Hadron loops effect on mass shifts of the charmed and charmed-strange spectra

Zhi-Yong Zhou*

Department of Physics, Southeast University, Nanjing 211189, People’s Republic of China

Zhiguang Xiao†

Interdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei, Anhui 230026, China(Received 31 May 2011; published 11 August 2011)

The hadron loop effect is conjectured to be important in understanding discrepancies between the

observed states in experiments and the theoretical expectations of the nonrelativistic potential model. We

present that in an easily operable procedure the hadron loop effect could shift the poles downwards to

reduce the differences and provide better descriptions of both the masses and the total widths, at least, of

the radial quantum number n ¼ 1 charmed and charmed-strange states. The 11P1 � 13P1 mixing

phenomena could be naturally explained due to their couplings with common channels. The newly

observed D states are also addressed, but there are still some problems that remain unclear.

DOI: 10.1103/PhysRevD.84.034023 PACS numbers: 12.39.Jh, 13.20.Fc, 13.75.Lb, 11.55.Fv

I. INTRODUCTION

Discoveries of more and more charmed or charmed-strange states in experiments attract great interest on thetheoretical side, because some members of them haveunexpected properties. In the Particle Data Group (PDG)table in Ref. [1], six lower charmed states, D0, D�ð2007Þ0,D�

0ð2400Þ0, D1ð2420Þ0, D1ð2430Þ0, D�2ð2460Þ0, and their

partners have already been established. Recently, someevidence of three new charmed states, Dð2550Þ, Dð2610Þ,and Dð2760Þ have been reported by the BABARCollaboration [2], whose features lead to intense discus-sions and theoretical suggestions of the further ex-perimental investigations [3–8]. There are also ninecharmed-strange states quoted in the PDG table amongwhich some states’ quantum numbers are undetermined.The mass spectra of these charmed and charmed-strangestates are roughly depicted in the predictions of the non-relativistic potential model in the classic work by Godfreyand Isgur (referred to as GI in the following) [9]. However,the observed masses are generally lower than the predictedones. For example, the biggest discrepancies happening inboth spectra are the 13P0 states. The D�

0ð2318Þ is about

80 MeV lower than the expectation, while theD�s0ð2317Þ is

about 160 MeV lower. There are a body of theoreticalefforts at solving this problem usually by changing therepresentation of the potential (see Refs. [10–14]and references therein). Lattice calculations have alsobeen made to explain the experiments [15,16]. However,the present systematic uncertainty of the Lattice calcula-tions does not allow determinations of the charmed mesonswith a precision less than several hundred MeV.

Another expectation to shed light on this problem is totake the coupled channel effects (also called hadron loop

effects) into account, which play an important role in under-standing the enigmatic light scalar spectrum and theirdecays [17,18]. In the light scalar spectrum, the strongattraction of opened or virtual channels may dramaticallyshift the poles of the bare states to different Riemann sheetsattached to the physical region, and the poles on unphysicalRiemann sheets appear as peaks or just humps of themodulus of scattering amplitudes in the experimentaldata. The mass shifts induced by the intermediate hadronloops have also been shown to present a better descriptionof the charmonium states [19–22]. The coupled channeleffects have already brought some insights into the natureof the charmed-strange DsJð2317Þ and some other states[23–27]. However, although this effect could explain someof the observed charmed or charmed-strange states, there isstill a concern that this effect may also exist in those statespreviously consistent with the theoretical expectation [28].In this paper, we will address this point by considering themass shifts, induced by hadron loops, of all the firmlyestablished charmed and charmed-strange states. Here wepropose an easily operable way, in which we use theimaginary part of the self-energy function calculated fromthe quark pair creation (QPC) model [29–31] in the disper-sion relation to obtain the analytically continued inversepropagator and extract the physical mass and width parame-ters, and then apply it to the charmed and charmed-strangespectra to interpret their masses and total decay widths in aconsistent way. It is found that the results of their massesand total widths are consistent with the experimental val-ues, at least for the nonradially excited states. This picturegives a natural explanation to the 11P1 � 13P1 mixing bythe coupling with the same channels instead of using aphenomenological mixing angle. This scheme has somesimilarities to the methods used by Heikkila et al. [19]and Pennington et al. [20] in their study of the charmoniumand bottomonium states, but there are significant differ-ences from them, as discussed in the text.

*zhouzhy@seu.edu.cn†xiaozg@ustc.edu.cn

PHYSICAL REVIEW D 84, 034023 (2011)

1550-7998=2011=84(3)=034023(8) 034023-1 � 2011 American Physical Society

The paper is organized as follows: In Sec. II, the mainscheme and how to model the decay channels are brieflyintroduced. The mixing mechanism is introduced inSec. III. Numerical procedures and results are discussedin Sec. IV. Section V is devoted to our conclusions andfurther discussions.

II. THE SCHEME

We start by considering a simple model at the hadronlevel, in which the inverse meson propagator, P�1ðsÞ,could be represented as [19,20]

P�1ðsÞ ¼ m20 � sþ�ðsÞ ¼ m2

0 � sþXn

�nðsÞ; (1)

where m0 is the mass of the bare q �q state and �nðsÞ is theself-energy function for the n-th decay channel. Here, thesum is over all the opened channels or including nearbyvirtual channels (‘‘just virtual’’). �nðsÞ is an analyticfunction with only a right-hand cut starting from the n-ththreshold sth;n, and so one can write its real part and

imaginary part through a dispersion relation

Re�nðsÞ ¼ 1

�PZ 1

sth;n

dzIm�nðzÞðz� sÞ ; (2)

where PRmeans the principal value integration. The pole

of PðsÞ on the unphysical Riemann sheet attached to thephysical region specifies its mass and total width of ameson by its position on the complex s plane, usuallydefined as spole ¼ ðMp � i�p=2Þ2.

One could recover a generalization of the familiar Breit-Wigner representation, usually used in experimental analy-ses, from Eq. (1), as

P�1ðsÞ ¼ mðsÞ2 � sþ imBW�totðsÞ; (3)

where mðsÞ2 ¼ m20 þ Re�ðsÞ is the ‘‘running squared

mass’’ and �totðsÞ ¼ Im�ðsÞ=mBW. mBW is determined atthe real axis where mðsÞ2 � s ¼ 0 is fulfilled. The massand width parameters in these two definitions give similarresults when one encounters a narrow resonance, but theydiffer when the resonance is broad or when there areseveral poles interacting with each other.

Based on the Cutkosky rule, the imaginary part of theself-energy function is expressed through the couplingsbetween the bare state and the coupled channels. Therelation could be pictorially expressed as Fig. 1.

Thus, one key ingredient of this scheme is to model thecoupling vertices in the calculation of the imaginary part ofthe self-energy function. The QPC model [29–31], alsoknown as the 3P0 model in the literature, turns out to be

applicable in explaining the Okubo-Zweig-Iizuka (OZI)allowed strong decays of a hadron into two other hadrons,which are expected to be the dominant decay modes of a

meson if they are allowed. It is not only because this modelhas proved to be successful but also because it couldprovide analytical expressions for the vertex functions,which are convenient for extracting the shifted poles inour scheme. Furthermore, the vertex functions have expo-nential factors which give a natural ultraviolet suppressionto the dispersion relation and we need not choose one byhand as in Ref. [20].Here, we just make a brief review of the main results of

the QPC model used in our calculation. (For a morecomplete review, see [32–34].) In the QPC model, themeson (with a quark q1 and an antiquark q2) decay occursby producing a quark (q3) and antiquark (q4) pair from thevacuum. In the nonrelativistic limit, the transition operatorcan be represented as

T¼�3�Xm

h1m1�mj00iZd3 ~p3d

3 ~p4�3ð ~p3þ ~p4Þ

�Ym1

�~p3� ~p4

2

��341�m�

340 !34

0 by3 ð ~p3Þdy4 ð ~p4Þ; (4)

where � is a dimensionless model parameter andYm1 ð ~pÞ �

plYml ð�p;�pÞ is a solid harmonic that gives the

momentum-space distribution of the created pair. Herethe spins and relative orbital angular momentum of thecreated quark and antiquark (referred to by subscripts 3and 4, respectively) are combined to give the pair the

overall JPC ¼ 0þþ quantum numbers. �340 ¼ ðu �uþ d �dþ

s�sÞ= ffiffiffi3

pand !34

0 ¼ �ij, where i and j are the SU(3)-color

indices of the created quark and antiquark. �341�m is a triplet

of spin.Define the S matrix for the meson decay A ! BC as

hBCjSjAi ¼ I� 2�i�ðEf � EiÞhBCjTjAi; (5)

and then

hBCjTjAi ¼ �3ð ~Pf � ~PiÞMMJAMJB

MJC : (6)

The amplitude turns out to be

FIG. 1. The imaginary part of the self-energy function.Rd�

means the integration over the phase space.

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MMJAMJB

MJC ð ~PÞ¼�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8EAEBEC

p XMLA

;MSA;MLB

;MSB;MLC

;MSC;m

hLAMLASAMSA jJAMJAihLBMLB

SBMSB jJBMJBi

�hLCMLCSCMSC jJCMJCih1m1�mj00ih�32

SCMSC�14SBMSB

j�12SAMSA

�341�mih�32

C �14B j�12

A �340 iIMLA

;m

MLB;MLC

ð ~PÞ: (7)

The spatial integral IMLA

;m

MLB;MLC

ð ~PÞ is given by

IMLA

;m

MLB;MLC

ð ~PÞ ¼Z

d3 ~kc �nBLBMLB

�� ~kþ �4

�1 þ�4

~P

�c �

nCLCMLC

�~k� �3

�2 þ�3

~P

�c nALAMLA

ð� ~kþ ~PÞYm1 ð ~kÞ; (8)

where we have taken ~P � ~PB ¼ � ~PC and�i is the mass of the i-th quark. c nALAMLAð ~kAÞ is the relative wave function of the

quarks in meson A in the momentum space.The recoupling of the spin matrix element can be written, in terms of the Wigner’s 9-j symbol, as [32]

h�32SCMSC

�14SBMSB

j�12SAMSA

�341�mi ¼ ½3ð2SB þ 1Þð2SC þ 1Þð2SA þ 1Þ�1=2X

S;MS

hSCMSCSBMSB jSMSihSMSjSAMSA ; 1;�mi

�8><>:1=2 1=2 SC

1=2 1=2 SB

SA 1 S

9>=>;: (9)

The flavor matrix element is

h�32C �14

B j�12A �34

0 i ¼ XI;I3

hIC; I3C; IBI3BjIAI3Ai½ð2IB þ 1Þð2IC þ 1Þð2IA þ 1Þ�1=28><>:I2 I3 IC

I1 I4 IB

IA 0 IA

9>=>;; (10)

where IiðI1; I2; I3; I4Þ is the isospin of the quark qi.The imaginary part of the self-energy function in the

dispersion relation, Eq. (2), could be expressed as

Im�A!BCðsÞ ¼ � �2

2JA þ 1

j ~PðsÞjffiffiffis

p

� XMJA

;MJB;MJC

jMMJA;MJB

;MJC ðsÞj2; (11)

where jPðsÞj is the three momentum of B and C in theircenter of mass frame. So,

jPðsÞjffiffiffis

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs� ðmB þmCÞ2Þðs� ðmB �mCÞ2Þ

p2s

: (12)

Care must be taken when Eq. (11) is continued to thecomplex s plane. Since what is used in this model is onlythe tree level amplitude, there is no right-hand cut for

MMJA;MJB

;MJC ðsÞ. Thus, the analytical continuation of theamplitude obeys Mðsþ i�Þ� ¼ Mðs� i�Þ ¼ Mðsþi�Þ. The physical amplitude with loop contributions shouldhave right-hand cuts, and, in principle, the analytical con-tinuation turns to be Mðsþ i�Þ� ¼ Mðs� i�Þ ¼Mnðsþ i�Þ by meeting the need of real analyticity.Mðsþ i�Þ means the amplitude on the physicalRiemann sheet (the first sheet, in language of the analyticS matrix theory), andMnðsþ i�Þ means the amplitude onthe unphysical Riemann sheet (the n-th sheet) attachedwith the physical region.

With the analytical expression of the imaginary part of thecoupled channel, one will be able to extract the poles or theBreit-Wigner parameters from the propagators by standardprocedures. In principle, all hadronic channels should con-tribute to themesonmass, as considered byHeikkila et al. instudying the charmonium states [19]. Even all the ‘‘virtual’’channels will contribute to the real parts of�ðsÞ and renor-malize the ‘‘bare’’ mass. Pennington et al. proposed that aonce-subtracted dispersion relation will suppress contribu-tions of the faraway virtual channels and make the picturesimpler [20]. Since what we consider here is only the massshifts, we could make a once-subtracted dispersion relationat some suitable point s¼ s0. It is reasonably expectedthat the lowest charmed state, D0, as a bound state, has themass defined by the potential model, uninfluenced by theeffect of the hadron loops. Its mass then essentially definesthe mass scale and thus fixes the subtracted point. So, we setthe subtracted point s0 ¼ m2

D0 or s0 ¼ ðmc þmuÞ2 in a

practical manner. The inverse of the meson propagator turnsout to be

P�1ðsÞ ¼ m2pot � sþX

n

s� s0�

Z 1

sth;n

dzIm�nðzÞ

ðz� s0Þðz� sÞ ;

(13)

wherempot is the bare mass of a certain meson defined in the

potential model.

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III. MIXING MECHANISM

In this scheme, all the states with the same spin-parityhave interference effects and could mix with each other.For example, the two J ¼ 1 states of the P wave areusually regarded as linear combinations of 3P1 and 1P1

assignments. Here in considering the coupled channeleffect, the mixing mechanism comes from the couplingvia common channels. It is also believed that the 23S1 and13D1 states mix with each other, similar to the interpreta-tion of the charmonium c ð3770Þ state [35].

The inverse of the propagator with two bare states mix-ing with each other reads

P�1ðsÞ ¼ M211ðsÞ M2

12ðsÞM2

12ðsÞ M222ðsÞ

� �� �a;bs

¼ m2bare;1 � sþ�11ðsÞ �12ðsÞ

�21ðsÞ m2bare;2 � sþ�22ðsÞ

!;

(14)

whereM2a;bðsÞ is the mass matrix and mbare;a represents the

mass parameter of the bare a state. The off-diagonal termsof the self-energy function is represented by the 1PI dia-gram for the two mixed states. The physical states shouldbe determined by the meson propagator matrix after diag-onalization

M2diagðsÞ ¼ ðsÞ�1M2

a;bðsÞðsÞ; (15)

where the mixing matrix ðsÞ satisfies ðsÞTðsÞ ¼ I, i.e.,ðsÞ is a complex orthogonal matrix since M2

a;bðsÞ is

symmetric. The ðsÞ matrix turns to be complex whenthe thresholds are open. The physical poles could be ex-tracted, in an equivalent way, by finding the zero points ofthe determinant of the inverse propagator, that is to solvedetðP�1ðsÞÞ ¼ 0.

IV. NUMERICAL ANALYSES

The bare masses of the related mesons are chosen at thevalues of the GI’s work [9]. As for the dimensionless

parameter, �, and the effective parameters in the QPCmodel to characterize the harmonic oscillator wave func-tions, we choose the same values as determined from thepotential in GI’s work for self-consistency. The constituentquark masses are Mc ¼ 1:628 GeV, Ms ¼ 0:419 GeV,and Mu ¼ 0:22 GeV. � ¼ 6:9 and the values of s arefrom Refs. [36,37]. The physical masses concerned in thefinal states are the average values in the PDG table. Therelative wave functions between the quarks in the mock-meson states are simple harmonic oscillator wave func-tions usually used in the QPC model calculation, whichbrings some uncertainties into the calculation, as discussedlater.There are some further explanations for the effective

parameters of c �u states. Godfrey and Isgur have onlypresented their results of n ¼ 1 S and P-wave charmedstates but not provided those of the D-wave and radialexcited states which are needed in our discussion of thenewly observed charmed states. We can only estimate thevalues by assuming that their ratios between the valuesof the c �u states are similar to the ratios in the results fromthe other research groups. For example, we find the ratiosof the values between different charmed states inRef. [38] and Ref. [6] are almost same. Thus, the valuesof c �u states used in our calculation, except those listed inRef. [37], are ð13DjÞ ¼ 0:44� 0:02 GeV, ð21S0Þ ¼0:47� 0:02 GeV, and ð23S1Þ ¼ 0:44� 0:02 GeV,respectively.The opened or nearby virtual two-body channels taken

into account in our calculations are all listed in Table I.Those channels with the � meson are not considered,because � meson is not regarded as a conventional q �qstate in the potential model [9]. It will quickly decay intotwo pions and the three-body decays usually present aminor contribution.The masses and widths are simultaneously determined,

as listed in Table II, for the charmed states, where wepresent the pole positions as well as the Breit-Wignerparameters for comparison. Remarkable improvements of

TABLE I. The channels of the charmed states considered in this paper.

Mode Channel 1�ð13S1Þ 0þð13P0Þ 1þð11P1Þ 1þð13P1Þ 2þð13P2Þ 0�ð21S0Þ 1�ð23S1Þ 1�ð13D1Þ 3�ð13D3Þ0� þ 0� D� h h h h h h

D� h h h

DsK h h h

1� þ 0� D�� h h h h h h h

D�� h h h

D�sK h h

0� þ 1� D h h

D! h h

0þ þ 0� D�0� h

1þðTÞ þ 0� D1ð2420Þ� h h h

1þðSÞ þ 0� D1ð2430Þ� h h h

2þ þ 0� D�2ð2460Þ� h h h

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the shifted masses of the already established charmedmesons could be found instantly. Furthermore, the totalwidths specified by twice of the imaginary part of the polepositions are also consistent in good quality with the valuesin the PDG table.

The Dð11S0Þ state is a long-lived particle in the stronginteraction and there is no opened strong channel, so weregard it to be well described as a bound state in thepotential model and choose its squared mass as the sub-traction point of the dispersion relations.

When we calculate the mass shift of the Dð13S1Þ, theD0�0 and Dþ�� threshold are both taken into account,because the Dþ�� threshold is at about 2009 MeV, just2 MeV higher than the observed D�ð2007Þ0. It is a typical‘‘just virtual’’ channel and in principle it will contribute asignificant mass shift to the bare state. If the coupling to theDþ�� threshold is excluded, the pole mass will only beshifted to about 2031 MeV using this set of parameters.

The pole of Dð13P0Þ is significantly shifted down to2275 MeV, which is 125 MeV down below the potentialmodel prediction. The pole width is about 250 MeV, whichis in accordance with the experimental value within errorbar.

The shifted pole mass favors the BABAR and Belleresults [39,40] over the FOCUS result [41]. The Dð11P1Þand Dð13P1Þ states stay close to each other and they bothhave the same quantum numbers JP ¼ 1þ and similardecay channels. The unmixed pole positions are atffiffiffiffiffiffiffiffiffiffiffiffisð11P1Þ

p ¼ 2387� i28 MeV andffiffiffiffiffiffiffiffiffiffiffiffisð13P1Þ

p ¼ 2427�i71 MeV, respectively. Both of the masses and the widthsof the unmixed 11P1 state have large differences from theexperimental values. It is the effect of their couplings withcommon channel D�� that significantly changes their polepositions to one narrower and the other broader. The polesdetermined by the zero points of the inverse propagatormatrix are at

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisð“11P1”Þ

p ¼ 2410� i7 MeV andffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisð“13P1”Þ

p ¼2377� i94 MeV respectively, which characterize thetwo observed states quite well. Their related Breit-Wigner parameters agree with the experimental values

better. It is interesting to mention that in this scheme themixing matrix, as a function of s, are complex-valued andit is not easy to find its relation with the mixing anglecommonly used in phenomenological analyses. Here, theapproximate values of the mixing matrix at about2410 MeV is

js¼ð2:41 GeVÞ2 ¼ �0:57þ 0:26i 0:87þ 0:17i0:87þ 0:17i 0:57� 0:26i

� �; (16)

whose imaginary parts are small compared with the realparts. If one just neglects the imaginary parts and definesthe mixing matrix as usual, one obtains

� sin� cos�cos� sin�

� �’ �0:57 0:87

0:87 0:57

� �; (17)

which means the mixing angle � ’ ð29� � 35�Þ. It is inagreement with the value � ’ 35:3� obtained by consider-ing the heavy quark symmetry [42].As for the other charmed states which are not quoted in

the PDG table, the evidences of several new charmedstates,Dð2550Þ,Dð2610Þ, andDð2760Þ, have been recentlyreported by the BABAR collaboration [2]. Several groupshave presented their tentative interpretations of the natureof these states [3–7], and we make a brief summary of theirconclusions here. Dð2550Þ is assigned to the 21S0, but itslarge decay width could not be explained by the QPCmodel, the chiral quark model, and the relativistic quarkmodel, so further experimental explorations were sug-gested. Although the potential model has predictedDð23S1Þ to be located at about 2640 MeV, the QPC modeland the chiral quark model also favor Dð2610Þ to be amixed state of Dð23S1Þ and Dð13D1Þ to interpret its largewidth. There are conflicting opinions about the assignmentof Dð2760Þ as the heavier mixed state of Dð23S1Þ andDð13D1Þ, or as Dð13D3Þ.In our calculation, the mass shift induced by the inter-

mediate states also reduces the pole masses of Dð21S0Þ,down to 2533 MeV, but its pole width is quite narrowcompared with the experimental value, as shown in

TABLE II. Compilation of the experimental masses and the total widths (the PDG average values [1]) of the charmed states, theBreit-Wigner parameters, the shifted pole positions and the mass spectrum in the GI’s model [9]. The experimental values of 21S0,23S1, and 13D1 are from Ref. [2]. Here we only list the neutral charmed states. The unit is MeV.

JPðn2sþ1LJÞ Expt. mass Expt. width mBW �BWffiffiffiffiffiffiffiffiffispole

p ¼ M� i�=2 GI mass

0�ð11S0Þ 1867 1880

1�ð13S1Þ 2007� 0:16 <2:1 2016 0.02 2016� 0:01i 2040

0þð13P0Þ 2318� 29 267� 40 2335 233 2275� 125i 2400

“1þð11P1Þ” 2422� 0:6 20� 1:7 2420 16 2410� 7i 2440

“1þð13P1Þ” 2427� 40 384þ130�110 2409 163 2377� 94i 2490

2þð13P2Þ 2463� 1 43� 3 2453 26 2452� 12i 2500

0�ð21S0Þ 2533� 14ð?Þ 128� 33ð?Þ 2534 25 2533� 12i 2580

1�ð23S1Þ 2608� 5ð?Þ 93� 19ð?Þ 2525 8 2523� 5i 2640

1�ð13D1Þ 2763� 5ð?Þ 61� 9ð?Þ 2730 120 2686� 66i 2820

3�ð13D3Þ (?) (?) 2735 9 2735� 4i 2830

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Table II. However, the mass of Dð23S1Þ is shifted toomuch down to about 2523 MeV due to many intermediatechannels opened. Actually, the pole is even shifted downabout 100 MeV below some thresholds, and its pole widthis fairly small as well. It seems to become a quasiboundstate due to its strong coupling with the D1� channel. Ofcourse, there is still some parameter space for the effec-tive parameter and the dimensionless coupling strengthparameter � to be tuned to reduce the mass shift tofit the experiment signal, because these parameters havesignificant uncertainties. In our opinion, one possiblereason why the result seems to be inaccurate is theuncertainty of the simple harmonic oscillator functionwe used to estimate the coupling vertices, which mighthave a larger tail than the realistic one in the high sregion, which will contribute to the mass shift throughthe dispersion relation. A more realistic wave functionsolved from the linear potential model could be morereliable to describe the meson property. However, usually,this kind of wave function does not have an analyticalrepresentation and it can not be easily continued into thecomplex s plane in our scheme. On the other hand, it istoo early to get any firm conclusion, since these states stillneed further experimental confirmations. The pole ofDð13D1Þ is shifted down to 2686� i66 MeV as well,whose Breit-Wigner mass is about 2730 MeV which iscloser to the mass of Dð2760Þ. Unlike the 11P1 � 13P1

case, with this set of parameters, the mixing mechanismdue to the coupling with their common channels does notchange their positions much. The unmixed pole ofDð13D1Þ could also be estimated at about 2735 MeVbut its width is narrow.

The discrepancies that happen in the charmed-strangespectrum could be well addressed qualitatively, owing totheir coupling with the opened thresholds and the nearbyvirtual OZI-allowed strong thresholds in Table III, as the

picture proposed by van Beveren and Rupp for explainingthe D�

sJð2317Þ state [23]. Some the thresholds are opened

as a result of the isospin breaking effects, e.g.,D�

sJð2317Þ ! Ds�0,D�

s�0, whose contributions are highly

suppressed by a factor of about ðmu �mdÞ=ðms � ðmu þmdÞ=2Þ 1=38, where the masses are the current quarkmasses. The coupling to such thresholds will contributetiny imaginary parts of the self-energy functions, whichhardly shift the mass of the state and only contribute to thedecay widths with an order of KeV. So we completelyneglect these thresholds with isospin breaking effects andthose OZI suppressed. When we choose the value of thedimensionless strength parameter � ¼ 6:9 for thecharmed-strange spectrum, the mass shifts will be a littlelarger. We change � to be around 5.5 and obtained theshifted masses of the charmed-strange S and P-wavestates, as listed in Table IV. One could regard this finetuning procedure as a ‘‘fit’’, because we only want to give aqualitative description for the charmed-strange spectrum.Indeed the � parameter in the QPC model, determined byfitting to experimental decay processes, usually has anuncertainty of about 30% [6,38]. Here we only list thepole positions, because they do not differ much with theBreit-Wigner parameters in this case as quasibound statesor narrow states.If the isospin breaking and other weak interaction chan-

nels are virtual, D�sð2112Þ and D�

sJð2317Þ are the bound

states when the bare 11S0 and 13S1 states are coupled to the

‘‘just virtual’’DK threshold. They show as the poles on thereal axis of the physical Riemann sheet. It is the couplingof the bare states to the lower isospin breaking Ds�

0

thresholds and the other weak thresholds that shift thepoles to the unphysical Riemann sheets when they areopen in reality. When the mixing of the 11P1 and 13P1

states is not considered, they are both the bound states at2478 MeV and 2493 MeV, respectively, below the D�

sK

TABLE III. The opened and nearby closed channels of the charmed-strange states consideredin this paper.

Mode Channel 1�ð13S1Þ 0þð13P0Þ 1þð11P1Þ 1þð13P1Þ 2þð13P2Þ0� þ 0� DK h h h

1� þ 0� D�K h h h

TABLE IV. Compilation of the experimental masses and the total widths (the PDG averagevalues [1] of the charmed-strange states. The unit is MeV.

JPðn2sþ1LJÞ Expt. mass Expt. widthffiffiffiffiffiffiffiffiffispole

p ¼ M� i�=2 GI mass

0�ð11S0Þ 1968 1980

1�ð13S1Þ 2112� 0:5 <1:9 2114� 0i 2130

0þð13P0Þ 2317� 0:6 <3:8 2358� 0i 2480

“1þð11P1Þ” 2459� 0:6 <3:5 2470� 0i 2530

“1þð13P1Þ” 2535� 0:2 <2:3 2508� 1i 2570

2þð13P2Þ 2573� 1 20� 5 2522� 7i 2590

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thresholds. The mixing owing to coupling with the com-mon D�

sK thresholds shifts the 11P1 downwards along thereal s axis, and the 13P1 moves upwards and crosses D�

sKthresholds into the complex s plane of unphysical Riemannsheet.

V. CONCLUSIONS

In this paper, we propose a simple procedure to extractthe pole positions or determine the Breit-Wigner parame-ters of the charmed states based on the parameters in thenonrelativistic potential model, by using the analyticalrepresentation of the QPC model to mimic the behaviorsof the imaginary part of the self-energy function of themeson propagator. Overall improvements could be foundbetween the pole positions or Breit-Wigner parametersand the well-established charmed and charmed-strangemesons. Several charmed-strange states could be regardedas the quasibound states, due to the coupling withnearby virtual OZI-allowed thresholds. In this model, the11P1 � 13P1 mixing is explained by the coupling withcommon channels and these resultant pole masses and

widths are consistent with the observed values. It is worthstressing that our calculation is the first one that system-atically addresses such a broad spectrum and the decays ofthe members by considering the coupled channel effects, asfar as we know. This calculation may help to improveour understanding the charmed and charmed-strangespectra.There are still some differences between the shifted pole

positions and the parameters of the newly observed states.Since at the present stage the statistics of the data is still notenough to make a firm determination, further experimentalevidences are required for a confirmation of these mesons.

ACKNOWLEDGMENTS

We are grateful to X. Liu and Z.-F. Sun for the valuablediscussions about the details of the QPC model. Z. X. issupported by the Fundamental Research Funds for theCentral Universities under Grant No. WK2030040020.Z. Z. is supported by the National Natural ScienceFoundation of China under Contract Nos. 10705009 and10875001.

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