P -wave charmed-strange mesons

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  • PHYSICAL REVIEW C 72, 065202 (2005)

    P-wave charmed-strange mesons

    Yukiko Yamada, Akira Suzuki, and Masashi KazuyamaDepartment of Physics, Tokyo University of Science, Shinjuku, Tokyo, Japan

    Masahiro KimuraDepartment of Electronics Engineering, Tokyo University of Science, Suwa, Nagano, Japan

    (Received 7 August 2005; published 9 December 2005)

    We examine charmed-strange mesons within the framework of the constituent quark model, focusing on thestates with L = 1. We are particularly interested in the mixing of two spin states that are involved in Ds1(2536)and the recently discovered DsJ (2460). We assume that these two mesons form a pair of states with J = 1.These spin states are mixed by a type of spin-orbit interaction that violates the total-spin conservation. Withoutassuming explicit forms for the interactions as functions of the interquark distance, we relate the matrix elementsof all relevant spin-dependent interactions to the mixing angle and the observed masses of the L = 1 quartet.We find that the spin-spin interaction, among various types of spin-dependent interactions, plays a particularlyinteresting role in determining the spin structure of Ds1(2536) and DsJ (2460).

    DOI: 10.1103/PhysRevC.72.065202 PACS number(s): 12.38.Bx, 12.39.Jh, 12.39.Pn, 14.40.Lb

    I. INTRODUCTION

    Recently a new charmed-strange meson, DsJ (2317), wasdiscovered by the BaBar Collaboration [1] and confirmedby the CLEO Collaboration [2]. The CLEO reported anothercharmed-strange meson called DsJ (2460). Both these mesonswere confirmed by the Belle Collaboration [3,4]. The massesand decay properties of DsJ (2317) and DsJ (2460) have beeninvestigated with two types of particular structures assumed forthem. One type is the ordinary qQ structure, and the other is anexotic structure such as the KD molecule [57,9] or tetra-quarkconfiguration [1013]. We will work with the former structurein this paper. Then, these new entries together with Ds1(2536)and Ds2(2573), which were discovered earlier, are expected toform a quartet with L = 1 (P states) of the cs (or sc) system.Given this expectation, Godfrey studied various propertiesof DsJ (2317) and DsJ (2460) [14,15], following the workdone prior to the discoveries of these mesons [16,17]. Also,decay modes of DsJ (2317) and DsJ (2460) were analyzed byColangero and De Fazio [18], Bardeen et al. [19], Mehen andSpringer [7], and Close and Swanson [8].

    With respect to the spin structure of these mesons, there arefour states, 1P1, 3P0, 3P1, and 3P2, in terms of the JLS bases.1

    While DsJ (2317) and Ds2(2573) can probably be assignedto 3P0 and 3P2, respectively, Ds1(2536) and DsJ (2460) areprobably mixtures of 1P1 and 3P1. The extent of the mixing canbe parametrized by a mixing angle [1417,20,21]. In additionto the masses of the mesons, the branching fractions for B

    Present address: Department of Physics, Kyushu University,Fukuoka, Japan.

    Present address: Department of Physics, Nagoya University,Nagoya, Japan.

    1We use the ordinary spectroscopic notation 2S+1LJ that is usedfor a two-particle system, where S,L, and J are total spin, orbitalangular momentum, and total angular momentum quantum numbers,respectively.

    DDsJ followed by the electromagnetic (EM) decays of DsJhave also been measured [3]. The mixing angles are closelyrelated to the EM decay rates of DsJ [15].

    The purpose of this paper is to examine the spin structureof the four mesons. We use the constituent quark model withthe interquark interactions that arise from the nonrelativisticexpansion of the QCD-inspired Fermi-Breit interaction. Wehave five types of interactions in the following sense. Inaddition to the spin-independent interaction that consists ofa confining potential and the color Coulomb interaction, wehave four types of spin-dependent interactions. They are thespin-spin, tensor, and two types of spin-orbit interactions, onwhich we elaborate in the next paragraph. The model is thesame as the one used by Godfrey et al. [16,17,20] except thatwe do not assume any explicit forms for the interactions asfunctions of the distance between the two quarks. We treat allspin-dependent interactions perturbatively.

    By the two types of spin-orbit interactions, we mean theones that are symmetric and antisymmetric with respect tothe interchange of the two quarks. We refer to the former asSLS and the latter as ASLS interactions. The SLS interactioncommutes with the total spin of the two quarks, whereasASLS interaction does not. The ASLS interaction violatesthe conservation of the total spin. This is the agent thatinduces the mixing of 1P1 and 3P1. The ASLS interaction isproportional to the mass difference between the quarks. Hence,its effect can be substantial when the mass difference is large,leading to a specific amount of mixing in the heavy quarklimit. This is indeed the case with the cs (or sc) system aswe will see. Historically, ASLS interaction effects were firstexamined for the -N interaction and hypernuclei [2224].Regarding the particular roles of spin-orbit interactions in qQsystems, we refer to a series of works by Schnitzer [25] andthe work by Cahn and Jackson [26] in addition to those citedalready [16,17,20].

    As we said above, we have five types of interactions. Onthe other hand, there are five pieces of experimental data now

    0556-2813/2005/72(6)/065202(7)/$23.00 065202-1 2005 The American Physical Society

    http://dx.doi.org/10.1103/PhysRevC.72.065202

  • YAMADA, SUZUKI, KAZUYAMA, AND KIMURA PHYSICAL REVIEW C 72, 065202 (2005)

    TABLE I. Summary of observed charmed-strange mesons.

    Label Mass (MeV) Assignment Year of(2S+1LJ ) discovery

    Ds 1968.3 0.5 1S0 1983 [28]Ds 2112.1 0.7 Probably 3S1 1987 [29]DsJ (2317)

    2317.4 0.9 Probably 3P0 2003 [3]DsJ (2460) 2459.3 1.3 ? 2003 [3]Ds1(2536) 2535.35 0.34 ? 1989 [30]Ds2(2573) 2572.4 1.5 Probably 3P2 1994 [31]

    available, which are the masses of the four mesons and thebranching ratio of the EM decays. [See Eq. (29).] The matrixelements of the five interactions (within the P-state sector) canbe determined such that the five pieces of the experimental dataare reproduced. At the same time, the spin structure of the fourmesons can be determined. In doing so, we do not have to knowthe radial dependence of the interactions. As it turns out, thespin-spin interaction, among the four types of spin-dependentinteractions, plays a particularly interesting role in relation tothe spin structure of Ds1(2536) and DsJ (2460).

    We begin Sec. II by defining a nonrelativistic model Hamil-tonian that incorporates relativistic corrections as variousspin-dependent interactions and proceed to determining thematrix elements of the interactions by using the mass spectraof the L = 1 quartet of charmed-strange mesons and the EMdecay widths of DsJ (2460). In Sec. III, we remark on theapproximations that we use. Discussions and a summary aregiven in the last section. In Table I we list the observedcharmed-strange mesons that we consider in this paper [27].

    II. HAMILTONIAN AND MIXING ANGLE

    We assume that the nonrelativistic scheme is appropriatefor the system, and relativistic corrections can be treatedas first-order perturbation. The nonrelativistic expansion ofthe Fermi-Breit interaction gives us the Hamiltonian for acharmed-strange meson in the form of

    H = H0 + Ss ScVS(r) + S12VT (r)+L SV (+)LS (r) + L (Ss Sc)V ()LS (r), (1)

    where Si is the spin operator of the strange quark when i = sand of the charmed quark when i = c, S = Ss + Sc, S12 is thetensor operator, and L the orbital angular momentum operator.The lowest-order terms in the nonrelativistic expansion are allin H0 which also contains a phenomenological potential toconfine the quarks. More explicitly, H0 reads as

    H0 = ms +mc + p2s

    2ms+ p

    2c

    2mc+ VC(r) + Vconf(r), (2)

    where mi and pi are the mass and momentum of quark i,respectively, VC is the color Coulomb interaction, and Vconfis the confinement potential. The last two terms of Eq. (1)are the SLS and ASLS interactions, respectively. The spatialfunctions attached to the operators in Eq. (1) can be expressedin terms of VC and Vconf [17,32]. However, we do not need

    such explicit expressions of these functions, as it will becomeclear shortly.

    We start with the eigenstates of H0 such that

    H0nJLS(r) = E(0)nLnJLS(r), (3)where

    nJLS(r) = RnL(r)J

    M=JCMYMJLS(, ). (4)

    Here CM are constants such that

    M |CM |2 = 1 and can bechosen as (2J + 1)1/2 since there is no preferable direction.We concentrate on the P states ofn = 1 with no radial node. Wedenote each of theL = 1 states with single index according to

    =

    1234

    corresponding to

    1P13P03P13P2

    . (5)

    Next we calculate the matrix elements of H in terms ofthe bases defined by Eqs. (3) and (4). Nonvanishing matrixelements are

    H11 = M0 34vS,H22 = M0 + 14vS 2vLS 4vT ,H33 = M0 + 14vS vLS + 2vT ,H44 = M0 + 14vS + vLS 25vT ,H13 = H31 =

    2,

    (6)

    where

    M0 =

    d3r J1S(r)H0J1S(r) = E(0)1 , (7)

    vS =

    0drr2VS(r)R

    21(r), (8)

    vLS =

    0drr2V

    (+)LS (r)R

    21(r), (9)

    vT =

    0drr2VT (r)R

    21(r), (10)

    =

    0drr2V

    ()LS (r)R

    21(r). (11)

    We choose the phases of the wave functions involved inEq. (11) such that is positive. Here we have suppressedsuffix n = 1 of the wave functions and the unperturbed P-stateenergy. We have ignored the tensor coupling of the 3P2 state tothe 3F2 state. We will remark on this point in the next section.Note that the ASLS interaction gives rise to = 0, whichcauses the mixing of 1P1 and 3P1.

    All of the matrix elements of the Hamiltonian that we needare parametrized in terms M0, vS, vLS, vT , and . These fiveparameters can be determined by the four observed masses andthe EM decay rates of DsJ (2460). We have no other adjustableparameters. In this context, we do not need explicit expressionsof the radial wave function nor the radial dependence of thepotential functions.

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  • P-WAVE CHARMED-STRANGE MESONS PHYSICAL REVIEW C 72, 065202 (2005)

    The diagonalization of H leads to four states whose massesare given by

    M+ = 12[2M0 12vS vLS + 2vT

    +{(vLS 2vT vS)2 + 82}1/2], (12)

    M2 = M0 + 14vS 2vLS 4vT , (13)M = 12

    [2M0 12vS vLS + 2vT

    {(vLS 2vT vS)2 + 82}1/2], (14)

    M4 = M0 + 14vS + vLS 25vT . (15)The second and fourth states with M2 and M4 are pure 3P0and 3P2 states, respectively. We identify them with DsJ (2317)and Ds2(2573). The other two states with M+ and M arecomposed of 1P1 and 3P1 states. We interpret them asDs1(2536)and DsJ (2460), respectively.

    Let us introduce a mixing angle that represents the extentof the mixing of 1P1 and 3P1 states inDs1(2536) andDsJ (2460).Following Godfrey and Isgur [16], we define by

    +(r) = 110(r) sin + 111(r) cos ,(r) = 110(r) cos + 111(r) sin ,

    (16)

    where + and are the eigenstates that correspond toDs1(2536) and DsJ (2460), respectively. The requirement thatthe energy eigenvalues for are M leads to

    tan(2 ) = 2

    2

    vS vLS + 2vT . (17)

    It is understood that lies in the interval of /2 0so that it conforms to the sign convention used in Ref. [16].Since /4 0 (or /2 /4) if (vS vLS +2vT ) 0 (or 0), we have 0 (or /2) as 0if (vS vLS + 2vT ) 0 (or 0). In other words, when (vS vLS + 2vT ) > 0 (or < 0), Ds1(2536) develops from the 3P1 (or1P1) state, whileDsJ (2460) develops from the 1P1 (or 3P1) statebecause of the ASLS interaction.

    We can express the five parameters M0, vS, vLS, vT , and in terms of the four observed masses and the mixing anglesuch that

    M0 = 14M+ + 14M + 112M2 + 512M4, (18)vS = 13 (1 2 cos(2 ))M+ 13 (1 + 2 cos(2 ))M

    + 19M2 + 59M4, (19)vLS = 18 (1 + cos(2 ))M+ 18 (1 cos(2 ))M

    16M2 + 512M4, (20)vT = 548 (1 + cos(2 ))M+ + 548 (1 cos(2 ))M

    536M2 572M4, (21) = 1

    2

    2(M+ M) sin(2 ). (22)

    Equation (18) states that the mass of the center of gravity ofthe l = 1 quartet is free from the spin-dependent interactionsinvolved in Eq. (1) in the lowest-order perturbation scheme.

    In order to determine the mixing angle, we consider EMdecays of DsJ (2460) to Ds and Ds . Generally the E1 decaywidth of a meson composed of quark 1 and antiquark 2 is

    given by

    (i f + ) = 4e2Q

    27k3(2Jf + 1) |f |r| i|2 Sif , (23)

    where eQ is the effective charge defined by

    eQ = m1e2 m2e1m1 +m2 , (24)

    k is the momentum of the emitted photon

    k = M2i M2f2Mi

    , (25)

    and

    Sif ={

    1 for a transition between triplet states,

    3 for a transition between singlet states,(26)

    is a statistical factor [33]. For the decays of DsJ , we have

    k ={

    322.7 MeV for the decay to Ds ,442.0 MeV for the decay to Ds,

    (27)

    and (2Jf + 1)Sif = 3 for both cases. Since only the 3P1 statein DsJ undergoes the transition to Ds and only the

    1P1 state toDs , the matrix element f |r|i is proportional to sin for thedecay to Ds and to cos for the decay to Ds [15]. Thus weobtain

    (DsJ Ds )(DsJ Ds ) =

    (322.7

    442.0

    )3tan2 . (28)

    The Belle Collaboration made the first observation of B DDsJ decays and reported the branching fractions for B DDsJ followed by the EM decays of DsJ [3]. Colangelo et al.analyzed the data to extract the ratio of branching fractions forthe EM decays ofDsJ (2460) toDs andDs [34]. They obtained

    Rexp [(DsJ Ds )(DsJ Ds )

    ]exp

    = 0.40 0.28. (29)

    The experimental value has the large statistical errors whichresults in a large uncertainty in determining the mixing angleas can be seen in Fig. 1. The numerical value is

    = 45.47.5+16.4 , (30)where the upper and lower increments are due to the positiveand negative corrections of the statistical errors in Rexp,respectively. This may be compared with 38 obtained byGodfrey and Kokoski [17], and 54.7 that emerges fromsin = 2/3 in the heavy quark limit [15,20].

    We can calculate M0, vS, vLS, vT , and throughEqs. (18)(22) by fitting the observed masses of Table I andthe mixing angle of Eq. (30). Again these quantities are subjectto uncertainties due to the statistical errors. Using the centralvalues of the observed masses, we obtain

    M0 = 2513.6 MeV, (31)vS = 21.013.1+27.5 MeV, (32)vLS = 61.4+2.55.2 MeV, (33)vT = 19.72.1+4.3 MeV, (34) = 26.91.04.1 MeV. (35)

    065202-3

  • YAMADA, SUZUKI, KAZUYAMA, AND KIMURA PHYSICAL REVIEW C 72, 065202 (2005)

    -60

    -40

    -20

    0

    0 0.2 0.4 0.6 0.8

    (degree)

    R exp

    FIG. 1. Variation of the mixing angle with Rexp. The dot-dashedline shows the value obtained from the central value of Rexp, andthe vertical dotted lines indicate the upper and lower values of Rexpallowed within the statistical errors.

    In Fig. 2 we show how the matrix elements vary when Rexpis varied within the statistical errors. Note that the matrixelement of the spin-spin interaction is particularly sensitiveto the variation of Rexp. Since the sign of (vS vLS + 2vT )determines the main spin states of M, it is interesting tosee the Rexp dependence of this quantity shown in Fig. 3. Wesee that (vS vLS + 2vT ) changes its sign from positive tonegative as Rexp passes over 0.39. If Rexp < 0.39 the mainspin states of Ds1(2536) and DsJ (2460) are, respectively,3P1 and 1P1. If Rexp exceeds 0.39, these two spin states areinterchanged.

    In the nonrelativistic expansion of the Fermi-Breit inter-action, the spin-spin interaction contains the derivative of thecolor Coulomb interaction. If the color Coulomb interaction

    0

    20

    40

    60

    80

    0 0.2 0.4 0.6 0.8

    Rexp

    Matrix Elements(MeV)

    vLS

    vS

    vT

    FIG. 2. Matrix elements calculated by applying the experimentalvalue of Rexp from Eqs. (18)(22) with Eq. (28). The dot-dashedline shows the value obtained from the central value of Rexp, andthe vertical dotted lines indicate the upper and lower values of Rexpallowed within the statistical errors.

    -40

    -20

    0

    20

    40

    60

    80

    0 0.2 0.4 0.6 0.8

    (vS-vLS+2vT)(MeV)

    Rexp

    FIG. 3. Matrix element (vS vLS + 2vT ) vs Rexp. The value atwhich the matrix element changes its sign is 0.39. The dot-dashedline shows the value obtained from the central value of Rexp, andthe vertical dotted lines indicate the upper and lower values of Rexpallowed within the statistical errors.

    is of the form of 1/r , the spin-spin interaction behaves likethe function near the origin. In that case, the matrix elementof the spin-spin interaction will vanish in P states becausethe P-state wave functions are strongly suppressed wherethe interaction acts. The real situation, however, is not sosimple. The singular spin-dependent interactions are smearedout because of the relativistic corrections [16,17] and theasymptotic freedom. The resultant spin-spin interaction willhave a well-behaved form at the origin. Consequently, thematrix element of the spin-spin interaction can become sizable.Its magnitude depends on the spatial form of the interactionwhich in turn depends on how one incorporates the relativis-tic corrections and the asymptotic freedom. Equation (32)is a constraint that the spin-spin interaction has to satisfy.

    Earlier we had experimental information on the effect ofthe spin-spin interaction on P states of heavy quark systemsonly from the charmonia. In first-order perturbation theory, wecan estimate the matrix element by calculating the differencebetween a weighted average of the masses of 3P states andthe mass of 1P state. [See Eq. (6).] For the cc system, if wecan regard hc(1P ) as the 1P0 state [27], we obtain 0.85 MeVfor this quantity. If one assumes that the spin-spin interactionis inversely proportional to the product of the quark massesand that the wave functions of the cc system and those ofthe charmed-strange mesons are the same, one obtains about3 MeV for the charmed-strange mesons. The value thatemerged from our analysis is much larger in magnitude thanthis value.

    Let us remark on the works of Godfrey and Isgur [16] andof Godfrey and Kokoski [17] in comparison with the presentwork. They used basically the same Hamiltonian that we usedand diagonalized it on the basis of the harmonic oscillatoreigenstates. They assumed explicit forms for the confinementpotential and the color Coulomb interaction in terms of whichthe spatial behavior of all spin-dependent interactions can beexpressed. They accomplished the relativistic corrections by

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  • P-WAVE CHARMED-STRANGE MESONS PHYSICAL REVIEW C 72, 065202 (2005)

    TABLE II. Matrix elements, mixing angle , and masses of the P-state charmed-strange mesons in the columns with the correspondingmeson symbols. They are given in MeV except for . In the first row, the central values of the masses reported by the Particle Data Group [27]are listed and we used them to obtain the matrix elements. Mixing angle was given by Eq. (30) with the statistical errors suppressed. Valuesof the masses and mixing angles in the second and fourth rows are predictions by the indicated authors. Matrix elements in each row werecalculated by substituting these quantities into Eqs. (18)(22). Numbers in parentheses in the third row are the matrix elements obtained inRef. [17].

    M0 vs vLS vT DsJ DsJ Ds1 Ds2

    This work 2513.6 21.0 61.4 19.7 26.9 45.4 2317.4 2459.3 2535.35 2572.4Godfrey and Kokoski [17] 2563 13 27 8 3 38 2480 2550 2560 2590

    (2564) (15) (27) (7) (3)Lucha and Schoberl [21] 2531 14 29 8 4 44.7 2446 2515 2517 2561

    introducing a smearing function which softens the singularbehavior of the spin-dependent interactions at the origin.As a consequence, a sizable contribution from the spin-spininteraction to the matrix element for the P state emerged. Theyfixed the parameters by fitting observed meson masses knownthen and predicted unobserved meson masses. Although theyworked beyond the perturbation theory, we thought it wouldbe interesting to estimate the matrix elements of the spin-dependent interactions perturbatively through Eqs. (18)(22)from the masses and the mixing angles that they obtained forcharmed-strange mesons.

    The results are summarized in Table II and compared withpreceding works by Godfrey and Kokoski [17] and Luchaand Schoberl [21]. The last four numbers under the mesonsymbols in the first row are the observed masses that we usedto evaluate the matrix elements in our analysis. The last fournumbers in the other rows are the predicted masses. The valuesin parentheses in the third row are the matrix elements obtainednonperturbatively in Ref. [17]. Note that these values are veryclose to the corresponding ones of the second row, showingthat our perturbative treatment is adequate.

    The masses of DsJ predicted in Refs. [17] and [21] aremuch larger than the experimental value. The matrix elementsof the SLS and tensor interactions come into the masses ofDsJ with a negative sign as seen in Eq. (13). In Refs. [17] and[21] the magnitudes of these matrix elements are very smallcompared with the ones that the experiments require. This iswhy they had approximately 110120 MeV larger masses forDsJ compared with the experimental value even when onecorrects the overestimate of the center of gravity M0 for theP-state masses. On the other hand, the matrix elements ofthe SLS and tensor interactions come with opposite signsfor the mass of Ds2. This moderates the overestimate of themass of Ds2. The mass differences between Ds1 and DsJ inRefs. [17] and [21] are very small as compared with 76 MeVof the experimental value. This is simply due to the featurethat the values of of Refs. [17] and [21] are much smallerthan the one implied by the experiments.

    III. VALIDITY OF THE APPROXIMATIONS USED

    Let us now discuss the approximations used inSec. II. First, we regarded all spin-dependent interactions as

    perturbation and obtained their matrix elements as given inEqs. (32)(35). A typical mass difference M0 between twoconsecutive principal states that emerges from H0 is probably400500 MeV. The values of vS, vLS, vT , and are muchsmaller than M0. This justifies our perturbative treatment ofthe spin-dependent interactions.

    Secondly, we ignored the tensor coupling of the 3P2state to the 3F2 state. The nonvanishing matrix element ofthe tensor interaction between these states gives rise to anadditive correction to H44 in Eq. (6) through the second-orderperturbation. Let us estimate the second-order correction. Notethat a quark in a state with L 1 feels the color Coulombinteraction much less than a quark in an S state. This isbecause the wave function of the former is much suppressednear the origin as compared with the wave function of thelatter. Therefore, the P and F state wave functions are notvery different from those emerging from the confinementpotential alone. Let us ignore the color Coulomb interactionin obtaining the P and F state wave functions and use theharmonic oscillator potential for Vconf . Then the radial parts ofnodeless P and F state wave functions are given by

    R1(r) =

    8

    3

    [()5

    ]1/4r er

    2/2, (36)

    R3(r) =

    32

    105

    [()9

    ]1/4r3 er

    2/2, (37)

    where is an oscillator constant and = (1/ms + 1/mc)1the reduced mass. Remember that is related to the massdifference between two consecutive principal states and,hence, M0. Since the tensor interaction can be expressedas

    VT (r) = VC(r) rV C (r)

    12msmcr(38)

    in terms of the color Coulomb interaction, the needed diagonaland off-diagonal matrix elements are given by

    3P2S12VT (r)3P2 = 845

    ()3

    msmc, (39)

    3P2

    S12VT (r)3F2 = 1615

    6

    35

    ()3

    msmc, (40)

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  • YAMADA, SUZUKI, KAZUYAMA, AND KIMURA PHYSICAL REVIEW C 72, 065202 (2005)

    where we used

    VC(r) = 43

    r(41)

    with the strong coupling constant . Thus we obtain

    3P2

    S12VT (r)3F23P2

    S12VT (r)3P2 = 6

    6

    35 2.5. (42)

    If we equate the denominator to the matrix element ofthe tensor operator times the quantity given in Eq. (34),that is,

    3P2S12VT (r)3P2 8 MeV, (43)

    an approximate magnitude of the off-diagonal element be-comes 3P2S12VT (r)3F2 20 MeV. (44)

    Since the energy difference between the 3P2 and 3F2 states isapproximately 2 1 GeV, the second-order correction willbe about 0.4 MeV, that is, about 5% of the diagonal elementfor the 3P2 state. Thus we conclude that the tensor coupling tothe 3F2 state will not appreciably change our result.

    IV. DISCUSSIONS AND SUMMARY

    We examined the P-state charmed-strange mesons, focus-ing on the mixing of 1P1 and 3P1 states in Ds1(2536) andDsJ (2460) that is caused by the antisymmetric spin-orbit(ASLS) interaction. We treated the spin-dependent interactionsthat arise from the nonrelativistic expansion of the Fermi-Breitinteraction perturbatively. We did not assume any explicitforms for the interactions as functions of the interquark dis-tance. We expressed the matrix elements of these interactionsin terms of the observed masses of the P-state quartet andthe mixing angle determined from the EM decay rates ofDsJ (2460).

    The EM decay rates have large statistical errors. If wevary the decay rates within the errors, the mixing angle varieswidely. The matrix elements of the spin-dependent interactionsalso vary accordingly. The matrix elements of the SLS, tensor,and ASLS interactions are relatively stable with the variationof the mixing angle, varying only within 20%. On the otherhand, the matrix element of the spin-spin interaction variesfrom 48.5 to 7.9 MeV when the mixing angle varies from oneend to the other as determined from the EM decay rates withthe statistical errors. Note that Godfrey and Kokoski obtainedfor the matrix element of the spin-spin interaction 15 MeV,which lies in this interval [17].

    The matrix element of the spin-spin interaction is particu-larly sensitive to the mixing angle and of crucial importance indetermining the dominant states of Ds1(2536) and DsJ (2460).With the large variation in the mixing angle, the dominantstate of Ds1(2536) is transferred from the 3P1 to the 1P1 stateand that of DsJ (2460) from the 1P1 to the 3P1 state. Thisimplies that the spin-spin interaction is the most importantamong the spin-dependent interactions for the determinationof the dominant states in Ds1(2536) and DsJ (2460). It will becrucial to their assignments, provided that other mechanismsfor the mixing such as the coupling to decay channels are lesssignificant than what we have discussed [3537].

    Our analysis is based on the branching fractions thatwere obtained from the first observation of B DDsJdecays by the Belle Collaboration. The analyses are ac-companied by large statistical errors, and so are the mixingangles that are extracted from the branching fractions. Forfurther discussion of the relationship between the mixingangle and the spin-dependent interactions, we need morerefined data on the branching fractions from the experimentalgroups.

    ACKNOWLEDGMENTS

    We are grateful to Yuki Nogami for helpful comments onthe manuscript. A.S. would like to thank Professor T. Udagawafor the warm hospitality extended to him at the University ofTexas at Austin where the last part of this work was done.

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