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
(Dated: October 29, 2018)
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 using the 3P0 model. After comparing the calculated decay pattern and total width with
the available data, we discuss the possible internal structure and quantum numbers of those charmed
baryons observed recently.
PACS numbers: 13.30.Eg, 12.39.Jh
I. INTRODUCTION
Babar and Belle collaborations observed sev-
eral excited charmed baryons: Λc(2880, 2940)
Ξc(2980, 3077)
+,0 and Ωc(2768)
0 last year [1, 2, 3, 4, 5],
which inspired several investigations of these states
in literature [6, 7, 8, 9]. We collect the experimental
information of these recently observed hadrons in Table
I. Their quantum numbers have not been determined
except Λc(2880)
+. In order to understand their struc-
tures using the present experimental information, we
study the strong decay pattern of the excited charmed
baryons systematically in this work. In the past decades,
there has been some research work on heavy baryons
[8, 11, 12].
The quantum numbers and decay widths of S-wave and
some P-wave charmed baryons are known [13]. We first
systematically analyze their strong decays in the frame-
work of the 3P0 strong decay model. Accordingly one
can extract the parameters and estimate the accuracy of
the 3P0 model when it’s applied in the charmed baryon
system. Then we go one step further and extend the
same formalism to study the decay patterns of these new
charmed baryons Λc(2880, 2940)
+, Ξ(2980, 3077)+,0 un-
der different assignments of their quantum numbers. Af-
ter comparing the theoretical results with the available
experimental data, we can learn their favorable quantum
numbers and assignments in the quark model.
State Mass and Width (MeV) Decay channels in experiments Other information
2881.5 ± 0.3, < 8 [10] Λcπ
2881.9 ± 0.1± 0.5 , 5.8± 1.5± 1.1 [1] D0p
JP favors 5
Λc(2880)
2881.2 ± 0.2+0.4
−0.3, 5.5
−0.5 ± 0.4 [2] Σ
⋆0,++
c (2520)π
Γ(Σ⋆c(2520)π
Γ(Σc(2455)π±)
= 0.225 ± 0.062 ± 0.025 [2]
2939. ± 1.3± 1.0, 17.5± 5.2± 5.9 [1] D0pΛc(2940)
2937.9 ± 1.0+1.8
−0.4, 10± 4± 5 [2] Σc(2455)
0,++π+,−
2967.1 ± 1.9± 1.0, 23.6± 2.8± 1.3 [3] Λ+c K
−π+Ξc(2980)
2978.5 ± 2.1± 2.0, 43.5± 7.5± 7.0 [4] Λ+c K
Ξc(2980)
0 2977.1 ± 8.8± 3.5, 43.5 [4] Λ+c K
3076.4 ± 0.7± 0.3, 6.2± 1.6± 0.5 [3] Λ+c K
−π+Ξc(3077)
3076.7 ± 0.9± 0.5, 6.2± 1.2± 0.8 [4] Λ+c K
Ξc(3077)
0 3082.8 ± 1.8± 1.5, 5.2± 3.1± 1.8 [4] Λ+c K
Ωc(2768)
0 2768.3 ± 3.0 [5] Ω0cγ J
P = 3
TABLE I: A summary of recently observed charmed baryons by Babar and Belle collaborations.
Very recently CDF collaboration reported four par-
ticles [14, 15], which are consistent with Σ±b and Σ
predicted in the quark model [16]. Their masses are
= 5808+2.0−2.3± 1.7 MeV, MΣ−
= 5816+1.0−1.0± 1.7MeV,
= 5829+1.6−1.8±1.7,MΣ∗−
= 5837+2.1−1.9±1.7MeV. The
http://arxiv.org/abs/0704.0075v3
mass splitting between Σb and Σ
b was discussed in Refs.
[17, 18] while the strong decays of Σ
b were studied in
Ref. [19]. As a byproduct, we also calculate the strong
decays of Σ
b and other S-wave bottom baryons in this
work.
This paper is organized as follows. We give a short the-
oretical review of S-wave, P-wave and D-wave charmed
baryons and introduce our notations for them in Section
II. Then we give a brief review of 3P0 model in Section
III. We present the strong decay amplitudes of charmed
baryons in Section IV. Section V is the numerical results.
The last section is our discussion and conclusion. Some
lengthy formulae are collected in the Appendix.
II. THE NOTATIONS AND CONVENTIONS OF
CHARMED BARYON
We first introduce our notations for the excited
charmed baryons. Inside a charmed baryon there are one
charm quark and two light quarks (u, d or s). It belongs
to either the symmetric 6F or antisymmetric 3̄F flavor
representation (see Fig. 1). For the S-wave charmed
baryons, the total color-flavor-spin wave function and
color wave function must be symmetric and antisymmet-
ric respectively. Hence the spin of the two light quarks
is S=1 for 6F or S=0 for 3̄F . The angular momentum
and parity of the S-wave charmed baryons are JP = 1
for 6F and J
P = 1
for 3̄F . The names of S-wave
charmed baryons are listed in Fig. 1, where we use the
star to denote 3
baryons and the prime to denote the
JP = 1
baryons in the 6F representation.
c (ssc)
(∗)++
c (uuc)
c (ddc) Ξ
′(∗)0
c (dsc)
c (udc) Ξ
′(∗)+
c (usc)
Ξ0c(dsc)
Λ+c (udc) Ξ
c (usc)
FIG. 1: The SU(3) flavor multiplets of charmed baryons
In Fig. 2 we introduce our notations and conventions
for the P-wave charmed baryons. lρ is the orbital angular
momentum between the two light quarks while lλ denotes
the orbital angular momentum between the charm quark
and the two light quark system. We use the prime to
label the ΞcJl baryons in the 6F representation and the
tilde to discriminate the baryons with lρ = 1 from that
with lλ = 1.
The notation for D-wave charmed baryons is more
complicated (see Fig. 3). Besides the prime, lρ and lλ
(a) lρ = 0, lλ = 1
(b) lρ = 1, lλ = 0
Jl = 1: Σc1(
) Ξ′c1(
Jl = 0: Σc0(
) Ξ′c0(
Jl = 2: Σc2(
) Ξ′c2(
fS(6): L = 1 ⊗ Sq1q2 = 1
fA(3̄): L = 1 ⊗ Sq1q2 = 0 =⇒ Jl = 1: Λc1(
) Ξc1(
Jl = 1: Λ̃c1(
) Ξ̃c1(
Jl = 0: Λ̃c0(
) Ξ̃c0(
Jl = 2: Λ̃c2(
) Ξ̃c2(
fA(3̄): L = 1 ⊗ Sq1q2 = 1
fS(6): L = 1 ⊗ Sq1q2 = 0 =⇒ Jl = 1: Σ̃c1(
) Ξ̃′c1(
FIG. 2: The notations for P-wave charmed baryons. fS(6F )
and fA(3̄F ) denote the SU(3) flavor representation. Sq1q2 is
the total spin of the two light quarks. L denotes the total
orbital angular momentum of charmed baryon system.
defined above, we use the hat and check to denote the
charmed baryons with lρ = 2 and lρ = 1 respectively. For
the baryons with lρ = 1 and lλ = 1, we use the super-
script L to denote the different total angular momentum
in Λ̌LcJl , Σ̌
and Ξ̌LcJl .
III. THE
3P0 MODEL
The 3P0 model was first proposed by Micu [20] and
further developed by Yaouanc et al. later [21, 22, 23].
Now this model is widely used to study the strong decays
of hadrons [24, 25, 26, 27, 28, 29, 30, 31].
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 qq̄ pair created from the vacuum
together with the qqq within the the initial baryon re-
group into the outgoing meson and baryon via the quark
rearrangement process. In the non-relativistic limit, the
transition operator is written as
T = −3γ
〈1 m; 1 −m|0 0〉
d3k4 d
3(k4 + k5)
k4 − k5
χ451,−m ϕ
4i(k4) d
5j(k5) (1)
where i and j are the color indices of the created quark
and anti-quark. ϕ450 = (uū+ dd̄ + ss̄)/
3 and ω450 = δij
for the flavor and color singlet respectively. χ451,−m is for
(a) lρ = 0, lλ = 2
(b) lρ = 2, lλ = 0
Jl = 2: Σc2(
) Ξ′c2(
Jl = 1: Σc1(
) Ξ′c1(
Jl = 3: Σc3(
) Ξ′c3(
fS(6): L = 2 ⊗ Sq1q2 = 1
fA(3̄): L = 2 ⊗ Sq1q2 = 0 =⇒ Jl = 2: Λc2(
) Ξc2(
Jl = 2: Σ̂c2(
) Ξ̂′c2(
Jl = 1: Σ̂c1(
) Ξ̂′c1(
Jl = 3: Σ̂c3(
) Ξ̂′c3(
fS(6): L = 2 ⊗ Sq1q2 = 1
fA(3̄): L = 2 ⊗ Sq1q2 = 0 =⇒ Jl = 2: Λ̂c2(
) Ξ̂c2(
(c) lρ = 1, lλ = 1
Jl = 1: Λ̌
) Ξ̌1c1(
Jl = 0: Λ̌
) Ξ̌1c0(
Jl = 2: Λ̌
) Ξ̌1c2(
L = 1 ⊗ Sq1q2 = 1
L = 0 ⊗ Sq1q2 = 1 =⇒ Jl = 1: Λ̌
) Ξ̌0c1(
)fA(3̄)
Jl = 2: Λ̌
) Ξ̌2c2(
Jl = 1: Λ̌
) Ξ̌2c1(
Jl = 3: Λ̌
) Ξ̌2c3(
L = 2 ⊗ Sq1q2 = 1
L = 0 ⊗ Sq1q2 = 0 =⇒ Jl = 0: Σ̌
) Ξ̌′0c0(
L = 1 ⊗ Sq1q2 = 0 =⇒ Jl = 1: Σ̌
) Ξ̌′1c1(
L = 2 ⊗ Sq1q2 = 0 =⇒ Jl = 2: Σ̌
) Ξ̌′2c2(
fS(6)
FIG. 3: The notations for the D-wave charmed baryons.
the spin triplet state. Ym1 (k) ≡ |k|Y m1 (θk, φk) is a solid
harmonic polynomial corresponding to the p-wave quark
pair. γ is a dimensionless constant related to the strength
of the quark pair creation from the vacuum, which was
extracted by fitting to data. The hadron and meson state
are defined as respectively according to the definition of
FIG. 4: The decay process of A → B +C in 3P0 model.
mock state [32]
|A(nA2SA+1LA JAMJA )(PA)〉
MLA ,MSA
〈LAMLASAMSA |JAMJA〉
d3k1d
3(k1 + k2 + k3−PA)
×ψnALAMLA(k1,k2,k3)χ
SAMSA
ϕ123A ω
×| q1(k1)q2(k2)q3(k3)〉,
|B(nB2SB+1LB JBMJB )(PB)〉
MLB ,MSB
〈LBMLBSBMSB |JBMJB 〉
d3kad
3(ka + kb−PB)ψnBLBMLB(ka,kb)
×χabSBMSBϕ
B | qa(ka)q̄b(kb)〉 (3)
and satisfy the normalization condition
〈A(PA)|A(P′A)〉 = 2EAδ3(PA −P
〈B(PB)|B(P′B)〉 = 2EBδ3(PB −P
B) . (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. ki(i = 1, 2, 3) are the momen-
tum of quarks in hadron A. ka and kb are the momentum
of the quark and antiquark in meson B. PA(B) represents
the momentum of state A(B). SA(B) and JA(B) denote
the total spin and the total angular momentum of state
A(B).
The S-matrix is defined as
〈f |S|i〉 = I − i2πδ(Ef − Ei)MMJAMJBMJC . (5)
The helicity amplitude of the process A→ B + C in the
center of mass frame of meson A is
MMJAMJBMJC (A→ BC)
8EAEBEC γ
MLA ,MSA ,
MLB ,MSB ,
MLC ,MSC ,m
〈LAMLASAMSA |JAMJA〉
×〈LBMLBSBMSB |JBMJB 〉〈LCMLCSCMSC |JCMJC 〉
×〈1 m; 1 −m| 0 0〉 〈χ235SCMSC χ
SBMSB
|χ123SAMSAχ
×〈ϕ235C ϕ14B |ϕ123A ϕ450 〉 I
MLA ,m
MLB ,MLC
(p) (6)
where the spatial integral I
MLA ,m
MLB ,MLC
(p) is defined as
MLA ,m
MLB ,MLC
d3k1d
3(k4 + k5)
×δ3(k1 + k2 + k3 −PA)δ3(k1 + k4 −PB)
×δ3(k2 + k3 + k5 −PC)
×ψ∗nBLBMLB(k1,k4)ψ
nCLCMLC
(k2,k3,k5)
×ψnALAMLA(k1,k2,k3) Y
(k4 − k5
. (7)
〈χ235SCMSC χ
SBMSB
|χ123SAMSAχ
1−m〉 and 〈ϕ235C ϕ14B |ϕ123A ϕ450 〉
denote the spin and flavor matrix element respectively.
The decay width of the process A→ B + C is
Γ = π2
2JA + 1
MJA ,MJB ,MJC
∣MMJAMJBMJC
where |p| is the momentum of the daughter baryon in
the parent’s center of mass frame. s = 1/(1 + δBC) is a
statistical factor which is needed if B and C are identical
particles.
IV. THE STRONG DECAYS OF CHARMED
BARYON
According to the 3P0 model, the decay occurs through
the recombination of the five quarks from the initial
charmed baryon and the created quark pair. So there
are three ways of regrouping:
A(q1, q2, c3) + P(q4, q̄5) → B(q2, q4, c3) + C(q1, q̄5), (8)
A(q1, q2, c3) + P(q4, q̄5) → B(q1, q4, c3) + C(q2, q̄5), (9)
A(q1, q2, c3) + P(q4, q̄5) → B(q1, q2, q4) + C(c3, q̄5)(10)
where qi and c3 denote the light quark and charm quark
respectively.
When the excited charmed baryon decays into a
charmed baryon plus a light meson as shown in Eq. (8)
and (9), the total decay amplitude reads
MMJAMJBMJC
= −2γ
8EAEBEC
m1,m3,m4,m
×〈J12M12; s3m3|JAMJA〉〈lρAmρA; lλAmλA|LAMLA〉〈LAMLA ; S1 2m1 2|J12M12〉
×〈s1m1; s2m2|S1 2m1 2〉 〈J14M14; s3m3|JB MJB 〉 〈lρBmρB; lλBmλB|LB MLB〉
×〈LBMLB ; S1 4m1 4|J14M14〉〈s1m1; s4m4|S1 4m1 4〉〈1m; 1 −m|00〉 〈s4m4; s5m5|1 −m〉
×〈LC MLC ; SC MC |JC MJC〉〈s2m2; s5m5|SCMC〉 × 〈φ
1,4,3
1,2,3
A 〉 × I
MLA ,m
MLB ,MLC
(p), (11)
where the pre-factor 2 in front of γ arises from the fact
that the amplitude from the Eq. (8) is the same as that
from Eq. (9).
The overlap integral in the momentum space is
MLA ,m
MLB ,MLC
= δ3(PB −PC)
d3p1d
B(lρB ,mρB, lλB,mλB)
×ψ∗C(LC MLC)Ym1
(p4 − p5
ψA(lρA,mρA, lλA,mλA).
Since all hadrons in the final states are S-wave in this
work, eq. (12) can be further expressed as
MLA ,m
MLB ,MLC
= δ3(PB −PC)Π(lρA,mρA, lλA,mλA,m), (13)
where we have used the harmonic oscillator wave func-
tions for both the meson and baryon. The expressions
of Π(lρA,mρA, lλA,mλA,m) for the decays of S-wave, P-
wave and D-wave charmed baryons are collected in the
Appendix. We also move the lengthy expressions of mo-
mentum space integration of S-wave, P-wave and D-wave
charmed baryons to the Appendix.
V. NUMERICAL RESULTS
The decay widths of charmed baryons from the 3P0
model involve several parameters: the strength of quark
pair creation from vacuum γ, the R value in the harmonic
oscillator wave function of meson and the αρ,λ in the
baryon wave functions. We follow the convention of Ref.
[34] and take γ = 13.4, which is considered as a universal
parameter in the 3P0 model. The R value of π and K
mesons is 2.1 GeV−1 [34] while it’s R = 2.3 GeV−1 for
the D meson [35]. αρ = αλ = 0.5 GeV for the proton and
Λ [31]. For S-wave charmed baryons, the parameters αρ
and αλ in the harmonic oscillator wave functions can be
fixed to reproduce the mass splitting through the contact
term in the potential model [33]. Their values are αρ =
0.6 GeV and αλ = 0.6 GeV. For P-wave and D-wave
charmed baryons, αρ and αλ are expected to lie in the
range 0.5 ∼ 0.7 GeV. In the following, our numerical
results are obtained with the typical values αρ = αλ =
0.6 GeV.
The strong decay widths of the S-wave charmed
baryons Σ++,+,0c (2455), Σ
∗++,+,0
c (2520) and Ξ
c (2645)
are listed in Table II. Accordingly the decay widths of
S-wave bottomed baryons are presented in Table III. Be-
cause Ξb, Ξ
b and Ξ
b have not been observed so far, their
masses are taken from the theoretical estimate in Ref.
[36], which are mΞb = 5805.7 MeV, mΞ′
= 5950 MeV
and mΞ∗
= 5966.1 MeV.
The quantum number and internal structure of the
following P-wave charmed baryons Λ+c (2593), Λ
c (2625),
Ξ+,0c (2790) and Ξ
c (2815) are relatively known exper-
imentally [13]. Their strong decay modes and widths
from the 3P0 model are collected in Table IV. The quan-
tum number of Σ++c (2800) is still unknown [13]. Thus
under different P-wave assignments of Σ++c (2800), we
present the strong decay widths of its possible decay
modes in Table V. In the heavy quark limit, the pro-
cess Σ++c (2800) → Λ+c π+ is forbidden if Σ++c (2800) is
assigned as Σc1(
), Σc1(
), Σ̃c1(
) and Σ̃c1(
which is observed in our calculation as can be seen from
Table V.
Λc(2880)
+ and Λc(2940)
+ are observed in the invariant
mass spectrum of D0p [1]. The first radial excitation of
Λc does not decay into D
0p from the 3P0 model. Hence
the possibility of Λc(2880)
+ and Λc(2940)
+ being a radial
excitation is excluded. We calculate their strong decays
assuming they are D-wave charmed baryons. The results
are shown in Table VI and VII.
With positive parity, Ξ(2980)+,0 and Ξ(3077)+,0 can
be either the first radially excited charmed baryons or the
D-wave charmed baryons. With different assumptions of
their quantum numbers we present their strong decay
widths in Table VIII, IX and Fig. 8.
The numerical results depend on the parameters αρ
and αλ in the harmonic oscillator wave functions of
the charmed baryons. We illustrate such a dependence
in Figs. 5, 6 and 7 using several typical decay chan-
nels: Σ++c (2455) → Λ+c π+, Λ+c (2593) → Σ++c (2455)π−
and Λ+c (2880) → Σ∗++c (2520)π−, where Σ++c (2455),
Λ+c (2593) and Λ
c (2880) are S-wave, P-wave and D-wave
baryons respectively.
0.70 0.50
 (GeV
 (GeV)
FIG. 5: The variation of the decay width of Σ++c (2455) →
Λ+c π
+ with αρ and αλ.
0.70 0.56
FIG. 6: The variation of decay width of Λ+c (2593) →
Σ++c (2455)π
− with αρ and αλ. Here Λ
c (2593) is assigned
as Λc1(
0.640
 (GeV)
FIG. 7: The variation of decay width of Λ+c (2880) →
Σ∗++c (2520)π
− with αρ and αλ. Here Λ
c (2880) is assigned
as Λc2(
TABLE II: The strong decay widths of S-wave charmed
baryons Σ++,+,0c (2455), Σ
∗++,+,0
c (2520) and Ξ
c (2645).
Here all results are in units of MeV.
JP Channel Width Total width (Exp) [13]
Σ++c (2455)
Λ+c π
+ 1.24 2.23 ± 0.30
Σ+c (2455)
Λ+c π
0 1.40 < 4.6
Σ0c(2455)
Λ+c π
− 1.24 2.2± 0.40
Σ∗++c (2520)
Λ+c π
+ 11.9 14.9 ± 1.9
Σ∗+c (2520)
Λ+c π
0 12.1 < 17
Σ∗0c (2520)
Λ+c π
− 11.9 16.1 ± 2.1
Ξ∗+c (2645)
Ξ+c π
0 0.64
Ξ∗+c (2645)
+ 0.49
< 3.1
Ξ∗0c (2645)
Ξ+c π
− 0.54
Ξ∗0c (2645)
0 0.54
< 5.5
VI. DISCUSSION AND CONCLUSION
At present it is still too difficult to calculate the strong
decay widths of hadrons from the first principles of QCD.
For this purpose, some phenomenological strong decay
models were proposed such as the 3P0 model, flux tube
model, QCD sum rule, lattice QCD etc, among which
only the first two approaches can be applied to the strong
decays of excited hadrons. To a large extent, the predic-
tions from the 3P0 and flux tube models roughly agree
with each other.
TABLE III: The strong decay widths of S-wave bottom
baryons Σb, Σ
b , Ξ
b and Ξ
b . Here all results are in units
of MeV.
JP Channel Width Experimental results [14]
+ 3.5
− 4.7
+ 7.5
− 9.2
Ξbπ 0.10 -
Ξbπ 0.85 -
TABLE IV: The decay widths of P-wave charmed baryons
Λ+c (2593, 2625) and Ξ
c (2790, 2815) with the fixed structure
and quantum number assignments. Here all results are in
units of MeV.
Assignment Channel Γ ΓExp [13]
Σ++c π
− 3.4
Σ+c π
0 6.4Λ
c (2593) Λc1(
+ 3.4
3.6+2.0
Σ++c π
− 1.9× 10−3 < 0.10
Σ+c π
0 2.6× 10−3 < 1.9Λ
c (2625) Λc1(
+ 1.9× 10−3 < 0.10
Ξ′+c π
0 5.0
Ξ+c (2790) Ξc1(
Ξ′0c π
+ 4.9
Ξ′+c π
− 5.2
Ξ0c(2790) Ξc1(
Ξ′0c π
0 5.1
Ξ⋆+c π
0 2.7
Ξ+c (2815) Ξc1(
Ξ⋆0c π
+ 2.6
< 3.5
Ξ⋆+c π
− 2.7
Ξ0c(2815) Ξc1(
Ξ⋆0c π
0 2.8
< 6.5
The 3P0 model possesses inherent uncertainties [21, 29,
34]. In certain cases, the result from the 3P0 model may
be a factor of 2 ∼ 3 off the experimental width. The un-
certainty source of the 3P0 model arises from the strength
of the quark pair creation from the vacuum γ, the approx-
imation of non-relativity, and assuming the simple har-
monic oscillator radial wave functions for the hadrons.
Even with the above uncertainty, the 3P0 model is still
the most systematic, effective and widely used framework
to study the hadron strong decays.
In this work, we have calculated the strong decay
widths of charmed baryons using the 3P0 model. Our
numerical results do not strongly depend on the param-
eters αρ and αλ as shown in Figs. 5, 6 and 7. Thus the
following qualitative features and conclusions remain es-
sentially unchanged with reasonable variations of αρ and
Our results for the S-wave charmed baryons
Σ++,+,0c (2455), Σ
∗++,+,0
c (2520) and Ξ
c (2645) are
roughly consistent with experimental data within the
inherent uncertainty of the 3P0 model. As a byproduct,
we have also calculated the strong decays of Σ±b and Σ
TABLE V: The decay widths of Σ++c (2800) in dif-
ferent P-wave charmed baryons assignments. R =
Σ+,++c π
+,0/Σ⋆+,++c π
+,0. The total width of Σ++c (2800) is
75+22
−17 MeV [13]. Here all results are in units of MeV.
Assignment Λ+c π
+ Σ+,++c π
+,0 Σ⋆+,++c π
) 307 0.0 0.0 -
) 0.0 296 0.4 740
) 0.0 0.7 220 3× 10−3
) 8.1 1.3 0.3 4.3
) 8.1 0.6 0.5 1.2
Σ̃c1(
) 0.0 75 69 1.1
Σ̃c1(
) 0.0 75 69 1.1
observed by CDF Collaboration recently. The numerical
results are consistent with the experimental values too.
The decay width of P-wave baryon Λ+c (2593) is three
times larger than the experimental value. With the
large experimental uncertainty and the inherent the-
oretical uncertainty of the the 3P0 model, such a
deviation is still acceptable. The decay widths of
Λ+c (2625) and Ξ
c (2790, 2815) are compatible with the
experimental upper bound. By comparing our results
with the experimental total width, we tend to exclude
the Σc0(
) assignment for Σ++c (2800). Since the
Σ++c (2800) is observed in Λ
+ channel [37], there are
only two assignments left for Σ++c (2800), i.e. Σc2(
or Σc2(
). More experimental information such as the
ratio
Γ[Σ++c (2800)→Σ
c (2800)→Σ
⋆+,++
c π+,0]
will be helpful in the deter-
mination of the quantum number of Σ++c (2800).
We have also calculated the strong decay widths of
newly observed Λc(2880, 2940)
+, Ξ(2980, 3077)+,0 as-
suming they are candidates of D-wave charmed baryons.
We find that the only possible assignment of Λc(2880)
Λ̌2c3(
) after considering both its total decay width and
the ratio Γ(Σ⋆cπ
±)/Γ(Σcπ
±), which agrees very well with
the indication from Belle experiment that Λc(2880)
+ fa-
vors JP = 5
by the analysis of the angular distribution
Unfortunately the experiment information about the
Λc(2940)
+, Ξ(2980, 3077)+,0 is scarce at present. From
their calculated decay widths, we can only exclude some
assignments which are marked with crosses in Tables
VII, VIII and IX. The decay width ratios of Λc(2940)
Ξ(2980, 3077)+,0 from the 3P0 model will be useful in
the identification of their quantum numbers in the fu-
ture since the inherent uncertainty cancels largely.
We have also discussed the strong decays of
Ξ(2980, 3077)+,0 assuming they are radial excitations.
Unfortunately the numerical results in Fig. 8 depend
quite strongly on the node of the spatial wave function
which is related to the parameters of the harmonic oscil-
lator wave functions as shown in Fig. 8. We are unable
to make strong conclusions here.
Appendix
A. The harmonic oscillator wave functions used in
our calculation
For the S-wave charmed baryon,
ψ(0, 0, 0, 0) = 33/4 (
4 exp
For the P-wave charmed baryon,
ψ(1,m, 0, 0) = −i 33/4
)1/2(
1/α2ρ
Ym1 (pρ)
, (15)
ψ(0, 0, 1,m) = −i 33/4
)1/2( 1
Ym1 (pλ)
. (16)
For the D-wave charmed baryon,
ψ(2,m, 0, 0) = 33/4
)1/2( 1
Ym2 (pρ)
,(17)
ψ(0, 0, 2,m) = 33/4
)1/2( 1
Ym2 (pλ)
,(18)
ψ(1,m, 1,m′) = −33/4
)1/2( 1
Ym1 (pρ)
)1/2( 1
1 (pλ)
× exp
. (19)
Here Yml (p) is the solid harmonic polynomial.
The ground state wave function of meson is
ψ(0, 0) =
R2(p2 − p5)2
. (20)
The the wave function of the first radially excited
charmed baryon ψ(nρ, nλ) reads as
ψ(1, 0)
= 33/4
π2αραλ
ψ(0, 1)
= 33/4
π2αραλ
where nρ and nλ denote the radial quantum number be-
tween the two light quarks and between heavy quark and
the two light quarks respectively. Here pρ =
(p1−p2)
and pλ =
(p1 + p2 − 2p3) for the above expressions.
All the above harmonic oscillator wave functions can be
normalized as
dp1dp2dp3|ψ|2 = 1.
B. The momentum space integration
The momentum space integration
Π(lρA,mρA, lλA,mλA,m) includes:
For the S-wave charmed baryon decay,
Π(0, 0, 0, 0, 0) = β|p| ∆0,0. (21)
For the P-wave charmed baryon decay,
Π(0, 0, 1, 0, 0) =
f2β|p|2 − ζ
∆0,1,
Π(0, 0, 1, 1,−1) = Π(0, 0, 1,−1, 1) = ζ
∆0,1,
Π(1, 0, 0, 0, 0) =
β̟|p|2 + 1
4λ1f1
∆1,0,
Π(1, 1, 0, 0,−1) = Π(1,−1, 0, 0, 1) = β̟|p|2 ∆1,0.
For the D-wave charmed baryon decay,
Π(0, 0, 2, 0, 0) = − f2
β |p|3 + ζ|p|
×∆0,2,
Π(0, 0, 2, 1,−1) = Π(0, 0, 2,−1, 1) =
|p|∆0,2,
Π(2, 0, 0, 0, 0) = −2
(β̟2|p|3 +
̟ |p|
2λ1f1
∆2,0,
Π(2, 1, 0, 0,−1) = Π(2,−1, 0, 0, 1)
2λ1f1
∆2,0,
Π(1, 0, 1, 0, 0) =
β̟|p|3 + 1
β + ζ̟
4λ1f1
)|p|+ f2
2λ1f1
∆1,1,
Π(1, 1, 1,−1, 0) = Π(1,−1, 1, 1, 0) =
4λ1f1
∆1,1,
Π(1, 0, 1, 1,−1) = Π(1, 0, 1,−1, 1) =
̟ ζ|p|
∆1,1,
Π(1, 1, 1, 0,−1) = Π(1,−1, 1, 0, 1)
4λ1f1
ζ)× f2
∆1,1.
For the strong decay of the radial excitation, the mo-
mentum space integrals denoted as Π(nρ, nλ) are:
Π(0, 1) =
4α2λf
k − 3β
2f21α
∆0,0,
Π(1, 0) =
(β̟2k3 −
3βα2ρ
k − λ2̟ζ
∆0,0.
where
R2, λ2 = −
R2, λ4 =
λ5 = −(
12α2λ
f1 = λ3 −
, f2 = λ5 −
2λ2λ4
f3 = λ6 −
, ζ =
4λ1f1
β = (1 +
3λ2f2 − 2
3λ4f1 + 2λ1f2
6λ1f1
∆0,0 = (
× exp
− (f3 −
)|p|2
∆0,1 = (
− (f3 −
)|p|2
∆1,0 = (
− (f3 −
)|p|2
∆0,2 = (
− (f3 −
)|p|2
∆2,0 = (
− (f3 −
)|p|2
∆1,1 = (
− (f3 −
)|p|2
− ( 3
In the above expressions, |p| reads as
|p| =
(m2A − (mB +mC)2)(m2A − (mB −mC)2)
Acknowledgments
C.C. thanks W.J. Fu for the help in the numeri-
cal calculation and Y.R. Liu and B. Zhang for useful
discussions. This project was supported by the Na-
tional Natural Science Foundation of China under Grants
10421503 and 10625521, Chinese Ministry of Educa-
tion and the China Postdoctoral Science foundation (No.
20060400376).
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TABLE VI: The decay widths of Λ+c (2880) with different D-wave assignments. All results are in units of MeV.
Assignment Σ0,+,++c π
+,0,− Σ⋆0,+,++c π
+,0,− Γ(Σ
Γ(Σcπ±)
D0p Remark
) 7.8 0.9 0.11 0.0 ×
) 0.06 5.34 89 0.0 ×
Λ̂c2(
) 78.3 59.1 0.75 0.0 ×
Λ̂c2(
) 78.3 59.1 0.75 0.0 ×
Λ̌0c1(
) 0.9 2.3 2.6 2.3 ×
Λ̌0c1(
) 0.22 6.0 27 2.3 ×
Λ̌1c0(
) 132 144 1.1 0.0 ×
Λ̌1c1(
) 66.3 18.0 0.27 150 ×
Λ̌1c1(
) 16.5 45.0 2.7 150 ×
Λ̌1c2(
) 82.8 9.0 0.10 0.0 ×
Λ̌1c2(
) 0.0 54.1 − 0.0 ×
Λ̌2c1(
) 25.7 8.1 0.32 64 ×
Λ̌2c1(
) 6.5 20.4 3.1 64 ×
Λ̌2c2(
) 57.9 14.2 0.24 0.0 ×
Λ̌2c2(
) 9.4 47.1 5.0 0.0 ×
Λ̌2c3(
) 10.8 5.5 0.51 12
Λ̌2c3(
) 6.1 7.4 1.2 12 ×
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TABLE VII: The decay widths of Λ+c (2940) with different D-wave assignments. Here all results are in units of MeV.
Assignment Σ0,+,++c π
+,0,− Σ⋆0,+,++c π
+,0,− Γ(Σ
Γ(Σcπ±)
D0p Remark
) 11.7 9.1 0.77 0.0 ×
) 0.2 9.1 46 0.0 ×
Λ̂c2(
) 170 150 0.88 0.0 ×
Λ̂c2(
) 170 150 0.88 0.0 ×
Λ̌0c1(
) 2.2 0.5 0.23 11
Λ̌0c1(
) 0.6 1.4 2.3 11
Λ̌1c0(
) 212 259 1.2 0.0 ×
Λ̌1c1(
) 106 32.4 0.31 340 ×
Λ̌1c1(
) 26.5 81.0 3.1 340 ×
Λ̌1c2(
) 142 16.2 0.11 0.0 ×
Λ̌1c2(
) 0.0 97.0 − 0.0 ×
Λ̌2c1(
) 34.5 12.6 0.37 95 ×
Λ̌2c1(
) 8.6 31.7 3.7 95 ×
Λ̌2c2(
) 77.7 27.7 0.36 0.0 ×
Λ̌2c2(
) 19.5 75.6 3.9 0.0 ×
Λ̌2c3(
) 22.2 12.9 0.58 49 ×
Λ̌2c3(
) 12.4 17.5 1.4 49 ×
0.2 0.4 0.6 0.8 1.0 1.2
 (GeV)
(3077): 
(1/2+) with (n =1, n =0)
0.4 0.6 0.8 1.0
(3077): 
(1/2+) with (n =0, n =1)
 (GeV)
0.2 0.4 0.6 0.8 1.0
 (GeV)
(3077): '
(1/2+) with (n =1, n =0)
(a) (b) (c)
0.4 0.5 0.6 0.7 0.8 0.9 1.0
 (GeV)
(3077): '
(1/2+) with (n =0, n =1)
0.2 0.4 0.6 0.8 1.0
(3077): 
(3/2+) with (n =1, n =0)
 (GeV)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
(3077): 
(3/2+) with (n =0, n =1)
 (GeV)
(d) (e) (f)
FIG. 8: The dependence of the total decay width of Ξc(3077) on the parameter αλ or αρ if Ξc(3077) is a radial excitation. In
these figures, we fix αρ = 0.6 GeV for the case with nλ = 1, and αλ = 0.6 GeV for nρ = 1. The situation of Ξc(2980) as a
radial excitation is very similar.
TABLE VIII: The decay widths of Ξ+c (2980) with different D-wave assignments. Here all results are in units of MeV.
Assignment Ξ0cπ
+ Ξ′0c π
+ Ξ⋆0c π
+ Σ++c k
− Λ+c k̄
0 Remark
) 0.0 1.1 0.11 0.37 0.0 ×
) 0.0 0.12 × 10−2 0.67 0.11 × 10−3 0.0 ×
Ξ′c1(
) 4.4 0.72 0.18 0.25 5.3
Ξ′c1(
) 4.4 0.18 0.46 0.062 5.3
Ξ′c2(
) 0.0 0.16 0.17 0.56 0.0 ×
Ξ′c2(
) 0.0 0.47 × 10−2 1.0 0.71 × 10−4 0.0 ×
Ξ′c3(
) 0.054 0.53 × 10−2 0.14 × 10−2 0.82 × 10−4 0.053 ×
Ξ′c3(
) 0.054 0.30 × 10−2 0.19 × 10−2 0.46 × 10−4 0.053 ×
Ξ̂c2(
) 0.0 9.5 6.1 0.61 0.0
Ξ̂c2(
) 0.0 9.5 6.1 0.61 0.0
Ξ̂′c1(
) 74 6.3 1.0 0.40 78 ×
Ξ̂′c1(
) 74 1.6 2.5 0.10 78 ×
Ξ̂′c2(
) 0.0 14 4.5 0.91 0.0
Ξ̂′c2(
) 0.0 6.3 7.1 0.40 0.0
Ξ̂′c3(
) 48 7.2 2.9 0.46 50 ×
Ξ̂′c3(
) 48 4.1 3.9 0.26 50 ×
) 0.0 0.30 1.4 1.3 0.0 ×
Ξ̌0c1(
) 1.0 0.40 0.46 1.7 0.46 ×
Ξ̌0c1(
) 1.0 0.10 1.2 0.43 0.46 ×
) 0.0 18 4.4 5.5 0.0
) 0.0 4.5 11 1.4 0.0
Ξ̌1c0(
) 0.0 18 18 5.5 0.0
Ξ̌1c1(
) 62 9.1 2.2 2.8 72 ×
Ξ̌1c1(
) 62 2.3 5.5 0.69 72 ×
Ξ̌1c2(
) 0.0 11 1.1 0.34 0.0 ×
Ξ̌1c2(
) 0.0 0.0 6.6 0.0 0.0 ×
) 0.0 5.6 1.8 2.4 0.0
) 0.0 1.7 4.32 0.24 0.0
Ξ̌2c1(
) 19 3.7 1.1 1.6 23
Ξ̌2c1(
) 19 0.93 2.6 0.40 23
Ξ̌2c2(
) 0.0 8.4 1.7 0.36 0.0
Ξ̌2c2(
) 0.0 1.2 6.0 0.16 0.0
Ξ̌2c3(
) 8.1 1.3 60 0.19 8.7 ×
Ξ̌2c3(
) 8.1 0.75 0.81 0.10 8.7
TABLE IX: The decay widths of Ξ+c (3077) with different D-wave assignments. Here all results are in units of MeV.
Assignment Ξ0cπ
+ Ξ′0c π
+ Ξ⋆0c π
+ Σ++c k
− Σ++c k
− Λ+c k̄
0 D+Λ Remark
) 0.0 2.1 0.30 0.73 0.054 0.0 0.0
) 0.0 0.037 1.7 0.42 × 10−2 0.32 0.0 0.0
Ξ′c1(
) 7.0 1.4 0.46 0.49 0.089 4.4 3.2
Ξ′c1(
) 7.0 0.36 1.1 0.12 0.22 4.4 3.2
Ξ′c2(
) 0.0 3.2 0.43 1.1 0.081 0.0 0.0
Ξ′c2(
) 0.0 0.025 × 10−2 2.5 0.28 × 10−2 0.48 0.0 0.0 ×
Ξ′c3(
) 0.19 0.029 0.012 0.32 × 10−2 0.32× 10−3 0.12 0.026 ×
Ξ′c3(
) 0.19 0.016 × 10−3 0.016 0.18 × 10−2 0.44× 10−3 0.12 0.026 ×
Ξ̂c2(
) 0.0 34 29 6.0 2.0 0.0 0.0 ×
Ξ̂c2(
) 0.0 34 29 6.0 2.0 0.0 0.0 ×
Ξ̂′c1(
) 201 23 4.8 4.0 0.33 130 38 ×
Ξ̂′c1(
) 201 5.7 12 1.0 0.83 130 38 ×
Ξ̂′c2(
) 0.0 51 22 8.9 1.5 0.0 0.0 ×
Ξ̂′c2(
) 0.0 23 34 4.0 2.3 0.0 0.0 ×
Ξ̂′c3(
) 129 26 14 4.5 0.94 84 25 ×
Ξ̂′c3(
) 129 15 19 2.6 0.13 84 25 ×
) 0.0 0.69 0.13 0.29 1.2 0.0 0.0
Ξ̌0c1(
) 15 0.92 0.044 0.39 0.38 11 0.64× 10−3 ×
Ξ̌0c1(
) 15 0.23 0.11 0.096 0.96 11 0.64× 10−3 ×
) 0.0 39 12 12 0.21 0.0 0.0 ×
) 0.0 9.9 30 3.0 5.2 0.0 0.0 ×
Ξ̌1c0(
) 0.0 39 47 12 8.3 0.0 0.0 ×
Ξ̌1c1(
) 110 20 5.9 6.1 1.0 69 42 ×
Ξ̌1c1(
) 110 5.0 15 1.5 2.6 69 42 ×
Ξ̌1c2(
) 0.0 25 3.0 7.6 0.52 0.0 0.0 ×
Ξ̌1c2(
) 0.0 0.0 18 0.0 3.1 0.0 0.0 ×
) 0.0 9.2 6.0 3.9 0.75 0.0 0.0
) 0.0 5.8 10 1.1 2.1 0.0 0.0
Ξ̌2c1(
) 22 6.1 2.3 2.6 0.54 14 15 ×
Ξ̌2c1(
) 22 1.5 5.6 0.64 1.3 14 15 ×
Ξ̌2c2(
) 0.0 14 5.2 5.8 0.77 0.0 0.0 ×
Ξ̌2c2(
) 0.0 3.9 14 0.74 3.0 0.0 0.0 ×
Ξ̌2c3(
) 21 4.4 2.5 0.85 0.23 14 4.3 ×
Ξ̌2c3(
) 21 2.5 3.4 0.48 0.31 14 4.3 ×
	Introduction
	The notations and conventions of charmed baryon
	The 3P0 model
	The strong decays of charmed baryon
	Numerical results
	Discussion and conclusion
	Appendix
	The harmonic oscillator wave functions used in our calculation
	The momentum space integration
	Acknowledgments
	References
