Analysis on Heavy Quarkonia Transitions with Pion Emission in Terms of the QCD Multipole Expansion and Determination of Mass Spectra of Hybrids
aa r X i v : . [ h e p - ph ] S e p Analysis on Heavy Quarkonia Transitions with Pion Emission inTerms of the QCD Multipole Expansion and Determination ofMass Spectra of Hybrids
Hong-Wei Ke, Jian Tang, Xi-Qing Hao and Xue-Qian Li
Department of Physics, Nankai University, Tianjin 300071, China
Abstract:One of the most important tasks in high energy physics is search for the ex-otic states, such as glueball, hybrid and multi-quark states. The transitions ψ ( ns ) → ψ ( ms ) + ππ and Υ( ns ) → Υ( ms ) + ππ attract great attentions be-cause they may reveal characteristics of hybrids. In this work, we analyze thosetransition modes in terms of the theoretical framework established by Yan andKuang. It is interesting to notice that the intermediate states between the twogluon-emissions are hybrids, therefore by fitting the data, we are able to deter-mine 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 cor-respond to any states measured in recent experiments, thus it may imply thatvery possibly, hybrids mix with regular quarkonia to constitute physical states.Comprehensive comparisons of the potentials for hybrids whose parameters areobtained in this scenario with the lattice results are presented.PACS numbers: 12.39.Mk, 13.20.Gd I. INTRODUCTION
In both the quark model and QCD which governs strong interaction, there is no any fundamental principleto prohibit existence of exotic hadron states such as glueball, hybrid and multi-quark states. In fact, toeventually understand the low energy behavior of QCD, one needs to find out such states. However, therecent research indicates that they may mix with the ordinary hadrons especially the quarkonia. Thus theyevade direct detection so far, even though many new resonances which have peculiar characteristics, havecontinuously been reported by various experimental collaborations. Theorists have proposed them to be puregluonic (glueball), quark-gluon (hybrid), and/or multi-quark (tetraqurk or pentaquark) structures which aredifferent from the regular valence quark structure of q ¯ q for meson and qqq for baryon. Since the quark modeland QCD theory advocate their existence, at least do not repel them, one should find them in experiments.However, even with many candidates of the exotic states, so far none of them have been confirmed yet.Moreover, the possible mixing of such exotic states with the regular mesons or baryons contaminates thesituation and would make a clear identification difficult, even though not impossible. From the theoreticaspect, one may try to help to clean the mist and find an effective way to do the job.The transition of heavy quarkonia such as ψ ( ns ) and Υ( ns ) to lower states ψ ( ms ) and Υ( ms ) ( m
03) was the ground state of | c ¯ cg > .Yan and Kuang used this postulate to carry out their estimation on the transition rates[2, 3]. For theintermediate hybrid states they used the phenomenological potential given by Buchm¨uller and Tye[7] tocalculate the widths of Υ(2 s ) → Υ(1 s ) ππ , Υ(3 s ) → Υ(1 s ) ππ ,Υ(3 s ) → Υ(2 s ) ππ . The theoretical predictionon the rate of Υ(2 s ) → Υ(1 s ) ππ and Υ(3 s ) → Υ(2 s ) ππ is roughly consistent with data[8], whereas thatfor Υ(3 s ) → Υ(1 s ) ππ obviously deviates from data. It is also noted that when they calculated the decaywidths, they need to invoke a cancellation among large numbers to obtain smaller physical quantities, thusthe calculations are very sensitive to the model parameters, i.e. a fine-tuning is unavoidable. RecentlyKuang [3] indicates that determining the proper intermediate hybrid states is crucial to predict the rates ofthe decay modes such as Υ(3 s ) → Υ(1 s ) ππ .There have been some models for evaluating the hybrid spectra, but there are several free parameters ineach model and one should determine them by fitting data. This leads to an embarrassing situation thatone has to determine at least one hybrid state, and then obtain the corresponding parameters in the model.Moreover, the recent studies indicate that hybrid may not exist as an independent physical state, but mixeswith regular quarkonia states, therefore the mass spectra listed on the data table are not the masses of apure hybrid, which are the eigenvalues of the Hamiltonian matrices. Therefore a crucial task is to determinethe mass spectra of pure hybrids, even though they are not physical eigenstates of the Hamiltonian matrices.Recently, thanks to the progress of measurements of the Babar [9] and Belle [10] collaborations, a remark-able amount of data on the transitions ψ ( ns )(Υ( ns )) → ψ ( ms )(Υ( ms )) + ππ have been accumulated andbecome more accurate. Since the large database is available, one may have a chance to use the data todetermine the mass spectra of hybrids.In this work, we apply the QCD multipole expansion method established by Yan and Kuang [2] and thepotential model given by several groups [11, 12, 13], to calculate the transition rates of ψ ( ns )(Υ( ns )) → ψ ( ms )(Υ( ms )) + ππ by keeping the potential model parameters free. Then by the typical method, namelyminimizing ¯ χ for the channels which have been well measured, we obtain the corresponding parameters,and then we go on predicting a few channels which are not been measured yet, finally with the potential wecan determine the masses of hybrids, at least the ground state.To make sense, we compare the potentials for hybrids whose parameters are obtained in this scenario withthe results of the lattice calculation. We find that if the parameters in the potential suggested by Allen etal.[13] adopt the values which are obtained in terms of our strategy, the potential satisfactorily coincideswith the lattice results.Our numerical results indicate that the ground states of pure hybrid | c ¯ cg > and | b ¯ bg > do not correspondto the physical states measured in recent experiments, the concrete numbers may somehow depend on theforms of the potential model adopted for the calculations (see the text). This may suggest that the purehybrids do not exist independently, but mix with regular mesons.After the introduction we present all the formulation in next section, where we only keep the necessaryexpressions for later calculations, but omitting some details which can be easily found in Yan and Kuang’spapers. Then we carry out our numerical analysis in term of the ¯ χ method. Comprehensive comparisonsof various potentials with the lattice results are presented. The last section is devoted to conclusion anddiscussion. Φ i Φ f MGE H π + + π − (2 π ) FIG. 1:
II. FORMULATIONA. The transition width
The theoretical framework about the QCD Multiploe Expandsion method is well established in Refs[2,3, 4, 5], and all the corresponding formulas are presented in their series of papers. Here we only make abrief introduction to the formulas for evaluating the widths which we are going to employ in this work.In Refs.[2, 3] the transition rate of a vector quarkonium into another vector quarkonium with a two-pionemission can be written as Γ( n I S → n F S ) = | C | G | f l,P I ,P F n I ,l I ,n F ,l F | (1)where | C | is a constant to be determined and it comes from the hadronization of gluons into pions, G is thephase space factor, f l,P I ,P F n I ,l I ,n F ,l F is the overlapping integration over the concerned hadronic wave functions,their concrete forms were given in [3] as f l,P I ,P F n I ,l I ,n F ,l F = X K R R F ( r ) r P F R ∗ Kl ( r ) r dr R R ∗ Kl ( r ′ ) r ′ P I R I ( r ′ ) r ′ dr ′ M I − E Kl , (2)where n I , n F are the principal quantum numbers of initial and final states, l I , l F are the angular momentaof the initial and final states, l is the angular momentum of the color-octet q ¯ q in the intermediate state, P I , P F are the indices related to the multipole radiation, for the E1 radiation P I , P F =1 and l = 1. R I , R F and R Kl are the radial wave functions of the initial and final states, M I is the mass of initial quarkoniumand E Kl is the energy eigenvalue of the intermediate hybrid state. B. The ¯ χ method The standard method adopted in analyzing data and extracting useful information is minimizing the ¯ χ and in our work, we hope to obtain the model parameters. When calculating ¯ χ , we would involve as manyas possible experimental measurements to make the fitted parameters more reasonable. Here we adopt theform of ¯ χ defined in [14] as ¯ χ = X i ( W thi − W expi ) (∆ W expi ) , (3)where i represents the i-th channel, W thi is the theoretical prediction on the width of channel i , W expi is thecorresponding experimentally measured value, ∆ W expi is the experimental error. W thi will be calculated in terms of the potential models with several free parameters which are describedin the following subsections, thus W thi is a function of the parameters. By minimizing ¯ χ , we would expectto determine the model parameters. Some details of our strategy will be depicted in subsection E. C. The phenomenological potential for the initial and final quarkonia
In this work, we adopt two different potentials for the initial and final heavy quarkonia and the intermediatehybrid states.The Cornell potential [15] is the most popular potential form to study heavy quarkonia. The potentialreads as V ( r ) = − κr + br, (4)usually in the literature many authors prefer to use α s instead of κ and it has a relation κ = α s ( r )3 , and α s ( r ) can be treated as a constant for the ¯ bb and ¯ cc quarkonia.The modifed Cornell potential: It may be more reasonable to choose a modified Cornell potential whichincludes a spin-related term [16], and the potential takes the form V ( r ) = − κr + br + V s ( r ) + V , (5)where the spin-related term V s is, V s = 8 πκ m q δ σ ( r ) −→ S q · −→ S ¯ q with δ σ ( r ) = ( σ √ π ) e − σ r , and V is the zero-point energy,( in Ref.[16] it was set to be zero), here we do not priori-assume it to be zero,but fix it by fitting the spectra of heavy quarkonia. D. The potential for hybrids
The intermediate state as discussed above is a hybrid state | q ¯ qg > and we need to obtain the spectraand wave-functions of the ground state and corresponding radially excited states. Yan and Kuang used thephenomenological potential given by Buchm¨uller and Tye [7] to evaluate the mass of the ground state ofhybrid, instead, in our work, we take some effective potential models which are based on the color-flux-tubemodel.Generally hybrids are labelled by the right-handed( n + m ) and left-handed( n − m ) transverse phonon modes N = ∞ X m =1 m ( n + m + n − m ) , and a characteristic quantity Λ as Λ = ∞ X m =1 ( n + m − n − m ) . All the details about the definitions and notations can be easily found in literature[11, 12, 13, 17, 18, 19].Various groups suggested different potential forms for the interaction between the quark and antiquark inthe hybrid state. We label them as Model 1, 2 and 3 respectively.In this work, we employ three potentials which are:Model 1 was suggested by Isgur and Paton [11] as V ( r ) = − κr + br + πr (1 − e − fb / r ) + V . (6)Model 2: Swanson and Szczepaniak[12] think that the Coulomb term in model 1 is not compatible withthe lattice results, so that they suggested an alternative effective potential as V ( r ) = br + πr (1 − e − fb / r ) . (7)To get a better fit to data, we add the zero-point energy V into Eq.(7), V ( r ) = br + πr (1 − e − fb / r ) + V . (8)Model 3: In model 1, the Coulomb piece is not proper, because the quark and antiquark in the hybridreside in a color-octet instead of a singlet (the meson case), the short-distance behavior should be repulsive(it is determined by the sign of the expectation value of the Casimir operator in octet). Thus Allen et al suggested the third model [13] and the corresponding potential form is V ( r ) = κ p ( br ) + 2 πb + V . (9)Because in these forms the authors do not consider the spin-related term (which we name as V s .), we canmodify the potential by adding a spin-related term V s , then the potential becomes: V ( r ) = V i + V s . (10)By this modification, one can investigate the spin-splitting effects. Generally, V s should have the same formas that in (5). E. Our strategy
The strategy of this work is that we will determine the concerned parameters in the potential (Eqs.(6),(8), (9) and Eq.(10)) by fitting the data of heavy quarkonia transitions.To obtain the concerned parameters in the potentials (Eqs.(6), (8), (9) and (10)) which specify the hybridssates, we use the method of minimizing ¯ χ defined in (3). Concretely, in Eq.(3), W thi is a function of theparameters κ, f, b, V and | C | , and following Ref.[11], we set f = 1, therefore ¯ χ is also a function of thoseparameters. Minimizing ¯ χ , one can fix the values of the corresponding parameters. Still for simplifying ourcomplicated numerical computations, we choose a special method, namely, we first pre-set a group of theparameters, and we calculate the hybrid spectra and wave-functions by solving the Schr¨odinger equation,then we determine | C | in Eq.(1) in terms of the well measured rate of ψ (2 S ) → J/ψ π π . With this | C | as a pre-determined value or say, a function of other parameters, we minimize ¯ χ to fix the values of the restof parameters κ, b, V .With all the parameters being fixed, we can determine the mass spectra of the hybrids which serve asthe intermediate states in the transitions of ψ ( ns )(Υ( ns )) → ψ ( ms )(Υ( ms )) + ππ . It is noted that thespectra determined in this scheme are not really the masses of physical states, unless the hybrids do not mixwith regular quarkonia. In other words, we would determine a diagonal element of the mass Hamiltonianmatrix, whose diagonalization would mix the hybrid and quarkonium and then determine the eigenvalues andeigen-functions corresponding to the physical masses and physical states which are measured in experiments. III. NUMERICAL RESULTS
To determine the model parameters in the potential, we need to fit the spectra of ψ ( ns ), η c (1 s ), η c (2 s )and Υ( ns ) and in this work, we only concern the ground states and radially excited states of c ¯ c , b ¯ b and c ¯ cg , b ¯ bg systems. A. Without the spin-related term V s The potentials for quarkonia (Eq.(4)) and hybrid (Eq.(6) (model 1), Eq.(8) (model 2) and Eq.(9) (model3)) do not include the spin-related term. In this work, we adopt the Cornell potential to calculate the spectraand wavefunctions of the regular heavy quarkonia. The concerned parameters in the Cornell potential havebeen given in literature as for the c ¯ c mesons, κ = 0 . , b = 0 . , m c = 1 .
84 GeV, whereas for the b ¯ b mesons, κ = 0 . , b = 0 . , m b = 5 .
17 GeV[2, 15]. It is also noted that to meet the measured spectraof charmonia and bottonia a zero-point energy V is needed.The potential for the hybrid takes three possible forms which are shown in Eq.(6), (8) and (9). We keepthe values m c = 1 .
84 GeV, m b = 5 .
17 GeV which are obtained by fitting the spectra of regular quarkonia | b ¯ b > (Υ( ns )) and | c ¯ c > ( ψ ( ns )) with the potential (4), but need to gain the values of the relevant parameters κ , b and V etc. by minimizing ¯ χ for the decays ψ ( ns )(Υ( ns )) → ψ ( ms )(Υ( ms )) + ππ . According to themeasured value for Γ( ψ (2 S ) → J/ψ π π ):Γ tot ( ψ (2 S )) = 337 ± B ( ψ (2 S ) → J/ψπ + π − ) = (31 . ± . B ( ψ (2 S ) → J/ψπ π ) = (16 . ± . C as a function of the potential parameters which exist in the three potentials ( Eqs.(6), (8) or(9)) and will be determined. It is noted that C is a factor related to the hadronization of gluons into twopions, so should be universal for both ψ and Υ decays. The parameters in the potentials are also universalfor the ¯ bb and ¯ cc cases except the masses are different.Then for Γ(Υ( nS ) → Υ( ms ) + ππ ) ( m < n ), we calculate W thi in terms of the three potential forms. Thecorresponding experimental values and errors are W expi and ∆ W expi given in the references which are shownin Table I. TABLE I: transition rate of Υ( nS ) → Υ( ms ) + ππ , (in unit of keV)decay mode Model 1 Model 2 Model 3 Experiment dataΥ(2 S ) → Υ(1 S ) ππ . ± . S ) → Υ(1 S ) ππ . ± . S ) → Υ(2 S ) ππ . ± . S ) → Υ(1 S ) ππ . ± . ± . S ) → Υ(2 S ) ππ . ± . By minimizing ¯ χ (eq.(3)), we finally get the potential parameters κ, b and V and the resultant ¯ χ =4.42for model 1, 13.69 for model 2 and 7.26 for model 3 . Then we obtain | C | = 100 . × − for model 1,259 . × − for model 2, and 121 . × − for model 3, the other parameters are listed in the followingtable (TableII). TABLE II: potential parameters for hybrid κ b (GeV ) V (GeV)Model 1 0.43 0.19 -0.85Model 2 - 0.15 -0.43Model 3 0.59 0.19 -0.85 With these potential parameters, we solve the Schr¨odinger equation to obtain the masses of ground hybridstates of | c ¯ cg > and | b ¯ bg > (Table V). It is noted that the resultant spectra depend on the potential forms.We will discuss this problem in the last section. TABLE III: the mass of hybrids(in units of GeV)Model 1 Model 2 Model 3 | c ¯ cg > | b ¯ bg > TABLE IV: prediction(in units of KeV)decay mode Model 1 Model 2 Model 3Υ(4 S ) → Υ(3 S ) ππ ψ (3 S ) → ψ (2 S ) ππ ψ (3 S ) → ψ (1 S ) ππ It is noted that values of Γ( ψ (3 S ) → ψ (1 S ) ππ ) predicted by models 1, 2 and 3 are quite apart, whileΓ(Υ(4 S ) → Υ(3 S ) ππ ) and Γ( ψ (3 S ) → ψ (2 S ) ππ ) predicted by all the three models are close. B. Comparison with the lattice results
To make sense, it would be helpful to compare the results obtained in our phenomenological work with thelattice results which are supposed to include both perturbative and non-perturbative QCD effects. Belowwe show comprehensive comparisons of our potentials with the lattice results.Following Refs.[12, 13, 17, 18, 20], the potentials shown in Fig.2 are specially scaled by V Σ + g (2 r ) which isthe potential for Σ + g (N=0) at 2 r = 5 GeV − (for the vertical axis of Fig.2.).In the three graphs of Fig. 2, we present comparisons of the three potentials (models, 1,2 and 3) with theparameters fixed in last subsections with the lattice results. In the graphs, the dots are the lattice values[20].It is emphasized that we obtain the potential by minimizing ¯ χ of the data on ψ ( ns )(Υ( ns )) → ψ ( ms )(Υ( ms )) + ππ , but do not fit the lattice values. Then our results, especially the third potentialcoincides with the lattice results extremely well. It may indicate that the physics description adopted inthis scenario is reasonable. It is also noted that by model 1, the short-distance behavior of the potentialis attractive and obviously distinct from the lattice results. This discrepancy was discussed above that thequark-antiquark system in hybrid should be a color-octet and short-distance interaction should be repulsive.The second potential (model 2) have the same trend as the lattice results, but have obvious deviations (seethe graph 2 of Fig. 2).1 Model 1 u+g r ( V ( r )- V + g ( r )) r/r Model 3 u +g r ( V ( r )- V + g ( r )) r/r Model 2 u +g r ( V ( r )- V + g ( r )) r/r FIG. 2:
C. With the spin-related terms V s For the regular quarkonia we adopt the non-relativistic potential(NR) Eq.(5) [16]. Since we add a zero-point energy V in the potential which can be seen as another free parameter (it is the same for both c ¯ c and b ¯ b quarkonia), we re-fit the spectra of the quarkonia to obtain the corresponding potential parameters inEq.(5). We list the resultant values of the parameters in Table V. In Table VI, we present the fitted spectraof c ¯ c and for a comparison, we also include the results given in Ref.[16] in the table. TABLE V: potential parameters for c ¯ cκ b (GeV ) m (GeV) σ (GeV ) V (GeV)0.67 0.16 1.78 1.6 -0.6TABLE VI: Eignvalues for c ¯ c in GeV J/ψ ψ (2 S ) ψ (3 S ) ψ (4 S ) η c (1 S ) η c (2 S )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 For the b ¯ b quarkonia, the corresponding parameters obtained by fitting data are listed in Table VII.By the parameters we predict m η b = 9 .
434 GeV, which is consistent with that given by [21].2
TABLE VII: potential parameters for b ¯ bκ b (GeV ) m (GeV) σ (GeV )) V (GeV)0.53 0.16 5.13 1.7 -0.60 Then we turn to the hybrid intermediate states.For the hybrids, by the observation made in the previous subsection one can conclude that the thirdpotential (model 3) better coincides with the lattice results, therefore, in this subsection when we includethe spin-related term to discuss spin-splitting case, we only adopt the third potential Eq.(9). It is reasonableto keep the values of m c , m b and σ to be the same as that we determined for pure q ¯ q quarkonia and we alsoset f = 1. Then following our strategy discussed in previous subsections, we obtain the potential parameterswhich are listed in the following table. TABLE VIII: potential parameters for hybrid κ ( c ¯ cg ) κ ( b ¯ bg ) b (GeV ) V (GeV)the best fitted values 0.54 0.40 0.24 -0.80 The fitted values and some predictions are also listed in Tables IX and X. We obtain | C | = 182 . × − , the mass of hybrids are 4.351GeV, 4.333 GeV for the spin-triplet and spin-singlet c ¯ c in the hybrid and10.916GeV, 10.913GeV for the spin-triplet and singlet b ¯ b respectively. Because of including the spin-relatedterm, the “ground states” with the q ¯ q (q=b or c) being in different spin structures would be slightly split.One can observe that the predicted Γ(Υ(4 S ) → Υ(3 S ) ππ ) and Γ( ψ (3 S ) → ψ (2 S ) ππ ) are slightly smallerthan that predicted in the models without the spin-related term, the future experiments may shed some lighton it, namely getting better understanding on the mechanisms which one can describe the hybrid structurebetter.We also calculate the transition rate of η ′ c → η c + π + π , our result is almost triple that obtained in Ref.[4]and it can be tested by the future experiments. It is noted that since we minimize ¯ χ , the decay widthsthat we obtain are different from the central values of the measured quantities. We list the widths we finallyobtained in the table IX.3 TABLE IX: Υ transition(in units of keV)decay mode widths (fit)Υ(2 S ) → Υ(1 S ) ππ S ) → Υ(1 S ) ππ S ) → Υ(2 S ) ππ S ) → Υ(1 S ) ππ S ) → Υ(2 S ) ππ S ) → Υ(3 S ) ππ ψ (3 S ) → ψ (2 S ) ππ ψ (3 S ) → J/ψππ η c (2 S ) → η c ππ IV. OUR CONCLUSION AND DISCUSSION
Search for exotic states which are allowed by the SU(3) quark model and QCD theory is very important forour understanding of the basic theory, but so far such states have not been found (or not firmly identified),thus it becomes an attractive task in high energy physics. No doubt, direct measurements on such exoticstates would provide definite information on them, however, it seems that most of the mysterious states mixwith mesons and baryons which have regular quark structures. Since they are hidden in the mixed states,they are not physical states and do not have physical masses, and it makes a clear identification of such exoticstates very difficult. In other words, they may only serve as a component of physical states. Even though,some phenomenological models, such as the color-flux-tube model, the bag model and the potential modeletc., are believed to properly describe their properties and determine their “masses”, in fact, if they mixwith the regular mesons or baryons, the resultant masses are only the diagonal elements of the Hamiltonianmatrix. For example, in the potential model, by solving Schr¨odinger equation, one obtains the eigen-energyand wave function, he only gets the element E = hyb < φ | H hyb | φ > hyb , where the subscript “hyb” denotesthe quantities corresponding to hybrids. Meanwhile, there is E = reg < φ | H reg | φ > reg corresponding tothe regular quark structure. If the two eigen-states are not far located, they may mix with each otherand provide an extra matrix element to the hamiltonian matrix, as E = E ∗ = hyb < φ | H mix | φ > reg .Unfortunately, there is not a reliable way to calculate the mixing matrix element. One may expect to4gain definite information about the hybrid states and maybe starting from there he can further study themechanism of the mixing.The theoretical framework established by Yan and Kuang confirms that the intermediate states betweentwo pion-emissions in the transition ψ ( ns )(Υ( ns )) → ψ ( ms )(Υ( ms )) + ππ , are hybrids which contain aquark-antiquark pair in color octet, and an extra valence gluon. Based on the color-flux-tube model, in80’s of last century Isgur and Paton suggested a potential model for the hybrid, and this greatly simplifiesthe discussion about hybrids and may offer an opportunity to study the regular quarkonium and hybrid ina unique framework. After their work, several other groups also proposed modified potentials to make abetter description on the hybrid states. When Yan and Kuang studied the transitions, there were not manydata available, i.e. most of the channels were not measured yet. Therefore they assumed that ψ (4 .
03) as theground state of charmed hybrids | c ¯ cg > and estimated the transition rates. Thanks to the great achievementsof the Babar and Belle collaborations, many such modes are measured with appreciable accuracy. Based onthe experimental data and the theoretical framework established by Yan and Kuang, we minimize the ¯ χ to obtain the model parameters in the potential for hybrid, and with them, we can estimate the masses ofthe ground states of hybrids. The theory of the QCD multi-expansion is based on the assumption that thehadronization of the emitted gluons can be factorized from the transition of Υ( ns )( ψ ( ns )) → Υ( ms )( ψ ( ms )).In fact, this factorization may be not complete if the non-perturbative QCD effects are invloved, namely thehigher twist contribution may somehow violate the factorization. However, as long as the non-perturbativeQCD effects are not too strong, this approximation should be acceptable within a certain tolerance range.Moreover, in our study, the non-factorization effects are partly involved in the parameter | C | of Eq. (1),and in our scheme it is also one of the free parameters which are fixed by fitting data. Indeed, it is implicitlyassumed that | C | is universal for all the processes, and it may cause some error. But it is believed thatsince the energy range does not change drastically, the error should controllable.In the calculations, we adopt the Cornell potential for the color-singlet q ¯ q (q=b or c) system and thepotentials suggested by Isgur and Paton (model 1)[11], by Swanson and Szczepaniak (model 2) [12] and byAllen et al (model 3) [13] to deal with the color-octet q ¯ q system, we add a spin-related term to the potentialfor hybrid (model 3 only) to investigate possible spin-splitting effects. The numerical results are slightlydifferent when this term is introduced. The masses of the ground state hybrids are 4.23 GeV for | c ¯ cg > and10.79 GeV for | b ¯ bg > which are estimated in terms of model 3. When the spin-related term is included,the results change to 4.351 GeV, 4.333 GeV for the spin-triplet and spin-singlet c ¯ c in the hybrid and 10.9165GeV, 10.913 GeV for the spin-triplet and singlet b ¯ b respectively. In other two models, the results are slightlydifferent. Indeed as aforementioned, a comprehensive comparison of the results with the lattice values, onemay be convinced that the model 3 may be the best choice at present. All the obtained masses are differentfrom the physical states measured in experiments, and it may imply that the hybrids mix with regularmesons.There are more data in the b-energy range than in charm-energy region. In fact, when we use the samemethod to calculate the transition ψ ( ns ) → ψ ( ms ) + ππ , with n and m being widely apart (say n=4, m=1etc.), the theoretical solutions are not stable and uncertainties are relatively large. It indicates that thereare still some defects in the theory which would be studied in our future works. Moreover, recently Shenand Guo [22] studies the processes in terms of the chiral perturbation theory and considered the final stateinteraction to fit the details of the ππ energy and angular distributions.The transition of higher excited states of quarkonia into lower ones (including the ground state) withoutflavor change but emitting photon or light mesons is believed to offer rich information on the hadron structureand governing dynamics, especially for the heavy quarkonia physics, for example, Brambilla et al .[23] studiedthe quarkonium radiative decays which are realized via electromagnetic interactions.Our studies indicate that the transitions of ψ ( ns )(Υ( ns )) → ψ ( ms )(Υ( ms )) + ππ may provide valuableinformation about the hybrid structures which have so far not been identified in experiments yet.Since we use the method of minimizing ¯ χ to achieve all the parameters in the potential model for hybrids,it certainly brings up some errors. 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