and the radial excitations of P-wave charmed-strange mesons

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  • Newly observed DsJ3040 and the radial excitations of P-wave charmed-strange mesonsZhi-Feng Sun () and Xiang Liu ()*

    School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China(Received 11 September 2009; published 30 October 2009)

    Inspired by the newly observed DsJ3040 state, in this work we systemically study the two-bodystrong decays of P-wave charmed-strange mesons with the first radial excitation. Under the assignment of

    1jP 12, i.e. the first radial excitation of Ds12460, we find that the width of DsJ3040 is close tothe lower limit of the BABAR measurement. This indicates that it is reasonable to interpret DsJ3040 asthe first radial excitation of Ds12460. Our calculation further predicts that 0 1 channels, e.g.,DK0, D0K, and Ds , are important for the search for DsJ3040. To help future experiments findthe remaining three P-wave charmed-strange mesons with the first radial excitation, we present the

    predictions for the strong decays of these three P-wave charmed-strange mesons.

    DOI: 10.1103/PhysRevD.80.074037 PACS numbers: 13.25.Ft

    I. INTRODUCTION

    With the new observation of the DsJ meson, the spec-trum of the charmed-strange state is becoming abundant.So far, there exist six established charmed-strange mesonsDs1968, Ds2112, Ds02317, Ds12460, Ds12536,and Ds22573 listed in the Particle Data Group (PDG)[1], which can be categorized as three doublets in terms ofthe heavy quark limit: H 0; 1 Ds1968;Ds2112, S 0; 1 Ds02317; Ds12460, andT 1; 2 Ds12536; Ds22573. Two years ago,a new charmed-strange meson Ds12710 with JP 1was first announced by the BABAR Collaboration [2] andconfirmed by the Belle Collaboration later [3]. Very re-cently the BABAR experiment found Ds12710 again inthe DK invariant mass spectrum [4]. Another newly ob-served charmed-strange meson is DsJ2860, which wasobserved in both DK [2] and DK channels [4]. Thephenomenological proposals of the quantum number ofDsJ2860 include JP 3 [5,6] and JP 0 [68]. Asindicated by the BABAR experiment, the JP 0 assign-ment for DsJ2860 is forbidden according to the parityconservation since the DK decay mode of DsJ2860 wasobserved in Ref. [4]. A series of theoretical work [513]relevant to Ds12710 and DsJ2860 were carried out.

    Besides the observations ofDs12710 andDsJ2860 byanalyzing the DK invariant mass spectrum in inclusiveee interactions [4], BABAR also announced a newcharmed-strange state DsJ3040 with the mass M 3044 8stat305 syst MeV and the width 239 35stat4642syst MeV [4]. The observation ofDsJ3040 not only makes the spectrum of the charmed-strange meson abundant (the mass spectrum of the ob-served charmed-strange mesons is listed in Fig. 1), butalso stimulates our interest in exploring its underlyingstructure.

    As indicated by the BABAR Collaboration, DsJ3040was only observed in the DK channel but not found in theDK decay mode. Thus, its possible quantum number in-cludes JP 1; 0; 2; . . . . Since Ds12710JP 1 isthe first radial excitation of Ds2112 and the mass ofDsJ3040 is far larger than that of Ds12710, we furtherexclude the 0 assignment, viz. the first radial excitation ofDs1968 for DsJ3040. In Ref. [14], Matsuki, Morii, andSudoh once predicted the mass of the c s state withn2s1LJ 23P1: m 3082 MeV, which is close to theexperimental value of the mass of DsJ3040. Thus, the 1assignment to DsJ3040, the first radial excitation ofDs12460, becomes the most possible.If DsJ3040 as the radial excitation of the P-wave

    charmed-strange state is true, further experiment has thepotential to search the remaining three radial excitations ofP-wave charmed-strange states. Thus, a systematical phe-nomenological study of the strong decay mode of the P-wave charmed-strange mesons with the first radial excita-tion is an important and interesting topic. By this study, we

    FIG. 1 (color online). The mass spectrum of the observedcharmed-strange mesons and the corresponding strong decaymodes observed in experiment [14].

    *Corresponding author.xiangliu@lzu.edu.cn

    PHYSICAL REVIEW D 80, 074037 (2009)

    1550-7998=2009=80(7)=074037(10) 074037-1 2009 The American Physical Society

    http://dx.doi.org/10.1103/PhysRevD.80.074037

  • will not only obtain the information for the decays of theseP-wave charmed-strange mesons, but also can test the 1quantum number assignment to DsJ3040 comparing thecalculated decay width with the experimental data.

    In this work, we will be dedicated to the study of thestrong decay modes of P-wave charmed-strange mesonswith the radial excitation by the 3P0 model [1521].Further, we will obtain the information of the order ofmagnitude of the strong decay modes of DsJ3040.

    The paper is organized as follows. In Sec. II, we brieflyreview the 3P0 model and present the formulation of thestrong decays of P-wave charmed-strange mesons with theradial excitation. Finally, the numerical result will beshown. Section III is a short summary.

    II. THE STRONG DECAY OF P-WAVE CHARMED-STRANGE MESONS WITH THE RADIAL

    EXCITATION

    Before illustrating the strong decay of P-wave charmed-strange mesons with the radial excitation, we first intro-duce the category of the heavy flavor meson.

    In the heavy quark limitmQ ! 1, the heavy quark playsthe role of a static color source to interact with the light partwithin the heavy flavor hadron. Thus, the spin of the heavyquark ~sQ can be separated from the total angular momen-

    tum J of the heavy flavor hadron. Furthermore, ~j ~sq ~L is a good quantum number, where ~sq and ~L denote the

    spin of the light part of the heavy flavor hadron and theorbital angular momentum between the light part and theheavy quark, respectively.

    Thus, the heavy mesons can be grouped into doubletsaccording to jP , which include the j 12 doublet0; 1 with the orbital angular momentum L 0, thej 12 doublet 0; 1, and the 32 doublet 1; 2 withL 1. For L 2 there exist 1; 2 and 2; 3 dou-blets with jP 32 and 52, respectively. As shown in Fig. 1,the states existing in the doublets 0; 1, 0; 1, and1; 2 are already filled with the observed charmed-strange mesons. Two 1 states existing as S and T arethe mixture of two basis states 11P1 and 1

    3P1 [22,23]

    j1; jPl 12ij1; jPl 32i !

    cos sin sin cos j11P1i

    j13P1i

    ;

    where one takes the mixing angle tan1 ffiffiffi2p 54:7 according to the estimate in the heavy quark limit.

    For P-wave charmed-strange mesons with the radialexcitation discussed in this work, one also categorizesthem as S 0; 1 and T 1; 2 doublets accord-ing to the above approach. Two 1 states are the mixture oftwo basis states 21P1 and 2

    3P1, which satisfy the relation

    j1; jPl 12ij1; jPl 32i !

    cos0 sin0

    sin0 cos0 j21P1i

    j23P1i

    :

    In this work, we approximately take 0 54:7.For distinguishing P-wave states with and without the

    first radial excitation, one labels four P-wave states withoutthe first radial excitation as 0S, 1S, 1T, and2T. Four P-wave states with the first radial excitationare named as 0S?, 1S?, 1T?, and 2T?.If we set the upper limit of the masses of P-wave states

    with the first radial excitation as 3.04 GeV, the kinemati-cally allowed decay modes of P-wave states with the firstradial excitation are listed in Table I. In the following, the3P0 model will be applied to calculate these strong decaysin Table I.

    A. A review of the QPC model

    The 3P0 model, also known as the quark pair creation(QPC) model, was first proposed by Micu in Ref. [15] tocalculate Okubo-Zweig-Iizuka (OZI) allowed strong de-cays of a meson. Later, this model was developed by theother theoretical groups [1621] and is successful whenapplied extensively to the calculation of the strong decay ofhadron [6,2233].In the QPC model, the heavy flavor meson decay occurs

    through a quark-antiquark pair production from the vac-uum, which is of the quantum number of the vacuum, i.e.0 [15,16]. For describing a strong decay process of thecharmed-strange meson Ac1 s2 ! Bc1 q3 C s2q4, one writes out the S-matrix

    hBCjSjAi I i2Ef EihBCjTjAi: (1)

    In the nonrelativistic limit, the transition operator T isdepicted as

    T 3Xm

    h1m; 1mj00iZ

    dk3dk43k3 k4

    Y1mk3 k4

    2

    341;m

    340 !

    340 d

    y3ik3by4jk4; (2)

    where i and j denote the SU3 color indices of the createdquark and antiquark. 340 u u d d ss=

    ffiffiffi3

    pand

    !340 1ffiffi3p 34 ( 1, 2, 3) correspond to flavor andcolor singlets, respectively. 341;m is a triplet state ofspin. Ymk jkjYmk;k is the th solid harmonicpolynomial. is a dimensionless constant which repre-sents the strength of the quark pair creation from thevacuum and can be extracted by fitting the data.For the convenience of the calculation, one usually takes

    the mock state to depict the meson [34]

    ZHI-FENG SUN AND XIANG LIU PHYSICAL REVIEW D 80, 074037 (2009)

    074037-2

  • jAn2S1LJMJ Ki ffiffiffiffiffiffi2E

    p XML;MS

    hLMLSMSjJMJiZ

    dk1dk23K k1 k2nLMLk1;k212SMS12!12jq1k1 q2k2i;

    (3)

    which satisfies the normalization conditions hAKjAK0i 2E3KK0. Here, nLMLk1;k2 is the spatial wavefunction describing the meson.

    Taking the center of the mass frame of the meson A, KA 0 and KB KC K, further we obtain a generalexpression of Eq. (1)

    hBCjTjAi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8EAEBECp XMLA ;MSA ;MLB ;MSB ;MLC ;MSC ;m

    h1m; 1mj00ihLAMLASAMSA jJAMJAihLBMLBSBMSB jJBMJBi

    hLCMLCSCMSC jJCMJCih13B 24C j12A 340 ih13SBMSB24SCMSC

    j12SAMSA341miI

    MLA ;m

    MLB ;MLCK: (4)

    The color matrix element h!13B !24C j!12A !340 i 1=3, which cancels out the factor 3 before in Eq. (2). h13B 24C j12A 340 iand h13SBMSB

    24SCMSC

    j12SAMSA341mi are the flavor matrix element and the spin matrix element, respectively. Here, the spatial

    TABLE I. The relevant strong decay modes of P-wave charmed-strange mesons with the firstradial excitation allowed by the conservation of the quantum number. Here denotes that thedecay modes are kinematically forbidden if setting the upper limit of the masses of P-wave stateswith the first radial excitation as 3.04 GeV. Since the 1 state in the 1; 2 doublet decays intoD via a D wave, it is very narrow and denoted as D12420 [1]. The 1 state in the 0; 1doublet decays into D via the S wave. Hence, it is very broad and denoted as D12430 [1].State Decay modes Decay channels

    0 0 DK0, D0K, Ds 01 1 DK0, D0K0 1

    0S? 1S 0 D124300K, D12430K0, Ds124601S 1 1T 0 D12420K0, D124200K1T 1 2 1 0 1 DK0, D0K, Ds 1 0 DK0, D0K, Ds 1 1 DK0, D0K0 0 D02400K0, D024000K, Ds02317

    1S?=1T? 0 1 1S 0 D124300K, D12430K0, Ds124601T 0 D12420K0, D124200K1S 1 1T 1 2 0 D22460K0, D224600K2 1 0 0 DK0, D0K, Ds 00 1 DK0, D0K, Ds 1 0 DK0, D0K, Ds 1 1 DK0, D0K0 1

    2T? 1S 0 D124300K, D12430K0, Ds124601S 1 1T 0 D12420K0, D124200K1T 1 2 0 D22460K0, D224600K2 1

    NEWLY OBSERVED DsJ3040 AND THE RADIAL . . . PHYSICAL REVIEW D 80, 074037 (2009)

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  • TABLE II. The expression of the partial wave amplitude for the strong decays of P-wave states with the first radial excitation. Here 2= ffiffiffiffiffiffi18p ,1=3 are for the strong decay involved in and 0 mesons, respectively, while 1= ffiffiffi3p is for the other strong decays,which are the result from the flavor matrix element.

    State Decay channel Partial wave amplitude

    0 0 M00 ffiffi2

    p3

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    p2I I0

    1 1 M00 ffiffi2

    p3ffiffi3

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp I0 2I0S? 1S 0 M11 cos

    ffiffi2

    p3

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    p2I1100 I0000 sin 23

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    pI1010 I0110

    1T 0 M11 sin ffiffi2

    p3

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    p2I1100 I0000 cos 23

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    pI1010 I0110

    0 1 M10 cos03ffiffi23

    q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    p2I I0 sin0 23 ffiffi3p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 2I I0

    M12 cos0 3ffiffi3

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 2I 2I0 sin0 23 ffiffi6p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp I I01 0 M10 cos03

    ffiffi23

    q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    p2I I0 sin0 23 ffiffi3p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 2I I0

    1S? M12 cos0 3ffiffi3

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 2I 2I0 sin0 23 ffiffi6p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp I I01 1 M10 cos0 23

    ffiffi13

    q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    p2I I0

    M22 sin023ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    pI I0

    0 0 M01 cos03ffiffi23

    q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    pI0000 2I0110 sin0 23 ffiffi3p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp I1100 I1010

    1S 0 M11 sin cos0 3ffiffi2

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 2I1100 2I1010 cos sin0 ffiffi2p3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp I1010 I0110 sin sin0 3

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    pI0000 I0110 I1100

    1T 0 M11 cos cos0 3ffiffi2

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 2I1100 2I1010 sin sin0 ffiffi2p3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp I1010 I0110 cos sin0 3

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    pI0000 I0110 I1100

    2 0 M21 cos0 3ffiffiffiffi30

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 4I0000 4I0110 6I1100 6I1010 sin0

    3ffiffiffiffi15

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 3I0000 3I0110 7I1100 2I10100 1 M10 sin03

    ffiffi23

    q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    p2I I0 cos0 23 ffiffi3p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 2I I0

    M12 sin0 3ffiffi3

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 2I 2I0 cos0 23 ffiffi6p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp I I01 0 M10 sin03

    ffiffi23

    q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    p2I I0 cos0 23 ffiffi3p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 2I I0

    1T? M12 sin0 3ffiffi3

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 2I 2I0 cos0 23 ffiffi6p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp I I01 1 M10 sin0 23

    ffiffi13

    q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    p2I I0

    M22 cos023ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    pI I0

    0 0 M01 sin03ffiffi23

    q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    pI0000 2I0110 cos0 23 ffiffi3p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp I1100 I1010

    1S 0 M11 sin sin0 3ffiffi2

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 2I1100 2I1010 cos cos0 ffiffi2p3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp I1010 I0110 sin cos0 3

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    pI0000 I0110 I1100

    1T 0 M11 cos sin0 3ffiffi2

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 2I1100 2I1010 sin cos0 ffiffi2p3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp I1010 I0110 cos cos0 3

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    pI0000 I0110 I1100

    2 0 M21 sin0 3ffiffiffiffi30

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 4I0000 4I0110 6I1100 6I1010 cos0

    3ffiffiffiffi15

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 3I0000 3I0110 7I1100 2I10100 0 M02 2

    3ffiffi5

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp I I00 1 M12 2ffiffiffiffi

    30p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp I I0

    1 0 M12 2ffiffiffiffi30

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp I I02T? 1 1 M20 23

    ffiffi23

    q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    p2I I0

    1S 0 M11 cos 15ffiffi2

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 4I0000 6I0110 4I1100 6I1010 sin15

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    p3I0000 7I0110 3I1100 2I1010

    1T 0 M11 sin 15ffiffi2

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp 4I0000 6I0110 4I1100 6I1010 cos15

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

    p3I0000 7I0110 3I1100 2I1010

    2 0 M21 5ffiffi3

    p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBECp I0000 I0110 I1100 4I1010

    ZHI-FENG SUN AND XIANG LIU PHYSICAL REVIEW D 80, 074037 (2009)

    074037-4

  • integral IMLA ;m

    MLB ;MLCK is

    IMLA ;m

    MLB ;MLCK

    Zdk1dk2dk3dk4

    3k1 k23k3 k42KB k1 k33KC k2 k4nBLBMLB k1;k3

    nCLCMLC k2;k4nALAMLA k1;k2Y1mk3 k4

    2

    ; (5)

    which describes the overlap of the initial meson (A) and thecreated pair with the two final mesons (B and C).

    In this work, we use the simple harmonic oscillator (HO)wave function to represent the radial portions of the mesonspace wave function. The wave functions corresponding tothe states with nL 1S, 1P, and 2P are, respectively,

    c n1;L0k R3=2

    3=4exp

    R

    2k2

    2

    ; (6)

    c n1;L1k i2ffiffiffi2

    3

    sR5=2

    1=4Ym1 k exp

    R

    2k2

    2

    ; (7)

    c n2;L1k i 2R5=2ffiffiffiffiffiffi

    15p

    1=45 2k2R2Ym1 k

    expR

    2k2

    2

    ; (8)

    which satisfy the normalizationRc n;Lkc n;Lkdk 1.

    Here the solid harmonic polynomialYm1 k ffiffiffiffiffi34

    qm k

    with 1 1=ffiffiffi2

    p;i= ffiffiffi2p ; 0 and 0 0; 0; 1. k

    mikj mjki=mi mj is the relative momentum be-tween the quark and the antiquark within a meson whenconsidering the quark mass difference. These HO wavefunctions are relevant to the calculation of the strong decayof P-wave states with the first radial excitation.

    The helicity amplitude satisfies the relation

    hBCjTjAi 3KB KC KAMMJAMJBMJC : (9)In terms of the partial wave amplitude, one obtains thepartial decay width

    2 jKjM2A

    XJL

    jMJLj2; (10)

    where jKj denotes the three momentum of the daughtermesons in the parents center of mass frame. The partialwave amplitude MJL is related to the helicity amplitudeMMJAMJBMJC via the Jacob-Wick formula [35]

    MJLA! BC ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2L 1p

    2JA 1X

    MJB ;MJC

    hL0JMJA jJAMJAi

    hJBMJBJCMJC jJMJAiMMJAMJBMJC K;(11)

    where J JB JC and JA JP JB JC L. The cal-culation of transition amplitude using the 3P0 model in-volves two parameters: the strength of the quark paircreation from vacuum and R in the harmonic oscillatorwave function. is a universal parameter which was al-ready fixed from other channels as indicated in Ref. [36].The value of R is chosen to reproduce the root mean square(rms) radius obtained by solving the Schrodinger equationwith the linear potential.

    B. The partial wave amplitude of two-body strongdecays of P-wave states with the first radial excitation

    With the preparation mentioned above, we obtain thepartial wave amplitude of the strong decays of the P-wavestates with the first radial excitation, which are listed inTable I. In Table II, the concrete expression of the partialwave amplitude is given. The details of the spatial integral

    IMLA ;m

    MLB ;MLCK are given in the Appendix.

    C. Numerical result

    The input parameters involved in the 3P0 model includethe strength of quark pair creation from the vacuum, the Rvalue in the HO wave function and the mass of the meson.One takes the strength of the quark pair creation from the

    vacuum as 6:3 [36], which is ffiffiffiffiffiffiffiffiffi96p times larger thanTABLE III. The R value in the HO wave function and the massrelevant to the strong decays listed in Table I. Here ( ) and (0)denote the charge of the meson.

    State Mass (MeV) [1] R (GeV1) [36]

    D 1869:62 1864.84 (0) 1.52Ds 1968:49 1.41D 2010:27 2006.97 (0) 1.85Ds 2112:3 1.69D02400 2403 2352 (0) 1.85Ds02317 2317:8 1.75Ds12460 2459:6 1.92D12430 2427 2427 (0) 2.00D12420 2423:4 2422.3 (0) 2.00D22460 2460:1 2461.1 (0) 2.22K 493:677 497.614 (0) 1.41K 891:66 896.00 (0) 2.08 547.853 1.410 957.66 1.41 1019.455 2.08

    NEWLY OBSERVED DsJ3040 AND THE RADIAL . . . PHYSICAL REVIEW D 80, 074037 (2009)

    074037-5

  • that adopted by the other theoretical groups [31,37]. The

    strength of ss creation satisfies s =ffiffiffi3

    p[17]. By re-

    producing the realistic rms radius by solving theSchrodinger equation with the linear potential, one can

    obtain the value of R in the HO wave function [36]. Themass and the R value used in this work are shown inTable III.If the mass of the charmed-strange meson with 0S? is

    2.837 GeV predicted in Ref. [14], there only exists thedecay channel 0S? ! 0 0, which is allowed bythe phase space. In Fig. 2, we give the dependence of thepartial decay widths of the strong decay of 0S? state onthe RA. Here, RA is the R value of the HO wave function ofthe charmed-strange state with 0S?. The minimum ofthe decay width around RA 1:7 GeV1 in Fig. 2 is due tothe node in the radial wave function of 0S?. The totaldecay width of the 0S? charmed-strange meson is108 MeV with RA 2:8 GeV1.In this work, we take the masses of 1S? and 1T?

    charmed-strange mesons as 3.044 GeV, which is the ex-perimental value of the mass of DsJ3040. Then, wecalculate the decay of DsJ3040 under the two assump-tions 1S? and 1T?. In Figs. 3 and 4, we present thenumerical results of the two charmed-strange mesons1S? and 1T?. The dependence of the total decaywidth of 1S on the RA is shown in Fig. 5. Here, RA

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    20

    40

    60

    80

    100

    120M

    eV

    D K0

    D0 K

    Ds

    All

    FIG. 2 (color online). The variation of the strong decay mode0S? ! 0 0 with RA.

    1.5 2 2.5 3 3.5 4

    R A GeV1

    0

    20

    40

    60

    80

    100

    120

    MeV

    a

    D K 0

    D0 K

    Ds

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    20

    40

    60

    80

    MeV

    b

    D K0

    D 0 KDs

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    10

    20

    30

    40

    MeV

    c

    D K 0

    D 0 K

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    2

    4

    6

    8

    MeV

    d

    D0 2400 K0

    D0 24000 K

    Ds0 2317

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    2

    4

    6

    8

    10

    12

    14

    MeV

    e

    D1 2430 K0

    D1 24300 K

    Ds1 2460

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    2

    4

    6

    8

    MeV

    f

    D1 2420 K0

    D1 24200 K

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    10

    20

    30

    40

    MeV

    g

    D2 2460 K0

    D2 24600 K

    total

    FIG. 3 (color online). The variation of the strong decays for (a) 1S? ! 0 1, (b) 1S? ! 1 0, (c) 1S? ! 1 1,(d) 1S? ! 0 0, (e) 1S? ! 1S 0, (f) 1S? ! 1T 0, and (g) 1S? ! 2 0 with the factor RA of theHO wave function of 1S?. Here the total partial decay width is labeled by total in the diagrams.

    ZHI-FENG SUN AND XIANG LIU PHYSICAL REVIEW D 80, 074037 (2009)

    074037-6

  • denotes the R value in the HO wave function ofDsJ3040. By comparing the calculated total decaywidth of DsJ3040 with that of the BABAR data, onefinds that the total decay width ( 204 MeV) ofDsJ3040 obtained by the 3P0 model reaches up to thelower limit of the experimental width of DsJ3040 whentaking RA as 2:8 GeV

    1. With increasing the RA up to3:5 GeV1, the total decay width is close to the centralvalue of the experimental width of DsJ3040. Thus,studying the decay width of DsJ3040 under the1S? assignment shows that the first radial excitationof the P-wave charmed-strange meson to DsJ3040, i.e.1S?, is suitable.

    The result of the partial decay widths of the 1S?charmed-strange meson corresponding to RA 2:8 GeV1 (see Fig. 3) indicates that 0 1 (DK0,D0K, and Ds ) and 1 0 (DK0, D0K, andDs ) are the dominant decay modes of DsJ3040,which can further explain why DsJ3040 was first ob-served in the DK decay channel. An experimental searchof DsJ3040 in the 0 1 channel (DK0, D0K,and Ds ) is encouraged in terms of the ratio

    1S? ! 0 11S? ! 1 0 0:79

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    5

    10

    15

    20

    25

    30

    35

    MeV

    a

    D K 0

    D0 K

    Ds

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    10

    20

    30

    40

    MeV

    b

    D K0

    D 0 K

    Ds

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    10

    20

    30

    40

    50

    60

    MeV

    c

    D K 0

    D 0 K

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    2

    4

    6

    8

    10

    12

    14

    MeV

    d

    D0 2400 K0

    D0 24000 K

    Ds0 2317

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    2

    4

    6

    8

    MeV

    e

    D1 2430 K0

    D1 24300 K

    Ds1 2460

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    MeV

    f

    D1 2420 K0

    D1 24200 K

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    0.2

    0.4

    0.6

    0.8

    MeV

    g

    D2 2460 K0

    D2 24600 K

    total

    FIG. 4 (color online). The variation of the strong decays for (a) 1T? ! 0 1, (b) 1T? ! 1 0,(c) 1T? ! 1 1, (d) 1T? ! 0 0, (e) 1T? ! 1S 0, (f) 1T? ! 1T 0, and(g) 1T? ! 2 0 with the factor RA of the HO wave function of 1T?. Here the total partial decay width is labeled bytotal in the diagrams.

    2 2.5 3 3.5 4

    RA GeV1

    125

    150

    175

    200

    225

    250

    275

    300

    MeV

    All

    Babar data

    FIG. 5 (color online). A comparison of the total decay width of1S? with BABAR data. Here the straight red line and theshaded yellow band are the central value for the error of the totalwidth of DsJ3040 measured by BABAR.

    NEWLY OBSERVED DsJ3040 AND THE RADIAL . . . PHYSICAL REVIEW D 80, 074037 (2009)

    074037-7

  • corresponding to RA 2:8 GeV1.Under the assignment of 1T? to DsJ3040, we can

    obtain the variation of the strong decays for 1T? !0 1, 1 0, 1 1, 0 0, 1S 0,1T 0, and 2 0 with the factor RA [R value of

    the HO wave function of 1T?] which is depicted inFig. 4. Furthermore, the dependence of the total decaywidth on the RA value is listed in Fig. 6. The total decaywidth of DsJ3040 is about 33.8 MeV with RA 2:8 GeV1, which is consistent with our knowledge, i.e.,

    1.5 2 2.5 3 3.5 4

    RA GeV1

    25

    50

    75

    100

    125

    150

    175

    tota

    lM

    eV

    tota

    lM

    eV

    1 T

    1.5 2 2.5 3 3.5 4

    RA GeV1

    25

    50

    75

    100

    125

    150

    175

    2002 T

    FIG. 6 (color online). The dependence of the total decay width of 1T? and 2T? states on RA.

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    5

    10

    15

    20

    25

    MeV

    a

    D K0

    D0 K

    Ds Ds '

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    5

    10

    15

    20

    25

    30

    MeV

    b

    D K 0

    D0 K

    Ds

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    5

    10

    15

    20

    25

    30

    35

    MeV

    c

    D K0

    D 0 K

    Ds

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    10

    20

    30

    40

    50

    60

    70

    MeV

    d

    D K 0

    D 0 K

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    0.5

    1

    1.5

    MeV

    e

    D1 2430 K0

    D1 24300 K

    Ds1 2460

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    5

    10

    15

    MeV

    f

    D1 2420 K0

    D1 24200 K

    total

    1.5 2 2.5 3 3.5 4

    RA GeV1

    0

    2

    4

    6

    8

    10

    MeV

    g

    D2 2460 K0

    D2 24600 K

    total

    FIG. 7 (color online). The variation of the strong decays for (a) 2T? ! 0 0, (b) 2T? ! 0 1,(c) 2T? ! 1 0, (d) 2T? ! 1 1, (e) 2T? ! 1S 0, (f) 2T? ! 1T 0, and(g) 2T? ! 2 0 with the factor RA of the HO wave function of 2T?. Here the total partial decay width is labeled bytotal in the diagrams.

    ZHI-FENG SUN AND XIANG LIU PHYSICAL REVIEW D 80, 074037 (2009)

    074037-8

  • the 1 state existing T doublet is of a narrow width. In fact,the result of the decay of the 1T? state further indicatesthat DsJ3040 cannot be explained as the 1T?charmed-strange meson.

    One also predicts that the partial decay widths corre-sponding to the decay channels 1T? ! 0 1, 1 0, 1 1, 0 0, 1S 0, 1T 0, and2 0 are 9:8 103, 6.3, 13.0, 10.1, 9:9 101, 3.5,and 1:3 101 MeV, respectively. These numerical re-sults show that 1 0, 1 1, 0 0, and 1T 0 channels are important when searching the 1T? statein experiment.

    The dependence of the strong decays 2T? ! 0 0, 0 1, 1 0, 1 1, 1S 0, 1T 0, and 2 0 on the factor RA [R value of the HO wavefunction of 2T?] is given in Fig. 7. Here, we take themass of 2T? as 3.157 GeV [14]. When taking RA 2:8 GeV1, the total decay width of 2T? is 87.9 MeV(see Fig. 6), and the partial decay widths (see Fig. 7),respectively, corresponding to 2T? ! 0 0, 0 1, 1 0, 1 1, 1S 0, 1T 0, and2 0 are 15.6, 0.49, 7.2, 49.3, 1.8, 13.2, and0.28 MeV, which show that 1 1, 0 0, 1T 0, and 1 0 are key decay channels to find the 2T?charmed-strange meson.

    III. SUMMARY

    Stimulated by the newly observed charmed-strange me-son DsJ3040, we systemically study the two-bodystrong decays of P-wave charmed-strange mesons withthe first radial excitation.

    Our numerical results show that DsJ3040 can becategorized as a 1 state in the S 0; 1 doubletwell, i.e. DsJ3040 is the first radial excitation ofDs12460. We suggest experimentalist to searchDsJ3040 by the 0 1 channel (DK0, D0K,and Ds ).

    In the past six years, BABAR and Belle experiments havemade big progress in searching for charmed-strange me-sons, which lets us believe that more charmed-strangemesons will be found in future experiments. IfDsJ3040 is the first radial excitation of Ds12460,there must exist three partners of DsJ3040, which arethe rest three P-wave charmed-strange mesons with thefirst radial excitation. In this work, we also study the strongdecays of the rest three P-wave charmed-strange mesonswith the first radial excitation. Our numerical result (seeSec. II C) will be helpful to instruct future experimentalsearch of the remaining three P-wave charmed-strangemesons with the first radial excitation.

    ACKNOWLEDGMENTS

    We are grateful to Professor Hai-Yang Cheng for sug-gestions and discussion. This project is supported by the

    National Natural Science Foundation of China under GrantNo. 10705001 and A Foundation for the Author ofNational Excellent Doctoral Dissertation of the PeoplesRepublic of China (FANEDD).Note added.When this manuscript was completed, a

    work of DsJ3040 appeared [38]. In this work, authorsinvestigated the Ds mesons by a semiclassic flux tubemodel and explained DsJ3040 as 1jP 12. In ourcase, we calculated the strong decays of DsJ3040 withthe assignment of the first radial excitation of Ds12460.By comparing the total decay width of DsJ3040 ob-tained by the 3P0 model with the BABAR data, we con-cluded that DsJ3040 is the first radial excitation ofDs12460, which is consistent with the conclusion of thestructure of DsJ3040 [38].

    APPENDIX

    According to the spatial integral in Eq. (5), one cancategorize the strong decays of P-wave charmed-strangemesons with the first radial excitation into two groups:2P ! 1S 1S and 2P ! 1P 1S.For the case of 2P ! 1S 1S, the spatial integral

    IMLA ;m

    MLB ;MLCis simplified as Im0n0 due to MLB MLC 0.

    According to the constraint from Eq. (11), we take thedirection of K along the z axis: K 0; 0; jKj. In thefollowing, we present the result of the spatial integral of2P ! 1S 1S listed in Table II:

    I I11 I11 2ffiffiffi2

    p3=2!1

    R5 5

    R7

    ; (A1)

    I0 I00 2 ffiffiffi2p 3=2!1

    1 jKj2

    R3

    R5

    21 2jKj2

    R5 31jKj

    2

    R5 5

    R7

    :

    (A2)

    The spatial integrals for 2P ! 1P 1S, which are in-volved in the expressions shown in Table II, include

    I0000 2ffiffiffi2

    p3=2!2

    1jKj3

    R3

    jKjR5

    3 1 22jKj2

    22 1jKj2 2 1jKj2

    3 1jKj2 53 1jKjR7

    6jKjR7

    ; (A3)

    NEWLY OBSERVED DsJ3040 AND THE RADIAL . . . PHYSICAL REVIEW D 80, 074037 (2009)

    074037-9

  • I0110 I0110 2ffiffiffi2

    p3=2!2jKj

    R5 7

    R7

    ; (A4)

    I1010 I1010 2 ffiffiffi2p 3=2!2

    1jKj

    R5 7 5jKj

    R7

    ;

    (A5)

    I1100 I1100 2 ffiffiffi2p 3=2!2

    jKj

    R5 7 5jKj

    R7

    :

    (A6)

    Here,

    R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2A R2B R2C

    q; m1

    m1 m3 ;

    m2m2 m4 ;

    R2B R2CffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2A R2B R2C

    q ;

    2 R2B2 R2C2 2;

    R;

    5 2R2A

    2jKj22R2A

    ;

    !1 3iR3ARARBRC3=2ffiffiffiffiffiffi15

    p11=4

    exp

    12

    2jKj2

    ;

    !2 ffiffiffi6

    pR2ARARBRC5=2ffiffiffi

    5p

    11=4RCexp

    12

    2jKj2

    :

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