Heavy-quark symmetry and the electromagnetic decays of excited charmed strange mesons

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  • PHYSICAL REVIEW D, VOLUME 70, 074014Heavy-quark symmetry and the electromagnetic decays of excited charmed strange mesons

    Thomas Mehen1,2,* and Roxanne P. Springer1,1Department of Physics, Duke University, Durham, North Carolina 27708, USA

    2Jefferson Laboratory, 12000 Jefferson Ave., Newport News, Virginia 23606, USA(Received 23 July 2004; published 14 October 2004)*ElectronicElectronic

    1550-7998=20Heavy-hadron chiral perturbation theory (HHPT) is applied to the decays of the even-paritycharmed strange mesons, Ds02317 and Ds12460. Heavy-quark spin-symmetry predicts the branch-ing fractions for the three electromagnetic decays of these states to the ground statesDs andDs in termsof a single parameter. The resulting predictions for two of the branching fractions are significantlyhigher than current upper limits from the CLEO experiment. Leading corrections to the branchingratios from chiral loop diagrams and spin-symmetry violating operators in the HHPT Lagrangian cannaturally account for this discrepancy. Finally the proposal that the Ds02317 (Ds12460) is a hadronicbound state of a DD meson and a kaon is considered. Leading order predictions for electromagneticbranching ratios in this molecular scenario are in very poor agreement with existing data.

    DOI: 10.1103/PhysRevD.70.074014 PACS numbers: 12.39.Hg, 12.39.FeI. INTRODUCTION

    The discovery of the Ds02317 [1] and Ds12460 [2]has revived interest in excited charmed mesons. Thedominant decay modes of these states are Ds02317 !Ds0 and Ds12460 ! Ds0, with widths less than7 MeV [2]. There is experimental evidence indicatingthat Ds02317 and Ds12460 are JP 0 and 1 states,respectively [3,4]. Had the masses of the 0 and 1 statesbeen above the threshold for the S-wave decay into Dmesons and kaons, as anticipated in quark model [5,6] aswell as lattice calculations [79], they would have hadwidths of a few hundred MeV. In reality, the unexpectedlylow masses make those decays kinematically impossible.The only available strong decay modes violate isospin,accounting for the narrow widths.

    The Ds02317 and Ds12460 can also decay electro-magnetically. The possible decays are Ds12460 ! Ds,Ds12460 ! Ds, and Ds02317 ! Ds. The decayDs02317 ! Ds is forbidden by angular momentumconservation. In the heavy-quark limit, both the threeelectromagnetic decays and the two strong decays arerelated by heavy-quark spin symmetry [10]. Belle hasobserved the decay Ds12460 ! Ds from Ds12460produced in the decays of B mesons [3] and from con-tinuum ee production [4]. The ratio of the electromag-netic branching fraction to the isospin violating one-piondecay reported by the experiment is

    BrDs12460 ! DsBrDs12460 ! Ds0

    0:38 0:11 0:04 30:55 0:13 0:08 4 : (1)

    In each case the first error is statistical and the secondsystematic. The other electromagnetic decays have notbeen observed. CLEO quotes the following bounds onthe branching fraction ratios [2]:address: mehen@phy.duke.eduaddress: rps@phy.duke.edu

    04=70(7)=074014(12)$22.50 70 0740BrDs12460 ! DsBrDs12460 ! Ds0

    < 0:16

    BrDs02317 ! DsBrDs02317 ! Ds0

    < 0:059:(2)

    (The BELLE collaboration quotes weaker lower boundsof 0.31 and 0.18, respectively, for these ratios [4].)

    In this paper the decays of the Ds02317 andDs12460 are analyzed using heavy-hadron chiral per-turbation theory (HHPT) [11]. HHPT is an effectivetheory applicable to the low energy strong and electro-magnetic interactions of particles containing a heavyquark. It incorporates the approximate heavy-quark andchiral symmetries of QCD. Corrections to leading orderpredictions can be computed in an expansion inQCD=mQ, M=, and p=, where mQ is the heavy-quark mass,M is a Goldstone boson mass, p is the typicalmomentum in the decay, and is the chiral symmetrybreaking scale.

    In Sec. II, the leading order HHPT predictions for thebranching ratios are derived:

    BrDs12460 ! DsBrDs12460 ! Ds0

    0:37 0:07

    BrDs02317 ! DsBrDs02317 ! Ds0

    0:13 0:03:(3)

    (Leading order calculations of strong and electromagneticdecays were first done in Refs. [1214].) These predic-tions deviate significantly from the CLEO limits. At next-to-leading order (NLO) there are O1=mQ suppressedheavy-quark spin-symmetry violating operators as wellas one-loop chiral corrections to the electromagneticdecays. Once these effects are included, predictions forthe ratios in Eq. (2) can be made consistent with thepresent experimental bounds with coupling constants inthe Lagrangian of natural size.14-1 2004 The American Physical Society

  • THOMAS MEHEN, ROXANNE P. SPRINGER PHYSICAL REVIEW D 70 074014The splitting between the even- and odd-parity dou-blets should be approximately the same for both bottomstrange and charmed strange mesons. Therefore it is likelythat the Bs even-parity states will be below threshold fordecay into kaons and narrow like their charm counter-parts. The calculations of this paper can also be applied tothe electromagnetic and strong decays of even-parity Bsmesons when these states are discovered.

    The leading order HHPT Lagrangian used in thispaper is invariant under nonlinearly realized chiralSU3L SU3R and no further assumptions are madeabout the mechanism of chiral symmetry breaking.Models that treat the Ds02317 and Ds12460 as the 0and 1 chiral partners of the ground state charm strangemesons are proposed in Refs. [13,1517]. In these models,referred to as parity-doubling models, the even-parityand odd-parity mesons are placed in a linear representa-tion of chiral SU3L SU3R. These fields couple in achirally invariant manner to a field that transforms inthe 3; 3 of SU3L SU3R. The field develops avacuum expectation value that breaks the chiral symme-try. The resulting nonlinear sigma model of Goldstonebosons coupled to heavy mesons has the same operatorsas the HHPT Lagrangian used in this paper. The as-sumed mechanism of chiral symmetry breaking inparity-doubling models predicts relationships betweencoupling constants in the HHPT Lagrangian. For ex-ample, the parity-doubling models predict that the hy-perfine splittings of the even- and odd-parity doublets areequal. This is in agreement with experimental observa-tions. Other relationships between coupling constants inHHPT are predicted [18,19] by the theory of algebraicrealizations of chiral symmetry [20], in which hadronsare placed in reducible representations of SU3L SU3R. QCD sum rules have also been used to calculatesome of the HHPT couplings [21]. When more data onthe electromagnetic decays of even-parity Ds and Bsmesons becomes available, the formulae derived in thispaper can be used to extract the relevant couplings andtest these theories.

    The low mass of the Ds02317 and Ds12460 hasprompted reexamination of quark models [2226] aswell as speculation that these states are exotic.Possibilities include DK molecules [2729], Ds mole-cules [30], and tetraquarks [28,3135]. Masses have beencalculated in lattice QCD [3638], heavy-quark effectivetheory (HQET) sum rules [39,40], and potential as wellas other models [22,2426,41,42]. The results of some ofthese papers are contradictory. For example, the latticecalculation of Ref. [36] yields a 0 0 mass splittingabout 120 MeV greater than experimentally observed,quotes errors of about 50 MeV, and argues this is evidencefor an exotic interpretation of the state. On the other hand,the lattice calculation of Ref. [37] obtains similar numeri-cal results but concludes that uncertainties in the calcu-074014lation are large enough to be consistent with aconventional cs P-wave state. Some quark model analy-ses [25,26,41] conclude that interpreting the states asconventional cs P-wave mesons naturally fits the ob-served data; others reach the opposite conclusion [22,42].

    There have been some attempts to determine the natureof theDs02317 andDs12460 from the observed patternof decays [23] as well as their production in b-hadrondecays [43 46]. Ref. [23] argues that the total width andelectromagnetic branching ratios can distinguish betweencs P-wave states and DK molecules, and gives predic-tions for these branching ratios calculated in the quarkmodel. Refs. [43,44] argue that the observed branchingfractions for B! Ds02317D and B! Ds12460Dare smaller then expected for cs P-wave states, suggest-ing that these states are exotic. These analyses assume anunproven (but plausible) factorization conjecture for thedecays as well as quark model estimates for theDs02317and Ds12460 decay constants, and have recently beenextended to b [45] and semileptonic Bs decays [46].

    Section III addresses the question of whether a modelindependent analysis of the decays can provide insightinto the nature of the Ds02317 and Ds12460. InHHPT the fields describing the 0 and 1 mesons areadded to the Lagrangian by hand. The only assumptionmade about these states is that the light degrees of free-dom in the hadron are in the 3 of SU3 and have jp 12.(In this paper, JP refers to the angular momentum andparity of a heavy meson, and jp to the angular momen-tum and parity of the light degrees of freedom.) Lightdegrees of freedom with these quantum numbers could bean s quark in an orbital P-wave or sq q quarks all in anS-wave. Therefore, a conventional quark model csP-wave state and an unconventional cs q q tetraquarkstate will be represented by fields having the same trans-formation properties in the HHPT Lagrangian. HHPTpredictions for the ratios in Eqs. (1)-(2) are valid foreither interpretation, and so cannot distinguish betweenthese two scenarios.

    However, if Ds02317 (Ds12460) is modeled as abound state of a D (D) meson and a kaon the predictionsfor the electromagnetic branching ratios will be different.In this scenario, instead of adding the even-parity heavy-quark doublet to the Lagrangian by hand, the dynamics ofthe theory containing only the ground state heavy-quarkdoublet and Goldstone bosons generate the observedDs02317 and Ds12460. This interpretation has beenpursued in Refs. [4751]. In this scenario the bindingenergy is only about 40 MeV, so the mesons in the had-ronic bound state are nonrelativistic. The decay rates canbe calculated by convolving the unknown nonrelativisticwavefunction with leading order HHPT amplitudes forDK ! Ds and DK ! Ds 0. Dependence on thebound state wavefunction drops out of the ratios inEqs. (1) and (2). The resulting predictions for these-2

  • HEAVY-QUARK SYMMETRY AND THE. . . PHYSICAL REVIEW D 70 074014branching ratios are much larger than experiment.Furthermore, the branching ratio for Ds12460 ! Dsis predicted to be the smallest of the three, in directconflict with experimental observations. ADK molecularinterpretation of the Ds02317 and Ds12460 is disfa-vored by the existing data on electromagnetic branchingfractions.

    II. ELECTROMAGNETIC AND STRONG DECAYSIN HHPT

    In the heavy-quark limit, hadrons containing a singleheavy quark fall into doublets of the SU2 heavy-quarkspin-symmetry group. Heavy hadrons can be classified bythe total angular momentum and parity quantum num-bers of their light degrees of freedom, jp. The groundstate doublet has jp 12 and therefore the mesons in thedoublet are 0 and 1 states. In HHPT, these states are074014combined into a single field [11]

    Ha 1 v62

    Ha Ha5; (4)

    where a is an SU3 index. In the charm sector, Haconsists of the D0; D; Ds c u; c d; cs pseudoscalarmesons and Ha are the D0; D; Ds vector mesons.The doublet with light degrees of freedom jp 12 con-sists of mesons whose quantum numbers are 0 and 1.These are combined into the field [52]

    Sa 1 v62

    Sa 5 Sa; (5)

    where the scalar states in the charm sector are Sa D0aand the axial vectors are Sa D1a.

    The relevant strong interaction terms in the HHPTchiral Lagrangian are [11,53]L f2

    8Tr@@y

    f2B04

    Trmq ymq TrHaiv DbaHb TrSaiv Dba %SH%abSb

    gTrHaHbA6 ba5 g0TrSaSbA6 ba5 hTrHaSbA6 ba5 h:c: H8

    TrHa)*Ha)*

    S8

    TrSa)*Sa)*: (6)The first two terms in Eq. (6) are the leading order chiralLagrangian for the octet of Goldstone bosons. f is theoctet meson decay constant. The conventions for defining in terms of meson fields, the chiral covariant deriva-tive, Dab, and the axial vector field, A

    ab, are identical to

    those of Ref. [54]. Heremq is the light quark mass matrix.The third and fourth terms in Eq. (6) contain the kineticterms for the fields Ha and Sa and the couplings to twoand more pions determined by chiral symmetry. Theparameter %SH is the residual mass of the Sa field. TheHa residual mass can be set to zero by an appropriatedefinition of the Ha field, and this convention is adoptedhere. Then %SH is the difference between the spin-averaged masses of the even- and odd-parity doublets inthe heavy quark limit. The second line contains thecouplings of Ha and Sa to the axial vector field A

    ab

    @ab=f :::. These terms are responsible for transi-tions involving a single pion. The couplings g, g0, and hare parameters that are not determined by the HHPTsymmetries. The last two terms in Eq. (6) are operatorsthat give rise to 1 0 and 1 0 hyperfine split-tings, which are H and S, respectively. Since the split-tings should vanish in the heavy-quark limit,S H 2QCD=mQ. The parameters S and H areindependent in HHPT so there is no relation between thehyperfine splitting in the even- and odd-parity doublets.In parity-doubling models, H S at tree level, inagreement with the observation that hyperfine splittingsare equal to within 2 MeV.

    Electromagnetic effects are incorporated by gaugingthe U1em subgroup of SU3L SU3R and addingterms to the Lagrangian involving the gauge invariantfield strength, F*. Gauging derivatives in Eq. (6) doesnot yield terms which can mediate the 0; 1 !0; 1 electromagnetic decays at tree level. The leadingcontribution to these decays comes from the operator

    L e~,4

    TrHaSb)*F*Q-ba; (7)

    where Q-ba 12 -Q-y -yQ-ba, -2 , and Q diag2=3;1=3;1=3 is the light quark electric chargematrix. A tree level calculation of the decay rates usingEq. (7) shows that

    !1a ! 1a 2

    3.e2q ~,

    2m1am1a

    jkj3

    !1a ! 0a 1

    3.e2q ~,

    2m0am1a

    jkj3

    !0a ! 1a .e2q ~,2m1am0a

    jkj3;

    (8)

    where eq is the electric charge of the light valence quark,. is the fine-structure constant, and ~, is the unknownparameter in Eq. (7). The three-momentum of the photon-3

  • TABLE I. Masses and widths of even-parity nonstrangecharmed mesons, DQJ , where Q is the electric charge.

    Experiment Particle(JP) Mass (MeV) Width (MeV)

    CLEO [55] D011 24615348 29011091Belle [56] D000 2308 36 276 66

    D011 2427 36 384130105FOCUS [57] D000 2407 41 240 81

    D0 0 2403 38 283 42

    THOMAS MEHEN, ROXANNE P. SPRINGER PHYSICAL REVIEW D 70 074014in the decay is k andmJPa is the mass of the heavy-mesonwith quantum numbers JPa . In the heavy-quark limit, themembers of each doublet are degenerate and the phasespace is the same for all three decays. If differences inphase space are neglected the decay rate ratios are!1a ! 1a :!1a ! 0a :!0a ! 1a 2:1:3.Differences in the phase space factors are formallyO1=mQ but in practice it is critical to include theseeffects to make sensible predictions. For the charmedstrange mesons using the physical masses gives

    !Ds12460 ! Ds ~, GeV215:6 keV!Ds12460 ! Ds ~, GeV218:7 keV!Ds02317 ! Ds ~, GeV25:6 keV:

    (9)

    The rates are then in the following ratios:

    !Ds12460 ! Ds:!Ds12460 ! Ds:!Ds02317 ! Ds 0:83:1:0:0:30: (10)

    Note that the rate for Ds12460 ! Ds, smallest in theexact heavy-quark limit, is actually the largest whenphase space effects are included since jkj is largest forthis decay.

    To compare with the ratio measured by Belle, theisospin violating strong decays must be calculated.These decays proceed through / 0 mixing. The resultis [14]

    !Ds12460 ! Ds0

    h202

    3f2mDs

    mDs12460E20jp0 j

    h2 17:0 keV if f f 130 MeV9:8 keV if f f/ 171 MeV

    !Ds02317 ! Ds0

    h202

    3f2mDs

    mDs02317E20jp0 j

    h2 16:9 keV if f f 130 MeV9:8 keV if f f/ 171 MeV

    ;

    (11)

    where 0 3p =2md mu=2ms md mu 0:01 isthe / 0 mixing angle. E0 and p0 are the energyand three momentum of the 0, respectively. At tree levelf f f/. The difference between the two predic-tions provides an estimate of the uncertainty due tohigher order SU3 violating effects.

    The branching fraction ratios in Eqs. (1) and (2) dependonly on the ratio ~,2=h2 at leading order in HHPT. Toobtain h2 separately, a measurement of an excited strongdecay width is needed. Currently h2 cannot be extracted074014from the strange sector because only loose experimentalbounds on !Ds02317 and !Ds12460 exist. Untilmeasurements of these widths are dramatically im-proved, h2 can be estimated using data on nonstrangeeven-parity D meson widths. CLEO [55] has observedpreliminary evidence for the D01 JP 1 meson. (Herethe superscript refers to the particle charge.) More re-cently, Belle [56] has reported observing even-parity D00JP 0 and D01 states. Finally, FOCUS [57] has ob-served broad structures in excess of background in theD and D0 invariant mass spectra. FOCUS doesnot claim to observe an excited charm resonance but doesfit the excess with a Breit-Wigner to determine the reso-nance properties required to explain their data. Themasses and widths reported by all three experimentsare collected in Table I. The experiments all quote severalerrors which have been combined in quadrature for sim-plicity. Note that the CLEO and Belle measurements ofthe D01 are consistent with each other while the centralvalue of the D00 mass obtained by FOCUS is 99 MeVhigher than the central value of the Belle measurement.Furthermore, the FOCUS D00 mass is actually greaterthan the mass of the Ds02317. If the effect observedby FOCUS is a scalar D resonance, it seems implausiblethat this resonance is related to the Ds02317 by SU3symmetry. Therefore, the FOCUS data will not be used toestimate h2. Even the masses obtained by CLEO andBelle are large compared to expectations based onSU3 symmetry. Combining the strange sector 0 0 and 1 1 mass splittings with SU3 symmetryleads to the prediction that the D00 mass is 2212 MeV andthe D01 mass is 2355 MeV [13].

    Applying the leading order expression for the decaywidths of the nonstrange 0 and 1 mesons

    !D01 h2

    2f2

    mD

    mD01E2jpj

    1

    2

    mD0

    mD01E20jp0 j

    !D00 h2

    2f2

    mD

    mD00E2jpj

    1

    2

    mD0

    mD00E20jp0 j

    ; (12)

    to the CLEO and Belle data yields h2 0:39 0:13 fromthe 0 decays and h2 0:49 0:14 from the 1 decay.The error in each case is obtained by adding in quadraturethe uncertainty in the decay rate from varying the mass-4

  • HEAVY-QUARK SYMMETRY AND THE. . . PHYSICAL REVIEW D 70 074014within the allowed range and the experimental error inthe decay rate. If the two results are averaged h2 0:44 0:11. This estimate of h2 is consistent with thebound h2 0:86 extracted from an Adler-Weisbergertype sum rule for B scattering [58]. (To obtain thisbound g 0:27 [54] is used in the result of Ref. [58].) Itis also consistent with a calculation of h 0:52 0:17obtained using QCD sum rules in Ref. [21].

    The lowest order HHPT prediction for the ratio mea-sured by the Belle collaboration is

    BrDs12460 ! DsBrDs12460 ! Ds0

    ~, GeV

    h

    2

    1:1 if f f 130 MeV1:9 if f f/ 171 MeV

    : (13)

    Averaging the results of the two Belle measurements, ~, GeV=h2 0:40 0:080:23 0:05 or j ~,j 0:42

    0:070:32 0:05 GeV1, where f ff f/. Theerror is estimated by first combining the statistical andsystematic errors in quadrature for each measurement,then combining the two measurements assuming theyare independent. The extracted values for ~, and h areconsistent with expectations based on naturalness.Plugging the value of ~, GeV=h2 into expressions forthe unobserved electromagnetic decays yields the follow-ing predictions for the branching fraction ratios

    BrDs12460 ! DsBrDs12460 ! Ds0

    0:37 0:07

    BrDs02317 ! DsBrDs02317 ! Ds0

    0:13 0:03:(14)

    Both predictions are in excess of bounds quoted by theCLEO experiment. Heavy-quark spin symmetry predictsbranching ratios for the electromagnetic decaysDs12460 ! Ds and Ds02317 ! Ds that are morethan a factor of 2 in excess of the experimental upperlimits.

    In the rest of this section the leading corrections to bothelectromagnetic and strong decays are analyzed. BecauseQCD=mc 1=3, corrections to heavy-quark spin-symmetry predictions can be rather large for charm had-rons. These corrections can be systematically analyzedusing HHPT. For example, the pattern of deviationsfrom heavy-quark spin-symmetry predictions for theone-pion decays of excited D-wave charm mesons [59]can be understood by analyzing the structure of spin-symmetry violating operators appearing at O1=mc inthe HHPT Lagrangian [60]. A ratio of decay widths forwhich the O1=mc correction vanishes agrees well withdata. There is another ratio for which the leading orderheavy-quark spin-symmetry prediction fails rather badly.In this case the leading O1=mc correction is multipliedby a large numerical coefficient. Thus HHPT is a useful074014tool for determining the robustness of predictions basedon heavy-quark spin symmetry.

    Spin-symmetry violating operators that contribute toS! H transitions must have the Dirac structureTrH)*S5 or TrH)*S.. Operators with HS con-serve spin symmetry, while operators withH)*S violatespin symmetry. Operators with HS and H5S areredundant since HS H 12 f; v6 gS vHS andH)*S H 12 f)*; v6 gS 2*.,v.H,5S, whileH5S 0. Spin-symmetry violating operators will beof the form TrH)*S!. ! must be . or 5 since thetrace vanishes for ! 1, while ! .5 and )., areredundant because

    TrH)*S.5 v.TrH)*S5TrH)*S)., iTrH)*Sv., v,.:

    Reparametrization invariance [61] forbids operators withderivatives acting on the H or S fields [53]. For S! Hdecays, the lowest dimension, parity conserving, spin-symmetry violating operators are

    L ieeQ~,0

    8mQTrHa)*Sa5F.,2*.,

    eeQ~,00

    8mQTrHa)*Sa.i@.F* h:c:: (15)

    The 1=mQ dependence (expected for any operator whichviolates heavy-quark spin symmetry) is explicit. Thefactors of i are required by time reversal invariance.The first operator in Eq. (15) and the leading operatorin Eq. (7) have mass dimension five. The second operatorin Eq. (15) has mass dimension six, so ~,00 has massdimension 1 and is expected to scale like 1= GeV1. Since 2)*@F.* )*@.F* for an abelianfield strength there is a unique way of contracting indicesin this operator. Note that there is a unique dimension six,spin-symmetry conserving operator proportional toTrHaSb)*Q-baiv @F*. This operator gives slight de-viations from the ratios in Eq. (10) since its contribution issuppressed by jkj= and jkj differs for the threedecays due to hyperfine splittings. These correctionsshould be smaller than corrections coming from operatorsin Eq. (15) so they are neglected in what follows.

    Power counting is used to determine the importance ofhigher dimension operators in the Lagrangian. HHPT isa double expansion in QCD=mQ and Q=, where Qpm mK. Two additional mass scales appearing inthe Lagrangian of Eq. (6) are the mass splitting betweenthe H and S doublet fields, %SH, and the hyperfine split-tings within each doublet. In the heavy-quark limit, the Sfield propagator is proportional to

    i2v k %SH : (16)-5

  • THOMAS MEHEN, ROXANNE P. SPRINGER PHYSICAL REVIEW D 70 074014If %SH were to scale as Q0, then the S propagator could beexpanded in powers of v k=%SH since v kQ. In thestrange sector loops receive important contributions frommomenta mK 495 MeV. Numerically, %SH 350 MeV in the strange quark sector and 430 MeVin the nonstrange sector, so expanding in v k=%SH is apoor approximation. Therefore, %SH Q is required. Thehyperfine splittings are also treated as Q since numeri-cally these splittings are 140 MeV which is m.

    There are SU3 violating corrections to the decayrates from operators such as TrHaSb)*F*Q-bcm-ca.These operators will give the same correction to all threeelectromagnetic decays in Eqs. (1) and (2), so their effectcan be absorbed into the definition of ~,. However, if onewere interested in relating the electromagnetic decays ofstrange and nonstrange heavy mesons these operatorsmust be included explicitly.

    The leading operator in Eq. (7) is order Q because ofthe derivative in F*. The first operator in Eq. (15) istreated as Q2 because it is suppressed by QCD=mQrelative to the leading operator. The second operator inEq. (15) has two derivatives and is also 1=mQ suppressed.It is treated as Q2. The correctness of this power count-FIG. 1. One-loop chiral corrections to the electromagneticdecays S! H in v A 0 gauge. Double lines are S mesons,solid lines are H mesons, dashed lines are Goldstone bosonsand the wavy line is the photon.

    074014ing is confirmed by the calculation of one-loop correc-tions to the decay, since in order to properly renormalizethese diagrams both counterterms in Eq. (15) are needed.In loop diagrams, integrals scale asQ4, the propagators ofthe H and S fields as Q1 and the propagators ofGoldstone bosons as Q2. The leading couplings ofthe H and S fields to kaons and pions are Q and thecouplings of the photon to the kaons and pions are Q.Calculations of the loop corrections are performed in v A 0 gauge where the leading coupling of the photon tothe heavy meson fields vanishes. Finally, there is an Q0coupling of two heavy-meson fields to a Goldstone bosonand photon which comes from gauging the derivativecouplings of the heavy-meson fields to Goldstone bosons.The HS vertex obtained by gauging the derivative onthe pion field in the leading HS coupling vanishes in v A 0 gauge. With these power counting rules, the loopdiagrams shown in Fig. 1 give an OQ2 contribution tothe S! H decays. Double lines are S fields, solid linesare H fields and the dotted lines are Goldstone bosons.ForDs decays the Goldstone bosons in these loops are Kand the virtual heavy mesons are neutral Ds.

    Including both the loop diagrams and tree level inser-tions of the operators in Eq. (15) yields:!1a ! 1a 12

    eq ~,

    eQ

    ~,0

    mQ Fm1a m1b ; m1a m1b ; jkj;M;

    !1a ! 0a 12

    eq ~,

    eQ ~,

    0

    mQ eQ

    ~,00jkV2mQ

    Fm1a m1b ; m1a m0b ; jkj;M;

    !0a ! 1a 12

    eq ~,

    eQ ~,0mQ

    eQ~,00jkj2mQ

    Fm0a m0b ; m0a m1b ; jkj;M;:

    (17)Here ! !NLO=!LO, where the !LO are given in Eq. (8).The SU3 index a refers to the external heavy mesonswhile the index b refers to the mesons inside the loop. Mis the mass of the virtual Goldstone boson. For heavy-strange decays the external particles are heavy-strangemesons; a 3, the Goldstone boson is a K, and theheavy mesons inside the loops are neutral heavy mesonswith b 1. ! is expanded to OQ. The functionF1;2; jkj;M; is given in the Appendix. The loopgraphs are regulated in dimensional regularization, coun-terterms are defined in the MS scheme and the dimen-sional regularization parameter is. All dependence iscanceled by the implicit dependence of the renormal-ized couplings ~,, ~,0, and ~,00.

    An NLO calculation of the electromagnetic branchingratios also requires O1=mc corrections to the decays!Ds02317 ! Ds0 and !Ds12460 ! Ds0. Theleading spin-symmetry violating operator contributingto these decays is

    L h0

    2mQTrHa)*Sb.A,ba2*.,: (18)

    Because of the 1=mQ suppression this operator is consid-ered OQ2. The one-loop diagrams contributing to S!-6

  • HEAVY-QUARK SYMMETRY AND THE. . . PHYSICAL REVIEW D 70 074014H transitions are subleading at OQ3. The decay ratesto NLO are

    !13 ! 13 0 h h

    0

    mQ

    2 02

    3f2m13m13

    E20jp0 j

    !03 ! 03 0 h 3 h

    0

    mQ

    2 02

    3f2m03m03

    E20 jp0 j:(19)074014Earlier in this section the data from D00 and D01 decays

    was averaged to extract h2. Including the leading correc-tion it is possible to fit h and h0 separately, extracting h 0:69 0:09 and h0=mc 0:019 0:034.

    The NLO expression for the branching fraction ratiosof heavy-strange mesons is obtained by combiningEq. (19) with Eq. (17) and (8). The result isBr13 ! 13 Br13 ! 13 0

    2.f2 ~,2

    902h2jkj3E20p0

    1 2h

    0

    hmQ 6eQ

    ~,0

    ~,mQ 6

    ~,Fm1a m1b ; m1a m1b ; jkj; mK ;

    Br13 ! 03 Br13 ! 13 0

    .f2 ~,2

    902h2m03m13

    jkj3E20p0

    1 2h

    0

    hmQ 6eQ

    ~,0

    ~,mQ 3eQ

    ~,00jkj~,mQ

    6~,Fm1a m1b ; m1a m0b ; jkj; mK ;

    Br03 ! 13 Br03 ! 03 0

    .f2 ~,2

    302h2m13m03

    jkj3E20p0

    1 6h

    0

    hmQ 6eQ

    ~,0

    ~,mQ 3eQ

    ~,00jkj~,mQ

    6~,Fm0a m0b ; m0a m1b ; jkj; mK ;

    ;

    (20)

    where eq has been set to es 1=3. Applying the formulae in Eq. (20) to the experimentally observed ratios gives

    BrDs12460 ! DsBrDs12460 ! Ds0

    1:58~,2

    h2

    1 1:43h

    0

    h 2:86

    ~,0

    ~, 0:18gh

    ~, 2:94

    0:700:58g

    0h~,

    < 0:16

    BrDs12460 ! DsBrDs12460 ! Ds0

    1:90~,2

    h2

    1 1:43h

    0

    h 2:86

    ~,0

    ~, 0:63

    ~,00

    ~, 0:03gh

    ~, 2:40

    0:730:59g

    0h~,

    0:44 0:09

    BrDs02317 ! DsBrDs02317 ! Ds0

    0:57~,2

    h2

    1 4:29h

    0

    h 2:86

    ~,0

    ~, 0:28

    ~,00

    ~, 0:37gh

    ~, 3:81

    0:900:75g

    0h~,

    < 0:059:

    (21)Here ~, and ~,00 are measured in units of GeV1 and h0

    in units of GeV. All other quantities are dimension-less. The charm quark mass is mc 1:4 GeV, the re-normalization scale is 1 GeV, and eQ ec 2=3.For the loop corrections with kaons the meson de-cay constant is f fK, while for the strong decaysf f/. (If f is used in the strong decays, thenthe branching fraction ratios in Eq. (20) should be multi-plied by f2=f2/ 0:58.) The masses used for the virtualnonstrange even-parity heavy mesons in the loopsare m01 2308 36 MeV and m11 2438 29 MeV,where the first number is the nonstrange 0 massmeasured by Belle and the second is the average ofthe nonstrange 1 mass measured by CLEO and Belle.The uncertainty in the coefficient of g0h= ~, in Eq. (21)is due to the uncertainty in the masses of the D00 andD01.

    The result depends on seven parameters:g; g0; h; h0; ~,; ~,0, and ~,00. The coupling g is constrainedto be 0:27:06:03 from a next-to-leading order HHPT analy-sis of D decays [54]. h and h0 are extracted from thenonstrange decays, leaving four unknown parameters.Since there are only three constraints coming from ex-periment, further analysis requires additional assump-tions to constrain the parameter space.

    To illustrate how the current data is consistent withnatural size parameters the following situation is consid-ered. The contribution from ~,00 is neglected since inEq. (21) ~,00 is multiplied by a coefficient that is muchsmaller than the coefficients multiplying h0 and ~,0. (Thesmallness of this coefficient is due to the factor jkj=mc.)g, h, h0, and the branching fraction ratio measured byBelle are set to their central values: 0.27, 0.69, 0:019mc,and 0.44, respectively. Ranges for the remaining parame-ters ( ~,; ~,0, and g0) are extracted by varying the branchingratios in Eq. (2) between zero and their upper limits.There are two solutions since the formulae for the elec-tromagnetic decay rate is quadratic in ~,. The results are-7

  • THOMAS MEHEN, ROXANNE P. SPRINGER PHYSICAL REVIEW D 70 0740140:70 ~, GeV 0:86 0:01 ~,0 0:010:32 g0 0:40;

    0:62 ~, GeV 0:46 0:01 ~,0 0:02 0:25 g0 0:16:

    (22)

    Note that the ranges quoted in Eq. (22) do not includeerrors due to the uncertainties in the parameters g, h, andh0 or the masses of the D00 and D

    01. h

    0 is highly uncertainbecause of the uncertainty in the masses and widths of theD00 and D

    01 used to extract it. The loop contribution

    proportional to g0h= ~, is also sensitive to the masses ofthe D00 and D

    01 that appear as intermediate states. The

    ranges given in Eq. (22) do not reflect these uncertaintiesand do not exhaust the possible parameter space. Instead,they are simply illustrative of natural size parametersconsistent with existing data.

    When more data on excited heavy meson systemsbecomes available, the formulae in Eq. (20) could beused to test models that make predictions for the parame-ters in HHPT. In parity-doubling models, g g0 andh 1 at tree level [13]. The authors of Ref. [13] note thath can be renormalized away from its tree level value andallow this parameter to vary in their analysis of strongdecays. The tree level result h 1 exceeds the valueextracted from excited nonstrange decays in HHPT.Another theoretical framework which makes similarpredictions for the coupling constants g, g0, and h is thealgebraic realization of chiral symmetry [20]. Applyingthis theory to heavy mesons [18,19] leads to the predic-tions g0 g and g2 h2 1. Using g 0:27 in thisrelation gives h2 0:93 which is also larger than ex-tracted from Eq. (19). While the predictions for h arenot in agreement with available data, the condition g g0 is consistent with available data but not required.

    Eventually the even-parity Bs states will be observedand all electromagnetic branching fractions for heavy-strange mesons will be measured. Then the parameterspace will be overconstrained and HHPT for excitedheavy mesons can be tested decisively. Furthermore, theextracted values for g, g0 and h can be compared withpredictions from parity-doubling models and algebraicrealizations of chiral symmetry. At the present time,observed violations of leading heavy-quark spin-symmetry predictions are consistent with what is ex-pected from loop effects and higher order operators ap-pearing in the HHPT Lagrangian.III. ELECTROMAGNETIC DECAYS AND D KMOLECULES

    The unexpectedly low masses of the Ds02317 andDs12460 have prompted speculation that these statesare unconventional. Two common proposals are that thesemesons are csq q tetraquarks or hadronic bound states ofD and K mesons. This section addresses the question of074014what the decays reveal about the internal structure of theDs02317 and Ds12460. In the analysis of the previoussection the only information about the states needed toconstruct the HHPT Lagrangian is the assumed SU3and jp quantum numbers of the light degrees of freedomin the hadrons. A constituent quark in a P-wave or anexotic with two light quarks and an antiquark both havejp 12. Both states are represented by a field like that inEq. (5). Analysis of electromagnetic and strong decayswithin HHPT is identical for both states, though thecoupling constants ~,; ~,0; h, etc., would be different forthe two states. Since these coupling constants are un-known in either case, the HHPT predictions for electro-magnetic and strong decays cannot distinguish betweenexotic csq q and conventional cs P-wave states. Of course,if theDs02317 andDs12460 are csq q states then in thequark model there should be distinct cs P-wave mesonswith the same quantum numbers. These states could bevery hard to detect, however, if they are above the DKthreshold. Mixing between the conventional and exoticmesons is also likely [28,62].

    However, if the Ds02317 Ds12460 is a bound stateof D and K mesons then the HHPT predictions forelectromagnetic and strong decays will be different. For ahadronic bound state of aD orD and a kaon, one could inprinciple calculate the bound state masses and other prop-erties from the HHPT Lagrangian with the field Haalone. There have been attempts to generate theDs02317 and Ds12460 as resonances in a unitarizedmeson model [47,48] as well as by solving Bethe-Salpeterequations in relativistic, unitarized chiral perturbationtheory [4951]. Producing a bound state requires resum-ming an infinite number of Feynman graphs in HHPTand the renormalization of these graphs requires intro-ducing higher order operators whose renormalized coef-ficients are unknown. Such a calculation will not beattempted in this paper. Instead theDK molecular picturewill be tested by simply assuming that strong forcesbetween D and K mesons give rise to the Ds02317and Ds12460 and determining what this implies for thedecay rates. If theDs02317 (Ds12460) are bound statesof DK then the characteristic momentum of the con-stituents is p 2Bp 190 MeV, where is the re-duced mass andB is the binding energy. TheDKmoleculecan then be modeled as a nonrelativistic bound state sincerelativistic corrections are suppressed by v2 p2=M2K 0:15. Strong and electromagnetic decays can be calcu-lated in terms of the unknown bound state wavefunction.Even without any knowledge of these wavefunctions it ispossible to make predictions for the decay ratios inEqs. (1) and (2). It turns out that these predictions dis-agree with data so interpreting Ds02317 and Ds12460as DK molecules is disfavored.

    Ref. [23] advocates using the radiative decays of theDs02317 andDs12460 to determine the nature of these-8

  • FIG. 2. Leading order diagrams forDK bound states decay-ing into Ds . The shaded oval represents the DK boundstate wavefunction.

    HEAVY-QUARK SYMMETRY AND THE. . . PHYSICAL REVIEW D 70 074014states and calculates radiative and strong decays within anonrelativistic quark model. Predictions for the branch-ing fraction ratios in Eqs. (1) and (2) are in the sameproportion as leading order heavy-quark symmetry pre-dictions, though they are approximately 45% larger thanthe leading order predictions obtained in Sec. II. Thequark model expectation for the total widths of theDs02317 and Ds12460 is O10 keV, consistent withEq. (11). However, the conclusion of Ref. [23] states that aDKmolecule should have a width ofO1 MeV and thatthe electromagnetic transitions should be absent. Theanalysis that follows is consistent with the first conclusionbut not the second. Below it is demonstrated that theelectromagnetic branching ratios of a DK moleculeare large and are in worse agreement with experimentthan the nonrelativistic quark model.

    The Ds02317 (Ds12460) is assumed to be an S-waveI 0 bound state of D and K mesons. The matrixelements for the electromagnetic decays of theDs02317 and Ds12460 are given by

    MDs02317 ! Ds 2

    mDs0

    s Z d3p23

    ~ DKp

    MDpKp ! Ds

    MDs12460 ! Ds 2

    mDs1

    s Z d3p23

    ~ DKp

    MDpKp ! Ds :(23)

    Here p is the three-momentum of the D meson in thebound state and ~ DKp is the bound state momentum-space wavefunction. Although calculation of the boundstate wavefunction is nonperturbative, the typical mo-mentum is small enough that the amplitudes MDK !074014Ds are perturbatively calculable in HHPT. The lead-ing order diagrams for the decay rates in Eq. (23) areshown in Fig. 2. The shaded oval on the left hand side ofthese Feynman diagrams represents the DK molecule.In Figs. 2(a) and 2(b) the dashed line is a K, and thevertex involving the photon comes from gauging theD Ds K coupling [Fig. 2(a)] or the K kineticterm [Fig. 2(b)]. In Figs. 2(c) and 2(d) the photon cou-pling comes from gauging the heavy-meson kinetic term.There are also diagrams like Figs. 2(c) and 2(d) where thephoton heavy-meson coupling comes from a term in theLagrangian proportional to TrHbHa)*Q-abF*, butthese only contribute in the P-wave channel.

    The graphs in Fig. 2(c) and 2(d) are nonvanishing inthe K0D channel, but are equal and opposite in sign sothe contribution in this channel vanishes. The graph inFig. 2(c) vanishes in the KD0 channel. The amplitudesareMDpKp ! Ds i mDmDsp 2egfpKpK p%

    pK p g% p

    %Kv

    v pK

    22*,7%v,2*3 2

    71

    MDpKp ! Ds mDmDsp 2egfpKpK p*

    pK p g* v

    p*Kv pK

    221*

    MDpKp ! Ds mDmDsp 2egfpKpK p*

    pK p g* v

    p*Kv pK

    223*: (24)Here pK and p are the kaon and photon four momen-tum, respectively. The polarization vectors for thephoton, D, and Ds are denoted 2, 21, and 23, respec-tively. It is easy to check that the amplitudes respectthe QED Ward identity. These expressions are in-serted into Eq. (23), pK is set to EKv

    p, p 0;p, and the matrix element is expanded to lowestorder in p. Because of the rotational symmetry of theS-wave wavefunction, ~ p, terms linear in p van-ish. There are corrections to the amplitudes from higherorders in chiral perturbation theory that are Om2K=2and relativistic corrections of Ov2. The errors inthe predictions for the decay rates could be as large as50%.-9

  • FIG. 3. Leading order diagram for DK bound states decay-ing intoDs 0. The dashed line from the bound state is a K, thedashed line in the final state is an / which mixes into a 0.

    THOMAS MEHEN, ROXANNE P. SPRINGER PHYSICAL REVIEW D 70 074014The results for the decay rates are

    !Ds12460 ! Ds 8g2.

    3f2

    mD0mDsm3Ds1

    j DK0j2jkj

    !Ds12460 ! Ds 4g2.

    3f2

    mD0mDsm3Ds1

    j DK0j2jkj

    !Ds02317 ! Ds 4g2.

    f2

    mD0mDsm3Ds0

    j DK0j2jkj:

    (25)

    Here DK0 DK0 is the wavefunction at the originfor the Ds02317Ds12460. In the heavy-quark limitthe partial width ratios are again !Ds12460 !Ds:!Ds12460 ! Ds:!Ds02317 ! Ds 2:1:3. However, the decay rates are proportional to jkjinstead of jkj3. This important difference in the kine-matic factors leads to a very different prediction for therelative sizes of the partial widths than obtained inEq. (10). In this case

    !Ds12460 ! Ds:!Ds12460 ! Ds:!Ds02317 ! Ds 1:57:1:R 1:58;

    where R j DK0j2=j DK0j2 is expected to be 1.In this scenario !Ds12460 ! Ds is the smallest de-cay rate rather than the largest.

    To compare with the measured branching ratios thestrong decays must also be calculated. The leading orderdiagrams are shown in Fig. 3. All three diagrams depictDK ! Ds / followed by / 0 mixing, which isrepresented by a cross on the dashed line in the final state.The vertex for the graph in Fig. 3(a) comes from thechirally covariant derivative in the heavy-meson kineticterm. This graph contributes in the S-wave channel whileFigs. 3(b) and 3(c) contribute to the P-wave channel only.The results for the decay rates are

    !Ds12460 ! Ds0

    3mK E0202

    4f4

    mDmDsm3Ds1

    j DK0j2jp0 j

    !Ds02317 ! Ds0

    3mK E0202

    4f4

    mDmDsm3Ds0

    j DK0j2jp0 j:

    (26)

    If f f 130 MeV then the bounds !Ds12460 7 MeV and !Ds02317 7 MeV imply j DK0j2 52 MeV3. If instead f f/ 171 MeV then074014j DK0j2 75 MeV3. In either case the bounds onthe wavefunctions are somewhat smaller than expected:j DK0j2 jpj3 190 MeV3. Since this is only anorder of magnitude estimate, the bounds on j DK0j2are not a problem for the DK molecular interpretation.However, they do imply that if the Ds02317 andDs12460 are DK molecules the states should not bemuch narrower than the present upper limits [23,27].

    Since the wavefunction squared cancels in the ratio ofstrong and electromagnetic decays the electromagneticbranching fractions can be predicted:

    BrDs12460 ! DsBrDs12460 ! Ds0

    3:23

    BrDs12460 ! DsBrDs12460 ! Ds0

    2:21

    BrDs02317 ! DsBrDs02317 ! Ds0

    2:96:

    (27)

    In this calculation . 1=137, 0 0:01, g 0:27, fK 159 MeV in the electromagnetic decays and f f/ 171 MeV in the strong decays. If instead f f 130 MeV is used in the strong decays the predictedbranching fraction ratios are smaller by a factor of 3.While the branching fraction ratios are quite sensitive tothe choice of f, in any case they are much too largecompared to experiment. Also, the relative sizes of thebranching fraction ratios are in disagreement with ex-periment, since the second branching fraction ratio inEq. (27) is predicted to be smallest, not largest. Notethat the possibility of these states being mixtures of quarklevel bound states and DK molecules [28,62] is also dis-favored since this would enhance the first and third ratiosin Eq. (27) relative to the second, whereas in reality theseratios are suppressed relative to the leading order predic-tion in Eq. (10).IV. CONCLUSIONS

    In this paper, corrections to electromagnetic and strongdecays of Ds02317 and Ds12460 are calculated inHHPT. The corrections depend on a number of un-known or poorly determined coupling constants. Thesepredictions can be consistent with Belle and CLEO datawith coupling constants of natural size. Serious tests ofthe HHPT description of the Ds02317 and Ds12460will require more data on the electromagnetic branching-10

  • HEAVY-QUARK SYMMETRY AND THE. . . PHYSICAL REVIEW D 70 074014ratios of the even-parity charmed strange mesons and thestrong decays of their nonstrange partners as well as thedecays of even-parity bottom strange mesons yet to beobserved. The work in this paper provides further stimu-lus for better experimental measurements of charmedstrange decays as well as discovery of their bottomstrange counterparts. Once better data becomes available,it would be interesting to test models of chiral symmetrybreaking which make specific predictions for the cou-pling constants appearing in the HHPT Lagrangian.

    This paper also tests the hypothesis that the Ds02317andDs12460 are molecular bound states ofDK andDKmolecules, respectively. In this scenario, these states aresufficiently nonrelativistic that HHPT can be used topredict the decay rates at lowest order. Furthermore,bound state wavefunctions cancel out of predictions forthe observed branching fraction ratios so absolute predic-tions can be made. These predictions are in much worseagreement with data than leading order HHPT predic-074014tions. Specifically, predictions for all the branching frac-tion ratios are larger than observed and the branchingfraction for the only observed electromagnetic decay ispredicted to be the smallest of the three possible decaysrather than the largest. Therefore, a molecular interpre-tation of these states is disfavored by available data onelectromagnetic decays.

    ACKNOWLEDGMENTS

    R. P. S. and T. M. are supported in part by DOE GrantNo. DE-FG02-96ER40945. T. M. is also supported in partby DOE Grant No. DE-AC05-84ER40150. T. M. wouldlike to thank the Aspen Center for Physics, where partof this work was completed.APPENDIX

    The function F1;2; jkj;M; isF1;2; jkj;M; gh

    82f2

    1

    3 ln

    2

    21

    1jkj2G1; jkj;M

    g

    0h82f2

    2 jkjln

    2

    22

    2 jkjjkj2H2; jkj;M

    ; (28)

    where

    G; jkj;M 2jkj jkj2lnM2

    2

    jkj2F1

    jkj

    M

    2jkjF1

    M

    M2

    F2

    M

    F2

    jkj

    M

    ;

    and

    H; jkj;M jkj2 2jkj jkj2lnM2

    2

    2 jkj2F1

    jkj

    M

    2F1

    M

    M2

    F2

    M

    F2

    jkj

    M

    :

    The functions F1;2x are given by:F1x 2

    1 x2

    p

    x

    2 arctan

    x

    1 x2p

    jxj< 1

    2x2 1

    p

    xlnx

    x2 1

    p jxj> 1

    F2x 2 arctan

    x

    1 x2p

    2 jxj< 1

    ln2x

    x2 1

    p jxj> 1: (29)-11

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