1 School of Physics, Shandong University, Jinan 250100, People’s Republic of China2 Institute of Theoretical Physics, Chinese Academy of Science, Beijing 100190, People’s Republic of China
Abstract We discuss the mass splittings for the S-wavetriply heavy pentaquark states with the QQQqq (Q =b, c; q = u, d, s) configuration which is a mirror structure ofQQqqq. The latter configuration is related with the nature ofPc(4380) observed by the LHCb Collaboration. The consid-ered pentaquark masses are estimated with a simple method.One finds that such states are probably not narrow even if theydo exist. This leaves room for molecule interpretation for astate around the low-lying threshold of a doubly heavy baryonand a heavy-light meson, e.g. �ccD, if it were observed. Asa by product, we conjecture that upper limits for the massesof the conventional triply heavy baryons can be determinedby the masses of the conventional doubly heavy baryons.
1 Introduction
Recently, the LHCb Collaboration confirmed the �cc baryon[1] which was predicted in the quark model (QM) decadesago [2] and first measured by the SELEX Collaborationin 2002 [3]. Excited conventional heavy hadrons are alsoobserved in recent years [4]. We are still on the way to confirmthe conventional structure with three heavy quarks. How-ever, the most exciting thing in hadron physics is that moreand more unexpected hadrons are detected [5–16] and thushadrons beyond the conventional quark model should exist.This indicates that we may identify the existence of multi-quark states before the confirmation of all the QM states. Forexample, the Pc(4380) and Pc(4450) states observed in theJ/ψN channel by the LHCb Collaboration [17] can be inter-preted as pentaquark states since their high masses are notnatural for conventional 3q baryons [18]. If a pentaquark canbe confirmed, the existence of other multiquark states shouldalso be possible. It is interesting to investigate theoreticallywhich multiquark systems allow bound states.
Since the pentaquark-like states Pc(4380) and Pc(4450)
are much higher than the J/ψN threshold and the J/ψNinteraction is not strong (a lattice simulation gives a valuearound 0.7 fm for the scattering length [19]), the interpre-tation with J/ψN scattering states seem not to be appro-priate. In the literature, various interpretations, such as�
or diquark-triquark pentaquarks, have been used to under-stand their nature [9–16]. Assigning them as qqqcc compactpentaquarks is another possible interpretation, where the ccis a color-octet state. In Refs. [20–22], investigations alongthis line were performed. It was found that interpreting thePc(4380) state as such a pentaquark can be accepted. Fur-ther investigations on this type configuration of pentaquarksshowed that a stable �-type hidden-charm state is also pos-sible [21,23]. In the present study, we are going to considera mirror-type structure by changing heavy (light) quarks tolight (heavy) quarks, pentquarks with the QQQqq configu-ration where QQQ is always a color-octet triply heavy state.In the following discussions, we use QQQ to denote ccc,bbb, ccb, or bbc. When one needs to distinguish the quarkcontents, we will also use QQQ to denote ccc or bbb andQQQ′ to denote ccb or bbc.
In the study of the udsQQ states with a colored uds, onefound that the quark-quark and quark-antiquark color-spininteractions can all be attractive [21]. Mixing effects amongdifferent color-spin structures further lowered the pentaquarkmasses and finally resulted in a �-type state which has arather low mass. The mass is not far from the value obtainedin Ref. [18] where molecule configuration was adopted andhidden-charm pentaquarks were proposed first. In the presentwork, we try to understand whether colored QQQ is alsohelpful to the formation of pentqaurks or not. If the answeris yes, a problem to distinguish the pentaquarks from con-ventional QQQ baryons may arise. If no, one expects thatconventional triply heavy baryons rather than pentaquarkswould be found experimentally first.
Table 1 Flavor contents of thepentaquarks we consider |12345〉 f |45〉 f = |nn〉 |45〉 f = |ns〉 |45〉 f = |sn〉 |45〉 f = |ss〉
|123〉 f = |ccc〉 |cccnn〉 |cccns〉 |cccsn〉 |cccss〉|123〉 f = |ccb〉 |ccbnn〉 |ccbns〉 |ccbsn〉 |ccbss〉|123〉 f = |bbc〉 |bbcnn〉 |bbcns〉 |bbcsn〉 |bbcss〉|123〉 f = |bbb〉 |bbbnn〉 |bbbns〉 |bbbsn〉 |bbbss〉
Up to now, theoretical studies have given the masses of theconventional QQQ baryons in various approaches, althoughthere is still no experimental evidence about them. Accordingto these calculations, the mass of the ground �ccc baryon,for example, is in the range of 4.6∼5.0 GeV [4,24–27]. Ifan additional quark-antiquark pair surrounds the heavy colorsource, there is a possibility that the system has a little highermass and it looks like an excited �ccc state. This possibilitywould be excluded if the QQQqq state has a much highermass. We would like to extract some information about thisproblem from the following investigations.
In this article, we consider the mass splittings of the com-pact triply heavy pentaquarks QQQqq (Q = b, c; q =u, d, s)1 and estimate their rough masses in the frameworkof a simple quark model. In the following Sect. 2, the wavefunctions will be constructed. In Sect. 3, we give the chro-momagnetic interaction (CMI) matrices for different typesof pentaquarks. In Sect. 4, we show our choice of relevantparameters and analyse the numerical results for the spec-trum. The last section is a short summary.
2 Construction of the wave functions
Before the calculation of CMI matrices for the consideredground pentaquark states, we need to construct all the spin-color wave functions. The configuration we consider containsthree heavy quarks (ccc, ccb, bbc, or bbb) and a light quark-antiquark pair (nn, ns, sn, or ss, n = u or d), which is simplydenoted as 12345. Table 1 lists the flavor contents of these16 systems.
Here, we use |S12, S123, S45, J = S12345〉 to denotethe possible spin wave functions. There are five states withJ = 1/2,
X1 = |1,1
2, 0,
1
2〉, X2 = |1,
1
2, 1,
1
2〉,
X3 = |1,3
2, 1,
1
2〉, X4 = |0,
1
2, 1,
1
2〉,
X5 = |0,1
2, 0,
1
2〉, (1)
1 Reference [28] presents a study of the triply heavy pentaquarks withthe QQQqq configuration.
four states with J = 3/2,
X6 = |1,1
2, 1,
3
2〉, X7 = |1,
3
2, 0,
3
2〉,
X8 = |1,3
2, 1,
3
2〉, X9 = |0,
1
2, 1,
3
2〉, (2)
and one state with J = 5/2
X10 = |1,3
2, 1,
5
2〉. (3)
Their explicit expressions are easy to get by using the SU (2)
C.G. coefficients.In color space, one uses |R12, R123, R45, R12345 = 1c〉 to
denote the wave functions. Then, we find two bases
C1 = |6, 8MS, 8, 1MS〉, C2 = |3, 8MA, 8, 1MA〉 (4)
for the present investigation, where the superscripts MS andMA mean that the first two quarks are symmetric and anti-symmetric, respectively. Their explicit expressions are thesame as those presented in Eq. (2) of Ref. [21].
Taking the Pauli principle into account when we combinethe bases in different spaces, one may obtain five types oftotal wave functions.
• Type A [Flavor = QQQqq , J = 1/2]:
�A1 = 1√
2
{[(QQ)0
6Q]128 (qq)0
8
} 12
1
− 1√2
{[(QQ)1
3Q]
128 (qq)0
8
} 12
1
= 1√2QQQqq ⊗ (C1 ⊗ X5 − C2 ⊗ X1),
�A2 = 1√
2
{[(QQ)0
6Q]128 (qq)1
8
} 12
1
− 1√2
{[(QQ)1
3Q]
128 (qq)1
8
} 12
1
= 1√2QQQqq ⊗ (C1 ⊗ X4 − C2 ⊗ X2); (5)
• Type B [Flavor = QQQqq , J = 3/2]:
�B1 = 1√
2
{[(QQ)0
6Q]128 (qq)1
8
} 32
1
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Eur. Phys. J. C (2019) 79 :87 Page 3 of 19 87
− 1√2
{[(QQ)1
3Q]
128 (qq)1
8
} 32
1
= 1√2QQQqq ⊗ (C1 ⊗ X9 − C2 ⊗ X6); (6)
• Type C [Flavor = QQQ′qq , J = 1/2]:
�C1 =
{[(QQ)0
6Q′]
128 (qq)0
8
} 12
1
= (QQQ′qq) ⊗ C1 ⊗ X5,
�C2 =
{[(QQ)0
6Q′]
128 (qq)1
8
} 12
1
= (QQQ′qq) ⊗ C1 ⊗ X4,
�C3 =
{[(QQ)1
3Q′]
128 (qq)0
8
} 12
1
= (QQQ′qq) ⊗ C2 ⊗ X1,
�C4 =
{[(QQ)1
3Q′]
128 (qq)1
8
} 12
1
= (QQQ′qq) ⊗ C2 ⊗ X2,
�C5 =
{[(QQ)1
3Q′]
328 (qq)1
8
} 12
1
= (QQQ′qq) ⊗ C2 ⊗ X3; (7)
• Type D [Flavor = QQQ′qq , J = 3/2]:
�D1 =
{[(QQ)0
6Q′]
128 (qq)1
8
} 32
1
= (QQQ′qq) ⊗ C1 ⊗ X9,
�D2 =
{[(QQ)1
3Q′]
128 (qq)1
8
} 32
1
= (QQQ′qq) ⊗ C2 ⊗ X6,
�D3 =
{[(QQ)1
3Q′]
328 (qq)0
8
} 32
1
= (QQQ′qq) ⊗ C2 ⊗ X7,
�D4 =
{[(QQ)1
3Q′]
328 (qq)1
8
} 32
1
= (QQQ′qq) ⊗ C2 ⊗ X8; (8)
• Type E [Flavor = QQQ′qq , J = 5/2]:
�E1 =
{[(QQ)1
3Q′]
328 (qq)1
8
} 52
1
= (QQQ′qq) ⊗ C2 ⊗ X10. (9)
Here, QQQ means ccc or bbb and QQQ′ means ccb orbbc. The superscripts (subscripts) for quarks indicate spins(representations in color space). In each type of pentaquarksystems, configuration mixing induced by the chromomag-netc interaction occurs. If we use �X to denote a mixed wavefunction for the type-X state, it can be written as a superpo-sition of different configurations,
�X =NX∑i=1
CXi �X
i , (10)
where the coefficientsCXi satisfy the normalization condition∑NX
i=1 |CXi |2 = 1 and NX = 2, 1, 5, 4, and 1 correspond
to X = A, B, C , D, and E , respectively. There are NX
independent �X .
3 Chromomagnetic interaction
The Hamiltonian for the mass calculation in the model reads
H =n∑
i=1
mi −∑i< j
3∑α=1
8∑β=1
Ci j
(σαi σα
j
) (λ
βi λ
βj
)
=n∑
i=1
mi + HCMI . (11)
Here, n is the number of (anti)quarks in the hadron and mi isthe effective quark mass for the i th quark by taking account ofeffects from kinetic energy, color confinement, and so on. Thecoefficient Ci j reflects the strength of the chromomagneticinteraction between the i th and j th quark components andis influenced by their masses. The Pauli matrix σα
i and Gell-
Mann matrix λβi =λ
βi (−λ∗β
i ) act on the spin and color wavefunctions of the i th quark (antiquark), respectively.
With the constructed bases of wave functions for the type-X pentaquarks, one can easily obtain the matrix element[HX
CMI ]kl = 〈�Xk |HCMI |�X
l 〉, where k, l = 1, 2, . . . NX .The calculation of all matrix elements gives five CMI matri-ces.
Comparing the present CMI matrices with those in Ref.[21], one finds that the above expressions form a subset ofthose for the qqqQQ case. The reason is that the flavor wavefunction of two identical heavy quarks must be symmetricwhile that of light quarks can also be antisymmetric. We willget the eigenvalues and eigenvectors of these matrices in thenumerical evaluation. For the type-X pentaquarks, their masssplittings are the differences between the NX eigenvalues.
We use masses of conventional hadrons to determine rele-vant parameters. For convenience, here we also present CMIexpressions for them [29],
HCMI (q1q2)J = 1 = 16
3C12,
HCMI (q1q2)J = 0 = −16C12,
HCMI (q1q2q3)J = 3/2 = 8
3(C12 + C23 + C13),
HCMI (q1q2q3)J = 1/2 = 8
3
[(C12 − 2C23 − 2C13)
√3(C23 − C13)√
3(C23 − C13) −3C12
],
(17)
where the two bases for the last matrix correspond to the caseof Jq1q2 = 1 and that of Jq1q2 = 0.
4 Numerical analysis
4.1 Parameter selection and estimation strategy
In order to estimate the masses of the possible pentaquarkstates, one needs to know the values of relevant mass param-eters and coupling strengths. We extract the values of Ci j ’sfrom the mass splittings of conventional hadrons. For exam-ple, Cnn = 1/16(m� − mN ) and Cnn = 3/64(mρ − mπ )
may be determined with Eq. (17). However, several effec-tive coupling constants, Css , Ccc, Cbb, Cbc, and Cbc, needto be assigned by models or assumptions. Although the �cc
state has been observed, the extraction of Ccc needs moremeasured baryon masses. We here simply adopt the assump-tion Ccc=Ccc for our evaluations. Similarly, we use Css=Css ,Cbb=Cbb, and Cbc=Cbc, where Cbc is obtained with the massdifference between B∗
c and Bc [30]. A variation of the valuesrelated with heavy quarks does not induce significant differ-ences [31–33]. Table 2 lists all the coupling strengths we willuse. At present, we further assume that these parameters canbe applied to different systems so that we may estimate themasses of the studied pentaquark states. In the last part ofthis section, we will check the effects caused by the adoptionof other values of coupling parameters.
Table 2 Relevant coupling parameters in units of MeV
Ccn = 4.0 Ccc = 5.3 Ccn = 6.6 Cnn = 29.8
Ccs = 4.5 Cbc = 3.3 Ccs = 6.7 Cns = 18.7
Cbn = 1.3 Cbb = 2.9 Cbn = 2.1 Css = 6.5
Cbs = 1.2 Cbs = 2.3
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Eur. Phys. J. C (2019) 79 :87 Page 5 of 19 87
Table 3 Used masses of theconventional hadrons in units ofMeV [38]. Since the spin of the�cc observed by LHCb may be1/2 or 3/2, we show results inboth cases. The adopted massesof other doubly heavy baryonsare taken from Refs. [39–42](�∗
cc is from Ref. [27]). Thevalues in parentheses areobtained with the parameters inTable 2
According to Eq. (11), the mass of a pentquark state in thechromomagnetic model is
M =5∑
i=1
mi + 〈HCMI 〉, (18)
where M and 〈HCMI 〉 are the mass of a pentaquark andthe corresponding eigenvalue of the choromomagnetic inter-action, respectively. By introducing a reference system, themass of the pentaquark state can be written as
M = (Mref − 〈HCMI 〉re f ) + 〈HCMI 〉. (19)
Here, Mref and 〈HCMI 〉re f are the mass of the referencesystem and the corresponding chromomagnetic interaction,respectively. Because none of QQQqq pentaquark states isobserved, we choose the threshold of a baryon-meson statewith the same quark content as Mref . If the simple modelcould give correct masses for all the hadron states, the abovetwo formulas should be equivalent. In fact, the model does notinvolve dynamics and the two approaches result in differentmultiquark masses. One may consult Refs. [21,34] for somediscussions on the difference.
If we adopt Eq. (18), one needs to know the effectivequark masses. Their values extracted from lowest conven-tional hadrons are mn = 361.7 MeV, ms = 540.3 MeV,mc = 1724.6 MeV, and mb = 5052.8 MeV. From Eq.(18) and the coupling parameters in Table 2, the massesof conventional hadrons may be evaluated. By comparingthem with the experimental measurements, one finds over-estimated theoretical results [34,35]. The multiquark massesare also generally overestimated [21,33,34,36,37]. Thus wemay treat the masses obtained in this method as theoreti-
cal upper limits. This fact indicates that attractions insidehadrons cannot be taken into account sufficiently in the sim-ple chromomagnetic model. A more reasonable approach isto adopt Eq. (19), where the necessary attractions for conven-tional hadrons have been implicitly included in their physicalmasses. In this approach, probably there are two choices onthe reference threshold. In this case, we present results withboth thresholds. To calculate the baryon-meson thresholds,we use the hadron masses shown in Table 3. The extractionof the above effective quark masses and coupling parametersalso relies on these numbers. Here, we assume that the spin of�cc is 1/2, which is presumed in Ref. [43]. If the spin of �cc is3/2 (consistent with the prediction in Refs. [39–42]), the mul-tiquark masses estimated with the threshold relating to �cc
would be shifted downward by 64 MeV. When determiningthresholds relating to other doubly heavy baryons, we adoptthe values obtained in Refs. [39–42]. However, the mass of�∗
cc in that reference was obtained by adding ∼100 MeV tothe mass of �∗
cc. Consider the fact that the quantum numbersof the observed �cc by LHCb have not been determined, weprefer to use another value obtained in Ref. [27]. A differentmass of the doubly heavy baryon (see Refs. [26,44] for a col-lection) will lead to a different mass of the pentaquark. Onemay compare the adopted mass with the value in Table 3 toestimate the mass difference. For example, if we got a pen-taquark mass M1 with M�bc = 6920 MeV while one wants toadopt M�∗
bc= 7065.7 MeV [27], then one would obtain the
mass of that pentaquark by adding (7065.7 − 6983.4) MeVto M1, where 6983.4 MeV is the mass of �∗
bc in Table 3. Inorder for further discussions, we also present a summary forthe theoretical investigations in the literature on the massesof triply heavy conventional baryons in Table 4.
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87 Page 6 of 19 Eur. Phys. J. C (2019) 79 :87
Table 4 The masses of triply heavy baryons in units of MeV in the literature. One may consult Refs. [26,44] for more results about the �ccc and�bbb
Based on the content of heavy quarks, one may classify thepentaquarks into four groups: cccqq , bbbqq , ccbqq , andbbcqq or two types: QQQqq and QQQ′qq . With all theparameters given above, we get the spectra of these pen-taquark systems which are shown in Tables 5, 6, 7 and 8.In these tables, the columns labeled with the title “Mass”list masses estimated with Eq. (18). The last one or twocolumns show results estimated with relevant baryon-mesonthresholds. For the QQQ′qq states, one may use (QQq)-(Q′q) or (QQ′q)-(Qq) type threshold for our purpose. Theresulting masses may have an uncertainty about 100 MeV,which is not a very large value if compared to the massaround 10 GeV. We present results with both type thresh-olds.
For the QQQnn states, we do not present the isospinindices in the results because the isoscalar and isovector casesare degenerate. If the isoscalar QQQnn and QQQss canmix, the resulting states would have different masses with theisovector QQQnn. Here, we do not consider this possibility.
In fact, one may also use the HQ-Hq type threshold to esti-mate the pentaquark masses, where the hadron Hq containsonly light quarks and HQ contains heavy quarks. From pre-vious investigations [21,34,35], we have seen that the multi-quark masses estimated with such thresholds can be treatedas theoretical lower limits. This feature should be relatedwith the attractions inside conventional hadrons that cannotbe taken into account in the present model. Since no evidencefor triply heavy baryons is reported, with this feature, we mayconversely set upper limits for the masses of the conventionaltriply heavy baryons. The formula is
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Eur. Phys. J. C (2019) 79 :87 Page 7 of 19 87
Table 5 Numerical results for the cccqq systems in units of MeV. The masses in the sixth column are estimated with Eq. (18) and those in the lastcolumn with the (ccq)-(cq) type threshold
System J P 〈HCM 〉 Eigenvalues Eigenvectors Mass �ccD
cccnn 12
−[
112.6 61.261.2 50.5
] [150.212.9
] [(0.85, 0.52)
(0.52,− 0.85)
] [60475910
] [57755638
]
32
−24.5 24.5 1 5922 5650
System J P 〈HCM 〉 Eigenvalues Eigenvectors Mass �ccDs
cccns 12
−[
90.4 61.861.8 58.5
] [138.310.7
] [(0.79, 0.61)
(0.61,− 0.79)
] [62146086
] [58645736
]
32
−31.5 31.5 1 6107 5757
System J P 〈HCM 〉 Eigenvalues Eigenvectors Mass �ccD
cccsn 12
−[
90.4 64.164.1 54.5
] [139.0
5.9
] [(0.80, 0.60)
(0.60,− 0.80)
] [62156082
] [58795745
]
32
−33.5 33.5 1 6109 5773
System J P 〈HCM 〉 Eigenvalues Eigenvectors Mass �ccDs
cccss 12
−[
66.0 64.764.7 63.3
] [129.3− 0.0
] [(0.71, 0.70)
(0.70,− 0.71)
] [63846254
] [59695840
]
32
−41.3 41.3 1 6296 5881
Table 6 Numerical results for the bbbqq systems in units of MeV. The masses in the sixth column are estimated with Eq. (18) and those in the lastcolumn with the (bbq)-(bq) type threshold
System J P 〈HCM 〉 Eigenvalues Eigenvectors Mass �bb B
bbbnn 12
−[
88.6 19.619.6 14.5
] [93.59.6
] [(0.97, 0.24)
(0.24,− 0.97)
] [1597515891
] [1550615422
]
32
−6.5 6.5 1 15888 15418
System J P 〈HCM 〉 Eigenvalues Eigenvectors Mass �bb Bs
bbbns 12
−[
66.4 20.820.8 23.2
] [74.814.8
] [(0.93, 0.37)
(0.37,− 0.93)
] [1613516075
] [1557815518
]
32
−13.2 13.2 1 16074 15516
System J P 〈HCM 〉 Eigenvalues Eigenvectors Mass �bb B
bbbsn 12
−[
66.4 19.119.1 22.5
] [73.515.4
] [(0.94, 0.35)
(0.35,− 0.94)
] [1613416076
] [1558415526
]
32
−13.5 13.5 1 16074 15524
System J P 〈HCM 〉 Eigenvalues Eigenvectors Mass �bb Bs
bbbss 12
−[
42.0 20.220.2 32.0
] [57.816.2
] [(0.79, 0.62)
(0.62,− 0.79)
] [1629716255
] [1566015618
]
32
−21.0 21.0 1 16260 15623
MHQQQ ≤[Mref − 〈HCMI 〉re f
]−
[MHqq − 〈HCMI 〉Hqq
]
+〈HCMI 〉HQQQ , (20)
where the value of [Mref − 〈HCMI 〉re f ] = [M − 〈HCMI 〉]may be calculated with definition or read out from Tables
5, 6, 7 and 8. The obtained upper limits can be differentif one considers different systems, but they do not rely onthe angular momentum if a system is given. Of course, thecorrectness of this conjecture needs to be confirmed by fur-ther measurements. From Eq. (17) and the masses in abovetables, we may set M�ccc ≤ (M�∗
cc+MD −Mω)+ 16
3 (Ccc −
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87 Page 8 of 19 Eur. Phys. J. C (2019) 79 :87
Table 7 Numerical results for the ccbqq systems in units of MeV. The masses in the sixth column are estimated with Eq. (18) and those in the lasttwo columns with the (ccq)-(bq) type and (bcq)-(cq) type thresholds, respectively
System J P 〈HCM 〉 Eigenvalues Eigenvectors Mass �cc B �bcD
Ccn + Cnn + 3Ccn) ≈ 5044 MeV if we consider the cccnnsystem. If the cccss system is used, we have M�ccc ≤(M�∗
cc+MDs −Mφ)+ 16
3 (Ccc −Ccs +Css +3Ccs) ≈ 4897MeV. The upper limits extracted with the cccns and cccsn are4975 MeV and 4989 MeV, respectively. One should adopt the
smallest value 4897 MeV as the final limit. Similarly, fromour parameters, we have
M�bbb ≤ (M�∗bb
+ MBs − Mφ)
+16
3(Cbb − Cbs + Css + 3Cbs) ≈ 14640 MeV,
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Eur. Phys. J. C (2019) 79 :87 Page 9 of 19 87
Table 8 Numerical results for the bbcqq systems in units of MeV. The masses in the sixth column are estimated with Eq. (18) and those in the lasttwo columns with the (bcq)-(bq) type and (bbq)-(cq) type thresholds, respectively
System J P 〈HCM 〉 Eigenvalues Eigenvectors Mass �bc B �bbD
and further M�ccb ≤ 8029 MeV and M�bbc ≤ 11261 MeV.These limits are much lower than the baryon masses obtained
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87 Page 10 of 19 Eur. Phys. J. C (2019) 79 :87
(ΞccD)1
(ΞccD∗)1,3
(Ξ∗ccD)3
(Ξ∗ccD
∗)1,3
(ΞccDs)1
(ΞccD∗s)1,3
(Ξ∗ccDs)3
(Ξ∗ccD
∗s)1,3
(ΩccD)1
(ΩccD∗)1,3
(Ω∗ccD)3
(Ω∗ccD
∗)1,3
(ΩccDs)1
(ΩccD∗s)1,3
(Ω∗ccDs)3
(Ω∗ccD
∗s)1,3
5638
5775
5650
5736
5864
5757 5745
5879
57735840
5969
5881
cccnn cccns cccsn cccss
: J=1/2: J=3/2
8868
8891
8975
90469087
8907
8961
8984
9053
8994
8960
8987
9063
91229165
90039054
9081
91309093
8982
9010
9084
91469188
9028907391079154
9118
9073
9108
9167
92239268
9124916192059235 9218
(ΞbcD)1
(ΞbcD∗)1,3
(Ξ′bcD)1
(Ξ′bcD
∗)1,3
(Ξ∗bcD)3
(Ξ∗bcD
∗)1,3,5
(ΞbcDs)1
(ΞbcD∗s)1,3
(Ξ′bcDs)1
(Ξ′bcD
∗s)1,3
(Ξ∗bcDs)3
(Ξ∗bcD
∗s)1,3,5
(ΩbcD)1
(ΩbcD∗)1,3
(Ω′bcD)1
(Ω′bcD
∗)1,3
(Ω∗bcD)3
(Ω∗bcD
∗)1,3,5
(ΩbcDs)1
(ΩbcD∗s)1,3
(Ω′bcDs)1
(Ω′bcD
∗s)1,3
(Ω∗bcDs)3
(Ω∗bcD
∗s)1,3,5
ccbnn ccbns ccbsn ccbss
: J=1/2: J=3/2: J=5/2
(a) cccqq (c) ccbqq
(ΞbbB)1
(ΞbbB∗)1,3
(Ξ∗bbB)3
(Ξ∗bbB
∗)1,3(ΞbbBs)1
(ΞbbB∗s )1,3
(Ξ∗bbBs)3
(Ξ∗bbB
∗s )1,3
(ΩbbB)1
(ΩbbB∗)1,3
(Ω∗bbB)3
(Ω∗bbB
∗)1,3
(ΩbbBs)1
(ΩbbB∗s )1,3
(Ω∗bbBs)3
(Ω∗bbB
∗s )1,3
15422
15506
15418
15518
15578
1551615526
15584
15524
15618
15660
15623
bbbnn bbbns bbbsn bbbss
: J=1/2: J=3/2
12055
12133
12186
12234
12274
12075
12169
12190
12247
12199
12148
12230
12277
12308
12349
12169
122671228812319
12298
12163
12240
12291
12320
12361
12182
12281
1230212330
12311
12256
1233612374
12397
12440
12276
1238012395
12406 12410
(ΞbcB)1
(ΞbbD)1
(ΞbbD∗)1,3
(Ξ∗bbD)3
(Ξ∗bbD
∗)1,3,5
(ΞbbDs)1
(Ξ∗bbDs)3
(ΞbcBs)1
(Ξ∗bbD
∗s)1,3,5
(ΩbbD)1
(Ω∗bbD)3
(Ω∗bbD
∗)1,3,5(ΩbcB)1(ΩbbD
∗)1,3
(ΩbcBs)1
(ΩbbDs)1
(Ω∗bbDs)3
(Ω∗bbD
∗s)1,3,5
bbcnn bbcns bbcsn bbcss
: J=1/2: J=3/2: J=5/2
(b) bbbqq (d) bbcqq
Fig. 1 Rough positions for the obtained triply heavy pentaquarks
with Eq. (18). Most of the masses obtained in the literature(Table 4) are consistent with such constraints. Of course,these upper limits rely on the input masses of the dou-bly heavy baryons, but are irrelevant with the pentaquarkmasses. In this sense, the masses of conventional triply heavybaryons are constrained by those of conventional doublyheavy baryons. On the other hand, if the experimentallyobserved triply heavy baryons have masses larger than suchvalues, one may conclude that they probably are not groundQQQ states.
In Fig. 1, we show the rough positions of the studied pen-taquark states and the thresholds of relevant (QQq)− (Qq),(QQq)−(Q′q), or (QQ′q)−(Qq) type decay patterns. Eachdiagram corresponds to a group of pentaquarks. We haveadopted the masses listed in the seventh columns of Tables5, 6, 7 and 8. From a study of dibaryon states [72], the addi-tional kinetic energy may be a possible effect in understand-ing why the predition of the H -dibaryon [73] is inconsistentwith experimental results. If quark dynamics are considered,probably reasonable pentaquarks have higher masses than
those obtained here. That is why we do not use the massesin the eighth columns of Tables 7 and 8 for the QQQ′qqsystems when plotting the diagrams. In Fig. 1, the spectrumrelies on the input masses of the doubly heavy baryons, butthe relative positions comparing to the thresholds relating tothe mass estimation is fixed in the present method. For exam-ple, the masses of the J = 5/2 ccbnn states (I = 1 andI = 0) may be changed if one uses a different input massof �bb but the distance between the threshold of �bb B andthem is fixed by the CMI matrices. Of course, the mass split-tings between the pentaquark states are also fixed only by theCMI matrices.
From Tables 5, 6, 7 and 8 and Fig. 1, one may under-stand some features for the spectra we study: (i). There isa hierarchy around 3200 MeV between the four groups ofpentaquarks, which is from the mass difference between thebottom and charm quarks. Within each group, the hierar-chy caused by the mass difference ms − mn is about 100MeV; (ii). For each system, the states with highest and low-est masses usually have the lowest spin J = 1/2; (iii).
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Eur. Phys. J. C (2019) 79 :87 Page 11 of 19 87
The QQQsn states generally have higher masses than theQQQns states, which is a result of the difference in cou-pling strengths, CQn �= CQs and CQn �= CQs .
If the considered compact pentaquarks exist, they willdecay into lower hadron states. The easiest decay mode,(QQq)+(Qq), (QQq)+(Q′q), or (QQ′q)+(Qq), wouldbe due to quark rearrangements. Figure 1 shows relevantthresholds for such a mode. When 2J (J is the spin of apentaquark) is equal to a subscript of the label for the baryon-meson channel, the decay into that channel through S-waveis allowed. The other decay mode is (QQQ) + (qq) or(QQQ′) + (qq). Now, the color of the three heavy quarksis changed by emitting a gluon. Because of the possible con-straints from the Pauli principle and/or angular momentum,the spin-flip of QQQ may happen and this decay mode issuppressed for some cases where the spin-flip of QQQ isforbidden in the heavy quark limit. We check this feature inthe following analyses.
4.3 Stability of various states
We explore the possible compact structure QQQqq in thisarticle, where QQQ is always a color octet state. Sincethe mass estimation for the pentaquark states depends oneffective quark masses or the masses of the unobserved dou-bly heavy baryons, what we obtain here are only the roughpositions of such states. More accurate values need furtherdynamical calculations. Before those studies, it is beneficialto discuss preliminarily the stability of such pentaquark statesand to find out possible interesting exotic states.
The basic idea is to consider the allowed two-body strongdecays and the size of the phase space for decay. From theabove results, all the pentaquark states seem to have rear-rangement decays and are probably not stable. Since hadronsare composed of quarks, the masses and decay propertiesare finally determined by the quark-quark interactions. It ishelpful to understand the hadron-level features from the innerinteractions. If heavy quark spin symmetry plays a role in thedecay processes [74–76], probably narrow pentaquarks arestill possible. We will check this possibility. In the study of theQQQQ systems [33], we checked the effective interactionbetween two heavy quarks in the mixing case by varying theeffective coupling strengths, through which we may roughlyguess whether the tetraquark states are stable or not. Here,we also perform a similar study.
4.3.1 The cccqq and bbbqq states
For these states, the spin of ccc or bbb (QQQ) is always 1/2from the symmetry consideration, irrespective of the totalspin of the system, 1/2 or 3/2. Since the spin of the color-singlet �ccc or �bbb is 3/2, the decay of such pentaquarksinto a triply heavy baryon plus a light meson involves an
emission of a chromomagnetic gluon and the spin-flip ofQQQ. In the heavy quark limit, such a decay process issuppressed. In principle, no symmetry principle suppressesthe decay into a doubly heavy baryon plus a heavy-lightmeson and one expects that narrow QQQqq states wouldnot exist if the pentaquark masses are high enough. To checkthis point, one may calculate overlapping factor between theinitial state wave function and the final state wave function.From the recoupling formula in the spin space, for the cccnncase (other QQQqq cases are similar), one has
(�ccD)J = 12 = 1
2(ccc)
12 (nn)0 − 1
2√
3(ccc)
12 (nn)1
+√
2
3(ccc)
32 (nn)1,
(�ccD∗)J = 1
2 = − 1
2√
3(ccc)
12 (nn)0 + 5
6(ccc)
12 (nn)1
+√
2
3(ccc)
32 (nn)1,
(�∗ccD
∗)J = 12 =
√2
3(ccc)
12 (nn)0 +
√2
3(ccc)
12 (nn)1
−1
3(ccc)
32 (nn)1;
(�∗ccD)J = 3
2 = − 1√3(ccc)
12 (nn)1
+1
2(ccc)
32 (nn)0 +
√15
6(ccc)
32 (nn)1,
(�ccD∗)J = 3
2 = 1
3(ccc)
12 (nn)1 − 1√
3(ccc)
32 (nn)0
+√
5
3(ccc)
32 (nn)1,
(�∗ccD
∗)J = 32 =
√5
3(ccc)
12 (nn)1 +
√15
6(ccc)
32 (nn)0
+1
6(ccc)
32 (nn)1; (22)
where the superscripts indicate the spins. Obviously, thereis no suppressed coupling from the comparison betweenthese wave functions and those in Eqs. (5) and (6). One mayroughly get relative coupling amplitudes between differentchannels with these wave functions, but could not derivethe absolute partial widths without knowing the couplingstrength. Therefore, the possible nature that the QQQqqstates have generally broad widths cannot be excluded fromthis coupling analysis.
In a multiquark state, the interactions between each pairof quark components together affect the final mass whichdetermines the phase space of decay. After the complicatedcoupling among different color-spin structures, the propertyof the interaction between two quark components becomesunclear. It is helpful to understand whether the effective inter-action is attractive or repulsive by introducing a measure.
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87 Page 12 of 19 Eur. Phys. J. C (2019) 79 :87
To find it, we may study the effects induced by the artifi-cial change of coupling strengths in the Hamiltonian. Whena coupling strength is reduced, the resulting mass may beincreased or decreased. If the effective interaction betweenthe considered components is attractive (repulsive), the masswould be shifted upward (downward) and vice versa. For thepurpose to see the effects, we may define a dimensionlessvariable
Ki j = �M
�Ci j, (23)
where �Ci j is the variation of a coupling strength in Eq.(11) and �M = �〈HCMI 〉 is the corresponding varia-tion of a pentaquark mass. Then the positive (negative) Ki j
corresponds to a repulsive (attractive) effective interactionbetween the i th and j th components. When �Ci j is smallenough, Ki j tends to be a constant (∂M)/(∂Ci j ). With theseconstants, we may write an equivalent expression for the pen-taquark mass as
M = M0 + 〈HCMI 〉 = M0 +∑i< j
Ki jCi j , (24)
where M0 is a constant for a given pentaquark system andits value can be obtained based on the estimation approach,Eq. (18) or (19). One should note that this formula does notmean that M is linearly related with Ci j .
Now we apply this definition of Ki j to the cccnn states.By reducing one relevant coupling strength to 99.99% ofits original value, we get a Ki j . Each time we change onlyone Ci j while others remain unchanged. The results are col-lected in Table 9 where the order of states is the same as thatin Table 5. A positive (negative) number indicates that thecorresponding color-spin interaction is effectively repulsive(attractive). In the bcbc or bbbb case, for example, one mayunderstand that a relatively stable tetraquark is favored ifthe effective bc or bb interaction is attractive. In the presentcase, not only one quark-quark interaction exists. Usually,it is hard to guess whether a pentaquark state is stable ornot just from the signs of Ki j . The sum of their contribu-tions gives the final mass shift. From Table 9, it is obviousthat the effective color-spin interactions between each pairof components in the high mass J = 1/2 cccnn states(I = 1 and I = 0) are repulsive and the states should notbe stable. By checking the CMI matrix in Eq. (12) where theinteraction between quark components can be attractive, oneunderstands that the mixing effect may change the natureof each interaction. In addition to judge the stability of apentaquark with Fig. 1, one may equivalently use Eq. (24) tojudge whether the decay into a (ccn)−(cn) channel happensor not. Since KccCcc + KcnCcn > 8/3Ccc + 16/3Ccn (CMIof �∗
cc) and KcnCcn > 16/3Ccn (CMI of D∗), the decayinto �∗
ccD∗ (and thus �ccD∗ and �ccD) through S-wave is
allowed. If this state exists, its width should not be narrow.
For a J = 3/2 state (I = 1 or I = 0), the nn and cn inter-actions are both attractive, which is obvious from Eq. (13)or (24), and the width of this state should be relatively nar-rower. From masses or effective interactions, S-wave �ccD∗and �∗
ccD decay channels are both opened. For the remain-ing J = 1/2 states, the attraction appears mainly betweena charm quark and a light quark. Now, the allowed S-wavedecay channels are �ccD and possibly �ccD∗. The sameKcc for different states reflects the fact that the spin of cccis always 3/2. From these discussions, it seems that all thesecccnn pentaquarks are not narrow states even if they do exist.
For the other cccqq states, only one isospin is allowed,either 1/2 or 0. The higher J = 1/2 cccns state hasthree S-wave channels �∗
ccD∗s , �ccD∗
s , and �ccDs and theJ = 3/2 state has two S-wave channels �ccD∗
s and�∗
ccDs . The allowed S-wave decay channels for the remain-ing J = 1/2 cccns state are �ccDs and possibly �ccD∗
s .Effective contributions from each pair of quark componentscan also be obtained from the values shown in Table 9. Inprinciple, these cccns states should not be narrow. The cccsnstates have similar decay properties, but Ds (D∗
s ) is replacedby D (D∗) and �cc (�∗
cc) is replaced by �cc (�∗cc). If one
replaces only �cc (�∗cc) by �cc (�∗
cc) in the decay productsof the cccns states, the decay patterns of the cccss states areobtained. From the above discussions, the widths of all thecccqq should not be narrow.
Similarly, one gets rough decay properties of the bbbqqstates. It is easy to understand that their widths should not benarrow either, since all of them have S-wave rearrangementdecay patterns and no suppressed channels are found.
If the interactions between light quark components havelarge contributions to the pentaquark mass, the state maybe low enough and probably stable, because the couplingconstant in the CMI model is proportional to 1/(mim j ).Two examples are the udscc state proposed in Refs. [21,23]and the QQqqq states studied in Ref. [34]. In the presentQQQqq case, it is easy to read out contributions from eachpair of interaction from Table 9. Obviously, the qq effectiveinteractions are not attractive enough (even repulsive). Theeffective interactions for heavy-heavy and heavy-light com-ponents do not provide enough attraction, either. Thus, themasses of QQQqq states are not low enough, which resultsin the conclusion that such pentaquark states are probablynot narrow even if they exist. Of course, further conclusionneeds accurate hadron masses and the determination of decayamplitudes.
4.3.2 The ccbqq and bbcqq states
Without the constraint from the Pauli principle, the spin ofccb or bbc (QQQ′) can be both 3/2 and 1/2. The resultingspectrum is more complicated. When the angular momentumof the pentaquark is 5/2, the spin of QQQ′ must be 3/2. The
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Table 9 Values of Ki j when reducing the coupling strength Ci j between the i th and j th quark components by 0.01%. The orders of states are thesame as those in Tables 5 and 6
State nn cn cn cc State nn bn bn bb
[cccnn]J = 12 1.27 6.97 3.32 10.00 [bbbnn]J = 1
2 1.84 3.09 2.31 10.00
0.06 −0.31 −9.99 10.00 −0.51 3.58 −8.98 10.00
[cccnn]J = 32 −0.67 −3.33 3.33 10.00 [bbbnn]J = 3
2 −0.67 −3.33 3.33 10.00
State ns cs cn cc State ns bs bn bb
[cccns]J = 12 1.00 8.09 3.09 10.00 [bbbns]J = 1
2 1.63 4.93 3.07 10.00
0.33 −1.42 −9.76 10.00 −0.29 1.73 −9.74 10.00
[cccns]J = 32 −0.67 −3.33 3.33 10.00 [bbbns]J = 3
2 −0.67 −3.33 3.33 10.00
State sn cn cs cc State sn bn bs bb
[cccsn]J = 12 1.03 7.99 3.12 10.00 [bbbsn]J = 1
2 1.67 4.60 2.97 10.00
0.31 −1.33 −9.79 10.00 −0.34 2.06 −9.64 10.00
[cccsn]J = 32 −0.67 −3.33 3.33 10.00 [bbbsn]J = 3
2 −0.67 −3.33 3.33 10.00
State ss cs cs cc State ss bs bs bb
[cccss]J = 12 0.69 9.04 2.51 10.00 [bbbss]J = 1
2 0.99 8.14 3.07 10.00
0.64 −2.37 −9.17 10.00 0.35 −1.47 −9.74 10.00
[cccss]J = 32 −0.67 −3.33 3.33 10.00 [bbbss]J = 3
2 −0.67 −3.33 3.33 10.00
decay of the pentaquark into �QQQ′ plus a light meson isforbidden or suppressed, but the decay into �∗
QQQ′ plus alight vector meson not. The latter decay is forbidden only bykinematics. When the angular momentum of the pentaquarkis 1/2 or 3/2, the spin of the inside QQQ′ can be both 1/2and 3/2. From the eigenvectors in Tables 7 and 8, the spin ofQQQ′ in some states is dominantly 1/2. The decays of the1/2 or 3/2 pentaquarks into �QQQ′ and a qq meson are notsuppressed, either. For the (QQq) + (Q′q) decay mode, itis easy to understand that each overlapping factor does notvanish from Eqs. (7), (8), and (22). (The nonvanishing over-lapping factor for the decay of a J = 5/2 pentaquark isobvious.) For the (QQ′q) + (Qq) decay mode, the nonvan-ishing overlapping factor can be understood similarly sincethe spin of QQ′ in a physical (QQ′q) state can be both 1 and0. Even if the spin of QQ′ in (QQ′q) is only zero, the factorstill does not vanish. Therefore, no symmetry requirementcan suppress the decay of the QQQ′qq states. Whether sucha state can decay or not depends only on kinematics.
To understand the effective quark-quark or quark-antiqu-ark color-spin interactions, we list Ki j of each pair of interac-tions in Table 10 where �Ci j = 0.00001Ci j . Contributionsto the pentaquark mass from each pair of components areeasy to obtain. For example, the effective cc or bb interac-tion is always repulsive. From the table, the interactions inall the highest states (J = 1/2) are repulsive and such statesare certainly not stable. For the lowest states (J = 1/2) in
each system, the chromomagnetic interactions except for ccor bb are all attractive. To understand whether the interactionsresult in low enough and thus possible narrow pentaquarks,what we need is to check whether the lowest states with var-ious spins can decay.
First, we focus on the lowest ccbnn states. The thresholdsof the decay mode containing a (bcn) baryon are generallylower than those of the mode containing a (ccn) baryon.For the case J = 5/2, the states (I = 1 and 0) proba-bly do not have the S-wave decay pattern �∗
ccB∗, but they
may decay into �∗bcD
∗ and �∗ccbρ or �∗
ccbω through S-wave.Their widths should not be narrow. For the lowest J = 3/2states (I = 1 and 0), the channels �bcD∗, �∗
bcD, and �′bcD
∗are all opened. They can also decay into �∗
ccbπ , �ccbρ, or�ccbω through S-wave. Their widths should not be narrow,either. For the lowest J = 1/2 states, the decay patterns�bcD∗, �bcD, �′
bcD, �ccbπ , �ccbω, etc. are all allowed,which should result in broad pentaquark states if they exist.Therefore, narrow ccbnn pentaquarks are not expected. How-ever, this conclusion depends on the pentaquark masses. Ifthe masses estimated with the �bcD are more reasonable, onemay conclude that relatively narrow J = 5/2 or J = 3/2pentaquark states are still possible.
Next, one may analyse similarly the cases ccbns, ccbsn,and ccbss. The basic conclusion is that relatively narrowJ = 5/2 pentaquarks are possible only when their massesare low enough.
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Table 10 Values of Ki j when reducing the coupling strength Ci j between the i th and j th quark components by 0.001%. The orders of states arethe same as those in Tables 7 and 8
State nn cn bn cn bn cc bc State nn cn bn cn bn bb bc
Finally, the bbcqq systems have similar spectra to theccbqq systems, but the thresholds of the decay mode con-taining a bbq baryon are lower than those of the mode con-taining a bcq baryon. If pentaquark masses shown in Fig. 1are reasonable, no narrow or stable bbcqq states should exist.
From the above results, one basically finds that all theQQQ′qq pentaquark states are probably not stable, althoughrelatively narrow states cannot be excluded completely. Thecolor-spin interaction itself does not lead to enough attrac-tion. Further theoretical and experimental investigations ondoubly heavy and triply heavy states may provide more infor-mation.
4.4 Discussions
In obtaining the rough masses of the studied pentaquarkstates, we have several uncertainties. First, the values of cou-pling constants Ci j in the model Hamiltonian are determinedfrom the conventional hadrons. Whether they can be used tomultiquark states is still an open question, as noticed in Refs.[77,78]. Secondly, the mass shifts from CMI energies for asystem and thus the pentaquark masses are not well deter-mined. If one uses the effective quark masses determinedfrom conventional hadrons, one obtains hadron masses withoverestimated values. To reduce the uncertainty, we phe-nomenologically adopt a reference hadron-hadron system forour purpose. Whether the selected system is appropriate ornot is also a question to be answered. At present, we tend touse a system with high threshold in order to include possi-ble additional effects like the additional kinetic energy [72].The uncertainty for the obtained pentaquark masses in thismethod will be checked by the future experimental measure-ments. Thirdly, the adopted model does not involve dynamicsand much information is hidden in constant parameters. Theparameters should be different from system to system. How-ever, as a preliminary work for multiquark system wherethe few-body problem is difficult to deal with, the presentinvestigation on basic features of multiquark spectra is stillhelpful for further theoretical and experimental studies onmultiquark properties. We want to emphasize that the esti-mated masses do not indicate that all of these states shouldbe bound. Whether such states exist or not needs the exper-
imental measurements to confirm. On the other side, if onecompact pentaquark state could be observed, the masses ofits partner states may be predicted with our results.
If the pentaquark masses shown in Fig. 1 are all reason-able, a question arises to distinguish a compact multiquarkstate from a molecule state. In understanding the nature ofthe Pc(4380) state, both the (cqq)(cq) molecule configura-tion and the (cc)8c(qqq)8c pentaquark configuration are notcontradicted with experimental measurements. The massesof the proposed udscc in these two configurations are alsoconsistent [18,21]. For the present QQQqq case, the situa-tion is different. Our study does not favor low mass cccqqor bbbqq pentaquarks while the investigations at the hadronlevel [79,80] indicate that the molecule states such as �ccDand �ccD∗ are possible. A study with the QCD sume rulemethod also favors the existence of such molecules [81]. If alow-lying pentaquark-like state around such thresholds wereobserved, the molecule interpretation would be preferred.However, for the possible states around the thresholds of�cc B and �cc B∗, one needs to use other information likebranching ratios to understand whether it is a compact pen-taquark or a molecule, which can be studied in future works.
As mentioned in the beginning of this section, we haveused the assumptions that Ccc = Ccc, Css = Css , Cbb =Cbb, and Cbc = Cbc because of lack of experimentaldata. This assumption certainly leads to uncertainties forthe estimated masses. To see the differences caused byassumption selection, one may try other methods in deter-mining the coupling parameters and compare the resultingmasses. First, we check the extraction of Ccc, Cbb, andCbc from baryon masses. Based on the formulas given inEq. (17), one cannot determine Ccc just from the massdifference between �cc and �∗
once the mass of another doubly charmed baryon is known.However, the obtained Ccc is either negative or unreason-ably large if M�∗
cc> M�cc = 3621.4 MeV or M�cc < 3850
MeV (see Ref. [26] for the rough value of M�cc in theo-retical investigations). One may also use (M� + 2M�ccc −2M�∗
cc− M�cc − 8Cnn)/8 to extract Ccc, but the reasonable
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(positive, not large) value is sensitive to the assignment forthe spin of �cc and the masses of the involved baryons. Toextract Cbb, the masses of two doubly bottom baryons are atleast needed, but similar situation to Ccc is found once quarkmodel predictions are used. For the value of Cbc, one mayextract it with (M�∗
ccb− M�ccb )/16 = (M�∗
bbc− M�bbc )/16
or from M�∗bc
− M�′bc
. If the mass difference M�∗ccb
− M�ccb
is around tens of MeV, the obtained Cbc is not far from theadopted 3.3 MeV. However, Cbc is sensitive to the mass dif-ference M�∗
bc− M�′
bc. One may also extract mc, mb, Ccc,
Cbb, and Cbc simultaneously by using a set of masses for thetriply heavy baryons shown in Table 4, but results sensitiveto baryon masses or unreasonable results are obtained. Theabove considerations indicate that the determination of Ccc,Cbc, and Cbb with baryon masses still has some problems.Secondly, we check the method Ci j = CH/(mim j ) usu-ally used in the literature, e.g. Refs. [31,32,82]. This methodallows us to estimate Css from Cnn . According to Ref. [31],we have Css = 9/25Cnn = 10.7 MeV. This value is compa-rable to that from Ref. [83], 3/8κss = 11.3 MeV, but largerthan that we have used. However, for Ccc, Cbb, and Cbc, theparameter CH is different from those for quark-antiquarkpairs and light diquarks [31]. The determination of CH inthis case needs more baryon masses and the above problemsare probably also be there. Thirdly, we check the estimationmethod Ci j = Ci jCnn/Cnn ≈ 2/3Ci j where Ci j means Ci j
for a quark-antiquark pair. By inspecting values we havedadopted, those we can determine from experimental data areconsistent with this method. Then one gets Css = 10.5 MeV,Ccc = 3.3 MeV, Cbb = 1.8 MeV, and Cbc = 2.0 MeV. Theobtained Css in this method is also consistent with that in thesecond method. We will compare estimated QQQqq massesusing these parameters with those using our assumptions.
By changing Ccc = Ccc = 5.3 MeV, Cbb = Cbb = 2.9MeV, Cbc = Cbc = 3.3 MeV, and Css = Css = 6.5 MeV to3.3 MeV, 1.8 MeV, 2.0 MeV, and 10.5 MeV, respectively,and re-calculating pentaquark masses, one finds that thisdoes not affect the results significantly: (1) the reductionof Ccc or Cbb results in ∼10 MeV lower masses for rele-vant systems and the effects caused by the change of Cbc
or Css are just around several MeV’s; and (2) the changeof all the coupling parameters simultaneously induces massshifts around 0∼20 MeV. In fact, the effects due to thechanges of Ci j can be roughly seen from the obtained Ki j ’salthough the value of Ki j is also affected by the couplingparameters. From Tables 9 and 10, the largest numbers areKcc = Kbb = 10. Since the variations of Ccc, Cbb, and Cbc
cannot be large, the mass uncertainties because of them areat most tens of MeV. For Css , the effects from its uncertaintyare not large, either, because the largest Kss is only around1. Therefore, from the above discussions, the approxima-tions we have adopted do not cause significant effects on theresults.
Now we move on to the question how to distinguishan orbitally excited QQQ structure from an isoscalar pen-taquark structure if a high mass QQQ-like state wasobserved. Since the conventional P-wave states and the pen-taquark states may have similar decay patterns, the theoreticalmass gaps between these two structures are useful. We cancheck the gaps between the upper limits for the masses ofground QQQ baryons and the lowest (QQn)(Qn) thresh-olds (�ccD ∼5490 MeV, �bb B ∼15370 MeV, �bcD ∼8690MeV, and �bbD ∼11960 MeV) above which pentaquarksprobably exist. They are around 590 MeV for the ccc case,730 MeV for the bbb case, 600 MeV for the ccb case, and650 MeV for the bbc case. According to the calculations inthe literature, the P-wave excitation energy for the ccc caseis around 120∼360 MeV [25,53,62,84], that for the bbbcase is around 120∼600 MeV [25,53,60,62], that for theccb case is around 130∼280 MeV [25,53,62], and that forthe bbc case is around 120∼250 MeV [25,39–42,53,62,71].Therefore, the P-wave QQQ states are expected to be belowthe (QQn)(Qn) thresholds and thus below the pentaquarks.Alternatively, in theoretical studies, the maximum masses ofP-wave states are [53]: (ccc) ∼ 5160 MeV, (bbb) ∼ 14976MeV, (ccb) ∼ 8432 MeV, and (bbc) ∼ 11762 MeV. It seemsthat one does not need to worry about the exotic nature of theobserved QQQ baryon once it is below relevant (QQn)(Qn)
thresholds. However, this argument is only applicable to thelowest P-wave QQQ states. If the P-wave QQQ states (e.g.radially excited states studied in Ref. [71]) are heavier, thecoupled channel effects due to the (QQn)(Qn) thresholdsprobably result in states with exotic properties. In this case,the situation would be similar to the X (3872). If our pen-taquark masses are overestimated and they are close to themasses of the P-wave QQQ states, the configuration mix-ing is also possible. In this case, relatively lower QQQ-likestates might exist. To identify the nature of such states needsinformation from production or decay properties. We have towait for future experimental searches for possible interestingphenomena in triply heavy baryon systems.
5 Summary
In this paper, we have investigated the mass splittings forthe pentaquark states with the configuration QQQqq ina chromomagnetic model, where QQQ can be color-octetccc, bbb, ccb, or bbc. The values of their masses cannotbe determined accurately since the model does not involvedynamics. We estimate roughly the masses with two meth-ods. Since no symmetry constraints suppress the S-waverearrangement decays into (QQq) + (Qq), whether thedecays happen or not depends only on kinematics. Althoughthe obtained results have uncertainties, it seems that allthe studied pentaquarks are above the respective lowest
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Eur. Phys. J. C (2019) 79 :87 Page 17 of 19 87
(QQq)-(Qq) type thresholds and thus do not have lowmasses and narrow widths. On the other hand, the proposedmolecules around the thresholds of �ccD, �ccD∗, �cc B,and �cc B∗ are still possible [79]. Based on our results, oncea state around the threshold of �ccD were observed, itsmolecular nature other than compact multiquark nature isfavored.
To see the important mixing effects from different color-spin structures, we express the mass of an obtained pen-taquark state with effective chromomagnetic interactions byvarying coupling constants in the model Hamiltonian. Therelevant coefficients Ki j ’s are tabulated in Tables 9 and 10.From these tables, it is easy to understand the contributionsto the pentaquark mass from each pair of quark interactions.When comparing with the CMI matrices in Sect. 3, one findsthat the mixing may change the interaction strengths signif-icantly (even signs). Actually, the value of Ki j can also beobtained with the eigenvectors in Tables 5, 6, 7 and 8 andthe CMI matrices in Sect. 3. With the explicit expressions,one can see how the mixing effects contribute to a state. Forexample, the effective cn interaction in the lower J = 1/2cccnn state (I = 1 or 0) reads
KcnCcn = 0.522 × 0 + (−0.85)2 ×(
−20
3
)Ccn + 0.52
×(−0.85) × 10√3Ccn × 2 = −9.92Ccn, (25)
which equals to −9.99Ccn if the error bars in the eigenvectorare included.
As a byproduct of the pentaquark study, we have obtaineda conjecture for the mass inequalities for conventional doublyheavy baryons and conventional triply heavy baryons. Suchrelations may be tested in the future experimental results.
We hope the present study may stimulate further inves-tigations about properties of conventional doubly or triplyheavy baryons and multiquark states on both the theoreticalside and the experimental side. Search for the exotic triplyheavy baryons can be performed at LHC.
Acknowledgements This project is supported by National Natural Sci-ence Foundation of China (Grant Nos. 11775130, 11775132, 11635009,11325525, 11875179) and the Natural Science Foundation of ShandongProvince (Grant No. ZR2017MA002).
Data Availability Statement This manuscript has no associated dataor the data will not be deposited. [Authors’ comment: Our study onlyinvolves simple theoretical calculations. The manuscript has no associ-ated data.]
Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.Funded by SCOAP3.
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