Strong decays of charmed baryons

Strong decays of charmed baryons

Chong Chen, Xiao-Lin Chen, Xiang Liu,* Wei-Zhen Deng, and Shi-Lin Zhu†

Department of physics, Peking University, Beijing, 100871, China(Received 5 April 2007; published 16 May 2007)

There has been important experimental progress in the sector of heavy baryons in the past several years.We study the strong decays of the S-wave, P-wave, D-wave, and radially excited charmed baryons usingthe 3P0 model. After comparing the calculated decay pattern and total width with the available data, wediscuss the possible internal structure and quantum numbers of those charmed baryons observed recently.

DOI: 10.1103/PhysRevD.75.094017 PACS numbers: 13.30.Eg, 12.39.Jh

I. INTRODUCTION

BABAR and Belle Collaborations observedseveral excited charmed baryons: c2880 2940,c2980 3077;0, and c27680 last year [1–5], whichinspired several investigations of these states in literature[6–9]. We collect the experimental information of theserecently observed hadrons in Table I. Their quantum num-bers have not been determined except c2880. In orderto understand their structures using the present experimen-tal information, we study the strong decay pattern of theexcited charmed baryons systematically in this work. In thepast decades, there has been some research work on heavybaryons [8,11,12].

The quantum numbers and decay widths of S-wave andsome P-wave charmed baryons are known [13]. We firstsystematically analyze their strong decays in the frame-work of the 3P0 strong decay model. Accordingly one canextract the parameters and estimate the accuracy of the 3P0

model when it is applied in the charmed baryon system.Then we go one step further and extend the same formal-ism to study the decay patterns of these new charmedbaryons c2880 2940, 2980 3077;0 under differentassignments of their quantum numbers. After comparingthe theoretical results with the available experimental data,we can learn their favorable quantum numbers and assign-ments in the quark model.

Very recently CDF Collaboration reported four par-ticles [14], which are consistent with b and b pre-dicted in the quark model [15]. Their massesare Mb

58082:02:3 1:7 MeV, Mb

58161:01:0

1:7 MeV, Mb 58291:6

1:8 1:7, Mb 58372:1

1:9

1:7 MeV. The mass splitting between b and b wasdiscussed in Refs. [16,17] while the strong decays ofb were studied in Ref. [18]. As a by-product, we alsocalculate the strong decays of b and other S-wavebottom baryons in this work.

This paper is organized as follows. We give a shorttheoretical review of S-wave, P-wave, and D-wavecharmed baryons and introduce our notations for them inSec. II. Then we give a brief review of the 3P0 model inSec. III. We present the strong decay amplitudes ofcharmed baryons in Sec. IV. Section V is the numericalresults. The last section is our discussion and conclusion.Some lengthy formulae are collected in the appendix.

II. THE NOTATIONS AND CONVENTIONS OFCHARMED BARYON

We first introduce our notations for the excited charmedbaryons. Inside a charmed baryon there are one charmquark and two light quarks (u, d, or s). It belongs to eitherthe symmetric 6F or antisymmetric 3F flavor representa-tion (see Fig. 1). For the S-wave charmed baryons, the totalcolor-flavor-spin wave function and color wave functionmust be symmetric and antisymmetric, respectively. Hencethe spin of the two light quarks is S 1 for 6F or S 0 for3F. The angular momentum and parity of the S-wavecharmed baryons are JP 1

2 or 3

2 for 6F and JP 1

2

for 3F. The names of the S-wave charmed baryons arelisted in Fig. 1, where we use the star to denote 3

2 baryons

and the prime to denote the JP 12 baryons in the 6F

representation.In Fig. 2 we introduce our notations and conventions for

the P-wave charmed baryons. l is the orbital angularmomentum between the two light quarks while l denotesthe orbital angular momentum between the charm quarkand the two light quark system. We use the prime to labelthe cJl baryons in the 6F representation and the tildeto discriminate the baryons with l 1 from that withl 1.

The notation for the D-wave charmed baryons is morecomplicated (see Fig. 3). Besides the prime, l and ldefined above, we use the hat and check to denote thecharmed baryons with l 2 and l 1, respectively.For the baryons with l 1 and l 1, we use the super-script L to denote the different total angular momentum inLcJl ,

LcJl , and

LcJl .

*Electronic address: [email protected]†Electronic address: [email protected]

PHYSICAL REVIEW D 75, 094017 (2007)

1550-7998=2007=75(9)=094017(13) 094017-1 © 2007 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.75.094017

III. THE 3P0 MODEL

The 3P0 model was first proposed by Micu [19] andfurther developed by Yaouanc et al. later [20–22]. Nowthis model is widely used to study the strong decays ofhadrons [23–30].

According to this model, a pair of quarks with JPC 0 is created from the vacuum when a hadron decays,which is shown in Fig. 4 for the baryon decay process A!B C. The new q q pair created from the vacuum togetherwith the qqq within the initial baryon regroup into theoutgoing meson and baryon via the quark rearrangementprocess. In the nonrelativistic limit, the transition operatoris written as

T 3Xm

h1m; 1mj00iZ

d3k4d3k53k4 k5

Ym1

k4 k5

2

45

1;m’450 !

450 by4ik4d

y5jk5; (1)

where i and j are the color indices of the created quark andantiquark. ’45

0 u u d d ss=3p

and !450 ij for

the flavor and color singlet, respectively. 451;m is for the

spin triplet state. Ym1 k jkjY

m1 k;k is a solid har-

monic polynomial corresponding to the p-wave quark pair. is a dimensionless constant related to the strength of thequark pair creation from the vacuum, which was extractedby fitting to data. The hadron and meson state are definedas, respectively, according to the definition of the mockstate [31]

TABLE I. A summary of recently observed charmed baryons by BABAR and Belle Collaborations.

State Mass and width (MeV) Decay channels in experiments Other information

c2880 2881:5 0:3, <8 [10] c JP favors 5

2 [2],

?c 2520

c2455 0:225 0:062 0:025 [2]2881:9 0:1 0:5, 5:8 1:5 1:1 [1] D0p2881:2 0:20:4

0:3, 5:50:70:5 0:4 [2] ?0;

c 2520;

c2940 2939: 1:3 1:0, 17:5 5:2 5:9 [1] D0p —2937:9 1:01:8

0:4, 10 4 5 [2] c24550;;

c2980 2967:1 1:9 1:0, 23:6 2:8 1:3 [3] c K —

2978:5 2:1 2:0, 43:5 7:5 7:0 [4] c K

c29800 2977:1 8:8 3:5, 43:5 [4] c K0S —

c3077 3076:4 0:7 0:3, 6:2 1:6 0:5 [3] c K —

3076:7 0:9 0:5, 6:2 1:2 0:8 [4] c K

c30770 3082:8 1:8 1:5, 5:2 3:1 1:8 [4] c K0S —

c27680 2768:3 3:0 [5] 0c JP 3

2

Ω(∗)0c (ssc )

Σ (∗)+ +c (uuc)

Σ (∗)0c (ddc) Ξ (∗)0

c (dsc)

Σ (∗)+c (udc) Ξ (∗)+

c (usc )

6Ξ0

c (dsc)

Λ+c (udc) Ξ+

c (usc )

3

FIG. 1. The SU(3) flavor multiplets of charmed baryons.

(a) lρ = 0 , lλ = 1

(b) lρ = 1 , lλ = 0

J l = 1 c1( 12

−, 3

2−

c1( 12

−, 3

2−

)

J l = 0 : Σ

: Σ

: Σ

: Σ

c0( 12

−c0 ( 1

2−

)

J l = 2 c2( 32

−, 5

2−

c2( 32

−, 5

2−

)

f S (6): L = 1 ⊗ S q1 q2 = 1

f A (3): L = 1 ⊗ S q1 q2 = 0 J l = 1 c1( 12

−, 3

2−) Ξ

) Ξ

) Ξ

) Ξ

c1 ( 12

−, 3

2−

)

J l = 1 : Λc1( 12

−, 3

2−

) Ξc1( 12

−, 3

2−

)

J l = 0 : Λc0( 12

−) Ξc0( 1

2−

)

J l = 2 : Λc2( 32

−, 5

2−

) Ξc2( 32

−, 5

2−

)

f A (3): L = 1 ⊗ S q1 q2 = 1

f S (6): L = 1 ⊗ S q1 q2 = 0 =⇒

=⇒

J l = 1 : Σ c1( 12

−, 3

2−

) Ξc1( 12

−, 3

2−

)

FIG. 2. The notations for P-wave charmed baryons. fS6F andfA3F denote the SU(3) flavor representation. Sq1q2

is the totalspin of the two light quarks. L denotes the total orbital angularmomentum of the charmed baryon system.

CHEN, CHEN, LIU, DENG, AND ZHU PHYSICAL REVIEW D 75, 094017 (2007)

094017-2

jAn2SA1A LAJAMJA

PAi 2EA

p XMLA

;MSA

hLAMLASAMSA jJAMJAiZ

d3k1d3k2d3k33k1 k2 k3 PA

nALAMLAk1;k2;k3

123SAMSA

’123A !123

A jq1k1q2k2q3k3i; (2)

jBnB2SB1LBJBMJB

PBi 2EB

p XMLB

;MSB

hLBMLBSBMSB jJBMJBiZ

d3kad3kb3ka kb PB nBLBMLBka;kb

abSBMSB’abB !

abB jqaka qbkbi (3)

and satisfy the normalization condition

(a) lρ = 0 , lλ = 2 (b) lρ = 2 , lλ = 0

J l c2( 32

+, 5

2+

c2( 32

+, 5

2+

)

J l = 1: Σ

= 2: Σ

= 3: Σ

c1( 12

+, 3

2+

) Ξ

) Ξ

) Ξ

) Ξ

c1( 12

+, 3

2+

)

J l c3( 52

+, 7

2+

c3( 52

+, 7

2+

)

f S (6): L = 2 ⊗ S q1 q2 = 1

f A (3): L = 2 ⊗ S q1 q2 = 0 =⇒ J l = 2: Λc2( 32

+, 5

2+

c2( 32

+, 5

2+

)

J l = 2: Σ c2( 32

+, 5

2+

) Ξc2( 32

+, 5

2+

)

J l = 1: Σ c1( 12

+, 3

2+

) Ξc1( 12

+, 3

2+

)

J l = 3: Σ c3( 52

+, 7

2+

) Ξc3( 52

+, 7

2+

)

f S (6): L = 2 ⊗ S q1 q2 = 1

f A (3): L = 2 ⊗ S q1 q2 = 0 =⇒ J l = 2: Λc2( 32

+, 5

2+

) Ξc2( 32

+, 5

2+

)

(c) lρ = 1 , lλ = 1

J l = 1: Λ1c1( 1

2+

, 32

+) Ξ1

c1( 12

+, 3

2+

)

J l = 0: Λ1c0( 1

2+

) Ξ1c0( 1

2+

)

J l = 2: Λ1c2( 3

2+

, 52

+) Ξ1

c2( 32

+, 5

2+

)

L = 1 ⊗ S q1 q2 = 1

L = 0 ⊗ S q1 q2 = 1 =⇒ J l = 1: Λ0c1( 1

2+

, 32

+) Ξ0

c1( 12

+, 3

2+

)f A (3)

J l = 2: Λ2c2( 3

2+

, 52

+) Ξ2

c2( 32

+, 5

2+

)

J l = 1: Λ2c1( 1

2+

, 32

+) Ξ2

c1( 12

+, 3

2+

)

J l = 3: Λ2c3( 5

2+

, 72

+) Ξ2

c3( 52

+, 7

2+

)

L = 2 ⊗ S q1 q2 = 1

L = 0 ⊗ S q1 q2 = 0 =⇒ J l = 0: Σ 0c0( 1

2+

) Ξ 0c0( 1

2+

)

L = 1 ⊗ S q1 q2 = 0 =⇒ J l = 1: Σ 1c1( 1

2+

, 32

+) Ξ 1

c1 ( 12

+, 3

2+

)

L = 2 ⊗ S q1 q2 = 0 =⇒ J l = 2: Σ 2c2( 3

2+

, 52

+) Ξ 2

c2 ( 32

+, 5

2+

)

f S (6)

FIG. 3. The notations for the D-wave charmed baryons.

FIG. 4. The decay process of A! B C in the 3P0 model.

STRONG DECAYS OF CHARMED BARYONS PHYSICAL REVIEW D 75, 094017 (2007)

094017-3

hAPAjAP0Ai 2EA3PA P0A; hBPBjBP0Bi 2EB3PB P0B: (4)

The subscripts 1, 2, 3 denote the quarks of parent hadron A. a and b refer to the quark and antiquark within the meson B,respectively. kii 1; 2; 3 are the momentum of quarks in hadron A. ka and kb are the momentum of the quark andantiquark in meson B. PAB represents the momentum of state AB. SAB and JAB denote the total spin and the totalangular momentum of state AB.

The S-matrix is defined as

hfjSjii I i2Ef EiMMJA

MJBMJC : (5)

The helicity amplitude of the process A! B C in the center of mass frame of meson A is

MMJAMJB

MJC A! BC 8EAEBEC

p

XMLA;MSA;

MLB;MSB;

MLC;MSC; m

hLAMLASAMSA jJAMJAihLBMLBSBMSB jJBMJBi

hLCMLCSCMSC jJCMJCih1m; 1mj00ih235SCMSC

14SBMSB

j123SAMSA

451mih’

235C ’14

B j’123A ’45

0 i

IMLA

;mMLB

;MLCp; (6)

where the spatial integral IMLA

;mMLB

;MLCp is defined as

IMLA

;mMLB

;MLCp

Zd3k1d3k2d3k3d3k4d3k53k4k53k1k2k3PA3k1k4PB3k2k3k5PC

nBLBMLBk1;k4 nCLCMLC

k2;k3;k5 nALAMLAk1;k2;k3Y

m1

k4k5

2

: (7)

h235SCMSC

14SBMSB

j123SAMSA

451mi and h’235

C ’14B j’

123A ’45

0 i de-note the spin and flavor matrix element, respectively.

The decay width of the process A! B C is

2 jpjM2A

s2JA 1

XMJA

;MJB;MJC

jMMJAMJB

MJC j2;

where jpj is the momentum of the daughter baryon in theparent’s center of mass frame. s 1=1 BC is a statis-tical factor which is needed if B and C are identicalparticles.

IV. THE STRONG DECAYS OF CHARMEDBARYON

According to the 3P0 model, the decay occurs throughthe recombination of the five quarks from the initialcharmed baryon and the created quark pair. So there are

three ways of regrouping:

A q1; q2; c3 P q4; q5 ! Bq2; q4; c3 Cq1; q5;

(8)

A q1; q2; c3 P q4; q5 ! Bq1; q4; c3 Cq2; q5;

(9)

A q1; q2; c3 P q4; q5 ! Bq1; q2; q4 Cc3; q5;

(10)

where qi and c3 denote the light quark and charm quark,respectively.

When the excited charmed baryon decays into acharmed baryon plus a light meson as shown in Eqs. (8)and (9), the total decay amplitude reads

MMJAMJB

MJC 28EAEBEC

p XMA

XMLA

XMB

XMLB

Xm1;m3;m4;m

hJ12M12; s3m3jJAMJAihlAmA; lAmAjLAMLAi

hLAMLA ; S12m12jJ12M12ihs1m1; s2m2jS12m12ihJ14M14; s3m3jJBMJBihlBmB; lBmBjLBMLBi

hLBMLB ; S14m14jJ14M14ihs1m1; s4m4jS14m14ih1m; 1mj00ihs4m4; s5m5j1mi

hLCMLC ; SCMCjJCMJCihs2m2; s5m5jSCMCi h1;4;3B 2;5

C j4;50 1;2;3

A i IMLA

;mMLB

;MLCp; (11)


094017-4

where the prefactor 2 in front of arises from the fact thatthe amplitude from Eq. (8) is the same as that from Eq. (9).

The overlap integral in the momentum space is

IMLA

;mMLB

;MLCp 3PB PC

Z

d3p1d3p2 BlB; mB; lB; mB

CLCMLCYm1

p4 p5

2

AlA; mA; lA; mA: (12)

Since all hadrons in the final states are S-wave in this work,Eq. (12) can be further expressed as

IMLA

;mMLB

;MLCp 3PB PClA; mA; lA; mA;m;

(13)

where we have used the harmonic oscillator wave functionsfor both the meson and baryon. The expressions oflA; mA; lA; mA;m for the decays of the S-wave,P-wave, and D-wave charmed baryons are collected inthe appendix. We also move the lengthy expressions ofmomentum space integration of the S-wave, P-wave, andD-wave charmed baryons to the appendix.

V. NUMERICAL RESULTS

The decay widths of charmed baryons from the 3P0model involve several parameters: the strength of quarkpair creation from vacuum , the R value in the harmonicoscillator wave function of the meson, and the ; in thebaryon wave functions. We follow the convention ofRef. [32] and take 13:4, which is considered as auniversal parameter in the 3P0 model. The R value of and K mesons is 2:1 GeV1 [32] while it is R 2:3 GeV1 for the D meson [33]. 0:5 GeVfor the proton and [30]. For the S-wave charmed bary-

ons, the parameters and in the harmonic oscillatorwave functions can be fixed to reproduce the mass splittingthrough the contact term in the potential model [34]. Theirvalues are 0:6 GeV and 0:6 GeV. For theP-wave and D-wave charmed baryons, and are

TABLE IV. The decay widths of P-wave charmed baryonsc 2593; 2625 and ;0c 2790; 2815 with the fixed structureand quantum number assignments. Here all results are in units ofMeV.

Assignment Channel exp [13]

c 2593 c112 c 3.4 3:62:0

1:3c

0 6.40c 3.4

c 2625 c132 c 1:9 103 <0:10

c 0 2:6 103 <1:90c 1:9 103 <0:10

c 2790 c112 0c

0 5.0 <1500c

4.9

0c2790 c1

12 0c

5.2 <1200c

0 5.1

c 2815 c132 ?

c 0 2.7 <3:5?0c 2.6

0c2815 c1

32 ?

c 2.7 <6:5?0c

0 2.8

TABLE III. The strong decay widths of S-wave bottom bary-ons b, b, 0b, and b. Here all results are in units of MeV.

JP Channel Width Experimental results [14]

b12 0

b 3.5 8

b12 0

b 4.7

b32 0

b 7.5 15

b32 0

b 9.2

0b12 b 0.10 —

b32 b 0.85 —

TABLE II. The strong decay widths of S-wave charmed bary-ons ;;0c 2455, ;;0c 2520, and ;0c 2645. Here allresults are in units of MeV.

JP Channel Width Total width (Exp) [13]

c 2455 12 c 1.24 2:23 0:30

c 2455 12 c

0 1.40 <4:60c2455 1

2 c

1.24 2:2 0:40

c 2520 32 c

11.9 14:9 1:9c 2520 3

2 c

0 12.1 <17

0c 2520 32 c 11.9 16:1 2:1

c 2645 32 c

0 0.64 <3:1c 2645 3

2 0

c 0.49

0c 2645 32 c

0.54 <5:50c 2645 3

2 0

c0 0.54

TABLE V. The decay widths of c 2800 in differentP-wave charmed baryons assignments. R ;c ;0=?;

c ;0. The total width of c 2800 is752217 MeV [13]. Here all results are in units of MeV.

Assignment c ;c ;0 ?;

c ;0 R

c012 307 0.0 0.0 —

c112 0.0 296 0.4 740

c132 0.0 0.7 220 3 103

c232 8.1 1.3 0.3 4.3

c252 8.1 0.6 0.5 1.2

~c112 0.0 75 69 1.1

~c132 0.0 75 69 1.1


094017-5

expected to lie in the range 0:5 0:7 GeV. In the follow-ing, our numerical results are obtained with the typicalvalues 0:6 GeV.

The strong decay widths of the S-wave charmed baryons;;0c 2455, ;;0c 2520, and ;0c 2645 arelisted in Table II. Accordingly the decay widths of theS-wave bottomed baryons are presented in Table III.

Because b, 0b, and b have not been observed so far,their masses are taken from the theoretical estimate inRef. [35], which are mb

5805:7 MeV, m0b

5950 MeV, and mb 5966:1 MeV.

The quantum number and internal structure of the fol-lowing P-wave charmed baryons c 2593, c 2625,;0c 2790, and ;0c 2815 are relatively known experi-

TABLE VI. The decay widths of c 2880 with different D-wave assignments. All results are in units of MeV.

Assignment 0;;c ;0; ?0;;

c ;0; ?c

c

D0p Remark

c232 7.8 0.9 0.11 0.0

c252 0.06 5.34 89 0.0

c232 78.3 59.1 75 0.0

c252 78.3 59.1 0.75 0.0

0c1

12 0.9 2.3 2.6 2.3

0c1

32 0.22 6.0 27 2.3

1c0

12 132 144 1.1 0.0

1c1

12 66.3 18.0 0.27 150

1c1

32 16.5 45.0 2.7 150

1c2

32 82.8 9.0 0.10 0.0

1c2

52 0.0 54.1 — 0.0

2c1

12 25.7 8.1 0.32 64

2c1

32 6.5 20.4 3.1 64

2c2

32 57.9 14.2 0.24 0.0

2c2

52 9.4 47.1 5.0 0.0

2c3

52 10.8 5.5 0.51 12

2c3

72 6.1 7.4 1.2 12

TABLE VII. The decay widths of c 2940 with different D-wave assignments. Here all results are in units of MeV.

Assignment 0;;c ;0; ?0;;

c ;0; ?c

cD0p Remark

c232 11.7 9.1 0.77 0.0

c252 0.2 9.1 46 0.0

c232 170 150 0.88 0.0

c252 170 150 0.88 0.0

0c1

12 2.2 0.5 0.23 11

0c1

32 0.6 1.4 2.3 11

1c0

12 212 259 1.2 0.0

1c1

12 106 32.4 0.31 340

1c1

32 26.5 81.0 3.1 340

1c2

32 142 16.2 0.11 0.0

1c2

52 0.0 97.0 — 0.0

2c1

12 34.5 12.6 0.37 95

2c1

32 8.6 31.7 3.7 95

2c2

32 77.7 27.7 0.36 0.0

2c2

52 19.5 75.6 3.9 0.0

2c3

52 22.2 12.9 0.58 49

2c3

72 12.4 17.5 1.4 49


094017-6

mentally [13]. Their strong decay modes and widths fromthe 3P0 model are collected in Table IV. The quantumnumber of c 2800 is still unknown [13]. Thus underdifferent P-wave assignments of c 2800, we presentthe strong decay widths of its possible decay modes inTable V. In the heavy quark limit, the processc 2800 ! c is forbidden if c 2800 is as-signed as c1

12, c1

32, ~c1

12, and ~c1

32, which

is observed in our calculation as can be seen from Table V.c2880 and c2940 are observed in the invariant

mass spectrum of D0p [1]. The first radial excitation of cdoes not decay into D0p from the 3P0 model. Hence the

possibility of c2880 and c2940 being a radialexcitation is excluded. We calculate their strong decaysassuming they are D-wave charmed baryons. The resultsare shown in Tables VI and VII.

With positive parity, 2980;0 and 3077;0 can beeither the first radially excited charmed baryons or theD-wave charmed baryons. With different assumptions oftheir quantum numbers we present their strong decaywidths in Tables VIII and IX and Fig. 5.

The numerical results depend on the parameters and in the harmonic oscillator wave functions of thecharmed baryons. We illustrate such a dependence in

TABLE VIII. The decay widths of c 2980 with different D-wave assignments. Here all results are in units of MeV.

Assignment 0c 00c ?0

c c k c k0 Remark

c232 0.0 1.1 0.11 0.37 0.0

c252 0.0 0:12 102 0.67 0:11 103 0.0

0c112 4.4 0.72 0.18 0.25 5.3

0c132 4.4 0.18 0.46 0.062 5.3

0c232 0.0 0.16 0.17 0.56 0.0

0c252 0.0 0:47 102 1.0 0:71 104 0.0

0c352 0.054 0:53 102 0:14 102 0:82 104 0.053

0c372 0.054 0:30 102 0:19 102 0:46 104 0.053

c232 0.0 9.5 6.1 0.61 0.0

c252 0.0 9.5 6.1 0.61 0.0

0c112 74 6.3 1.0 0.40 78

0c1

32 74 1.6 2.5 0.10 78

0c232 0.0 14 4.5 0.91 0.0

0c2

52 0.0 6.3 7.1 0.40 0.0

0c3

52 48 7.2 2.9 0.46 50

0c3

72 48 4.1 3.9 0.26 50

00c012 0.0 0.30 1.4 1.3 0.0

0c1

12 1.0 0.40 0.46 1.7 0.46

0c1

32 1.0 0.10 1.2 0.43 0.46

01c1

12 0.0 18 4.4 5.5 0.0

01c132 0.0 4.5 11 1.4 0.0

1c0

12 0.0 18 18 5.5 0.0

1c1

12 6.2 9.1 2.2 2.8 7.2

1c1

32 6.2 2.3 5.5 0.69 72

1c2

32 0.0 11 1.1 0.34 0.0

1c2

52 0.0 0.0 6.6 0.0 0.0

02c232 0.0 5.6 1.8 2.4 0.0

02c2

52 0.0 1.7 4.32 0.24 0.0

2c1

12 19 3.7 1.1 1.6 23

2c1

32 19 0.93 2.6 0.40 23

2c2

32 0.0 8.4 1.7 0.36 0.0

2c2

52 0.0 1.2 6.0 0.16 0.0

2c3

52 8.1 1.3 60 0.19 8.7

2c3

72 8.1 0.75 0.81 0.10 8.7


094017-7

Figs. 6–8 using several typical decay channels:c 2455 ! c

, c 2593 ! c 2455, andc 2880 ! c 2520, where c 2455,c 2593, and c 2880 are S-wave, P-wave, andD-wave baryons, respectively.

VI. DISCUSSION AND CONCLUSION

At the present it is still too difficult to calculate thestrong decay widths of hadrons from the first principlesof QCD. For this purpose, some phenomenological strongdecay models were proposed such as the 3P0 model, flux

tube model, QCD sum rule, lattice QCD, etc., amongwhich only the first two approaches can be applied to thestrong decays of excited hadrons. To a large extent, thepredictions from the 3P0 and flux tube models roughlyagree with each other.

The 3P0 model possesses inherent uncertainties[20,28,32]. In certain cases, the result from the 3P0 modelmay be a factor of 2 3 off the experimental width. Theuncertainty source of the 3P0 model arises from thestrength of the quark pair creation from the vacuum ,the approximation of nonrelativity, and assuming the sim-

TABLE IX. The decay widths of c 3077 with different D-wave assignments. Here all results are in units of MeV.

Assignment 0c 00c

?0c

c k c k c k0 D Remark

c232 0.0 2.1 0.30 0.73 0.054 0.0 0.0

c252 0.0 0.037 1.7 0:42 102 0.32 0.0 0.0

0c112 7.0 1.4 0.46 0.49 0.089 4.4 3.2

0c132 7.0 0.36 1.1 0.12 0.22 4.4 3.2

0c232 0.0 3.2 0.43 1.1 0.081 0.0 0.0

0c252 0.0 0:025 102 2.5 0:28 102 0.48 0.0 0.0

0c352 0.19 0.029 0.012 0:32 102 0:32 103 0.12 0.026

0c372 0.19 0:016 103 0.016 0:18 102 0:44 103 0.12 0.026

c232 0.0 34 29 6.0 2.0 0.0 0.0

c252 0.0 34 29 6.0 2.0 0.0 0.0

0c1

12 201 23 4.8 4.0 0.33 130 38

0c1

32 201 5.7 12 1.0 0.83 130 38

0c2

32 0.0 51 22 8.9 1.5 0.0 0.0

0c252 0.0 23 34 4.0 2.3 0.0 0.0

0c3

52 129 26 14 4.5 0.94 84 25

0c372 129 15 19 2.6 0.13 84 25

00c0

12 0.0 0.69 0.13 0.29 1.2 0.0 0.0

0c1

12 15 0.92 0.044 0.39 0.38 11 0:64 103

0c1

32 15 0.23 0.11 0.096 0.96 11 0:64 103

01c112 0.0 39 12 12 0.21 0.0 0.0

01c1

32 0.0 9.9 30 3.0 5.2 0.0 0.0

1c0

12 0.0 39 47 12 8.3 0.0 0.0

1c1

12 110 20 5.9 6.1 1.0 69 42

1c1

32 110 5.0 15 1.5 2.6 69 42

1c2

32 0.0 25 3.0 7.6 0.52 0.0 0.0

1c2

52 0.0 0.0 18 0.0 3.1 0.0 0.0

02c2

32 0.0 9.2 6.0 3.9 0.75 0.0 0.0

02c252 0.0 5.8 10 1.1 2.1 0.0 0.0

2c1

12 22 6.1 2.3 2.6 0.54 14 15

2c1

32 22 1.5 5.6 0.64 1.3 14 15

2c2

32 0.0 14 5.2 5.8 0.77 0.0 0.0

2c2

52 0.0 3.9 14 0.74 3.0 0.0 0.0

2c3

52 21 4.4 2.5 0.85 0.23 14 4.3

2c3

72 21 2.5 3.4 0.48 0.31 14 4.3


094017-8

(a) (b) (c)

(d) (e) (f)

FIG. 5. The dependence of the total decay width of c3077 on the parameter or if c3077 is a radial excitation. In thesefigures, we fix 0:6 GeV for the case with n 1, and 0:6 GeV for n 1. The situation of c2980 as a radial excitationis very similar.

FIG. 6. The variation of the decay width of c 2455 !c

with and .

FIG. 7. The variation of the decay width of c 2593 !c 2455 with and . Here c 2593 is assigned asc1

12.


094017-9

ple harmonic oscillator radial wave functions for the had-rons. Even with the above uncertainty, the 3P0 model is stillthe most systematic, effective, and widely used frameworkto study the hadron strong decays.

In this work, we have calculated the strong decay widthsof charmed baryons using the 3P0 model. Our numericalresults do not strongly depend on the parameters andas shown in Figs. 6–8. Thus the following qualitativefeatures and conclusions remain essentially unchangedwith reasonable variations of and .

Our results for the S-wave charmed baryons;;0c 2455, ;;0c 2520, and ;0c 2645 areroughly consistent with experimental data within the in-herent uncertainty of the 3P0 model. As a by-product, wehave also calculated the strong decays of b and bobserved by CDF Collaboration recently. The numericalresults are consistent with the experimental values too.

The decay width of the P-wave baryon c 2593 is 3times larger than the experimental value. With the largeexperimental uncertainty and the inherent theoretical un-certainty of the 3P0 model, such a deviation is still accept-able. The decay widths of c 2625 and ;0c 2790; 2815are compatible with the experimental upper bound. Bycomparing our results with the experimental total width,we tend to exclude the c0

12 assignment for c 2800.

Since the c 2800 is observed in the c channel[36], there are only two assignments left for c 2800,i.e. c2

32 or c2

52. More experimental information

such as the ratio c 2800!;c ;0c 2800!?;

c ;0will be helpful in

the determination of the quantum number of c 2800.We have also calculated the strong decay widths of

newly observed c2880; 2940, 2980; 3077;0 as-

suming they are candidates of D-wave charmed baryons.We find that the only possible assignment of c2880 is2c3

52 after considering both its total decay width and

the ratio ?c=c, which agrees very well with

the indication from the Belle experiment that c2880

favors JP 52 by the analysis of the angular distribution

[2].Unfortunately the experiment information about the

c2940, 2980; 3077;0 is scarce at the present.From their calculated decay widths, we can only excludesome assignments which are marked with crosses inTables VII, VIII, and IX. The decay width ratios ofc2940, 2980; 3077;0 from the 3P0 model willbe useful in the identification of their quantum numbersin the future since the inherent uncertainty cancels largely.

We have also discussed the strong decays of2980; 3077;0 assuming they are radial excitations.Unfortunately the numerical results in Fig. 5 depend quitestrongly on the node of the spatial wave function which isrelated to the parameters of the harmonic oscillator wavefunctions as shown in Fig. 5. We are unable to make strongconclusions here.

ACKNOWLEDGMENTS

C. C. thanks W. J. Fu for the help in the numericalcalculation and Y. R. Liu and B. Zhang for useful discus-sions. This project was supported by the NationalNatural Science Foundation of China under GrantNo. 10421503 and No. 10625521, Chinese Ministry ofEducation and the China Postdoctoral Science foundation(No. 20060400376).

APPENDIX

1. The harmonic oscillator wave functions used in ourcalculation

For the S-wave charmed baryon,

0; 0; 0; 0 33=4

1

2

3=4

1

2

3=4

exp

p2

22

p2

22

: (A1)

For the P-wave charmed baryon,

1; m; 0; 0 i33=4

8

3p

1=21=2

5=4Ym

1 p

1

2

3=4

exp

p2

22

p2

22

; (A2)

0; 0; 1; m i33=4

8

3p

1=2

1

2

5=4

Ym1 p

1

2

3=4

exp

p2

22

p2

22

: (A3)

FIG. 8. The variation of the decay width of c 2880 !c 2520 with and . Here c 2880 is assignedas c2

52.


094017-10

For the D-wave charmed baryon,

2; m; 0; 0 33=4

16

15p

1=2

1

2

7=4

Ym2 p

1

2

3=4

exp

p2

22

p2

22

; (A4)

0; 0; 2; m 33=4

16

15p

1=2

1

2

7=4

Ym2 p

1

2

3=4

exp

p2

22

p2

22

; (A5)

1; m; 1; m0 33=4

8

3p

1=2

1

2

5=4

Ym1 p

8

3p

1=2

1

2

5=4

Ym01 p

exp

p2

22

p2

22

: (A6)

Here Yml p is the solid harmonic polynomial.

The ground state wave function of the meson is

0; 0 R2

3=4

expR2p2 p5

2

8

: (A7)

The wave function of the first radially excited charmedbaryon n; n reads as

1; 0 33=4

2

3

s 1

2

3=2

3

2

p2

2

exp

p2

22

p2

22

;

0; 1 33=4

2

3

s 1

2

3=2

3

2

p2

2

exp

p2

22

p2

22

;

where n and n denote the radial quantum number be-tween the two light quarks and between the heavy quark

and the two light quarks, respectively. Here p 12

qp1

p2 and p 16

qp1 p2 2p3 for the above expres-

sions. All the above harmonic oscillator wave functionscan be normalized as

Rdp1dp2dp3j j2 1.

2. The momentum space integration

The momentum space integration lA; mA; lA;mA;m includes:

For the S-wave charmed baryon decay,

0; 0; 0; 0; 0 jpj0;0: (A8)

For the P-wave charmed baryon decay,

0; 0; 1; 0; 0 1

2f1f2jpj2 0;1;

0; 0; 1; 1;1 0; 0; 1;1; 1

2f10;1;

1; 0; 0; 0; 0 $jpj2

1

22p1

241f1

1;0;

1; 1; 0; 0;1 1;1; 0; 0; 1 $jpj21;0:

For the D-wave charmed baryon decay,

0; 0; 2; 0; 0 f2

f21

f2

2jpj3 jpj

0;2; 0; 0; 2; 1;1 0; 0; 2;1; 1

3pf2

2f21

jpj0;2;

2; 0; 0; 0; 0 2$2jpj3

12p1

$jpj 2

21f1$

2;0;

2; 1; 0; 0;1 2;1; 0; 0; 1

12p1

2

21f1

2;0;

1; 0; 1; 0; 0 f2

2f1$jpj3

1

2f1

221

$2f2

41f1

jpj

f2

42p1f1

jpj

1;1;

1; 1; 1;1; 0 1;1; 1; 1; 0 2

41f1jpj

1;1; 1; 0; 1; 1;1 1; 0; 1;1; 1

1

2f1$jpj

1;1;

1; 1; 1; 0;1 1;1; 1; 0; 1

1

22p1

2

41f1f2

2f1jpj

1;1:


094017-11

For the strong decay of the radial excitation, the momentum space integrals denoted as n; n are:

0; 1

2

3

s

f22

42f

21

k3 32k

3

2f12

kf2

2f21

2

k

0;0;

1; 0

2

3

s 1

2

$2k3

32

2k

321

k

2p$

21k

2$31

k2

2

421

0;0;

where

1 1

2

1

4R2; 2

1

23p R2; 3

1

2

1

12R2; 4

12p2

1

22p R2;

5

16p2

1

26p R2

; 6

1

42

1

122

1

8R2; f1 3

22

41; f2 5

224

41;

f3 6 2

4

41;

2

22p1

16p ; $

2f2

41f14

21;

1

3p2f2 2

3p4f1 21f2

46p1f1

;

and

0;0

1

2

3=4

1

2

3=4R2

3=42

1f1

3=2

exp

f3

f22

4f1

jpj2

3

4

s 1

2

3=4

1

2

3=4;

0;1

1

2

3=4

1

2

3=4R2

3=42

1f1

3=2

exp

f3

f22

4f1

jpj2

3i4

1

2

3=4

8

3p

1=2

1

2

5=4;

1;0

1

2

3=4

1

2

3=4R2

3=42

1f1

3=2

exp

f3

f22

4f1

jpj2

3i4

8

3p

1=2

1

2

5=4

1

2

3=4;

0;2

1

2

3=4

1

2

3=4R2

3=42

1f1

3=2

exp

f3

f22

4f1

jpj2

15p

8

1

2

3=4

16

15p

1=2

1

2

7=4;

2;0

1

2

3=4

1

2

3=4R2

3=42

1f1

3=2

exp

f3

f22

4f1

jpj2

15p

8

16

15p

1=2

1

2

7=4

1

2

3=4;

1;1

1

2

3=4

1

2

3=4R2

3=42

1f1

3=2

exp

f3

f22

4f1

jpj2

3

4

3=2 8

3p

1

2

2

5=4:

In the above expressions, jpj reads as

jpj

m2

A mB mC2m2

A mB mC2

q2mA

:

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Documents

Strong decays of charmed baryons