta }_c(2S){\pi }^{+}{\pi }^{-})$$

$$F^{LL}$$

$$F^{LR}$$

$${\Lambda }_{\overline{MS}}^{(f=4)}$$

$$m_{\pi }^{\pm }$$

$$\displaystyle\lambda=0.22506\pm 0.00050,\ \ A=0.811\pm 0.026\ \ \bar{\rho}=0.124_{-0.018}^{+0.019},\ \ \bar{\eta}=0.356\pm 0.011.\ \$$

$$\frac{d\mathcal{B}}{d\omega }={\tau }_B\frac{\omega |\overrightarrow{p_1}||\overrightarrow{p_3}|}{4(2\pi {)}^3{m}_B^3}|\mathcal{A}{|}^2,$$

$$|\overrightarrow{p_1}|=\frac{1}{2}\sqrt{{\omega }^2-4m_{{\pi }^{\pm }}^2},|\overrightarrow{p_3}|=\frac{1}{2\omega }\sqrt{\left[m_B^2-(\omega +m_{{\eta }_c(2S)}{)}^2\right]\left[m_B^2-(\omega -m_{{\eta }_c(2S)}{)}^2\right]}.$$

$$\mathcal{B}(B_s^0\to {\eta }_c(2S)f_0(980)[f_0(980)\to {\pi }^{+}{\pi }^{-}])$$

$$\mathcal{B}(B_s^0\to {\eta }_c(2S)f_0(1500)[f_0(1500)\to {\pi }^{+}{\pi }^{-}])$$

$$\mathcal{B}(B_s^0\to {\eta }_c(2S)({\pi }^{+}{\pi }^{-}{)}_S)$$

$$\mathcal{B}(B^0\to {\eta }_c(2S)f_0(500)[f_0(500)\to {\pi }^{+}{\pi }^{-}]{)}_{(\mathrm{BW})}$$

$$\mathcal{B}(B^0\to {\eta }_c(2S)f_0(500)[f_0(500)\to {\pi }^{+}{\pi }^{-}]{)}_{(\mathrm{Bugg})}$$

Full text

Key words: PQCD factorization approach, Two-pion distribution amplitudes, Quasi-two-body decay

I Introduction

The study of the three-body hadronic $B$ meson decays can help us understand the standard model and search for the possible effects of new physics. Experimentally, quite a number of channels have been measured by collaborations like BaBar prl90-091801 ; prd70-092001 ; prd72-072003 ; prd80-112001 ; prd79-072006 ; prd78-052005 , Belle prd75-012006 ; prl96-251803 ; prd71-092003 ; prd79-072004 and LHCb  prd90-112004 ; prl112-011801 ; prl111-101801 ; prd86-052006 ; prd89-092006 ; prd92-032002 ; prd87-052001 ; npb-871403 ; prd90-012003 ; plb742-38 ; 1702.08048 . Theoretically, there are several approaches working in this field, for instance, the QCD factorization  plb622-207 ; prd74-114009 ; prd79-094005 ; prd81-094033 ; appb42-2013 ; prd66-054015 ; prd72-094003 ; prd76-094006 ; prd88-114014 ; prd89-074025 ; prd94-094015 ; prd89-094007 ; npb899-247 ; epjc75-536 ; prd87-076007 ; 1512.09284 ; prd70-034033 , the perturbative QCD (PQCD) approach plb561-258 ; prd70-054006 ; prd89-074031 ; prd91-094024 ; plb763-29 ; epjc76-675 ; 1611.08786 ; prd95-056008 ; 1701.02941 , and some methods based on symmetry principles plb564-90 ; prd72-075013 ; prd72-094031 ; prd84-056002 ; prd84-034040 ; plb727-136 ; plb726-337 ; prd89-074043 ; plb728-579 ; ijmpa29-1450011 ; prd91-014029 ; prd89-094013 ; prd92-054010 . The aim of those studies is to understand the the resonant and nonresonant contributions, as well as the final state interactions (FSIs) 1512.09284 ; prd89-094013 in three-body $B$ decays. But it is still in the early stage for both the theoretical studies and the experimental measurements in studying those decays.

The PQCD factorization approach is one of the major theoretical frameworks to deal with the two-body hadronic $B$ meson decays plb504-6 ; ppnp51-85 . Very recently, some three-body hadronic $B$ meson decays have been studied by employing the PQCD factorization approach, for example in Refs. plb561-258 ; prd70-054006 ; prd89-074031 ; prd91-094024 ; plb763-29 ; epjc76-675 ; 1701.02941 ; 1611.08786 ; prd95-056008 . For the cases of three-body decays, however, the previous PQCD approach plb504-6 ; ppnp51-85 should be modified by introducing the two-meson distribution amplitudes fp42-101 ; prl81-1782 ; npb555-231 ; sjnp38-289 to describe the selected pair of final state mesons due to the following reason discussed in plb504-6 ; ppnp51-85 : the contribution from the direct evaluation of hard b-quark decay kernels containing two virtual gluons is generally power suppressed, and the dominant contribution comes most possibly from the region where the two energetic light mesons are almost collimating to each other with an invariant mass below $O(\bar{\Lambda}m_{B})$( $\bar{\Lambda}=m_{B}-m_{b}$, means the $B$ meson and $b$ quark mass difference). Then, the typical PQCD factorization formula with the crucial nonperturbative input of two-hadron distribution amplitudes for a $B\to h_{1}h_{2}h_{3}$ decay amplitude can be written symbolically in the form of

[TABLE]

Here the hard kernel $H(x_{i},b_{i},t)$ contains the contributions from one hard gluon exchange diagrams only, the nonperturbative inputs $\phi_{B}(x,b)$, $\phi_{h_{1}h_{2}}(z,\omega)$, $\phi_{h_{3}}(x_{3},b_{3})$ are the distribution amplitudes for the $B$ meson, the $h_{1}$-$h_{2}$ pair and the $h_{3}$ meson respectively, while the symbols $\otimes$ mean the convolution integration over the variables of the momentum fractions $(x,z,x_{3})$ and the conjugate space coordinates $b_{i}$ of $k_{\rm iT}$. With the help of the two-pion distribution amplitudes, many works have been done for quasi-two-body decays, the parameters in the $S$-wave and $P$-wave two-pion distribution amplitudes have been fixed in Refs. prd91-094024 ; plb763-29 . Based these work, we have studied the $S$-wave resonance contributions to the decays $B^{0}_{(s)}\to\eta_{c}\pi^{+}\pi^{-}$ epjc76-675 , $B^{0}_{s}\to\psi(2s)\pi^{+}\pi^{-}$ 1701.02941 , and the $P$-wave resonance ($\rho(770)$) contributions to $B^{0}_{(s)}\to(D/P)\rho\to(D/P)\pi\pi$ decays 1611.08786 ; prd95-056008 with $D$ represents the charmed $D$ mesons and the $P$ stands for the light pseudoscalar mesons: $\pi,K,\eta$ or $\eta^{\prime}$.

Up to now, several decay modes of the $B$ and $B_{s}$ mesons to the charmonium state plus pion pair, like $B^{0}\to J/\psi\pi^{+}\pi^{-}$ prl90-091801 ; prd87-052001 ; prd90-012003 ; plb742-38 , $B_{s}^{0}\to J/\psi\pi^{+}\pi^{-}$  prd86-052006 ; prd89-092006 , $B_{(s)}^{0}\to\psi(2S)\pi^{+}\pi^{-}$npb-871403 and $B_{s}^{0}\to\eta_{c}\pi^{+}\pi^{-}$ 1702.08048 , have been measured by BaBar and LHCb Collaboration. With the continuous running of the LHCb experiment, more data of such $B/B_{s}$ decays with the inclusion of various excited charmonium states ( $\eta_{c}(2S)$ etc.) will be collected. It is therefore interesting to study such decay modes theoretically. In this work, we will study the $S$-wave resonance contributions to $B^{0}_{(s)}\to\eta_{c}{(2S)}f_{0}(X)\to\eta_{c}(2S)\pi^{+}\pi^{-}$ decays and give our predictions for the branching fractions of the considered decay modes.

This paper is organized as follows. In Sec. II, we give a brief introduction for the theoretical framework. The numerical values, some discussions and the conclusions will be given in last two sections.

II The theoretical framework

In the $B^{0}_{(s)}\to\eta_{c}{(2S)}\pi^{+}\pi^{-}$ decays, by using of the light-cone coordinates and in the rest frame of $B^{0}_{(s)}$ meson, the momentum of $B^{0}_{(s)}$, the pion pair and $\eta_{c}{(2S)}$ could be chosen as

[TABLE]

where $\eta=\omega^{2}/[(1-r^{2})m^{2}_{B}]$, $r=m_{\eta_{c}{(2S)}}/m_{B}$ and $\omega^{2}=p^{2}$ means the squared invariant mass of the pion pair. The momenta for the spectators in the ${B^{0}_{(s)}}$ meson, the pion pair, and the ${\eta_{c}{(2S)}}$ meson read as

[TABLE]

where the momentum fractions $x_{B}$, $z$, and $x_{3}$ run from zero to unity.

The $S$-wave two-pion distribution amplitudes can be written as prd91-094024 ; plb730-336

[TABLE]

with $n_{+}=(1,0,{\bf 0}_{T})$, $n_{-}=(0,1,{\bf 0}_{T})$ and the $\pi^{+}$ meson momentum fraction $\zeta=p^{+}_{1}/p^{+}$. Their asymptotic forms are parameterized as prd91-094024

[TABLE]

with the time-like scalar form factor $F_{s}(w^{2})$ and the Gegenbauer coefficient $a_{2}^{I=0}=0.2\pm 0.2$.

The expressions of the time-like scalar form factor $F_{s}(\omega^{2})$ associated with the $s\bar{s}$ component of both $f_{0}(980)$ and $f_{0}(1500)$, and $d\bar{d}$ component of $f_{0}(500)$ can be found in Ref. prd91-094024 . Following the LHCb collaboration prd86-052006 ; prd89-092006 ; prd87-052001 ; prd90-012003 , the Breit$-$Wigner (BW) formula for the $f_{0}(500)$ and $f_{0}(1500)$ resonances will be used to parameterize the time-like scalar form factors in the two-pion distribution amplitudes, which include both the resonant and non-resonant contributions of the $\pi\pi$ pair. For $f_{0}(980)$, however, the Flatté model plb63-228 will be used since $f_{0}(980)$ is close to the $K\bar{K}$ threshold and the BW formula does not work well for this meson  jpg34-151 ; plb63-228 . We know that there exist some disputations about the nature of the meson $f_{0}(500)$ due to its wide shape. Following the same treatment of $f_{0}(500)$ as LHCb collaboration prd92-032002 , we here also parameterize its contribution to the scalar form factor in the Bugg resonant line-shape jpg34-151

[TABLE]

with the following relevant parameters

[TABLE]

In the numerical calculation, we set $m_{r}$ = $0.953~{}GeV$, $s_{A}=0.41\ m_{\pi}^{2}$, $b_{1}=1.302~{}GeV$, $b_{2}=0.340~{}GeV^{-1}$, $A=2.426~{}GeV^{2}$ and $g_{4\pi}=0.011~{}GeV$ jpg34-151 . The phase-space factors of the decay channels $\pi\pi$, $KK$ and $\eta\eta$ are defined as $\rho_{i}(s)=\sqrt{1-4m^{2}_{i}/s}$ with $i=1,2,3$ for $\pi,K$ and $\eta$ respectively. It is worth of mentioning that another description of pion-pion form factors were introduced in Ref. prd83-074004 ; prd89-053015 .

For the $B^{0}_{(s)}$ mesons, we use the same distribution amplitudes $\phi_{B}(x,b)$ in the $b$ space as being used for example in Ref. epjc76-675 ,

[TABLE]

The distribution amplitude is chosen as

[TABLE]

In the numerical calculation, we also use the shape parameter $\omega_{B}=0.40\pm 0.04$ GeV with $f_{B}=0.19$ GeV for $B^{0}$ decays, and $\omega_{B_{s}}=0.50\pm 0.05$ GeV with $f_{B_{s}}=0.236$ GeV for $B^{0}_{s}$ decays  epjc76-675 .

As the first radial excitation of the $\eta_{c}$ charmonium ground state, $\eta_{c}(2S)$ is observed firstly by the Belle collaboration in $B$ decays prl89-102001 ; prl92-142002 . The harmonic-oscillator wave function with the principal quantum number $n=2$ and the orbital angular momentum $l=0$ is defined as prd71-114008

[TABLE]

The asymptotic models for the twist-2 distribution amplitudes $\psi^{v}$, and the twist-3 distribution amplitudes $\psi^{s}$ for the radially excited $\eta_{c}(2S)$ is parameterized as epjc75-293

[TABLE]

with the function $\mathcal{T}(x)=1-4b^{2}m_{c}wx\bar{x}+m_{c}(x-\bar{x})^{2}/(wx\bar{x})$ and the same normalization conditions as the $B^{0}_{(s)}$ mesons: $\int_{0}^{1}\Psi^{i}(x,b=0)dx=f_{\eta_{c}(2S)}/(2\sqrt{6})$. And we also choose $f_{\eta_{c}(2S)}=0.243^{+0.079}_{-0.111}$ GeV and $w=0.2\pm 0.1$ GeV as in Ref. epjc75-293 .

In the PQCD factorization approach, there are four kinds of emission Feynman diagrams for the $B^{0}_{(s)}\to\eta_{c}(2S)\pi^{+}\pi^{-}$ as illustrated in FIG. 1, where (a) and (b) are factorizable diagrams, while (c) and (d) are the non-factorizable ones. We will use $F^{LL},F^{LR},F^{SP}$ and $M^{LL},M^{LR},M^{SP}$ to describe the contributions of the factorizable [FIG. 1(a) and 1(b)] and non-factorizable [FIG. 1(c) and 1(d)] emission diagrams with the $(V-A)(V-A),(V-A)(V+A)$, and $(S-P)(S+P)$ currents, respectively. The total decay amplitudes for the considered decays can therefore be written as

[TABLE]

where $C_{i}(\mu)(i=1,...,10)$ are Wilson coefficients at the renormalization scale $\mu$. For simplicity, we denote the distribution amplitudes $\Phi_{v\nu=-}^{I=0}(z,\zeta,\omega^{2})$ $[\Phi_{s}^{I=0}(z,\zeta,\omega^{2}),\Phi_{t\nu=+}^{I=0}(z,\zeta,\omega^{2})]$ by $\phi_{0}$ $(\phi_{s},\phi_{\sigma})$ below. From Fig. 1(a) and 1(b), we find

[TABLE]

with a color factor $C_{F}=4/3$. The explicit expressions of the hard functions $h_{a}$ and $h_{b}$, the evolution factors $E_{e}(t_{i})$ including the Sudakov exponents and the hard scales $(t_{a},t_{b})$ can be found for example in Ref. prd91-094024 . Following the same procedure, one can obtain the explicit expressions for decay amplitude $M^{LL}$, $M^{LR}$ and $M^{SP}$ from the evaluation of Fig. 1(c) and 1(d).

III Numerical results

In numerical calculations, the following input parameters are used implicitly. The QCD scale, masses and decay constants are in unit of GeV  PDG2016 :

[TABLE]

The Wolfenstein parameters for the CKM matrix elements read as PDG2016

[TABLE]

The differential branching ratio for the $B^{0}_{(s)}\to\eta_{c}{(2S)}\pi^{+}\pi^{-}$ decay can be written as prd91-094024

[TABLE]

with the $B^{0}_{(s)}$ meson mean lifetime $\tau_{B}$. The kinematic variables $|\overrightarrow{p_{1}}|$ and $|\overrightarrow{p_{3}}|$ denote the magnitudes of the $\pi^{+}$ and $\eta_{c}{(2S)}$ momenta in the center-of-mass frame of the pion pair,

[TABLE]

From our numerical calculations, we find the following results:

• •

In Fig. 2(a), we show the differential branching ratios $d{\cal B}/d\omega$ for $B^{0}_{s}\to{\eta_{c}(2S)}\pi^{+}\pi^{-}$ decay, where the solid curve and the dots curve shows the contribution from $f_{0}(980)$ and $f_{0}(1500)$ is taken into account, respectively. In Fig. 2(b), we show the $\omega$-dependence of the differential decay rate $d{\cal B}/d\omega$ when the BW model (solid curve) and the Bugg’s model (dots curve) are employed. The allowed region of $\omega$ is $4m_{\pi}^{2}\leq\omega^{2}\leq(M_{B}-m_{\eta_{c}(2S)})^{2}$.

• •

For the decays $B^{0}_{s}\to\eta_{c}{(2S)}f_{0}(X)\to\eta_{c}(2S)\pi^{+}\pi^{-}$, when the contribution from $f_{0}(980)$ and $f_{0}(1500)$ are included respectively, the PQCD predictions for the branching ratios ${\cal B}(B^{0}_{s}\to\eta_{c}{(2S)}f_{0}(X)\to\eta_{c}(2S)\pi^{+}\pi^{-})$ are of the form of

[TABLE]

where the first two errors come from the uncertainty $\omega_{B_{s}}=0.50\pm 0.05$ GeV and $a^{I=0}_{2}=0.2\pm 0.2$, the last two errors are from $w=0.2\pm 0.1$ GeV and $f_{\eta_{c}(2S)}=0.243^{+0.079}_{-0.111}$ GeV ( the parameters in the wave function of $\eta_{c}(2S)$). The errors from the uncertainties of other input parameters, for instance the CKM matrix elements, are very small and have been neglected.

By taking into account the S-wave contributions from $f_{0}(980)$ and $f_{0}(1500)$ simultaneously, we find the PQCD prediction for the total branching ratio:

[TABLE]

It is easy to see that the dominant contribution comes from the resonance $f_{0}(980)$ $(82.0\%)$, while the constructive interference between $f_{0}(980)$ and $f_{0}(1500)$ provide $\sim 13\%$ enhancement to the total decay rate. One can read out this information from Fig. 2(a) approximately. When compared with the previous study for $B^{0}_{s}\to\eta_{c}(\pi^{+}\pi^{-})_{s}$ in Ref. epjc76-675 , we find that ${\cal B}(B\to\eta_{c}(2S)[\pi^{+}\pi^{-}]_{s}):{\cal B}(B\to\eta_{c}[\pi^{+}\pi^{-}]_{s})\approx 1:2$.

• •

For $B^{0}\to\eta_{c}{(2S)}f_{0}(500)\to\eta_{c}{(2S)}\pi^{+}\pi^{-}$ decay, the PQCD predictions based on the BW model or the Bugg’s model for the parametrization of the wide $f_{0}(500)$ are the following:

[TABLE]

where the major errors have been added in quadrature. One can see easily that the PQCD predictions obtained by employing the BW model or the Bugg’s model are very similar, the difference is only about $10\%$.

• •

Based on our previous studies of the quasi-two-body B meson decays involving $\rho$ meson plb763-29 , we get to know that the main contribution lies indeed in the region around the pole mass of the $\rho$ resonance. Because $\Gamma_{\eta_{c}(2S)}\approx 11.3$ MeV is much narrow than $\Gamma_{\rho}\approx 149$ MeV, it is reasonable for us to assume that the possible effect due to the narrow width of $\eta_{c}(2S)$ is very small and can be neglected safely.

IV Summary

In summary, we studied the quasi-two-body $B^{0}_{(s)}\to\eta_{c}{(2S)}(\pi^{+}\pi^{-})_{S}$ decays in the PQCD factorization approach by introducing the $S$-wave two-pion distribution amplitudes. For $B^{0}_{s}\to\eta_{c}{(2S)}f_{0}(X)\to\eta_{c}{(2S)}\pi^{+}\pi^{-}$ decay, the contributions from the $S$-wave resonance $f_{0}(980)$ and $f_{0}(1500)$ were taken into account, but the $f_{0}(980)$ provide the dominant contribution to the PQCD prediction: ${\cal B}(B^{0}_{s}\to\eta_{c}{(2S)}(\pi^{+}\pi^{-})_{S})=\left(2.67^{+1.78}_{-1.08}\right)\times 10^{-5}$. For $B^{0}\to\eta_{c}{(2S)}f_{0}(X)\to\eta_{c}{(2S)}\pi^{+}\pi^{-}$ decay, the contribution from $f_{0}(500)$ was taken into account, the PQCD prediction for its decay rate is $\left(1.40^{+0.92}_{-0.56}\right)\times 10^{-6}$ in the BW model or $\left(1.53^{+0.97}_{-0.61}\right)\times 10^{-6}$ in the Bugg’s model. These PQCD predictions for the branching ratios of the considered decays can be measured and tested at the near future LHCb and/or Belle-II experiments.

Acknowledgements.

Many thanks to Hsiang-nan Li, Cai-Dian Lü, Xin Liu, Rui Zhou and Wei Wang for valuable discussions. This work was supported by the National Natural Science Foundation of China under the No. 11235005 and No. 11547038.