JPC ¼ 1�� hidden charm resonances

J. Segovia, A.M. Yasser,* D. R. Entem, and F. Fernandez

Grupo de Fısica Nuclear and IUFFyM, Universidad de Salamanca, E-37008 Salamanca, Spain(Received 7 October 2008; published 31 December 2008)

The recent B factories measurements of new states exhibit, in most of the decays, charmonium-like

resonances. Among them we study the JPC ¼ 1�� hidden charm resonances in order to assert if they are

q �q states or more complicated structures. The energy spectrum, electromagnetic, and strong decays are

studied in a constituent quark model which has been able to reproduce the hadronic phenomenology and

the baryon-baryon interaction. Most of the reported states are nicely interpreted as q �q pairs. The new

c ð4360Þ state is found as the 4S state and the c ð4415Þ as the 3D which differs from the standard

assignment. Some of the states, such as the Yð4260Þ, cannot be accommodated as q �q pairs.

DOI: 10.1103/PhysRevD.78.114033 PACS numbers: 12.39.Jh, 13.20.Gd, 21.45.�v

I. INTRODUCTION

Today it is widely believed that the standard model ofelementary particles is a well-established theory althoughthe strong interaction is not yet fully under control. Thenon-Abelian gauge theory of strong interactions (QCD) isable to make very precise predictions at high momentumtransfer but still it is not able to reproduce the hadronicproperties when momentum is low. Heavy quark-antiquarksystems are the ideal laboratory to obtain some hints on thebehavior of QCD in this regime.

Recently the field has experienced a very fast develop-ment with a wealth of new data coming from the B facto-ries and other dedicated experiments. A series ofobservations of charmonium-like states, called collectivelyXYZ states, has challenged our understanding of nonper-turbative QCD. These discoveries have triggered a lot ofspeculations about their nature because the number ofstates is greater than one can accommodate in a q �q scheme.Then the possibility of hybrids, molecules, or tetraquarkshas been considered to explain its properties.

In this work we concentrate on the study of the JPC ¼1�� family in order to establish which states can beexplained as q �q pairs and which do not fit in this scheme.The advantage of this set of states lies on the fact that all ofthem have a clear assignment of its JPC value because theyare produced via eþe� annihilation. B factories can inves-tigate a large number of these states by looking into eventswhere the initial state radiation brings the eþe� center ofmass energy down to the ‘‘particle mass.’’ Alternativelydedicated eþe� machines like CESR and BEPC can scandirectly the center of mass energy of interest.

Until the year 2005 five states appeared as well estab-lished in the PDG [1], namely, the J=c , c ð2SÞ, c ð3770Þ,c ð4040Þ, c ð4160Þ, and c ð4415Þ. In that year a new statewas reported by BABAR [2], the Yð4260Þ decaying toJ=c�þ�� soon confirmed by CLEO [3]. Two years later,

investigating whether the Yð4260Þ decayed toc ð2SÞ�þ��, BABAR discovered a new 1�� state, theYð4360Þ [4]. Recently Belle has confirmed all these statesand found two more. The first one Yð4660Þ [5] was notclearly visible in the BABAR data due to the limited statis-tic. The second appears when confirming the Yð4260Þ [6].The Belle experiment indicates that a fit using two inter-fering Breit-Wigner shapes describes the data better thanusing only the Yð4260Þ. The new structure appears atM ¼4008� 40þ114

�28 MeV with a width � ¼ 226� 44�87 MeV. Very recently a combined fit of the BABAR andBelle [7] data gives the best measurement of the Yð4360Þand Yð4660Þ states. With respect to this last state a newpeak with a significance of 8:8� has been observed [8] inthe �þ

c ��c invariant mass distribution of the eþe� !

�þc �

�c process with a mass and width ofM ¼ 4634� 8�

5 MeV and � ¼ 92� 40� 21 MeV suggesting a new1�� state in this energy region.In the meantime the knowledge of the properties of the

PDG states has been also improved and new values fromthe masses, total widths, and leptonic widths have beenreported for the c ð3770Þ, c ð4040Þ, c ð4160Þ, and c ð4415Þstates [9].Finally, one should mention that a new state, the

X(4160), has been observed by the Belle collaboration inthe process eþe� ! J=cD� �D� [10]. It has been specu-lated that this state is the known JPC ¼ 1�� c ð4160Þ.However, a careful study of its properties could rule outthis interpretation assigning a new state to theD-wave spinsinglet charmonium state 1D2ð2DÞ [11].In this work we address the description of the 1��

charmonium states in the framework of the constituentquark model developed in Ref. [12] which successfullydescribes deuteron properties, NN phase-shifts, and had-ron phenomenology [12,13]. We do not only study theenergy spectrum but also the strong and electromagneticdecay widths. The Schrodinger equation is solved by acoupled-channel method which allows to study the possi-bility of S�D couplings. One important characteristic ofthis model is the use of a screened confinement potential

*Permanent address: Phys. Dept., Faculty of Science at Qena,South Valley University, Egypt.

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1550-7998=2008=78(11)=114033(11) 114033-1 � 2008 The American Physical Society

which has been able to reproduce the high excited states inlight meson spectroscopy [14]. This fact is important be-cause we want to describe the high excitation region of thecharmonium spectrum.

The problem has been partially studied by other authorsin different models. Thus Refs. [15,16] describe propertiesof states just below 4.4 GeV. The two new states Yð4360Þand Yð4660Þ are studied in [17] using a different model.However, a unified description of all the states still lacks. Arecent review of the experimental situation can be found in[18].

The paper is organized as follows. After this introduc-tion, in Sec. II we explain the bases of our model.Section III is devoted to the description of the differentdecay mechanisms. The results for the spectrum and decaywidths are presented in Sec. IV. Finally, in Sec. V, wediscuss the recently measured J=c ! 3�width. The paperends with some concluding remarks.

II. THE CONSTITUENT QUARK MODEL

Confinement is one of the crucial aspects of QCD. Colorcharges are confined inside the hadrons and this explainsthe fact that fractional charges (quarks) have not beenobserved, until now, as isolated. It is general knowledgethat multigluon exchanges produce an attractive linearrising potential with the distance which confine the quarks.This idea has been confirmed, but not rigorously proved, byquenched lattice gauge Wilson loop calculations for heavyvalence quark systems. However, sea quarks are also im-portant ingredients of the strong interaction dynamics.When included in the lattice calculations they contributeto the screening of the rising potential at low momenta andeventually to the breaking of the quark-antiquark bindingstring. This fact, which has been observed in nf ¼ 2 lattice

QCD [19], will be important when we study high excitedq �q states or the strong decays of hadrons through lightquark-antiquark pair creation.

Besides confinement the other crucial aspect of QCD isthe spontaneous breaking of chiral symmetry.Independently of the mechanism of this breaking, a con-sequence is that the nearly massless ‘‘bare’’ or ‘‘current’’light quarks develop a dynamical, momentum dependentmassMðpÞ, withMð0Þ � 300 MeV for the u and d quarks,namely, the constituent mass. To compensate this new termin the Hamiltonian a new interaction term (Goldstoneboson exchanges) should appear between constituentquarks.

One may wonder why we speak about light quarks andchiral symmetry breaking when we are faced with thedescription of heavy quark mesons which, due to theirlarge current mass (mc � 1500–1700 MeV), break explic-itly the chiral symmetry of the QCD Lagrangian. Thereason for that lies in the fact that it is widely believedthat confinement is flavor independent. Therefore the in-

teractions, which largely determine the high energy quark-onium spectrum should be constrained by the light sector.Based on these two cornerstones our model can be

summarized in the following way. The picture of theQCD vacuum as a dilute medium of instantons [20] ex-plains nicely such symmetry breaking. The quark interactswith fermionic zero modes of the individual instantons inthe medium and therefore the light quark propagator getsmodified acquiring a momentum dependent mass whichdrops to zero for momentum lighter than the inverse of theaverage instanton size ��.The momentum dependent mass acts as a natural cutoff

of the theory. In the domain of momentum k < 1= �� asimple Lagrangian invariant under chiral transformationscan be derived as [20]

L ¼ �c ði��@� �MU�5Þc ; (1)

where U�5 ¼ expði�a�a�5=f�Þ, �a denotes the pseudo-scalar fields ð ~�;K; �8Þ and M is the constituent quarkmass.Beyond the chiral symmetry breaking scale one expects

the dynamics being governed by QCD perturbative effects.There are consequences of the one-gluon fluctuationsaround the instanton vacuum and we take it into accountthrough the qqg coupling

L gqq ¼ iffiffiffiffiffiffiffiffiffiffiffiffi4��s

p�c��G

�c �cc ; (2)

where �c are the SUð3Þ color matrices and G�c is the gluon

field.Expressions for the nonrelativistic potential coming

from the Lagrangian (2) are given in the appendix.Finally, one should incorporate another nonperturbativeeffect, namely, the confinement. Such a term can be physi-cally interpreted in a picture in which the quark and theantiquark are linked by a one-dimensional color flux tube.The spontaneous creation of light quark pairs may give riseto a breakup of the color flux tube [19]. This can betranslated into a screened potential [21] in such a waythat the potential saturates at the same interquark distance.We use the parametrization

VCONð~rijÞ ¼ f�acð1� e��crijÞ þ �gð ~�ci � ~�c

jÞ; (3)

where� is a global constant to fit the origin of energies. Atshort distances this potential presents a linear behavior

with an effective confinement strength a ¼ �ac�cð ~�ci �

~�cjÞ while it becomes constant at large distances. This

type of potential shows a mass threshold defined by

Vthr ¼ f�ac þ�gð ~�ci � ~�c

jÞ þ 2mc: (4)

No q �q bound states can be found for energies higher thanthis threshold and the system suffers a transition from acolor string configuration between two static color sourcesinto a pair of static mesons due to the breaking of the color

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flux tube and the most favored subsequent decay intohadrons.

To find the quark-antiquark bound states with this inter-action we solve the Schrodinger equation using theGaussian expansion method [22]. In this method the radialwave functions solution of the Schrodinger equation areexpanded in terms of basis functions

R�ðrÞ ¼Xnmax

n¼1

c�nGnlðrÞ; (5)

where � refers to the channel quantum numbers. Thecoefficients c�n and the eigenenergy E are determinedfrom the Rayleigh-Ritz variational principle

Xnmax

n¼1

�ðT�

n0n � EN�n0nÞc�n þX

�0V��0n0n c

�0n ¼ 0

�; (6)

where the operators T�n0n andN

�n0n are diagonal and the only

operator which mixes the different channels is the potential

V��0n0n .

A crucial problem of the variational methods is how tochoose the radial functions G

nlðrÞ in order to have a

minimal, but enough, number of basis functions.Following [22] we employ Gaussian trial functions whoseranges are in geometric progression. The geometric pro-gression is useful in optimizing the ranges with a smallnumber of free parameters. Moreover the distribution ofthe Gaussian ranges in geometric progression is dense atsmall ranges, which is well suited for making the wavefunction correlate with short range potentials. The fastdamping of the Gaussian tail is not a real problem sincewe can choose the maximal range much longer than thehadronic size.

III. STRONG, LEPTONIC, AND RADIATIVEDECAYS

Although masses of meson states are relevant informa-tion about its structure, a more complete description can beachieved studying mesonic decays. In this way we arechecking particular regions of the wave function and notan average over all the meson size as in the calculation ofthe mass spectrum.

The dominant contribution to the decay of quarkoniumstates into lepton pairs proceed via a single virtual photon,as long as the mass of the initial meson state is sufficientlysmall so that the contribution of a virtual Z can be ignored.

The knowledge of the leptonic width of higher charmo-nium states is important for several reasons. First of all itallows to test the wave function at very short range.Moreover it can help to distinguish between conventionalc �c mesons and multiquark structures which have muchsmaller dielectron widths [23].

The leptonic width of n3S1 charmonium including ra-diative QCD corrections is given by [24]

�ðn3S1 ! eþe�Þ ¼ 4�2e2cjRnð0Þj2M2

n

�1� 16�s

3�

�; (7)

where ec ¼ 2=3. Similarly for D-wave c �c states, the lead-ing order decay width into eþe� is given by [25]

�ðn3D1 ! eþe�Þ ¼ 25�2e2c2m4

cM2n

jR00nð0Þj2: (8)

Meson strong decay is a complex nonperturbative pro-cess that still has not been described from first principles.Instead phenomenological models have been developed todeal with this problem. The most popular are the Cornellmodel, the flux-tube model, and the 3P0 model. The

Cornell model [26] assumes that the strong decay takesplace through pair creation from the linear confining po-tential, whereas in the flux tube [27] and in the 3P0 [28]

model the quark-antiquark pair is created from the vacuum.Both are similar but the flux-tube model takes into accountthe dynamics of the flux tubes by including the overlaps ofthe flux tube of the initial meson with those of the twooutgoing mesons. All these models describe reasonablywell the experimental data [29], and we will use for sim-plicity the 3P0 model.The model was first proposed by Micu [28] and further

developed by Le Yaouanc et al. [30]. To describe themeson decay process A ! Bþ C it assumes that a quarkand an antiquark is created with JPC ¼ 0þþ quantumnumbers. The created q �q pair together with the q �q pairin the original meson regroup in the two outgoing mesonsvia a quark rearrangement process. Then, the transitionoperator is given by [31]

T ¼ �3�X�

Zd3pd3p0ð3Þðpþ p0Þ

��Y1

�p� p0

2

�by�ðpÞdy�ðp0Þ

�C¼1;I¼0;S¼1;J¼0

; (9)

where � (� ¼ ��) are the quark (antiquark) quantum num-bers and � is a dimensionless constant that denotes thestrength of the q �q pair creation from the vacuum.Defining the S-matrix as

hfjSjii ¼ I þ ið2�Þ44ðpf � piÞM; (10)

where M is the decay amplitude of the process A ! BþC, the decay width in terms of the partial wave amplitudeusing the relativistic phase space is

� ¼ 2�EBEC

kMA

XJL

jMJLj2; (11)

where k is the on-shell relative momentum of the decayingmesons.The study of radiative decays is an important tool to

determine the internal charge structure of hadrons and itsquantum numbers. These decays produce lower charmo-nium states and allow to study their wave functions. The

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partial width for the E1 transitions between states n2Sþ1LJ

and n02S0þ1L0J0 is given by [32]

�E1ðn2Sþ1LJ ! n02S0þ1L0J0 Þ

¼ 4�e2ck3

3ð2J0 þ 1ÞSEfiSS0 jEfij2

Ef

Mi

; (12)

where SEfi is the statistical factor

SEfi ¼ maxðL; L0Þ�J 1 J0L0 S L

�2; (13)

k is the photon momentum given by k ¼ ðM2i �M2

fÞ=2Mi,

Mi (Mf) is the mass of the initial (final) meson state, and

Ef is the energy of the final state. If the full momentum

dependence is retained the overlap integral Efi is

E fi ¼ 3

k

Z 1

0R�0 ðrÞ

�kr

2j0

�kr

2

�� j1

�kr

2

��R�ðrÞr2dr;

(14)

where � (�0) are the initial (final) meson quantumnumbers.

Although the M1 transitions are in general weaker thanE1 rates, they are nonetheless important because theyallow to access spin states which cannot be reached bythe E1. The M1 width is given by

�M1ðn2Sþ1LJ ! n02S0þ1L0J0 Þ

¼ 4�e2ck3

3m2c

ð2J0 þ 1ÞSMfijMfij2Ef

Mi

; (15)

where we use the same notation as in the E1 case and now

SMfi ¼ 6ð2Sþ 1Þð2S0 þ 1Þ�J 1 J0S0 L S

�2

��

1 1=2 1=21=2 S0 S

�2

(16)

and

M fi ¼Z 1

0R�0 ðrÞj0

�kr

2

�R�ðrÞr2dr: (17)

IV. DISCUSSION OF THE c �c STATES

As stated in the introduction, most of the model parame-ters are constrained by the meson light quark sector. Thenwe take the same parameters as Ref. [12] fine-tuning theconfinement parameters to include the high excited lightmeson states. The resulting parameters are shown inTable I.

In Table II we compare the calculated spectrum with theexperimental data. Most of them are taken from [1] exceptthe ones corresponding to c ð4360Þ and c ð4660Þ which aretaken from the recent combined fit [7]. For these two stateswe have already adopted the standard notation for the 1��states instead of the Y. As one can see the agreement with

the experimental data is remarkable except for two states:the Yð4008Þ and the Yð4260Þ which does not fit in thisscheme. The leptonic widths are compared in Table IIIwith the recent data of the BES collaboration and, onceagain, the agreement is impressive.Let us discuss in more detail the results of Tables II and

III. The four well established c �c states above the D �Dthreshold c ð3770Þ, c ð4040Þ, c ð4160Þ, and c ð4415Þ aregenerally assumed to be a mixture of 3S1 and

3D1 states in

TABLE I. Quark model parameters.

Quark mass mcðMeVÞ 1763

Confinement acðMeVÞ 507.4

�c (fm�1) 0.576

�ðMeVÞ 184.432

as 0.81

OGE �0 2.118

�0 (fm�1) 0.113

�0ðMeVÞ 36.976

r0 (fm) 0.181

rg (fm) 0.259

TABLE II. Masses in MeV of JPC ¼ 1�� c �c mesons. (nL)refers to the dominant partial wave and QM denotes the resultsof the model.

(nL) States QM Exp.

(1S) J=c 3096 3096:916� 0:011 [1]

(2S) c ð2SÞ 3703 3686:09� 0:04 [1]

(1D) c ð3770Þ 3796 3772� 1:1 [1]

Yð4008Þ 4008� 40 [6]

(3S) c ð4040Þ 4097 4039� 1 [1]

(2D) c ð4160Þ 4153 4153� 3 [1]

Yð4260Þ 4260� 10 [2]

(4S) c ð4360Þ 4389 4355þ9�10 � 9 [7]

(3D) c ð4415Þ 4426 4421� 4 [1]ð5SÞð4DÞ

� �c ð4660Þ 4614

4641

� �4661þ9

�8 � 6 [7]

TABLE III. Leptonic decay widths in keV. The experimentaldata are taken from [9] except those denoted with �, which aretaken from [1].

(nL) States QM (MeV) �QM �exp

(1S) J=c 3096 3.93 5:55� 0:14�(2S) c ð2SÞ 3703 1.78 2:43� 0:05�(1D) c ð3770Þ 3796 0.22 0:22� 0:05(3S) c ð4040Þ 4097 1.11 0:83� 0:20(2D) c ð4160Þ 4153 0.30 0:48� 0:22(4S) c ð4360Þ 4389 0.78

(3D) c ð4415Þ 4426 0.33 0:35� 0:12ð5SÞð4DÞ

� �c ð4660Þ 4614

4641

� �0:570:31

� �

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order to reproduce the leptonic widths. The mixing angleranges from � ¼ �17 [15] to � ¼ 34 [33]. In our modelthe mixing is not fitted to the experimental data but drivenby the tensor piece of the quark-antiquark interaction. InTable IV we show the different components of these states.One can see that all are almost pure states, either 3S1 or3D1.

One striking feature of our model is the new assignmentof the c ð4415Þ. Usually this state has been assigned as a 4Sstate. Our particular choice of the potential includes thenew c ð4360Þ as a 4S state between the well establishedc ð4160Þ and c ð4415Þ which are both predicted as Dstates. Whether or not this assignment is correct can betested with the eþe� leptonic widths. From Table III onecan see that the width of the 4S state is 0.78 keV, whereasthe experimental value for the c ð4415Þ is �eþe� ¼ 0:35�0:12 keV, in excellent agreement with the result for the 3Dstate (0.33 keV). This new assignment contradicts theresult of Ref. [17] which identifies the Yð4360Þ with a33D1 c �c state, although with a mass (Mð3D1Þ ¼4455 MeV) far from the experimental result. The measure-ment of the leptonic width for the c ð4360Þ is very impor-tant and would clarify the situation.

Above the c ð4415Þ we found two states with massesM ¼ 4614 MeV (53S1) andM ¼ 4641 MeV (43D1). Bothare in the region of the Yð4660Þ although we will postponea possible assignment until the discussion of the strongdecays. No more states are predicted below 4700 MeV,although we are far from the threshold of our potential(M ¼ 5248 MeV). For completeness, we give in Table Vthe next four states and the most important decayproperties.

In Tables VI, VII, VIII, and IX we show the strong decaywidths of the states discussed above. The notation D1D2

includes both D1�D2 and �D1D2 channels. For the kinemat-

ics we use experimental masses whenever they are avail-able. We only show charge meson channels where the massdifference gives sizeable effects. The parameter � of the3P0 model is adjusted to reproduce the decay rate

c ð3770Þ ! DþD� which is the only decay rate well es-tablished. One can see that the general trends of the totalwidth are well reproduced. There are two particular casesin which the theoretical results exceed the experimentaldata. The first case is the c ð4415Þ where we predict atotal width of 133 MeV, while the PDG [1] average valueis 62� 20 MeV. However, one should mention thatthe experimental data are clustered around two values( � 100 MeV and � 50 MeV) corresponding the lowerone to very old measurements. If we compare our resultswith the recent experimental result of Seth et al. [34] (� ¼119� 16 MeV) we are within the experimental error bar.The second result which disagrees with the experimental

data is the corresponding to the pair of states in the vicinity

TABLE V. Predicted c �c states between 4700 and 5000 MeV.

(nL)

QM

(MeV)

�eþe�

ðkeVÞStrong

decay mode

�strong

ðMeVÞ6S 4791 0.42 D�D�

2 28.6

5D 4810 0.28 D�D� 22.9

7S 4929 0.32 D�D� 11.2

6D 4943 0.24 D�D� 18.7

TABLE IV. 3S1 and 3D1 channel probability in (%).

State QM (MeV) P 3S1

P 3D1

J=c 3096 99.959 0.041

c ð2SÞ 3703 99.958 0.042

c ð3770Þ 3796 0.032 99.968

c ð4040Þ 4097 99.935 0.065

c ð4160Þ 4153 0.060 99.940

c ð4360Þ 4389 99.908 0.092

c ð4415Þ 4426 0.089 99.911

c ð4660Þ 4614 99.884 0.116

c ð4660Þ 4641 0.114 99.886

TABLE VI. Open flavor strong decay widths. The experimen-tal values are from [1] except those denoted with �, which aretaken from [7].

Meson State Mode �QM (MeV) �exp (MeV)

c ð3770Þ 13D1 DþD� 9.5 9:5� 1:4D0 �D0 12.7 12:8� 1:8DD 22.2 22:4� 2:5total 22.2 26:3� 1:9

c ð4040Þ 33S1 DD 3.9

DD� 18.6

D�þD�� 28.6

D�0 �D�0 39.9

D�D� 68.8

DsDs 1.7

total 93.0 80� 10

c ð4160Þ 23D1 DD 19.0

DD� 1.9

D�D� 70.0

DsDs 0.2

DsD�s 5.8

total 96.9 103� 8

c ð4360Þ 43S1 DD 6.7

DD� 6.9

D�D� 7.4

DD�0 0.0

DD1 43.0

DD01 0.7

DD�2 22.7

DsDs 6:4� 10�2

DsD�s 1.6

D�sD

�s 0.8

DsD�s0 0.0

total 89.9 103þ17�15 � 11�

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of 4660 MeV. Both widths are larger than the experimentalresult. The smallest total width of the Yð4660Þ favors the43D1 option for this state although interference betweenthe two states can be the origin of the small experimentalwidth.

One should mention that all the JPC ¼ 1�� statescouple strongly to the DD channels. Recently DD, DD�,and D�D� decay channels, among others, have been mea-sured using initial state radiation in eþe� annihilation [35–

37]. The shape of the mass spectrum is complicated, withseveral maxima and minima, in most of the channels.Reference [35] shows a local maximum at 4160 MeVenergy in the D�þD�� cross section which disappears inthe DþD�� channel. In our results the branching ratio

c ð4160Þ ! DþD��

c ð4160Þ ! D�þD�� � 1:3� 10�2 (18)

could explain this fact. For the rest of the states morestatistics are definitely needed to discuss our results.As a general trend the branching ratios for the c ð4660Þ

seems to be a little bit higher than the one obtained in theexperimental data. That can be due to the fact that the 3P0

model we use has only one parameter � which we fit to thec ð3770Þ ! DþD� decay width. For that reason, themodel may be too simple to describe the branching ratiodetails for decays far from the threshold.Our calculation strongly constrains the Yð4008Þ and

Yð4260Þ structure. Leaving aside the possibility to be1�� c �c states one can think in a tetraquark or molecularstructure. If one of these states was a tetraquark or mo-lecular state with I ¼ 0, JPC ¼ 1�� it would be coupled tothe q �q states with the same quantum numbers and modifyits spectrum as in the case of the light scalar mesons [38].However, the rest of the 1�� states are in agreement withthe experimental data and this possibility should be dis-regarded. One attractive option is that both states were

TABLE VII. Open flavor strong decay widths (continuation).See the discussion in the text for the value of the experimentaldata.

Meson State Mode �QM (MeV) �exp (MeV)

c ð4415Þ 33D1 DD 12.6

DD� 4.9

D�D� 27.3

DD�0 0.0

DD1 54.5

DD01 5.5

DD�2 19.7

D�D�0 5.9

DsDs 0.3

DsD�s 0.6

D�sD

�s 1.8

DsD�s0 0.0

total 133:1 62� 20

TABLE VIII. Open flavor decay widths (continuation). Herethe experimental value is from [7].

Meson State Mode �the (MeV) �exp (MeV)

c ð4660Þ 53S1 DD 5.5

DD� 22.0

D�D� 13.1

DD�0 0.0

DD1 2.9

DD01 3.3

DD�2 2:0� 10�3

D�D�0 5.8

D�D1 28.1

D�D01 9.5

D�D�2 67.3

DsDs 0.8

DsD�s 0.3

D�sD

�s 1.0

DsD�s0 0.0

DsDs1 1.3

DsD0s1 0.5

DsD�s2 0.2

D�sD

�s0 2.4

D�sDs1 3:3� 10�2

D�sD

0s1 0.5

D�s0D

�s0 0.1

total 164.6 42þ17�12 � 6�

TABLE IX. Open flavor decay widths (continuation). Here theexperimental value is from [7].

Meson State Mode �QM (MeV) �exp (MeV)

c ð4660Þ 43D1 DD 9.1

DD� 6.3

D�D� 31.9

DD�0 0.0

DD1 1.8

DD01 0.8

DD�2 5:1� 10�7

D�D�0 2.9

D�D1 23.7

D�D01 6.7

D�D�2 18.3

DsDs 0.8

DsD�s 5:0� 10�4

D�sD

�s 0.3

DsD�s0 0.0

DsDs1 3.0

DsD0s1 0.4

DsD�s2 7:2� 10�2

D�sD

�s0 1.0

D�sDs1 0.1

D�sD

0s1 0.1

D�s0D

�s0 0.2

total 107.5 42þ17�12 � 6�

SEGOVIA, YASSER, ENTEM, AND FERNANDEZ PHYSICAL REVIEW D 78, 114033 (2008)

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isovector multiquarks favored by the fact that the one pionexchange potential is attractive in this channel for thequark-antiquark light system. This analysis coincideswith those of Ref. [39].

Finally, in Tables X, XI, XII, and XIII we quote the E1and M1 radiative transitions. The results are in generalagreement with the scarce experimental results. CLEOhas recently reported [40] results on c ð3770Þ ! � cJ

with J ¼ 0, 1 and an upper limit for J ¼ 2. Also upperlimits are placed on the decays c ð4040Þ ! � cJ andc ð4160Þ ! � cJ. Our results slightly overestimate theexperimental results which may indicate that coupled-channel corrections may be significant in this case.

Radiative transitions can be used to elucidate the struc-ture of the c ð4660Þ looking to the ratios

c ð53S1Þ ! c0ð13P0Þ�c ð23S1Þ ! c0ð13P0Þ�

� 0:3;

c ð43D1Þ ! c0ð13P0Þ�c ð23S1Þ ! c0ð13P0Þ�

� 1:6;

(19)

where the predicted values differ in a factor 5.

Finally, the formula for the E1 transitions can be used tocalculate the branching fraction of the process c ð2SÞ !J=c�� through � cJ. In Table XIV we compare ourresults with those of Ref. [41]. We reproduce not onlythe tendency of the experimental data but the agreementof the absolute value is also good.

V. THE J=c ! 3� WIDTH

Recently the CLEO collaboration has measured thebranching ratio of the J=c ! 3� decay using thec ð2SÞ ! �þ��J=c events acquired with the CLEO-C

TABLE X. E1 radiative transitions. The experimental data arefrom [1].

Initial

meson

Final

meson Bexp BQM

�QM

ðkeVÞc ð23S1Þ c2ð13P2Þ ð8:1� 0:4Þ � 10�2 10:0� 10�2 28.6

c1ð13P1Þ ð8:8� 0:4Þ � 10�2 11:5� 10�2 33.0

c0ð13P0Þ ð9:3� 0:4Þ � 10�2 10:1� 10�2 28.8

c ð13D1Þ c2ð13P2Þ <9:0� 10�4 1:2� 10�4 3.3

c1ð13P1Þ ð2:9� 0:6Þ � 10�3 3:2� 10�3 89.7

c0ð13P0Þ ð7:3� 0:9Þ � 10�3 8:0� 10�3 221.7

c ð33S1Þ c2ð13P2Þ <1:7� 10�2 1:3� 10�4 10.1

c1ð13P1Þ <1:1� 10�2 1:6� 10�4 12.8

c0ð13P0Þ 11.7

c2ð23P2Þ 46.6

c1ð23P1Þ 13.6

c0ð23P0Þ 10.3

c ð23D1Þ c2ð13P2Þ <1:3� 10�2 9:7� 10�6 1.0

c1ð13P1Þ <7� 10�3 4:0� 10�4 41.4

c0ð13P0Þ 113.6

c2ð23P2Þ 6.4

c1ð23P1Þ 87.4

c0ð23P0Þ 133.7

c ð43S1Þ c2ð13P2Þ 14.1

c1ð13P1Þ 14.9

c0ð13P0Þ 11.6

c2ð23P2Þ 60.6

c1ð23P1Þ 32.4

c0ð23P0Þ 18.9

c2ð33P2Þ 66.1

c1ð33P1Þ 49.7

c0ð33P0Þ 26.0

TABLE XI. E1 radiative transitions (continuation). No experi-mental data are available.

Initial meson Final meson �QMðkeVÞc ð33D1Þ c2ð13P2Þ 0.4

c1ð13P1Þ 22.7

c0ð13P0Þ 67.2

c2ð23P2Þ 2.0

c1ð23P1Þ 45.5

c0ð23P0Þ 85.5

c2ð33P2Þ 2.4

c1ð33P1Þ 60.0

c0ð33P0Þ 89.0

c ð53S1Þ c2ð13P2Þ 10.8

c1ð13P1Þ 11.2

c0ð13P0Þ 8.7

c2ð23P2Þ 42.8

c1ð23P1Þ 25.3

c0ð23P0Þ 14.9

c2ð33P2Þ 38.6

c1ð33P1Þ 32.8

c0ð33P0Þ 18.9

c2ð43P2Þ 55.5

c1ð43P1Þ 41.8

c0ð43P0Þ 22.1

TABLE XII. E1 radiative transitions (continuation). No ex-perimental data are available.

Initial meson Final meson �QMðkeVÞc ð43D1Þ c2ð13P2Þ 0.2

c1ð13P1Þ 15.3

c0ð13P0Þ 46.5

c2ð23P2Þ 1.2

c1ð23P1Þ 32.8

c0ð23P0Þ 65.2

c2ð33P2Þ 1.3

c1ð33P1Þ 41.4

c0ð33P0Þ 70.5

c2ð43P2Þ 1.9

c1ð43P1Þ 49.4

c0ð43P0Þ 69.4

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114033-7

detector operating at the CESR eþe� collider. A signal of6� significance was found with branching fraction B3� ¼ð1:2� 0:3� 0:2Þ � 10�5 [42]. This value is almost a fac-tor 5 below the one expected from the J=c ! eþe� decayignoring QCD corrections which suggests that these cor-rections can be large. Then the understanding of this dis-agreement is very important because it can shed some lighton the knowledge of the behavior of QCD loop expansionwhen its coefficients are large.

The expression for the width of the J=c ! 3� can becalculated in the formalism developed by Glover andMorgan for one boson decay to three photons [43]. Thecalculation is analogous to the decay of the orthopositro-nium (o-Ps) to three photons which has the same quantumnumbers. This rate has been extensively studied in the pastmainly due to the discrepancies that, from time to time,appeared between the successive improvements of theexperimental data and the theoretical corrections. Thetheoretical results from o-Ps decay can be written as

�0 ¼ 16

9

ð�2 � 9Þm2

�3e6j’ð0Þj2: (20)

This formula coincides with the old Ore and Powell [44]expression if one takes for the wave function at the originthe hydrogenlike wave function value.To obtain the J=c ! 3�width from the expression (20)

one has to use eq ¼ 2=3 instead of e ¼ �1 and multiply by

a factor 3 to take into account the three quark colors,leading to the final expression

�0ðJ=c ! 3�Þ ¼ 4

3�

ð�2 � 9Þm2

q

�3e6qjR10ð0Þj2; (21)

where we have separated the radial part from the total wavefunction. Using the wave function calculated in Sec. IV weobtain B3� ¼ 3:4� 10�5 where we have used the experi-

mental value for the total width. This value is almost threetimes the experimental result.One loop QCD corrections can be adapted from the

expressions for o-Ps decay. This correction has been cal-culated in Ref. [45] using the formalism of Ref. [43] in theframework of the nonrelativistic QED. In Fig. 1 we showthe diagrams that enter in the one loop correction to thedecay amplitude. Following [45] we refer to the six dia-grams as the outer vertex (OV), inner vertex (IV), doublevertex (DV), self-energy (SE), ladder (L), and annihilationvertex (A). The separated contributions of each diagramcan be found in [45]. Collecting all the contributionstogether we arrive at

� ¼ �0

�1þ A

��s

�

��; (22)

where the best value for A is given by [45]

A ¼ �10:286 606: (23)

To adapt this diagram to the QCD case one has to

include a factor �c

2 in each vertex diagram which contrib-

utes with a global factor 4=3. Furthermore, the diagram Adoes not contribute to the QCD case due to color conser-vation. Summing up all the contributions one arrives at theexpression

TABLE XIII. M1 radiative transitions. The experimental dataare from [1].

Initial

meson

Final

meson Bexp BQM

�QM

ðkeVÞc ð13S1Þ �cð11S0Þ ð1:3� 0:4Þ � 10�2 2:1� 10�2 2.0

c ð23S1Þ �cð21S0Þ <2:0� 10�3 5:5� 10�4 0.2

�cð11S0Þ ð3:0� 0:5Þ � 10�3 2:5� 10�2 7.2

c ð33S1Þ �cð31S0Þ 4:6� 10�3

�cð21S0Þ 1.6

�cð11S0Þ 6.6

c ð43S1Þ �cð41S0Þ 4:6� 10�2

�cð31S0Þ 1.2

�cð21S0Þ 3.5

�cð11S0Þ 7.2

c ð53S1Þ �cð51S0Þ 2:3� 10�2

�cð41S0Þ 0.6

�cð31S0Þ 1.7

�cð21S0Þ 3.5

�cð11S0Þ 6.3

TABLE XIV. Branching ratio for the decays c ð2SÞ !�ð�J=c Þ cJ

. Experimental data are from [41].

Mode Bexpð%Þ BQMð%Þ�ð�J=c Þ c0

0:125� 0:007� 0:013 0.122

�ð�J=c Þ c13:56� 0:03� 0:12 3.455

�ð�J=c Þ c21:95� 0:02� 0:07 1.658

OV IV DV SE

AL

FIG. 1. One loop diagrams that contribute to the decay ampli-tude at first order in �s.

SEGOVIA, YASSER, ENTEM, AND FERNANDEZ PHYSICAL REVIEW D 78, 114033 (2008)

114033-8

� ¼ �0

�1þ AQCD

��s

�

��(24)

with

AQCD ¼ �12:630 (25)

which coincides with a similar expression in Ref. [46].Now, using the value of the �sðmcÞ coupling constant at

the charm quark mass (�s ¼ 0:288) the first correction tothe width cancels with the zeroth order approximationgiven a theoretical prediction compatible with zero.

Because of the strong cancellation between the first twoterms in Eq. (24), the most important contribution to thedecay rate comes from the ð�s

� Þ2 term. This correction has

two different sources. The first one arises when we squarethe one loop corrections to the annihilation amplitude. Thesecond one corresponds to the leading order two loopcorrections. This part cannot be straightforwardly deducedfrom the two loop QED corrections due to the non-Abeliancharacter of QCD, and its calculation is beyond the scopeof the present work.

However, we can naively estimate the first contributionas

B 1;QCD ¼�AQCD

2

�2: (26)

In the o-Ps case Burichenko has shown [47] that thisestimate is in fact a lower bound for the contribution underconsideration. Thus we estimate the J=c ! 3� decay rateas

�ðJ=c ! 3�Þ¼ 3:17� 10�3

�1� 12:630

��s

�

�þ 39:879

��s

�

�2�

(27)

expressed in keV.With this new correction the theoretical value is now

B3� ¼ 0:6� 10�5 lower than the experimental data but

near the experimental error bar.This result shows the importance of the high order

corrections in this particular decay. The leading order ofthe two loop corrections, which are almost of the same sizeas the one coming from the square of one loop correctionsin the orthopositronium decay [48] should contribute toobtain more precise results.

VI. SUMMARY

We have considered the properties of the JPC ¼ 1��charmonium sector using a constituent quark model con-strained by the light meson spectra. An important charac-teristic of the model is the use of a screened confinementpotential which improves the description of the high ex-cited meson states. We confirm that, besides the wellestablished J=c and c states, the c ð4360Þ and the

c ð4660Þ can be considered as 1�� charmonium states.The c ð4360Þ is assigned as the 4S state and so thec ð4415Þ is the 3D state which differs from the standardassignment. However, the Yð4008Þ and the Yð4260Þ statescannot be accommodated in the spectrum and probably itsstructure is more complex than a simple c �c pair. All thestates described by the model are almost pure states eithern3S1 or n3D1. Nevertheless, the leptonic width of all ofthem are consistent with the experimental data due to theflattening of the confinement potential. We have studiedthe E1 and M1 radiative transitions and the open-charmstrong decays using the 3P0 model in detail. Almost all the

total strong widths of the different states agree with theexperiment and the discrepancies are discussed. Thec ð4360Þ is predicted to dominantly decay into DD1 andDD�

2. The c ð4660Þwill decay predominantly intoD�D�2 in

the case of a 53S1 assignment but the preferred channelwould be D�D� and D�D1 if its structure corresponds witha 43D1 configuration. Finally, we have shown that theJ=c ! 3� is an ideal system to study high order QCDcorrections because in this decay the first order in ð�s

� Þalmost cancels the leading order. We have partially esti-mated the second order corrections in ð�s

� Þ and show that

they are in reasonable agreement with the recent reportedexperimental data.

ACKNOWLEDGMENTS

This work has been partially funded by Ministerio deCiencia y Tecnologıa under Contract No. FPA2007-65748and by Junta de Castilla y Leon under Contract No. SA-106A07. A.M.Y. would like to acknowledge the SouthValley University and Higher Education Ministry ofEgypt for financial support.

APPENDIX: THE QUARK-QUARK POTENTIAL

In this appendix we give explicit expressions for thedifferent pieces of the quark-quark potentials. They containcentral, tensor, and spin-orbit terms that will be grouped forconsistency.

1. Gluon exchange interaction

The central part of the OGE reads

VCOGEð~rijÞ ¼

1

4�sð ~�c

i � ~�cj�1

rij� 1

6mimj

ð ~�i � ~�jÞe�rij=r0ð�Þ

rijr20ð�Þ

�;

(A1)

where the delta function is regularized in a suitable wayand r0ð�Þ ¼ r0

�nn

�ijscales with the reduced mass as ex-

pected for a Coulombic system.The noncentral potentials, tensor, and spin orbit, have

the form

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114033-9

VTOGEð ~rijÞ ¼ � 1

16

�s

mimj

ð ~�ci � ~�c

j�1

r3ij� e�rij=rgð�Þ

rij

�1

r2ijþ 1

3r2gð�Þ þ1

rijrgð�Þ��

Sij;

VSOOGEð ~rijÞ ¼ � 1

16

�s

m2i m

2j

ð ~�ci � ~�c

j�1

r3ij� e�rij=rgð�Þ

r3ij

�1þ rij

rgð�Þ��

� ½ððmi þmjÞ2 þ 2mimjÞð ~Sþ � ~LÞ þ ðm2j �m2

i Þð ~S� � ~LÞ�;(A2)

where ~S� ¼ ~Si � ~Sj and rgð�Þ ¼ rg�nn

�ijscales as the cen-

tral term. Finally, we use an effective scale dependentstrong coupling constant given by

�sð�Þ ¼ �0

lnð�2þ�20

�20

Þ; (A3)

where� is the reduced mass of the q �q pair and �0,�0, and�0 are parameters of the model [12].

2. Confinement

The screened confinement potential is written as

VCCONð~rijÞ ¼ ½�acð1� e��crijÞ þ ��ð ~�c

i � ~�cjÞ: (A4)

The spin-orbit contribution is an arbitrary combination ofscalar and vector terms

VSOCONð ~rijÞ ¼ �ð ~�c

i � ~�cjÞac�ce

��crij

4m2i m

2jrij

½ððm2i þm2

j Þð1� 2asÞ

þ 4mimjð1� asÞÞð ~Sþ � ~LÞþ ðm2

j �m2i Þð1� 2asÞð ~S� � ~LÞ�; (A5)

where as would control the ratio between them.

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