Hidden-charm pentaquarks and their hidden-bottom and $B_c$-like partner states

0.71 MB·

by  Jing Wu

Hidden-charm pentaquarks and their hidden-bottom and B -like partner states
c
Jing Wu1 and Yan-Rui Liu1,2∗
1School of Physics and Key Laboratory of Particle Physics and Particle
Irradiation (MOE), Shandong University, Jinan 250100, China
2Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, CAS, Beijing 100190, China
Kan Chen3,4 and Xiang Liu3,4†
3School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
4Research Center for Hadron and CSR Physics, Lanzhou University
and Institute of Modern Physics of CAS, Lanzhou 730000, China
Shi-Lin Zhu5,6,7‡
5School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China
6Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
7
1 7Center of High Energy Physics, Peking University, Beijing 100871, China
0 (Dated: January 19, 2017)
2
In the framework of the color-magnetic interaction, we have systematically studied the mass

n splittingsofthepossiblehidden-charmpentaquarksqqqcc¯(q=u,d,s)wherethethreelightquarks
a are in acolor-octet state. We findthat i) the LHCbPc states fallin the mass regionof the studied
J system; ii) most pentaquarks should be broad states since their S-wave open-charm decays are

8 allowedwhile thelowest stateis theJP = 21− Λ-like pentaquarkwith probablythe suppressedηcΛ
1
decaymodeonly;andiii)theJP = 5− statesdonotdecaythroughS-waveandtheirwidthsarenot
2
] sobroad. ThemassesandwidthsofthetwoLHCbPc baryonsarecompatiblewithsuchpentaquark
h states. Wealsoexplorethehidden-bottomandBc-likepartnersofthehidden-charmstatesandfind
p the possible existence of the pentaquarks which are lower than the relevant hadronic molecules.
-
p
PACSnumbers: 14.20.Pt,12.39.Jh
e
h
[

2
v I. INTRODUCTION
3
7
In2015, theLHCbCollaboration[1]reportedtwopentaquark-likeresonancesP (4380)andP (4450)intheprocess
c c
8
Λ0 J/ψK−p with the same decay mode J/ψp. The decay channel indicates that their minimal quark content is
3 b →
nnncc¯ (n = u,d). The resonance parameters are M = 4380 8 29 MeV, Γ = 205 18 86 MeV
0 Pc(4380) ± ± Pc(4380) ± ±
1. and MPc(4450) = 4449.8±1.7±2.5 MeV, ΓPc(4450) = 39±5±19 MeV. The preferred angular momenta are 23 and
0 52, respectively and their P parities are opposite. Later, the Pc(4380) and Pc(4450) were confirmed by the reanalysis
7 with a model-independent method [2]. Recently, these two P states were also observed in the Λ0 J/ψpπ− decay
c b →
1 [3].
v: Infact,thetheoreticalexplorationofthehidden-charmpentaquarkswasperformedbeforetheobservationoftwoPc
Xi states by LHCb. In Refs. [4, 5], the authors predicted two Nc∗c¯ states and four Λ∗cc¯ states, where their masses, decay
behaviors and production properties were given in a coupled-channel unitary approach. Possible molecular states
r composed of a charmed baryon and an anticharmed meson were systematically studied with the one-boson-exchange
a
(OBE) model in Ref. [6] and the chiral quark model in Ref. [7]. More investigations can be found in Refs. [8–15]. In
addition, Li and Liu indicated the existence of hidden-charm pentaquarks by the analysis of a global group structure
[16].
After the announcement of the P (4380) and P (4450) states by LHCb, these two P states were interpreted as
c c c
Σ D¯∗, Σ∗D¯, or Σ∗D¯∗ molecules [17–28], bound states or resonances of charmonium and nucleon [29–32], diquark-
c c c
diquark-antiquark states [33–38], diquark-triquark states [39, 40], compact pentaquark states [41–43], kinematical
effects due to χ p rescattering [44], due to triangle singularity [45–47], or due to a D¯-soliton [48], or a bound state
c1
of the colored baryon and meson [49]. Their decay and production properties were studied in Refs. [50–66].
Electronicaddress: [email protected]
∗
Electronicaddress: [email protected]
†
Electronicaddress: [email protected]
‡
2
The observation of these P resonances also stimulated the arguments for more possible pentaquarks [67–71].
c
Productions of another N∗ and Λ∗ were discussed in Ref. [72] and Refs. [73–75], respectively. For the detailed
cc¯
overview on the hidden-charm pentaquarks, the readers may refer to Refs. [76, 77].
The dynamical calculations of the bound states are relatively easier if one treats the system as two clusters. The
investigation at the quark level is also simplified when one assumes the existence of substructures in a five-body
system. If a hidden-charm pentaquark really exists, its spin partners with the same flavor content should also
exist. The existence of substructures certainly results in less pentaquarks. Needless to say, configurations with
various substructures (baryon-meson, diquark-diquark-antiquark, or diquark-triquark) lead to different results. From
symmetryconsideration,aphysicalpentaquarkstateshouldbeamixtureofalltheseconfigurationswithvariouscolor
structures. We here would like to explore a pentaquark structure without the assumption of its substructure.
ThemassesoftheP ’sarebothabovethethresholdofJ/ψp. BecausetheinteractionbetweentheJ/ψ andnucleon
c
is very weak, the scattering resonances in this channel are not appropriate interpretations for the observed P states.
c
We focus on the possible pentaquark configurations where either the three light quark qqq or the cc¯pair is a color
octetstate. WeinvestigatewhetherthelowerP statecanbeassignedasatightlyboundfive-quarkstateandexplore
c
its possible partner states. Recently, there appeared a preliminary quark model study on the hidden color-octet uud
baryons [43].
In principle, a dynamical calculation for a five-body problem is needed in order to calculate their masses. In this
work, wecalculatetheirmasssplittingswithasimplecolor-magneticinteractionfromtheone-gluon-exchange(OGE)
potential. For example, the ∆+ baryon and the proton have the same quark content and color structure and their
massdifferencemainlyarisesfromthecolor-magneticinteraction. Withthecalculatedmasssplittingsandareference
threshold, one can estimate the pentaquark masses roughly.
This paper is organized as follows. In Sec. II, we construct the flavor color spin wave functions of the hidden-
⊗ ⊗
charm pentaquark states and calculate the matrix elements for the color magnetic interaction in the symmetric limit.
Then we consider the flavor breaking case in Sec. III and give numerical results in Sec. IV. In Sec. V, we explore the
heavier pentaquarks. We discuss our results and summarize in the final section.
II. WAVE FUNCTIONS AND COLOR-MAGNETIC INTERACTION
The color-magnetic Hamiltonian reads
(cid:88)
H = m +H ,
i CM
i
(cid:88)
H = C λ λ σ σ , (1)
CM ij i j i j
− · ·
i<j
where the i-th Gell-Mann matrix λ should be replaced with λ∗ for an antiquark. In the Hamiltonian, m is the
i − i i
effective mass of the i-th quark and C δ(r ) /(m m ) is the effective coupling constant between the i-th quark
ij ij i j
∼ (cid:104) (cid:105)
and the j-th quark. The values of the parameters for light quarks and those for heavy quarks are different and they
will be extracted from the known hadron masses. For the hidden-charm systems, we have four types of coupling
parameters C , C , C , and C with q = u,d,s. More parameters need to be determined for the other qqqQQ¯
qq qc qc¯ cc¯
pentaquarks (Q = b,c). In Ref. [78], we have estimated the mass of another not-yet-observed but plausible exotic
meson T withthissimplemodel. InRefs. [79–81], wediscussedthemasssplittingsfortheQQQ¯Q¯, csc¯s¯, and QQQ¯q¯
cc
systems, respectively.
To calculate the required matrix elements of the color-magnetic interaction (CMI), we here construct the flavor-
color-spin wave functions of the ground state pentaquark systems. These wave functions will also be useful in the
study of other properties of the pentaquak states in quark models. In Ref. [8], a study with the color-magnetic
interaction is also involved but the wave functions are constructed with flavor SU(4) symmetry. Now we consider
flavor SU(3) symmetry and treat the heavy (anti)quark as a flavor singlet state.
Because the three light quarks qqq must obey Pauli principle, it is convenient to discuss the constraint with flavor-
spin SU(6) symmetry. The three-quark colorless ground baryons belong to the symmetric [3]=56 representation.
fs
ItsSU(3) SU(2) decompositiongives(10,4)+(8,2)andthereforeflavorsingletbaryonin3 3 3=10+8+8+1is
f s
⊗ ⊗ ⊗
forbidden. Nowthecolor-octetqqq mustbelongtothemixed[21]=70SU(6) representation. TheSU(3) SU(2)
fs f s
⊗
decomposition gives (10,2)+(8,4)+(8,2)+(1,2). So the flavor singlet pentaquark is allowed and we have two flavor
octets with different spins. There is no symmetry constraint for the heavy quark pair and one finally gets three
pentaquark decuplets with J =1/2, 1/2, and 3/2, three octets with J =1/2, 1/2, and 3/2, four octets with J =3/2,
1/2, 3/2, and 5/2, and three singlets with J =1/2, 1/2, and 3/2.
To get a totally antisymmetric qqq wave function, we need the components presented in Tab. I and need to make
appropriate combinations. The notation MS (MA) means that the first two quarks are symmetric (antisymmetric)
3
Multiplet Space Wave function Wave function
10f Color φMS φMA
Spin χMA χMS
[111]cs Flavor FS FS
1f Color φMS φMA
Spin χMS χMA
[3]cs Flavor FA FA
8f(1) Color φMS φMA
Spin χS χS
[21]cs Flavor FMA FMS
8f(2) Color φMS φMA
Spin χMS χMA χMS χMA
[21]cs Flavor FMA FMS FMS FMA
TABLE I: Flavor multiplets and wave functions of the colored qqq in different spaces. Young diagrams for the color-spin
SU(6)cs are also given in the first column.
1[(rg+gr)g 2ggr] 1[(rg+gr)r 2rrg] 1[(gr rg)g] 1[(gr rg)r]
−√6 − √6 − √2 − √2 −
1 [(gb+bg)r+(rb
−2√+3br)g−2(rg+gr)b] −√16[(rb+br)r−2rrb] 12[(gb−bg)r+(rb−br)g] √12[(rb−br)r]
−√16[(gb+bg)g−2ggb] 12[(gb+bg)r*−(rb+br)g] √12[(gb−bg)g] 2√13[(rgbb)g−b*g2)(rrg+(bgrr)b]
− − −
1[(gb+bg)b 2bbg] 1[(rb+br)b 2bbr] 1[(gb bg)b] 1[(rb br)b]
√6 − √6 − √2 − √2 −
φMS φMA
FIG. 1: Color wave functions of the light quarks.
g¯b r¯b
1 (gg¯ rr¯)
√2 − rg¯
gr¯ *
− 1 (rr¯+gg¯ 2b¯b)
−√6 −
br¯ bg¯
−
FIG. 2: Color wave functions of the heavy quark pair.
when they are exchanged in corresponding space and the superscript S (A) means that the wave function is totally
symmetric (antisymmetric). When one combines the color and spin (or flavor and color, or flavor and spin), a wave
function with a required symmetry is determined by the relative sign between different components. For example,
(φMS χMS +φMA χMA) is symmetric for the exchange of color and spin indices simultaneously for any two
⊗ ⊗
quarks. If one uses a minus sign, the wave function is symmetric only for the first two quarks, i.e. a mixed MS
type color-spin wave function. One has to adopt a self-consistent convention for the SU(3) Clebsch-Gordan (C.G.)
coefficientswhenconstructingthewavefunctions. Weheretaketheconventionconvenientforuse[82,83]. Forclarity,
we present the color wave functions of qqq in Fig. 1 and those of cc¯in Fig. 2. The flavor octet wave functions are
easy to obtain with the replacements r u, g d, and b s. The spin wave functions for the spin-half case are
χMS = √1 [( + ) 2 ], χMS =→√1 [( →+ ) 2→ ], χMA = √1 ( ) , and χMA = √1 ( ) .
↑ − 6 ↑↓ ↓↑ ↑− ↑↑↓ ↓ 6 ↑↓ ↓↑ ↓− ↓↓↑ ↑ 2 ↑↓−↓↑ ↑ ↓ 2 ↑↓−↓↑ ↓
There is no confusion for the totally symmetric flavor and spin wave functions and we do not show them explicitly.
4
With explicit calculation, we find the totally antisymmetric wave functions of the colored qqq states and show them
in Tab. II.
Multiplet Flavor-color-spin wave function
10f √12[FS⊗(φMS⊗χMA−φMA⊗χMS)]
1f √12[FA⊗(φMS⊗χMS+φMA⊗χMA)]
8f(1) √12[(FMS⊗φMA−FMA⊗φMS)⊗χS]
8f(2) 12[(FMS⊗χMA+FMA⊗χMS)⊗φMS
+(FMS χMS FMA χMA) φMA]
⊗ − ⊗ ⊗
TABLE II: Antisymmetric wave function for a color-octet qqq state.
With the above wave functions and the C.G. coefficients of SU(3) [82, 83] and SU(2), one can construct the
pentaquark wave functions. We here only show the color part
(cid:34)
1
φMS,MA = pMS,MA(br¯) nMS,MA(bg¯) Σ+MS,MA(gr¯)
penta 2√2 − C − C − C
+Σ−MS,MA(rg¯) Ξ0MS,MA(g¯b)+Ξ−MS,MA(r¯b)
C − C C
(cid:35)
1 1
Σ0MS,MA(gg¯ rr¯)+ ΛMS,MA(rr¯+gg¯ 2b¯b) , (2)
−√2 C − √6 C −
wherethebaryonsymbolswiththesubscript’C’areborrowedfromflavoroctetandrepresentthecolorwavefunctions
in Fig. 1. The structure of the full wave functions is the same as that in Tab. II by adding a subscript “penta” to
each wave function.
Since the color-spin interaction is the same for baryons in the same flavor multiplet in the SU(3) limit, it is enough
to consider only pentaquarks with flavor content uuucc¯in decuplet, uudcc¯in octet, and udscc¯in singlet. After some
calculations we get the results for the H as follows,
CM
(cid:104) (cid:105)
(cid:16) 1(cid:17)
10 : H =10C +2C , for S =0,J =
f CM qq cc¯ cc¯
(cid:104) (cid:105) 2
2 20 (cid:16) 1(cid:17)
H =10C C (C C ), for S =1,J =
CM qq cc¯ qc qc¯ cc¯
(cid:104) (cid:105) − 3 − 3 − 2
2 10 (cid:16) 3(cid:17)
H =10C C + (C C ), for S =1,J = , (3)
CM qq cc¯ qc qc¯ cc¯
(cid:104) (cid:105) − 3 3 − 2
(cid:16) 1(cid:17)
1 : H = 14C +2C , for S =0,J =
f CM qq cc¯ cc¯
(cid:104) (cid:105) − 2
2 4 (cid:16) 1(cid:17)
H = 14C C (C +11C ), for S =1,J =
CM qq cc¯ qc qc¯ cc¯
(cid:104) (cid:105) − − 3 − 3 2
2 2 (cid:16) 3(cid:17)
H = 14C C + (C +11C ), for S =1,J = , (4)
CM qq cc¯ qc qc¯ cc¯
(cid:104) (cid:105) − − 3 3 2
(cid:16) 3(cid:17)
8 (1) : H =2C +2C , for S =0,J =
f CM qq cc¯ cc¯
(cid:104) (cid:105) 2
2 (cid:16) 1(cid:17)
H =2C C 10(C +C ), for S =1,J =
CM qq cc¯ qc qc¯ cc¯
(cid:104) (cid:105) − 3 − 2
2 (cid:16) 3(cid:17)
H =2C C 4(C +C ), for S =1,J =
CM qq cc¯ qc qc¯ cc¯
(cid:104) (cid:105) − 3 − 2
2 (cid:16) 5(cid:17)
H =2C C +6(C +C ), for S =1,J = , (5)
CM qq cc¯ qc qc¯ cc¯
(cid:104) (cid:105) − 3 2
(cid:16) 1(cid:17)
8 (2) : H = 2C +2C , for S =0,J =
f CM qq cc¯ cc¯
(cid:104) (cid:105) − 2
2 (cid:16) 1(cid:17)
H = 2C C 4(C +C ), for S =1,J =
CM qq cc¯ qc qc¯ cc¯
(cid:104) (cid:105) − − 3 − 2
2 (cid:16) 3(cid:17)
H = 2C C +2(C +C ), for S =1,J = . (6)
CM qq cc¯ qc qc¯ cc¯
(cid:104) (cid:105) − − 3 2
5
One may confirm the part for C with the formula [84, 85]
qq
(cid:68)(cid:88) (cid:69) (cid:104) 4 (cid:105)
(λ λ )(σ σ ) = 8N + S(S+1)+2C [SU(3) ] 4C [SU(6) ] , (7)
i j i j 2 c 2 cs
· · − 3 −
i<j
whereN =3,S = 1 or 3, andC [SU(g)]isthequadraticCasimiroperatorspecifiedbytheYoungdiagram[f ,...,f ]
2 2 2 1 g
(cid:34) (cid:35)
1 (cid:88) N2
C [SU(g)]= f (f 2i+g+1) . (8)
2 i i
2 − − g
i
The Young diagram for color symmetry is [21] and those for color-spin SU(6) symmetry can be found in Tab. I.
c cs
The part for C can be verified with the formula
cc¯
(cid:68) (cid:69) (cid:104) 8(cid:105)(cid:104) 3(cid:105)
(λ λ )(σ σ ) =4 C [SU(3) ] S (S +1) . (9)
4 5 4 5 2 c cc¯ cc¯
· · − 3 − 2
However, it is problematic to discuss mass splittings for pentaquarks with Eqs. (3)–(6) directly because of violations
of the heavy quark spin symmetry (HQSS) and the flavor SU(3) symmetry.
The heavy quark symmetry is strict in the limit m , which leads to the irrelevance of the heavy quark spin
c
→ ∞
(and flavor) for the interaction between a heavy quark and a light quark. In this limit, the interaction within the
heavy quark pair is also irrelevant with their spin (but not the flavor) and the spin-flip between the S =0 case and
cc¯
the S = 1 case is suppressed. The color-magnetic interaction obviously violates HQSS. This means that only C
cc¯ qq
terms in Eqs. (3)–(6) are important in the heavy quark limit and there are four degenerate multiplets with the mass
ordering: 10 , 8 (1), 8 (2), and 1 from high to low.
f f f f
After the heavy quark mass correction is included, all the terms involving m in Eqs. (3)–(6) contribute. Since
c
now the spin-flip between the S = 0 case and the S = 1 case is considered, the mixing between states with the
cc¯ cc¯
same J occurs, which results from the term proportional to 1/m m . Then one should determine the final H ’s
c q CM
(cid:104) (cid:105)
for mixed states by diagonalizing the specified matrix.
Usually, the flavor mixing between different multiplet representations occurs once the symmetry breaking is con-
sidered. In the present case, even in the SU(3) limit, the mixing between the two octets is nonvanishing, which
f
complicates the color magnetic interactions. To be convenient, we now collect the averages of the CMI in a matrix
formforthennncc¯(n=u,d)andssscc¯cases. Fortheothercasescontainingthesquark, weshowresultsinthenext
section.
For I = 3 nnncc¯states (3 baryons in 10 ),
2 f
10 2
H = 10C + (C C ) C ,
(cid:104) CM(cid:105)J=23 nn 3 nc− nc¯ − 3 cc¯
(cid:32) (cid:33)
H = 10Cnn− 230(Cnc−Cnc¯)− 32Ccc¯ √103(Cnc+Cnc¯) . (10)
(cid:104) CM(cid:105)J=21 10C +2C
nn cc¯
One gets similar expressions for the I =0 ssscc¯states (3 baryons in 10 ) by replacing n with s.
f
For I =1/2 nnncc¯states (7 baryons in 8 ), the results read
f
2
H = 2C +6(C +C ) C ,
(cid:104) CM(cid:105)J=25 nn nc nc¯ − 3 cc¯
 √ 
2C 4(C +C ) 2C 2√15(C C ) 2 10(C 4C )
nn− nc nc¯ − 3 cc¯ nc− nc¯ − √3 nc− nc¯
(cid:104)HCM(cid:105)J=32 =  2(Cnn+Ccc¯) 236(Cnc+4Cnc¯) ,
2C +2(C +C ) 2C
− nn nc nc¯ − 3 cc¯
 
2Cnn−10(Cnc+Cnc¯)− 32Ccc¯ −√43(Cnc+4Cnc¯) −43(Cnc−4Cnc¯)
(cid:104)HCM(cid:105)J=21 =  2(−Cnn+Ccc¯) 2√3(Cnc−Cnc¯) , (11)
2C 4(C +C ) 2C
− nn− nc nc¯ − 3 cc¯
(cid:16) (cid:17)T (cid:16)
where the bases for J = 23 and J = 12 are 8f(1)[Scc¯=1], 8f(1)[Scc¯=0], 8f(2)[Scc¯=1] and 8f(1)[Scc¯=1], 8f(2)[Scc¯=0],
(cid:17)T
8f(2)[Scc¯=1] , respectively.
6
III. SU(3)f BREAKING
Up to now, we have not considered the SU(3) breaking. Once the mass difference between the strange quark and
f
the u,d quarks is included, the general mixing between flavor multiplets appears. Such an effect is included in the
color-magnetic interaction and we now discuss this case. The systems we need to consider additionally are nnscc¯
and ssncc¯. They are classified into two categories according to the symmetry for the first two quarks in flavor space:
symmetricnnscc¯(I =1)andssncc¯(I =0)andantisymmetricnnscc¯(I =0). Inthefollowing,weusethesymbollike
[(qqq(cid:48))MA(cc¯)0]J to denote the base states. In this example, the subscript MS means that the color representation
MS 8
forthe(qqq(cid:48))is8MS andthecolorwavefunctionisφMS. Thesubscript8isthecolorrepresentationforthe(cc¯). The
superscript MA means that the spin wave function for the (qqq(cid:48)) is χMA and the spin is 1/2. The superscript 0 (J)
indicates the spin of the (cc¯) (pentaquark).
The calculation method can be found in Refs. [86, 87]. We first give the results for the symmetric category. For
the case J = 5,
2
2
H = (4C C +7C +2C +2C +7C C ). (12)
(cid:104) CM(cid:105)J=52 3 12− 13 14 15 34 35− 45
(cid:16)
There is only one base state [(qqq(cid:48))SMA(cc¯)18]52. For the case J = 32, we use the base vector [(qqq(cid:48))SMA(cc¯)18]32,
(cid:17)T
[(qqq(cid:48))SMA(cc¯)08]32, [(qqq(cid:48))MMSA(cc¯)18]32, [(qqq(cid:48))MMAS(cc¯)08]32 and get
 √ √ 
2(3µ 2α 2γ) 2√15(β+δ) 2 5(α 2γ) 5(13α 15β)
9 − − 9 −√9 − √21 −
 2(8λ 4ν 7µ) 2 3(β 2δ) 3(15α 13β) 
(cid:104)HCM(cid:105)J=32 = 3 − − 92(39ν+2−α−γ) 211(42µ2−1 42ν+−13α−15β), (13)
1 (14λ+13γ+15δ)
21
whereα=7C +2C ,β =7C 2C ,γ =2C +7C ,δ =2C 7C ,µ=4C C C ,ν =4C +2C C ,
14 15 14 15 34 35 34 35 12 13 45 12 13 45
and λ=6C C . For the cas−e J = 1, we have − − − −
12− 45 2
 √ √ √ √ 
2(3µ 5α 5γ) 2 2( α+2γ) 2 6( β+2δ) 2(13α 15β) 6(15α 13β)
9 − − 9 − 9 − (cid:32)21 − (cid:33) −21 −
 29(3ν−4α+2γ) 2√93(2β−δ) 221 2113µα−+2115νβ −√213(15α−13β)
(cid:104)HCM(cid:105)J=12 = 2(8λ 8µ 3ν) √3−(15α 13β) 2(µ ν) 
 3 − − 221−(72λ1 −13γ−−15δ) √213(15γ−+13δ) 
2(2C +C )
12 45
(14)
(cid:16) (cid:17)T
with the base vector [(qqq(cid:48))SMA(cc¯)18]12, [(qqq(cid:48))MMSA(cc¯)18]12, [(qqq(cid:48))MMSA(cc¯)08]21, [(qqq(cid:48))MMAS(cc¯)18]21, [(qqq(cid:48))MMAS(cc¯)08]23 .
Now we present the results for the antisymmetric category. For the case J = 25, the base state is [(qqq(cid:48))SMS(cc¯)18]25
and the matrix element is
2
H = ( 2C +5C +5C +10C +4C C C ). (15)
(cid:104) CM(cid:105)J=52 3 − 12 13 14 15 34− 35− 45
(cid:16) (cid:17)T
FortheJ = 32 case,weusethebasevector [(qqq(cid:48))SMS(cc¯)18]23,[(qqq(cid:48))SMS(cc¯)08]23,[(qqq(cid:48))MMSS(cc¯)18]32,[(qqq(cid:48))MMAA(cc¯)08]23
and the obtained matrix is
 √ √ √ 
2(3µ(cid:48)+10α(cid:48)+2δ(cid:48)) 2 15(5β(cid:48)+γ(cid:48)) 2 5( 5α(cid:48)+2δ(cid:48)) 5(α(cid:48) 3β(cid:48))
−9 9 9√ − √3 −
 2 (11µ(cid:48)+8ν(cid:48) 4λ(cid:48)) 2 3(5β(cid:48) 2γ(cid:48)) 3(3α(cid:48) β(cid:48)) 
(cid:104)HCM(cid:105)J=32 = 15 − 92(190α(cid:48)−δ−(cid:48)−3ν(cid:48)) 115(6µ(cid:48)−36ν(cid:48)+−5α(cid:48)−15β(cid:48)),(16)
1(4λ(cid:48) 15γ(cid:48)+13δ(cid:48))
−6 −
where α(cid:48) = C + 2C , β(cid:48) = C 2C , γ(cid:48) = 4C + C , δ(cid:48) = 4C C , µ(cid:48) = 2C 5C + C , ν(cid:48) =
14 15 14 15 34 35 34 35 12 13 45
− − −
7
2C +10C +C , and λ(cid:48) =12C +C . For the J = 1 case, we have
12 13 45 12 45 2
2
H =
(cid:104) CM(cid:105)J=12 −9 ×
(3µ(cid:48)+25α(cid:48)+5δ(cid:48)) √2(5α(cid:48) 2δ(cid:48)) √6(5β(cid:48) 2γ(cid:48)) √3 (3β(cid:48) α(cid:48)) 3√6(3α(cid:48) β(cid:48)) 
− − (cid:32) 2 − (cid:33) 2 −
 (20α(cid:48) 2δ(cid:48)+3ν(cid:48)) √3(10β(cid:48) γ(cid:48)) 3 3ν(cid:48)−3µ(cid:48) 3√3(3α(cid:48) β(cid:48)) 
 − − − 5 +5α(cid:48) 15β(cid:48) 2 − 
 √ − 
 35(4λ(cid:48)−16µ(cid:48)−3ν(cid:48)) 3(2λ32(cid:48)3+(31α5γ(cid:48)−(cid:48) β1(cid:48))3δ(cid:48)) 3√359(1(ν3(cid:48)γ−(cid:48) µ1(cid:48))5δ(cid:48)) 
2 − 4 −
3(9λ(cid:48) 16µ(cid:48) 8ν(cid:48))
5 − −
(17)
(cid:16) (cid:17)T
with the base vector [(qqq(cid:48))SMS(cc¯)18]12, [(qqq(cid:48))MMSS(cc¯)18]21, [(qqq(cid:48))MMSS(cc¯)08]21, [(qqq(cid:48))MMAA(cc¯)18]21, [(qqq(cid:48))MMAA(cc¯)08]32 .
One may use the matrices in this section to numerically reproduce the H ’s for the nnncc¯ systems after
CM
diagonalization. For the J = 5 case, both Eq. (12) and Eq. (15) give the sa(cid:104)me fo(cid:105)rmula and thus the same result
2
when q(cid:48) = q = n. For the case J = 3, Eq. (13) and Eq. (16) result in different eigenvalues by assuming q(cid:48) = q = n.
2
However, one finds that the common numbers of the two sets of eigenvalues are just the results for the I = 1 nnncc¯
2
systems. The remaining value given by Eq. (13) is the result for the I = 3 nnncc¯system while that given by Eq.
2
(16) can be thought as a forbidden number because of the Pauli principle. The J = 1 case has similar features with
2
the J = 3 case. Probably these features can be used to simplify the calculation for multiquark systems.
2
IV. NUMERICAL RESULTS FOR THE HIDDEN-CHARM SYSTEMS
BycalculatingtheCMImatrixelementsforgroundstatebaryonsandmesons[88],weextracttheeffectivecoupling
parameters presented in Tab. III. In determining C , one may also use the mass difference between Λ and Σ . The
cn c c
resulting pentaquark masses would be around 10 MeV lower, which is a not a large number in the present method of
estimation. Sincewealsodiscusshidden-bottomandB -likepentaquarkstates,Tab. IIIdisplaysrelevantparameters,
c
too. Because there is no experimental data for the B∗ meson, we determine the value of C to be 3.3 MeV from a
c bc¯
quark model calculation mBc∗ −mBc =70 MeV [89].
TABLE III: The effective coupling parameters extracted from the mass differences between ground hadrons.
Hadron CMI Hadron CMI Parameter(MeV)
N 8Cnn ∆ 8Cnn Cnn =18.4
−
Σ 38Cnn− 332Cns Σ∗ 83Cnn+ 136Cns Cns =12.4
Ξ0 83(Css−4Cns) Ξ∗0 38(Css+Cns)
Ω 8Css Css =6.5
Λ 8Cnn
−
D −16Ccn¯ D∗ 136Ccn¯ Ccn¯ =6.7
Ds −16Ccs¯ Ds∗ 136Ccs¯ Ccs¯=6.7
B −16Cbn¯ B∗ 136Cbn¯ Cbn¯=2.1
Bs −16Cbs¯ B∗ 136Cbs¯ Cbs¯=2.3
ηc −16Ccc¯ J/ψ 136Ccc¯ Ccc¯=5.3
ηb −16Cb¯b Υ 136Cb¯b Cb¯b =2.9
Σc 83Cnn− 332Ccn Σ∗c 83Cnn+ 136Ccn Ccn =4.0
Ξ(cid:48)c 83Cns− 136Ccn− 136Ccs Ξ∗c 83Cns+ 83Ccn+ 38Ccs Ccs =4.8
Σb 83Cnn− 332Cbn Σ∗b 83Cnn+ 136Cbn Cbn =1.3
Ξ(cid:48)b 83Cns− 136Cbn− 136Cbs Ξ∗b 83Cns+ 83Cbn+ 38Cbs Cbs =1.2
In the simple model used in the present study, the mass splittings of the pentaquark states mainly rely on the
(cid:80)
coupling parameters. The masses can be roughly estimated with the formula M = m + H or from a
i i (cid:104) CM(cid:105)
reference mass M = M H + H . In the latter scheme, the reference system has the same quark
ref CM ref CM
−(cid:104) (cid:105) (cid:104) (cid:105)
8
content with the pentaquark system and the dependence on the effective quark masses is partially canceled. We will
show results in both schemes. Here, the effective quark masses are m =361.8 MeV, m =540.4 MeV, m =1724.8
n s c
MeV, and m = 5052.9 MeV, which are also extracted from the ground hadrons. In Ref. [80], it is illustrated that
b
these effective quark masses result in overestimated hadron masses and one may treat the values as theoretical upper
limits. Then we mainly focus on the second scheme and use various meson-baryon thresholds as reference masses.
A. The nnncc¯system
In this case, there are two types of thresholds we may use: (charmonium)+(light baryon) and (charmed
baryon)+(charmed meson). However, the CMI matrix elements (cid:104)HCM(cid:105)(J/ψp) =−119 MeV and (cid:104)HCM(cid:105)(ΣcD¯) =−102
MeV are not consistent with the thresholds (4035 MeV for J/ψp and 4320 MeV for Σ D¯). This indicates that one
c
cannot eliminate completely the quark mass effects with the reference mass scheme. The reason is that the model
does not involve dynamics and the contributions from the other terms in the potential model are related with the
systemstructure. Forexample,theadditionalkineticenergycanprobablyshifttheestimatedmasstoamorephysical
value [90]. To understand which threshold is more reasonable, one needs detailed calculation in a future work. Here,
we estimate pentaquark masses with both types of thresholds. The numerical results are given in Tab. IV. The use
of the J/ψN, J/ψ∆, η N, and η ∆ thresholds gives similar values and that of Σ D¯, Σ D¯∗, Σ∗D¯, and Σ∗D¯∗ gives
c c c c c c
similar values. The reference threshold Λ D¯ or Λ D¯∗ results in around 10 MeV lower masses than Σ D¯ does. We do
c c c
not present these results in the table.
TABLEIV:CalculatedCMI’sandestimatedpentaquarkmassesofthennncc¯systemsinunitsofMeV.Themassesintheforth
column are calculated with the effective quark masses and are theoretical upper limits.
nnncc¯(I = 3)
2
JP HCM Eigenvalue Mass (J/ψ∆) (ΣcD¯)
(cid:104) (cid:105)
3− 171.5 171.5 4706.5 4325.5 4591.1
2 (cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33)
198.5 61.8 258.3 4793.3 4412.4 4677.9
1
−
2 61.8 194.6 134.7 4669.7 4288.8 4554.3
nnncc¯(I = 1)
2
JP HCM Eigenvalue Mass (J/ψN) (ΣcD¯)
(cid:104) (cid:105)
5− 97.5 97.5 4632.5 4251.6 4517.1
2          
9.5 20.9 48.1 83.1 4451.9 4071.0 4336.5
− − −
32− −20.9 47.4 50.3   74.8  4609.8 4228.9 4494.4
48.1 50.3 18.9 27.3 4562.3 4181.4 4446.9
 −         
73.7 71.1 30.4 133.0 4402.0 4021.1 4286.6
− − −
21− −71.1 −26.2 −9.4   −80.8  4454.2 4073.3 4338.8
30.4 9.4 83.1 30.8 4565.8 4184.9 4450.4
− −
Fig. 3 (a) shows relative positions for these pentaquarks when one adopts the threshold of (Σ D¯) as a reference.
c
Wealsoplotallthethresholdsoftherelatedrearrangementdecaypatterns,i.e. J/ψN,J/ψ∆,η N,η ∆,Σ D¯,Σ D¯∗,
c c c c
Σ∗D¯, Σ∗D¯∗, Λ D¯, and Λ D¯∗. The decays may occur through the S- or D-wave interactions and each pentaquark
c c c c
with JP = 1−, 3−, or 5− can decay to these channels from the parity conservation and the angular momentum
2 2 2
conservation. Theisospinconservationreducesthenumberofdecaychannelsandwelabeltheisospin,forconvenience,
in the subscripts of the meson-baryon states. Once the considered state is an initial pentaquark plotted with dashed
(solid) line, it can decay into meson-baryon channels having the subscript 3 (1). Of course, whether the decay can
2 2
happen or not is also kinematically constrained by the pentaquark mass, which depends on models. Contrary to
the light quark case, the decay for the hidden-charm pentaquarks may also get constraints from the heavy quark
symmetry. For the states with (I,J)=(3,3) and (I,J)=(1,5), the cc¯spin is always 1. Their decays into η ∆ and
2 2 2 2 c
η N, respectively, involve the heavy quark spin-flip and are suppressed. In fact, all the hidden-charm decay channels
c
ofthestudiedpentaquarksareprobablysuppressedbecausethetransitionfromacoloredcc¯toacolorlesscc¯isahigh
order correction of 1/m . With these considerations in mind, it is easy to judge which channels can be used to search
c
for such unobserved pentaquark states.
From the results in Fig. 3 (a), it is obvious that the heaviest state is a decuplet baryon with J = 1. The lightest
2
statebelongingtotheflavoroctetalso has thespinJ = 1. TheobservedP (4380)isjustbelowthethresholdofΣ∗D¯
2 c c
9
4678
4758
4591
(J/ψΔ)3/24444234583557904 444344394747 4517 ((((((ΛΣΣΣΣηcccc∗c∗cΔDDDDD¯¯¯¯¯)∗)∗)∗3)1)1)/1/1/12/2/2/2,2,233,,/3/32/2/22 (Ξ((Ξ(Σcη∗cD¯ccDD¯Σ∗))∗s00)),,111144444444465433451398794592194735 44444444664534557277849427472483 44664129 (((((((ΣΞΣΣΛΞΞ∗cc(cid:2)c(cid:2)cc∗c∗cDDDDDDD¯¯¯s∗∗)s∗∗ss∗)0)))))10,00111,,,11(J/ψΣ∗)1
(ΛcD¯)1/2 ((JΞ/cDψ¯Σ)0),11
(ΛcDs)0
(J/ψN)1/2 4188 ((ηJc/Σψ)Λ1)0
(ηcN)1/2 (ηcΛ)0
12− 23− 25− 12− 23− 52−
(a) I = 3 (dashed) and I = 1 (solid) nnncc¯states (b) I =1 (dashed) and I =0 (solid) nnscc¯states
2 2
4823
4946
(Ω∗cD¯∗)
(Ξ∗cDs∗)
4737
4693 4698 4716 ((ΩΞc(cid:2)cDD¯∗s∗)) (Ω∗cDs∗)
4659 4639 (Ω∗cD¯),(J/ψΞ∗) 4860
(Ξ∗cDs)
(ΩcD¯) 4552 4545 ((ΞΞc(cid:2)cDDss∗)) 4813 (ΩcDs∗)
(ηcΞ∗) (J/ψΩ)
4483
(ΞcDs) (Ω∗cDs)
(J/ψΞ)
(ΩcDs)
(ηcΞ) (ηcΩ)
12− 23− 25− 12− 23−
(c) I = 1 (solid) ssncc¯states (d) I =0 (solid) ssscc¯system
2
FIG.3: Relativepositionsfortheobtainedqqqcc¯pentaquarkstates. Thedottedlinesindicatevariousmeson-baryonthresholds
and the long solid line in(a) indicates the observed isospin-half Pc(4380). When a number in the subscript of a meson-baryon
state is equal to the isospin of an initial state, the decay for the initial state into that meson-baryon channel through S- or
D-wave is allowed. We adopt the masses estimated with the reference thresholds of ΣcD¯ (a), ΞcD¯ (b), ΞcDs (c), and ΩcDs
(d). The masses are all in units of MeV.
andfallsinthemassrangeofthestudiedsystem. Thus, iftheestimatedmassesarereasonable, theinterpretationfor
theP (4380)asatightlyboundpentaquarkwithcoloredcc¯isnotexcluded, althoughthepresentstudycannotgivea
c
preferred spin. The partner states decaying into J/ψp in the mass region (4280 4520 MeV) are also possible. Above
∼
the 4520 MeV, the observation of pentaquark-like baryons in the invariant mass of J/ψ∆ is possible, too. All these
pentaquarkshaveopen-charmdecaychannelsandareprobablybroadstates. However,theirdecaysintohidden-charm
channelsshouldhaveasmallfraction. ThefeatureofthebroadwidthdoesnotcontradictwiththeobservedP (4380).
c
The branching ratios for various decay channels will be crucial information to understand its nature. Note that the
decays of the JP = 5− state (slightly below the Σ∗D¯∗ threshold) into the channels in the figure are all through D
2 c
wave. The width of this state should not be so broad. If our result is overestimated, both the mass and the width
seem not to be contradicted with those of the P (4450), although the parity is opposite to the preferred P = +. In
c
the literature, various calculations also find a JP = 5− state below the Σ∗D¯∗ threshold. If the parity of the P (4450)
2 c c
is really +, search for such a negative parity state is also strongly called for. It is difficult to understand the nature
of the P (4450) without further investigations.
c
Now we discuss the masses estimated with the J/ψN threshold. As mentioned in Ref. [80], the obtained masses
seem to be underestimated and can be treated as lower limits in this simple model. The argument is based on the
(cid:80)
formulae M = m + H and M =M H + H , where the effective quark masses are assumed
i i (cid:104) CM(cid:105) ref −(cid:104) CM(cid:105)ref (cid:104) CM(cid:105)
to be equal for various hadrons. Of course this assumption leads to uncertainty for the hadron mass estimation. We
illustratethisuncertaintywiththeJ/ψN threshold. Becausetheusedm =1724.8MeVgivesoverestimatedm ,it
c J/ψ
10
isnotsurprisingthattheformerformularesultsinoverestimatedmasses,whereacharm-anticharmpairexists. When
using the latter formula, in principle, one should have Mth = 2m +3m + H in order to cancel the quark
ref c n (cid:104) CM(cid:105)ref
mass dependence in the former formula. Because we have adopted M = m +m = 2mψ +3mN + H
ref J/ψ N c n (cid:104) CM(cid:105)ref
but mψ <m and mN =m =361.8 MeV, theresulting masses are underestimated. Even in this low mass limit, the
c c n n
I = 1 (I = 3) states above the threshold of Λ D¯ (Σ D¯) should be broad.
2 2 c c
Fromtheabovearguments,probablythemassesestimatedwiththeΣ D¯ thresholdaremorereasonable. Ifthisisthe
c
case, more hidden-charm pentaquarks that can decay into J/ψ∆ or J/ψp are allowed. After a dynamical calculation
withmorepotentialtermsisperformedinafuturework,onecangetmoreinformationforthemassesofthesehidden-
charm pentaquarks. From the obtained masses, various thresholds in Fig. 3 (a), and the above arguments, most
pentaquarks are probably broad states with a small fraction for the hidden-charm decays. This feature should be
different fromthemoleculepicture, wherethe hidden-charmdecays throughrearrangement mechanisms areprobably
not suppressed. The exceptional case is for the JP = 5− state. Its decays are through D wave and its width is
2
probably not so broad. If the LHCb hidden-charm pentaquark states are confirmed, models of rearrangement decays
needtobeconstructedononehand,andthemeasurementofvariousbranchingratiosandthesearchformoreproposed
states, on the other hand, are strongly called for. Of course, the interference effects around the mass region would
make the experimental analysis more difficult.
B. The nnscc¯system
TABLEV:CalculatedCMI’sandestimatedpentaquarkmassesofthennscc¯systemsinunitsofMeV.Themassesintheforth
column are calculated with the effective quark masses and are theoretical upper limits.
nnscc¯(I =1)
JP HCM Eigenvalue Mass (J/ψΣ) (ΣcDs) (ΞcD¯)
(cid:104) (cid:105)
5− 102.5 102.5 4816.1 4443.8 4625.6 4641.8
2
           
6.2 19.5 35.6 34.0 132.8 4846.4 4474.0 4655.9 4672.0
− −
 19.5 51.4 34.3 35.6   87.3  4800.9 4428.5 4610.4 4626.5
32− −35.6 34.3 67.9 59.2  65.7 4647.9 4275.5 4457.4 4473.5
− −
34.0 35.6 59.2 78.4 37.2 4750.8 4378.4 4560.3 4576.5
 −           
71.5 22.5 48.6 21.5 50.3 219.1 4932.7 4560.3 4742.2 4758.4
− − −
 22.5 50.4 25.6 104.8 35.6  120.8 4592.8 4220.5 4402.3 4418.5
 − −  −         
12− −48.6 25.6 76.2 −35.6 −74.4  92.4  4806.0 4433.6 4615.5 4631.6
 21.5 104.8 35.6 53.4 29.9   51.9  4765.5 4393.1 4575.0 4591.1
− −
50.3 35.6 74.4 29.9 84.2 49.9 4663.7 4291.3 4473.2 4489.3
− − − −
nnscc¯(I =0)
JP HCM Eigenvalue Mass (J/ψΛ) (ΛcDs) (ΞcD¯)
(cid:104) (cid:105)
5− 79.6 79.6 4793.2 4411.1 4588.8 4618.9
2
           
31.0 18.2 30.8 34.0 157.5 4556.1 4174.0 4351.7 4381.7
− − − −
 18.2 27.4 38.0 35.6   95.8  4617.8 4235.8 4413.4 4443.5
32− −30.8 38.0 −74.8 59.2   −58.2  4771.8 4389.7 4567.4 4597.5
− − − −
34.0 35.6 59.2 113.1 3.6 4717.2 4335.2 4512.8 4542.9
 − −           
97.3 19.5 53.8 21.5 50.3 351.3 4362.3 3980.3 4157.9 4188.0
− − − −
 19.5 182.5 46.1 104.8 35.6   165.4 4548.2 4166.1 4343.8 4373.8
− − − − −  −         
12−  53.8 −46.1 −96.6 −35.6 −74.4  −142.2 4571.4 4189.4 4367.0 4397.1
 21.5 104.8 35.6 226.1 43.1   96.2  4617.4 4235.4 4413.0 4443.1
− − − − −
50.3 35.6 74.4 43.1 136.6 16.0 4729.6 4347.5 4525.2 4555.2
− − − − −
This case is related with the Σ-like or Λ-like baryons with an excited charm-anticharm pair. In Refs. [4, 5], the
Λ-like molecules above 4.2 GeV were predicted. Higher states around 4.6 GeV are also proposed in Refs. [73, 75].
From a calculation with the one-meson-exchange model [71], several (charmed baryon)-(charmed strange meson) and
(charmed strange baryon)-(charmed meson) type molecules above 4.5 GeV are possible. Now we discuss the mass
spectrum of a compact structure.