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Newly observed DsJð3040Þ and the radial excitations of P-wave charmed-strange mesons

Zhi-Feng Sun (孙志峰) and Xiang Liu (刘翔)*

School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China(Received 11 September 2009; published 30 October 2009)

Inspired by the newly observed DsJð3040Þþ state, in this work we systemically study the two-body

strong decays of P-wave charmed-strange mesons with the first radial excitation. Under the assignment of

1þðjP ¼ 12þÞ, i.e. the first radial excitation of Ds1ð2460Þþ, we find that the width of DsJð3040Þþ is close to

the lower limit of the BABAR measurement. This indicates that it is reasonable to interpret DsJð3040Þþ as

the first radial excitation of Ds1ð2460Þþ. Our calculation further predicts that 0� þ 1� channels, e.g.,

DþK�0, D0K�þ, and Dþs �, are important for the search for DsJð3040Þþ. To help future experiments find

the remaining three P-wave charmed-strange mesons with the first radial excitation, we present the

predictions for the strong decays of these three P-wave charmed-strange mesons.

DOI: 10.1103/PhysRevD.80.074037 PACS numbers: 13.25.Ft

I. INTRODUCTION

With the new observation of the DsJ meson, the spec-trum of the charmed-strange state is becoming abundant.So far, there exist six established charmed-strange mesonsDsð1968Þ, D�

sð2112Þ, D�s0ð2317Þ, Ds1ð2460Þ, Ds1ð2536Þ,

and D�s2ð2573Þ listed in the Particle Data Group (PDG)

[1], which can be categorized as three doublets in terms ofthe heavy quark limit: H ¼ ð0�; 1�Þ ¼ ðDsð1968Þ;D�

sð2112ÞÞ, S ¼ ð0þ; 1þÞ ¼ ðD�s0ð2317Þ; Ds1ð2460ÞÞ, and

T ¼ ð1þ; 2þÞ ¼ ðDs1ð2536Þ; D�s2ð2573ÞÞ. Two years ago,

a new charmed-strange meson D�s1ð2710Þ with JP ¼ 1�

was first announced by the BABAR Collaboration [2] andconfirmed by the Belle Collaboration later [3]. Very re-cently the BABAR experiment found D�

s1ð2710Þ again in

the D�K invariant mass spectrum [4]. Another newly ob-served charmed-strange meson is D�

sJð2860Þ, which wasobserved in both DK [2] and D�K channels [4]. Thephenomenological proposals of the quantum number ofD�

sJð2860Þ include JP ¼ 3� [5,6] and JP ¼ 0þ [6–8]. Asindicated by the BABAR experiment, the JP ¼ 0þ assign-ment for D�

sJð2860Þ is forbidden according to the parityconservation since the D�K decay mode of D�

sJð2860Þ wasobserved in Ref. [4]. A series of theoretical work [5–13]relevant to D�

s1ð2710Þ and D�sJð2860Þ were carried out.

Besides the observations ofD�s1ð2710Þ andD�

sJð2860Þ byanalyzing the D�K invariant mass spectrum in inclusiveeþe� interactions [4], BABAR also announced a newcharmed-strange state DsJð3040Þ with the mass M ¼3044� 8ðstatÞþ30

�5 ðsystÞ MeV and the width � ¼239� 35ðstatÞþ46

�42ðsystÞ MeV [4]. The observation of

DsJð3040Þ not only makes the spectrum of the charmed-strange meson abundant (the mass spectrum of the ob-served charmed-strange mesons is listed in Fig. 1), butalso stimulates our interest in exploring its underlyingstructure.

As indicated by the BABAR Collaboration, DsJð3040Þwas only observed in the D�K channel but not found in theDK decay mode. Thus, its possible quantum number in-cludes JP ¼ 1þ; 0�; 2�; . . . . Since Ds1ð2710ÞðJP ¼ 1�Þ isthe first radial excitation of D�

sð2112Þ and the mass ofDsJð3040Þ is far larger than that of Ds1ð2710Þ, we furtherexclude the 0� assignment, viz. the first radial excitation ofDsð1968Þ for DsJð3040Þ. In Ref. [14], Matsuki, Morii, andSudoh once predicted the mass of the c �s state withn2sþ1LJ ¼ 23P1: m ¼ 3082 MeV, which is close to theexperimental value of the mass of DsJð3040Þ. Thus, the 1þassignment to DsJð3040Þ, the first radial excitation ofDs1ð2460Þ, becomes the most possible.If DsJð3040Þ as the radial excitation of the P-wave

charmed-strange state is true, further experiment has thepotential to search the remaining three radial excitations ofP-wave charmed-strange states. Thus, a systematical phe-nomenological study of the strong decay mode of the P-wave charmed-strange mesons with the first radial excita-tion is an important and interesting topic. By this study, we

FIG. 1 (color online). The mass spectrum of the observedcharmed-strange mesons and the corresponding strong decaymodes observed in experiment [1–4].

*Corresponding [email protected]

PHYSICAL REVIEW D 80, 074037 (2009)

1550-7998=2009=80(7)=074037(10) 074037-1 � 2009 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.80.074037

will not only obtain the information for the decays of theseP-wave charmed-strange mesons, but also can test the 1þquantum number assignment to DsJð3040Þ comparing thecalculated decay width with the experimental data.

In this work, we will be dedicated to the study of thestrong decay modes of P-wave charmed-strange mesonswith the radial excitation by the 3P0 model [15–21].

Further, we will obtain the information of the order ofmagnitude of the strong decay modes of DsJð3040Þ.

The paper is organized as follows. In Sec. II, we brieflyreview the 3P0 model and present the formulation of the

strong decays of P-wave charmed-strange mesons with theradial excitation. Finally, the numerical result will beshown. Section III is a short summary.

II. THE STRONG DECAY OF P-WAVE CHARMED-STRANGE MESONS WITH THE RADIAL

EXCITATION

Before illustrating the strong decay of P-wave charmed-strange mesons with the radial excitation, we first intro-duce the category of the heavy flavor meson.

In the heavy quark limitmQ ! 1, the heavy quark plays

the role of a static color source to interact with the light partwithin the heavy flavor hadron. Thus, the spin of the heavyquark ~sQ can be separated from the total angular momen-

tum J of the heavy flavor hadron. Furthermore, ~j‘ ¼ ~sq þ~L is a good quantum number, where ~sq and ~L denote the

spin of the light part of the heavy flavor hadron and theorbital angular momentum between the light part and theheavy quark, respectively.

Thus, the heavy mesons can be grouped into doubletsaccording to jP‘ , which include the j‘ ¼ 1

2� doublet

ð0�; 1�Þ with the orbital angular momentum L ¼ 0, thej‘ ¼ 1

2þ doublet ð0þ; 1þÞ, and the 3

2þ doublet ð1þ; 2þÞ with

L ¼ 1. For L ¼ 2 there exist ð1�; 2�Þ and ð2�; 3�Þ dou-blets with jP‘ ¼ 3

2� and 5

2�, respectively. As shown in Fig. 1,

the states existing in the doublets ð0�; 1�Þ, ð0þ; 1þÞ, andð1þ; 2þÞ are already filled with the observed charmed-strange mesons. Two 1þ states existing as S and T arethe mixture of two basis states 11P1 and 13P1 [22,23]

j1þ; jPl ¼ 12þi

j1þ; jPl ¼ 32þi

!¼ cos� sin�

� sin� cos�

� � j11P1ij13P1i

� �;

where one takes the mixing angle � ¼ �tan�1ffiffiffi2

p ¼�54:7� according to the estimate in the heavy quark limit.

For P-wave charmed-strange mesons with the radialexcitation discussed in this work, one also categorizesthem as S ¼ ð0þ; 1þÞ and T ¼ ð1þ; 2þÞ doublets accord-ing to the above approach. Two 1þ states are the mixture oftwo basis states 21P1 and 23P1, which satisfy the relation

j1þ; jPl ¼ 12þi

j1þ; jPl ¼ 32þi

!¼ cos�0 sin�0

� sin�0 cos�0� � j21P1i

j23P1i� �

:

In this work, we approximately take �0 ¼ � ¼ �54:7�.For distinguishing P-wave states with and without the

first radial excitation, one labels four P-wave states withoutthe first radial excitation as 0þðSÞ, 1þðSÞ, 1þðTÞ, and2þðTÞ. Four P-wave states with the first radial excitationare named as 0þðS?Þ, 1þðS?Þ, 1þðT?Þ, and 2þðT?Þ.If we set the upper limit of the masses of P-wave states

with the first radial excitation as 3.04 GeV, the kinemati-cally allowed decay modes of P-wave states with the firstradial excitation are listed in Table I. In the following, the3P0 model will be applied to calculate these strong decays

in Table I.

A. A review of the QPC model

The 3P0 model, also known as the quark pair creation

(QPC) model, was first proposed by Micu in Ref. [15] tocalculate Okubo-Zweig-Iizuka (OZI) allowed strong de-cays of a meson. Later, this model was developed by theother theoretical groups [16–21] and is successful whenapplied extensively to the calculation of the strong decay ofhadron [6,22–33].In the QPC model, the heavy flavor meson decay occurs

through a quark-antiquark pair production from the vac-uum, which is of the quantum number of the vacuum, i.e.0þþ [15,16]. For describing a strong decay process of thecharmed-strange meson Aðcð1Þ �sð2ÞÞ ! Bðcð1Þ �qð3ÞÞ þCð �sð2Þqð4ÞÞ, one writes out the S-matrix

hBCjSjAi ¼ I� i2��ðEf � EiÞhBCjTjAi: (1)

In the nonrelativistic limit, the transition operator T isdepicted as

T ¼ �3�Xm

h1m; 1�mj00iZ

dk3dk4�3ðk3 þ k4Þ

�Y1m

�k3 � k4

2

��341;�m’

340 !34

0 dy3iðk3Þby4jðk4Þ; (2)

where i and j denote the SUð3Þ color indices of the createdquark and antiquark. ’34

0 ¼ ðu �uþ d �dþ s�sÞ= ffiffiffi3

pand

!340 ¼ 1ffiffi

3p ��3�4

(� ¼ 1, 2, 3) correspond to flavor and

color singlets, respectively. �341;�m is a triplet state of

spin. Y‘mðkÞ � jkj‘Y‘mð�k;�kÞ is the ‘th solid harmonicpolynomial. � is a dimensionless constant which repre-sents the strength of the quark pair creation from thevacuum and can be extracted by fitting the data.For the convenience of the calculation, one usually takes

the mock state to depict the meson [34]

ZHI-FENG SUN AND XIANG LIU PHYSICAL REVIEW D 80, 074037 (2009)

074037-2

jAðn2Sþ1LJMJÞðKÞi ¼ ffiffiffiffiffiffi

2Ep X

ML;MS

hLMLSMSjJMJiZ

dk1dk2�3ðK� k1 � k2Þ�nLML

ðk1;k2Þ�12SMS

’12!12jq1ðk1Þ �q2ðk2Þi;

(3)

which satisfies the normalization conditions hAðKÞjAðK0Þi ¼ 2E�3ðK�K0Þ. Here, �nLMLðk1;k2Þ is the spatial wave

function describing the meson.Taking the center of the mass frame of the meson A, KA ¼ 0 and KB ¼ �KC ¼ K, further we obtain a general

expression of Eq. (1)

hBCjTjAi ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8EAEBEC

p�

XMLA

;MSA;MLB

;MSB;MLC

;MSC;m

h1m; 1�mj00ihLAMLASAMSA jJAMJAihLBMLB

SBMSB jJBMJBi

� hLCMLCSCMSC jJCMJCih’13

B ’24C j’12

A ’340 ih�13

SBMSB�24SCMSC

j�12SAMSA

�341�miI

MLA;m

MLB;MLC

ðKÞ: (4)

The color matrix element h!13B !24

C j!12A !34

0 i ¼ 1=3, which cancels out the factor 3 before � in Eq. (2). h’13B ’24

C j’12A ’34

0 iand h�13

SBMSB�24SCMSC

j�12SAMSA

�341�mi are the flavor matrix element and the spin matrix element, respectively. Here, the spatial

TABLE I. The relevant strong decay modes of P-wave charmed-strange mesons with the firstradial excitation allowed by the conservation of the quantum number. Here � denotes that thedecay modes are kinematically forbidden if setting the upper limit of the masses of P-wave stateswith the first radial excitation as 3.04 GeV. Since the 1þ state in the ð1þ; 2þÞ doublet decays intoD�� via a D wave, it is very narrow and denoted as D1ð2420Þ [1]. The 1þ state in the ð0þ; 1þÞdoublet decays into D�� via the S wave. Hence, it is very broad and denoted as D1ð2430Þ [1].State Decay modes Decay channels

0� þ 0� DþK0, D0Kþ, Dþs �

ð0Þ1� þ 1� D�þK�0, D�0K�þ0þ þ 1� �

0þðS?Þ 1þðSÞ þ 0� D1ð2430Þ0Kþ, D1ð2430ÞþK0, Ds1ð2460Þþ�1þðSÞ þ 1� �1þðTÞ þ 0� D1ð2420ÞþK0, D1ð2420Þ0Kþ1þðTÞ þ 1� �2þ þ 1� �0� þ 1� DþK�0, D0K�þ, Dþ

s �1� þ 0� D�þK0, D�0Kþ, D�þ

s �1� þ 1� D�þK�0, D�0K�þ0þ þ 0� D�

0ð2400ÞþK0, D�0ð2400Þ0Kþ, D�

s0ð2317Þþ�1þðS?Þ=1þðT?Þ 0þ þ 1� �

1þðSÞ þ 0� D1ð2430Þ0Kþ, D1ð2430ÞþK0, Ds1ð2460Þþ�1þðTÞ þ 0� D1ð2420ÞþK0, D1ð2420Þ0Kþ1þðSÞ þ 1� �1þðTÞ þ 1� �2þ þ 0� D�

2ð2460ÞþK0, D�2ð2460Þ0Kþ

2þ þ 1� �0� þ 0� DþK0, D0Kþ, Dþ

s �ð0Þ

0� þ 1� DþK�0, D0K�þ, Dþs �

1� þ 0� D�þK0, D�0Kþ, D�þs �

1� þ 1� D�þK�0, D�0K�þ0þ þ 1� �

2þðT?Þ 1þðSÞ þ 0� D1ð2430Þ0Kþ, D1ð2430ÞþK0, Ds1ð2460Þþ�1þðSÞ þ 1� �1þðTÞ þ 0� D1ð2420ÞþK0, D1ð2420Þ0Kþ1þðTÞ þ 1� �2þ þ 0� D�

2ð2460ÞþK0, D�2ð2460Þ0Kþ

2þ þ 1� �

NEWLY OBSERVED DsJð3040Þ AND THE RADIAL . . . PHYSICAL REVIEW D 80, 074037 (2009)

074037-3

TABLE II. The expression of the partial wave amplitude for the strong decays of P-wave states with the first radial excitation. Here� ¼ 2=

ffiffiffiffiffiffi18

p,�1=3 are for the strong decay involved in � and �0 mesons, respectively, while � ¼ 1=

ffiffiffi3

pis for the other strong decays,

which are the result from the flavor matrix element.

State Decay channel Partial wave amplitude

0� þ 0� M00 ¼ �ffiffi2

p3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

p�½2I� � I0�

1� þ 1� M00 ¼ �ffiffi2

p3ffiffi3

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

p�½I0 � 2I��

0þðS?Þ 1þðSÞ þ 0� M11 ¼ cos�½� �ffiffi2

p3


p�ð2I1�1

00 � I0000Þ� þ sin�½� 2�3


p�ðI1010 � I0110Þ�

1þðTÞ þ 0� M11 ¼ � sin�½� �ffiffi2

p3


p�ð2I1�1

00 � I0000Þ� þ cos�½� 2�3


p�ðI1010 � I0110Þ�

0� þ 1� M10 ¼ cos�0½�3ffiffi23

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC

p�ð2I� � I0Þ� þ sin�0½� 2�

3ffiffi3


p�ð2I� � I0Þ�

M12 ¼ cos�0½ �3ffiffi3


p�ð2I� þ 2I0Þ� þ sin�0½ 2�

3ffiffi6


p�ðI� þ I0Þ�

1� þ 0� M10 ¼ cos�0½�3ffiffi23


p�ð2I� � I0Þ� þ sin�0½� 2�

3ffiffi3


p�ð2I� � I0Þ�

1þðS?Þ M12 ¼ cos�0½ �3ffiffi3


p�ð2I� þ 2I0Þ� þ sin�0½ 2�

3ffiffi6



1� þ 1� M10 ¼ cos�0½� 2�3

ffiffi13


p�ð2I� � I0Þ�

M22 ¼ sin�0½2�3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC


0þ þ 0� M01 ¼ cos�0½�3ffiffi23


p�ðI0000 þ 2I0110Þ� þ sin�0½ 2�

3ffiffi3


p�ðI1�1

00 þ I1010Þ�1þðSÞ þ 0� M11 ¼ sin� cos�0½ �

3ffiffi2


p�ð2I1�1

00 þ 2I1010Þ� þ cos� sin�0½� �ffiffi2

p3


p�ðI1010 � I0110Þ�

þ sin� sin�0½� �3


p�ð�I0000 � I0110 þ I1�1

00 Þ�1þðTÞ þ 0� M11 ¼ cos� cos�0½ �

3ffiffi2


p�ð2I1�1

00 þ 2I1010Þ� � sin� sin�0½� �ffiffi2

p3


p�ðI1010 � I0110Þ�

þ cos� sin�0½� �3


p�ð�I0000 � I0110 þ I1�1

00 Þ�2þ þ 0� M21 ¼ cos�0½ �

3ffiffiffiffi30


p�ð4I0000 � 4I0110 � 6I1�1

00 þ 6I1010Þ�þ sin�0½� �

3ffiffiffiffi15


p�ð3I0000 � 3I0110 � 7I1�1

00 þ 2I1010Þ�

0� þ 1� M10 ¼ � sin�0½�3ffiffi23


p�ð2I� � I0Þ� þ cos�0½� 2�

3ffiffi3


p�ð2I� � I0Þ�

M12 ¼ � sin�0½ �3ffiffi3


p�ð2I� þ 2I0Þ� þ cos�0½ 2�

3ffiffi6



1� þ 0� M10 ¼ � sin�0½�3ffiffi23


p�ð2I� � I0Þ� þ cos�0½� 2�

3ffiffi3


p�ð2I� � I0Þ�

1þðT?Þ M12 ¼ � sin�0½ �3ffiffi3


p�ð2I� þ 2I0Þ� þ cos�0½ 2�

3ffiffi6



1� þ 1� M10 ¼ � sin�0½� 2�3

ffiffi13


p�ð2I� � I0Þ�

M22 ¼ cos�0½2�3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEAEBEC


0þ þ 0� M01 ¼ � sin�0½�3ffiffi23


p�ðI0000 þ 2I0110Þ� þ cos�0½ 2�

3ffiffi3


p�ðI1�1

00 þ I1010Þ�1þðSÞ þ 0� M11 ¼ � sin� sin�0½ �

3ffiffi2


p�ð2I1�1

00 þ 2I1010Þ� þ cos� cos�0½� �ffiffi2

p3


p�ðI1010 � I0110Þ�

þ sin� cos�0½� �3


p�ð�I0000 � I0110 þ I1�1

00 Þ�1þðTÞ þ 0� M11 ¼ � cos� sin�0½ �

3ffiffi2


p�ð2I1�1

00 þ 2I1010Þ� � sin� cos�0½� �ffiffi2

p3


p�ðI1010 � I0110Þ�

þ cos� cos�0½� �3


p�ð�I0000 � I0110 þ I1�1

00 Þ�2þ þ 0� M21 ¼ � sin�0½ �

3ffiffiffiffi30


p�ð4I0000 � 4I0110 � 6I1�1

00 þ 6I1010Þ�þ cos�0½� �

3ffiffiffiffi15


p�ð3I0000 � 3I0110 � 7I1�1

00 þ 2I1010Þ�

0� þ 0� M02 ¼ 2�3ffiffi5


p�½I� þ I0�

0� þ 1� M12 ¼ 2�ffiffiffiffi30


p�½I� þ I0�

1� þ 0� M12 ¼ 2�ffiffiffiffi30


p�½I� þ I0�

2þðT?Þ 1� þ 1� M20 ¼ 2�3

ffiffi23


p�½2I� � I0�

1þðSÞ þ 0� M11 ¼ cos�½ �15ffiffi2


p�ð4I0000 þ 6I0110 þ 4I1�1

00 þ 6I1010Þ�þ sin�½�15


p�ð3I0000 þ 7I0110 þ 3I1�1

00 þ 2I1010Þ�1þðTÞ þ 0� M11 ¼ � sin�½ �

15ffiffi2


p�ð4I0000 þ 6I0110 þ 4I1�1

00 þ 6I1010Þ�þ cos�½�15


p�ð3I0000 þ 7I0110 þ 3I1�1

00 þ 2I1010Þ�2þ þ 0� M21 ¼ �

5ffiffi3


p�½I0000 � I0110 þ I1�1

00 þ 4I1010�


074037-4

integral IMLA

;m

MLB;MLC

ðKÞ is

IMLA

;m

MLB;MLC

ðKÞ ¼Z

dk1dk2dk3dk4�3ðk1 þ k2Þ�3ðk3 þ k4Þ�2ðKB � k1 � k3Þ�3ðKC � k2 � k4Þ��

nBLBMLBðk1;k3Þ

��nCLCMLC

ðk2;k4Þ�nALAMLAðk1;k2ÞY1m

�k3 � k4

2

�; (5)

which describes the overlap of the initial meson (A) and thecreated pair with the two final mesons (B and C).

In this work, we use the simple harmonic oscillator (HO)wave function to represent the radial portions of the mesonspace wave function. The wave functions corresponding tothe states with nL ¼ 1S, 1P, and 2P are, respectively,

c n¼1;L¼0ðkÞ ¼ R3=2

�3=4exp

��R2k2

2

�; (6)

c n¼1;L¼1ðkÞ ¼ �i2

ffiffiffi2

3

sR5=2

�1=4Ym

1 ðkÞ exp��R2k2

2

�; (7)

c n¼2;L¼1ðkÞ ¼ i2R5=2ffiffiffiffiffiffi15

p�1=4

ð5� 2k2R2ÞYm1 ðkÞ

� exp

��R2k2

2

�; (8)

which satisfy the normalizationRc �

n;LðkÞc n;LðkÞdk ¼ 1.

Here the solid harmonic polynomialYm1 ðkÞ ¼

ffiffiffiffiffi34�

q��m � k

with �1 ¼ ð�1=ffiffiffi2

p;�i=

ffiffiffi2

p; 0Þ and 0 ¼ ð0; 0; 1Þ. k ¼

ðmikj �mjkiÞ=ðmi þmjÞ is the relative momentum be-

tween the quark and the antiquark within a meson whenconsidering the quark mass difference. These HO wavefunctions are relevant to the calculation of the strong decayof P-wave states with the first radial excitation.

The helicity amplitude satisfies the relation

hBCjTjAi ¼ �3ðKB þKC �KAÞMMJAMJB

MJC : (9)

In terms of the partial wave amplitude, one obtains thepartial decay width

� ¼ �2 jKjM2

A

XJL

jMJLj2; (10)

where jKj denotes the three momentum of the daughtermesons in the parent’s center of mass frame. The partialwave amplitude MJL is related to the helicity amplitude

MMJAMJB

MJC via the Jacob-Wick formula [35]

MJLðA! BCÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Lþ 1

p2JAþ 1

XMJB

;MJC

hL0JMJA jJAMJAi

� hJBMJBJCMJC jJMJAiMMJAMJB

MJC ðKÞ;(11)

where J ¼ JB þ JC and JA þ JP ¼ JB þ JC þL. The cal-culation of transition amplitude using the 3P0 model in-

volves two parameters: the strength of the quark paircreation from vacuum � and R in the harmonic oscillatorwave function. � is a universal parameter which was al-ready fixed from other channels as indicated in Ref. [36].The value of R is chosen to reproduce the root mean square(rms) radius obtained by solving the Schrodinger equationwith the linear potential.

B. The partial wave amplitude of two-body strongdecays of P-wave states with the first radial excitation

With the preparation mentioned above, we obtain thepartial wave amplitude of the strong decays of the P-wavestates with the first radial excitation, which are listed inTable I. In Table II, the concrete expression of the partialwave amplitude is given. The details of the spatial integral

IMLA

;m

MLB;MLC

ðKÞ are given in the Appendix.

C. Numerical result

The input parameters involved in the 3P0 model include

the strength of quark pair creation from the vacuum, the Rvalue in the HO wave function and the mass of the meson.One takes the strength of the quark pair creation from the

vacuum as � ¼ 6:3 [36], which isffiffiffiffiffiffiffiffiffi96�

ptimes larger than

TABLE III. The R value in the HO wave function and the massrelevant to the strong decays listed in Table I. Here (� ) and (0)denote the charge of the meson.

State Mass (MeV) [1] R (GeV�1) [36]

D 1869:62 ð�Þ 1864.84 (0) 1.52

Ds 1968:49 ð�Þ 1.41

D� 2010:27 ð�Þ 2006.97 (0) 1.85

D�s 2112:3 ð�Þ 1.69

D�0ð2400Þ 2403 ð�Þ 2352 (0) 1.85

D�s0ð2317Þ 2317:8 ð�Þ 1.75

Ds1ð2460Þ 2459:6 ð�Þ 1.92

D1ð2430Þ 2427 ð�Þ 2427 (0) 2.00

D1ð2420Þ 2423:4 ð�Þ 2422.3 (0) 2.00

D�2ð2460Þ 2460:1 ð�Þ 2461.1 (0) 2.22

K 493:677 ð�Þ 497.614 (0) 1.41

K� 891:66 ð�Þ 896.00 (0) 2.08

� 547.853 1.41

�0 957.66 1.41

� 1019.455 2.08


074037-5

that adopted by the other theoretical groups [31,37]. The

strength of s�s creation satisfies �s ¼ �=ffiffiffi3

p[17]. By re-

producing the realistic rms radius by solving theSchrodinger equation with the linear potential, one can

obtain the value of R in the HO wave function [36]. Themass and the R value used in this work are shown inTable III.If the mass of the charmed-strange meson with 0þðS?Þ is

2.837 GeV predicted in Ref. [14], there only exists thedecay channel 0þðS?Þ ! 0� þ 0�, which is allowed bythe phase space. In Fig. 2, we give the dependence of thepartial decay widths of the strong decay of 0þðS?Þ state onthe RA. Here, RA is the R value of the HO wave function ofthe charmed-strange state with 0þðS?Þ. The minimum ofthe decay width around RA ¼ 1:7 GeV�1 in Fig. 2 is due tothe node in the radial wave function of 0þðS?Þ. The totaldecay width of the 0þðS?Þ charmed-strange meson is108 MeV with RA ¼ 2:8 GeV�1.In this work, we take the masses of 1þðS?Þ and 1þðT?Þ

charmed-strange mesons as 3.044 GeV, which is the ex-perimental value of the mass of DsJð3040Þþ. Then, wecalculate the decay of DsJð3040Þþ under the two assump-tions 1þðS?Þ and 1þðT?Þ. In Figs. 3 and 4, we present thenumerical results of the two charmed-strange mesons1þðS?Þ and 1þðT?Þ. The dependence of the total decaywidth of 1þðSÞ on the RA is shown in Fig. 5. Here, RA

1.5 2 2.5 3 3.5 4

RA GeV 1

0

20

40

60

80

100

120M

eV

D K0

D0 K

Ds η

All

FIG. 2 (color online). The variation of the strong decay mode0þðS?Þ ! 0� þ 0� with RA.

1.5 2 2.5 3 3.5 4

R A GeV 1

0

20

40

60

80

100

120

MeV

a

D K 0

D0 K

Ds φ

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

20

40

60

80

MeV

b

D K0

D 0 KDs η

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

10

20

30

40

MeV

c

D K 0

D 0 K

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

2

4

6

8

MeV

d

D0 2400 K0

D0 2400 0 K

Ds0 2317 η

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

2

4

6

8

10

12

14

MeV

e

D1 2430 K0

D1 2430 0 K

Ds1 2460 η

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

2

4

6

8

MeV

f

D1 2420 K0

D1 2420 0 K

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

10

20

30

40

MeV

g

D2 2460 K0

D2 2460 0 K

total

FIG. 3 (color online). The variation of the strong decays for (a) 1þðS?Þ ! 0� þ 1�, (b) 1þðS?Þ ! 1� þ 0�, (c) 1þðS?Þ ! 1� þ 1�,(d) 1þðS?Þ ! 0þ þ 0�, (e) 1þðS?Þ ! 1þðSÞ þ 0�, (f) 1þðS?Þ ! 1þðTÞ þ 0�, and (g) 1þðS?Þ ! 2þ þ 0� with the factor RA of theHO wave function of 1þðS?Þ. Here the total partial decay width is labeled by ‘‘total’’ in the diagrams.


074037-6

denotes the R value in the HO wave function ofDsJð3040Þþ. By comparing the calculated total decaywidth of DsJð3040Þþ with that of the BABAR data, onefinds that the total decay width ( 204 MeV) ofDsJð3040Þþ obtained by the 3P0 model reaches up to the

lower limit of the experimental width of DsJð3040Þþ whentaking RA as 2:8 GeV�1. With increasing the RA up to3:5 GeV�1, the total decay width is close to the centralvalue of the experimental width of DsJð3040Þþ. Thus,studying the decay width of DsJð3040Þþ under the1þðS?Þ assignment shows that the first radial excitationof the P-wave charmed-strange meson to DsJð3040Þþ, i.e.1þðS?Þ, is suitable.

The result of the partial decay widths of the 1þðS?Þcharmed-strange meson corresponding to RA ¼2:8 GeV�1 (see Fig. 3) indicates that 0� þ 1� (DþK�0,D0K�þ, and Dþ

s �) and 1� þ 0� (D�þK0, D�0Kþ, andD�þ

s �) are the dominant decay modes of DsJð3040Þþ,which can further explain why DsJð3040Þþ was first ob-served in the D�K decay channel. An experimental searchof DsJð3040Þþ in the 0� þ 1� channel (DþK�0, D0K�þ,and Dþ

s �) is encouraged in terms of the ratio

�ð1þðS?Þ ! 0� þ 1�Þ�ð1þðS?Þ ! 1� þ 0�Þ 0:79

1.5 2 2.5 3 3.5 4

RA GeV 1

0

5

10

15

20

25

30

35

MeV

a

D K 0

D0 K

Ds φ

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

10

20

30

40

MeV

b

D K0

D 0 K

Ds η

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

10

20

30

40

50

60

MeV

c

D K 0

D 0 K

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

2

4

6

8

10

12

14

MeV

d

D0 2400 K0

D0 2400 0 KDs0 2317 η

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

2

4

6

8

MeV

e

D1 2430 K0

D1 2430 0 K

Ds1 2460 η

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

0.5

1

1.5

2

2.5

3

3.5

MeV

f

D1 2420 K0

D1 2420 0 K

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

0.2

0.4

0.6

0.8

MeV

g

D2 2460 K0

D2 2460 0 K

total

FIG. 4 (color online). The variation of the strong decays for (a) 1þðT?Þ ! 0� þ 1�, (b) 1þðT?Þ ! 1� þ 0�,(c) 1þðT?Þ ! 1� þ 1�, (d) 1þðT?Þ ! 0þ þ 0�, (e) 1þðT?Þ ! 1þðSÞ þ 0�, (f) 1þðT?Þ ! 1þðTÞ þ 0�, and(g) 1þðT?Þ ! 2þ þ 0� with the factor RA of the HO wave function of 1þðT?Þ. Here the total partial decay width is labeled by‘‘total’’ in the diagrams.

2 2.5 3 3.5 4

RA GeV 1

125

150

175

200

225

250

275

300

MeV

All

Babar data

FIG. 5 (color online). A comparison of the total decay width of1þðS?Þ with BABAR data. Here the straight red line and theshaded yellow band are the central value for the error of the totalwidth of DsJð3040Þ measured by BABAR.


074037-7

corresponding to RA ¼ 2:8 GeV�1.Under the assignment of 1þðT?Þ to DsJð3040Þ, we can

obtain the variation of the strong decays for 1þðT?Þ !0� þ 1�, 1� þ 0�, 1� þ 1�, 0þ þ 0�, 1þðSÞ þ 0�,1þðTÞ þ 0�, and 2þ þ 0� with the factor RA [R value of

the HO wave function of 1þðT?Þ] which is depicted inFig. 4. Furthermore, the dependence of the total decaywidth on the RA value is listed in Fig. 6. The total decaywidth of DsJð3040Þþ is about 33.8 MeV with RA ¼2:8 GeV�1, which is consistent with our knowledge, i.e.,

1.5 2 2.5 3 3.5 4

RA GeV 1

25

50

75

100

125

150

175

tota

lM

eV

tota

lM

eV

1 T

1.5 2 2.5 3 3.5 4

RA GeV 1

25

50

75

100

125

150

175

2002 T

FIG. 6 (color online). The dependence of the total decay width of 1þðT?Þ and 2þðT?Þ states on RA.

1.5 2 2.5 3 3.5 4

RA GeV 1

0

5

10

15

20

25

MeV

a

D K0

D0 K

Ds ηDs η '

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

5

10

15

20

25

30

MeV

b

D K 0

D0 K

Ds φ

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

5

10

15

20

25

30

35

MeV

c

D K0

D 0 K

Ds η

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

10

20

30

40

50

60

70

MeV

d

D K 0

D 0 K

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

0.5

1

1.5

MeV

e

D1 2430 K0

D1 2430 0 KDs1 2460 η

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

5

10

15

MeV

f

D1 2420 K0

D1 2420 0 K

total

1.5 2 2.5 3 3.5 4

RA GeV 1

0

2

4

6

8

10

MeV

g

D2 2460 K0

D2 2460 0 K

total

FIG. 7 (color online). The variation of the strong decays for (a) 2þðT?Þ ! 0� þ 0�, (b) 2þðT?Þ ! 0� þ 1�,(c) 2þðT?Þ ! 1� þ 0�, (d) 2þðT?Þ ! 1� þ 1�, (e) 2þðT?Þ ! 1þðSÞ þ 0�, (f) 2þðT?Þ ! 1þðTÞ þ 0�, and(g) 2þðT?Þ ! 2þ þ 0� with the factor RA of the HO wave function of 2þðT?Þ. Here the total partial decay width is labeled by‘‘total’’ in the diagrams.


074037-8

the 1þ state existing T doublet is of a narrow width. In fact,the result of the decay of the 1þðT?Þ state further indicatesthat DsJð3040Þþ cannot be explained as the 1þðT?Þcharmed-strange meson.

One also predicts that the partial decay widths corre-sponding to the decay channels 1þðT?Þ ! 0� þ 1�, 1� þ0�, 1� þ 1�, 0þ þ 0�, 1þðSÞ þ 0�, 1þðTÞ þ 0�, and2þ þ 0� are 9:8� 10�3, 6.3, 13.0, 10.1, 9:9� 10�1, 3.5,and 1:3� 10�1 MeV, respectively. These numerical re-sults show that 1� þ 0�, 1� þ 1�, 0þ þ 0�, and 1þðTÞ þ0� channels are important when searching the 1þðT?Þ statein experiment.

The dependence of the strong decays 2þðT?Þ ! 0� þ0�, 0� þ 1�, 1� þ 0�, 1� þ 1�, 1þðSÞ þ 0�, 1þðTÞ þ0�, and 2þ þ 0� on the factor RA [R value of the HO wavefunction of 2þðT?Þ] is given in Fig. 7. Here, we take themass of 2þðT?Þ as 3.157 GeV [14]. When taking RA ¼2:8 GeV�1, the total decay width of 2þðT?Þ is 87.9 MeV(see Fig. 6), and the partial decay widths (see Fig. 7),respectively, corresponding to 2þðT?Þ ! 0� þ 0�, 0� þ1�, 1� þ 0�, 1� þ 1�, 1þðSÞ þ 0�, 1þðTÞ þ 0�, and2þ þ 0� are 15.6, 0.49, 7.2, 49.3, 1.8, 13.2, and0.28 MeV, which show that 1� þ 1�, 0� þ 0�, 1þðTÞ þ0�, and 1� þ 0� are key decay channels to find the 2þðT?Þcharmed-strange meson.

III. SUMMARY

Stimulated by the newly observed charmed-strange me-son DsJð3040Þþ, we systemically study the two-bodystrong decays of P-wave charmed-strange mesons withthe first radial excitation.

Our numerical results show that DsJð3040Þþ can becategorized as a 1þ state in the S ¼ ð0þ; 1þÞ doubletwell, i.e. DsJð3040Þþ is the first radial excitation ofDs1ð2460Þþ. We suggest experimentalist to searchDsJð3040Þþ by the 0� þ 1� channel (DþK�0, D0K�þ,and Dþ

s �).In the past six years, BABAR and Belle experiments have

made big progress in searching for charmed-strange me-sons, which lets us believe that more charmed-strangemesons will be found in future experiments. IfDsJð3040Þþ is the first radial excitation of Ds1ð2460Þþ,there must exist three partners of DsJð3040Þþ, which arethe rest three P-wave charmed-strange mesons with thefirst radial excitation. In this work, we also study the strongdecays of the rest three P-wave charmed-strange mesonswith the first radial excitation. Our numerical result (seeSec. II C) will be helpful to instruct future experimentalsearch of the remaining three P-wave charmed-strangemesons with the first radial excitation.

ACKNOWLEDGMENTS

We are grateful to Professor Hai-Yang Cheng for sug-gestions and discussion. This project is supported by the

National Natural Science Foundation of China under GrantNo. 10705001 and A Foundation for the Author ofNational Excellent Doctoral Dissertation of the People’sRepublic of China (FANEDD).Note added.—When this manuscript was completed, a

work of DsJð3040Þþ appeared [38]. In this work, authorsinvestigated the Ds mesons by a semiclassic flux tubemodel and explained DsJð3040Þþ as 1þðjP ¼ 1

2þÞ. In our

case, we calculated the strong decays of DsJð3040Þþ withthe assignment of the first radial excitation of Ds1ð2460Þ.By comparing the total decay width of DsJð3040Þþ ob-tained by the 3P0 model with the BABAR data, we con-

cluded that DsJð3040Þþ is the first radial excitation ofDs1ð2460Þ, which is consistent with the conclusion of thestructure of DsJð3040Þþ [38].

APPENDIX

According to the spatial integral in Eq. (5), one cancategorize the strong decays of P-wave charmed-strangemesons with the first radial excitation into two groups:2P ! 1Sþ 1S and 2P ! 1Pþ 1S.For the case of 2P ! 1Sþ 1S, the spatial integral

IMLA

;m

MLB;MLC

is simplified as Im0n0 due to MLB¼ MLC

¼ 0.

According to the constraint from Eq. (11), we take thedirection of K along the z axis: K ¼ ð0; 0; jKjÞ. In thefollowing, we present the result of the spatial integral of2P ! 1Sþ 1S listed in Table II:

I� ¼ I1�1 ¼ I�11 ¼ 2ffiffiffi2

p�3=2!1

�

R5þ 5

R7

�; (A1)

I0 ¼ I00

¼ 2ffiffiffi2

p�3=2!1

��ð1� �ÞjKj2

R3�

R5

þ 2�ð1� 2�ÞjKj2R5

þ 3�ð1��ÞjKj2R5

� 5

R7

�:

(A2)

The spatial integrals for 2P ! 1Pþ 1S, which are in-volved in the expressions shown in Table II, include

I0000 ¼ �2ffiffiffi2

p�3=2!2

��ð��Þð�� 1ÞjKj3

R3

þ jKjR5

½ð3�� 1Þ þ 2�2ð��ÞjKj2

þ 2�2ð�� 1ÞjKj2 þ 2�ð��Þð�� 1ÞjKj2

þ 3�ð��Þð�� 1ÞjKj2� þ 5ð3�� 1ÞjKjR7

þ 6�jKjR7

�; (A3)


074037-9

I0110 ¼ I0�1�10 ¼ �2

ffiffiffi2

p�3=2!2�jKj

�

R5þ 7

R7

�; (A4)

I1010 ¼ I�10�10

¼ �2ffiffiffi2

p�3=2!2

�ð�� 1ÞjKj

R5þ ð7�� 5ÞjKj

R7

�;

(A5)

I1�100 ¼ I�11

00

¼ 2ffiffiffi2

p�3=2!2

�ð��ÞjKj

R5þ ð7�� 5�ÞjKj

R7

�:

(A6)

Here,

R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2A þ R2

B þ R2C

q; � ¼ m1

m1 þm3

;

� ¼ m2

m2 þm4

; � ¼ R2B�þ R2

C�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2A þ R2

B þ R2C

q ;

2 ¼ R2B�

2 þ R2C�

2 � �2; � ¼ �

R;

¼ � 5� 2R2A�

2jKj22R2

A

;

!1 ¼ � 3iR3AðRARBRCÞ3=2ffiffiffiffiffiffi15

p�11=4

exp

�� 1

2 2jKj2

�;

!2 ¼ffiffiffi6

pR2AðRARBRCÞ5=2ffiffiffi5

p�11=4RC

exp

�� 1

2 2jKj2

�:

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