Ds12536 decays and the properties of P-wave charmed strange mesonsJ. Segovia, A.M. Yasser,* D. R. Entem, and F. Fernandez
Grupo de Fsica Nuclear and IUFFyM, Universidad de Salamanca, E-37008 Salamanca, Spain(Received 1 June 2009; published 18 September 2009)
Recently the Belle collaboration has measured a new decay channel for the charmed strange meson
Ds1 2536Ds1 2536 ! DK together with an angular analysis of the Ds1 2536 ! DK0Sdecay. We study this reaction in a constituent quark model which has been able to reproduce the hadronic
phenomenology and the baryon-baryon interaction. The reported branching fractions and the properties of
the Ds1 2536 state are nicely reproduced. Some consequences on the structure of the P-wave mesonsare discussed.
DOI: 10.1103/PhysRevD.80.054017 PACS numbers: 14.40.Lb, 12.39.Jh, 13.25.Ft
The Ds P-wave mesons have been revealed as an ex-cellent system to test low-momentum QCD. The combina-tion of a heavy and a light quark allows us to makeapproximate predictions based on the assumption of heavyquark symmetry (HQS). In this limit, the dynamics of thesystem is driven by the light quark spin and the heavy actsas a spectator.
More relevant, however, are the unexpected propertiesshown by the experiments. In 2003, the BABAR collabora-tion observed  the Ds02317 state. It was soon con-firmed by the CLEO collaboration , which reportedanother charm strange meson called Ds12460. Both me-sons were also measured by the Belle collaboration [3,4].Their results were consistent with the spin-parity assign-ment of JP 0 for the Ds02317 and JP 1 for theDs12460.
Following HQS, the light quark of the cs system ischaracterized by its total angular momentum jq sq L, where sq is the light quark spin and L the orbital angular
momentum. The total angular momentum of the meson J isobtained by coupling jq to the heavy quark spin SQ. Then
the P-wave mesons can be grouped into two doubletscharacterized by jq 1=2 with JP 0, 1 and jq 3=2 with JP 1, 2. In the infinite heavy quark masslimit the doublets are degenerated. Moreover, the strongdecays of the DsJ jq 3=2 proceed only throughD-waves while the DsJ jq 1=2 decays only throughS-waves. The decay to a D wave will be suppressed bythe barrier factor, which behaves as q2L1 where q is therelative momentum of the two decaying mesons.Therefore, the states decaying through D waves are ex-pected to be narrower than those decaying in S waves,which are expected to be broad.
Although some of the properties of the jq 3=2 statesare consistent with the data of theDs12536 andDs22573
discovered earlier , the observed properties of theDs02317 andDs12460 did not agree with the theoreticalpredictions for the jq 1=2 states.Recently, new data related with theDs12536meson has
appeared. The BABAR collaboration has performed a highprecision measurement of the Ds12536 decay width ob-taining a value of 1:03 0:05 0:12 MeV .Furthermore, the Belle collaboration has reported the firstobservation of the Ds12536 ! DK decay mea-suring the branching fraction 
Ds12536 ! DKDs12536 ! DK0
3:27 0:18 0:37%:(1)
They also measured the ratio of the D and S wave ampli-tudes in theDs12536 ! DK0 decay finding a value of0:72 0:05 0:01. These results contradict in some sensethe predictions of HQS because, although the Ds12536state is narrow, its S-wave decay amplitude is sizable,which suggests strong cancellations in the decayamplitudes.All these properties make the DsJ P-wave states an
interesting system to study not only the meson masses, asis usually done (see Refs. [8,9] and references therein), butalso its strong decays in a model without heavy quarkapproximations.In this work we will use the model of Ref.  to study
the reaction rates of theDs12536 ! DK decay aswell as the angular decomposition of the Ds12536 !DK0 in order to gain insight into the structure of theP-wave charm strange mesons. As the D pair in thefinal state is the only D combination that cannot comefrom aD resonance, we will describe the reaction througha virtual D0 meson since MD0
II. MODEL HAMILTONIAN AND P-WAVEMESONS
We will work in the framework of the nonrelativisticquark model in which quarks carry a constituent mass. Inspite of its name, the model incorporates some relativisticcorrections in the potential through the spin-spin and spin-orbit terms but not in the kinetic energy. As it is wellknown, in these models the light-quark momentum is ofthe order of the constituent mass. However, it is also widelyaccepted that nonrelativistic quark models are able toreproduce the meson spectra with a similar quality as thosewhich use semirelativistic kinematics . This shows thatthe relativistic effects may be incorporated in an effectiveway into the model parameters.
The picture of QCD vacuum as a dilute medium ofinstantons explains nicely why at low energy light quarksbehave as particles with a dynamical mass of the order of300 MeV . This dynamical mass appears as a conse-quence of the breaking of the original chiral symmetry ofthe QCD Lagrangian. In the instanton liquid, light quarksinteract with fermionic zero modes of the individual in-stantons and the quark propagator gets modified by amomentum-dependent mass which drops off for momen-tum lighter than the inverse of the average instanton size.To compensate, the mass term in the Hamiltonian newinteractions appears between constituent quarks, namely,the Goldstone boson exchange interactions. Beyond thechiral symmetry breaking scale, quark dynamics is gov-erned by QCD perturbative effects. There are consequen-ces of the one-gluon fluctuation around the instantonvacuum and we take it into account by the nonrelativisticexpansion of the QCD inspired the Fermi-Breit interaction.For the heavy quarks, chiral symmetry is explicitly brokenby its large current mass and they do not couple to theGoldstone bosons. However, one also can assign to thesequarks an effective mass due to the gluon dressing. All ofthese interactions have been discussed in Ref. , and werefer the reader to it for further details. We will only writethe spin-orbit interaction coming from the one gluon ex-change for later discussions
VSOOGE ~rij 1
mi mj2 2mimj
~S ~L m2j m2i ~S ~L; (2)
where ~S ~Si ~Sj and rg rg nnij scales with thereduced mass of the interacting particles.
Both heavy and light quarks are confined into the mesonwhich guarantees the nonexistence of isolated colorcharges. Such a term can be physically interpreted in apicture in which the quark and the antiquark are linked by aone-dimensional color flux-tube. The spontaneous creation
of light-quark pairs may give rise to a breakup of the colorflux-tube . This can be translated into a screenedpotential  in such a way that the potential saturates atthe same interquark distance. One important questionabout the confinement is its covariance properties. Thisaspect is discussed in Ref. , and we will consider aconfinement spin-orbit contribution as a combination ofscalar and vector terms
VSOCON ~rij ~ci ~cjacce
m2i m2j 1 2as
4mimj1 as ~S ~L m2j m2i 1 2as ~S ~L; (3)
where as controls the ratio between them.For the low-lying positive parity excitations, any quark
model predicts four states 1P1,3P0,
3P1, and3P2 in terms
of the JLS basis. As charge conjugation is not well definedin the heavy-light sector, 1P1 and
3P1 states are mixed. Afirst approximation to the mixing can be obtained in theheavy quark limit. As stated above, in this limit the mesonproperties are characterized by the dynamics of the lightquark. For P-waves the spin of the light quark couples withthe orbital angular momentum giving two degeneratedjq 3=2 states with JP 2 and JP 1 and two de-generated jq 1=2 states with JP 1 and JP 0.These states are given by
j1=2; 0i j3P0i (4)
j1=2; 1i ffiffi23
j3=2; 1i ffiffi13
j3=2; 2i j3P2i: (7)Moreover, this assumption predicts that the jq 3=2, 1state should be narrow as it is experimentally . Thedegeneration is approximately fulfilled in the jq 3=2sector but the new measured Ds02317 and Ds12460states contradict this first approximation. When we includethe charm quark finite mass corrections, the mixing be-tween the 1P1 and
3P1 states is induced by the antisym-metric term of the spin-orbit interaction. However, eventhis mixing is unable to reproduce the experimental data asone can see in Table I, where the results for the low-lyingpositive parity excitations 1P1,
3P2 calcu-lated in our model are shown.The small experimental mass of the Ds02317 has been
attributed to several mechanisms. The existence of a tetra-quark structure with JP 0 and mass M 2731 MeV=c2 is used in Refs. [8,14] to explain not onlythe Ds02317 but also the DsJ 2860  as mixed states
SEGOVIA et al. PHYSICAL REVIEW D 80, 054017 (2009)
of cs states and the tetraquark. The same mechanism hasbeen invoked to explain the mass of the Ds12460.
III. THE Ds12536 DECAYSMeson strong decay is a complex nonperturbative pro-
cess that still has not been described from first principles.Instead, phenomenological models have been developed todeal with this problem. The most popular are the Cornellmodel , the flux tube model , and the 3P0 model. The Cornell model assumes that the strong decaytakes place through pair creation from the linear confiningpotential, whereas in the flux tube and in the 3P0 model thequark-antiquark pair is created from the vacuum. Bothmodels are similar but the flux tube model takes intoaccount the dynamics of the flux tubes by including theoverlap of the flux tube of the initial meson with those ofthe two outgoing mesons. All these models describe rea-sonably well the experimental data , and we will usefor simplicity the 3P0 model.
The model was first proposed by Micu  and furtherdeveloped by Le Yaouanc et al. . To describe themeson decay process A ! B C it assumes that a quarkand an antiquark are created with JPC 0 quantumnumbers. The created q q pair together with the q q pairin the original meson regroups in the two outgoing mesonsvia a quark rearrangement process. Then, the transitionoperator is given by 
where are the quark (antiquark) quantum num-bers and is a dimensionless constant that denotes thestrength of the q q pair creation from the vacuum.
Defining the S-matrix as
hfjSjii I i244pf piM; (9)where M is the decay amplitude of the process A ! BC, the decay width in terms of the partial wave amplitude is
ZdkEi EfjMJLA!BCkj2 (10)
using the relativistic phase space one arrives to the finalexpression
where k0 is the on-shell relative momentum of the decay-ing mesons.The reaction Ds12536 ! DK is characterized
by the fact that the pair D in the final state is the onlyD combination that cannot come from a D resonancemaking this channel different from the usual Ds12536 !DK. The D0 meson can only decay into D virtuallysince MD0
kmax ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2A MB1 MB2 MC2M2A MB1 MB2 MC2
IV. RESULTS AND DISCUSSION
To describe the P-waveDs system wewill use the modelof Refs. [10,23]. In this model a tetraquark c sn n state hasbeen calculated in Ref.  with I 0 and JP 1 andmass M 2841 MeV=c2. This state should be coupled tothe cs Ds states.
Working in the HQS limit, the csn n tetraquark has threedifferent spin states, j0 1=2i, j1 1=2i, and j1 3=2i, where thefirst index denotes the spin of the n n pair and the seconddenotes the coupling with the s spin. Although we use the3P0 model to calculate the meson decay widths, a descrip-tion of the coupling between the Ds meson and the tetra-quark based on this model is beyond the scope of thepresent calculation. However, we will use it here to selectthe dominant couplings and parametrize the vertex as aconstant CS. The model assumes that the n n pair created isin a J 0 state which means that the Ds states will onlycouple with the first tetraquark component which has spin1=2 for the three light quarks. In the HQS limit, the heavyquark is a spectator and the angular momentum of the lightquarks has to be conserved so that the tetraquark will onlycouple to the c s jq 1=2 state.
For that reason, we couple the tetraquark structure withthe jq 1=2cs Ds state. This choice differs from the oneperformed in Ref. , where the tetraquark is only coupledto the 1P1 state and not to the
3P1. However, this choice hasseveral advantages; it has the correct heavy quark limit; itmay reproduce the narrow width of theDs12536 state; itis in agreement with the experimental situation, which tellsus that the prediction of the heavy quark limit is reasonablefor the jq 3=2 state but not for the jq 1=2 one.
In this case we diagonalize the matrix
qCS Mcsn n
where M3P1 2571:5 MeV, M1P
1 2576:0 MeV, and
Mcsn n 2841 MeV are the masses of the states withoutcouplings, the CSO 19:6 MeV is the coupling inducedby the antisymmetric spin-orbit interaction calculatedwithin the model, and CS is the parameter that gives thecoupling between the jq 1=2 component of the 3P1 and1P1 states and the tetraquark. The value of the parameterCS 224 MeV is fitted to the mass of the Ds12460. Weget the three eigenstates shown in Table II. There we alsoshow the probabilities of the three components for eachstate and the relative phases between different components.
We now calculate the different decay widths for theDs12536 state of Table II. As expected, the DK decaywidth is narrow 0:46 MeV. As the DK decay is sup-pressed the total width would be mainly given by the DKchannel and is in the order of the experimental valueexp 1:03 0:05 0:12 MeV measured by BABAR. Of course the value strongly depends on the 3P0 strength parameter that we have taken from a previousstudy of strong decays in charmonium . It also dependson the fact that we have only coupled the 1=2 state with thetetraquark, making the remaining state a purest 3=2, whichmakes it narrower. If we would include an small couplingbetween the 3=2 state and the tetraquark, our Ds12536will be broader.There are two other experimental data that does not
depend on the parameter, namely, the branching ratio 
R1 Ds12536 ! D0K
Ds12536 ! DK0 1:27 0:21 (19)
and the ratio of S wave over the full width for the DK0decay 
R2 SDs12536 ! DK0
Ds12536 ! DK0 0:72 0:05 0:01:
The first branching ratio should be 1 if the isospin symme-try was exact. However, the charge symmetry breaking inthe phase space makes it different from this value. Theeffect is sizable since the Ds12536 is close to the DKthreshold, and, for this reason, it also depends on the detailsof the Ds1 wave function. We get for this ratio the value
R1 1:31 in good agreement with the experimental one.In the HQS limit the branching R2 should be zero
because the decay of jq 3=2 state would go only throughD-wave. In our case we get a value of R2 0:66 close tothe experimental value. The fact that our result is smallerthan the experimental one indicates that the probability ofthe jq 3=2 state is too high, which is in agreement withthe fact that we get a too narrow state.
TABLE II. Masses and probability distributions for the threeeigenstates obtained from the coupling of the Ds and tetraquarkstates. The relative sign to the tetraquark component is alsoshown.
MMeV S3P1 P3P1 S1P1 P1P1 Scsn n Pcsn n2459 - 55.7 - 18.8 25.52557 27.7 - 72.1 0.22973 16.6 9.1 74.3
SEGOVIA et al. PHYSICAL REVIEW D 80, 054017 (2009)
Finally, we calculate the branching
R3 Ds12536 ! DK
Ds12536 ! DK0 3:27 0:18 0:37%: (21)
The reaction in the numerator goes through a virtualD0 asexplained previously and for that reason the branching issmall. We get the value R3 4:00%.
All these results for the width and the ratios R1, R2, andR3 are summarized in Table III, where we also show, forthe sake of completeness, the results for the two 1 stateswithout coupling to the csn n tetraquark (Table I) wherenone of these two states agree with the full set of experi-mental values.
As summary, we have calculated some of the Ds12536decays in the framework of a constituent quark model andusing the 3P0 model as the decay mechanism. These de-cays pose very demanding constraints to the Ds1 wave
function. We have coupled the cs jq 1=2 componentwith the tetraquark state of mass 2841 MeV. We got theDs12536 as a mixture of 1P1 and 3P1 states close to thejq 3=2 which is crucial to reproduce simultaneously itsnarrow width and the ratio of the S andD-wave amplitudesin the DK0 decay. Also, the decay Ds12536 !DK through a virtual D0 is well reproduced withinthe model.Finally, a new 1 state with an important component of
csn n tetraquark structure is predicted at 2973 MeV.
This work has been partially funded by Ministerio deCiencia y Tecnologa under Contract No. FPA2007-65748,by Junta de Castilla y Leon under Contract No. SA-106A07 and GR12, by the European Community-Research Infrastructure Integrating Activity Study ofStrongly Interacting Matter (HadronPhysics2 GrantNo. 227431) and by the Spanish Ingenio-Consolider2010 Program CPAN (CSD2007-00042). A.M.Y. wouldlike to acknowledge the South Valley University andHigher Education Ministry of Egypt for financial support.
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TABLE III. Width and the 3 branching ratios defined in thetext. The first row shows the experimental data and the secondshows our results for the physical Ds1 2536 state given inTable II. For completeness we give in the last two rows theresults for the two 1 cs states in Table I.
M (MeV) (MeV) R1 R2 R3 (%)
Exp. 1.03 1.27 0.72 3.27
2557 0.46 1.31 0.66 4.00
2593 88 1.09 1.00 3.73
2554 5.2 1.11 0.97 3.75
Ds1 2536 DECAYS AND THE PROPERTIES . . . PHYSICAL REVIEW D 80, 054017 (2009)