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  and Ds12460 has revived interest in excited charmed mesons. Thedominant decay modes of these states are Ds02317 !Ds0 and Ds12460 ! Ds0, with widths less than7 MeV . 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 , 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 . Belle hasobserved the decay Ds12460 ! Ds from Ds12460produced in the decays of B mesons  and from con-tinuum ee production . 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 :address: firstname.lastname@example.org: email@example.com
04=70(7)=074014(12)$22.50 70 0740BrDs12460 ! DsBrDs12460 ! Ds0
BrDs02317 ! DsBrDs02317 ! Ds0
(The BELLE collaboration quotes weaker lower boundsof 0.31 and 0.18, respectively, for these ratios .)
In this paper the decays of the Ds02317 andDs12460 are analyzed using heavy-hadron chiral per-turbation theory (HHPT) . 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
BrDs02317 ! DsBrDs02317 ! Ds0
(Leading order calculations of strong and electromagneticdecays were first done in Refs. .) 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 , in which hadronsare placed in reducible representations of SU3L SU3R. QCD sum rules have also been used to calculatesome of the HHPT couplings . 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  aswell as speculation that these states are exotic.Possibilities include DK molecules , Ds mole-cules , and tetraquarks [28,3135]. Masses have beencalculated in lattice QCD , 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.  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.  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  as well as their production in b-hadrondecays [43 46]. Ref.  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  and semileptonic Bs decays .
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. . 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 
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 
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
Trmq ymq TrHaiv DbaHb TrSaiv Dba %SH%abSb
gTrHaHbA6 ba5 g0TrSaSbA6 ba5 hTrHaSbA6 ba5 h:c: H8
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. . 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=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
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
!1a ! 0a 1
!0a ! 1a .e2q ~,2m1am0a
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  D011 24615348 29011091Belle  D000 2308 36 276 66
D011 2427 36 384130105FOCUS  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...