DOI 10.1140/epja/i2014-14076-y

Regular Article – Theoretical Physics

Eur. Phys. J. A (2014) 50: 76 THE EUROPEANPHYSICAL JOURNAL A

Strong decays of χcJ(2P) and χcJ(3P)

Hui Wang1, Youchang Yang2, and Jialun Ping1,a

1 Department of Physics, Nanjing Normal University and Jiangsu Key Laboratory for Numerical Simulation of Large ScaleComplex Systems, Nanjing 210023, China

2 Department of Physics, Zunyi Normal College, Zunyi 563002, China

Received: 16 February 2014 / Revised: 22 March 2014Published online: 21 April 2014 – c© Societa Italiana di Fisica / Springer-Verlag 2014Communicated by Xin-Nian Wang

Abstract. In the framework of the chiral quark model, the mass spectrum of χcJ(nP ) (J = 0, 1, 2,n = 1, 2, 3) is studied with the Gaussian expansion method. Using the wave functions obtained in the studyof mass spectrum, the open charm two-body strong decay widths of these states are calculated by using the3P0 model. The results show that the masses of χcJ(1P ) and χc2(2P ) are consistent with the experimentaldata. But the strong decay width of χc2(2P ) is three times that of the experimental value. The decay widthof χc1(2P ) is sensitive to its mass. In the quark-antiquark picture, the width is about 385MeV. However,if the channel coupling effects shift its mass to 3872MeV, its decay width will be around 1 MeV. Thepossibility of assigning the state X(3872) as χc1(2P ) cannot be excluded. To assign X(3915) as χc0(2P ) isdisfavored, due to the unmatching of decay channel. For the χcJ(3P ) states, no states have been assigned.The possible candidates of χc0(3P ) are X(4160) and X(4140). Their masses are close to the theoreticalones. The experimental branching ratio of X(4160), Γ (X(4160) → DD)/Γ (X(4160) → D∗D∗) < 0.09 iscompatible with that of χc0(3P ), 0.07. However the broad decay width of X(4160) cannot be explained bythe open charm two-body decay. To assign X(4140) as χc0(3P ) is also possible, due to the compatibilityof the total decay width, the further measurement of decay modes of X(4140) are expected to justify theassignment.

1 Introduction

With the development of experimental techniques, thehighly excited states of hadrons can be accessed now.In the charm sector, since the Belle Collaboration re-ported the first observation of X(3872) [1], a lot ofcharmonium-like states, so-called “XY Z” states, havebeen reported [2]. To understand the properties of thesestates has drawn great attention in both theoretical andexperimental research fields of hadron physics.

For the P -wave charmonia, there are one spin-singletstate hc(3525), three established 1P -wave spin-tripletstates χc0(3415), χc1(3510) and χc2(3556), and one 2P -wave spin-triplet state X(3930) as χc2(2P ) in the com-pilation of the Particle Data Group (PDG) [3–5]. Inthe latest compilation of the PDG, X(3915) is as-signed as χc0(2P ) [6]. Then, where are the χc1(2P ) andχcJ(3P )(J = 0, 1, 2) states? Meanwhile, there are severalreported states in this mass region which are not assigned,for example, X(3872), X(3940), X(4140), X(4160), etc.The quantum numbers of X(3872) are fixed recently,IG(JPC) = 0+(1++), so it is a good candidate for

a e-mail: jlping@njnu.edu.cn

χc1(2P ) [7]. The Godfrey-Isgur(GI) relativized potentialmodel [8] and the nonrelativistic potential model [9] gavethe mass of χc1(2P ) around 3925–3953MeV, which is alittle higher than the experimental mass of X(3872), andits open charm two-body strong decay width of χc0(2P )is much larger than that of X(3872). Hence it is difficultto assign X(3872) to χc1(2P ). However, the channel cou-pling with the open charm decay channels can shift themass of χc1(2P ) to the threshold of DD∗ [10–12], andthe decay width will be reduced considerably. So the as-signment cannot be excluded, further study is needed. Forχc2(2P ) state, the experimental mass is 3927.2± 2.6MeVwith decay width Γ = 24 ± 6MeV. The quark modelcalculation gave 3972–3979MeV with open charm two-body decay width 80MeV [8,9]. The theoretical mass isa little higher than the experimental one and the de-cay width is about 3 times of the experimental one. Thechannel coupling effect, which is missing in the calcula-tions, may suppress the energy [13] and reduce the de-cay width. In this way, the assignment of X(3930) asχc2(2P ) is supported in the quark model. According toref. [6], X(3915) is considered as χc0(2P ). In the quarkmodel [9], the mass of χc0(2P ) is 3852–3916MeV, and the

Page 2 of 6 Eur. Phys. J. A (2014) 50: 76

open charm decay width is 30MeV, they are close to theexperimental data 3918.4 ± 1.9MeV and 20 ± 5MeV [3].Dian-Yong Chen et al.also supported the assignment ofX(3915) as χc0(2P ) [14]. But F.K. Guo et al. [15] arguedthat X(3915) was unlikely to be the charmonium stateχc0(2P ) due to the unmatching of decay width, althoughtheir masses were compatible with each other.

From the well-established spectrum of χbJ , one cansee a clear pattern. For 1P -wave spin-triplet states, themass difference between χb1 and χb0 is about 33MeV, andthat between χb2 and χb0 is about 53MeV. And for 2Pstates, the two mass differences are reduced to 2/3 of thatfor 1P states, 23MeV and 36MeV. It is expected thatthe same pattern will appear in charm sector accordingto the heavy-flavor symmetry. The three well-established1P states follow the pattern well, where the mass dif-ferences between χc1 and χc0, and between χc2 and χc1,are about 96MeV and 141MeV, respectively. However, ifX(3915) and X(3930) are assigned to be χc0 and χc2, themass difference, 9MeV, is too small compared with theexpected one 2/3 ∗ 141 = 94MeV. In the quark model,the mass of χc2 can be reproduced well, while the massof χc0 is far below 3915MeV. The argument for thewell behavior of the 1P and 2P χbJ spectrum is thatthe masses of these states are all below the threshold ofBB, whereas for the 2P χcJ states, the decay channel toDD is open. However, according to the channel couplingcalculations [13,16], the masses of cc states will be shifteddown and the the mass shifts for all states with same N,Lare equal when cc states are coupled to open-charm DDchannels. Further study is needed to clarify the situation.

To validate the assignment of χcJ(nP ), the more rigor-ous way is to calculate the decay width of the states. In thepresent work we study the open charm two-body strongdecay widths of all the χcJ(nP )(J = 0, 1, 2, n = 2, 3)mesons systematically in the quark model. The spectrumof these P -wave mesons are obtained by using a high-precision few-body method, Gaussian expansion method(GEM) which has been successfully applied in nuclearphysics and hadron physics [17], in the framework of chi-ral quark model [18]. Another advantage of GEM is thatall the interactions are treated equally rather than someinteractions: spin-orbit and tensor terms, are treated per-turbatively in other approaches. The decay amplitudes toall open charm two-body modes that are nominally acces-sible are derived within the 3P0 model. In the numericalevaluation of decay widths, the wave functions obtainedin the study of meson spectrum, rather than the simpleharmonic oscillator one, are used to calculate the tran-sition matrix elements. It is expected to validate the as-signment of the P -wave charmonia and to provide usefulinformation for experimental searching for the still missingstates.

This paper is organized as follows: the 3P0 decay modelis briefly reviewed in sect. 2. In sect. 3, the chiral quarkmodel and wave functions of meson are presented. Thenumerical results of spectrum and the open charm two-body decay widths of χcJ(nP )(J = 0, 1, 2, n = 2, 3) areobtained and presented with discussions in sect. 4. Thelast section is a short summary.

Fig. 1. The two possible diagrams contributing to the decayA → B + C in the 3P0 model.

2 A review of the 3P0 model

In this work, the 3P0 model, or the quark pair creationmodel, is used to calculate the open charm two-bodystrong decay widths of χcJ(nP )(J = 0, 1, 2, n = 2, 3).The model has been widely used and gives a rathergood description of open flavor two-body strong decaywidths of hadrons [19–24]. In the 3P0 model, the mesondecay occurs via a quark-antiquark pair production fromthe vacuum, which is depicted in fig. 1. The quantumnumbers of the created quark pair are of the vacuum,JPC = 0++. In the nonrelativistic limit, the transitionoperator is introduced by [19]

T = −3 γ∑

m

〈1m1 − m|00〉∫

dp3dp4δ3(p3 + p4)

×Ym1

(p3 − p4

2

)χ34

1−mφ340 ω34

0 b†3(p3)d†4(p4), (1)

where γ is a dimensionless constant which denotes thestrength of the quark pair creation. In the presentcalculation, γn = 6.95 for uu, dd pairs and γs = γn/

√3

for ss pair [9]. The created pair is characterized by acolor-singlet wave function ω34

0 = (RR + GG + BB)/√

3,a flavor-singlet function φ34

0 = (uu + dd + ss)/√

3, and aspin-triplet function χ34

1−m and the orbital wave functionYm

l (p) ≡ |p|lY ml (θp, φp), the l-th solid spherical harmonic

polynomial. The “mock state” (a meson is defined tobe a collection of free quarks with the wave function ofthe bound quarks in the physical meson [25]) is usedto describe the meson A with the spatial wave functionψnALAMLA

(pA) in the momentum representation

|A(nA2SA+1LAJAMJA

)(PA)〉 ≡√

2EAφ12A ω12

A

×∑

MLA,MSA

〈LAMLASAMSA

|JAMJA〉χ12

SAMSA

×∫

dpAψnALAMLA(pA)|q1(p1)q2(p2)〉. (2)

The subscripts 1 and 2 in eq. (2) refer to the quarkand the antiquark within the meson A, respectively. m1

(m2) is the mass of quark (antiquark) with momentump1(p2), PA = p1 + p2 is the meson’s momentum,pA = (m2p1 − m1p2)/(m1 + m2) is the relative momen-tum between the quark and the antiquark within themeson A. SA = Sq1 +Sq2 is the total spin. JA = LA +SA

denotes the total angular momentum.

Eur. Phys. J. A (2014) 50: 76 Page 3 of 6

For the A → B +C process, the S-matrix is defined as

〈BC|S|A〉 = I − 2πiδ(EA − EB − EC)〈BC|T |A〉, (3)

with

〈BC|T |A〉 = δ3(PA − PB − PC)MMJAMJB

MJC , (4)

where MMJAMJB

MJC is the helicity amplitude of A → B+C. In the center-of-mass frame of meson A, PA = 0, andPB=−PC=P. Then, MMJA

MJBMJC can be evaluated as

MMJAMJB

MJC (P) = γ√

8EAEBEC

∑

MLA,MSA

,MLB

,MSB,

MLC,MSC

,m

〈LAMLASAMSA

|JAMJA〉〈LBMLB

SBMSB|JBMJB

〉〈LCMLC

SCMSC|JCMJC

〉〈1m1 − m|00〉〈χ14

SBMSBχ32

SCMSC|χ12

SAMSAχ34

1−m〉[〈φ14

B φ32C |φ12

A φ340 〉

IMLA,m

MLB,MLC

(P,m1,m2,m3) + (−1)1+SA+SB+SC

〈φ32B φ14

C |φ12A φ34

0 〉IMLA,m

MLB,MLC

(−P,m2,m1,m3)], (5)

with the momentum space integral,

IMLA,m

MLB,MLC

(P,m1,m2,m3) =∫

dp

ψ∗nBLBMLB

( m3m1+m3

P + p)ψ∗nCLCMLC

( m3m2+m3

P + p)

ψnALAMLA(P + p)Ym

1 (p), (6)

where p = (m4p3 − m3p4)/(m3 + m4), the relative mo-mentum between the created quark and antiquark, m3

(m4) is the mass of the created quark q3 (antiquark q4).〈χ14

SBMSBχ32

SCMSC|χ12

SAMSAχ34

1−m〉 and 〈φ14B φ32

C |φ12A φ34

0 〉 arethe overlaps of spin and flavor wave functions, respectively.

The partial decay width can be written as

Γ = π2 |P|M2

A

∑

JL

∣∣∣MJL∣∣∣2

. (7)

The partial wave amplitude MJL can be related to thehelicity amplitude MMJA

MJBMJC by the Jacob-Wick for-

mula [26,27],

MJL(A → BC) =√

2L + 12JA + 1

∑

MJB,MJC

〈L0JMJA|JAMJA

〉

×〈JBMJBJCMJC

|JMJA〉MMJA

MJBMJC (P), (8)

where J = JB + JC , JA = JB + JC + L, and MJA=

MJB+ MJC

, P is the three-momentum of the daughterhadrons in the center-of-mass frame of initial meson,

P =

√(M2

A − (M2B + M2

C))(M2A − (M2

B − M2C))

2MA. (9)

3 Wave functions

Considering all the degrees of freedom of quarks, the totalwave function of a meson can be written as

ΨJMJ= [ψL(r)χS ]JMJ

φ(qq)ω0(qq),

[ψL(r)χS ]JMJ=

∑

ML,MS

〈LMLSMS |JMJ 〉ψLML(r)χSMS

,

(10)

where 〈LMLSMS |JMJ 〉 is the Clebsh-Gordan coefficient,χSMS

, φ(qq) and ω0(qq) are spin, flavor and color wavefunction of meson, respectively. The orbital part ψLML

(r)can be expanded in Gaussian basis functions [17]

ψLML(r) =

kmax∑

k=1

cLkφGLMk(r), (11)

φGLMk(r) = NLkrL exp

(−νkr2

)Y L

ML(r). (12)

The normalization constant NLk is

NLk =

[2L+2(2νk)L+ 3

2

√π(2L + 1)!!

] 12

(k = 1 ∼ kmax). (13)

The Gaussian size parameters are in geometric progres-sion,

νk =1r2k

, rk = r1ak−1 (k = 1 ∼ kmax). (14)

The expansion coefficients cLk and the eigenenergy E ofthe system are determined by solving the Schrodingerequation with the Rayleigh-Ritz variational principle.Note that the set

{φG

LMk; k = 1 ∼ kmax

}is a nonorthogo-

nal set, so the equation to be solved is a general eigenequa-tion:

Hc = ENc, (15)

where H and N are Hamiltonian and overlap matrices,respectively. The Hamiltonian of the chiral quark modelfor meson is taken from ref. [18],

Hqq(r) = m1 + m2 +p2

r

2μ+ V12

V12 = V C12 + V OGE

12 + V π12 + V K

12 + V η12 + V σ

12, (16)

where m1 (m2) is the mass of quark (antiquark), pr is therelative momentum between quark and antiquark. Vij isthe quark-quark interaction, and their detailed form canbe found in ref. [18]. Note that the color confinement po-tential is a screened one,

V C12 = λ1 · λ2

[−ac(1 − e−μcr) + Δ

], (17)

the channel coupling effect of DD is taken into accountpartly, according to the study of ref. [13].

Page 4 of 6 Eur. Phys. J. A (2014) 50: 76

Table 1. The masses of the mesons obtained by solving theSchrodinger equation with GEM (unit: MeV).

Meson GEM Ref. [9] Expt. [3]

D, D 1878 – 1864.80±0.14

D+, D− 1878 – 1869.57±0.16

D∗+, D∗− 2005 – 2010.22±0.16

D∗, D∗ 2005 – 2006.93±0.16

Ds, Ds 1968 – 1969.0±1.4

D∗s , D∗

s 2104 – 2112.3±0.5

J/ψ(1S) 3096 3090 3096.916±0.011

ηc(1S) 2989 2982 2983.7±0.7

ψ′(2S) 3684 3672 3686.109+0.012−0.014

η′c(2S) 3627 3630 3639.4±1.3

χc0(1P ) 3430 3424 3414.75±0.31

χc1(1P ) 3491 3505 3510.66±0.07

χc2(1P ) 3523 3556 3556.20±0.09

χc0(2P ) 3868 3852 –

χc1(2P ) 3911 3925 –

χc2(2P ) 3935 3972 3927.2±2.6

χc0(3P ) 4172 4202 –

χc1(3P ) 4204 4271 –

χc2(3P ) 4222 4317 –

4 Numerical calculation

The masses of the mesons and the corresponding wavefunctions are obtained by solving the general eigenequa-tion (15). All the parameters except the running strongcoupling constants are taken from ref. [18]. The strongcoupling constant is fine-tuned for the spectrum of char-monium, αs = 0.445 for D, D,D+,D−,D∗, D∗,D∗+,D∗−,αs = 0.375 for Ds, Ds,D

∗s , D∗

s and αs = 0.294 for χcJ(nP )(J = 0, 1, 2, n = 1, 2, 3). The masses of open charm, 1S-,2S- and P -wave χcJ(nP )(J = 0, 1, 2, n = 1, 2, 3) charmo-nia are shown in table 1.

From table 1, we can see that the mass splitting ratio,R = m(χc2)−m(χc1)

m(χc2)−m(χc0)= 0.35, in χcJ (J = 0, 1, 2)(1P ), is

getting close to experimental data, 0.32, although it isstill a little high (if the spin-orbit interaction is taken asperturbation, the ratio will be 0.67). For the 2P states,the mass of χc2(2P ) is very close to the experimentalvalue, while the mass of χc0(2P ) is still too smallerthan the experimental value if X(3915) is assigned asχc0(2P ) state. For the 3P states, the theoretical ratioR = 0.36, and it is almost the same as that for the1P states. No experimental data are available so far. Themasses of the highly excited states (3P states) in ourcalculation are lower than the ones in ref. [9], because thescreened color confinement is used.

To justify the assignment, the decay width is more im-portant. The open charm two-body strong decay modesand decay channels of χcJ(nP )(J = 0, 1, 2, n = 2, 3) al-lowed by the phase space and OZI law are listed in table 2.

Fig. 2. The dependence of the open charm two-body decaywidth on the mass of initial meson.

The open charm two-body decay widths of χcJ(2P )(J =0, 1, 2) and χcJ(3P )(J = 0, 1, 2) are calculated and alsoshown in the fourth column of table 2. In calculating thedecay widths, the theoretical masses of mesons involvedand the corresponding wave functions obtained in solvingthe Schrodinger equations are used. In this way, the cal-culation of the widths is more self consistent than mostof the previous works, where the SHO wave functions areused. Because the decay width is sensitive to the massesof mesons, especially around the threshold of the decay,the results of using experimental masses of mesons in cal-culating the decay widths are also shown in table 2 (fifthcolumn).

From table 2, one can see that the open charm two-body decay width of χc0(2P ) is around 48MeV in ourcalculation, which is not far from the experimental valueof X(3915), 20± 5MeV, and 30MeV in ref. [9]. Many re-searches got even larger decay widths for χc0(2P ), 105–143MeV in ref. [15], ∼ 140MeV in ref. [28], and ∼200MeV in ref. [29]. But if the experimental masses ofthe initial meson and the final mesons are used, the decaywidth will be reduced to 0.12MeV in our calculation. Thephase space plays an important role here. However, themain decay channels of χc0(2P ) and X(3915) are different.The decay of χc0(2P ) to DD∗ is forbidden by the angularmomentum conservation. So the decay width of χc0(2P )only comes from the contribution of DD. Whereas thedecay of X(3915) to DD∗ was seen, the decay to DDwas not, in the experiments. Therefore the assignment ofX(3915) to χc0(2P ) is disfavored in the quark model.

In our calculation, the decay width of χc1(2P ) to DD∗

is very large, 385MeV. Other works also obtained a largewidth, around 165MeV [9]. The large width prevents usfrom assigning X(3872) to χc1(2P ) due to its small width,Γ < 1MeV, of X(3872). However, coupling χc1(2P ) tothe open charm decay channels will shift the mass of thestate down and decrease the decay width. Figure 2 showsthe dependence of the open charm two-body decay widthon the mass of initial meson. If the mass of χc1(2P ) isshifted to 3872.0MeV, which is just above the threshold

Eur. Phys. J. A (2014) 50: 76 Page 5 of 6

Table 2. The open charm two-body strong decay modes and decay widths of the charmonia χcJ(nP )(J = 0, 1, 2, n = 2, 3)allowed by the OZI rule and phase space (unit: MeV). The widths in the 4th column are derived with the theoretical masses ofthe involved mesons and the experimental masses of mesons are used to derive the decay widths in the 5th column.

State Decay mode Decay channel Γa Γb Ref. [9]

χc0(2P ) 0− + 0− DD, D+D− 48.37 0.12 30

χc1(2P ) 0− + 1− DD∗, D∗D, D∗+D−, D+D∗− 385.01 see text 165

χc2(2P ) 0− + 0− DD, D+D− 34.00 33.40 42

D+S D−

S – – 0.7

0− + 1− DD∗, D∗D, D∗+D−, D+D∗− 33.88 33.87 37

Total 67.88 67.27 80

χc0(3P ) 0− + 0− DD, D+D− 0.84 0.40 0.5

D+s D−

s 1.30 1.30 6.8

1− + 1− D∗D∗, D∗+D∗− 11.50 13.03 43

Total 13.64 14.73 51

χc1(3P ) 0− + 1− DD∗, D∗D, D∗+D−, D+D∗− 4.06 4.66 6.9

D+s D∗−

s , D∗+s D−

s 3.60 3.60 9.7

1− + 1− D∗D∗, D∗+D∗− 8.45 9.47 19

D∗+S D∗−

S – – 2.7

Total 16.11 17.73 39

χc2(3P ) 0− + 0− DD, D+D− 4.52 4.22 8.0

D+s D−

s 0.13 0.13 0.8

0− + 1− DD∗, D∗D, D∗+D−, D+D∗− 9.25 10.45 2.4

D+s D∗−

s , D∗+s D−

s 0.87 0.87 11

1− + 1− D∗D∗, D∗+D∗− 16.64 17.50 24

D∗+s D∗−

s 1.51 1.51 7.2

Total 32.92 34.68 66(a)

(a)The partial decay widths to DD1 and DD′

1 have been added.

due to coupling effect, the decay width will be ∼ 1MeV.So the possibility of assigning X(3872) as χc1(2P ) cannotbe excluded. Further study on the coupling channel effectis needed to clarify the situation.

For the χc2(2P ) state, the mass is very close to the ex-perimental value of X(3930), but the calculated width,∼ 68MeV, is about 3 times that of the experimentalvalue. The obtained width is consistent with the resultof ref. [9], 80MeV. Again, it is expected that the couplingeffect of open charm channels can shift the mass a littlelower and decrease the decay width. From the calcula-tions of refs. [9,30], one can see that a variable quark paircreation strength γ will describe the open flavor two-bodydecay better. Taking all these arguments into account, theassignment of X(3930) to χc2(2P ) is possible.

For the 3P states, the decay to all the open charmchannels are possible. From the results, we can see thatthe main decay mode of these states is 1− + 1−, andthe total decay widths are around 15MeV (for J = 0or 1) and 30MeV (for J = 2), which is 2–4 timeslower than those in ref. [9]. The difference comes fromthe fact that the different wave functions are used intwo approaches. The SHO functions are not a good ap-proximation of the wave functions of the excited states.

For the χc2(3P ) state, the theoretical mass of ref.[9] is sohigh that the decay channels DD1,DD′

1 are open, whichare not shown in the table. So the total open charm de-cay width in ref. [9] amounts to 66MeV. So far, there isno state assigned as 3P states. Reference [31] proposedthat X(4160) may be assigned to be χc0(3P ). In our cal-culation, the branching ratio Γ (χc0(3P )→DD)

Γ (χc0(3P )→D∗D∗)= 0.073

and it is just in the range of the experimental data ofX(4160), < 0.09 [3]. The upper limit of another branch-ing ratio of X(4160) Γ (X(4160)→D∗D+c.c.)

Γ (X(4160)→D∗D∗), 0.22, is also

given experimentally. In theory, this ratio is 0 due to thefact that the decay of χc0(3P ) to D∗D is forbidden. Theanother problem is the total decay width of the states,Γt(X(4160)) = 139+111

−61 ±21MeV, is much larger than theopen charm two-body decay width of χc0(3P ), 32.92MeV.So one has to find a way to give a large decay width toχc0(3P ) before assigning X(4160) as χc0(3P ). Anotherstate X(4140) can also be a candidate of χc0(3P ), the to-tal decay width of X(4140), 11.7 ± 3.7, is consistent withthe theoretical one, 12.77MeV. The problem is that theopen charm two-body decay of X(4140) has not been seen.Further measurement of the decay modes is expected tojustify the assignment.

Page 6 of 6 Eur. Phys. J. A (2014) 50: 76

5 Summary

In this work, we study the mass spectrum of χcJ(2P )(J = 0, 1, 2) and χcJ(3P )(J = 0, 1, 2) with Gaussianexpansion method in the framework of the chiral quarkmodel and calculate the open charm two-body strong de-cays of them with the 3P0 model. The results show thatthe masses of χcJ(1P )(J = 0, 1, 2) and χc2(2P ) are con-sistent with the experimental data. The decay width ofχc2(2P ) is higher than the experiment value of X(3930),but it is still possible to assign X(3930) as χc2(2P ) bytaking into account of the coupling channel effect. For theχc1(2P ) state, it has a wide decay width due to the domi-nance of the S-wave decay. To assign X(3872) as χc1(2P ),a mechanism is needed to shift the mass of χc1(2P ) to theDD∗ threshold, where the open charm decay width willbecome very small. Channel coupling to open charm chan-nels is possible to shift the mass of the states downward.The further calculation is needed to clarify the situation.For the assignment of X(3915) as χc0(2P ), it is not sup-ported in the present work due to the unmatched massand the dominant decay channel. Although two branch-ing ratios of χc0(3P ) is within the limit of experimentaldata of X(4160), the large total decay width of X(4160)should be explained to make this assignment possible. Theopen charm two-body decay is not enough. X(4140) maybe a good candidate of χc0(3P ) due to the agreement ofthe total decay width. To justify the assignment, furthermeasurement to find out the main decay mode of the stateis expected.

Because of the opening of open charm decay, thespectra of χcJ(nP ) (J = 0, 1, 2, n ≥ 2) are in a mess.The conventional quark model seems to fail in describ-ing the excited spectrum of charmonium. To develop thequark model, the effect of open charm channels needsto be taken into account. For open flavor two-body de-cay model, the 3P0 model, improvement is also neededbecause of the sensitive behavior of the decay width tothe mass of initial meson. The study of the properties ofχcJ(nP ) (J = 0, 1, 2, n ≥ 2) is helpful for understandingthe possible exotic, XY Z states.

The work is supported partly by the National Science Founda-tion of China under Grant Nos. 11175088, 11205091, 11035006and 11265017.

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