Analysis on heavy quarkonia transitions with pion emission in terms of the QCD multipoleexpansion and determination of mass spectra of hybrids

Hong-Wei Ke, Jian Tang, Xi-Qing Hao, and Xue-Qian LiDepartment of Physics, Nankai University, Tianjin 300071, China

(Received 21 June 2007; published 25 October 2007)

One of the most important tasks in high energy physics is the search for the exotic states, such asglueball, hybrid, and multiquark states. The transitions �ns� ! �ms� � �� and ��ns� ! ��ms� � ��attract great attention because they may reveal characteristics of hybrids. In this work, we analyze thosetransition modes in terms of the theoretical framework established by Yan and Kuang. It is interesting tonotice that the intermediate states between the two gluon emissions are hybrids, therefore by fitting thedata, we are able to determine the mass spectra of hybrids. The ground hybrid states are predicted as4.23 GeV (for charmonium) and 10.79 GeV (for bottonium) which do not correspond to any statesmeasured in recent experiments, thus it may imply that, very possibly, hybrids mix with regular quarkoniato constitute physical states. Comprehensive comparisons of the potentials for hybrids whose parametersare obtained in this scenario with the lattice results are presented.

DOI: 10.1103/PhysRevD.76.074035 PACS numbers: 12.39.Mk, 13.20.Gd

I. INTRODUCTION

In both the quark model and QCD which governs stronginteraction, there is no fundamental principle to prohibitthe existence of exotic hadron states such as glueball,hybrid, and multiquark states. In fact, to eventually under-stand the low energy behavior of QCD, one needs to findout such states. However, the recent research indicates thatthey may mix with the ordinary hadrons, especially thequarkonia. Thus they have evaded direct detection so far,even though many new resonances which have peculiarcharacteristics have continually been reported by variousexperimental collaborations. Theorists have proposed themto be pure gluonic (glueball), quark-gluon (hybrid), and/ormultiquark (tetraquark or pentaquark) structures which aredifferent from the regular valence quark structure of q �q formeson and qqq for baryon. Since the quark model andQCD theory advocate their existence, or at least do notrepel them, one should find them in experiments. However,even with many candidates of the exotic states, so far noneof them have been confirmed yet. Moreover, the possiblemixing of such exotic states with the regular mesons orbaryons contaminates the situation and would make a clearidentification difficult, even though not impossible. Fromthe theoretic aspect, one may try to help to clean the mistand find an effective way to do the job.

The transition of heavy quarkonia such as �ns� and��ns� to lower states �ms� and ��ms� (m< n) with twopions being emitted, provides an ideal laboratory to studythe spectra of hybrids. In the transitions �ns����ns�� ! �ms����ms�� � �� (m< n), the momentum transfer isnot large and usually the perturbative method does notapply. The QCD multipole expansion (QCDME) methodsuggested by Gottfried, Yan, and Kuang [1–5] well solvesthe light-meson emission problem. In the picture of themultipole expansion, two gluons are emitted which are notdescribed as energetic particles, but a chromofield of TM

or TE modes, then the two gluons which constitute a colorsinglet hadronize into light hadrons [6]. It is worth empha-sizing again that the two gluons are not free gluons in thesense of the perturbative quantum field theory, but a field inthe QCDME. It is easy to understand that such transition isdominated by the E1-E1 mode, while the M1-M1 mode issuppressed for the heavy quarkonia case.

Since two gluons are successively emitted, there existsan intermediate state where the quark-antiquark pair re-sides in a color octet. The color octet q� �q and a color-octet gluon constitute a color-singlet hybrid state.Therefore, in the framework, a key point is to determinethe spectra of the hybrid states jq �qgi where q can be eitherb or c in our case. Because of the lack of enough data to fixthe ground state of hybrid mesons, Buchmuller and Tye [7]assumed that the observed �4:03� was the ground state ofjc �cgi.

Yan and Kuang used this postulate to carry out theirestimation on the transition rates [2,3]. For the intermedi-ate hybrid states they used the phenomenological potentialgiven by Buchmuller and Tye [7] to calculate the widthsof ��2s� ! ��1s���, ��3s� ! ��1s���, ��3s� !��2s���. The theoretical prediction on the rate of��2s� ! ��1s��� and ��3s� ! ��2s��� is roughly con-sistent with the data [8], whereas that for ��3s� !��1s��� obviously deviates from the data. It is also notedthat when they calculated the decay widths, they needed toinvoke a cancellation among large numbers to obtainsmaller physical quantities, thus the calculations are verysensitive to the model parameters, i.e., a fine-tuning isunavoidable. Recently Kuang [3] indicated that determin-ing the proper intermediate hybrid states is crucial topredict the rates of the decay modes such as ��3s� !��1s���.

There have been some models for evaluating the hybridspectra, but there are several free parameters in each model

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and one should determine them by fitting data. This leadsto an embarrassing situation that one has to determine atleast one hybrid state, and then obtain the correspondingparameters in the model. Moreover, the recent studiesindicate that hybrid may not exist as an independent physi-cal state, but mixes with regular quarkonia states, thereforethe mass spectra listed on the data table are not the massesof a pure hybrid, which are the eigenvalues of theHamiltonian matrices. Therefore a crucial task is to deter-mine the mass spectra of pure hybrids, even though theyare not physical eigenstates of the Hamiltonian matrices.

Recently, thanks to the progress of measurements of theBABAR [9] and Belle [10] Collaborations, a remarkableamount of data on the transitions �ns����ns�� ! �ms����ms�� � �� have been accumulated and becomemore accurate. Since the large database is available, onemay have a chance to use the data to determine the massspectra of hybrids.

In this work, we apply the QCDME method establishedby Yan and Kuang [2] and the potential model given byseveral groups [11–13] to calculate the transition rates of �ns����ns�� ! �ms����ms�� � �� by keeping the po-tential model parameters free. Then by the typical method,namely, minimizing ��2 for the channels which have beenwell measured, we obtain the corresponding parameters,and then we go on predicting a few channels which havenot been measured yet. Finally, with the potential we candetermine the masses of hybrids, at least the ground state.

For clarity, we compare the potentials for hybrids whoseparameters are obtained in this scenario with the results ofthe lattice calculation. We find that if the parameters in thepotential suggested by Allen et al. [13] adopt the valueswhich are obtained in terms of our strategy, the potentialsatisfactorily coincides with the lattice results.

Our numerical results indicate that the ground states ofpure hybrid jc �cgi and jb �bgi do not correspond to thephysical states measured in recent experiments, the con-crete numbers may somehow depend on the forms of thepotential model adopted for the calculations (see the text).This may suggest that the pure hybrids do not exist inde-pendently, but mix with regular mesons.

After the introduction we present all the formulation inSec. II, where we only keep the necessary expressions forlater calculations, but omit some details which can beeasily found in Yan and Kuang’s papers. Then we carryout our numerical analysis in term of the ��2 method inSec. III where comprehensive comparisons of various po-tentials with the lattice results are presented. Section IV isdevoted to our conclusion and discussion.

II. FORMULATION

A. Transition width

The theoretical framework about the QCDME method iswell established in Refs. [2–5], and all the correspondingformulas are presented in their series of papers. Here weonly make a brief introduction to the formulas for evaluat-ing the widths which we are going to employ in this work.In Refs. [2,3] the transition rate of a vector quarkoniuminto another vector quarkonium with a two-pion emissioncan be written as

��nI3S1 ! nF

3S1� � jC1j2Gjfl;PI;PFnI;lI ;nF;lF

j2; (1)

where jC1j2 is a constant to be determined and it comes

from the hadronization of gluons into pions, G is the phasespace factor, fl;PI;PFnI;lI ;nF;lF

is the overlapping integration overthe concerned hadronic wave functions, their concreteforms were given in [3] as

fl;PI;PFnI;lI ;nF;lF�XK

RRF�r�r

PFR�Kl�r�r2dr

RR�Kl�r

0�r0PIRI�r0�r02dr0

MI � EKl; (2)

where nI; nF are the principal quantum numbers of initialand final states, lI; lF are the angular momenta of the initialand final states, l is the angular momentum of the coloroctet q �q in the intermediate state, PI; PF are the indicesrelated to the multipole radiation, for the E1 radiation PI,PF � 1 and l � 1. RI, RF, and RKl are the radial wavefunctions of the initial and final states, MI is the mass ofinitial quarkonium, and EKl is the energy eigenvalue of theintermediate hybrid state.

B. ��2 method

The standard method adopted in analyzing data andextracting useful information is minimizing the ��2, andin our work we hope to obtain the model parameters. Whencalculating ��2, we would involve as many as possible

experimental measurements to make the fitted parametersmore reasonable. Here we adopt the form of ��2 defined in[14] as

�� 2 �Xi

�Wthi �W

expi �

2

��Wexpi �

2 ; (3)

where i represents the ith channel, Wthi is the theoretical

prediction on the width of channel i, Wexpi is the corre-

sponding experimentally measured value, �Wexpi is the

experimental error.Wthi will be calculated in terms of the potential models

with several free parameters which are described in thefollowing subsections, thus Wth

i is a function of the pa-rameters. By minimizing ��2, we would expect to determine

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the model parameters. Some details of our strategy will bedepicted in Sec. II E.

C. Phenomenological potential for the initial and finalquarkonia

In this work, we adopt two different potentials for theinitial and final heavy quarkonia and the intermediatehybrid states.

The Cornell potential [15] is the most popular potentialform to study heavy quarkonia. The potential reads as

V�r� � ��r� br; (4)

usually in the literature many authors prefer to use �sinstead of � and it has a relation � � 4�s�r�

3 , and �s�r�can be treated as a constant for the �bb and �cc quarkonia.

The modified Cornell potential: It may be more reason-able to choose a modified Cornell potential which includesa spin-related term [16], and the potential takes the form

V�r� � ��r� br� Vs�r� � V0; (5)

where the spin-related term Vs is

Vs �8��

3m2q���r� ~Sq � ~S �q;

with

���r� ��������p

�3e��

2r2;

and V0 is the zero-point energy (in Ref. [16] it was set to bezero); here we do not a priori assume it to be zero, but fix itby fitting the spectra of heavy quarkonia.

D. Potential for hybrids

The intermediate state as discussed above is a hybridstate jq �qgi and we need to obtain the spectra and wavefunctions of the ground state and corresponding radiallyexcited states. Yan and Kuang used the phenomenologicalpotential given by Buchmuller and Tye [7] to evaluate themass of the ground state of hybrid; instead, in our work, wetake some effective potential models which are based onthe color-flux-tube model.

Generally hybrids are labeled by the right-handed (n�m)and left-handed (n�m) transverse phonon modes

N �X1m�1

m�n�m � n�m�;

and a characteristic quantity � as

� �X1m�1

�n�m � n�m�:

All the details about the definitions and notations can beeasily found in the literature [11–13,17–19].

Various groups suggested different potential forms forthe interaction between the quark and antiquark in thehybrid state. We label them as model 1, 2, and 3,respectively.

In this work, we employ three potentials which areModel 1 was suggested by Isgur and Paton [11] as

V�r� � ��r� br�

�r�1� e�fb

1=2r� � V0: (6)

Model 2: Swanson and Szczepaniak [12] think that theCoulomb term in model 1 is not compatible with the latticeresults, so they suggested an alternative effective potentialas

V�r� � br��r�1� e�fb

1=2r�: (7)

To get a better fit to data, we add the zero-point energyV0 into Eq. (7),

V�r� � br��r�1� e�fb

1=2r� � V0: (8)

Model 3: In model 1, the Coulomb piece is not properbecause the quark and antiquark in the hybrid reside in acolor octet instead of a singlet (the meson case), the short-distance behavior should be repulsive (it is determined bythe sign of the expectation value of the Casimir operator inoctet). Thus Allen et al. suggested the third model [13] andthe corresponding potential form is

V�r� ��8�

���������������������������br�2 � 2�b

q� V0: (9)

Because in these forms the authors do not consider thespin-related term (which we name as Vs), we can modifythe potential by adding a spin-related term Vs, then thepotential becomes

V�r� � Vi � Vs: (10)

By this modification, one can investigate the spin-splittingeffects. Generally, Vs should have the same form as that in(5).

E. Our strategy

The strategy of this work is that we will determine theconcerned parameters in the potential [Eqs. (6) and (8)–(10)] by fitting the data of heavy quarkonia transitions.

To obtain the concerned parameters in the potentials[Eqs. (6) and (8)–(10)] which specify the hybrids sates,we use the method of minimizing ��2 defined in (3).Concretely, in Eq. (3), Wth

i is a function of the parameters�, f, b, V0, and jC1j

2, and following Ref. [11], we set f �1, therefore ��2 is also a function of those parameters.Minimizing ��2, one can fix the values of the correspondingparameters. Still for simplifying our complicated numeri-cal computations, we choose a special method; namely, wefirst preset a group of the parameters, and we calculate thehybrid spectra and wave functions by solving the

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Schrodinger equation, then we determine jC1j2 in Eq. (1) in

terms of the well-measured rate of �2S� ! J= ��. Withthis jC1j

2 as a predetermined value or, say, a function ofother parameters, we minimize ��2 to fix the values of therest of parameters �, b, V0.

With all the parameters fixed, we can determine the massspectra of the hybrids which serve as the intermediatestates in the transitions of �ns����ns�� ! �ms�����ms�� � ��. It is noted that the spectra determined inthis scheme are not really the masses of physical states,unless the hybrids do not mix with regular quarkonia. Inother words, we would determine a diagonal element of themass Hamiltonian matrix, whose diagonalization wouldmix the hybrid and quarkonium and then determine theeigenvalues and eigenfunctions corresponding to thephysical masses and physical states which are measuredin experiments.

III. NUMERICAL RESULTS

To determine the model parameters in the potential, weneed to fit the spectra of �ns�, �c�1s�, �c�2s�, and ��ns�,and in this work, we only concern the ground states andradially excited states of c �c; b �b and c �cg; b �bg systems.

A. Without the spin-related term Vs

The potentials for quarkonia [Eq. (4)] and hybrid[Eq. (6) (model 1), Eq. (8) (model 2), and Eq. (9) (model 3)]do not include the spin-related term. In this work, we adoptthe Cornell potential to calculate the spectra and wavefunctions of the regular heavy quarkonia. The concernedparameters in the Cornell potential have been given in theliterature as for the c �c mesons, � � 0:52, b � 0:18 GeV2,mc � 1:84 GeV, whereas for the b �b mesons, � � 0:48,b � 0:18 GeV2, mb � 5:17 GeV [2,15]. It is also notedthat to meet the measured spectra of charmonia and botto-nia a zero-point energy V0 is needed.

The potential for the hybrid takes three possible formswhich are shown in Eqs. (6), (8), and (9). We keep thevalues mc � 1:84 GeV, mb � 5:17 GeV which are ob-tained by fitting the spectra of regular quarkonia jb �bi[��ns�] and jc �ci [ �ns�] with the potential (4), but needto gain the values of the relevant parameters �, b, and V0,etc. by minimizing ��2 for the decays �ns����ns�� ! �ms����ms�� � ��. According to the measured valuefor �� �2S� ! J= ���

�tot� �2S�� � 337 13 keV;

B� �2S� ! J= ����� � �31:8 0:6�%;

B� �2S� ! J= �0�0� � �16:46 0:35�%

we express C21 as a function of the potential parameters

which exist in the three potentials [Eqs. (6) and (8), or (9)]and will be determined. It is noted thatC2

1 is a factor relatedto the hadronization of gluons into two pions, so should beuniversal for both and � decays. The parameters in thepotentials are also universal for the �bb and �cc cases exceptthe masses are different.

Then for ����nS� ! ��ms� � ��� (m< n), we calcu-late Wth

i in terms of the three potential forms. The corre-sponding experimental values and errors are Wexp

i and�Wexp

i given in the references which are shown in Table I.By minimizing ��2 [Eq. (3)], we finally get the potential

parameters �, b, and V0 and the resultant ��2 � 4:42 formodel 1, 13.69 for model 2, and 7.26 for model 3. Then weobtain jC1j

2 � 100:39� 10�6 for model 1, 259:24� 10�6

for model 2, and 121:78� 10�6 for model 3. The otherparameters are listed in Table II.

With these potential parameters, we solve theSchrodinger equation to obtain the masses of ground hy-brid states of jc �cgi and jb �bgi (Table III). It is noted that theresultant spectra depend on the potential forms. We willdiscuss this problem in Sec. IV.

We also make a prediction on the rates which have notbeen measured yet (Table IV).

It is noted that values of �� �3S� ! �1S���� pre-dicted by models 1, 2, and 3 are quite apart, while����4S� ! ��3S���� and �� �3S� ! �2S���� pre-dicted by all the three models are close.

B. Comparison with the lattice results

To clarify, it would be helpful to compare the resultsobtained in our phenomenological work with the latticeresults which are supposed to include both perturbative andnonperturbative QCD effects. Below we show comprehen-sive comparisons of our potentials with the lattice results.

Following Refs. [12,13,17,18,20], the potentials shownin Fig. 1 are specially scaled by V��g �2r0� which is thepotential for ��g (N � 0) at 2r0 � 5 GeV�1 (for the ver-tical axis of Fig. 1).

TABLE I. Transition rate of ��nS� ! ��ms� � �� (in keV).

Decay mode Model 1 Model 2 Model 3 Experiment data

��2S� ! ��1S��� 9.36 9.28 8.69 12:0 1:8��3S� ! ��1S��� 1.81 1.67 1.85 1:72 0:35��3S� ! ��2S��� 0.86 0.76 0.86 1:26 0:40��4S� ! ��1S��� 3.87 3.43 4.14 3:7 0:6 0:7 [10]��4S� ! ��2S��� 1.83 0.2 1.44 2:7 0:8 [9]

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In the three graphs of Fig. 1, we present comparisons ofthe three potentials (models 1, 2, and 3) with the parame-ters fixed in the previous subsections with the lattice re-sults. In the graphs, the dots are the lattice values [20].

It is emphasized that we obtain the potential by mini-mizing ��2 of the data on �ns����ns�� ! �ms�����ms�� � ��, but do not fit the lattice values. Then ourresults, especially the third potential, coincides with thelattice results extremely well. It may indicate that thephysics description adopted in this scenario is reasonable.

It is also noted that by model 1, the short-distance behaviorof the potential is attractive and obviously distinct from thelattice results. This discrepancy was discussed above thatthe quark-antiquark system in hybrid should be a coloroctet and short-distance interaction should be repulsive.The second potential (model 2) has the same trend as thelattice results, but has obvious deviations (see graph 2 ofFig. 1).

C. With the spin-related terms VsFor the regular quarkonia we adopt the nonrelativistic

potential Eq. (5) [16]. Since we add a zero-point energy V0

in the potential which can be seen as another free parame-ter (it is the same for both c �c and b �b quarkonia), we refitthe spectra of the quarkonia to obtain the correspondingpotential parameters in Eq. (5). We list the resultant valuesof the parameters in Table V. In Table VI, we present thefitted spectra of c �c and, for a comparison, we also includethe results given in Ref. [16] in the table.

For the b �b quarkonia, the corresponding parametersobtained by fitting data are listed in Table VII.

By the parameters we predict m�b � 9:434 GeV, whichis consistent with that given by [21].

Then we turn to the hybrid intermediate states.For the hybrids, by the observation made in the previous

subsection one can conclude that the third potential(model 3) better coincides with the lattice results; there-fore, in this subsection when we include the spin-relatedterm to discuss the spin-splitting case, we only adopt thethird potential Eq. (9). It is reasonable to keep the values ofmc, mb, and � to be the same as that we determined forpure q �q quarkonia and we also set f � 1. Then followingour strategy discussed in previous subsections, we obtainthe potential parameters which are listed in Table VIII.

The fitted values and some predictions are also listed inTables IX and X. We obtain

jC1j2 � 182:12� 10�6;

TABLE IV. Prediction (in keV).

Decay mode Model 1 Model 2 Model 3

��4S� ! ��3S��� 0.60 0.57 0.61 �3S� ! �2S��� 14.96 14.45 14.83 �3S� ! �1S��� 589.91 72.34 424.22

TABLE III. The mass of hybrids (in GeV).

Model 1 Model 2 Model 3

jc �cgi 4.099 4.549 4.226jb �bgi 10.560 11.137 10.789

TABLE V. Potential parameters for c �c.

� b (GeV2) m (GeV) � (GeV2) V0 (GeV)

0.67 0.16 1.78 1.6 �0:6

TABLE II. Potential parameters for hybrid.

� b (GeV2) V0 (GeV)

Model 1 0.43 0.19 �0:85Model 2 � � � 0.15 �0:43Model 3 0.59 0.19 �0:85

FIG. 1 (color online). Comparisons between the potentials we fit with the lattice results.

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the mass of hybrids are 4.351, 4.333 GeV for the spin-triplet and spin-singlet c �c in the hybrid and 10.916,10.913 GeV for the spin-triplet and singlet b �b, respectively.Because of including the spin-related term, the ‘‘groundstates’’ with the q �q (q � b or c) being in different spinstructures would be slightly split.

One can observe that the predicted ����4S� !��3S���� and �� �3S� ! �2S���� are slightly smallerthan that predicted in the models without the spin-relatedterm, the future experiments may shed some light on it,namely, getting better understanding on the mechanismswith which one can describe the hybrid structure better.

We also calculate the transition rate of �0c ! �c � ���. Our result is almost triple that obtained in Ref. [4] and itcan be tested by the future experiments. It is noted thatsince we minimize ��2, the decay widths that we obtain aredifferent from the central values of the measured quanti-ties. We list the widths we finally obtained in the Table IX.

IV. CONCLUSION AND DISCUSSION

Search for exotic states which are allowed by the SU(3)quark model and QCD theory is very important for ourunderstanding of the basic theory, but so far such stateshave not been found (or not firmly identified), thus itbecomes an attractive task in high energy physics. Nodoubt, direct measurements on such exotic states wouldprovide definite information on them; however, it seemsthat most of the mysterious states mix with mesons andbaryons which have regular quark structures. Since theyare hidden in the mixed states, they are not physical statesand do not have physical masses, and it makes a clearidentification of such exotic states very difficult. In otherwords, they may only serve as a component of physicalstates. Even though some phenomenological models, suchas the color-flux-tube model, the bag model, and the po-tential model, etc., are believed to properly describe theirproperties and determine their ‘‘masses,’’ in fact, if theymix with the regular mesons or baryons, the resultantmasses are only the diagonal elements of the Hamil-tonian matrix. For example, in the potential model, bysolving the Schrodinger equation, one obtains the eigene-nergy and wave function, and only gets the element E11 �

hybh�jHhybj�ihyb, where the subscript ‘‘hyb’’ denotes thequantities corresponding to hybrids. Meanwhile, there isE22 � regh�jHregj�ireg corresponding to the regular quarkstructure. If the two eigenstates are not far located, theymay mix with each other and provide an extra matrixelement to the Hamiltonian matrix, as E12 � E�21 �

regh�jHmixj�ireg. Unfortunately, there is not a reliableway to calculate the mixing matrix element. One mayexpect to gain definite information about the hybrid statesand maybe starting from there one can further study themechanism of the mixing.

The theoretical framework established by Yan andKuang confirms that the intermediate states between two-pion emissions in the transition �ns����ns�� ! �ms�����ms�� � �� are hybrids which contain a quark-antiquark pair in color octet, and an extra valence gluon.Based on the color-flux-tube model, in the 1980s Isgur andPaton suggested a potential model for the hybrid, and thisgreatly simplifies the discussion about hybrids and mayoffer an opportunity to study the regular quarkonium andhybrid in a unique framework. After their work, severalother groups also proposed modified potentials to make abetter description of the hybrid states. When Yan andKuang studied the transitions, there were not many dataavailable, i.e., most of the channels were not measured yet.Therefore they assumed that �4:03� as the ground state ofcharmed hybrids jc �cgi and estimated the transition rates.Thanks to the great achievements of the BABAR and BelleCollaborations, many such modes are measured with ap-preciable accuracy. Based on the experimental data and thetheoretical framework established by Yan and Kuang, we

TABLE VII. Potential parameters for b �b.

� b (GeV2) m (GeV) � (GeV2) V0 (GeV)

0.53 0.16 5.13 1.7 �0:60

TABLE VIII. Potential parameters for hybrid.

��c �cg� ��b �bg� b (GeV2) V0 (GeV)

Best fitted values 0.54 0.40 0.24 �0:80

TABLE IX. � transition (in keV).

Decay mode Widths (fit)

��2S� ! ��1S��� 8.73��2S� ! ��1S��� 1.94��3S� ! ��2S��� 0.69��4S� ! ��1S��� 4.10��4S� ! ��2S��� 1.88

TABLE X. Prediction (in keV).

Decay mode Widths of prediction

��4S� ! ��3S��� 0.36 �3S� ! �2S��� 8.84 �3S� ! J= �� 12.38�c�2S� ! �c�� 335.66

TABLE VI. Eigenvalues for c �c in GeV.

J= �2S� �3S� �4S� �c�1S� �c�2S�

Ref. [16] 3.090 3.672 4.072 4.406 2.982 3.630This work 3.097 3.687 4.093 4.433 2.971 3.634

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minimize the ��2 to obtain the model parameters in thepotential for hybrid, and with them, we can estimate themasses of the ground states of hybrids. The theory of theQCD multiexpansion is based on the assumption that thehadronization of the emitted gluons can be factorized fromthe transition of ��ns�� �ns�� ! ��ms�� �ms��. In fact,this factorization may not be complete if the nonperturba-tive QCD effects are involved; namely, the higher twistcontribution may somehow violate the factorization.However, as long as the nonperturbative QCD effects arenot too strong, this approximation should be acceptablewithin a certain tolerance range. Moreover, in our study,the nonfactorization effects are partly involved in the pa-rameter jC1j

2 of Eq. (1), and in our scheme it is also one ofthe free parameters which is fixed by fitting data. Indeed, itis implicitly assumed that jC1j

2 is universal for all theprocesses, and it may cause some error. But it is believedthat since the energy range does not change drastically, theerror should be controllable.

In the calculations, we adopt the Cornell potential for thecolor-singlet q �q (q � b or c) system and the potentialssuggested by Isgur and Paton (model 1)[11], by Swansonand Szczepaniak (model 2) [12], and by Allen et al.(model 3) [13] to deal with the color-octet q �q system, weadd a spin-related term to the potential for hybrid (model 3only) to investigate possible spin-splitting effects. Thenumerical results are slightly different when this term isintroduced. The masses of the ground state hybrids are4.23 GeV for jc �cgi and 10.79 GeV for jb �bgi which areestimated in terms of model 3. When the spin-related termis included, the results change to 4.351, 4.333 GeV for thespin-triplet and spin-singlet c �c in the hybrid and 10.916,10.913 GeV for the spin-triplet and singlet b �b, respectively.In the other two models, the results are slightly different.Indeed, as mentioned previously in the comprehensivecomparison of the results with the lattice values, one maybe convinced that model 3 may be the best choice atpresent. All the obtained masses are different from thephysical states measured in experiments, and it may implythat the hybrids mix with regular mesons.

There are more data in the b-energy range than in thecharm-energy region. In fact, when we use the samemethod to calculate the transition �ns� ! �ms� � ��,with n and m being far apart (say n � 4, m � 1, etc.), thetheoretical solutions are not stable and uncertainties arerelatively large. It indicates that there are still some defectsin the theory which would be studied in our future works.Moreover, recently Guo et al. [22] studied the processes interms of the chiral perturbation theory and considered thefinal state interaction to fit the details of the �� energy andangular distributions.

The transition of higher excited states of quarkonia intolower ones (including the ground state) without flavorchange but emitting photon or light mesons is believed tooffer rich information on the hadron structure and govern-ing dynamics, especially for the heavy quarkonia physics.For example, Brambilla et al. [23] studied the quarkoniumradiative decays which are realized via electromagneticinteractions.

Our studies indicate that the transitions of �ns�����ns�� ! �ms����ms�� � �� may provide valuable in-formation about the hybrid structures which have so far notbeen identified in experiments.

Since we use the method of minimizing ��2 to achieve allthe parameters in the potential model for hybrids, it cer-tainly brings up some errors. It is a common method forboth experimentalists and theorists to analyze data andobtain useful information. Definitely, the more data areavailable, the more accurate the results would be.Therefore, more data are very necessary, especially thedata on the families which are one of the research fieldsof the BES III and CLEOc.

ACKNOWLEDGMENTS

This work is supported by the National Natural ScienceFoundation of China (NNSFC), under ContractNo. 10475042.

[1] K. Gottfried, Phys. Rev. Lett. 40, 598 (1978).[2] Y. Kuang and T. Yan, Phys. Rev. D 24, 2874 (1981).[3] Y. Kuang, Front. Phys. China 1, 19 (2006).[4] T. Yan, Phys. Rev. D 22, 1652 (1980).[5] Y. Kuang, Y. Yi, and B. Fu, Phys. Rev. D 42, 2300

(1990).[6] L. Brown and R. Cahn, Phys. Rev. Lett. 35, 1 (1975).[7] W. Buchmuller and H. Tye, Phys. Rev. Lett. 44, 850

(1980).[8] W. Yao et al. (Partical Data Group), J. Phys. G 33, 1

(2006).

[9] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett.96, 232001 (2006).

[10] A. Sokolov et al. (Belle Collaboration), Phys. Rev. D 75,071103 (2007).

[11] N. Isgur and J. Paton, Phys. Rev. D 31, 2910 (1985).[12] E. Swanson and A. Szczepaniak, Phys. Rev. D 59, 014035

(1998).[13] T. Allen, M. Olsson, and S. Veseli, Phys. Lett. B 434, 110

(1998).[14] C. Chiang, M. Gronau, J. Rosner, and D. Suprun, Phys.

Rev. D 70, 034020 (2004).

ANALYSIS ON HEAVY QUARKONIA TRANSITIONS WITH . . . PHYSICAL REVIEW D 76, 074035 (2007)

074035-7

[15] E. Eichten, K. Gottfried, T. Kinashita, K. Lane, andT. Yan, Phys. Rev. D 17, 3090 (1978); 21, 203 (1980).

[16] T. Barnes, S. Godfrey, and E. Swanson, Phys. Rev. D 72,054026 (2005).

[17] F. Buisseret and V. Mathieu, Eur. Phys. J. A 29, 343(2006).

[18] A. Szczepaniak and P. Krupinski, Phys. Rev. D 73, 116002(2006); P. Guo, A. Szczepaniak, G. Galata, A. Vassallo,and E. Santopina, arXiv:0707.3156.

[19] G. Bali and A. Pineda, Phys. Rev. D 69, 094001 (2004).[20] K. Juge, J. Kuti, and C. Morningstar, Nucl. Phys. B,

Proc.Suppl. 63, 326 (1998); Phys. Rev. Lett. 90, 161601(2003).

[21] G. Hao, Y. Jia, C. Qiao, and P. Sun, J. High Energy Phys.

02 (2007) 057; D. Ebert, R. Faustov, and V. Galkin, Phys.Rev. D 67, 014027 (2003); S. Recksiegel and Y. Sumino,Phys. Lett. B 578, 369 (2004); B. Kniehl, A. Penin,A. Pineda, V. Smirnov, and M. Steinhauser, Phys. Rev.Lett. 92, 242001 (2004); A. Gray, I. Allison, C. Davies, E.Gulez, G. Lepage, J. Shigemitsu, and M. Wingate, Phys.Rev. D 72, 094507 (2005).

[22] F. Guo, P. Shen, H. Chiang, and R. Ping, Nucl. Phys.A761, 269 (2005); F. Guo, P. Shen, and H. Chiang, Phys.Rev. D 74, 014011 (2006); F. Guo, P. Shen, H. Chiang, andR. Ping, arXiv:hep-ph/0601120.

[23] N. Brambilla, Y. Jia, and A. Vairo, Phys. Rev. D 73,054005 (2006); N. Brambilla et al., arXiv:hep-ph/0412158.

HONG-WEI KE, JIAN TANG, XI-QING HAO, AND XUE-QIAN LI PHYSICAL REVIEW D 76, 074035 (2007)

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