Are the X (4160) and X (3915) charmonium states?
aa r X i v : . [ h e p - ph ] D ec Are the X (4160) and X (3915) charmonium states? You-chang Yang , , Zurong Xia , Jialun Ping ∗ Department of Physics, Nanjing Normal University, Nanjing 210097, P. R. China Department of Physics, Zunyi Normal College, Zunyi 563002, P. R. China
Inspired by the newly observed X (4160) and X (3915) states, we analyze the mass spectrum ofthese states in different quark models and calculate their strong decay widths by the P model.According to the mass spectrum of charmonium states predicted by the potential model, the states χ (3 P ) , χ (3 P ) , η c (2 D ) , η c (4 S ) all can be candidates for the X (4160). However, only thedecay width of the state η c (2 D ) in our calculation is in good agreement with the data reportedby Belle and the decay of η c (2 D ) → D ¯ D , which is not seen in experiment, is also forbidden.Therefore, it is reasonable to interpret the charmonium state η c (2 D ) as the state X (4160). Forthe state X (3915), although the mass of χ (2 P ) is compatible with the experimental value, thecalculated strong decay width is much larger than experimental data. Hence, the assignment of X (3915) to charmonium state χ (2 P ) is disfavored in our calculation. PACS numbers: 14.40.Pq, 13.25.Gv, 12.38.Lg
I. INTRODUCTION
Many new charmonium like states, the so-called
XY Z mesons, have been reported by Belle and BaBar collab-orations in recent years. Some of these states can beunderstood as conventional mesons that are comprisedof only pure c ¯ c quark pair. However, most of the XY Z states do not match well the mass spectrum of c ¯ c pre-dicted by the QCD-motivated potential models. By con-sidering the effects of virtual mesons loop [1, 2, 3, 4] andcolor screening [6], the masses of some excited charmo-nium states are smaller than it calculated by conventionalquark model. Therefore, some XY Z states [2] may bestill compatible with the mass spectrum of charmonium.However, the state X (3872) [2, 4, 5] is probably the mostrobust of all the charmonium like objects.Last year, Belle collaborations reported a new charmo-nium like state, the X (4160) [7], in the processes e + e − → J/ψD ( ∗ ) ¯ D ( ∗ ) with a significance of 5.1 σ . It has the mass M = 4156 +25 − ±
15 MeV, and width Γ = 139 +111 − ± e + e − → J/ψD ¯ D , e + e − → J/ψD ∗ ¯ D , and e + e − → J/ψD ∗ ¯ D ∗ , The upperlimits of the branch ratios of X (4160) are given as, B D ¯ D ( X (4160)) / B D ∗ ¯ D ∗ ( X (4160)) < . , B D ∗ ¯ D ( X (4160)) / B D ∗ ¯ D ∗ ( X (4160)) < . . The X (4160) has possible charge parity C = + mostly,since the photon γ and J/ψ have J P C = 1 −− , and e + e − → γ → J/ψX (4160) is a main process. Hencethe X (4160) can have J P C = 0 − + , 0 ++ , 1 − + , 2 − + ,1 ++ , ++ , . . . . In Ref.[12], Chao discussed the possi-ble interpretation of the X (4160) in view of produc-tion rate in e + e − → J/ψX (4160). He believes thatthe charmonium states 4 S , P may be assigned tothe state X (4160) by analogy with the cross section of ∗ Electronic address: [email protected] e + e − → J/ψη c (1 S )( η c (2 S ) χ c (1 P )), while the 2 D [14]can not be rule out. According to the mass spectrum [6]predicted by the potential model with color screening, Liand Chao also give some arguments about the χ (3 P )as an assignment for the X (4160).Using the vector-vector interaction within the frame-work of the hidden gauge formalism, Molina and Oset[15] suggested that the X (4160) is a molecular state of D ∗ s ¯ D ∗ s with J P C = 2 ++ .Very recently, Refs.[8, 9, 10, 11] reported the newestcharmonium like state, the X (3915), which is observedby Belle in γγ → ωJ/ψ with a statistical significance of7.5 σ . It has the mass and width M = 3914 ± ± , Γ = 28 ± +2 − MeV . Belle collaborations determine the X (3915) productionrate Γ γγ ( X (3915)) B ( X (3915) → ωJ/ψ ) = 69 ± +7 − eV and Γ γγ ( X (3915)) B ( X (3915) → ωJ/ψ ) = 21 ± +2 − eV for J P = 0 + or 2 + , respectively. Because the partialwidth of this state to γγ or ωJ/ψ is too large, it is veryunlikely to be a charmonium state analyzed by Yuan [9].The X (3915) also has the charge parity C = +, be-cause it is observed in the process of γγ → ωJ/ψ . InRef.[21], Liu et al. argued that the χ (2 P ) can be as-signed to the X (3915) if taking R = 1 . ∼ .
85 GeV − inthe SHO (the simple harmonic oscillator wave functions).Up to now, the interpretation of the X (4160) and X (3915) is still unclear. The states χ (3 P ), χ (3 P ), η c (2 D ) listed in Table I all can be interpreted asthe X (4160) just on mass level. Which charmoniumstate is an assignment for the X (4160)? One can an-swer this question in different ways. We study the X (4160) and X (3915) via strong decay by the P model[16, 17, 18, 19] in this work. In following discussion, wetake the χ (3 P ) , χ (3 P ) , η c (2 D ) , η c (4 S ) and χ (2 P ) as candidates of the X (4160) and X (3915), re-spectively.The paper is organized as follows. In the next sectionwe take a review of the P model. Sect. III devotes TABLE I: Theoretical mass spectrum of the charmonium can-didates for the X (4160) and X (3915). The mass are in units ofMeV. The results are taken from Ref.[6] with color screeningpotential model, and Ref.[13] including Nonrelativistic poten-tial and Godfrey-Isgur relativized potential model.State χ (2 P ) η c (4 S ) χ (3 P ) χ (3 P ) η c (2 D ) J PC ++ − + ++ ++ − + Ref.[6] SCR 3842 4250 4131 4178 4099Ref.[13] NR 3852 4384 4202 4271 4158Ref.[13] GI 3916 4425 4292 4317 4208 to discuss the possible strong decay channels and givesthe corresponding amplitudes of the candidates for the X (4160) and X (3915). In Sect. IV we present and ana-lyze the results obtained by the P model. Finally, thesummary of the present work is given in the last section. II. A REVIEW OF THE P MODEL OF MESONDECAY
Fig.1 The two possible diagrams contributing to A → B + C in the P model. The P decay model, also known as the Quark-PairCreation model (QPC), was originally introduced byMicu[16] and further developed by Le Yaouanc, Ack-leh, Roberts et al .[17, 18, 19]. It is applicable to OZI (Okubo, Zweig and Iizuka) rule allowed strong decaysof a hadron into two other hadrons, which are expectedto be the dominant decay modes of a hadron. Due tothe P model gives a good description of many observedpartial widths of the hadrons, it has been widely used toevaluate the strong decays of mesons and baryons com-posed of u, d, s, c, b quarks [20, 21, 22, 23, 24, 25, 26,27, 28, 29, 30, 31, 32]. The P model of strong decaysassumes that quark-antiquark pair are created with vac-uum quantum number J P C = 0 ++ [16]. The diagrams ofall possible decay process A → B + C of meson are shownin Fig.1. In many cases only one of them contributes tothe strong decay of meson.The transition operator of this model takes T = − γ X m h m − m | i Z d p d p δ ( p + p ) × Y m ( p − p χ − m φ ω b † ( p ) d † ( p ) , (1)where γ , which is a dimensionless parameter, repre-sents the probability of the quark-antiquark pair createdfrom the vacuum and can be extracted by fitting ob-served experimental data. φ = ( u ¯ u + d ¯ d + s ¯ s ) / √ ω = ( R ¯ R + G ¯ G + B ¯ B ) / √ χ , − m is a spin-triplet state. Y ml ( p ) ≡ | p | l Y ml ( θ p , φ p ) is the l th solid harmonic poly-nomial that reflects the momentum-space distribution ofthe created quark-antiquark pair. b † ( p ), d † ( p ) are thecreation operators of the quark and antiquark, respec-tively.In general, the mock state is adopted to describe themeson with the spatial wave function ψ n A L A M LA ( p , p )in the momentum representation [33]. | A ( n A S A +1 L A J A M JA )( P A ) i ≡ p E A X M LA ,M SA h L A M L A S A M S A | J A M J A i× Z d p A ψ n A L A M LA ( p , p ) χ S A M SA φ A ω A | q ( p )¯ q ( p ) i , (2)with the normalization conditions h A ( n A S A +1 L A J A M JA )( P A ) | A ( n A S A +1 L A J A M JA )( P ′ A ) i = 2 E A δ ( P A − P ′ A ) . (3)where n A represent the radial quantum number of themeson A composed of q , ¯ q with momentum p and p . E A is the total energy, P A is the momentum ofthe meson A and p A = ( m p − m p ) / ( m + m ) isthe relative momentum between quark and antiquark. S A = s q + s q , J A = L A + S A stand for the total spinand total angular momentum, respectively. L A is the relative orbital angular momentum between q and ¯ q . h L A M L A S A M S A | J A M J A i denotes a Clebsch-Gordan co-efficient, and χ S A M SA , φ A and ω A are the spin, flavorand color wave functions, respectively.The S -matrix of the process A → B + C is defined by h BC | S | A i = I − πiδ ( E A − E B − E C ) h BC | T | A i , (4)with h BC | T | A i = δ ( P A − P B − P C ) M M JA M JB M JC , (5)where M M JA M JB M JC is the helicity amplitude of A → B + C . In the center of mass frame of meson A , P A = 0,and M M JA M JB M JC can be written as M M JA M JB M JC ( P ) = γ p E A E B E C X M LA ,M SA ,M LB ,M SB ,M LC ,M SC ,m h L A M L A S A M S A | J A M J A ih L B M L B S B M S B | J B M J B i×h L C M L C S C M S C | J C M J C ih m − m | ih χ S B M SB χ S C M SC | χ S A M SA χ − m i× [ h φ B φ C | φ A φ iI M LA ,mM LB ,M LC ( P , m , m , m )+( − S A + S B + S C h φ B φ C | φ A φ iI M LA ,mM LB ,M LC ( − P , m , m , m )] , (6)with the momentum space integral, I M LA ,mM LB ,M LC ( P , m , m , m ) = Z d p ψ ∗ n B L B M LB ( m m m P + p ) ψ ∗ n C L C M LC ( m m m P + p ) ψ n A L A M LA ( P + p ) Y m ( p ) , (7)where P = P B = − P C , p = p , m is the mass of thecreated quark q ; h χ S B M SB χ S C M SC | χ S A M SA χ − m i and h φ B φ C | φ A φ i are the overlap of spin and flavor wavefunction, respectively.The spin overlap in terms of Winger’s 9 j symbol canbe given by h χ S B M SB χ S C M SC | χ S A M SA χ − m i = X S,M S h S B M S B S C M S C | SM S ih S A M S A − m | SM S i× ( − S C +1 p S A + 1)(2 S B + 1)(2 S C + 1) ×
12 12 S A
12 12 S B S C S . (8)Generally, one takes the simple harmonic oscillator(SHO) approximation for the meson space wave func-tions in Eq. (7). In momentum-space, the SHO wavefunction readsΨ nLM L ( p ) = ( − n ( − i ) L R L + s n !Γ( n + L + ) × exp (cid:18) − R p (cid:19) L L + n (cid:0) R p (cid:1) Y LM L ( p ) , (9)with Y LM L ( p ) = | p | L Y LM L (Ω p ). Here R denotes theSHO wave function scale parameter; p represents the rel-ative momentum between the quark and the antiquarkwithin a meson; L L + n (cid:0) R p (cid:1) is an associated Laguerrepolynomial. The decay width for the process A → B + C in termsof the helicity amplitude isΓ = π | P | M A J A + 1 X M JMA ,M JMB ,M JMC (cid:12)(cid:12)(cid:12) M M JA M JB M JC (cid:12)(cid:12)(cid:12) . For comparing with experiments, M M JA M JB M JC ( P )can be converted into the partial amplitude via theJacob-Wick formula [34] M JL ( A → BC ) = √ L + 12 J A + 1 X M JB ,M JC h L JM J A | J A M J A i×h J B M J B J C M J C | JM J A iM M JA M JB M JC ( P ) , (10)where J = J B + J C , J A = J B + J C + L ,and M J A = M J B + M J C . Then the decay width in terms of the partialwave amplitude is taken as,Γ = π | P | M A X JL (cid:12)(cid:12)(cid:12) M JL (cid:12)(cid:12)(cid:12) , (11)where | P | , as mentioned above, is the three momentumof the outgoing meson in the rest frame of meson A . Ac-cording to the calculation of 2-body phase space, one canget | P | = p [ M A − ( M B + M C ) ][ M A − ( M B − M C ) ]2 M A , where M A , M B , and M C are the masses of the meson A , B , and C , respectively. III. THE POSSIBLE STRONG DECAYCHANNELS AND AMPLITUDES OF THECANDIDATES FOR THE X (4160) AND X (3915) As analyzed in section I, we consider the η c (4 S ), χ (3 P ), χ (3 P ), η c (2 D ) as the possible candidatesof the X (4160), and assume that the upper limit of themass is 4156 MeV observed by Belle. For the X(3915),one chooses charmonium state χ (2 P ) with mass 3916MeV. According to the P model discussed in the abovesection, the OZI rule allows open-charm strong decay andcorresponding amplitudes of possible charmonium statesare listed in Tables II and III. We replace I +1 − , , I − , with I ± and I , , with I , in Table III, respectively. Thedetails of the spatial integral about I ± ( P ) and I , ( P )are given in the Appendix. TABLE II: The OZI rule and phase space allowed open-charmstrong decay modes of the possible charmonium states for theX(4160) and X(3915).State J PC Decay mode Decay channel η c (4 S ) 0 − + − + 1 − D ¯ D ∗ , D + s D ∗− s − + 1 − D ∗ ¯ D ∗ χ (3 P ) 0 ++ − + 0 − D ¯ D, D + s D − s − + 1 − D ∗ ¯ D ∗ χ (3 P ) 1 ++ − + 1 − D ¯ D ∗ , D + s D ∗− s − + 1 − D ∗ ¯ D ∗ η c (2 D ) 2 − + − + 1 − D ¯ D ∗ , D + s D ∗− s − + 1 − D ∗ ¯ D ∗ χ (2 P ) 0 ++ − + 0 − D ¯ D TABLE III: The partial wave amplitude for the strong de-cays of relevant charmonium state. The element of flavormatrix h φ B φ C | φ A φ i = 1 / √ E = γ √ E A E B E C in this table.State decay channel Decay amplitude η c (4 S ) 0 − + 1 − M = √ EI − + 1 − M = EI χ (3 P ) 0 − + 0 − M = √ √ E (cid:0) I − I ± (cid:1) − + 1 − M = √ E (cid:0) I − I ± (cid:1) M = E (cid:0) I + I ± (cid:1) χ (3 P ) 0 − + 1 − M = E (cid:0) I − I ± (cid:1) M = √ E (cid:0) I + I ± (cid:1) − + 1 − M = √ E (cid:0) I + I ± (cid:1) η c (2 D ) 0 − + 1 − M = E (cid:0) √ I ± − I (cid:1) − + 1 − M = √ E (cid:0) √ I ± − I (cid:1) χ (2 P ) 0 − + 0 − M = √ √ E (cid:0) I − I ± (cid:1) IV. NUMERICAL RESULTS AND DISCUSSION
There are several parameters should be input to cal-culate the strong decay in the P model. In the presentwork, the masses of constituent quarks are taken as m u = m d = 0 .
22 GeV, m s = 0 .
419 GeV, m c = 1 . γ = 6 . γ is higher than that used in Ref. [37] bya factor of √ π due to different field theory conven-tions. The strength of s ¯ s creation satisfies γ s = γ/ √ R values of D, D ∗ , D s , D ∗ s in the SHO are shown in Table IV, which are obtainedby the calculation of the nonrelativistic quark model withCoulomb item, linear confinement and smeared hyperfineinteractions. TABLE IV: The parameters relevant to the two-body strongdecays of the charmonium state in the P model.State Mass (MeV) [35] R (GeV − ) [36] D . ± ) 1864.84(0) 1.52 D ∗ . ± ) 2006.97(0) 1.85 D s . ± ) 1.41 D ∗ s . ± ) 1.69 First of all, we study the strong decay of the χ (3 P )which is discussed by Chao and Li in Refs.[6, 12] from theproduction process of e + e − → J/ψ + X (4160) and themass spectrum is obtained by the potential model withcolor screening. Using the method of Numerov algorithm[39], we also obtain the mass 4149 MeV by the samepotential and parameters in Ref. [6]. Usually, the widthof strong decay is sensitive [20, 21, 22, 24, 27, 32] to the R value in the SHO. Here the reasonable value of R isobtained by fitting the wave function obtained by solvingthe schr¨odinger equation [6].Through the Fourier transform, the Eq. (9) turns intoΨ nLM L ( r ) = R nL ( r ) Y LM L (Ω r ) , (12)with the radial wave function R nL ( r ) = R − ( L + ) s n !Γ( n + L + ) × exp (cid:18) − R − r (cid:19) r L L L + n (cid:0) R − r (cid:1) . (13)The wave function u ( r ) = r R nL ( r ) of charmonium state3 P is shown in Fig.2. Using Eq.(13) to fit the wave func-tion got by Numerov algorithm method (the wave func-tion is denoted as ’NAWF’ in the following), we can getthe R = 2 . ∼ .
98 GeV − . u (r) r (fm) By Numerov algorithm R -1A = 2 fm -1 (SHO) R -1A = 1.7 fm -1 (SHO) Fig.2 The wave function of charmonium state 3 P . The χ (3 P ) has decay channels of 0 ++ → − + 0 − with S -wave and 0 ++ → − + 1 − with S -, D -wave, whilethe 0 ++ → − + 1 − is forbidden. Therefore, it can decayinto D ¯ D, D s D s , D ∗ ¯ D ∗ , which are allowed by the phasespace. In Fig.3, we show the dependence of the partialwidths of the strong decay of the χ (3 P ) on the R A .Taking R A = 2 . ∼ .
98 GeV − discussed above, thetotal width ranges from 105 to 143 MeV which falls inthe range of experimental data. However, the dominatecontribution comes from the χ (3 P ) → DD which isinconsistent with the experimental result. So the assign-ment of the charmonium state χ (3 P ) to the X (4160)is disfavored. W i d t h ( M e V ) R A (GeV -1 ) DD D*D*(S-wave) D*D*(D-wave) D s D s Total
Fig.3 The possible strong decay of the χ (3 P ). The η c (4 S ) is mostly like the X (4160) for it hashigh production cross sections in the process of e + e − → J/ψ + X (4160) discussed by Chao [12]. However, it is dif-ficult to understand why the predicted mass 4250 MeV[6], 4384, 4425 MeV [13] are much higher than 4156 MeV.By considering the effect of the meson loops [40], themass may be lower than that of Refs.[6, 13]. Here, weassume the mass of the η c (4 S ) is 4156 MeV. The main decay channels of the η c (4 S ) are 0 − + → − + 1 − and0 − + → − + 1 − with P -wave between outgoing mesons.Obviously, the 0 − + → − + 0 − is forbidden. The decaywidth of main decay channels are shown in Fig.4. Thetotal width can only reach up to about 25 MeV with R A around 2.9 GeV, which is obtained by fitting to NAWF ofthe η c (4 S ). It is about 3 times smaller than the lowerlimit of the experimental result of the X (4160). Since theresults of some hadron states predicted by the P modelmay be a factor of 2 ∼ η c (4 S ) cannot beexcluded. The ratio of main decay channel D ¯ D ∗ , D ∗ ¯ D ∗ is B ( η c (4 S ) → D ¯ D ∗ ) B ( η c (4 S ) → D ∗ ¯ D ∗ ) = 1 . . (14)It is much larger than the 0 .
22 reported by Belle. Ifone takes the η c (4 S ) as an assignment of X(4160), theprecision measurement of the ratio between the width ofthe D ¯ D ∗ and D ∗ ¯ D ∗ is necessary in further experiment. W i d t h ( M e V ) R A (GeV -1 ) DD* D*D* DsDs Total Fig.4 The possible strong decay of the η c (4 S ). Because the χ (3 P ) has quantum number J P C =1 ++ and mass 4178 MeV, it is also a possible candidateof the X (4160). 1 ++ → − + 1 − and 1 ++ → − + 1 − with S - and D -wave are the main decay channels of the χ (3 P ). Fig.5 shows our results in the P model. Tak-ing R A = 2 . ∼ .
98 GeV − , the total width is consistentwith the range of the X (4160). However, the dominantdecay is χ (3 P ) → D ¯ D ∗ while the decay width has onlya few MeV for the χ (3 P ) → D ∗ ¯ D ∗ channel, which isinconsistent with the experimental data. Therefore, re-garding the X (4160) as the χ (3 P ) state is impossible. W i d t h ( M e V ) R A (GeV -1 ) DD* D*D* DsDs* Total Fig.5 The possible strong decay of the χ (3 P ). The another possible candidate of the X (4160) is thecharmonium state η c (2 D ). Firstly, it has quantumnumber J P C = 2 − + and mass 4099 MeV [6], 4158 MeV[13] which are compatible with the result of Belle. Sec-ondly, the ψ (4160)[35] is known to be the good candi-date of the ψ (2 D ) with J P C = 1 −− , which is discussedin detail by Chao [12]. So the X (4160) may be the D-wave spin-singlet charmonium state D (2 D ). Thirdly, η c (2 D ) decaying into D ¯ D is forbidden, and this decayis also not seen by Belle.For the strong decay of the η c (2 D ), it has 2 − + → − + 1 − and 2 − + → − + 1 − decay channels with P -wave between outgoing mesons. In this case, final states D ¯ D ∗ , D s ¯ D ∗ s and D ∗ ¯ D ∗ are phase space allowed. InFig.6, we present the numerical results of main decaychannels for the η c (2 D ). By fitting the NAWF of the η c (2 D ), we get R A = 2 . ∼ . − . The totaldecay width of the η c (2 D ) falls in the range of the X (4160) released by Belle. Taking the reasonable R A value of the SHO, the ratio of the main decay channel D ¯ D ∗ , D ∗ ¯ D ∗ is B ( η c (2 D ) → D ¯ D ∗ ) B ( η c (2 D )) → D ∗ ¯ D ∗ ) = 1 . ∼ .
76 (15)and shown in Fig.7. However, the result is somewhatlarger than the B D ∗ ¯ D ( X (4160)) / B D ∗ ¯ D ∗ ( X (4160)) < . γ of the quark pair creation from vacuum.To sum up, the η c (2 D ) is a better candidate for the X (4160) in the present calculation. DD* D*D* DsDs* Total W i d t h ( M e V ) R A (GeV -1 ) Fig.6 The possible strong decay of the η c (2 D ). B r( DD * ) / B r( D * D * ) R A (GeV -1 ) Fig.7 The ration of B ( η c (2 D ) → D ¯ D ∗ ) B ( η c (2 D )) → D ∗ ¯ D ∗ ) with R A value of theSHO. The X (3915), which was observed by Belle in γγ → ωJ/ψ with a statical significance of 7 . σ [8], is the mostrecent addition to the collection of the XY Z states. Ac-cording to the Table I predicted by potential model, theexcited charmonium state χ (2 P ) is a good candidatefor the X (3915), due to it has mass M = 3914 ± ± J P C = 0 ++ .The χ (2 P ) has only the strong decay channel0 ++ → − + 0 − allowed by phase space. The widthof χ (2 P ) → D ¯ D with R A of the SHO is presentedin Fig.8. The total width ranges from 132 to 187 MeVwith R A = 2 . ∼ . − fitted to the NAWF of the χ (2 P ). It is much larger than the Γ = 28 ± +2 − MeVreported by Refs. [8, 9, 10]. Therefore, the X (3915) isunlikely to be the charmonium state χ (2 P ) althoughthe mass is compatible with the X (3915). W i d t h ( M e V ) R A (GeV -1 ) DD Fig.8 The possible strong decay of the χ (2 P ). V. SUMMERY
In summary, we have discussed the possible interpreta-tions of the X (4160) observed by Belle collaborations in e + e − → J/ψ + X (4160) followed by X (4160) → D ∗ ¯ D ∗ .We also study the newest state X (3915) observed by Bellein the process γγ → J/ψω [8].In quark models, the masses of the charmonium states: χ (3 P ), χ (3 P ), η c (2 D ) are all around 4156 MeV.By taking the effect of virtual mesons loop [40] into ac-count, the η c (4 S ) may also has mass around 4156 MeV.All the four states have charge parity C = + which arecompatible with the X (4160) observed by Belle.For the strong decay of the χ (3 P ), the dominantstrong decay is χ (3 P ) → D ¯ D while χ (3 P ) → D ∗ ¯ D ∗ contributes to the total width only a little in the reason-able R in the SHO. It is contrast to the experimentalresult. Thus the excited charmonium state χ (3 P ) dis-favor the X (4160).The η c (4 S ) can not decay into D ¯ D and may has high production rate [12] in e + e − → J/ψ + η c (4 S ) processby analogy with e + e − → J/ψ + η c (1 S )( η c (2 S ) χ c (1 P )).However, the total width in present work is lower thanthe experimental data of the X (4160).The main strong decay channel of the χ (3 P ) is D ¯ D ∗ while D ∗ ¯ D ∗ is only a few MeV. It is inconsistent withthe results of Belle. Therefore, taking the χ (3 P ) as anassignment for the X (4160) is impossible.The η c (2 D ) can not decay to D ¯ D which is also notseen in the experiment. The total width of the η c (2 D )match well with the data of the X (4160) in our calcu-lation. So, the η c (2 D ) is a good candidate for the X (4160), for it is not only the mass but also the strongdecay are well compatible with the results observed byBelle, although the excited charmonium state η c (4 S )can not be rule out as an assignment for the X (4160).We also give the ratio of B ( η c (2 D ) → D ¯ D ∗ ) B ( η c (2 D )) → D ∗ ¯ D ∗ ) which isindependent on the parameter γ in the P model. Thenumerical result is somewhat larger than the experimen-tal data. Therefore, we suggest Belle, BaBar and otherexperimental collaborations to measure it to confirm thisstate.By assuming the X (3915) is the χ (2 P ), the strongdecay of the state is calculated. From our numerical re-sults, we think this assumption is unacceptable. Due tothe partial width of the X (3915) to γγ or ωJ/ψ is toolarge, Yuan [9] also believes that it is very unlikely to bea charmonium state. Thus, It is necessary to do morestudy to understand the properties of the X (3915). Acknowledgments
You-chang Yang would like to thank Xin Liu for use-ful discussion. The work is supported partly by theNational Science Foundation of China under ContractNo.10775072 and the Research Fund for the DoctoralProgram of Higher Education of China under Grant No.20070319007, No. 1243211601028.
Appendix
The spatial overlap I M LA ,mM LB ,M LC ( P , m , m , m ) is simplified as I n ′ m ′ ( P ) in present work due to M L B = M L C = 0.According to the Eq. (7), the concrete calculations of the integration are trivial after choosing the direction of P along z axis [34]. We list all expressions of I ± , I used in Table IIIIn the case of 2 P → S + 1 SI ± = I − = I − = i √ √ π / ∆ (cid:16) R / A R / B R / C (cid:17) exp (cid:18) − ζ P (cid:19) (10 R A + ∆ ( − P R A (1 + λ ) )) I = − i √ √ π / ∆ (cid:16) R / A R / B R / C (cid:17) exp (cid:18) − ζ P (cid:19) (10 R A + ∆ ( − P (1 + λ )( − λ + 2 R A (3 + λ (8 + ∆ P (1 + λ ) ))))) . (16)For 2 D → S + 1 SI ± = I − = I − = 2 √ √ π / ∆ (cid:16) R / A R / B R / C (cid:17) exp (cid:18) − ζ P (cid:19) P (1 + λ )(14 R A + ∆ ( − P R A (1 + λ ) )) I = − √ π / ∆ (cid:16) R / A R / B R / C (cid:17) exp (cid:18) − ζ P (cid:19) P (1 + λ )(28 R A + ∆ ( −
14 + P (1 + λ )( − λ + 2 R A (4 + λ (11 + ∆ P (1 + λ ) ))))) . (17)For 3 P → S + 1 SI ± = I − = I − = i √ √ π / ∆ (cid:16) R / A R / B R / C (cid:17) exp (cid:18) − ζ P (cid:19) (140 R A + 28 ∆ R A ( − P R A (1 + λ ) )+ ∆ (35 − P R A (1 + λ ) + 4 P R A (1 + λ ) )) I = − i √ √ π / ∆ (cid:16) R / A R / B R / C (cid:17) exp (cid:18) − ζ P (cid:19) (35 R A + 14 ∆ P λ (1 + λ )(35 − P R A (1 + λ ) + 4 P R A (1 + λ ) ) + 7 ∆ R A ( − P R A (1 + λ )(6 + 11 λ ))+ 14 ∆ (35 − P R A (1 + λ )(3 + 8 λ ) + 4 P R A (1 + λ ) (5 + 19 λ ))) . (18)For 4 S → S + 1 SI = 12 √ π / ∆ (cid:16) R / A R / B R / C (cid:17) exp (cid:18) − ζ P (cid:19) P (840 R A (2 + 3 λ ) + ∆ λ ( −