Ds1ð2536Þþ decays and the properties of P-wave charmed strange mesons

J. Segovia, A.M. Yasser,* D. R. Entem, and F. Fernandez

Grupo de Fısica 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 ð2536ÞþðDs1 ð2536Þþ ! Dþ��KþÞ together with an angular analysis of the Ds1 ð2536Þþ ! D�þK0S

decay. 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 ð2536Þþ state are nicely reproduced. Some consequences on the structure of the P-wave mesons

are discussed.

DOI: 10.1103/PhysRevD.80.054017 PACS numbers: 14.40.Lb, 12.39.Jh, 13.25.Ft

I. INTRODUCTION

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 [1] the D�

s0ð2317Þ state. It was soon con-

firmed by the CLEO collaboration [2], which reportedanother charm strange meson called Ds1ð2460Þ. Both me-

sons were also measured by the Belle collaboration [3,4].Their results were consistent with the spin-parity assign-ment of JP ¼ 0þ for the D�

s0ð2317Þ and JP ¼ 1þ for the

Ds1ð2460Þ.Following HQS, the light quark of the c�s system is

characterized by its total angular momentum jq ¼ sq þL, 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 ¼ 1=2 with JP ¼ 0þ, 1þ and jq ¼3=2 with JP ¼ 1þ, 2þ. In the infinite heavy quark masslimit the doublets are degenerated. Moreover, the strongdecays of the DsJ ðjq ¼ 3=2Þ proceed only through

D-waves while the DsJ ðjq ¼ 1=2Þ decays only through

S-waves. The decay to a D wave will be suppressed bythe barrier factor, which behaves as q2Lþ1 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 ¼ 3=2 states

are consistent with the data of theDs1ð2536Þ andDs2ð2573Þ

discovered earlier [5], the observed properties of theD�

s0ð2317Þ andDs1ð2460Þ did not agree with the theoreticalpredictions for the jq ¼ 1=2 states.

Recently, new data related with theDs1ð2536Þmeson has

appeared. The BABAR collaboration has performed a highprecision measurement of the Ds1ð2536Þ decay width ob-

taining a value of 1:03� 0:05� 0:12 MeV [6].Furthermore, the Belle collaboration has reported the firstobservation of the Ds1ð2536Þþ ! Dþ��Kþ decay mea-

suring the branching fraction [7]

Ds1ð2536Þþ ! Dþ��Kþ

Ds1ð2536Þþ ! D�þK0 ¼ ð3:27� 0:18� 0:37Þ%:

(1)

They also measured the ratio of the D and S wave ampli-tudes in theDs1ð2536Þþ ! D�þK0 decay finding a value of

0:72� 0:05� 0:01. These results contradict in some sensethe predictions of HQS because, although the Ds1ð2536Þstate 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. [10] to study

the reaction rates of theDs1ð2536Þþ ! Dþ��Kþ decay as

well as the angular decomposition of the Ds1ð2536Þþ !D�þK0 in order to gain insight into the structure of theP-wave charm strange mesons. As the Dþ�� pair in thefinal state is the only D� combination that cannot comefrom aD� resonance, we will describe the reaction througha virtual D�0 meson since MD�0 <MDþ þM�� .The paper is organized as follows. In Sec. II we define

the model Hamiltonian and the P-wave meson structure.Section III is devoted to the description of the strongdecays. Results and comments are given in Sec. IV andfinally we give a summary in Sec. V.

*Permanent address: Physics Department, Faculty of Scienceat Qena, South Valley University Egypt.

PHYSICAL REVIEW D 80, 054017 (2009)

1550-7998=2009=80(5)=054017(5) 054017-1 � 2009 The American Physical Society

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 [10]. 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 [11]. 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. [10], 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

VSOOGEð ~rijÞ ¼ � 1

16

�s

m2i m

2j

ð ~�ci � ~�c

j�1

r3ij� e�rij=rgð�Þ

r3ij

��1þ rij

rgð�Þ��

½ððmi þmjÞ2 þ 2mimjÞ

� ð ~Sþ � ~LÞ þ ðm2j �m2

i Þð ~S� � ~LÞ�; (2)

where ~S� ¼ ~Si � ~Sj and rgð�Þ ¼ rg�nn

�ijscales with the

reduced mass of the interacting particles.Both heavy and light quarks are confined into the meson

which 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 [12]. This can be translated into a screenedpotential [13] 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. [10], and we will consider aconfinement spin-orbit contribution as a combination ofscalar and vector terms

VSOCONð ~rijÞ ¼ �ð ~�c

i � ~�cjÞac�ce

��crij

4m2i m

2jrij

½ððm2i þm2

j Þð1� 2asÞ

þ 4mimjð1� asÞÞð ~Sþ � ~LÞþ ðm2

j �m2i Þð1� 2asÞð ~S� � ~LÞ�; (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 ¼ 3=2 states with JP ¼ 2þ and JP ¼ 1þ and two de-

generated jq ¼ 1=2 states with JP ¼ 1þ and JP ¼ 0þ.These states are given by

j1=2; 0þi ¼ j3P0i (4)

j1=2; 1þi ¼ffiffi23

qj3P1i þ

ffiffi13

qj1P1i (5)

j3=2; 1þi ¼ffiffi13

qj3P1i �

ffiffi23

qj1P1i (6)

j3=2; 2þi ¼ j3P2i: (7)

Moreover, this assumption predicts that the jq ¼ 3=2, 1þ

state should be narrow as it is experimentally [6]. Thedegeneration is approximately fulfilled in the jq ¼ 3=2

sector but the new measured D�s0ð2317Þ and Ds1ð2460Þ

states 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,

3P0,3P1, and

3P2 calcu-

lated in our model are shown.The small experimental mass of the D�

s0ð2317Þ has beenattributed to several mechanisms. The existence of a tetra-quark structure with JP ¼ 0þ and mass M ¼2731 MeV=c2 is used in Refs. [8,14] to explain not onlythe D�

s0ð2317Þ but also the DsJ ð2860Þ [15] as mixed states

SEGOVIA et al. PHYSICAL REVIEW D 80, 054017 (2009)

054017-2

of c�s states and the tetraquark. The same mechanism hasbeen invoked to explain the mass of the Ds1ð2460Þ.

III. THE Ds1ð2536Þþ DECAYS

Meson 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 [16], the flux tube model [17], and the 3P0 model

[18]. 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 the

quark-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 [19], and we will usefor simplicity the 3P0 model.

The model was first proposed by Micu [18] and furtherdeveloped by Le Yaouanc et al. [20]. To describe themeson decay process A ! Bþ C it assumes that a quarkand an antiquark are created with JPC ¼ 0þþ 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 [21]

T ¼ �3�X�

Zd3pd3p0�ð3Þðpþ p0Þ

��Y1

�p� p0

2

�by�ðpÞdy�ðp0Þ

�C¼1;I¼0;S¼1;J¼0

; (8)

where �ð� ¼ ��Þ 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 ¼ Iþ ið2�Þ4�4ðpf � piÞM; (9)

where M is the decay amplitude of the process A ! BþC, the decay width in terms of the partial wave amplitude is

� ¼ 2�XJL

Zdk�ðEi � EfÞjMJL

A!BCðkÞj2 (10)

using the relativistic phase space one arrives to the finalexpression

� ¼ 2�EBEC

k0MA

XJL

jMJLA!BCðk0Þj2; (11)

where k0 is the on-shell relative momentum of the decay-ing mesons.The reaction Ds1ð2536Þþ ! Dþ��Kþ is characterized

by the fact that the pair Dþ�� in the final state is the onlyD� combination that cannot come from a D� resonancemaking this channel different from the usual Ds1ð2536Þ !D�K. The D�0 meson can only decay into Dþ�� virtuallysince MD�0 <MDþ þM�� . Then, to describe this decay,we need to modify the intermediate D� propagator takinginto account that it is not an stable particle but a resonancewith a width that allows the decay to take place. Followinga similar treatment as in Ref. [22], we include the width ofthe meson by replacing the Dirac � function in Eq. (10).One may regard the �-function as arising from the narrow-width limit of the energy denominator

1

MA � EB � EC � i�¼ P

1

MA � EB � EC

þ i��ðMA � EB � ECÞ; (12)

where � is related with the total width of the unstableintermediate state. For finite width �B

1

MA � EB � EC � i �B

2

¼ MA � EB � EC þ i �B

2

ðMA � EB � ECÞ2 þ �2B

4

(13)

and so the �-function should be replaced by

�ðMA � EB � ECÞ ! �B

2�½ðMA � EB � ECÞ2 þ �2B

4 �:

(14)

For the decay B ! B1B2, we multiply by the branching

B ðB ! B1B2Þ ¼�B!B1B2

ðkÞ�B

(15)

so, neglecting the momentum dependence of the totalwidth of the B meson �B, the width for the decay A !ðB1B2ÞC is given by

�A!ðB1B2ÞC ¼ XJL

Z kmax

0dk

�B!B1B2ðkÞ

½ðMA � EB � ECÞ2 þ �2B

4 �� jMJL

A!BCðkÞj2; (16)

where kmax is the maximum relative momentum for the BCsystem allowed by the three-body decay A ! ðB1B2ÞC andis given by

TABLE I. Results for the masses (in MeV) of the low-lyingpositive parity c�s states in the quark model (QM).

JP State QM Experimental data

0þ D�s0 ð2317Þ 2511 2317:4� 0:9

1þ Ds1 ð2460Þ 2593 2459:3� 1:31þ Ds1 ð2536Þ 2554 2535:3� 0:62þ Ds2 ð2573Þ 2592 2572:4� 1:5

Ds1 ð2536Þþ DECAYS AND THE PROPERTIES . . . PHYSICAL REVIEW D 80, 054017 (2009)

054017-3

kmax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½M2

A � ðMB1þMB2

þMCÞ2�½M2A � ðMB1

þMB2�MCÞ2�

q2MA

: (17)

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. [9] with I ¼ 0 and JP ¼ 1þ andmass M ¼ 2841 MeV=c2. This state should be coupled tothe c�s Ds states.

Working in the HQS limit, the c�sn �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 ¼ 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 ¼ 1=2 state.

For that reason, we couple the tetraquark structure withthe jq ¼ 1=2c�s Ds state. This choice differs from the one

performed in Ref. [9], 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 theDs1ð2536Þþ state; it

is in agreement with the experimental situation, which tellsus that the prediction of the heavy quark limit is reasonablefor the jq ¼ 3=2 state but not for the jq ¼ 1=2 one.

In this case we diagonalize the matrix

M ¼M3P

1CSO

ffiffi23

qCS

CSO M1P1

ffiffi13

qCSffiffi

23

qCS

ffiffi13

qCS Mc�sn �n

0BBBB@

1CCCCA; (18)

where M3P1¼ 2571:5 MeV, M1P

1¼ 2576:0 MeV, and

Mc�sn �n ¼ 2841 MeV are the masses of the states withoutcouplings, the CSO ¼ 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 ¼ 1=2 component of the 3P1 and1P1 states and the tetraquark. The value of the parameterCS ¼ 224 MeV is fitted to the mass of the Ds1ð2460Þ. We

get 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 theDs1ð2536Þþ state of Table II. As expected, the D�K decay

width is narrow � ¼ 0:46 MeV. As the DK decay is sup-pressed the total width would be mainly given by the D�Kchannel and is in the order of the experimental value�exp ¼ 1:03� 0:05� 0:12 MeV measured by BABAR

[6]. Of course the value strongly depends on the 3P0 �strength parameter that we have taken from a previousstudy of strong decays in charmonium [23]. 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 Ds1ð2536Þwill be broader.There are two other experimental data that does not

depend on the � parameter, namely, the branching ratio [5]

R1 ¼ �ðDs1ð2536Þþ ! D�0KþÞ�ðDs1ð2536Þþ ! D�þK0Þ ¼ 1:27� 0:21 (19)

and the ratio of S wave over the full width for the D�þK0

decay [7]

R2 ¼ �SðDs1ð2536Þþ ! D�þK0Þ�ðDs1ð2536Þþ ! D�þK0Þ ¼ 0:72� 0:05� 0:01:

(20)

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 Ds1ð2536Þþ is close to the D�Kthreshold, and, for this reason, it also depends on the detailsof the Ds1 wave function. We get for this ratio the value

R1 ¼ 1:31 in good agreement with the experimental one.In the HQS limit the branching R2 should be zero

because the decay of jq ¼ 3=2 state would go only through

D-wave. In our case we get a value of R2 ¼ 0:66 close tothe experimental value. The fact that our result is smallerthan the experimental one indicates that the probability ofthe jq ¼ 3=2 state is too high, which is in agreement with

the 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.

MðMeVÞ Sð3P1Þ Pð3P1Þ Sð1P1Þ Pð1P1Þ Sðc�sn �nÞ Pðc�sn �nÞ2459 - 55.7 - 18.8 þ 25.5

2557 þ 27.7 - 72.1 þ 0.2

2973 þ 16.6 þ 9.1 þ 74.3

SEGOVIA et al. PHYSICAL REVIEW D 80, 054017 (2009)

054017-4

Finally, we calculate the branching

R3 ¼ �ðDs1ð2536Þþ ! Dþ��KþÞ�ðDs1ð2536Þþ ! D�þK0Þ

¼ ð3:27� 0:18� 0:37Þ%: (21)

The reaction in the numerator goes through a virtualD�0 asexplained previously and for that reason the branching issmall. We get the value R3 ¼ 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 1þ stateswithout coupling to the c�sn �n tetraquark (Table I) wherenone of these two states agree with the full set of experi-mental values.

V. SUMMARY

As summary, we have calculated some of the Ds1ð2536Þdecays 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 c�s jq ¼ 1=2 component

with the tetraquark state of mass 2841 MeV. We got theDs1ð2536Þ as a mixture of 1P1 and 3P1 states close to the

jq ¼ 3=2 which is crucial to reproduce simultaneously its

narrow width and the ratio of the S andD-wave amplitudesin the D�þK0 decay. Also, the decay Ds1ð2536Þþ !Dþ��Kþ through a virtual D�0 is well reproduced withinthe model.Finally, a new 1þ state with an important component of

c�sn �n tetraquark structure is predicted at 2973 MeV.

ACKNOWLEDGMENTS

This work has been partially funded by Ministerio deCiencia y Tecnologıa 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.

[1] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett.90, 242001 (2003).

[2] D. Besson et al. (CLEO Collaboration), Phys. Rev. D 68,032002 (2003).

[3] P. Krokovny et al. (Belle Collaboration), Phys. Rev. Lett.91, 262002 (2003).

[4] Y. Mikami et al. (Belle Collaboration), Phys. Rev. Lett.92, 012002 (2004).

[5] C. Amsler et al. (Particle Data Group), Phys. Lett. B 667, 1(2008).

[6] B. Aubert et al. (BABAR Collaboration), arXiv:hep-ex/0607084.

[7] V. Balagura et al. (Belle Collaboration), Phys. Rev. D 77,032001 (2008).

[8] E. S. Swanson, Phys. Rep. 429, 243 (2006).[9] J. Vijande, F. Fernandez, and A. Valcarce, Phys. Rev. D

73, 034002 (2006).[10] J. Vijande, F. Fernandez, and A. Valcarce, J. Phys. G 31,

481 (2005).[11] D. Diakonov, Prog. Part. Nucl. Phys. 51, 173 (2003).

[12] G. S. Bali, H. Neff, T. Dussel, T. Lippert, and K. Schilling,Phys. Rev. D 71, 114513 (2005).

[13] K. D. Born, E. Laermann, N. Pirch, T. F. Walsh, and P.M.Zerwas, Phys. Rev. D 40, 1653 (1989).

[14] J. Vijande, A. Valcarce, and F. Fernandez, Phys. Rev. D79, 037501 (2009).

[15] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett.97, 222001 (2006).

[16] E. Eichten et al., Phys. Rev. D 17, 3090 (1978).[17] R. Kokoski and N. Isgur, Phys. Rev. D 35, 907 (1987).[18] L. Micu, Nucl. Phys. B10, 521 (1969).[19] E. S. Ackleh, T. Barnes, and E. S. Swanson, Phys. Rev. D

54, 6811 (1996).[20] A. Le Yaouanc, L. Olivier, O. Pene, and J. C. Raynal,

Phys. Rev. D 8, 2223 (1973).[21] R. Bonnaz and B. Silvestre-Brac, Few-Body Syst. 27, 163

(1999).[22] S. Capstick and W. Roberts, Phys. Rev. D 49, 4570 (1994).[23] J. Segovia, A.M. Yasser, D. R. Entem, and F. Fernandez,

Phys. Rev. D 78, 114033 (2008).

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 ð2536Þ state given in

Table II. For completeness we give in the last two rows theresults for the two 1þ c�s 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 ð2536Þþ DECAYS AND THE PROPERTIES . . . PHYSICAL REVIEW D 80, 054017 (2009)

054017-5