Determining the effective Wilson coefficient $a_2$ in terms of $BR(B_s\to \eta_c\phi)$ and evaluating $BR(B_s\to \eta_cf_0(980))$

TL;DR

This paper investigates $B_s$ meson decays into $ o \,eta_c$ and other states, extracting the Wilson coefficient $a_2$ using a light-front quark model, and concludes that $f_0(980)$ is likely a tetraquark or molecular state based on experimental data.

Contribution

The study provides a theoretical calculation of hadronic matrix elements and extracts the Wilson coefficient $a_2$, offering insights into the quark structure of $f_0(980)$ and its implications for $B_s$ decay predictions.

Findings

Theoretical predictions do not match experimental data regardless of mixing adjustments.

Recent measurements suggest $f_0(980)$ is predominantly a four-quark state.

The Wilson coefficient $a_2$ is extracted within the light-front quark model framework.

Abstract

In this work, we investigate decays of $B_s\to \eta_c\phi$ and $B_s\to \eta_cf_0(980)$ in a theoretical framework. The calculation is based on the postulation that $f_0(980)$ and $f_0(500)$ are mixtures of pure quark states ${1\over\sqrt 2} (\bar uu+\bar dd)$ and $\bar ss$. The hadronic matrix elements for $B_s\to \phi$ and $B_s\to f_0(980)$ are calculated in the light-front quark model and the important Wilson coefficient $a_2$ which is closely related to non-perturbative QCD is extracted. However, our numerical results indicate that no matter how to adjust the mixing parameter to reconcile contributions of $f_0(980)$ and $f_0(500)$, one cannot make the theoretical prediction on $B_s\to \eta_c+\pi^+\pi^-$ to meet the data. Moreover, the new measurement of $BR(B_s\to J/\psi+f_0(500))<1.7\times 10^{-6}$ also negates the mixture scenario. Thus, we conclude that the recent data suggest that $f_0(980)$ is a four quark state ( tetraquark or $K\bar K$ molecule ), at least the fraction of its pure quark constituents is small.

Tables

Table 1. Table 1: the parameters F 1 ​ ( 0 ) , a , b subscript 𝐹 1 0 𝑎 𝑏 F_{1}(0),\ a,\ b are defined in Eq. ( 11 ).

$F\left(q^2\right)$	$F_1\left(0\right)$	$a$	$b$
$A_0$	0.292	1.590	1.794
$A_1$	0.247	1.068	0.310
$A_2$	0.226	1.764	1.172
$V$	0.303	1.949	1.410
$F_1$	0.239cos$\theta$	1.690	0.917
$F_0$	0.239cos$\theta$	0.514	0.236

Table 2. Table 2: the B s → f 0 ​ ( 980 ) → subscript 𝐵 𝑠 subscript 𝑓 0 980 B_{s}\to f_{0}(980) form factor F 0 ​ ( q 2 = 0 ) subscript 𝐹 0 superscript 𝑞 2 0 F_{0}(q^{2}=0) with cos ⁡ θ 𝜃 \cos\theta =1.

	CLFD/DR	PQCD	QCDSR	LCQCDSR	this work
$F_0$	0.40/0.29	0.35	0.12	0.238	0.239

Equations

$$\mathcal{L}=c_1\bar{c}{\gamma }_{\mu }(1-{\gamma }_5)b\bar{s}{\gamma }^{\mu }(1-{\gamma }_5)c+c_2\bar{c}{\gamma }_{\mu }(1-{\gamma }_5)c\bar{s}{\gamma }^{\mu }(1-{\gamma }_5)b,$$

$$<{\eta }_c\varphi (f_0(980)|\mathcal{L}|B_s>$$

$$\mathcal{A}=\frac{G_F{V}_{cs}^{\ast }{V}_{bc}{a}_2}{\sqrt{2}}⟨{\eta }_c\varphi |(\bar{c}c{)}_{V-A}(\bar{s}b{)}_{V-A}|B_s⟩=\frac{G_F{V}_{cs}^{\ast }{V}_{bc}{a}_2}{\sqrt{2}}⟨{\eta }_c|(\bar{c}c{)}_{V-A}|0⟩⟨\varphi |(\bar{s}b{)}_{V-A}|B_s⟩,$$

$$⟨V(P^{\prime \prime },{\epsilon }^{\prime \prime })|V_{\mu }|P(P^{\prime })⟩$$

$$⟨V(P^{\prime \prime },{\epsilon }^{\prime \prime })|A_{\mu }|P(P^{\prime })⟩$$

$$A_3^{PV}(q^2)=\frac{M^{\prime }+M^{\prime \prime }}{2M^{\prime \prime }}{A}_1^{PV}(q^2)-\frac{M^{\prime }-M^{\prime \prime }}{2M^{\prime \prime }}{A}_2^{PV}(q^2),$$

$$V(q^2)=$$

$$\mathcal{A}=\frac{G_F{V}_{cs}^{\ast }{V}_{bc}{a}_2}{\sqrt{2}}⟨{\eta }_c{f}_0(980)|(\bar{c}c{)}_{V-A}(\bar{s}b{)}_{V-A}|B_s⟩=\frac{G_F{V}_{cs}^{\ast }{V}_{bc}{a}_2}{\sqrt{2}}⟨{\eta }_c|(\bar{c}c{)}_{V-A}|0⟩⟨f_0(980)|(\bar{s}b{)}_{V-A}|B_s⟩.$$

$$⟨S(P^{\prime \prime })|A_{\mu }|P(P^{\prime })⟩=i\left[u_{+}(q^2)p_{\mu }+u_{-}(q^2)q_{\mu }\right].$$

$$u_{+}(q^2)=$$

$$⟨S(P^{\prime \prime })|A_{\mu }|P(P^{\prime })⟩=-i\left\{F_1(q^2)[{\mathcal{P}}_{\mu }-\frac{m_{B_s}^2-m_{f_0(980)}^2}{q^2}{q}_{\mu }]+F_0(q^2)\frac{m_{B_s}^2-m_{f_0(980)}^2}{q^2}{q}_{\mu }\right\}.$$

$$F(q^2)=\frac{F(0)}{\left[1-a\left(\frac{q^2}{M_{B_s}^2}\right)+b{\left(\frac{q^2}{M_{B_s}^2}\right)}^2\right]}.$$

$$⟨{\eta }_c(p)|A_{\mu }|0⟩=i{f}_{{\eta }_c}{p}_{\mu }.$$

$${p^{\prime }}_1^{+}=x_1{P^{\prime }}^{+},{p^{\prime }}_2^{+}=x_2{P}^{+},$$

$${p^{\prime }}_{1\perp }=x_1{P^{\prime }}_{\perp }+p_{\perp }^{\prime },{p^{\prime }}_{2\perp }=x_2{P^{\prime }}_{\perp }-p_{\perp }^{\prime },$$

$${M^{\prime }}_0^2=\frac{{p^{\prime }}_{\perp }^2+{m^{\prime }}_1^2}{x_1}+\frac{{p^{\prime }}_{\perp }^2+{m^{\prime }}_2^2}{x_2},$$

$$\widetilde{{M^{\prime }}_0}=\sqrt{{M^{\prime }}_0^2-({m^{\prime }}_1-{m^{\prime }}_2{)}^2},$$

$$h_p^{\prime }=(M^{\prime 2}-M_0^{\prime 2})\sqrt{\frac{x_1{x}_2}{N_c}}\frac{1}{\sqrt{2}\widetilde{M_0^{\prime }}}{\phi }^{\prime },$$

$$h_s^{\prime }=(M^{\prime 2}-M_0^{\prime 2})\sqrt{\frac{x_1{x}_2}{N_c}}\frac{\widetilde{M_0^{\prime }}}{2\sqrt{6}{M}_0^{\prime }}{\phi }_p^{\prime },$$

$$\widehat{N_1^{\prime }}=x_1(M^{\prime 2}-M_0^{\prime 2}),$$

$$\widehat{N_1^{\prime \prime }}=x_1(M^{\prime \prime 2}-M_0^{\prime \prime 2}).$$

$${\phi }^{\prime }=4(\frac{\pi }{{\beta }^2}{)}^{3/4}\sqrt{\frac{d{p^{\prime }}_z}{d{x}_2}}\exp (-\frac{{p^{\prime }}_z^2+{p^{\prime }}_{\perp }^2}{2{\beta }^2}),{\phi }_p^{\prime }=\sqrt{2/\beta }{\phi }^{\prime },$$

Full text

I Introduction

The values of $BR(B_{s}\to\eta_{c}\phi)=(5.01\pm 0.53\pm 0.27\pm 0.63)\times 10^{-4}$ and $BR(B_{s}\to\eta_{c}\pi^{+}\pi^{-})=1.76\pm 0.59\pm 0.12\pm 0.29)\times 10^{-4}$ recently measured by the LHCb collaboration Aaij:2017hfc have stimulated new vigor for studying the hadron structures and the decay mechanism which is closely related to the non-perturbative QCD effects. Based on data, the Collaboration suggests that the $\pi^{+}\pi^{-}$ pair in $B_{s}\to\eta_{c}\pi^{+}\pi^{-}$ arises from the decay of $f_{0}(980)$. To understand the data and look for some hints about involved physics, corresponding theoretical calculations are needed. The traditional scheme is using the heavy quark effective theory (HQET) Isgur:1989vq ; Georgi:1990um and naive factorization which is an old issue, but still applicable in parallel to the fancy theories such as SCET and others.

The subprocess is $b\to c\bar{c}s$, and at the tree level, the main contribution is the internal $W-$emission while the light quark serves as a spectator. For a completeness, let us briefly retrospect the standard procedures of applying HQET. In the HQET, the corresponding lagrangian is written as

[TABLE]

where $c_{1}={1\over 2}(c_{+}+c_{-})$ and $c_{2}={1\over 2}(c_{+}-c_{-})$ and $c_{+},c_{-}$ are obtained by means of the re-normalization group equation (RGE). Sandwiching the lagrangian between the initial and final states, we have

[TABLE]

It is noted that the $c_{1}O_{1}$ term contributes to the decay process via a color-re-arrangement. Naively, one can expect $a_{2}=c_{2}+1/3c_{1}$ by the color rearrangement. However, it was pointed out by some authors Bauer:1986bm ; Cheng:1986an ; Li:1988hr “the sub-leading order in $1/N_{c}$ includes not only the next-to-leading vacuum-insertion contribution but also the nonperturbative QCD correction” . Keeping the factorization form, one should replace $a_{2}=c_{2}+c_{1}/3$ by $a_{2}=c_{2}+c_{1}/3+\epsilon_{a}/2$ where $\epsilon_{a}$ is a parameter(with Cheng’s notationCheng:1986an ). Even though one can calculate $\epsilon_{a}$ in terms of some models Li:1988hr , the result is not accurate, therefore, generally one should phenomenologically fix it by fitting the well measured data. Our work is exactly along the line. This issue was first discussed in Ref.Bauer:1986bm . In fact, $a_{2}$ includes some non-perturbative QCD effects so it is not universal for the different channels of the D or B decaysCheng:1994fr as shown above. Definitely, determining the value of $a_{2}$ based on data fitting one can obtain information about non-perturbative physics. In Ref.Cheng:1994fr $a_{2}=0.23\pm 0.06$ was fixed by fitting $BR(B\to D^{(*)}\pi(\rho))$ . In this work we instead use $B_{s}\to\eta_{c}\phi$ to extract the corresponding $a_{2}$ value. Then we evaluate $BR(B_{s}\to\eta_{c}f_{0}(980))$ in terms of the newly obtained $a_{2}$. It is worth of noticing that the derivation is based on the postulation that $f_{0}(980)$ is of a pure $\bar{q}q$ structure ($q$ stands as u,d and s quarks). We will come to this issue for some details in the last section.

In order to calculate the decay width under the factorization assumption one needs to evaluate the hadronic transition matrix element between two mesons. Since the transition is governed by the non-perturbative QCD effects, so far one has to invoke certain phenomenological models. In this work, we employ the light-front quark model(LFQM). This relativistic model has been thoroughly discussed in literatures Jaus:1999zv ; Cheng:2003sm and applied to study several hadronic transition processesWei:2009nc ; Ke:2012wa ; Ke:2013yka ; Choi:2007se ; Hwang:2006cua ; Ke:2010vn ; Ke:2011jf ; Ke:2013zs ; Ke:2011mu . The results obtained in this framework qualitatively agree with the data for all the concerned processes.

For the transitions $B_{s}\to\eta_{c}\phi$ and $B_{s}\to\eta_{c}f_{0}(980)$ one needs to evaluate hadronic matrix elements $B_{s}\to\phi$ and $B_{s}\to f_{0}(980)$. The structure of $f_{0}(980)$ is still not very clear yet, for example, JaffeJaffe:1976ig suggested $f_{0}(980)$ to be a four-quark state, instead, since the resonance is close to the $K\bar{K}$ threshold a $K\bar{K}$ molecular structure was considered by Weinstein and IsgurWeinstein:1982gc . However, the regular $s\bar{s}$ structure for $f_{0}(980)$ still cannot be ruled out Scadron:1982eg ; Klabucar:2001gr ; Cheng:2002ai . In this paper the scalar meson $f_{0}(980)$ is regarded as a conventional mixture of ${1\over\sqrt{2}}(\bar{u}u+\bar{d}d)$ and $s\bar{s}$.

In Ref.Cheng:2003sm the authors studied the formula of $0^{-}\to 1^{-}$ and $0^{-}\to 0^{+}$ in the LFQM. Actually, the $0^{-}\to 1^{-}$ hadronic matrix element can be parameterized by four form factors $A_{0}$, $A_{1}$, $A_{2}$ and $V$ and whereas for $0^{-}\to 0^{+}$ transition it can be parameterized by two form factors $F_{0}$ and $F_{1}$. Their detailed expressions obtained in LFQM can be found in Ref. Cheng:2003sm . In this work, we will calculate these form factors numerically. With the form factors one can further evaluate the transition widths of $B_{s}\to\eta_{c}\phi$ and $B_{s}\to\eta_{c}f_{0}(980)$. In this model the Gaussian-type wave functions are often used to depict the spatial distribution of the inner constituents in the hadrons. There exists a free parameter $\beta$ in the wave-function beside the masses of the constituents. One should fix it by comparing the decay constant of the involved meson which is either theoretically calculated in LFQM with data.

This paper is organized as following: after this introduction, we list all relevant formulas in Sec.II, and then in Sec. III, we present our numerical results along with all inputs which are needed for the numerical computations. In the last section we draw our conclusion and make a brief discussion.

II The formulas for the decays of $B_{s}\to\eta_{c}\phi$ and $B_{s}\to\eta_{c}f_{0}(980)$ in LFQM

The leading contributions to $B_{s}\to\eta_{c}\phi$ and $B_{s}\to\eta_{c}f_{0}(980)$ are shown in Fig.1. We will discuss them respectively in the following text.

II.1 $B_{s}\rightarrow\phi$ transition in the LFQM

The decay proceeds via $b\to\bar{c}c\bar{s}$ at tree level which is an internal $W$-emissionAaij:2017hfc process. The hadronic matrix element is factorized asBauer:1986bm

[TABLE]

where $a_{2}$ is the factor introduced in the introduction. It is also noted the first term in Eq.(2) $<\eta_{c}\phi(f_{0}(980))|\bar{c}\gamma_{\mu}(1-\gamma_{5})c|0><0|\bar{s}\gamma^{\mu}(1-\gamma_{5})b|B_{s}>$ can be re-organized via the crossing symmetry to a new form which indeed corresponds to a process where a $q\bar{q}$ pair annihilates into an $c\bar{c}$ pair. It is very suppressed, so we ignore this term in later calculations.

The transition $B_{s}\rightarrow\phi$ is a typical process and the involved form factors are defined as

[TABLE]

with

[TABLE]

where $M^{\prime}(M^{\prime\prime})$ and $P^{\prime}(P^{\prime\prime})$ are the masses and momenta of the vector (pseudoscalar) states. We also set $\mathcal{P}=P^{\prime}+P^{\prime\prime}$ and $q=P^{\prime}-P^{\prime\prime}$.

In Ref.Cheng:2003sm the authors deduce all the expressions for the form factors $A_{0}$, $A_{1}$, $A_{2}$ and $V$ in the covariant LFQM. For example

[TABLE]

where $m^{\prime}_{1},m^{\prime\prime}_{1}$ and $m_{2}$ are the corresponding quark masses, $M^{\prime}$ and $M^{\prime\prime}$ are the masses of the initial and final mesons respectively. The wave functions are included in $h^{\prime}_{p}$ and $h^{\prime\prime}_{v}$ and they are usually chosen to be Gaussian-type and the parameter $\beta$ in the Gaussian wave function is closely related to the confinement scale and is expected to be of order $\Lambda_{\rm QCD}$. $N^{\prime}_{1}$ and $N^{\prime\prime}_{1}$ come from the propagators of the inner quark or antiquark of the mesons. $N_{c}=3$ is the color factor. The notations $N^{\prime}_{1}$, $N^{\prime\prime}_{1}$, $h^{\prime}_{p}$ and $h^{\prime\prime}_{v}$ are given in the appendix.

One can refer to Eqs.(32) and (B4) of Ref.Cheng:2003sm for finding the explicit expressions of $A_{0}$, $A_{1}$ and $A_{2}$ and the corresponding derivations.

II.2 The transition $B_{s}\to f_{0}(980)$

The amplitude for $B_{s}\to\eta_{c}f_{0}(980)$ is

[TABLE]

$B_{s}\rightarrow f_{0}(980)$ is a typical $P\rightarrow S$ transition process. The form factors for $P\rightarrow S$ are defined as

[TABLE]

As an example, the explicit expression of $u_{+}$ is presented as

[TABLE]

where $h^{\prime\prime}_{s}$, $M^{\prime}_{0}$ and $M^{\prime\prime 2}_{0}$ are given in the appendix. The explicit expression of $u_{-}(q^{2})$ is formulated in Ref. Cheng:2003sm .

As postulated, $f_{0}(980)$ is a pure $q\bar{q}$ state and its quark structure is a superposition state as $|f_{0}(980)>=\sin\theta|{1\over\sqrt{2}}(u\bar{u}+d\bar{d})>+\cos\theta|s\bar{s}>$. Since strange quark $s$ in $B_{s}$ can directly transit into the final scalar meson as a spectator, one can notice that only $s\bar{s}$ component of $f_{0}(980)$ contributes to the transition $B_{s}\to f_{0}(980)$. In Ref.ElBennich:2008xy ; Li:2008tk ; Ghahramany:2009zz ; Colangelo:2010bg the transition was studied using Covariant Light-Front Dynamics (CLFD), Dispersion Relations (DR), PQCD approach, QCD sum rules (QCDSR) and light-cone QCD sum rules (LCQCDSR). In those articlesElBennich:2008xy ; Li:2008tk ; Ghahramany:2009zz ; Colangelo:2010bg the form factors of the transition are defined as

[TABLE]

There are two relations $F_{1}=-u_{+}(q^{2})$ and $F_{0}=-[u_{+}(q^{2})+\frac{q^{2}}{m_{B_{s}}^{2}-m^{2}_{f_{0}(980)}}u_{-}(q^{2})]$ which associate the conventional form factors used in literature with that we introduced above.

II.3 Extension of the form factors to the physical region and the decay constant of $\eta_{c}$

As discussed in Ref.Cheng:2003sm the form factors are calculated in the space-like region with $q^{+}=0$, thus to obtain the physical amplitudes an extension to the time-like region is needed. To make the extension one should write out an analytical expressions for these form factors, and in Ref.Cheng:2003sm a three-parameter form was suggested

[TABLE]

where $F(q^{2})$ denotes all $A_{1}{(q^{2})}$, $A_{2}{(q^{2})}$, $A_{3}{(q^{2})}$, $V{(q^{2})}$, $F_{1}{(q^{2})}$ and $F_{0}{(q^{2})}$. $F(0)$ is the value of $F(q^{2})$ at $q^{2}=0$. In the scheme of LFQM one can calculate $F(q^{2})$ for the space-like region ($q^{2}<0$), then through Eq.(11) $a$ and $b$ can be solved out. When we apply that expression of $F(q^{2})$ for $q^{2}>0$ with the same $a$ and $b$, the form factors are extrapolated to the time-like physical regions. That is a natural analytical extension.

In the two processes, there is a unique matrix element $\langle\eta_{c}|(\bar{c}c)_{V-A}|0\rangle$ which determines the decay constant of $\eta_{c}$ and

[TABLE]

Some mesons’ decay constants can be fixed by fitting data, whereas others must be calculated in terms of phenomenological models or the lattice because no data are available so far. Here the case for $f_{\eta_{c}}$ belongs to the latter.

In this scheme $\langle\eta_{c}|(\bar{c}c)_{V-A}|0\rangle$ is factorized out from the hadronic matrix element and is independent of the matrix element $\langle f_{0}(980)|(\bar{s}b)_{V-A}|B_{s}\rangle$. Moreover, if replacing $\langle\eta_{c}|(\bar{c}c)_{V-A}|0\rangle$ by $\langle J/\psi|(\bar{c}c)_{V-A}|0\rangle$ which is related to the decay constant $f_{J/\psi}$, one can study the transition $B_{s}\to J/\psi f_{0}(980)$.

III Numerical results

In this work, $m_{s}=0.37$ GeV, $m_{c}=1.4$ GeV and $m_{b}=4.64$ GeV are adopted according to Ref. Cheng:2003sm . $V_{cs}$ and $V_{bc}$ are taken from the databook PDG12 . The parameter $\beta$ in the wave function is fixed by calculating the corresponding decay constant and comparing it with dataCheng:2003sm . For the vector meson $\phi$ one can extract the decay constant ($227.7\pm 1.2)$ MeV from the data $BR(\phi\to e^{+}e^{-})((2.954\pm 0.030)\times 10^{-4})$PDG12 and then $\beta_{\phi}=(0.3001\pm 0.0010)$ GeV is achieved. For the pesudoscalar meson $B_{s}$ its decay constant ($228.4\pm 3.2$ MeV) coming from the lattice resultAoki:2016frl is used and we obtain $\beta_{B_{s}}=(0.6165\pm 0.0072)$ GeV.

In order to calculate the relevant form factors we need to know $\beta^{s}_{f_{0}}$. For a scalar meson, as long as the masses of the valence quark and antiquark are equal, due to a symmetry with respect to $x_{1}$ and $x_{2}$ which are their shares of momenta in the meson, the decay constant becomes zero as it should be. It is shown by the integral over $x_{1}$ and $x_{2}$ in the framework of LFQM Cheng:2003sm . Following Ref.Cheng:2003sm ; Ke:2009ed , we set $\beta^{s}_{f_{0}}=0.3$ in our numerical computations. The mixing parameter $\theta$ takes a value of $(56\pm 6)^{\circ}$ which was fixed by fitting the branching ratio of $D_{s}\to f_{0}(980)e^{+}\nu_{e}$Ke:2009ed and then the decay constant is $f_{\eta_{c}}=(387\pm 7)$ MeVBecirevic:2013bsa . It is also noted, when the semileptonic decay of $D_{s}\to f_{0}(980)+e^{+}\nu_{e}$ was measured by the CLEO collaboration, there were no data on $D_{s}\to f_{0}(500)+e^{+}\nu_{e}$ available, therefore based on the mixing postulation, such mixing angle was obtained by fitting only the data of $B_{s}\to f_{0}(980)e^{+}\nu_{e}$. Later in this work, we will show that the recent measurements on non-leptonic decays of $B_{s}\to f_{0}(980)+X$ and $B_{s}\to f_{0}(500)+X$ disagree with the mixing picture. We will give more discussions in the last section.

In Tab.1 we present the parameters in those form factors when all the input parameters are taking the central values given elsewhere. In Ref.ElBennich:2008xy ; Li:2008tk ; Ghahramany:2009zz ; Colangelo:2010bg the transition $B_{s}\to f_{0}(980)$ were also studied and we collect the results in Tab.2. Our prediction is close to the value -0.238 obtained by the authors of Colangelo:2010bg which includes the next-to-leading order corrections.

At first we explore whether using the value $a_{2}$ ($0.23\pm 0.06$) fixed in Ref.Cheng:1994fr the predicted decay width can meet the present data. With all the form factors and parameters as given above, we obtain the branching ratio $BR(B_{s}\to\eta_{c}\phi)=(2.795\pm 1.652)\times 10^{-4}$ where the errors come from the uncertainties of $\beta_{B_{s}},\,\beta_{\phi},\,f_{\eta_{c}}$ and $a_{2}$, but mainly from $a_{2}$. Apparently the estimate is smaller than the data $(5.01\pm 0.53\pm 0.27\pm 0.63)\times 10^{-4}$, but as indicated above, the theoretical errors are relatively large, so within a 2$\sigma$ tolerance, one still can count them as being consistent. If we deliberately vary the parameter $a_{2}$ within a reasonable range, as setting $a_{2}=0.308\pm 0.029$ the branching ratio $BR(B_{s}\to\eta_{c}\phi)$ becomes $(5.012\pm 0.863)\times 10^{-4}$ which is satisfactorily consistent with data.

Using the new value of $a_{2}$ let us evaluate the branching ratio of $B_{s}\to\eta_{c}+f_{0}(980)$ and we obtain $BR(B_{s}\to\eta_{c}f_{0}(980))=(1.591\pm 0.568)\times 10^{-4}$. If one applies this result to make a theoretical prediction on the branching ratio of $B_{s}\to\eta_{c}\pi^{+}\pi^{-}$ by assuming the $\pi^{+}\pi^{-}$ pair fully coming from an on-shell $f_{0}(980)$, he will notice that the prediction is consistent with the present measured value of $BR(B_{s}\to\eta_{c}\pi^{+}\pi^{-})$. It seems that the $\pi^{+}\pi^{-}$ pair in $B_{s}\to\eta_{c}\pi^{+}\pi^{-}$ mainly comes from $f_{0}(980)$. But a discrepancy immediately emerges. In Ref.Scadron:1982eg ; Klabucar:2001gr ; Cheng:2002ai the authors suggest that the scalar $f_{0}(500)(\sigma)$ is the complemental state of $f_{0}(980)$. Thus the $s\bar{s}$ component of $f_{0}(500)$ which dominantly decays into $\pi\pi$ pairs, would play the same role as that of $f_{0}(980)$. If simply setting $\theta=0$, we calculate the branching ratio of $B_{s}\to\eta_{c}0^{+}(s\bar{s})$ ( i.e $B_{s}\to\eta_{c}\pi^{+}\pi^{-}$ ) again. In that case we obtain $BR(B_{s}\to\eta_{c}\pi^{+}\pi^{-})=(5.089\pm 1.022)\times 10^{-4}$ which is about three larger than the data. This would raise a conflict between theoretical prediction and experimental data.

Using the decay constant $f_{\psi}=416.3\pm 5.3$ MeVColangelo:2010bg we also estimate $BR(B_{s}\to J/\psi f_{0}(980))=(1.727\pm 0.615)\times 10^{-4}$ which is slightly larger than the data $(1.19\pm 0.22)\times 10^{-4}$PDG2016 , it seems OK, but at the quark level, we have theoretically evaluate $BR(B_{s}\to J/\psi 0^{+}(s\bar{s}))$ and gain it as $(5.523\pm 1.103)\times 10^{-4}$ which leads $BR(B_{s}\to J/\psi f_{0}(500))$ to be much larger than the upper limit $BR(B_{s}\to J/\psi f_{0}(500))<1.7\times 10^{-6}$PDG2016 .

One possibility to pave the gap between theoretical prediction and data is to assume an exotic structure for $f_{0}(980)$, namely is a $K\bar{K}$ molecule state or tetraquark or a mixture of them. Using data of LHC whose integrated luminosity reaches 3 fb1 the structure of $\bar{B}^{0}_{s}\to J/\psi\pi^{+}\pi^{-}$ was studiedAaij:2014emv and the mixing angle $\theta<7.7^{\circ}$(at 90$\%$ C.L.) which is consistent with the prediction of the tetraquark modelStone:2013eaa ; Fleischer:2011au . Apparently if the upper-limit of the mixing angle is confirmed our prediction on $BR(B_{s}\to\eta_{c}f_{0}(980))$ and $BR(B_{s}\to J/\psi f_{0}(980))$ will be at least twice larger than the data so the $q\bar{q}$ structure is disfavored.

IV Summary

In this work based on the postulation that $f_{0}(980)$ and $f_{0}(500)$ are mixture of $\frac{1}{\sqrt{2}}(\bar{u}u+\bar{d}d)$ and $\bar{s}s$ we evaluate the decay widths of $B_{s}\to\eta_{c}\phi$ and $B_{s}\to\eta_{c}f_{0}(980)$ in LFQM. At the quark level the two transitions proceed dominantly through an internal $W-$ emission sub-process $b\to\bar{c}c\bar{s}$. By the factorization assumption the hadronic matrix element can factorized into a simple transition matrix element multiplying by the decay constant of the involved pseudoscalar meson. In this scenario the effective Wilson coefficient factor $a_{2}$ plays a crucial role. By the naive factorization $a_{2}$ is just related to $c_{2}+c_{1}/3$ due to the color rearrangement. However such naive combination is only a rough approximation because some nonperturbative QCD effects would get involved for a complete color rearrangement. The new contribution is not universal for $B$ or $D$ decays. Thus extracting the value of $a_{2}$ will provide us with information about the nonperturbative QCD effects in the corresponding decays and even more.

In order to calculate the decay widths of $B_{s}\to\eta_{c}\phi$ and $B_{s}\to\eta_{c}f_{0}(980)$ one needs to compute the transition hadronic matrix elements $B_{s}\to\phi$ ($0^{-}\to 1^{-}$) and $B_{s}\to f_{0}(980)$ ($0^{-}\to 0^{+}$) which can be parametrized by several form factors. The phenomenological model LFQM is employed to calculate these form factors in this work. With the form factors and all the input parameters we evaluate the rate of $B_{s}\to\eta_{c}\phi$ and obtain the value as $\mathcal{BR}(B_{s}\to\eta_{c}\phi)=(2.795\pm 1.652)\times 10^{-4}$ as $a_{2}$ taking value of $0.23\pm 0.06$ as an input. If one admits that $a_{2}$ is a free parameter, he can vary it to be $0.308\pm 0.029$ and the obtained result is compatible with the data.

Using the new $a_{2}$ we evaluate the branching ratio of $B_{s}\to\eta_{c}f_{0}(980)$ with $\theta=56\pm 6^{\circ}$ and obtain it as $BR(B_{s}\to\eta_{c}f_{0}(980))=(1.591\pm 0.568)\times 10^{-4}$ which is almost consistent with the present data. It seems the $\pi^{+}\pi^{-}$ only comes from $f_{0}(980)$. However, this assumption brings up unacceptable consequence, that since $f_{0}(500)$ contains a large fraction of $\bar{s}s$ (proportional to $\sin^{2}\theta$), the contribution of $B_{s}\to\eta_{c}f_{0}(500)\to\eta_{c}\pi^{+}\pi^{-}$ becomes un-tolerably large as $(3.498\pm 1.249)\times 10^{-4}$, this number would lead to the branching ratio of $B_{s}\to\eta_{c}\pi^{+}\pi^{-}$ to be roughly $5\times 10^{-4}$ which is roughly 3 times larger than the measured value.

Moreover, the recent measurements indicate the branching ratio of $B_{s}\to J/\psi+f_{0}(980)$ is $1.19\times 10^{-4}$ while $BR(B_{s}\to J/\psi+f_{0}(500)<1.7\times 10^{-6}$. The data imply that if the mixture scenario is correct, the mixing angle should be smaller than $7.7^{\circ}$ instead of the large $56^{\circ}$. In other words $BR(B_{s}\to J/\psi+f_{0}(500)<1.7\times 10^{-6}$ implies that the fraction of $\bar{s}s$ in $f_{0}(500)$ should be very tiny.

If we accept the small mixing angle $\theta\sim 7.7^{\circ}$, we obtain $BR(B_{s}\to\eta_{c}f_{0}(980)\to\eta_{c}\pi^{+}\pi^{-})$ to be $4.998\times 10^{-4}$, namely is consistent with the allegation that the final $\pi^{+}\pi^{-}$ pair in $B_{s}\to\eta_{c}\pi^{+}\pi^{-}$ is totally from $f_{0}(980)$, however, the theoretical picture is surely disagreed by the data.

It is an obvious contradiction that in the mixing scenario, no matter what value the mixing angle is adopted, the calculated branching ratio for $B_{s}\to\eta_{c}\pi^{+}\pi^{-}$ is at least 3 times larger than the data.

A synthesis of the measured branching ratio of $BR(B_{s}\to\eta_{c}\pi^{+}\pi^{-})\sim 1.76\times 10^{-4}$ and the data $BR(B_{s}\to J/\psi f_{0}(500))<7.7^{\circ}$ determines no room for a subprocess $B_{s}\to\eta_{c}f_{0}(500)\to\eta_{c}\pi^{+}\pi^{-}$. Namely if the mixing scenario is adopted, no matter choosing what value for the mixing angle, one cannot let the theoretically prediction meet the data.

Therefore, under a complete consideration, one should draw a conclusion that the main contents of $f_{0}(980)$ are not a mixture of $(\bar{u}u+\bar{d}d)/\sqrt{2}$ and $\bar{s}s$, but could be a four quark state: $K\bar{K}$ molecule as Isgur et al. suggested or a tetraquark.

We suggest the experimentalists to carry out a more precise measurement on the $B_{s}\to\eta_{c}\pi^{+}\pi^{-}$ where the invariant mass of $\pi^{+}\pi^{-}$ would clearly tell us if $\pi^{+}\pi^{-}$ mainly come from $f_{0}(980)$.

Acknowledgement

This work is supported by the National Natural Science Foundation of China (NNSFC) under the contract No. 11375128 and 11675082.

Appendix A Notations

Here we list some variables appearing in the context. The incoming meson in Fig. 1 has the momentum $P^{\prime}=p^{\prime}_{1}+p^{\prime}_{2}$ where $p^{\prime}_{1}$ and $p^{\prime}_{2}$ are the momenta of the off-shell quark and antiquark and

[TABLE]

with $x_{i}$ and $p^{\prime}_{\perp}$ are internal variables and $x_{1}+x_{2}=1$.

The variables ${M^{\prime}}_{0}$, $\tilde{{M^{\prime}}_{0}}$, $h_{p}^{\prime}$, $h_{s}^{\prime}$, $\hat{N_{1}^{\prime}}$ and $\hat{N_{1}^{\prime\prime}}$ are defined as

[TABLE]

where

[TABLE]

with ${p^{\prime}}_{z}=\frac{x_{2}{M^{\prime}}_{0}}{2}-\frac{{m^{\prime}}_{2}^{2}+{p^{\prime}}^{2}_{\perp}}{2x_{2}{M^{\prime}}_{0}}$.