Determining the effective Wilson coefficient in terms of and evaluating
Abstract
In this work, we investigate decays of and in a theoretical framework. The calculation is based on the postulation that and are mixtures of pure quark states and . The hadronic matrix elements for and are calculated in the lightfront quark model and the important Wilson coefficient which is closely related to nonperturbative QCD is extracted. However, our numerical results indicate that no matter how to adjust the mixing parameter to reconcile contributions of and , one cannot make the theoretical prediction on to meet the data. Moreover, the new measurement of also negates the mixture scenario. Thus, we conclude that the recent data suggest that is a four quark state ( tetraquark or molecule ), at least the fraction of its pure quark constituents is small.
pacs:
13.25.Hw, 14.40.Cs, 12.39.KiI Introduction
The values of and 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 nonperturbative QCD effects. Based on data, the Collaboration suggests that the pair in arises from the decay of . 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 , and at the tree level, the main contribution is the internal 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
(1) 
where and and are obtained by means of the renormalization group equation (RGE). Sandwiching the lagrangian between the initial and final states, we have
(2)  
It is noted that the term contributes to the decay process via a colorrearrangement. Naively, one can expect by the color rearrangement. However, it was pointed out by some authors Bauer:1986bm (); Cheng:1986an (); Li:1988hr ()“the subleading order in includes not only the nexttoleading vacuuminsertion contribution but also the nonperturbative QCD correction” . Keeping the factorization form, one should replace by where is a parameter(with Cheng’s notationCheng:1986an ()). Even though one can calculate 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, includes some nonperturbative 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 based on data fitting one can obtain information about nonperturbative physics. In Ref.Cheng:1994fr () was fixed by fitting . In this work we instead use to extract the corresponding value. Then we evaluate in terms of the newly obtained . It is worth of noticing that the derivation is based on the postulation that is of a pure structure ( 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 nonperturbative QCD effects, so far one has to invoke certain phenomenological models. In this work, we employ the lightfront 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 and one needs to evaluate hadronic matrix elements and . The structure of is still not very clear yet, for example, JaffeJaffe:1976ig () suggested to be a fourquark state, instead, since the resonance is close to the threshold a molecular structure was considered by Weinstein and IsgurWeinstein:1982gc (). However, the regular structure for still cannot be ruled out Scadron:1982eg (); Klabucar:2001gr (); Cheng:2002ai (). In this paper the scalar meson is regarded as a conventional mixture of and .
In Ref.Cheng:2003sm () the authors studied the formula of and in the LFQM. Actually, the hadronic matrix element can be parameterized by four form factors , , and and whereas for transition it can be parameterized by two form factors and . 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 and . In this model the Gaussiantype wave functions are often used to depict the spatial distribution of the inner constituents in the hadrons. There exists a free parameter in the wavefunction 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 and in LFQM
The leading contributions to and are shown in Fig.1. We will discuss them respectively in the following text.

ii.1 transition in the LFQM
The decay proceeds via at tree level which is an internal emissionAaij:2017hfc () process. The hadronic matrix element is factorized asBauer:1986bm ()
(3) 
where is the factor introduced in the introduction. It is also noted the first term in Eq.(2) can be reorganized via the crossing symmetry to a new form which indeed corresponds to a process where a pair annihilates into an pair. It is very suppressed, so we ignore this term in later calculations.
The transition is a typical process and the involved form factors are defined as
(4) 
with
(5) 
where and are the masses and momenta of the vector (pseudoscalar) states. We also set and .
In Ref.Cheng:2003sm () the authors deduce all the expressions for the form factors , , and in the covariant LFQM. For example
where and are the corresponding quark masses, and are the masses of the initial and final mesons respectively. The wave functions are included in and and they are usually chosen to be Gaussiantype and the parameter in the Gaussian wave function is closely related to the confinement scale and is expected to be of order . and come from the propagators of the inner quark or antiquark of the mesons. is the color factor. The notations , , and are given in the appendix.
One can refer to Eqs.(32) and (B4) of Ref.Cheng:2003sm () for finding the explicit expressions of , and and the corresponding derivations.
ii.2 The transition
The amplitude for is
(7) 
is a typical transition process. The form factors for are defined as
(8) 
As an example, the explicit expression of is presented as
(9)  
where , and are given in the appendix. The explicit expression of is formulated in Ref. Cheng:2003sm ().
As postulated, is a pure state and its quark structure is a superposition state as . Since strange quark in can directly transit into the final scalar meson as a spectator, one can notice that only component of contributes to the transition . In Ref.ElBennich:2008xy (); Li:2008tk (); Ghahramany:2009zz (); Colangelo:2010bg () the transition was studied using Covariant LightFront Dynamics (CLFD), Dispersion Relations (DR), PQCD approach, QCD sum rules (QCDSR) and lightcone QCD sum rules (LCQCDSR). In those articlesElBennich:2008xy (); Li:2008tk (); Ghahramany:2009zz (); Colangelo:2010bg () the form factors of the transition are defined as
(10) 
There are two relations and 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
As discussed in Ref.Cheng:2003sm () the form factors are calculated in the spacelike region with , thus to obtain the physical amplitudes an extension to the timelike region is needed. To make the extension one should write out an analytical expressions for these form factors, and in Ref.Cheng:2003sm () a threeparameter form was suggested
(11) 
where denotes all , , , , and . is the value of at . In the scheme of LFQM one can calculate for the spacelike region (), then through Eq.(11) and can be solved out. When we apply that expression of for with the same and , the form factors are extrapolated to the timelike physical regions. That is a natural analytical extension.
In the two processes, there is a unique matrix element which determines the decay constant of and
(12) 
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 belongs to the latter.
In this scheme is factorized out from the hadronic matrix element and is independent of the matrix element . Moreover, if replacing by which is related to the decay constant , one can study the transition .
Iii Numerical results
In this work, GeV, GeV and GeV are adopted according to Ref. Cheng:2003sm (). and are taken from the databook PDG12 (). The parameter in the wave function is fixed by calculating the corresponding decay constant and comparing it with dataCheng:2003sm (). For the vector meson one can extract the decay constant ( MeV from the data PDG12 () and then GeV is achieved. For the pesudoscalar meson its decay constant ( MeV) coming from the lattice resultAoki:2016frl () is used and we obtain GeV.
In order to calculate the relevant form factors we need to know . For a scalar meson, as long as the masses of the valence quark and antiquark are equal, due to a symmetry with respect to and 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 and in the framework of LFQM Cheng:2003sm (). Following Ref.Cheng:2003sm (); Ke:2009ed (), we set in our numerical computations. The mixing parameter takes a value of which was fixed by fitting the branching ratio of Ke:2009ed () and then the decay constant is MeVBecirevic:2013bsa (). It is also noted, when the semileptonic decay of was measured by the CLEO collaboration, there were no data on available, therefore based on the mixing postulation, such mixing angle was obtained by fitting only the data of . Later in this work, we will show that the recent measurements on nonleptonic decays of and disagree with the mixing picture. We will give more discussions in the last section.
0.292  1.590  1.794  
0.247  1.068  0.310  
0.226  1.764  1.172  
0.303  1.949  1.410  
0.239cos  1.690  0.917  
0.239cos  0.514  0.236 
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 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 nexttoleading order corrections.
CLFD/DR  PQCD  QCDSR  LCQCDSR  this work  
0.40/0.29  0.35  0.12  0.238  0.239 
At first we explore whether using the value () 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 where the errors come from the uncertainties of and , but mainly from . Apparently the estimate is smaller than the data , but as indicated above, the theoretical errors are relatively large, so within a 2 tolerance, one still can count them as being consistent. If we deliberately vary the parameter within a reasonable range, as setting the branching ratio becomes which is satisfactorily consistent with data.
Using the new value of let us evaluate the branching ratio of and we obtain . If one applies this result to make a theoretical prediction on the branching ratio of by assuming the pair fully coming from an onshell , he will notice that the prediction is consistent with the present measured value of . It seems that the pair in mainly comes from . But a discrepancy immediately emerges. In Ref.Scadron:1982eg (); Klabucar:2001gr (); Cheng:2002ai () the authors suggest that the scalar is the complemental state of . Thus the component of which dominantly decays into pairs, would play the same role as that of . If simply setting , we calculate the branching ratio of ( i.e ) again. In that case we obtain which is about three larger than the data. This would raise a conflict between theoretical prediction and experimental data.
Using the decay constant MeVColangelo:2010bg () we also estimate which is slightly larger than the data PDG2016 (), it seems OK, but at the quark level, we have theoretically evaluate and gain it as which leads to be much larger than the upper limit PDG2016 ().
One possibility to pave the gap between theoretical prediction and data is to assume an exotic structure for , namely is a molecule state or tetraquark or a mixture of them. Using data of LHC whose integrated luminosity reaches 3 fb the structure of was studiedAaij:2014emv () and the mixing angle (at 90 C.L.) which is consistent with the prediction of the tetraquark modelStone:2013eaa (); Fleischer:2011au (). Apparently if the upperlimit of the mixing angle is confirmed our prediction on and will be at least twice larger than the data so the structure is disfavored.
Iv Summary
In this work based on the postulation that and are mixture of and we evaluate the decay widths of and in LFQM. At the quark level the two transitions proceed dominantly through an internal emission subprocess . 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 plays a crucial role. By the naive factorization is just related to 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 or decays. Thus extracting the value of 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 and one needs to compute the transition hadronic matrix elements () and () 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 and obtain the value as as taking value of as an input. If one admits that is a free parameter, he can vary it to be and the obtained result is compatible with the data.
Using the new we evaluate the branching ratio of with and obtain it as which is almost consistent with the present data. It seems the only comes from . However, this assumption brings up unacceptable consequence, that since contains a large fraction of (proportional to ), the contribution of becomes untolerably large as , this number would lead to the branching ratio of to be roughly which is roughly 3 times larger than the measured value.
Moreover, the recent measurements indicate the branching ratio of is while . The data imply that if the mixture scenario is correct, the mixing angle should be smaller than instead of the large . In other words implies that the fraction of in should be very tiny.
If we accept the small mixing angle , we obtain to be , namely is consistent with the allegation that the final pair in is totally from , 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 is at least 3 times larger than the data.
A synthesis of the measured branching ratio of and the data determines no room for a subprocess . 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 are not a mixture of and , but could be a four quark state: molecule as Isgur et al. suggested or a tetraquark.
We suggest the experimentalists to carry out a more precise measurement on the where the invariant mass of would clearly tell us if mainly come from .
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 where and are the momenta of the offshell quark and antiquark and
(13) 
with and are internal variables and .
The variables , , , , and are defined as
(14) 
where
(15) 
with .
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