Towards the assignment for the 41S0 meson nonet

De-Min Li* and Shan Zhou

Department of Physics, Zhengzhou University, Zhengzhou, Henan 450052, People’s Republic of China(Received 20 May 2008; published 15 September 2008)

The strong decays of the �ð2070Þ, �ð2010Þ, �ð2100Þ, �ð2190Þ, and �ð2225Þ as the 41S0 quark-

antiquark states are investigated in the framework of the 3P0 meson decay model. It is found that the

�ð2070Þ, �ð2100Þ, and �ð2225Þ appear to be the convincing 41S0 q �q states, while the assignment of the

�ð2010Þ and �ð2190Þ as the 41S0 isoscalar states is not favored by their widths. In the presence of the

�ð2070Þ, �ð2100Þ, and �ð2225Þ being the members of the 41S0 meson nonet, the mass of the 41S0 kaon is

phenomenologically determined to be about 2153 MeV. The width of this unobserved kaon is expected to

be about 197 MeV in the 3P0 decay model.

DOI: 10.1103/PhysRevD.78.054013 PACS numbers: 14.40.Cs, 12.39.Jh

I. INTRODUCTION

From PDG2006 [1], the 11S0 meson nonet (�, �, �0, andK) as well as the 21S0 members [�ð1300Þ, �ð1295Þ, and�ð1475Þ] have been well established. In Ref. [2], wesuggested that the �ð1800Þ, Kð1830Þ, together with theXð1835Þ and �ð1760Þ observed by the BES Collaboration[3,4], constitute the 31S0 meson nonet. More recently, we

argued that the �ð2225Þ with a mass of ð2240þ30þ30�20�20Þ MeV

and a width of ð190� 30þ40�60Þ MeV observed by the BES

Collaboration [5] could be the 41S0 s�s in Ref. [6], where

the other members of the 41S0 meson nonet were not

discussed. In the present work, we shall address the pos-sible assignment for the 41S0 meson nonet.

With the assignment of the �ð2225Þ as the s�smember ofthe 41S0 meson nonet, one can expect that both the iso-

vector and other isoscalar members of the 41S0 meson

nonet should be lighter than the �ð2225Þ. Experimentally,in the mass region 2000–2225MeV, the pseudoscalar states�ð2070Þ (mass: 2070� 35 MeV; width: 310þ100

�50 MeV),�ð2010Þ (mass: 2010þ35

�60 MeV; width: 270� 60 MeV),�ð2100Þ (mass: 2103� 50 MeV; width: 187� 75 MeV),and �ð2190Þ (mass: 2190� 50 MeV; width: 850�100 MeV) are reported [1]. Theoretically, some predictedvalues for the �ð41S0Þ mass are 2.15 GeV by QCD sum

rules [7,8], 2.009 GeV by the spectrum integral equation[9], 2.193 GeV by a covariant quark model [10], 2.039 GeVby a relativistic independent quark model [11], and2.07 GeV by Regge phenomenology [12]. In addition, themass of the third radial excitation of the� is predicted to beabout 2.267 GeV by a covariant quark model [10] or2.1 GeV by Regge phenomenology [12]. The �ð2070Þmass is similar to the predicted �ð41S0Þ mass, and all the

masses of the �ð2010Þ, �ð2100Þ, and �ð2190Þ are close tothe predicted mass range of the third radial excitation of the�. Only the mass information of these states is insufficientto classify them. The main purpose of this work is to

discuss whether these reported pseudoscalar states can beassigned as the members of the 41S0 meson nonet or not by

investigating their decay properties in the 3P0 meson decay

model.The organization of this paper is as follows. In Sec. II,

we give a brief review of the 3P0 decay model (for a

detailed review, see e.g. Refs. [13–16].) In Secs. III andIV, the decay widths of the �ð2070Þ, �ð2010Þ, �ð2100Þ,�ð2190Þ, and �ð2225Þ as the 41S0 q �q state are presented.

The decay widths of the 41S0 kaon are predicted in Sec. V,and the summary and conclusion are given in Sec. VI.

II. THE 3P0 MESON DECAY MODEL

The 3P0 decay model, also known as the quark-pair

creation model, was originally introduced by Micu [17]and further developed by Le Yaouanc et al. [13]. The 3P0

decay model, which (in several variants) is the standardmodel for strong decays, at least for mesons in the initialstate, has been widely used to evaluate the strong decays ofhadrons [18–27], since it gives a good description of manyof the observed decay amplitudes and partial widths of thehadrons. The main assumption of the 3P0 decay model is

that strong decays take place via the creation of a 3P0

quark-antiquark pair from the vacuum. The newly pro-duced quark-antiquark pair, together with the q �q withinthe initial meson, regroups into two outgoing mesons in allpossible quark rearrangements, which corresponds to thetwo decay diagrams shown in Fig. 1 for the meson decayprocess A! Bþ C.

FIG. 1. The two possible diagrams contributing to A! Bþ Cin the 3P0 model.*lidm@zzu.edu.cn

PHYSICAL REVIEW D 78, 054013 (2008)

1550-7998=2008=78(5)=054013(7) 054013-1 � 2008 The American Physical Society

The transition operator T of the decay A! BC in the3P0 model is given by

T ¼ �3�Xm

h1m1�mj00iZd3 ~p3d

3 ~p4�3ð ~p3 þ ~p4Þ

�Ym1

�~p3 � ~p4

2

��341�m�

340 !

340 b

y3 ð ~p3Þdy4 ð ~p4Þ; (1)

where � is a dimensionless parameter representing theprobability of the quark-antiquark pair q3 �q4 with JPC ¼

0þþ creation from the vacuum, and ~p3 and ~p4 are themomenta of the created quark q3 and antiquark �q4, respec-tively. �34

0 , !340 , and �34

1;�m are the flavor, color, and spin

wave functions of the q3 �q4, respectively. The solid har-monic polynomial Ym

1 ð ~pÞ � jpj1Ym1 ð�p;�pÞ reflects the

momentum-space distribution of the q3 �q4.For the meson wave function, we adopt the mock meson

jAðn2SAþ1A LAJAMJA

Þð ~PAÞi defined by [28]

jAðn2SAþ1A LAJAMJA

Þð ~PAÞi �ffiffiffiffiffiffiffiffiffi2EA

p XMLA

;MSA

hLAMLASAMSA jJAMJAiZd3 ~pA nALAMLA

ð ~pAÞ�12SAMSA

�12A !

12A

���������q1

�m1

m1 þm2

~PA þ ~pA

��q2

�m2

m1 þm2

~PA � ~pA

��; (2)

where m1 and m2 are the masses of the quark q1 with a momentum of ~p1 and the antiquark �q2 with a momentum of ~p2,respectively. nA is the radial quantum number of the meson A composed of q1 �q2. ~SA ¼ ~sq1 þ ~s �q2 ,

~JA ¼ ~LA þ ~SA, ~sq1 ( ~s �q2)

is the spin of q1 ( �q2), and ~LA is the relative orbital angular momentum between q1 and �q2. ~PA ¼ ~p1 þ ~p2, ~pA ¼ m1 ~p1�m1 ~p2

m1þm2.

hLAMLASAMSA jJAMJAi is a Clebsch-Gordan coefficient, and EA is the total energy of the meson A. �12SAMSA

, �12A , !12

A , and

nALAMLAð ~pAÞ are the spin, flavor, color, and space wave functions of the meson A, respectively. The mock meson satisfies

the normalization condition

hAðn2SAþ1A LAJAMJA

Þð ~PAÞjAðn2SAþ1A LAJAMJA

Þð ~P0AÞi ¼ 2EA�

3ð ~PA � ~P0AÞ: (3)

The S matrix of the process A! BC is defined by

hBCjSjAi ¼ I � 2�i�ðEA � EB � ECÞhBCjTjAi; (4)

with

hBCjTjAi ¼ �3ð ~PA � ~PB � ~PCÞMMJAMJB

MJC ; (5)

where MMJAMJB

MJC is the helicity amplitude of A! BC. In the center-of-mass frame of meson A, MMJAMJB

MJC can bewritten as

MMJAMJB

MJC ð ~PÞ ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8EAEBEC

p XMLA

;MSA;MLB

;MSB;MLC

;MSC;m

hLAMLASAMSA jJAMJAihLBMLBSBMSB jJBMJBi

� hLCMLCSCMSC jJCMJCih1m1�mj00ih�14SBMSB

�32SCMSC

j�12SAMSA

�341�mi½f1Ið ~P;m1; m2; m3Þ

þ ð�1Þ1þSAþSBþSCf2Ið� ~P;m2; m1; m3Þ�; (6)

with f1 ¼ h�14B �

32C j�12

A �340 i and f2 ¼ h�32

B �14C j�12

A �340 i, corresponding to the contributions from Figs. 1(a) and 1(b),

respectively, and

Ið ~P;m1; m2; m3Þ ¼Zd3 ~p �

nBLBMLB

�m3

m1 þm2

~PB þ ~p

� �nCLCMLC

�m3

m2 þm3

~PB þ ~p

� nALAMLA

ð ~PB þ ~pÞYm1 ð ~pÞ; (7)

where ~P ¼ ~PB ¼ � ~PC, ~p ¼ ~p3, and m3 is the mass of the created quark q3.The spin overlap in terms of Wigner’s 9j symbol can be given by

h�14SBMSB

�32SCMSC

j�12SAMSA

�341�mi ¼

XS;MS

hSBMSBSCMSC jSMSihSAMSA1�mjSMSið�1ÞSCþ1

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð2SA þ 1Þð2SB þ 1Þð2SC þ 1Þ

q 8><>:

12

12 SA

12

12 1

SB SC S

9>=>;: (8)

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In order to compare with the experiment conventionally,MMJA

MJBMJC ð ~PÞ can be converted into the partial ampli-

tude by a recoupling calculation [29],

M LSð ~PÞ ¼ XMJB

;MJC;MS;ML

hLMLSMSjJAMJAi

� hJBMJBJCMJC jSMSi�Zd�Y�

LMLMMJA

MJBMJC ð ~PÞ: (9)

If we consider the relativistic phase space, the decay width�ðA! BCÞ in terms of the partial wave amplitudes is

�ðA! BCÞ ¼ �P

4M2A

XLS

jMLSj2: (10)

Here P ¼ j ~Pj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½M2

A�ðMBþMCÞ2�½M2

A�ðMB�MCÞ2�

p2MA

, and MA,MB, and MC are the masses of mesons A, B, and C,respectively.

The decay width can be derived analytically if thesimple harmonic oscillator (SHO) approximation for themeson space wave functions is used. In momentum space,the SHO wave function is

nLMLð ~pÞ ¼ RSHO

nL ðpÞYLMLð�pÞ; (11)

where the radial wave function is given by

RSHOnL ¼ ð�1Þnð�iÞL

�3=2

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2n!

�ðnþ Lþ 32Þ

s �p

�

�Le�ðp2=2�2ÞLLþð1=2Þ

n

�p2

�2

�:

(12)

Here � is the SHO wave function scale parameter, and

LLþð1=2Þn ðp2

�2Þ is an associated Laguerre polynomial.

The SHOwave functions cannot be regarded as realistic;however, they are a de facto standard for many nonrelativ-istic quark model calculations. Moreover, the more realis-tic space wave functions, such as those obtained fromCoulomb, plus the linear potential model, do not alwaysresult in systematic improvements due to the inherentuncertainties of the 3P0 decay model itself [19,20,22].

The SHO wave function approximation is commonly em-ployed in the 3P0 decay model in the literature. In the

present work, the SHO wave function approximation forthe meson space wave functions is taken.

Under the SHO wave function approximation, the pa-rameters used in the 3P0 decay model involve the q �q pair

production strength parameter �, the SHO wave functionscale parameter �, and the masses of the constituentquarks. In the present work, we take � ¼ 8:77 and �A ¼�B ¼ �C ¼ � ¼ 0:4 GeV, the values recently obtained

by fitting 32 experimentally well-determined decay rateswith the 3P0 decay model1, and mu ¼ md ¼ 0:33 GeV,ms ¼ 0:55 GeV [26]. The meson masses used to determinethe phase space and final state momenta are2 M� ¼138 MeV, MK ¼ 496 MeV, M� ¼ 548 MeV, M�0 ¼958 MeV, M ¼ 776 MeV, MK� ¼ 894 MeV, M! ¼783 MeV, M� ¼ 1019 MeV, Ma2ð1320Þ ¼ 1318 MeV,

MK�2ð1430Þ ¼1429MeV, Mf2ð1270Þ ¼1275MeV, Mf0

2ð1525Þ ¼

1525 MeV, M�ð1300Þ ¼ 1240 MeV, Ma0ð1450Þ ¼1474 MeV, MK�

0ð1430Þ ¼ 1414 MeV, Mf0ð1370Þ ¼

1370 MeV, Mð1450Þ ¼ 1459 MeV, M!ð1420Þ ¼1420 MeV, MK�ð1580Þ ¼ 1580 MeV, Mð1700Þ ¼1720 MeV, MK�ð1680Þ ¼ 1717 MeV, and MK�

3ð1780Þ ¼

1776 MeV.

III. DECAYS OF THE �ð2070ÞFrom (10), the numerical values of the partial decay

widths of the �ð2070Þ as the 41S0 isovector state are listedin Table I. The initial state mass is set to 2070 MeV.Table I indicates that the total width of the �ð2070Þ as

the 41S0 isovector state predicted by the 3P0 decay model

is about 278 MeV, consistent with the observationð310þ100

�50 Þ MeV within errors, and the dominant decay

modes are expected to be �ð1300Þ, ð1450Þ�, andf2ð1270Þ�. Also, in order to check the dependence of thetheoretical result on the initial state mass, the predictedtotal width of the �ð2070Þ is shown in Fig. 2 as a functionof the initial state mass. Figure 2 shows that when theinitial state mass varies from 2035 to 2105 MeV, the totalwidth of the 41S0 isovector state varies from about 230 to

TABLE I. Decays of the �ð2070Þ as the 41S0 isovector state inthe 3P0 model.

Mode �i (MeV) Mode �i (MeV)

! 3.1 � 5.9

�ð1300Þ 52.0 ð1700Þ� 3.6

ð1450Þ� 112.5 f2ð1270Þ� 48.0

f0ð1370Þ� 0.3 a2ð1320Þ� 8.8

a0ð1450Þ� 5.1 KK� 8.5

K�K� 14.9 K�0ð1430ÞK 10.3

K�2ð1430ÞK 4.6

�thy ¼ 277:6 MeV, �expt ¼ 310þ100�50 MeV

1Our value of � is higher than that used by Ref. [26] (0.505) bya factor of

ffiffiffiffiffiffiffiffiffi96�

p, due to different field conventions, constant

factors in T, etc. The calculated results of the widths are, ofcourse, unaffected.

2The assignment of the K�ð1410Þ as the 23S1 kaon is problem-atic [25,30]. The quark model [31] and other phenomenologicalapproaches [32] consistently suggest that the 23S1 kaon has amass of about 1580 MeV; here we take 1580 MeVas the mass ofthe 23S1 kaon [K�ð1580Þ]. Also, we assume that the a0ð1450Þ,K�

0ð1430Þ, and f0ð1370Þ are the ground scalar mesons as inRefs. [23–25].

TOWARDS THE ASSIGNMENT FOR THE 41S0 MESON . . . PHYSICAL REVIEW D 78, 054013 (2008)

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320 MeV, generally in accord with the width range of the�ð2070Þ. Both the mass and width of the �ð2070Þ areconsistent with the predicted 41S0 isovector state, which

therefore suggests that the assignment of the �ð2070Þ asthe 41S0 isovector state seems plausible.

IV. DECAYS OF THE �ð2010Þ, �ð2100Þ, �ð2190Þ,AND �ð2225Þ

In the presence of the �ð2225Þ, we shall discuss thepossibility of the �ð2010Þ, �ð2100Þ, or �ð2190Þ beinganother isoscalar member. It is well known that in a mesonnonet, the two physical isoscalar states can mix. The mix-ing of the two isoscalar states can be parametrized as

�ðxÞ ¼ cos�n �n� sin�s�s; (13)

�ð2225Þ ¼ sin�n �nþ cos�s�s; (14)

where n �n ¼ ðu �uþ d �dÞ= ffiffiffi2

pand s�s are the pure 41S0 non-

strange and strange states, respectively, and �ðxÞ denotesthe �ð2010Þ, �ð2100Þ, or �ð2190Þ.According to (10), the partial widths of �ðxÞ and

�ð2225Þ become, with mixing,

�ð�ðxÞ ! BCÞ ¼ �P

4M2�ðxÞ

XLS

j cos�MLSn �n!BC

� sin�MLSs�s!BCj2; (15)

�ð�ð2225Þ ! BCÞ ¼ �P

4M2�ð2225Þ

XLS

j sin�MLSn �n!BC

þ cos�MLSs�s!BCj2: (16)

Based on (15) and (16), the predicted total widths of the�ð2010Þ, �ð2100Þ, �ð2190Þ, and �ð2225Þ are shown inFig. 3 as functions of the initial state mass and the mixingangle �. From Fig. 3, one can see that, with the variationsof the initial state mass and�, only the measured widths ofthe �ð2100Þ and �ð2225Þ can possibly be reasonably re-produced in the 3P0 model. We therefore suggest that the

assignment of the �ð2010Þ and �ð2190ÞÞ as the 41S0 iso-

scalar states seems unfavorable. We shall focus on thepossibility of the �ð2100Þ being the partner of the�ð2225Þ. Taking M�ð2100Þ ¼ 2103 MeV and M�ð2225Þ ¼2240 MeV, we list the numerical values of the partialdecay widths of the �ð2100Þ and �ð2225Þ in Table II.The variation of the theoretical total widths of the�ð2100Þ and �ð2225Þ with the mixing angle � is shownin Fig. 4.From Fig. 4, we find that if the�ð2100Þ–�ð2225Þmixing

angle � lies in the range from about�0:6 toþ0:7 radians,both the measured widths of the �ð2100Þ and �ð2225Þ canbe reasonably reproduced. In order to check whether the

FIG. 2. The predicted total width of the �ð2070Þ as the 41S0isovector versus the initial state mass.

FIG. 3 (color online). The predicted total widths of the �ð2010Þ, �ð2190Þ, �ð2100Þ, and �ð2225Þ as the 41S0 isoscalar states versusthe initial state mass and the mixing angle �.

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possibility of �0:6 � � � þ0:7 radians exists or not,below we shall estimate the �ð2100Þ–�ð2225Þ mixingangle phenomenologically.

In the n �n and s�s bases, the mass-squared matrix describ-ing the �ð2100Þ and �ð2225Þ mixing can be written as[33,34]

M2 ¼ M2n �n þ 2Am

ffiffiffi2

pAmXffiffiffi

2pAmX M2

s�s þ AmX2

!; (17)

whereMn �n andMs �s are the masses of the states n �n and s�s,respectively, Am denotes the total annihilation strength ofthe q �q pair for the light flavors u and d, and X describes theSUð3Þ-breaking ratio of the nonstrange and strange quarkmasses via the constituent quark mass ratio mu=ms. Themasses of the two physical states �ð2100Þ and �ð2225Þ can

be related to the matrix M2 by the unitary matrix

U ¼ cos� � sin�sin� cos�

� �;

UM2Uy ¼ M2�ð2100Þ 0

0 M2�ð2225Þ

!: (18)

n �n is the orthogonal partner of the �ð41S0Þ, the isovector

state of the 41S0 meson nonet, and one can expect that n �ndegenerates with �ð41S0Þ in effective quark masses; here

we take Mn �n ¼ M�ð41S0Þ ¼ M�ð2070Þ. With the help of the

Gell-Mann-Okubo mass formula M2s�s ¼ 2M2

Kð41S0Þ �M2

n �n

[35], the following relations can be derived from (18),

8X2ðM2Kð41S0Þ

�M2�ð2070ÞÞ2 ¼ ½4M2

Kð41S0Þ� ð2� X2ÞM2

�ð2070Þ � ð2þ X2ÞM2�ð2100Þ�

� ½ð2� X2ÞM2�ð2070Þ þ ð2þ X2ÞM2

�ð2225Þ � 4M2Kð41S

0Þ�; (19)

TABLE II. Decays of the �ð2100Þ and �ð2225Þ as the 41S0 isoscalar states in the 3P0 model. c � cos�, s � sin�.

�ð2100Þ �ð2225ÞMode �i (MeV) �i (MeV)

2:1c2 1:4s2

!! 0:9c2 0:3s2

�� 9:7s2 20:1c2

a2ð1320Þ� 143:7c2 158:8s2

a0ð1450Þ� 0:1c2 4:5s2

KK� 7:4c2 � 15:0csþ 7:6s2 14:4c2 þ 12:7csþ 2:8s2

K�K� 15:4c2 � 13:4csþ 2:9s2 0:8c2 � 6:6csþ 13:6s2

KK�0ð1430Þ 8:8c2 � 7:9csþ 1:8s2 2:5c2 � 4:7csþ 2:2s2

KK�2ð1430Þ 7:7c2 � 31:1csþ 31:3s2 69:4c2 þ 91:9csþ 30:4s2

KK�ð1580Þ 5:4c2 þ 17:2csþ 13:6s2 90:7c2 � 158:1csþ 68:9s2

KK�ð1680Þ 1:6c2 � 1:4csþ 0:3s2

�thy ¼ 191:5c2 � 50:2csþ 66:9s2 �thy ¼ 199:4c2 � 66:2csþ 283:2s2

�expt ¼ 187� 75 �expt ¼ 190� 30þ40�60

FIG. 4 (color online). The predicted total widths of the �ð2100Þand �ð2225Þ as the 41S0 isoscalar states versus the mixing angle

�.

TOWARDS THE ASSIGNMENT FOR THE 41S0 MESON . . . PHYSICAL REVIEW D 78, 054013 (2008)

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Am ¼ðM2

�ð2225Þ � 2M2Kð41S

0Þ þM2

�ð2070ÞÞðM2�ð2100Þ � 2M2

Kð41S0Þ þM2

�ð2070ÞÞ2ðM2

�ð2070Þ �M2Kð41S0Þ

ÞX2: (20)

If the SUð3Þ-breaking effect is not considered, i.e., X ¼ 1,relation (19) can be reduced to Schwinger’s original nonetmass formula [36]. Taking X ¼ mu=ms ¼ 0:33=0:55 ¼0:6, from (19) and (20) we have

MKð41S0Þ ¼ 2:153 GeV; Am ¼ 0:07 GeV2: (21)

Based on the values of the above parameters involved in(17), the unitary matrix U can be given by

U ¼ cos� � sin�sin� cos�

� �¼ þ0:995 �0:104

þ0:104 þ0:995

� �; (22)

which gives� ¼ þ0:1 radians, just lying in the range fromabout �0:6 to þ0:7 radians. From Table II, this estimatedmixing angle leads to �thyð�ð2100ÞÞ ¼ 185:2 MeV and

�thyð�ð2225ÞÞ ¼ 193:7 MeV, both in good agreement

with the experimental results. The �ð2100Þ and �ð2225Þ,together with the �ð2070Þ, therefore appear to be theconvincing 41S0 states.

V. DECAYS OF THE 41S0 KAON

The above two sections show that in the presence of the�ð2070Þ, �ð2100Þ, and �ð2225Þ belonging to the 41S0meson nonet, the total widths of these three states can benaturally accounted for in the 3P0 decay model, and the

41S0 kaon is expected to have a mass of about 2153 MeV

by the mass formula (19). Below, the Kð2150Þ denotes the41S0 kaon. We note that the K, Kð1460Þ,3 Kð1830Þ, andKð2150Þ approximately populate a common trajectory, asshown in Fig. 5. The quasilinear trajectories at the (n,mass-squared) plots are able to describe the light mesonswith a good accuracy [12]. Figure 5 therefore indicates thatthe Kð1460Þ, Kð1830Þ, and Kð2150Þ could be good candi-dates for the 21S0, 3

1S0, and 41S0 kaons, respectively.The Kð2150Þ is not related to any current experimental

candidate. The predicted decay widths of the Kð2150Þ arelisted in Table III. The initial state mass is set to 2153MeV.The total width of the Kð2150Þ is predicted to be about197 MeV, and the dominant decay modes of the Kð2150Þare expected to be K�

2ð1430Þ�, K�ð1580Þ�, ð1450ÞK, anda2ð1320ÞK. These results could be of use in looking for thecandidate for the 41S0 kaon experimentally.

VI. SUMMARYAND CONCLUSION

With the assignment of the �ð2225Þ recently observedby the BES Collaboration as the s�s member of the 41S0meson nonet, the possibility of the �ð2070Þ, �ð2010Þ,

�ð2100Þ, and �ð2190Þ being the 41S0 q �q states is dis-

cussed. With respect to the �ð2070Þ, its assignment tothe 41S0 isovector state is favored not only by its mass,

but also by its width. The assignment of the �ð2010Þ and�ð2190Þ as the 41S0 isoscalar states is not favored by their

widths. Both the widths of the �ð2100Þ and �ð2225Þ can bereasonably reproduced with the mixing angle lying in therange from about�0:6 toþ0:7 radians. The assignment ofthe �ð2070Þ, �ð2100Þ, and �ð2225Þ as the members of the41S0 meson nonet not only leads to the �ð2100Þ–�ð2225Þmixing angle being about þ0:1 radians, which naturallyaccounts for the widths of the �ð2100Þ and �ð2225Þ, butalso shows that the 41S0 kaon has a mass of about

2153 MeV. The K, Kð1460Þ, Kð1830Þ, and Kð2150Þ ap-proximately populate a common (n, mass-squared) trajec-tory. We tend to conclude that the observed pseudoscalarstates �ð2070Þ, �ð2100Þ, �ð2225Þ, together with the un-observed Kð2150Þ, appear to be good candidates for the

FIG. 5. The (n, mass-squared) trajectory for the K.

TABLE III. Decays of the Kð2150Þ as the 41S0 isodoublet inthe 3P0 model.

Mode �i (MeV) Mode �i (MeV)

K 3.9 !K 1.3

�K 1.3 K� 0.08

!K� 0.04 �K� 12.0

�K� 4.5 �K� 1.5

�0K� 0.09 K�0ð1430Þ� 1.6

K�2ð1430Þ� 29.7 K�ð1580Þ� 34.5

K�ð1680Þ� 2.5 K�3ð1780Þ� 1.0

�ð1300ÞK� 6.3 ð1450ÞK 42.8

!ð1420ÞK 14.6 a2ð1320ÞK 26.7

f2ð1270ÞK 9.9 f02ð1525ÞK 2.9

�thy ¼ 197:2 MeV3The Kð1460Þ mass is taken to be 1400 MeV, as reported by

[37].

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members of the 41S0 meson nonet. The Kð2150Þ width is

predicted to be about 197 MeV, and the dominant decaymodes of the Kð2150Þ are expected to be K�

2ð1430Þ�,K�ð1580Þ�, ð1450ÞK, and a2ð1320ÞK. These resultscould be of use in looking for the candidate for the 41S0kaon experimentally.

ACKNOWLEDGMENTS

This work is supported in part by HANCET underContract No. 2006HANCET-02, and by the Program forYouthful Teachers in University of Henan Province.

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