charmed and charmed-strange mesons

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  • Possible 2S and 1D charmed and charmed-strange mesons

    Bing Chen,* Ling Yuan, and Ailin Zhang

    Department of Physics, Shanghai University, Shanghai 200444, China(Received 22 February 2011; published 10 June 2011)

    Possible 2S and 1D excited D and Ds states are studied, the charmed states D25500, D2600,D27500, and D2760 newly observed by the BABAR Collaboration are analyzed. The masses of thesestates are explored within the Regge trajectory phenomenology, and the strong decay widths are computed

    within the heavy-quark effective theory. Both the mass and the decay width indicate that D25500 isa good candidate for 21S0. The strong decay property of D

    2600 and Ds12700 is described well bypure 23S1 states. If a mixing between 2

    3S1 and 13D1 does exist, the mixing angle is not large and 2

    3S1 is

    predominant. D2760 and DsJ2860 are possibly the 13D3 D, and Ds, respectively. D27500 andD2760 seem two different states, and D27500 is very possibly the 1D2; 52 though the possibilityof 1D2; 32 has not been excluded. There may exist an unobserved meson DsJ2850 correspondingto DsJ2860.DOI: 10.1103/PhysRevD.83.114025 PACS numbers: 13.25.Ft, 11.30.Hv, 12.39.Hg

    I. INTRODUCTION

    The properties of 2S and 1D Q q mesons have beenstudied for a long time. However, no such higher excitedQ q state has been established for lack of experimentaldata. In the past years, some higher excited charmed orcharmed-strange states were reported though most of themhave not yet been pinned down [1]. It will be useful tostudy the possible 2S and 1D charmed and charmed-strange mesons systemically in time.

    The first possible charmed radial excitation,D02640, was reported by DELPHI [2]. This state isdifficult to be understood as a charmed radially excitedstate for the observed decaying channel D anddecay width

  • more models is required. In this paper, the method pre-sented by Eichten et al. (EHQs method) [22] is employedto study the strong decay of the heavy-light mesons. Wewill label them with the notation nLJP; jq in most cases,where n is the radial quantum number, L is the orbitalangular momentum, JP refers to the total angular momen-tum and parity, and jq is the total angular momentum of the

    light degrees of freedom.The paper is organized as follows. In Sec. II, the spec-

    trum of 2S and 1D Ds and D will be examined within theRegge trajectory phenomenology. In Sec. III, the two-bodystrong decay of these states will be explored with EHQsmethod. Finally, we present our conclusions and discus-sions in Sec. IV.

    II. MASS SPECTRUM IN REGGE TRAJECTORIES

    Linearity of Regge trajectories (RTs) is an importantobservation in particle physics [23]. In the relativizedquark model [24], the RTs for normal mesons are linear.For Q q mesons, the approximately linear, parallel, andequidistant RTs were obtained both in J;M2 and innr;M2 planes in the framework of a QCD-motivatedrelativistic quark model [25].

    However, when RTs are reconstructed with the experi-mental data, the linearity is always approximate. For orbi-tally excited states, Tang and Norbury plotted many RTsof mesons and indicated that the RTs are nonlinear andintersecting [26]:

    M2 aJ2 bJ c; (1)where the coefficients a, b, c are fixed by the experi-mental data, and jaj jbj [26]. The coefficients areusually different for different RTs.

    For radially excited light q q mesons, Anisovich et al.systematically studied the trajectories on the planes n;M2in the mass region up to M< 2400 MeV [27]. The RTson n;M2 plots behave as

    M2 M20 n 12; (2)where M0 is the mass of the basic meson, n is the radialquantum number, and 2 is the slope parameter of thetrajectory.

    Possible 1S and 2S D and Ds states are listed in Table I,where yD0s2635 is the predicted mass of 2S1; 12 Dsmeson. It is easy to notice that these candidates for 1S and2S meet well with the trajectories on the n;M2 plotaccording to Eq. (2). The narrow charmed-strange state

    DsJ2632 is located around the mass region of2S, Ds. However, the exotic relative branching ratioD0K=Ds 0:14 0:06 excludes its 2S1; 12possibility. Therefore, we denote the 2S1; 12 Ds mesonwith yD0s2635. As indicated in Ref. [12], the 2P candi-date DsJ3040 meets well with the trajectory on then;M2 plot.The measured masses of D27500, D2760, and

    DsJ2860 seem a little lower than most theoretical pre-dictions of the 1D states [24,25,28]. In Fig. 1, nonlinearRTs of D and Ds states consisting of 1

    3S11, 13P22,and 13D33 were reconstructed, where the polynomialfits indicate jaj jbj. In a relativistic flux tube model, aratio bhl=bll 2 was obtained at the lowest order [29],where bhl is the coefficient for the heavy-light mesonand bll is the coefficient for the light-light meson inEq. (1). The bll (about 0.701.60) has been obtained inRef. [26]. The fitted bhl of D and Ds in Fig. 1 is about 2.74and 3.03, respectively. Obviously, the fitted ratio is con-sistent with the theoretical prediction.Through the analysis of the spectrum only, D25500,

    D2600, and Ds12700 are very likely the first radiallyexcited D and Ds states, and D

    2760 and DsJ2860 arelikely the 13D3 states.However, as is well known, the RTs can only give a

    preliminary analysis of the observed states, the investi-gation of the decay widths and the ratios of branchingfractions will be more useful to shed light on the under-lying properties of these states.

    III. DECAY WIDTH IN EHQS FORMULA

    As is well known, in the heavy-quark symmetrytheory, the heavy-light mesons degenerate in jPq , i.e.,

    two orbital ground states form a spin doublet 1S0; 1with jPq 12 , and the decay amplitude satisfies certainsymmetry relations due to the heavy-quark symmetry [30].

    TABLE I. 1S and 2S D and Ds mesons are shown.

    States 0; 12 1; 12 0; 12 1; 122S D25500 D12600 yD0s2635 Ds127001S D1869 D20070 Ds1968 Ds21122 (GeV2) 2.97 2.78 3.07 2.88

    1.0 1.5 2.0 2.5 3.0

    4

    5

    6

    7

    8

    J

    M2

    GeV

    2

    FIG. 1. Nonlinear RTs of the D and Ds triplet with N, S 1.The polynomial fits are M2 0:23J2 2:74J 1:53 GeV2and M2 0:29J2 3:03J 1:72 GeV2, respectively.

    BING CHEN, LING YUAN, AND AILIN ZHANG PHYSICAL REVIEW D 83, 114025 (2011)

    114025-2

  • The decay properties of heavy-light mesons have beenstudied in detail in the heavy-quark effective theory.When 1=mQ corrections to heavy-quark symmetry predic-tions for strong decay are ignored, the decays of the twomesons in one doublet are governed by the same transitionstrength [4,22,30,31]. As mentioned above, the concisemethod presented by Eichten et al. [22] is employed tostudy the decays of D and Ds mesons.

    In the decay of an excited heavy-light meson H, char-acterized by nLJP; jq, to a heavy-light meson H0[n0L0J0P0 ; j0q] and a light hadron hwith spin sh and orbitalangular momentum l relative to H0, the two-body strongdecay width is written as [4,22]

    H!H0h CsQ;j0q;J0jh;jq;J 2Fjq;j

    0q

    jh;l0p2l1 exp

    p

    2

    62

    ; (3)

    where

    CsQ;j

    0q;J

    0jh;J;jq

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2J0 12jq 1

    qf sQ j

    0q J

    0jh J jq

    g

    and ~jh ~sh ~l. F jq;j0q

    jh;l0 is the transition strength, and p

    is the momentum of decay products in the rest frame of H.The coefficients C depend only upon the total angularmomentum jh of the light hadron, and not separately onits spin sh and the orbital angular momentum l of the decay.The 6 j symbols of the coefficients C exhibit the heavy-quark symmetry in the strong decays of heavy-light me-sons [4,32]. The flavor factor for different decay channelscan be found in Ref. [28].

    The value of parameter is important to the decaywidth. In Ref. [22], the momentum scale was assumeduniversally 1 GeV, which implies 0:41 GeV. Inthis work, the optimum value of is taken as 0.38 GeV.It is consistent with the harmonic oscillator parameter(0.350.50 GeV) which usually appears in thepseudoscalar-meson emission model [24], the chiral quarkmodel [14,33,34], and the 3P0 model [3538].Because of lack of measurements of partial widths in

    the charmed states, the decay width of K mesons [i.e.K11270 ! K] was used to fix the transition strengthin Ref. [22]. c and b quarks are much heavier than u, d, ands quarks, so the open charm or bottom mesons providebetter place to test EHQs formula. Systematic studiesof S- and P-wave heavy-light meons (D, B, Ds, and Bsmesons) by EHQs formula have been presented inRef. [39].The EHQs formula is also obtained by the 3P0 model

    where a unitary rotation between the bases of Q q mesonsJ2; j2q; s2Q; Jz and q q mesons J2; L2; S2; Jz has been per-formed [39]. In this way, the transition strength F

    jq;j0q

    jh;l0

    obtained in the 3P0 model includes only two parameters:the dimensionless parameter and the harmonic oscillatorparameter [39]. In fact, the nodal Gaussian form factorobtained by the 3P0 model has been used for the transition

    strength Fjq;j

    0q

    jh;l0 to interpret D02640 in terms of EHQs

    formula [4].

    The relevant transition strengths Fjq;j

    0q

    jh;l0 used in this

    paper are given in Table II. Some expressions in the tablecan be found in Refs. [3638], and others are obtained in

    TABLE II. The transition strength Fjq;j

    0q

    jh;l0, where the sign P denotes a light pseudoscalar-

    meson or a light vector meson is shown.

    nLjPq ! nLjPq P F jq;j0q

    jh;l0 Polynomial of p=

    2S12 ! 1S12 0 F 1=2;1=21;1 0 52

    34121 215 p

    2

    22

    2S12 ! 1P12 0 F 1=2;1=20;0 0 1233 1 79 p2

    2 227 p

    4

    42

    2S12 ! 1P32 0 F 1=2;3=22;2 0 132

    37141 239 p

    2

    22

    1D32 ! 1S12 0 F 3=2;1=21;1 0 5234 12 1 215 p2

    22

    1D32 ! 1S12 1 F 3=2;1=21;1 0 22

    34121 215 p

    2

    22

    1D32 ! 1P12 0 F 3=2;1=22;2 0 537 14 1 215 p2

    22

    1D32 ! 1P32 0 F 3=2;3=20;0 0 22533

    1 518 p2

    2 1135 p

    4

    42

    F 3=2;3=22;2 0 132375 14 1 239 p2

    22

    1D52 ! 1S12 0 F 5=2;1=23;3 0 23

    36516

    1D52 ! 1S12 1 F 5=2;1=23;3 0 25

    37516

    F 5=2;1=22;1 0 2434 12 1 215 p2

    22

    1D52 ! 1P12 0 F 5=2;1=22;2 0 22537

    141 115 p

    2

    22

    1D52 ! 1P32 0 F 5=2;3=22;2 0 257375

    141 142 p

    2

    22

    F 5=2;3=24;4 0 243857 18

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  • the 3P0 model in detail in Ref. [39]. For these transitionstrengths, a constant

    G 1=22 210

    34

    ~MB ~MC~MA

    1

    (4)

    was omitted. Here the phase space normalization ofKokoski and Isgur is employed [24,38]. ~MA, ~MB, ~MCare the mock-meson masses of A, B, C, respectively.The constant G absorbs the dimensionless parameter inthe 3P0 model. The variation of the constant G with themock-meson masses ~Mi is slow.

    In the analysis that follows, the decay widths of possible2S and 1D D and Ds states are computed in terms ofEq. (3).

    A. 21S0 or [2S0; 12]D25500 observed in the decay channel D is a

    good candidate for a 21S0 charmed meson. Following theprocedure in Ref. [39], we take the decay width ofD224600 as an input and obtain the d-wave transitionstrength F 3=2;1=22;2 0 0:964 GeV4, where

    F 3=2;1=22;2 0 G22

    341

    4:

    All the other transition strengths Fjq;j

    0q

    jh;l0 in Table II

    could be fixed easily once the mock-meson masses ~Mieffect has been taken into account. According to ourcomputation [39], the total decay width of D25500 isabout 124.1 MeV. The dominating decay mode is the Dchannel with D 121:0 MeV, and the decaywidth of another allowed D02400 channel is 3.1 MeV[the mass of D02400 is taken as 2318 MeV [1]].

    These results agree well with the experiments. It ex-plains the fact that D25500 was first observed in D[18]. In Fig. 2, the variation of the decay width with is plotted. Obviously, the observed decay width of

    D25500 is well obtained in the reasonable region of (0.350.42 GeV).In Ds states, the mass of the 2

    1S0 state is predictedaround 2635 20 MeV [a little smaller than the thresholdof D and D02400K], and DK is the only two-bodystrong decay channel. Our result for this decay channel isDK 82:2 15:1 MeV, so it is impossible that theobserved DsJ2632 is 21S0.

    B. Mixing states of 23S1 and 13D1

    The predicted masses of 23S1, D are almost about26002640 MeV, and the masses of 23S1, Ds are almostabout 27102730 MeV (Table III) [24,25,4042]. Thespectrum and the helicity-angle distributions suggest thatD2600 is the 23S1 [18]. In our analysis, it is possible toexplain both D2600 and Ds12700 as the pure 23S1states. In this case, the variations of the branching fractionsand decay widths with are given in Figs. 3 and 4,respectively. Obviously, theoretical decay widths and ra-tios in the reasonable region of are consistent with theexperimental data.In the charmonium system, c 2S and c 3770 are

    two orthogonal partners of mixtures of 23S1 and 13D1

    with JPC 1 [43]. This mixing scheme has also beenemployed to explain the decay width and the ratio ofbranching fractions of Ds12700 and DsJ2860 [14].If this mixing does exist, there are two orthogonal partners(JP 1) of D and Ds. They can be denoted as

    D 2550 0

    D 2550 0 : 130 12stat 13syst

    0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42

    80

    100

    120

    140

    160

    180

    200

    GeV

    Dec

    ayW

    idth

    MeV

    FIG. 2 (color online). The decay width versus , whereD25500 is taken as a pure 21S0 (green line) state. Thedashed line refers to central values of the decay width givenby experiment.

    TABLE III. D, Ds masses of the states 23S1 and 1

    3D1predicted in different models are shown (MeV).

    States Ref. [11] Ref. [24] Ref. [25] Ref. [40] Ref. [41]

    D123S1 2640 2632 2620 2636D01 13D1 2820 2788 2710 2740Ds123S1 2711 2730 2731 2730 2714D0s113D1 2784 2900 2913 2820 2804

    D K DK

    D D

    Ds1 2700 : 0.91 0.13stat 0.12syst

    D 2600 0 : 0.32 0.02stat 0.09syst

    0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.420.0

    0.5

    1.0

    1.5

    GeV

    Bra

    nchi

    ngR

    atio

    FIG. 3 (color online). Branching ratios of D26000 andDs12700 with . The dashed lines refer to central values ofdecay width given by experiment.

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  • jSD1iL cosj23S1i sinj13D1i;jSD1iR sinj23S1i cosj13D1i: (5)

    Details for the estimate of the decay width are given inthe Appendix where D2600 is identified with thejSD1iL of D.

    To proceed our analysis, the masses of pure 23S1 and13D1 obtained in Refs. [24,41] are used. For 2

    3S1, themasses from these two groups are almost the same. For13D1, the mass given by Ref. [24] is much larger than thatin Ref. [41] (Table III).

    When mixing angles are treated as free variables,the decay widths and ratios of D26000 and Ds12700dependence on them are presented in Figs. 5 and 6,respectively. In the figures, the red lines and the blue lines

    result from the predicted masses of the pure 23S1 and 13D1

    in Ref. [24] and in Ref. [41], respectively. However, whenD26000 and Ds12700 are identified with the jSD1iLof D and Ds, respectively, the mixing angles are fixed(Table IV). The mixing angle does not seem to bestrongly dependent on the masses input of 13D1. Themixing angles determined from two different massesinput are used as the reasonable boundaries of the vari-ables. Obviously, the ratio D=D of D26000 andDK=DK of Ds12700 in the reasonable region agreewell with experiments. The decay widths are a little largerthan the experimental data.In summary, both D26000 and Ds12700 can be

    explained as the pure 23S1 states. If the mixing between23S1 and 1

    3D1 exists, the mixing angle is not large and23S1 is predominant.The decay channels D27600 ! D and

    DsJ2860 ! D0K have been observed. However, it isdifficult to identify D27600 and DsJ2860 with the

    D 2600 0: 93 6 13

    21250

    50

    100

    150

    200

    Dec

    ayW

    idth

    MeV

    D 2600 0: 0.32 0.02stat 0.09syst

    80 60 40 20 0 20 40 60 80

    0.0

    0.5

    1.0

    1.5

    2.0

    Mixing Angle Degree

    Rat

    ioD

    D

    DPB

    GI

    DPB

    GI

    FIG. 5 (color online). Decay width and ratio D=D ofD26000 in the diagram method. The horizontal dashed linesrefer to central values of the decay width given by experiment.

    Ds1 2700 : 125 30

    13 184060

    80

    100

    120

    140

    160

    180

    Dec

    ayW

    idth

    MeV

    Ds1 2700 : 0.91 0.13stat 0.12syst

    80 60 40 20 0 20 40 60 800.0

    0.5

    1.0

    1.5

    2.0

    Mixing Angle Degree

    Rat

    ioD

    KD

    K

    DPB

    GI

    DPB

    GI

    FIG. 6 (color online). Decay width and ratio DK=DK ofDs12700 in the diagram method. The horizontal dashed linesrefer to central values of decay width given by experiment.

    TABLE IV. Mixing angles determined by masses input inRefs. [24,41] are shown.

    D123S1 13D1 Ds123S1 13D1Theoretical

    prediction

    (Theo.)

    Experimental

    data

    Theoretical

    prediction

    Experimental

    data

    Ref. [24] 2640 2608 2730 2709

    2820 2851 (Theo.) 2900 2921 (Theo.)

    21 18Ref. [41] 2636 2608 2714 2709

    2740 2767 (Theo.) 2804 2809 (Theo.)

    25 13

    Ds1 2700

    D 2600 0

    Ds1 2700 : 125 30

    D 2600 0 : 93 6 13

    0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.420

    50

    100

    150

    200

    250

    300

    350

    GeV

    Dec

    ayW

    idth

    MeV

    FIG. 4 (color online). Decay widths of D26000 andDs12700 with . The dashed lines refer to central values ofdecay width given by experiment.

    POSSIBLE 2S AND 1D CHARMED AND CHARMED- . . . PHYSICAL REVIEW D 83, 114025 (2011)

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  • jSD1iR of D and Ds, respectively. In other words, if theyare the orthogonal partners of D26000 and Ds12700,respectively, the decay width ofDsJ2860 is broader than200 MeV and the decay width of D27600) is broaderthan 110 MeV. These decay widths are much broader thanthe experimental results.

    C. 13D3 or [1D3; 52]D2760 and DsJ2860 are very possibly the 13D3 D

    and Ds, respectively.D27600 was observed in the decay channel D

    and was suggested to be a D-wave charmed meson [18].If D27600 has the same JP with the 13D1, it would havea broad width through the mixing scheme mentionedabove.

    Under the assumption that both D2760 andDsJ2860 are the 13D3 states, their partial widths andtotal decaywidths are given in Table V. The predicted decaywidths of them are in accord with experimental results.

    D27500 has mass close to D2760, if these twostates are the same state of 13D3, the predicted ratioD2760 ! D=D2760 ! D 1:78 (seeTable IV) is much larger than the observedBD27600!D=BD27500!D0:420:050:11. This fact supports the suggestion thatD27500and D2760 are two different charmed states [18,20].

    For DsJ2860, the predicted DsJ2860 ! DK=DsJ2860 ! DK 0:43 is much smaller than theexperimental

    BDsJ2860!DKBDsJ2860!DK 1:10 0:15stat 0:19syst.

    It is apparent that the mass gaps of the correspondingground state between D and Ds are about 100 MeV [1].The mass gap between Ds12700 and D2600, and themass gap betweenDsJ2860 andD2760 are also about100 MeV. The mass gap supports the suggestion thatDs12700 is a similar state as D2600 with the sameJP. Therefore, there should exist a charmed-strangeDsJ2850 which has the same JP; jq of D27500with mass close to DsJ2860.

    D. 1D2; 32 and 1D2; 52D27500 was observed in D and is possibly a

    1D2; 32 or 1D2; 52, there exists similar assignmentfor the suggested DsJ2850. The partial widths of sometwo-body decay modes of D27500 and DsJ2850 in thetwo possible assignments have been computed and pre-sented in Table VI.If DsJ2850 is the 1D2; 32, the predicted

    ratio of branching fraction BDsJ2850 ! DK=BDsJ2860 ! DK is about 2.42. Theoretical predictionsof the decay width and the ratio of branching fractionBD27600 ! D=BD27500 ! D? 0:52of D27500 are in accord with experiment.If D27500 and DsJ2850 are the 1D2; 52,

    D27500, D27600 and DsJ2850, DsJ2860 formthe 1D2; 3 doublet of D and Ds, respectively.For charmed mesons D27500 and D27600, we ob-

    tained BD052!D=BD052!D?0:82,which is a little larger than the observed BD27600 !D=BD27500 ! D 0:42 0:05 0:11.We obtained BDsJ52!DK=BDsJ52!DK0:92for the charmed-strange mesons DsJ2850 andDsJ2860, and the observed BD

    sJ2860!DK

    BDsJ2860!DK 1:100:15stat 0:19syst. Theoretical predictions are in accordwith experiments within the uncertainties of the 3P0 model.In our computation, a spin counting has been used.The two states in the doublet 1D2; 3 [1D2; 52 and

    1D3; 52] have masses close to each other while theirmass splitting is comparable to the uncertainty of theirmasses, it will be difficult to distinguish these two statesthrough the channel of D and D. In this case, thepartial width of D observed by experiment is the totalone of D27500 and D27600. However, the state1D2; 52 decays through the Pwave and the F wave while

    TABLE V. Two-body strong decays of the states 13D3 areshown.

    Modesa

    i (MeV) Modesbi (MeV) Modes

    bi (MeV)

    D K 12.3 D 12.4 Ds K 0.9D K 28.4 D 22.0 D

    s K 0.1

    Ds 0.6 D 0.2 D012430 1.1Ds 3.0 D 0.8 D12420 0.4D K 0.5 D 0.1 D22460 1.3Ds ! 0.2 D ! 0 atotal 44.9

    btotal 39.3

    Experimental

    data

    48 7 Experimentaldata

    60:9 8:7

    adecay modes of DsJ2860.bthose of D2760.

    TABLE VI. Two-body strong decays of the states 2; 32 and2; 52 are shown.Modesa 2; 32 2; 52 Modesb 2; 32 2; 52D 58.9 20.0 D K 96.2 19.2D 5.4 0.2 Ds 21.7 0.9Ds K 8.6 0.2 D K 4.3 18.0D 1.9 9.2 D ! 0.7 3.3 Ds ! 2.7 13.3

    D02400 0.6 10.9 D02400K 0.2 0.2D012430 0.2 1.4 D12420 0.5 1.4 D22460 1.2 0.3 ytotal (MeV) 77.9 47.9

    ztotal (MeV) 125.1 51.6

    Experimental

    data 71 17 Experimental

    data

    adecay modes of D27500.bDsJ2850, respectively.

    BING CHEN, LING YUAN, AND AILIN ZHANG PHYSICAL REVIEW D 83, 114025 (2011)

    114025-6

  • the state 1D3; 52 can only decay through the F wave.Therefore, the widths of decay channels D and D! ofD27500 would much broader than those of D27600.The observation of the channels D and D! in forthcom-ing experiments will be useful to pin down these states.

    IV. CONCLUSIONS AND DISCUSSIONS

    In this work, we study the possible 2S and 1D D and Dsstates, especially the four new D candidates observed bythe BABAR Collaboration. Both the mass and the decaywidth indicate that D25500 is a good candidate of the21S0 charmed state. The 2

    1S0 Ds meson is predicted tohave mass about 2635 20 MeV and decay width about82:2 15:1 MeV. The observedDsJ2632 seems impos-sible the 21S0 Ds meson if it exists.

    D2600 andDs12700 can be explained as pure 23S1states. If the mixing between 23S1 and 1

    3D1 exists, themixing angle is not large and 23S1 is predominant. Theresults indicate that the mixing angle is not stronglydependent on the input mass of 13D1. Our analysis doesnot support the possibility that D2760 and DsJ2860are the orthogonal partners of D2600 and Ds12700,respectively.

    If an unobserved meson, corresponding to D27500,DsJ2850 exists, more measurement of DsJ2860is required. D2760 and DsJ2860 could be identifiedwith the 13D3 D and Ds states, respectively. D27500and D2760 favor to form the doublet 1D2; 3.The possibility that D27500 is the 1D2; 32 state hasnot been excluded, so the observation of the channels Dand D! would be important for the identification ofD27500 and D27600.

    ACKNOWLEDGMENTS

    Bing Chen thanks Professor Tom Steele for usefuldiscussions. This work is supported by the NationalNatural Science Foundation of China under GrantsNo. 10775093 and No. 11075102. Bing Chen is also sup-ported by Shanghai University under the GraduatesInnovation Fund SHUCX092016 of its contract: No. A.16-0101-09-543.

    APPENDIX

    When one considers the two-body strong decay ofD26000 in the mixing scheme [Eq. (5)], the Eq. (3)should be written as

    H!H0h pGXLS

    j cosC1P LS23S1!H0hx1ex21=12

    sinC2P LS13D1!H0hx2ex

    22=12j2: (A1)

    where

    x1p1 ; x2p2;

    pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

    D26000 mH0 mh2m2D26000mH0 mh2q

    2mD26000;

    p1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

    23S1mH0 mh2m223S1mH0 mh2

    q2m23S1

    ;

    p2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

    13D1mH0 mh2m213D1mH0 mh2

    q2m13D1

    ;

    C1 and C2 are coefficients for different decay channels(Table VII). P LSx are the channel-dependentpolynomials.For 23S1:

    23S1 !1S0 1S03S1 1S0 :

    5

    34x1

    1 2

    15x21

    ;

    23S1 ! 1P1;

    1

    2

    1S0: 1

    21=237=2

    1 7

    9x21

    2

    27x41

    ;

    23S1 !(1P

    1; 32

    1S0

    3P2 1S0:

    13

    311=2x21

    1 2

    39x21

    :

    For 13D1:

    13D1 !1S0 1S03S1 1S0 :

    51=221=2

    34x2

    1 2

    15x22

    ;

    13D1 ! 1P1;

    1

    2

    1S0: 5

    1=2

    311=2x22

    1 2

    15x22

    ;

    13D1!1P1;

    3

    2

    1S0

    8>:251=2

    37=2

    1 518x22 1135x42

    13311=2

    x22

    1 239x22

    ;

    13D1 ! 3P2 1S0: 13

    311=251=2x22

    1 2

    39x22

    :

    TABLE VII. The coefficients for different decay channels inheavy-quark effective theory. S, P, and D refer to S, P, andD-wave decays.

    C123S1 C213D11S0 1S0

    ffiffi13

    qP

    ffiffi23

    qP

    3S1 1S0 ffiffi23

    qP

    ffiffi13

    qP

    1P1; 12 1S0 1 S 1 D1P1; 32 1S0

    ( ffiffi12

    q

    D 1 S

    ffiffi12

    qD

    3P2 1S0 ffiffi12

    qD

    ffiffi12

    qD

    POSSIBLE 2S AND 1D CHARMED AND CHARMED- . . . PHYSICAL REVIEW D 83, 114025 (2011)

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