Mass of Xð3872Þ in the relativistic quark model

Xiaozhao Chen,1 Bin Wang,1 Xiaoya Li,1 Xiaoqiang Zeng,1 Shaolan Yu,1 and Xiaofu Lu1,2,3,*1Department of Physics, Sichuan University, Chengdu, 610064, China

2Institute of Theoretical Physics, The Chinese Academy of Sciences, Beijing 100080, China3CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China

(Received 18 November 2008; revised manuscript received 17 April 2009; published 5 June 2009)

Assuming that the narrow state Xð3872Þ is a molecule state consisting of primarily D0 �D�0, we calculateits mass within the constituent quark model. The attractive potential between D0 and �D�0 including one-�exchange and one-� exchange is considered. We introduce the interaction between the heavy meson and

light meson through the heavy meson form factor describing the heavy meson structure. The obtained

mass of X is consistent with the experimental value. Adopting this method, we study the various molecule

states with hidden bottom quark.

DOI: 10.1103/PhysRevD.79.114006 PACS numbers: 12.40.Yx, 12.39.Ki, 14.40.Gx

I. INTRODUCTION

The narrow state Xð3872Þwas discovered in 2003 by theBelle Collaboration [1]. The narrow state was then con-firmed by the CDF [2], D0 [3], and BABAR [4]collaborations.

The investigation of the structure of X is of great sig-nificance, while the structure has not been determined. Upto now, many theoretical interpretations have been pro-posed, including a loosely bound molecule state of D0 �D�0[5,6], a diquark-antidiquark bound state [7,8], a hybridcharmonium state [9], a tetraquark state [10], and a excitedcharmonium state, etc. Originally, the molecule state hy-pothesis was proposed for the reason that the mass of X liesclose to D0 �D�0 threshold. [5,6] In addition, the analysis ofthe decay angular distributions (JPC ¼ 1þþ for Xð3872Þ)[11] strongly sustains this hypothesis, and implies thatD0 �D�0 is the most likely bound state in relative S wave.However, in previous works, [5,6] the effective interactionpotential between D0 and �D�0 is obscure, for it containssome parameters that are dubious, and these heavy mesonsin the molecule state are considered as pointlike objects.

In this work, we assume that the narrow state is amolecule state and calculate the interaction potential be-tween D0 and �D�0 without an extra parameter. If themolecule state is composed of two heavy mesons, it canbe regarded as a two-body system. Because of the parityconservation, in this system the interaction potential be-tween D0 and �D�0 can be derived from one-� exchangeand one-� exchange if only one particle exchange is con-sidered. The internal structure of these heavy mesons istaken into account by the calculated heavy meson formfactor and its effect will be included in the interactionbetween the heavy meson and light meson. In order toobtain the form factor, we calculate the matrix elementof the quark current between heavy meson states, as inRefs. [12,13]. In this approach, first of all, we have to knowthe wave functions of two heavy mesons.

On the other hand, according to the assumption that D0,�D�0 are composed of two constituent quarks, the heavymeson wave functions can be obtained by solving Bethe-Salpeter equation. The Bethe-Salpeter (BS) amplitudes ofthese heavy mesons were obtained [14–17]. Then fromthese BS amplitudes we can calculate the heavy mesonform factors. The attractive potential between two heavymesons can be calculated from the form factors and theselight meson propagators. Finally, numerically solving therelativistic Schrodinger-like equation with this attractivepotential, we obtain the mass of the molecule state.Our paper is presented as follows: In Sec. II, we show

the quasipotential equation of the Schrodinger type thatdescribes the molecule state. In Sec. III, we show how toget the heavy meson wave functions from the Bethe-Salpeter amplitudes calculated in previous works. Usingthese wave functions, we calculate the heavy meson formfactors. Then the interaction potential and the mass of Xare calculated. Section IV presents our conclusion.

II. THE QUASIPOTENTIAL EQUATION

As above shown, we assume that the Xð3872Þ is amolecule state, a bound state composed of a heavy pseu-doscalar meson (D0) and a heavy vector meson ( �D�0). Inthe quasipotential approach the interaction of two heavymesons in the bound state is described by the moleculestate wave function, which satisfies the quasipotentialequation [18] of the Schrodinger type [19]�

b2ðMÞ2�R

� p02

2�R

��ðp0Þ ¼

Z d3q0

ð2�Þ3 Vðp0;q0;MÞ�ðq0Þ; (1)

where the relativistic reduced mass is

�R ¼ E1E2

E1 þ E2

¼ M4 � ðM21 �M2

2Þ24M3

; (2)

and E1, E2 are given by

E1 ¼ M2 �M22 þM2

1

2M; E2 ¼ M2 �M2

1 þM22

2M: (3)

*luxiaofu@gmail.com; corresponding author

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Here, M ¼ E1 þ E2 is the molecule state mass, M1;2 rep-

resent the masses of the pseudoscalar meson (D0) and thevector meson ( �D�0), and p0 is their relative momentum. Inthe center-of-mass system the relative momentum squaredon mass shell reads

b2ðMÞ ¼ ½M2 � ðM1 þM2Þ2�½M2 � ðM1 �M2Þ2�4M2

: (4)

The kernel Vðp0;q0;MÞ in Eq. (1) is the quasipotentialoperator of the meson-meson interaction. With the help ofthe off-mass-shell scattering amplitude projected onto thepositive energy states, the quasipotential is constructed. Toconstruct the quasipotential of the meson-meson interac-tion, we consider that the effective interaction is derivedfrom the usual one light meson ð�;�Þ exchange. In thequasipotential approach, it is necessary that the final par-ticle is same to the initial onewhose momentum is differentfrom the former’s, as shown in Fig. 1. Therefore, from theparity conservation, the coupling term in the Lagrangiandescribing the interaction between the heavy and lightmeson cannot contain the one-� exchange. In this paper,we consider one-� exchange and one-� exchange betweentwo heavy mesons to construct the quasipotential.

III. THE INTERACTION BETWEEN HEAVYMESONS

In quark model mesons are the simplest bound statescomposed of a dressed-quark and a dressed-antiquark. TheBethe-Salpeter equation is recognized as a exact method todeal with the two-body bound state, and this method wasused to investigate the properties of heavy mesons [14–17].The Bethe-Salpeter amplitudes of the heavy mesons havethe form: [17]

For heavy pseudoscalar mesons (Pse), the BS amplitudeis

�Pseðp;PÞ ¼ 1

NPsei�5’Hðp2Þ; (5)

and for heavy vector mesons (V), the amplitude is

�V�ðp;PÞ ¼ 1

NV

��� þ P�

� � PM2

V

�’Hðp2Þ; (6)

where P is the momentum of the heavy meson, which is setas P ¼ ð0; 0; 0; iMHÞ in the rest frame, p becomes therelative momentum between quark and antiquark, N isthe normalization, and ’Hðp2Þ ¼ expð�p2=!2

HÞ. Theseparable Ansatz that describes the dressed-quark propa-gators was used to solve the Bethe-Salpeter equation.Though the Ansatz is a phenomenological approximation,its algebraic form combines the effects of confinement anddynamical chiral symmetry breaking with free-particlebehavior at large momentum. [14–17] Therefore, the BSamplitudes obtained with this ansatz contain a great deal ofinformation describing two quarks in mesons, and theseparameters in these amplitudes can not be adjusted freely.In the constituent quark model, the heavy meson D iscomposed of a heavy quark c and a light antiquark �u. Forthe heavy meson D, we take !D ¼ !D� ¼ 1:81 Gev as inRef. [17]. Then the Bethe-Salpeter wave functions areobtained

� ¼�S1ðpþ PÞ�Pseðp;PÞS2ðpÞ for pseudosclar meson

S1ðpþ PÞ�V�ðp;PÞS2ðpÞ for vector meson

;

(7)

where S1ðpþ PÞ ¼ �i��pþ��P�im1

and S2ðpÞ ¼ �i��p�im2

rep-

resent two quark propagators in the constituent quarkmodel, m1;2 are the constituent quark masses.

As defined in Refs. [20,21], the heavy meson formfactors can be calculated from the wave functions of theseheavy mesons. In Eq. (7), the BS wave function is ex-pressed as a 4� 4matrix. This matrix can be written in the16 linear independent matrices as 1, ��, ���, ���5, �5.The BS wave function can be written as

� ¼ �S þ�V��

� þ�T���

�� þ�AV� ���5 þ�Pse�5:

(8)

The corresponding wave function in the quasipotentialapproach can be obtained from Eq. (8):

σ, ρ

FIG. 1. The light meson exchange between two heavy mesons.

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(1) for heavy pseudoscalar meson

�PseðpÞ ¼ 1

4trf�5�g

¼ 1

4tr

��5

�i

� � pþ � � P� im1

i

NPse

� �5’D0ðp2Þ �i

� � p� im2

�

¼ i

NPse’D0ðp2Þ 1

p2 þm21

1

p2 þm22

� ðp2 þm1m2Þ; (9)

(2) for heavy vector meson.The vector meson wave function is a three-vector, itscomponents are

�Vi ðpÞ ¼

1

4trf�i�g

¼ 1

4tr

��i

�i

� � pþ � � P� im1

1

NV

���i þ Pi

� � PM2

�D�0

�’ �D�0ðp2Þ �i

� � p� im2

�

¼ 1

NV ’ �D�0ðp2Þ 1

p2 þm21

1

p2 þm22

� ðp2i þm1m2Þ i ¼ 1; 2; 3; (10)

then the three-vector wave function is equal to

�VðpÞ ¼ffiffiffi3

pNV ’ �D�0ðp2Þ 1

p2 þm21

1

p2 þm22

��p2

3þm1m2

�p; (11)

where p is the unit momentum.Firstly, we calculate the interaction potential derived

from one-� exchange. Because the heavy meson is con-sidered to have internal structure, in order to determine itsinteraction with light scalar meson, it is necessary tocalculate the form factors of heavy mesons which are asfollows: [12,13]

(1) heavy Pse

hPseðQÞjJjPseðPÞi ¼ h�ðk2Þ; (12)

(2) heavy vector meson (V)

hVðQÞjJjVðPÞi ¼ �½"�ðQÞ � "ðPÞ�h1ðk2Þþ h2ðk2Þ 1

MV

f½"�ðQÞ � P�

þ ½"ðPÞ �Q�g þ h3ðk2Þ 1

M2V

� ½"�ðQÞ � P�½"ðPÞ �Q�; (13)

where k ¼ Q� P is the momentum of the lightmeson and "ðPÞ is the polarization vector of theheavy vector meson.

In the quasipotential approach, the matrix element of thequark current J ¼ �qq between the heavy meson states (H)has the form [20,21]

hHðQÞjJð0ÞjHðPÞi ¼Z d3qd3p

ð2�Þ6��HQðqÞ�ðp;qÞ�H

P ðpÞ;(14)

where �ðp;qÞ is the two-particle vertex function, and �HQ

are the wave functions of pseudoscalar meson and vectormeson projected onto the positive energy states of quarksand boosted to the moving reference frame with momen-tumQ. The vertex function � in the impulse approximationis shown in Fig. 2. The corresponding vertex function ofthe quark-meson interaction is given by

�ð1Þðp;qÞ ¼ �uq1ðq1Þuq1ðp1Þð2�Þ3�ðq2 � p2Þþ �v �q2ðq2Þv �q2ðp2Þð2�Þ3�ðq1 � p1Þ; (15)

where uðpÞ and vðpÞ are spinors

uq1ðpÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ðpÞ þm1

21ðpÞ

s1��p

1ðpÞþm1

!�;

v�q2ðpÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðpÞ þm2

22ðpÞ

s��p

2ðpÞþm2

1

� ��;

(16)

with 1;2ðpÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 þm2

1;2

q, m1 ¼ mc, m2 ¼ m �u, and

[20,21]

p1;2 ¼ 1;2ðpÞ P

MH

�X3i¼1

nðiÞðPÞpi;

MH ¼ 1ðpÞ þ 2ðpÞ;

q1;2 ¼ 1;2ðqÞ Q

MH

�X3i¼1

nðiÞðQÞqi;

MH ¼ 1ðqÞ þ 2ðqÞ;and nðiÞ are three four vectors defined by

FIG. 2. The vertex function � in the impulse approximation.The light meson interaction with c quark and u antiquark isshown.

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nðiÞ�ðPÞ ¼�Pi

MH

; �ij þ PiPj

MH½EHðPÞ þMH��;

EHðPÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP2 þM2

H

q:

The first term in Eq. (15) represents the light scalar mesoninteraction with the c quark in the heavy meson, while thesecond term is its interaction with the u antiquark.

Substituting the vertex function �ð1Þ given by Eq. (15)into the matrix element (14) and comparing the resultingexpressions with the form factor decompositions (12) and(13), we obtain

h�ðk2Þ ¼ h1ðk2Þ ¼ h2ðk2Þ ¼ F1ðk2Þ; h3ðk2Þ ¼ 0;

F1ðk2Þ ¼Z d3p

ð2�Þ3��H

�pþ 22ðpÞ

EH þMH

k

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ðpÞ þm1

1ðpþ kÞ þm1

s

��1ðpþ kÞ � 1ðpÞ þ 2m1

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ðpþ kÞ1ðpÞ

p� pk

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ðpþ kÞ1ðpÞ

p ð1ðpÞ þm1Þ��HðpÞ

� ð1$ 2Þ; (17)

where �H are the heavy meson wave functions, expressedas Eqs. (9) and (11). The functions F1ðk2Þ for the pseudo-scalar meson D0 and the vector meson �D�0 are shown inFig. 3.

Applying the Feynman rules, we obtain the interactionpotential derived from one-� exchange, as shown in Fig. 1

V1ðk2Þ ¼ �FPse1 ðk2Þ 1

k2 þm2�

FV1 ðk2Þ; (18)

the superscript of form factor represents the � meson

interaction with the heavy pseudoscalar or vector meson,respectively.Now, we consider one-� exchange between two heavy

mesons. The quark current in Eq. (12) and (13) is replacedby J� ¼ �q��q, and the form factors become [12,13]

(1) heavy Pse

hPseðQÞjJ�jPseðPÞi ¼ h�ðk2ÞðPþQÞ�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEPseMPse

p ; (19)

(2) heavy vector meson (V)

hVðQÞjJ�jVðPÞi¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

EVMV

p��½"�ðQÞ �"ðPÞ�h1ðk2ÞðPþQÞ�

þh2ðk2Þf½"�ðQÞ �P�"�ðPÞþ ½"ðPÞ �Q�"��ðQÞgþh3ðk2Þ 1

M2V

½"�ðQÞ �P�½"ðPÞ �Q�ðPþQÞ��:

(20)

The matrix element of this quark current betweenthe heavy meson states has the form

hHðQÞjJ�ð0ÞjHðPÞi

¼Z d3qd3p

ð2�Þ6��HQðqÞ��ðp;qÞ�H

P ðpÞ; (21)

and the corresponding vertex function is given by

�ð1Þ� ðp;qÞ ¼ �uq1ðq1Þ��uq1ðp1Þð2�Þ3�ðq2 � p2Þ

þ �v �q2ðq2Þ��v �q2ðp2Þð2�Þ3�ðq1 � p1Þ:(22)

Adopting the same procedure as above, we obtainthe form factor corresponding to one-� exchange

F2ðk2Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiEHMH

pEH þMH

Z d3p

ð2�Þ3��H

�pþ 22ðpÞ

EH þMH

k

�

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ðpÞ þm1

1ðpþ kÞ þm1

s �1ðpþ kÞ þ 1ðpÞ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ðpþ kÞ1ðpÞ

pþ pk

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ðpþ kÞ1ðpÞ

p ð1ðpÞ þm1Þ��HðpÞ

þ ð1$ 2Þ; (23)

and the interaction potential derived from one-�exchange is

V2ðk2Þ ¼ �FPse2 ðk2Þ 1

k2 þm2�

FV2 ðk2Þ: (24)

The functions F2ðk2Þ for the pseudoscalar mesonD0 and the vector meson �D�0 are shown in Fig. 4.Finally, the total interaction potential between twoheavy mesons includes the scalar potential and the

FIG. 3. The form factors for the vertexes of heavy mesonscoupling to � meson. The solid line represents the pseudoscalarmeson D0; the dashed line represents the vector meson �D�0.

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vector potential

Vðk2Þ ¼ V1ðk2Þ þ V2ðk2Þ: (25)

The constituent quark masses mc ¼ 1:55 Gev, mu ¼0:33 Gev, and the meson masses m� ¼ 0:56 Gev, m� ¼0:77 Gev,mD0 ¼ 1:865 Gev,m �D�0 ¼ 2:007 Gev. Then thequasipotential Eq. (1) is solved numerically with the po-tential Vðk2Þ, and the calculated molecule state mass is3.871 Gev, while in the experiments the mass of X is3.872 Gev. [22]

Additionally, applying the method as above, we inves-tigate the S-wave molecule states composed of two heavymesons, which are composed of bottom quark and lightquark d. Then the constituent quark masses mb ¼4:88 Gev, md ¼ 0:33 Gev, the meson masses mB0 ¼5:279 Gev, mB� ¼ 5:325 Gev, and !B ¼ !B� ¼1:81 Gev [17]. In the case of the molecule state composedof two pseudoscalar mesons B0 �B0 the interaction potentialbetween two heavy mesons is

Vðk2Þ ¼ �FPse1 ðk2Þ 1

k2 þm2�

FPse1 ðk2Þ � FPse

2 ðk2Þ

� 1

k2 þm2�

FPse2 ðk2Þ: (26)

Furthermore, in the molecule states composed of twovector mesons B� �B� the interaction potential is

Vðk2Þ ¼ �FV1 ðk2Þ 1

k2 þm2�

FV1 ðk2Þ � FV

2 ðk2Þ

� 1

k2 þm2�

FV2 ðk2Þ: (27)

In Table I we compare our results for masses of moleculestates with other theoretical predictions [23,24]. Then wefind that the molecule states B0 �B0 and B0 �B� lie very closed

to their corresponding thresholds so that it is difficult toobserve these bound states in experiments. This predictionis evidently different from the four-quark state model[23,24] where these bound states are expected to lie con-siderably below the B0 �B0 and B0 �B� thresholds, respec-tively. Besides, we find that the molecule stats B� �B� lieabove the threshold, which is consistent with Ref. [23]. InRefs. [23,24], the authors calculated the masses of theground states of tetraquarks with two bottom quarks as-suming the diquark-antidiquark structure. Therefore, thedifference between our results and the above quoted papersis derived from two distinct models.In this work, assuming the meson-meson structure of the

molecule state, we reduce the relativistic four-body prob-lem to two relativistic two-body problems. We consider thestructure of the heavy mesons and at the same time weapply the form factor of heavy mesons to discuss theinteraction between meson and meson in the moleculestate. The previous works have approximatively solvedthe first problem through solving the Bethe-Salpeter equa-tion about heavy mesons, and their results can be verifiedby experiments. While solving the second problem, weadopt the credible heavy meson wave functions from theseobtained Bethe-Salpeter amplitudes to calculate the inter-action potential between two heavy mesons, which is themain difference between our approach and these otherworks [23,24] investigating the enigmatic mesonXð3872Þ. Besides, we also study the composite states oftwo heavy mesons composed of bottom quark and lightquark.

IV. CONCLUSION

In this work, we assume that the narrow state Xð3872Þ isa molecule state and calculate its mass. These mesons inthe molecule state are considered to have an internalstructure. Using the wave functions of heavy mesons ob-tained from previous works, we calculate heavy-meson-light-meson interaction form factors, which describe theinternal structure of heavy meson. Then the interactionpotential between two heavy mesons is calculated andthe quasipotential equation is solved numerically. Thecalculated mass of the narrow state is consistent with theexperimental value, which gives further grounds for themolecule hypothesis of X. Using this method we inves-tigate the various composite states of heavy mesons B0 andB� in the molecule picture.

FIG. 4. The form factors for the vertexes of heavy mesonscoupling to � meson. The solid line represents the pseudoscalarmeson D0; the dashed line represents the vector meson �D�0.

TABLE I. Masses of bottom meson molecule states (in Gev).

Composite State JPC This work Ref. [23] Ref. [24]

B0 �B0 0þþ 10.548 10.471

B0 �B� 1þ� 10.594 10.502 10.492

B� �B� 0þþ 10.671 10.648 10.473

B� �B� 1þ� 10.733 10.657 10.494

B� �B� 2þþ 10.857 10.673 10.534

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ACKNOWLEDGMENTS

This work was supported in part by the National NaturalScience Foundation of China under Grant No. 10335012

and the key research plan of theoretical physics and crossscience of China under Grant No. 90503011.

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