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)

    074037-3

  • 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 3ffif...