PHYSICAL REVIEW D 99, 014035 (2019)

Hidden-bottom pentaquarks

† ‡ Gang Yang,1,* Jialun Ping,2, and Jorge Segovia3, 1Department of Physics and State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing 100084, People’s Republic of China 2Department of Physics and Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing 210023, People’s Republic of China 3Institut de Física d’Altes Energies (IFAE) and Barcelona Institute of Science and Technology (BIST), Universitat Autònoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain

(Received 19 September 2018; published 25 January 2019)

The LHCb Collaboration has recently reported strong evidences of the existence of pentaquark states in þ þ the hidden-charm baryon sector, the so-called Pcð4380Þ and Pcð4450Þ signals. Five-quark bound states in the hidden-charm sector were explored by us using, for the quark-quark interaction, a chiral quark model which successfully explains meson and baryon phenomenology, from the light to the heavy quark sector. We extend herein such study into the hidden-bottom pentaquark sector, analyzing possible bound-states P 1 3 5 1 3 with spin-parity quantum numbers J ¼ 2 , 2 and 2 , and in the 2 and 2 isospin sectors. We do not find positive parity hidden-bottom pentaquark states; however, several candidates with negative parity are found ΣðÞ ¯ ðÞ with dominant baryon-meson structures b B . Their inner structures have been also analyzed with the computation of the distance among any pair of quarks within the bound-state. This exercise reflects that molecular-type bound-states are favored when only color-singlet configurations are considered in the coupled-channels calculation whereas some deeply-bound compact pentaquarks can be found when hidden-color configurations are added. Finally, our findings resemble the ones found in the hidden-charm sector but, as expected, we find in the hidden-bottom sector larger binding energies and bigger contributions of the hidden-color configurations.

DOI: 10.1103/PhysRevD.99.014035

I. INTRODUCTION In 2015, the LHCb Collaboration observed two ψ After decades of experimental and theoretical studies of hidden-charm pentaquark states in the J= p invariant Λ0 → ψ − hadrons, the conventional picture of mesons and baryons mass spectrum of the b J= K p decay [1]. One is ð4380Þþ ð4380 8 29Þ as, respectively, quark-antiquark and 3-quark bound states Pc with a mass of MeV and a ð205 18 86Þ ð4450Þþ is being left behind. On one hand, quantum chromody- width of MeV, and another is Pc ð4449 8 1 7 2 5Þ namics (QCD), the non-Abelian quantum field theory of with a mass of . . . MeV and a width of ð39 5 19Þ MeV. The preferred JP assignments are of the strong interactions, does not prevent to have exotic 3 5 hadrons such as glueballs, quark-gluon hybrids, and multi- opposite parity, with one state having spin 2 and the other 2. ð4380Þþ ð4450Þþ quark systems. On the other hand, more than two dozens of The discovery of the Pc and Pc has nontraditional charmonium- and bottomonium-like states, triggered many theoretical works on this kind of multiquark the so-called XYZ mesons, have been observed in the last systems. The interested reader is directed to the recent 15 years at B-factories (BABAR, Belle and CLEO), τ-charm review [2] on hidden-charm pentaquark and tetraquark facilities (CLEO-c and BESIII) and also proton-(anti) states in order to have a global picture of the current progress; however, one can highlight those theoretical proton colliders (CDF, D0, LHCb, ATLAS and CMS). þ þ studies of the Pcð4380Þ and Pcð4450Þ in which different kind of quark arrangements are used such as diquark- – – *[email protected] triquark [3 5], diquark-diquark-antiquark [3,6 11], and † [email protected] meson-baryon molecule [3,12–23]. It is also noteworthy ‡ [email protected] that some recent investigations have considered other possible physical mechanisms as the origin of the exper- Published by the American Physical Society under the terms of imental signals like kinematic effects and triangle singu- the Creative Commons Attribution 4.0 International license. – Further distribution of this work must maintain attribution to larities [24 28]. the author(s) and the published article’s title, journal citation, The observation of hadrons containing valence c-quarks and DOI. Funded by SCOAP3. is historically followed by the identification of similar

2470-0010=2019=99(1)=014035(11) 014035-1 Published by the American Physical Society GANG YANG, JIALUN PING, and JORGE SEGOVIA PHYS. REV. D 99, 014035 (2019)

¯ ¯ structures with b-quark content. Therefore, it is natural dominant ΣcD and ΣcD Fock-state components were also to expect a subsequent observation of the bottom analogues found in the region about 4.3–4.5 GeV. þ þ of the Pcð4380Þ and Pcð4450Þ resonances, if they All the details about our computational framework eventually exist. The LHCb Collaboration has recently will be described later but let us sketch here some of made an attempt (with negative result) to search for its main features. Our chiral quark model (ChQM) is pentaquark states containing a single b-quark, that decays based on the fact that chiral symmetry is spontaneously weakly via the b → ccs¯ transition, in the final states broken in QCD and, among other consequences, it pro- J=ψKþπ−p, J=ψK−π−p, J=ψK−πþp, and J=ψϕp [29]. vides a constituent quark mass to the light quarks. To Therefore, reports about similar explorations in other restore the chiral symmetry in the QCD Lagrangian, bottom pentaquark sectors like the hidden-bottom one, Goldstone-boson exchange interactions appear between should be expected in the near future. the light quarks. This fact is encoded in a phenomeno- Theoretical investigations of the spectrum of hidden- logical potential which already contains the perturbative bottom pentaquarks as well as their electromagnetic, one-gluon exchange (OGE) interaction and a nonperturba- strong and weak decays help in the experimental hunt tive linear-screened confining term.1 It is worthwhile to mentioned above. In addition to this, further theoretical note that chiral symmetry is explicitly broken in the heavy studies supply complementary information on the internal quark sector and this translates in our formalism to the fact structure and interquark interactions of pentaquarks with that the interaction terms between light-light, light-heavy ð4380Þþ heavy quark content. In Ref. [30], besides the Pc and heavy-heavy quarks are not the same, i.e., while state, the possible existence of hidden-bottom penta- Goldstone-boson exchanges are considered when the two quarks with a mass around 11.08–11.11 GeV and quan- quarks are light, they do not appear in the other two − tum numbers JP ¼ 3=2 was emphasized; it was also configurations: light-heavy and heavy-heavy; however, the indicated that there may exist some loosely-bound one-gluon exchange and confining potentials are flavor molecular-type pentaquarks in other heavy quark sectors. blindness. Ming-Zhu Liu et al. used heavy-quark symmetry argu- The five-body bound state problem is solved by means of ð4380Þ ð4450Þ ments to find the partners of the Pc and Pc the Gaußian expansion method (GEM) [43] which provides [31,32]. A quark model study of the baryo-quarkonium enough accuracy and simplifies the subsequent evaluation picture for hidden-charm and -bottom pentaquarks was of the matrix elements. As it is well know, the quark model recently released in Ref. [33] concluding that hidden- parameters are crucial in order to describe particular bottom pentaquarks are more likely to form than their physical observables. We have used values that have hidden-charm counterparts. The baryo-quarkonium pic- been fitted before through hadron [44–49], hadron-hadron ture has been also explored within a nonrelativistic [50–54] and multiquark [40,55,56] phenomenology. effective field theory approach in Refs. [34–36]. Finally, The structure of the present manuscript is organized in see also Refs. [37,38] for more information on the the following way. In Sec. II the ChQM, pentaquark wave- properties of the charmed and bottom pentaquark states functions and GEM are briefly presented and discussed. using the coupled-channel unitary approach, as well as Section III is devoted to the analysis and discussion on the Refs. [24,26,28,39] for illuminating discussions on the obtained results. We summarize and give some prospects structure of pentaquarks and their possible relation with in Sec. IV. triangle singularities. We study herein, within a chiral quark model formalism, II. THEORETICAL FRAMEWORK the possibility of having pentaquark bound-states in the P ¼ 1 3 hidden-bottom sector with quantum numbers J 2 , 2 Although lattice QCD (LQCD) has made an impressive 5 1 3 and 2 , and in the 2 and 2 isospin sectors. Their inner progress on understanding multiquark systems [57,58] and structures are also obtained by computing the distance the hadron-hadron interaction [59–61], the QCD-inspired among any pair of quarks within the bound-state. This quark models are still the main tool to shed some light on work is a natural extension of the analysis performed in the nature of the multiquark candidates observed by Ref. [40] in which similar structures were studied but in the experimentalists. þ hidden-charm sector. In Ref. [40], the Pcð4380Þ was The general form of our five-body Hamiltonian is given ¯ suggested to be a bound state of ΣcD with quantum by [40] P ¼ 3− ð4450Þþ numbers J 2 whereas the nature of the Pc structure was not clearly established because, despite of X5 ⃗2 X5 ¼ þ pi − þ ð⃗ Þ ð Þ having a couple of possible candidates attending to the H mi TCM V rij ; 1 2mi agreement between theoretical and experimental masses, i¼1 j>i¼1 there was an inconsistency between the parity of the state determined experimentally and those predicted theoreti- 1The interested reader is referred to Refs. [41,42] for detailed cally. Further pentaquark bound-states which contain reviews on the naive quark model in which this work is based.

014035-2 HIDDEN-BOTTOM PENTAQUARKS PHYS. REV. D 99, 014035 (2019)

– where TCM is the center-of-mass kinetic energy and the The central terms of the chiral quark (anti-)quark two-body potential interaction can be written as 2 2 2 Vðr⃗ Þ¼V ðr⃗ ÞþV ðr⃗ ÞþVχðr⃗ Þ; ð2Þ Λ ij CON ij OGE ij ij ð⃗ Þ¼gch mπ π ð Þ Vπ rij 2 2 mπ Y mπrij 4π 12mimj Λπ − mπ includes the color-confining, one-gluon exchange and Λ3 X3 Goldstone-boson exchange interactions. Note herein that π a a − YðΛπr Þ ðσ⃗ σ⃗Þ ðλ λ Þ; ð Þ the potential could contain central, spin-spin, spin-orbit and 3 ij i · j i · j 7 mπ ¼1 tensor contributions; only the first two will be considered a attending the goal of the present manuscript and for clarity 2 Λ2 Λ in our discussion. ð⃗ Þ¼− gch σ ð Þ − σ ðΛ Þ Vσ rij 2 2 mσ Y mσrij Y σrij ; Color confinement should be encoded in the non- 4π Λσ − mσ mσ Abelian character of QCD. Studies of lattice-regularized ð8Þ QCD have demonstrated that multigluon exchanges pro- duce an attractive linearly rising potential proportional to the distance between infinite-heavy quarks [62]. However, 2 2 Λ2 ð⃗ Þ¼gch mK K ð Þ the spontaneous creation of light-quark pairs from the QCD VK rij 4π 12 Λ2 − 2 mK Y mKrij mimj K mK vacuum may give rise at the same scale to a breakup of the 3 X7 created color flux-tube [62]. We have tried to mimic these Λ − K YðΛ r Þ ðσ⃗ · σ⃗Þ ðλa · λaÞ; ð9Þ two phenomenological observations by the expression: 3 K ij i j i j mK a¼4

−μ r ⃗c ⃗c V ðr⃗ Þ¼½−a ð1 − e c ij ÞþΔðλ · λ Þ; ð3Þ 2 2 2 CON ij c i j mη Λη ð⃗ Þ¼gch ð Þ Vη rij 4π 12 2 2 mη Y mηrij μ mimj Λη − mη where ac and c are model parameters, and the SU(3) color Gell-Mann matrices are denoted as λc. One can Λ3 − η ðΛ Þ ðσ⃗ σ⃗Þ½ θ ðλ8 λ8Þ see in Eq. (3) that the potential is linear at short inter- 3 Y ηrij i · j cos p i · j mη quark distances with an effective confinement strength ⃗c ⃗c − sin θp; ð10Þ σ ¼ −acμcðλi · λj Þ, while it becomes constant at large distances. ð Þ The one-gluon exchange potential is given by where Y x is the standard Yukawa function defined by YðxÞ¼e−x=x. We consider the physical η meson instead of − ðμÞ θ λa 1 1 1 rij=r0 the octet one and so we introduce the angle p. The are ð⃗ Þ¼ α ðλ⃗c λ⃗cÞ − ðσ⃗ σ⃗Þ e VOGE rij 4 s i · j 6 i · j 2 ; the SU(3) flavor Gell-Mann matrices. Taken from their rij mimj rijr0ðμÞ experimental values, mπ, mK and mη are the masses of the ð Þ 4 SU(3) Goldstone bosons. The value of mσ is determined 2 ≃ 2 þ 4 2 through the PCAC relation mσ mπ mu;d [63]. Finally, where mi is the quark mass and the Pauli matrices are σ⃗ the chiral coupling constant, gch, is determined from the denoted by . The contact term of the central potential has πNN coupling constant through been regularized as

2 2 2 −r =r0 1 e ij g 9 gπ mu;d δð⃗ Þ ∼ ð Þ ch ¼ NN ; ð11Þ rij 2 ; 5 4π 25 4π 2 4πr0 rij mN ðμ Þ¼ˆ μ μ with r0 ij r0= ij a regulator that depends on ij, the which assumes that flavor SU(3) is an exact symmetry only reduced mass of the quark–(anti-)quark pair. broken by the different mass of the strange quark. The wide energy range needed to provide a consistent The model parameters have been fixed in advance description of mesons and baryons from light to heavy reproducing hadron [44–49], hadron-hadron [50–54] and quark sectors requires an effective scale-dependent strong multiquark [40,55,56] phenomenology. For clarity, the coupling constant. We use the frozen coupling constant of, ones involved in this calculation are listed in Table I. for instance, Ref. [42] They were used in Ref. [40] to study possible hidden- charm pentaquark bound-states with quantum numbers α0 α ðμ Þ¼ ð Þ P ¼ 1 ð1Þ 1 ð3Þ 1 ð5Þ s ij μ2 þμ2 ; 6 IJ 2 2 , 2 2 and 2 2 ; moreover, their pro- ij 0 ln 2 Λ0 perties were compared with those associated with the hidden-charm pentaquark signals observed by the LHCb in which α0, μ0 and Λ0 are parameters of the model. Collaboration in Ref. [1].

014035-3 GANG YANG, JIALUN PING, and JORGE SEGOVIA PHYS. REV. D 99, 014035 (2019)

TABLE I. Quark model parameters. B1−1 ¼ ddb; ð16Þ Quark masses m ¼ m (MeV) 313 u d 1 mb (MeV) 5100 −1 B00 ¼ pffiffiffi ðud − duÞb; ð17Þ Goldstone bosons Λπ ¼ Λσ (fm ) 4.20 2 −1 Λη (fm ) 5.20 2 ð4πÞ 1 gch= 0.54 1 B1 1 ¼ pffiffiffi ð2uud − udu − duuÞ; ð18Þ θPð°Þ −15 2;2 6 Confinement ac (MeV) 430 μ (fm−1) 0.70 1 c 2 ¼ pffiffiffi ð − Þ ð Þ Δ (MeV) 181.10 B1 1 ud du u; 19 2;2 2 α0 2.118 −1 Λ0 (fm ) 0.113 ¯ M1 1 ¼ bu; ð20Þ OGE μ0 (MeV) 36.976 2;2 rˆ0 (MeV fm) 28.170 ¯ M1 1 ¼ bd; ð21Þ 2;−2 The pentaquark wave function is a product of four terms: ¯ M00 ¼ bb: ð22Þ color, flavor, spin and space wave functions. Concerning the color degree-of-freedom, multiquark systems have Consequently, the flavor wave-functions for the 5-quark richer structure than the conventional mesons and baryons. system with isospin I ¼ 1=2 or 3=2 are For instance, the 5-quark wave function must be colorless rffiffiffi rffiffiffi but the way of reaching this condition can be done through f1 2 1 χ ð5Þ¼ 1 1 − 1 1 ð Þ either a color-singlet or a hidden-color channel or both. The 1 1 B11M ;− B10M ; ; 23 2;2 3 2 2 3 2 2 authors of Refs. [64,65] assert that it is enough to consider the color singlet channel when all possible excited states of f2 χ ð5Þ¼ 1 1 ð Þ 1 1 B00M ; ; 24 a system are included. However, a more economical way of 2;2 2 2 computing is considering both, the color singlet wave f3 1 function: χ1 1 ð5Þ¼B1 1M00; ð25Þ 2;2 2;2 1 χc ¼ pffiffiffiffiffi ðrgb − rbg þ gbr − grb þ brg − bgrÞ f4 2 1 χ1 1 ð5Þ¼B1 1M00; ð26Þ 18 2;2 2;2 ð¯ þ ¯ þ ¯ Þ ð Þ × rr gg bb ; 12 f1 χ ð5Þ¼ 3 3 ð Þ 3 3 B ; M00; 27 2;2 2 2 and the hidden-color one: f2 χ3 3 ð5Þ¼B1 1M1 1; ð28Þ 1 ; ; 2;2 χc ¼ pffiffiffi ðχk χ − χk χ − χk χ þ χk χ 2 2 k 8 3;1 2;8 3;2 2;7 3;3 2;6 3;4 2;5 where the third component of the isospin is set to be equal þ χk χ − χk χ − χk χ þ χk χ Þ ð Þ 3;5 2;4 3;6 2;3 3;7 2;2 3;8 2;1 ; 13 to the total one without loss of generality because there is no interaction in the Hamiltonian that can distinguish such ¼ 2ð3Þ where k is an index which stands for the symmetric component. (antisymmetric) configuration of two quarks in the 3-quark We consider herein 5-quark bound states with total spin subcluster. All color configurations have been used herein, 1 2 5 2 þ ranging from = to = . Since our Hamiltonian does not as in the case of the Pc hidden-charm pentaquarks studied have any spin-orbital coupling dependent potential, we can in Ref. [40]. þ assume that third component of the spin is equal to the total In analogy to the study of the Pc -type bound states in one without loss of generality. Our spin wave function is Ref. [40], we assume that the flavor wave function of given by: the uudbb¯ system is composed by ðudbÞðbu¯ ÞþðuubÞðbd¯ Þ ¯ rffiffiffi rffiffiffi and ðuudÞðbbÞ configurations. According to the SU(2) 1 1 χσ1ð5Þ¼ χσ ð3Þχσ − χσ ð3Þχσ symmetry in isospin space, the flavor wave functions for 1 1 3 −1 11 3 1 10 2;2 6 2; 2 3 2;2 the subclusters mentioned above are given by: rffiffiffi 1 σ σ þ χ3 3ð3Þχ1−1 ð29Þ B11 ¼ uub; ð14Þ 2 2;2 rffiffiffi rffiffiffi 1 σ2 1 σ1 σ 2 σ1 σ B10 ¼ pffiffiffi ðud þ duÞb; ð Þ χ ð5Þ¼ χ ð3Þχ − χ ð3Þχ ð Þ 15 1 1 1 1 10 1 −1 11 30 2 2;2 3 2;2 3 2; 2

014035-4 HIDDEN-BOTTOM PENTAQUARKS PHYS. REV. D 99, 014035 (2019)

rffiffiffi rffiffiffi ϕ’ 1 2 functions, s in Eq. (39), by a superposition of infinitesi- χσ3ð5Þ¼ χσ2ð3Þχσ − χσ2 ð3Þχσ ð Þ 1 1 1 1 10 1 −1 11 31 mally-shifted Gaussians (ISG) [43]: 2;2 3 2;2 3 2; 2 2 l −νnr σ4 σ1 σ ϕnlmðr⃗Þ¼Nnlr e YlmðrˆÞ χ1 1 ð5Þ¼χ1 1 ð3Þχ00 ð32Þ 2;2 2;2 Xkmax 1 −ν ð⃗−ε ⃗ Þ2 ¼ n r Dlm;k ð Þ σ5 σ2 σ Nnllim l Clm;ke : 44 χ1 1 ð5Þ¼χ1 1 ð3Þχ00 ð33Þ ε→0 ðν εÞ 2;2 2;2 n k¼1 for S ¼ 1=2, and where the limit ε → 0 must be carried out after the matrix rffiffiffi rffiffiffi elements have been calculated analytically. This new set of σ1 3 σ σ 2 σ σ basis functions makes the calculation of 5-body matrix χ3 3 ð5Þ¼ χ3 3ð3Þχ10 − χ3 1ð3Þχ11 ð34Þ 2;2 5 2;2 5 2;2 elements easier without the laborious Racah algebra [43]. Moreover, all the advantages of using Gaußians remain χσ2ð5Þ¼χσ ð3Þχσ ð Þ with the new basis functions. 3;3 3;3 00 35 2 2 2 2 Finally, in order to fulfill the Pauli principle, the σ3 σ1 σ complete antisymmetric wave function is written as χ3 3 ð5Þ¼χ1 1 ð3Þχ11 ð36Þ 2;2 2;2 σ Ψ ¼ A½½ψ χ i ð5Þ χfχc; ð45Þ σ4 σ2 σ JM;i;j;k;n L S JMJ j k χ3 3 ð5Þ¼χ1 1 ð3Þχ11 ð37Þ 2;2 2;2 where A is the antisymmetry operator of the 5-quark for S ¼ 3=2, and system. This is needed because we have constructed an

σ1 σ σ antisymmetric wave function for only two quarks of the χ5 5 ð5Þ¼χ3 3ð3Þχ11 ð38Þ 2;2 2;2 3-quark subcluster, the remaining (anti)quarks of the system have been added to the wave function by simply for S ¼ 5=2. These expressions can be obtained easily considering the appropriate Clebsch-Gordan coefficients. considering the 3-quark and quark-antiquark subclusters Moreover, the antisymmetry operator A has six terms but and using SU(2) algebra. They were derived in Ref. [40] for since we are considering that the uudbb¯ system is made the hidden-charm pentaquarks. by the quark arrangements ðudbÞðbu¯ ÞþðuubÞðbd¯ Þ and Among the different methods to solve the Schrödinger- ðuudÞðbb¯ Þ,wehave like 5-body bound state equation, we use the Rayleigh-Ritz variational principle which is one of the most extended A1 ¼ 1 − ð15Þ − ð25Þ; ð46Þ tools to solve eigenvalue problems due to its simplicity and flexibility. However, it is of great importance how to choose for the ðudbÞðbu¯ ÞþðuubÞðbd¯ Þ configuration, and the basis on which to expand the wave function. The spatial wave function of a 5-quark system is written as follows: A2 ¼ 1 − ð13Þ − ð23Þ; ð47Þ ψ ¼ ½½½ϕ ðρ⃗Þϕ ðλ⃗Þ ϕ ð⃗Þ ϕ ð ⃗Þ ð Þ ¯ LM n1l1 n2l2 n3l3 r 0 n4l4 R ; 39 ð Þð Þ L l l LML for the uud bb structure. where the internal Jacobi coordinates are defined as III. RESULTS

ρ⃗¼ x⃗1 − x⃗2; ð40Þ In the present calculation, we investigate the possible ¯ lowest-lying states of the uudbb pentaquark system taking ¯ ¯ ¯ ⃗ m1x⃗1 þ m2x⃗2 into account the ðudbÞðbuÞþðuubÞðbdÞ and ðuudÞðbbÞ λ ¼ x⃗3 − ; ð41Þ m1 þ m2 configurations in which the considered baryons have always positive parity and the open- and hidden-bottom P − − 2 r⃗¼ x⃗4 − x⃗5; ð42Þ mesons are either pseudoscalars ðJ ¼ 0 Þ or vectors ð1 Þ. This means that, in our approach, a pentaquark state with m1x⃗1 þ m2x⃗2 þ m3x⃗3 m4x⃗4 þ m5x⃗5 positive parity should have at least one unity of angular R⃗¼ − : ð43Þ m1 þ m2 þ m3 m4 þ m5 momentum: L ¼ 1, whereas the negative parity states have L ¼ 0. Reference [40] showed that positive parity This choice is convenient because the center-of-mass L ¼ 1 hidden-charm pentaquark states are always above its kinetic term TCM can be completely eliminated for a nonrelativistic system. 2There may exist other baryon-meson structures which contain In order to make the calculation tractable, even for excited hadrons such as χb1Nð940Þ, ϒð1SÞNð1440Þ and so on; all complicated interactions, we replace the orbital wave of them are beyond the scope of the present calculation.

014035-5 GANG YANG, JIALUN PING, and JORGE SEGOVIA PHYS. REV. D 99, 014035 (2019)

TABLE II. All possible channels for hidden-bottom pentaquark TABLE III. Lowest-lying states of hidden-bottom pentaquarks P 1 1− systems with negative parity. with quantum numbers IðJ Þ¼2 ð2 Þ. First column: channel in which a bound state appears, we show in parenthesis the 1 3 I ¼ 2 I ¼ 2 experimental value, in MeV, of the noninteracting baryon-meson σ f σ f threshold; second column: color-singlet (S), hidden-color (H) and P χ i χ j χc χ i χ j χc J Index J ; I ; k Channel J ; I ; k Channel coupled-channels (S þ H) calculation; third column: theoretical ½i; j; k½i; j; k mass, in MeV, of the pentaquark state; fourth column: its binding 1− ½4 5; 3 4; 1 ð η Þ1 ½1; 1; 1 ðΔϒÞ1 energy, in MeV, considering the theoretical baryon-meson thresh- 2 1 ; ; N b ½4 5; 3 4; 2 3 8 ½1; 1; 3 8 old; fifth column: again the pentaquark’s mass, in MeV, but re- 2 ; ; ; ðNηbÞ ðΔϒÞ ½2 3; 3 4; 1 1 ½4; 2; 1 ¯ 1 scaled attending to the experimental baryon-meson threshold. 3 ; ; ðNϒÞ ðΣbBÞ 8 8 The percentages of color-singlet (S) and hidden-color (H) 4 ½2; 3; 3; 4; 2; 3 ðNϒÞ ½4; 5; 2; 2; 3 ðΣ B¯ Þ b channels are also given when the coupled-channels calculation 5 ½5; 2; 1 ðΛ B¯ Þ1 ½2; 2; 1 ðΣ B¯ Þ1 b b is performed. The baryon-meson channels that do not appear here 6 ½4; 5; 2; 2; 3 ðΛ B¯ Þ8 ½2; 3; 2; 2; 3 ðΣ B¯ Þ8 b b have been also considered in the computation but no bound states 7 ½3; 2; 1 ðΛ ¯ Þ1 ½1; 2; 1 ðΣ ¯ Þ1 bB bB were found. ½2 3; 2; 2 3 ðΛ ¯ Þ8 ½1; 2; 3 ðΣ ¯ Þ8 8 ; ; bB bB ½4; 1; 1 ¯ 1 0 9 ðΣbBÞ Channel Color MEB M 10 ½4; 5; 1; 2; 3 ðΣ ¯ Þ8 bB Σ B¯ S 11 080 −15 11 074 ½2; 1; 1 ðΣ ¯ Þ1 b 11 bB (11 089) H 11 364 þ269 11 358 ½2 3; 1; 2 3 ðΣ ¯ Þ8 12 ; ; bB S þ H 11 078 −17 11 072 ½1; 1; 1 ðΣ ¯ Þ1 13 bB Percentage (S;H): 98.5%; 1.5% 14 ½1; 1; 3 ðΣB¯ Þ8 ¯ −21 b ΣbB S 11 115 11 113 3− ½3 4; 3 4; 1 ð ϒÞ1 ½2; 1; 1 ðΔη Þ1 2 1 ; ; N b (11 134) H 11 257 þ121 11 255 ½3 4; 3 4; 2 3 8 ½2; 1; 3 8 þ −93 2 ; ; ; ðNϒÞ ðΔηbÞ S H 11 043 11 041 ½4; 2; 1 ¯ 1 ½1; 1; 1 1 3 ðΛbB Þ ðΔϒÞ Percentage (S;H): 57.9%; 42.1% ½3 4; 2; 2 3 ¯ 8 ½1; 1; 3 8 Σ ¯ −26 4 ; ; ðΛbB Þ ðΔϒÞ bB S 11 127 11 128 ½3; 1; 1 ¯ 1 ½3; 2; 1 ¯ 1 −232 5 ðΣbB Þ ðΣbB Þ (11 154) H 10 921 10 922 ½3 4; 1; 2 3 ¯ 8 ½3 4; 2; 2 3 ¯ 8 þ −292 6 ; ; ðΣbB Þ ; ; ðΣbB Þ S H 10 861 10 862 ½2; 1; 1 ðΣ ¯ Þ1 ½2; 2; 1 ðΣ ¯ Þ1 Percentage (S;H): 15.8%; 84.2% 7 bB bB ½2; 1; 3 ðΣ ¯ Þ8 ½2; 2; 3 ðΣ ¯ Þ8 8 bB bB ½1; 1; 1 ðΣ ¯ Þ1 ½1; 2; 1 ðΣ ¯ Þ1 9 bB bB ½1; 1; 3 ðΣ ¯ Þ8 ½1; 2; 3 ðΣ ¯ Þ8 1 5− 3 3− 3 5− 3 10 bB bB 2 ð2 Þ, 2 ð2 Þ and 2 ð2 Þ, respectively. In each Table, the first 5− ½1; 1; 1 ðΣ ¯ Þ1 ½1; 1; 1 ðΔϒÞ1 2 1 bB column shows the baryon-meson channel in which a ½1; 1; 3 ðΣ ¯ Þ8 ½1; 1; 3 ðΔϒÞ8 2 bB bound state appears, it also indicates in parenthesis the ½1; 2; 1 ðΣ ¯ Þ1 3 bB experimental value of the noninteracting baryon-meson ½1; 2; 3 ðΣ ¯ Þ8 4 bB threshold; the second column refers to color-singlet (S), hidden-color (H) and coupled-channels (S þ H) calcula- tions; the third and fourth columns show the theoretical corresponding noninteracting baryon-meson threshold mass and binding energy of the pentaquark bound-state; and the same situation is found within the hidden-bottom and the fifth column presents the theoretical mass of the sector. pentaquark state but rescaled attending to the experimental ¼ 0 For negative parity L hidden-bottom pentaquarks baryon-meson threshold, this is in order to avoid theoretical we assume that the angular momenta l1, l2, l3, l4, which uncertainties coming from the quark model prediction of appear in Eq. (39), are 0. In this way, the total angular the baryon and meson spectra. The percentages of color- momentum, J, coincides with the total spin, S, and can take singlet (S) and hidden-color (H) channels are also given values 1=2, 3=2 and 5=2. The possible baryon-meson when the coupled-channels calculation is performed. For channels which are under consideration in the computation the channels in which a bound-state is found, we show in are listed in Table II, they have been grouped according to Table VIII a calculation of all possible quark-quark dis- total spin and isospin. The third and fifth columns of σ tances in order to get some insight about the kind of ðχ i Þ Table II show the necessary basis combination in spin J , pentaquark we are dealing with: molecular or compact. ðχfj Þ ðχcÞ We proceed now to describe in detail our theoretical flavor I , and color k degrees-of-freedom. The phy- sical channels with color-singlet (labeled with the super- findings: IðJPÞ¼1 ð1−Þ index 1) and hidden-color (labeled with the superindex 8) The 2 2 channel: Among all the possible η Υ Λ ¯ Λ ¯ Σ ¯ configurations are listed in the fourth and sixth columns. baryon-meson channels: N b, N , bB, bB , bB, Tables ranging from III–VII summarize our findings 3 P 3 1− about the possible existence of lowest-lying hidden-bottom A table associated with the IðJ Þ¼2 ð2 Þ sector is not shown P 1 1− 1 3− pentaquarks with quantum numbers IðJ Þ¼2 ð2 Þ, 2 ð2 Þ, because no bound-states were found.

014035-6 HIDDEN-BOTTOM PENTAQUARKS PHYS. REV. D 99, 014035 (2019)

TABLE IV. Lowest-lying states of hidden-bottom pentaquarks TABLE VI. Lowest-lying states of hidden-bottom pentaquarks P 1 3− P 3 3− with quantum numbers IðJ Þ¼2 ð2 Þ. First column: channel in with quantum numbers IðJ Þ¼2 ð2 Þ. First column: channel in which a bound state appears, we show in parenthesis the which a bound state appears, we show in parenthesis the experimental value, in MeV, of the noninteracting baryon-meson experimental value, in MeV, of the noninteracting baryon-meson threshold; second column: color-singlet (S), hidden-color (H) and threshold; second column: color-singlet (S), hidden-color (H) and coupled-channels (S þ H) calculation; third column: theoretical coupled-channels (S þ H) calculation; third column: theoretical mass, in MeV, of the pentaquark state; fourth column: its binding mass, in MeV, of the pentaquark state; fourth column: its binding energy, in MeV, considering the theoretical baryon-meson thresh- energy, in MeV, considering the theoretical baryon-meson thresh- old; fifth column: again the pentaquark’s mass, in MeV, but re- old; fifth column: again the pentaquark’s mass, in MeV, but scaled attending to the experimental baryon-meson threshold. rescaled attending to the experimental baryon-meson threshold. The percentages of color-singlet (S) and hidden-color (H) The percentages of color-singlet (S) and hidden-color (H) channels are also given when the coupled-channels calculation channels are also given when the coupled-channels calculation is performed. The baryon-meson channels that do not appear here is performed. The baryon-meson channels that do not appear here have been also considered in the computation but no bound states have been also considered in the computation but no bound states were found. were found.

0 0 Channel Color MEB M Channel Color MEB M ¯ −12 ¯ ΣbB S 11 124 11 122 ΣbB S 11 136 0 11 134 (11 134) H 11 476 þ340 11 475 (11 134) H 11 310 þ174 11 308 S þ H 11 122 −14 11 120 S þ H 11 021 −115 11 019 Percentage (S;H): 99.6%; 0.4% Percentage (S;H): 64.7%; 35.3% Σ ¯ −15 Σ ¯ bB S 11 097 11 094 bB S 11 112 0 11 109 (11 109) H 11 175 þ63 11 172 (11 109) H 11 041 −71 11 038 S þ H 11 045 −67 11042 S þ H 10 999 −113 10 996 Percentage (S;H): 55.5%; 44.5% Percentage (S;H): 18.4%; 81.6% Σ ¯ −15 Σ ¯ bB S 11 138 11 139 bB S 11 153 0 11 154 (11 154) H 11 051 −102 11 052 (11 154) H 11 102 −51 11 103 S þ H 10 958 −195 10 959 S þ H 11 048 −105 11 049 Percentage (S;H): 22.2%; 77.8% Percentage (S;H): 15.7%; 84.3%

Σ ¯ Σ ¯ bB and bB ; only the last three point to the possibility −26 MeV, respectively. This motivates the possibility of having bound states. In particular, when considering of finding molecular-type baryon-meson structures around Σ ¯ Σ ¯ ðÞ only the color-singlet configuration of bB, bB and the Σ B¯ ðÞ thresholds. One can see in Table III that the ΣB¯ the binding energies are −15 MeV, −21 MeV and b ¯ b binding energy is slightly larger for ΣbB ðEB ¼ −17 MeVÞ

TABLE V. Lowest-lying states of hidden-bottom pentaquarks TABLE VII. Lowest-lying states of hidden-bottom pentaquarks ð PÞ¼1 ð5−Þ P 3 5− with quantum numbers I J 2 2 . First column: channel in with quantum numbers IðJ Þ¼2 ð2 Þ. First column: channel in which a bound state appears, we show in parenthesis the which a bound state appears, we show in parenthesis the experimental value, in MeV, of the noninteracting baryon-meson experimental value, in MeV, of the noninteracting baryon-meson threshold; second column: color-singlet (S), hidden-color (H) and threshold; second column: color-singlet (S), hidden-color (H) and coupled-channels (S þ H) calculation; third column: theoretical coupled-channels (S þ H) calculation; third column: theoretical mass, in MeV, of the pentaquark state; fourth column: its binding mass, in MeV, of the pentaquark state; fourth column: its binding energy, in MeV, considering the theoretical baryon-meson thresh- energy, in MeV, considering the theoretical baryon-meson thresh- old; fifth column: again the pentaquark’s mass, in MeV, but old; fifth column: again the pentaquark’s mass, in MeV, but rescaled attending to the experimental baryon-meson threshold. rescaled attending to the experimental baryon-meson threshold. The percentages of color-singlet (S) and hidden-color (H) The percentages of color-singlet (S) and hidden-color (H) channels are also given when the coupled-channels calculation channels are also given when the coupled-channels calculation is performed. The baryon-meson channels that do not appear here is performed. The baryon-meson channels that do not appear here have been also considered in the computation but no bound states have been also considered in the computation but no bound states were found. were found.

0 0 Channel Color MEB M Channel Color MEB M Σ ¯ −12 Σ ¯ −101 bB S 11 141 11 151 bB S 11 052 11 053 (11 154) H 11 547 þ394 11 548 (11 154) H 10 974 −179 10 975 S þ H 11 140 −13 11 141 S þ H 10 931 −222 10 932 Percentage (S;H): 99.6%; 0.4% Percentage (S;H): 19.9%; 80.1%

014035-7 GANG YANG, JIALUN PING, and JORGE SEGOVIA PHYS. REV. D 99, 014035 (2019)

TABLE VIII. The distance, in fm, between any two quarks of again the possibility of finding molecular-type baryon- the found pentaquark bound-states. ΣðÞ ¯ ðÞ meson structures around the b B thresholds. If one P incorporates in the coupled-channels calculation the hid- IðJ Þ Channel Mixing rqq rqQ rqQ¯ rQQ¯ den-color configurations, the situation in the IðJPÞ¼1 ð3−Þ 1 ð1−Þ Σ ¯ 2 2 2 2 bB S 1.17 0.87 1.02 1.00 ð PÞ¼1 ð1−Þ þ channel is quite similar with respect the I J 2 2 one. S H 1.13 0.84 0.98 0.94 ¯ ¯ While the ΣbB bound-state modifies slightly its mass and ΣbB S 1.09 0.81 0.92 0.82 S þ H 0.94 0.70 0.71 0.34 points to a molecular-type structure (see Table VIII for ΣB¯ S 1.06 0.79 0.88 0.75 comparing interquark distances) the other two states found b Σ ¯ Σ ¯ S þ H 0.91 0.71 0.70 0.24 in bB and bB channels appear to be tightly bound with 1 3− Σ ¯ −67 −195 2 ð2 Þ bB S 1.23 0.90 1.09 1.09 binding energies MeV and MeV, respectively. S þ H 1.21 0.90 1.07 1.07 One can see in Table VIII that, for the deeply-bound states, Σ ¯ the distance between the heavy quark-antiquark pair bB S 1.18 0.88 1.04 1.01 S þ H 0.98 0.74 0.74 0.34 reduces considerably and the other interquark distances, Σ ¯ bB S 1.17 0.87 1.02 0.97 despite becoming smaller, are much larger and of the same S þ H 0.95 0.72 0.72 0.25 order of magnitude. This could point to a possible compact 1 ð5−Þ Σ ¯ 2 2 bB S 1.25 0.92 1.11 1.13 multiquark nature of these states as explained above. It is S þ H 1.25 0.92 1.11 1.11 also interesting to mention herein that the contribution to 3 3− Σ ¯ þ 2 ð2 Þ bB S H 1.02 0.78 0.77 0.27 the wave function of the hidden color configuration is Σ ¯ þ Σ ¯ bB S H 1.02 0.84 0.82 0.26 negligible for the bB channel, slightly subdominant in Σ ¯ þ Σ ¯ Σ ¯ bB S H 1.05 0.83 0.81 0.26 the bB case and about 80% for the bB channel. In 3 ð5−Þ Σ ¯ ð4380Þþ 2 2 bB S 1.03 0.86 0.86 0.29 Ref. [40], we assigned to the Pc signal observed S þ H 1.00 0.86 0.84 0.26 by the LHCb Collaboration [1] a bound state found in the P 1 3− ¯ IðJ Þ¼2 ð2 Þ ΣcD channel. Its hidden-bottom pentaquark Σ ¯ when the hidden-color configuration is incorporated in the partner would be the bound state in the bB channel shown calculation; in fact, its contribution to the hadron’swave in Table IV. Both states present similar characteristics function is pretty small, 1.5%. This small change could beyond the typical differences associated with having indicate that the state is of molecular-type and, in fact, all different heavy quark content. The quantum numbers ð4450Þþ interquark distances shown in Table VIII are very similar, assignment of the Pc signal is not yet clear and ∼1 fm, pointing to a relatively extended hadron. The different possibilities are currently discussed in the liter- situation is quite different for the other two bound states ature [2], this reason avoids us to comment herein on what found in the Σ B¯ and ΣB¯ channels. One can see in would be the hidden-bottom pentaquark partner of b b this state. Table III that the binding energy becomes very large The IðJPÞ¼1 ð5−Þ channel: The ΣB¯ channel is the when the hidden-color configuration is incorporated: E ¼ 2 2 b B only baryon-meson structure needed to be considered (see −93 MeV for Σ B¯ and E ¼ −292 MeV for ΣB¯ . These b B b Table II). When the computation is performed taking into deeply bound states are usually associated with compact account only the singlet-color configuration, a slightly multiquark structures. One can observe in Table VIII that bound state is found with a binding energy of −12 MeV the distance between the two heavy quarks reduces con- (see now Table V). If the hidden-color channel is incorpo- siderably when the hidden-color configuration is incorpo- rated to the coupled-channels calculation the binding rated, indicating that a compact heavy quark-antiquark core energy increases just by 1 MeV. This indicates that the is formed and surrounded by light quarks. In comparison effect of this channel is very small, i.e., the contribution of with our study of hidden-charm pentaquarks of Ref. [40],a the hidden-color configuration to the bound-state wave similar trend is observed but, as expected, we find in the function is negligible, 0.4%. As one can see in Table VIII, hidden-bottom sector larger binding energies and bigger the distance between any pair of quarks is around 1 fm contributions of the hidden-color configurations. As an which is a common feature within our framework for all ðΣ ¯ Þ8 example of the last feature, we have 42.1% for bB and bound states dominated by the color-singlet configuration. ðΣ ¯ Þ8 ðΣ ¯ Þ8 84.2% for bB which compare with 32.6% for cD Again, we remark herein that this peculiarity could point to ðΣ ¯ Þ8 and 77% for cD . have a state of molecular nature. IðJPÞ¼1 ð3−Þ P 3 3− The 2 2 channel: There exist bound-states in The IðJ Þ¼2 ð2 Þ channel: Table VI indicates that no Σ ¯ Σ ¯ Σ ¯ the bB , bB and bB configurations but no signal of bound states are found in any baryon-meson configuration binding is found for the baryon-meson channels NΥ and when color-singlet arrangements are the only ones consid- ¯ ΛbB . Looking at Table IV, one can realize that the binding ered in the coupled-channels calculation. If hidden-color energies are −12 MeV, −15 MeV and −15 MeV for the clusters are added to the computation, a bound state appears Σ ¯ Σ ¯ Σ ¯ Σ ¯ Σ ¯ Σ ¯ bB , bB and bB channels, respectively, when consid- in the bB , bB and bB channels. The bound state ¯ ering only the color-singlet configuration. This motivates found in the ΣbB channel is quite sensitive to the

014035-8 HIDDEN-BOTTOM PENTAQUARKS PHYS. REV. D 99, 014035 (2019) numerical setup, making our prediction not very trustable. of Goldstone-boson exchange interactions between light However, the other two bound-states are very stable against quarks that are encoded in a phenomenological potential numerical checks and also reflect similar features: (i) the which already contains the perturbative one-gluon ðΣ ¯ Þ ðΣ ¯ Þ bB - and bB -type states are deeply bound with exchange and the nonperturbative linear-screened confin- binding energies around 100 MeV, (ii) the hidden-color ing terms. Note also that the model parameters have been configurations contribute ∼80% to the wave function, and fitted in the past through hadron, hadron-hadron and (iii) the distance of the bb¯-pair is much smaller than the multiquark phenomenology. Among the different methods others (see Table VIII) indicating that there is a quark- to solve the Schrödinger equation in order to find 5-body antiquark core surrounded by three light quarks. bound states, we use the Gaußian expansion method which P 3 5− is accurate enough and allows us to compute straightfor- IðJ Þ¼2 ð2 Þ channel: Only two baryon-meson chan- nels contribute to this case: ΔΥð1SÞ and ΣB¯ . As in all wardly the different matrix elements. b We have not found any positive parity hidden-bottom cases studied before, we do not find any bound state in the ΔΥð1 Þ pentaquark state within the scanned quantum numbers: S configuration. However, a bound-state is found in ¼ 1 3 5 ¼ 1 3 Σ ¯ J 2, 2, 2 and I 2, 2. However, several hidden-bottom the bB channel when considering either the singlet- or hidden-color configurations; the coupling between them pentaquark bound states with negative parity have been identified. These are characterized by the following just increases the binding energy of the state. We can see in þ Table VII that, in the complete coupled-channels calcu- features: (i) bottom-baryon open-bottom meson such as ΣðÞ ¯ ðÞ lation, the bound state has a binding energy of around b B configurations are the dominant ones, (ii) molecu- 200 MeV and the singlet-color configuration is subdomi- lar-type bound-states are favored when only color-singlet nant, contributing 20% to the formation of the hadron. arrangements are considered in the coupled-channels cal- ¯ Table VIII reflects again that this state is a heavy quark- culation, (iii) structures in which a compact bb-pair is antiquark core surrounded by light quarks. surrounded by three light quarks appear frequently when hidden-color configurations are added to the calculation, IV. EPILOGUE (iv) slightly bound states are found when the singlet-color configuration dominates over the hidden-color one whereas ð4380Þþ ð4450Þþ The Pc and Pc structures were discov- deeply bound states appear when the roles of the color ered by the LHCb Collaboration in 2015. They have configurations are reversed. captured the interest of many theorists because their It is worthwhile to highlight here that the hidden-bottom þ possible hidden-charm pentaquark composition since they pentaquark partner of the Pcð4380Þ signal observed by the were observed in the J=ψp invariant mass spectrum of the LHCb Collaboration would be a bound state in the ΣB¯ 0 − b Λ → J=ψK p decay. The measurement of hadrons con- P 1 3− b channel with quantum numbers IðJ Þ¼2 ð2 Þ and a mass taining valence c-quarks has been historically followed around 11.04–11.09 GeV. In the complete coupled- by the identification of similar structures with b-quark channels calculation, both singlet- and hidden-color con- content. Therefore, it is reasonable to expect a subsequent figurations play an important role contributing almost ð4380Þþ observation of the bottom analogues of the Pc and equally to the formation of the state, 55% and 45% ð4450Þþ Pc resonances. respectively. We have avoided to comment on what would In Ref. [40], within a chiral quark model formalism, the be the hidden-bottom pentaquark partner of the P ð4450Þþ þ ¯ c Pcð4380Þ was suggested to be a bound state of ΣcD with signal because the quantum numbers assignment of this P 3− þ quantum numbers J ¼ 2 . The nature of the Pcð4450Þ state is still under discussion. signal was not clearly established because, despite of Finally, we have recently learned from Refs. [66,67] having a couple of possible candidates attending to the that stable doubly open-bottom tetraquarks should exist. agreement between theoretical and experimental masses, This is also supported by the early works of Meng-Lin Du there was an inconsistency between the parity of the state et al. [68] and Wei Chen et al. [69] using QCD sum determined experimentally and those predicted theoreti- rules approach, and by few lattice-QCD computations in cally. Further pentaquark bound-states which contain dom- Refs. [70–72]. Therefore, an important next step in our ¯ ¯ investigation of possible bound states in pentaquark sys- inant ΣcD and ΣcD Fock-state components were also found in the region about 4.3–4.5 GeV. tems will be the computation of doubly open-bottom The work presented herein constitutes a natural exten- pentaquarks. This could be extended to other flavored sion of the analysis performed in Ref. [40].Wehave double-heavy pentaquark states as done already for doubly analyzed the possibility of having pentaquark bound- charmed tetraquarks in Refs. [73,74] using a different states in the hidden-bottom sector with quantum numbers phenomenological framework. P ¼ 1 3 5 1 3 J 2 , 2 and 2 , and in the 2 and 2 isospin sectors. Their ACKNOWLEDGMENTS inner structure have been also studied by computing the distance among any pair of quarks within the bound-state. G. Y. would like to thank L. He for his support and The chiral quark model used is based on the existence informative discussions. Work partially financed by:

014035-9 GANG YANG, JIALUN PING, and JORGE SEGOVIA PHYS. REV. D 99, 014035 (2019)

National Natural Science Foundation of China under Cierva-Incorporación programme with Grant Agreement Grants No. 11535005 and No. 11775118; European No. IJCI-2016-30028; and by Spanish Ministerio de Union’s Horizon 2020 research and innovation pro- Economía, Industria y Competitividad under Contracts gramme under the Marie Skłodowska-Curie Grant No. FPA2014-55613-P, No. FPA2017-86989-P and Agreement No. 665919; Spanish MINECO’s Juan de la No. SEV-2016-0588.

[1] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 115, [29] R. Aaij et al. (LHCb Collaboration), Phys. Rev. D 97, 072001 (2015). 032010 (2018). [2] H.-X. Chen, W. Chen, X. Liu, and S.-L. Zhu, Phys. Rep. [30] Y. Shimizu, D. Suenaga, and M. Harada, Phys. Rev. D 93, 639, 1 (2016). 114003 (2016). [3] G.-J. Wang, R. Chen, L. Ma, X. Liu, and S.-L. Zhu, Phys. [31] M.-Z. Liu, F.-Z. Peng, M. S. Snchez, and M. P. Valderrama, Rev. D 94, 094018 (2016). Phys. Rev. D 98, 114030 (2018). [4] R. Zhu and C.-F. Qiao, Phys. Lett. B 756, 259 (2016). [32] M.-Z. Liu, T.-W. Wu, J.-J. Xie, M. P. Valderrama, and L.-S. [5] R. F. Lebed, Phys. Lett. B 749, 454 (2015). Geng, Phys. Rev. D 98, 014014 (2018). [6] V. V. Anisovich, M. A. Matveev, J. Nyiri, A. V. Sarantsev, [33] M. N. Anwar, M. A. Bedolla, J. Ferretti, and E. Santopinto, and A. N. Semenova, arXiv:1507.07652. arXiv:1807.01207. [7] L. Maiani, A. D. Polosa, and V. Riquer, Phys. Lett. B 749, [34] J. T. Castell and G. Krein, Phys. Rev. D 98, 014029 289 (2015). (2018). [8] R. Ghosh, A. Bhattacharya, and B. Chakrabarti, Phys. Part. [35] N. Brambilla, V. Shtabovenko, J. T. Castell, and A. Vairo, Nucl. Lett. 14, 550 (2017). Phys. Rev. D 95, 116004 (2017). [9] Z.-G. Wang and T. Huang, Eur. Phys. J. C 76, 43 (2016). [36] N. Brambilla, G. Krein, J. T. Castell, and A. Vairo, Phys. [10] Z.-G. Wang, Eur. Phys. J. C 76, 70 (2016). Rev. D 93, 054002 (2016). [11] Z.-G. Wang, Nucl. Phys. B913, 163 (2016). [37] J.-J. Wu, R. Molina, E. Oset, and B. S. Zou, Phys. Rev. Lett. [12] L. Roca, J. Nieves, and E. Oset, Phys. Rev. D 92, 094003 105, 232001 (2010). (2015). [38] J.-J. Wu and B. S. Zou, Phys. Lett. B 709, 70 (2012). [13] R. Chen, X. Liu, X.-Q. Li, and S.-L. Zhu, Phys. Rev. Lett. [39] F.-K. Guo, U. G. Meiner, J. Nieves, and Z. Yang, Eur. Phys. 115, 132002 (2015). J. A 52, 318 (2016). [14] H. Huang, C. Deng, J. Ping, and F. Wang, Eur. Phys. J. C 76, [40] G. Yang and J. Ping, Phys. Rev. D 95, 014010 624 (2016). (2017). [15] U.-G. Meiner and J. A. Oller, Phys. Lett. B 751,59 [41] A. Valcarce, H. Garcilazo, F. Fernandez, and P. Gonzalez, (2015). Rep. Prog. Phys. 68, 965 (2005). [16] C. W. Xiao and U. G. Meiner, Phys. Rev. D 92, 114002 [42] J. Segovia, D. R. Entem, F. Fernandez, and E. Hernandez, (2015). Int. J. Mod. Phys. E 22, 1330026 (2013). [17] J. He, Phys. Lett. B 753, 547 (2016). [43] E. Hiyama, Y. Kino, and M. Kamimura, Prog. Part. Nucl. [18] H.-X. Chen, W. Chen, X. Liu, T. G. Steele, and S.-L. Zhu, Phys. 51, 223 (2003). Phys. Rev. Lett. 115, 172001 (2015). [44] A. Valcarce, F. Fernandez, P. Gonzalez, and V. Vento, Phys. [19] R. Chen, X. Liu, and S.-L. Zhu, Nucl. Phys. A954, 406 Lett. B 367, 35 (1996). (2016). [45] J. Vijande, F. Fernandez, and A. Valcarce, J. Phys. G 31, 481 [20] Y. Yamaguchi and E. Santopinto, Phys. Rev. D 96, 014018 (2005). (2017). [46] J. Segovia, D. R. Entem, and F. Fernandez, Phys. Lett. B [21] J. He, Phys. Rev. D 95, 074004 (2017). 662, 33 (2008). [22] P. G. Ortega, D. R. Entem, and F. Fernndez, Phys. Lett. B [47] J. Segovia, A. M. Yasser, D. R. Entem, and F. Fernandez, 764, 207 (2017). Phys. Rev. D 78, 114033 (2008). [23] K. Azizi, Y. Sarac, and H. Sundu, Phys. Rev. D 96, 094030 [48] P. G. Ortega, J. Segovia, D. R. Entem, and F. Fernndez, (2017). Phys. Rev. D 94, 114018 (2016). [24] F.-K. Guo, U.-G. Meiner, W. Wang, and Z. Yang, Phys. Rev. [49] G. Yang, J. Ping, and J. Segovia, Few-Body Syst. 59, 113 D 92, 071502 (2015). (2018). [25] M. Mikhasenko, arXiv:1507.06552. [50] F. Fernandez, A. Valcarce, U. Straub, and A. Faessler, [26] X.-H. Liu, Q. Wang, and Q. Zhao, Phys. Lett. B 757, 231 J. Phys. G 19, 2013 (1993). (2016). [51] A. Valcarce, F. Fernandez, A. Buchmann, and A. Faessler, [27] X.-H. Liu and M. Oka, Nucl. Phys. A954, 352 (2016). Phys. Rev. C 50, 2246 (1994). [28] M. Bayar, F. Aceti, F.-K. Guo, and E. Oset, Phys. Rev. D 94, [52] P. G. Ortega, J. Segovia, D. R. Entem, and F. Fernandez, 074039 (2016). Phys. Rev. D 81, 054023 (2010).

014035-10 HIDDEN-BOTTOM PENTAQUARKS PHYS. REV. D 99, 014035 (2019)

[53] P. G. Ortega, J. Segovia, D. R. Entem, and F. Fernandez, [65] J. Vijande, A. Valcarce, and N. Barnea, Phys. Rev. D 79, Phys. Rev. D 94, 074037 (2016). 074010 (2009). [54] P. G. Ortega, J. Segovia, D. R. Entem, and F. Fernndez, [66] E. J. Eichten and C. Quigg, Phys. Rev. Lett. 119, 202002 Phys. Rev. D 95, 034010 (2017). (2017). [55] J. Vijande, A. Valcarce, and K. Tsushima, Phys. Rev. D 74, [67] M. Karliner and J. L. Rosner, Phys. Rev. Lett. 119, 202001 054018 (2006). (2017). [56] G. Yang and J. Ping, Phys. Rev. D 97, 034023 (2018). [68] M.-L. Du, W. Chen, X.-L. Chen, and S.-L. Zhu, Phys. [57] C. Alexandrou, P. De Forcrand, and A. Tsapalis, Phys. Rev. D 87, 014003 (2013). Rev. D 65, 054503 (2002). [69] W. Chen, T. G. Steele, and S.-L. Zhu, Phys. Rev. D 89, [58] F. Okiharu, H. Suganuma, and T. T. Takahashi, Phys. Rev. 054037 (2014). Lett. 94, 192001 (2005). [70] A. Francis, R. J. Hudspith, R. Lewis, and K. Maltman, [59] S. Prelovsek, C. B. Lang, L. Leskovec, and D. Mohler, Phys. Rev. Lett. 118, 142001 (2017). Phys. Rev. D 91, 014504 (2015). [71] P. Bicudo and M. Wagner (European Twisted Mass), Phys. [60] C. B. Lang, L. Leskovec, D. Mohler, S. Prelovsek, and Rev. D 87, 114511 (2013). R. M. Woloshyn, Phys. Rev. D 90, 034510 (2014). [72] P. Bicudo, K. Cichy, A. Peters, B. Wagenbach, and M. [61] R. A. Briceno, J. J. Dudek, and R. D. Young, Rev. Mod. Wagner, Phys. Rev. D 92, 014507 (2015). Phys. 90, 025001 (2018). [73] A. Esposito, M. Papinutto, A. Pilloni, A. D. Polosa, and N. [62] G. S. Bali, H. Neff, T. Duessel, T. Lippert, and K. Schilling Tantalo, Phys. Rev. D 88, 054029 (2013). (SESAM Collaboration), Phys. Rev. D 71, 114513 (2005). [74] A. L. Guerrieri, M. Papinutto, A. Pilloni, A. D. Polosa, [63] M. D. Scadron, Phys. Rev. D 26, 239 (1982). and N. Tantalo, Proc. Sci., LATTICE2014 (2015) 106 [64] M. Harvey, Nucl. Phys. A352, 326 (1981). [arXiv:1411.2247].

014035-11