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QQss tetraquarks in the chiral quark model
Gang Yang ,1,* Jialun Ping,2,† and Jorge Segovia 3,‡
1Department of Physics, Zhejiang Normal University, Jinhua 321004, China2Department of Physics and Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex
Systems, Nanjing Normal University, Nanjing 210023, People’s Republic of China3Departamento de Sistemas Físicos, Químicos y Naturales, Universidad Pablo de Olavide,
E-41013 Sevilla, Spain
(Received 13 July 2020; accepted 2 September 2020; published 24 September 2020)
The low-lying S-wave QQss (Q ¼ c, b) tetraquark states with IðJPÞ ¼ 0ð0þÞ, 0ð1þÞ and 0ð2þÞ aresystematically investigated in the framework of complex scaling range of chiral quark model. Everystructure including meson-meson, diquark-antidiquark and K-type configurations, and all possible colorchannels in four-body sector are considered by means of a commonly extended variational approach,Gaussian expansion method. Within the studied mass region, several narrow and wide resonance states areobtained for ccss and bbss tetraquarks with IðJPÞ ¼ 0ð0þÞ and 0ð2þÞ. Meanwhile, narrow resonances forcbss tetraquarks are also found in IðJPÞ ¼ 0ð0þÞ, 0ð1þÞ and 0ð2þÞ states. The possibility of findingtetraquark structures with masses ∼0.6 GeV above the non-interacting hadron-hadron thresholds (and notclose to them) is a feature related with the absence of the one-pion exchange interaction in theQQs s sectorand thus it could deserve experimental confirmation.
DOI: 10.1103/PhysRevD.102.054023
I. INTRODUCTION
We are witnessing in the last two decades of a bigexperimental effort for understanding the heavy-flavorquark sectors of both meson and baryon systems. Manyexperiments have been settled worldwide such asB-factories (BABAR, Belle, and CLEO), τ-charm facilities(CLEO-c and BES) and hadron-hadron colliders (CDF, D0,LHCb, ATLAS, and CMS), providing a sustained progressin the field with new measurements of conventional andexotic heavy-flavored hadrons.Within the baryon sector, and attending mostly to the
spectrum, five excited Ωc baryons were announced threeyears ago by the LHCb collaboration in the Ξþ
c K− massspectrum [1] and, very recently, the same collaboration hasreported additional four narrow excited states of the Ωb
system in the Ξ0bK
− mass spectrum [2]. In 2019, twoexcited bottom baryons, Λ0
bð6146Þ and Λ0bð6152Þ, were
discovered in the LHCb experiment [3]. Later on, theLHCb collaboration also announced one more Λ0
b baryon
around 6070 MeV in the Λ0bπ
þπ− invariant mass spectrum[4], which is consistent with the reported one of the CMScollaboration [5]. Additionally, three excited Ξ0
c states wereannounced by the LHCb collaboration in the Λþ
c K− massspectrum [6].All of these newly discovered baryons undoubtedly
complement the scarce data on heavy flavor baryons inthe Review of Particle Physics (RPP) of the Particle DataGroup (PDG) [7]. Furthermore, these experimental findingstrigger a large number of theoretical investigations. Thethree-quark structure of the new Ωc baryons has beenclaimed by QCD sum rules [8] and different potentialmodels [9–11]. Also, the description of the Ωb signals asP-wave conventional baryons is preferred by phenomeno-logical quark model approach [12,13], heavy quark effec-tive theory [14] and QCD sum rules [15]. Meanwhile, thebaryon-meson molecular interpretation has been suggestedfor the excited Ωb baryons in Ref. [16]. The Λ0
bð6072Þ,Λ0bð6146Þ, and Λ0
bð6152Þ have been identified as radialand angular excitations within QCD sum rules [17–20]and chiral quark models [21,22]. However, the DΛ − DΣmolecular configurations have been also suggested forthese states in Ref. [23].Apart from conventional heavy flavored baryons, there
are limited results on open-bottom mesons and detailedstudies of the open-charm ones were not undertaken untillarge datasets were obtained by CLEO at discrete energypoints and by the B-factory experiments using radiativereturns to obtain a continuous exposure of the mass region.
*[email protected]†[email protected]‡[email protected]
Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI. Funded by SCOAP3.
PHYSICAL REVIEW D 102, 054023 (2020)
2470-0010=2020=102(5)=054023(18) 054023-1 Published by the American Physical Society
The picture that has emerged is complex due to the manythresholds in the region. This resembles the experimentalsituation found in the heavy quarkonium spectrum with theobservation of more than two dozens of unconventionalcharmonium- and bottomoniumlike states, the so-calledXYZ mesons. However, successful observations of 6 newconventional heavy quarkonium states (4 cc and 2 bb) havestill been made.Exotic states such as tetraquarks and pentaquarks have
lastly received considerable attention within the hadronphysics community. Related with the first structures, thebest known is the Xð3872Þ, which was observed in 2003 asan extremely narrow peak in the Bþ → Kþðπþπ−J=ψÞchannel and at exactly the D0D�0 threshold [24,25], and itis suspected to be a cncn (n ¼ u or d quark) tetraquarkstate whose features resemble those of a molecule, butsome experimental findings forbid to discard a morecompact, diquark-antidiquark, component or even somecc trace in its wave function. On the other hand, thereare no doubts of the tetraquark character of the Zc’s [26,27]and Zb’s [28,29] states due to its nonzero charge. The mostprominent examples of the second mentioned structuresare the hidden-charm pentaquarks Pþ
c ð4312Þ, Pþc ð4380Þ,
Pþc ð4440Þ, and Pþ
c ð4457Þ reported in 2015 and 2019by the LHCb collaboration in the Λ0
b decay, Λ0b →
J=ψK−p [30,31].The discussion about the nature of these exotic signals
are carried out by various theoretical approaches. Inparticular, the three newly announced hidden-charm pen-taquarks, Pþ
c ð4312Þ, Pþc ð4440Þ, and Pþ
c ð4457Þ are favoredto be molecular states of ΣcD� in, for instance, effectivefield theories [32,33], QCD sum rules [34], phenomeno-logical potential models [35–40], heavy quark spin sym-metry formalisms [41,42] and heavy hadron chiralperturbation theory [43]. Moreover, their photo-production[44,45] and decay properties [46] have been also discussed.As for the other types of pentaquarks, bound states ofthe Qqqqq system are not found within a constituent quarkmodel [47]. Using the same approach, several narrowdouble-heavy pentaquark states are found to be possiblein the systematical investigations of Refs. [48–50].Moreover, within the one-boson-exchange model, possibletriple-charm molecular pentaquarks ΞccDð�Þ are suggested[51]. In the tetraquark sector, double-heavy tetraquarksare studied using QCD sum rules [52], quark models[53,54] and even lattice-regularized QCD computations[55]. Besides, theoretical techniques such as diffusionMonte Carlo [56], Bethe-Salpeter equation [57], QCDsum rules [58,59] and effective phenomenological models[60–64] have recently contributed to the investigationsof fully heavy tetraquarks QQQQ. Some reviews onboth tetraquark and pentaquark systems can be found inRefs. [65,66].Our QCD-inspired chiral quark model explained suc-
cessfully the nature of the Pþc states in Ref. [67], even
before the last updated data reported by the LHCbcollaboration [30]. Based on such fact, the hidden-bottom[68] and double-charm pentaquarks [50] were systemati-cally investigated within the same theoretical framework,finding several bound states and resonances. Reference [54]reported results on the double-heavy tetraquarks QQqqðQ ¼ c, b and q ¼ u; dÞ, its natural extension should be theQQss tetraquark sector with the hope of finding eitherbound or resonance states. In order to do so, we haverecently established a complex scaling range formalismof the chiral quark model which allows us to determine(if exist) simultaneously scattering, resonance and boundstates. We shall study herein the QQss tetraquarks in thespin-parity channels JP ¼ 0þ, 1þ and 2þ, and in theisoscalar sector I ¼ 0. Another relevant feature of ourstudy is that all configurations: meson-meson, diquark-antidiquark and K-type for four-body systems are consid-ered; moreover, every possible color channel is taken intoaccount, too. Finally, the Rayleigh-Ritz variational methodis employed in dealing with the spatial wave functions oftetraquark states, which are expanded by means of the well-known Gaussian expansion method (GEM) of Ref. [69].The present manuscript is arranged as follows. Section II
is devoted to briefly describe our theoretical approachwhich includes the complex-range formulation of thechiral quark model and the discussion of the QQsswave-functions. Section III is devoted to the analysisand discussion of the obtained results. The summary andsome prospects are presented in Sec. IV.
II. THEORETICAL FRAMEWORK
The complex scaling method (CSM) applied to our chiralquark model has been already explained in Refs. [50,54].The general form of the four-body complex Hamiltonian isgiven by
HðθÞ ¼X4i¼1
�mi þ
p2i
2mi
�− TCM þ
X4j>i¼1
VðrijeiθÞ; ð1Þ
where the center-of-mass kinetic energy, TCM, is subtractedwithout loss of generality since we focus on the internalrelative motions of quarks inside the multiquark system.The interplay is of two-body potential which includescolor-confining, VCON, one-gluon exchange, VOGE, andGoldstone-boson exchange, Vχ , respectively,
VðrijeiθÞ ¼ VCONðrijeiθÞ þ VOGEðrijeiθÞ þ VχðrijeiθÞ:ð2Þ
In this work, we focus on the low-lying positive parityQQs s tetraquark states of S-wave, and hence only thecentral and spin-spin terms of the potentials shall beconsidered.
GANG YANG, JIALUN PING, and JORGE SEGOVIA PHYS. REV. D 102, 054023 (2020)
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By transforming the coordinates of relative motionsbetween quarks as rij → rijeiθ, the complex scaledSchrödinger equation
½HðθÞ − EðθÞ�ΨJMðθÞ ¼ 0; ð3Þ
is solved, giving eigenenergies that can be classified intothree kinds of poles: bound, resonance and scatteringones, in a complex energy plane according to the so-calledABC theorem [70,71]. In particular, the resonance pole isindependent of the rotated angle θ, i.e., it is fixed above thecontinuum cut line with a resonance’s width Γ ¼ −2ImðEÞ.The scattering state is just aligned along the cut line with a2θ rotated angle, whereas a bound state is always located onthe real axis below its corresponding threshold.The two-body potentials in Eq. (2) mimic the most
important features of QCD at low and intermediate ener-gies. First, color confinement should be encoded in the non-Abelian character of QCD. It has been demonstrated bylattice-QCD that multigluon exchanges produce an attrac-tive linearly rising potential proportional to the distancebetween infinite-heavy quarks [72]. However, the sponta-neous creation of light-quark pairs from the QCD vacuummay give rise at the same scale to a breakup of the createdcolor flux-tube [72]. Therefore, the following expressionwhen θ ¼ 0° is used for the confinement potential:
VCONðrijeiθÞ ¼ ½−acð1 − e−μcrijeiθÞ þ Δ�ðλci · λcjÞ; ð4Þ
where ac, μc, and Δ are model parameters, and theSU(3) color Gell-Mann matrices are denoted as λc. Onecan see in Eq. (4) that the potential is linear at shortinter-quark distances with an effective confinementstrength σ ¼ −acμcðλci · λcjÞ, while VCON becomes constant
ðΔ − acÞðλci · λcjÞ at large distances.Second, the QCD’s asymptotic freedom is expressed
phenomenologically by the Fermi-Breit reduction of theone-gluon exchange interaction which, in the case ofhadron systems with ≥3 quarks, consists on a Coulombterm supplemented by a chromomagnetic contact interac-tion given by
VOGEðrijeiθÞ ¼1
4αsðλci · λcjÞ
�1
rijeiθ
−1
6mimjðσi · σjÞ
e−rijeiθ=r0ðμÞ
rijeiθr20ðμÞ�; ð5Þ
where mi and σ are the quark mass and the Pauli matrices,respectively. The contact term of the central potential incomplex range has been regularized as
δðrijeiθÞ ∼1
4πr20
e−rijeiθ=r0
rijeiθ; ð6Þ
with r0 ¼ r0ðμijÞ a regulator which depends on the reducedmass of the quark–(anti)quark pair [73].The QCD-inspired effective scale-dependent strong cou-
pling constant, αs, offers a consistent description of mesonsand baryons from light to heavy quark sectors in wideenergy range, and we use the frozen coupling constant of,for instance, Ref. [73]
αsðμijÞ ¼α0
ln�μ2ijþμ2
0
Λ20
� ; ð7Þ
in which α0, μ0, and Λ0 are parameters of the model.Thirdly, the Goldstone-boson exchange interactions
between light quarks, and constituent quark masses, appearbecause the dynamical breaking of chiral symmetry.Therefore, the following two terms of the chiral potentialmust be taken into account between the ðssÞ-pair for QQsstetraquarks:
VσðrijeiθÞ ¼ −g2ch4π
Λ2σ
Λ2σ −m2
σmσ
×
�YðmσrijeiθÞ −
Λσ
mσYðΛσrijeiθÞ
�; ð8Þ
VηðrijeiθÞ ¼g2ch4π
m2η
12mimj
Λ2η
Λ2η −m2
ηmη
×
�YðmηrijeiθÞ −
Λ3η
m3ηYðΛηrijeiθÞ
�
× ðσi · σjÞ½cos θpðλ8i · λ8jÞ − sin θp�; ð9Þ
where YðxÞ ¼ e−x=x is the standard Yukawa function. Thepion- and kaon-exchange interactions do not appearbecause no up- and down-quarks are considered herein.Furthermore, the physical η meson is taken into account byintroducing the angle θp. The λa are the SU(3) flavor Gell-Mann matrices. The value of mσ is determined through thePCAC relation m2
σ ≃m2π þ 4m2
u;d [74], whereas the mass ofthe η meson is taken from their experimental value [7].Finally, the chiral coupling constant, gch, is determinedfrom the πNN coupling constant through
g2ch4π
¼ 9
25
g2πNN
4π
m2u;d
m2N; ð10Þ
which assumes that flavor SU(3) is an exact symmetry, onlybroken by the different mass of the strange quark.The model parameters, which are listed in Table I, have
been fixed in advance reproducing hadron [75–84], hadron-hadron [85–89] and multiquark [11,67,68,90] phenomenol-ogy. Additionally, in order to assist on our analysis of theQQss tetraquarks in the following section, Table II lists thetheoretical and experimental masses of the ground state and
QQS S TETRAQUARKS IN THE CHIRAL … PHYS. REV. D 102, 054023 (2020)
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its first radial excitation (if available) for the Dð�Þþs and Bð�Þ
s
mesons. Besides, their mean-square radii are collected inTable II.Figure 1 shows six kinds of configurations for double-
heavy tetraquarks QQss (Q ¼ c, b). In particular, Fig. 1(a)is the meson-meson (MM) structure, Fig. 1(b) is thediquark-antidiquark (DA) one, and the other K-type con-figurations are from panels (c) to (f). All of them, and theircouplings, are considered in our investigation. However, forthe purpose of solving a manageable 4-body problem, theK-type configurations are restricted to the case in which thetwo heavy quarks of QQs s tetraquarks are identical. It isimportant to note herein that just one configuration wouldbe enough for the calculation, if all radial and orbitalexcited states were taken into account; however, this isobviously much less efficient and thus an economic way isto combine the different configurations in the ground stateto perform the calculation.Four fundamental degrees of freedom at the quark level:
color, spin, flavor, and space are generally accepted byQCD theory and the multiquark system’s wave function isan internal product of color, spin, flavor, and space terms.
First, concerning the color degree-of-freedom, plenty ofcolor structures in multiquark systems will be available withrespect those of conventional hadrons (qq mesons and qqqbaryons). The colorless wave function of a 4-quark system indimeson configuration, i.e., as illustrated in Fig. 1(a), can beobtained by either a color-singlet or a hidden-color channelor both. However, this is not the unique way for the authorsof Refs. [91,92], who assert that it is enough to consider thecolor singlet channel when all possible excited states of asystem are included.1 The SUð3Þcolor wave functions of acolor-singlet (two coupled color-singlet clusters, 1c ⊗ 1c)and hidden-color (two coupled color-octet clusters, 8c ⊗ 8c)channels are given by, respectively,
χc1 ¼1
3ðrrþ ggþ bbÞ × ðrrþ ggþ bbÞ; ð11Þ
χc2 ¼ffiffiffi2
p
12ð3brrbþ 3grrgþ 3bggbþ 3gbbgþ 3rggr
þ 3rbbrþ 2rrrrþ 2ggggþ 2bbbb − rrgg
− ggrr − bbgg − bbrr − ggbb − rrbbÞ: ð12Þ
In addition, the color wave functions of the diquark-antidiquark structure shown in Fig. 1(b) are χc3 (color
TABLE I. Model parameters.
Quark masses ms (MeV) 555mc (MeV) 1752mb (MeV) 5100
Goldstone bosons Λσ (fm−1) 4.20Λη (fm−1) 5.20g2ch=ð4πÞ 0.54θPð°Þ −15
Confinement ac (MeV) 430μc (fm−1Þ 0.70Δ (MeV) 181.10α0 2.118Λ0 (fm−1) 0.113
OGE μ0 (MeV) 36.976r0 (MeV fm) 28.170
TABLE II. Theoretical and experimental masses of Dð�Þþs and
Bð�Þs mesons; their mean-square radii are also shown.
Meson nLphr2iThe. (fm) MThe. (MeV) MExp. (MeV)
Dþs 1S 0.47 1989 1969
2S 1.06 2703 …
D�þs 1S 0.55 2116 2112
2S 1.14 2767 …
B0s 1S 0.47 5355 5367
2S 1.01 6017 …
B�s 1S 0.50 5400 5415
2S 1.04 6042 …
3
2
1 s
Q 4Q
s 3
2
1 s
Q 4Q
s
(a) (b)
3
2
1 s
Q 4Q
s 3
2
1 s
Q 4Q
s
(c) (d)
3
2
1 s
Q 4Q
s 3
2
1 s
Q 4Q
s
(e) (f)
FIG. 1. Six types of configurations in QQs s (Q ¼ c, b)tetraquarks. Panel (a) is the meson-meson configuration, panel(b) is diquark-antidiquark, one and the K-type structures are frompanel (c) to (f).
1After a comparison, a more economical way of computingthrough considering all the possible color structures and theircoupling is preferred.
GANG YANG, JIALUN PING, and JORGE SEGOVIA PHYS. REV. D 102, 054023 (2020)
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triplet-antitriplet clusters, 3c ⊗ 3c) and χc4 (color sextet-antisextet clusters, 6c ⊗ 6c), respectively:
χc3 ¼ffiffiffi3
p
6ðrrgg − grrgþ ggrr − rggrþ rrbb
− brrbþ bbrr − rbbrþ ggbb − bggb
þ bbgg − gbbgÞ; ð13Þ
χc4 ¼ffiffiffi6
p
12ð2rrrrþ 2ggggþ 2bbbbþ rrggþ grrg
þ ggrrþ rggrþ rrbbþ brrbþ bbrr
þ rbbrþ ggbbþ bggbþ bbggþ gbbgÞ: ð14Þ
Meanwhile, the colorless wave functions of the K-typestructures shown in Fig. 1(c) to 1(f) are obtained byfollowing standard coupling algebra within the SUð3Þ colorgroup2:
(i) K1-type of Fig. 1(c): ½C½21�½11�;½1�C
½221�½21�;½11�C
½222�½221�;½1��5;
½C½111�½11�;½1�C
½221�½111�;½11�C
½222�½221�;½1��6;
(ii) K2-type of Fig. 1(d): ½C½111�½11�;½1�C
½211�½111�;½1�C
½222�½211�;½11��7;
½C½21�½11�;½1�C
½211�½21�;½1�C
½222�½211�;½11��8;
(iii) K3-type of Fig. 1(e): ½C½2�½1�;½1�C
½211�½2�;½11�C
½222�½211�;½11��9;
[C½11�½1�;½1�C
½211�½11�;½11�C
½222�½211�;½11��10;
(iv) K4-type of Fig. 1(f): ½C½22�½11�;½11�C
½221�½22�;½1�C
½222�½221�;½1��11;
½C½211�½11�;½11�C
½221�½211�;½1�C
½222�½221�;½1��12.
These group chains will generate the following K-type colorwave functions whose subscripts correspond to those num-bers above:
χ5c ¼ χ2c ð15Þ
χc6 ¼ χc1; ð16Þ
χc7 ¼ χc1; ð17Þ
χ8c ¼ χ2c ð18Þ
χc9 ¼1
2ffiffiffi6
p ðrbbrþ rrbbþ gbbgþ ggbbþ rggrþ rrgg
þ bbggþ bggbþ ggrrþ grrgþ bbrrþ brrbÞ
þ 1ffiffiffi6
p ðrrrrþ ggggþ bbbbÞ; ð19Þ
χc10 ¼1
2ffiffiffi3
p ðrbbr − rrbbþ gbbg − ggbbþ rggr − rrgg
− bbggþ bggb − ggrrþ grrg − bbrrþ brrbÞ;ð20Þ
χc11 ¼ χc9; ð21Þχc12 ¼ −χc10: ð22Þ
As for the flavor degree-of-freedom, since the quarkcontent of the tetraquark systems considered herein are twoheavy quarks, (Q ¼ c, b), and two strange antiquarks, s,only the isoscalar sector, I ¼ 0, will be discussed. Theflavor wave functions denoted as χfiI;MI
, with the superscripti ¼ 1, 2, and 3 referring to ccss, bbss, and cbss systems,can be written as
χf10;0 ¼ scsc; ð23Þ
χf20;0 ¼ sbsb; ð24Þ
χf30;0 ¼ scsb; ð25Þwhere, in this case, the third component of the isospinMI isequal to the value of total one I.The total spin S of tetraquark states ranges from 0 to 2.
All of them shall be considered and, since there is notany spin-orbit potential, the third component ðMSÞ can beset to be equal to the total one without loss of generality.Therefore, our spin wave functions χσiS;MS
are given by
χσl10;0ð4Þ ¼ χσ00χσ00; ð26Þ
χσl20;0ð4Þ ¼1ffiffiffi3
p ðχσ11χσ1;−1 − χσ10χσ10 þ χσ1;−1χ
σ11Þ; ð27Þ
χσl30;0ð4Þ ¼1ffiffiffi2
p�� ffiffiffi
2
3
rχσ11χ
σ12;−1
2
−ffiffiffi1
3
rχσ10χ
σ12;12
�χσ12;−1
2
−� ffiffiffi
1
3
rχσ10χ
σ12;−1
2
−ffiffiffi2
3
rχσ1;−1χ
σ12;12
�χσ12;12
�; ð28Þ
χσl40;0ð4Þ ¼1ffiffiffi2
p�χσ00χ
σ12;12
χσ12;−1
2
− χσ00χσ12;−1
2
χσ12;12
�; ð29Þ
χσm1
1;1 ð4Þ ¼ χσ00χσ11; ð30Þ
χσm2
1;1 ð4Þ ¼ χσ11χσ00; ð31Þ
χσm3
1;1 ð4Þ ¼1ffiffiffi2
p ðχσ11χσ10 − χσ10χσ11Þ; ð32Þ
χσm4
1;1 ð4Þ ¼ffiffiffi3
4
rχσ11χ
σ12;12
χσ12;−1
2
−ffiffiffiffiffi1
12
rχσ11χ
σ12;−1
2
χσ12;12
−ffiffiffi1
6
rχσ10χ
σ12;12
χσ12;12
; ð33Þ2The group chain of K-type is obtained in sequence of quark
number. Moreover, each quark and antiquark is represented,respectively, with [1] and [11] in the group theory.
QQS S TETRAQUARKS IN THE CHIRAL … PHYS. REV. D 102, 054023 (2020)
054023-5
χσm5
1;1 ð4Þ ¼� ffiffiffi
2
3
rχσ11χ
σ12;−1
2
−ffiffiffi1
3
rχσ10χ
σ12;12
�χσ12;12
; ð34Þ
χσm6
1;1 ð4Þ ¼ χσ00χσ12;12
χσ12;12
; ð35Þ
χσ12;2ð4Þ ¼ χσ11χσ11: ð36Þ
The superscripts l1;…; l4 and m1;…; m6 are numberingthe spin wave function for each configuration of tetra-quark states, their specific values are shown in Table III.Furthermore, these expressions are obtained by consideringthe coupling of two subcluster spin wave functions withSU(2) algebra, and the necessary bases read as
χσ11 ¼ χσ12;12
χσ12;12
; ð37Þ
χσ1;−1 ¼ χσ12;−1
2
χσ12;−1
2
; ð38Þ
χσ10 ¼1ffiffiffi2
p�χσ12;12
χσ12;−1
2
þ χσ12;−1
2
χσ12;12
�; ð39Þ
χσ00 ¼1ffiffiffi2
p�χσ12;12
χσ12;−1
2
− χσ12;−1
2
χσ12;12
�: ð40Þ
Among the different methods to solve the Schrödinger-like 4-body bound state equation, we use the Rayleigh-Ritzvariational principle which is one of the most extendedtools to solve eigenvalue problems because its simplicityand flexibility. Meanwhile, the choice of basis to expandthe wave function solution is of great importance. Withinthe CSM, the spatial wave function can be written asfollows
ψLMLðθÞ ¼ ½½ϕn1l1ðρeiθÞϕn2l2ðλeiθÞ�lϕn3l3ðReiθÞ�LML
; ð41Þ
where the internal Jacobi coordinates for the meson-mesonconfiguration [Fig. 1(a)] are defined as
ρ ¼ x1 − x2; ð42Þ
λ ¼ x3 − x4; ð43Þ
R ¼ m1x1 þm2x2m1 þm2
−m3x3 þm4x4m3 þm4
; ð44Þ
and for the diquark-antdiquark one [Fig. 1(b)] are
ρ ¼ x1 − x3; ð45Þ
λ ¼ x2 − x4; ð46Þ
R ¼ m1x1 þm3x3m1 þm3
−m2x2 þm4x4m2 þm4
: ð47Þ
The Jacobi coordinates for the remaining K-type configu-rations shown in Fig. 1, panels (c) to (f), are (i, j, k, l areaccording to the definitions of each configuration in Fig. 1):
ρ ¼ xi − xj; ð48Þ
λ ¼ xk −mixi þmjxjmi þmj
; ð49Þ
R ¼ xl −mixi þmjxj þmkxk
mi þmj þmk: ð50Þ
Obviously, the center-of-mass kinetic term TCM can becompletely eliminated for a nonrelativistic system whenusing these sets of coordinates.Avery efficient method to solve the bound-state problem
of a few-body system is the Gaussian expansion method(GEM) [69], which has been successfully applied by us inother multiquark systems [50,54,67,68]. The Gaussianbasis in each relative coordinate is taken with geometricprogression in the size parameter.3 Therefore, the form ofthe orbital wave functions, ϕ’s, in Eq. (41) is
ϕnlmðreiθÞ ¼ NnlðreiθÞle−νnðreiθÞ2YlmðrÞ: ð51Þ
As one can see, the Jacobi coordinates are all transformedwith a common scaling angle θ in the complex scalingmethod. In this way, both bound states and resonances canbe described simultaneously within one scheme. Moreover,only S-wave state of double-heavy tetraquarks are inves-tigated in this work and thus no laborious Racah algebra isneeded during matrix elements calculation.Finally, in order to fulfill the Pauli principle, the
complete wave function is written as
ΨJMJ;I;i;j;kðθÞ ¼ A½½ψLðθÞχσiS ð4Þ�JMJχfjI χ
ck�; ð52Þ
TABLE III. Index of spin wave function from Eq. (26) to (36),their numbers are listed in the column of each configuration,respectively.
Dimeson Diquark-antidiquark K1 K2 K3 K4
l1 1 3l2 2 4l3 5 7 9 11l4 6 8 10 12
m1 1 4m2 2 5m3 3 6m4 7 10 13 16m5 8 11 14 17m6 9 12 15 18
3The details on Gaussian parameters and how they are fixedcan be found in Ref. [67].
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whereA is the antisymmetry operator ofQQs s tetraquarkswhen considering interchange between identical particles(s s, cc and bb). This is necessary because the completewave function of the 4-quark system is constructed fromtwo sub-clusters: meson-meson, diquark-antidiquark andK-type structures. In particular, when the two heavy quarksare of the same flavor (QQ ¼ cc or bb), the operator Awith the quark arrangements sQsQ is defined as
A ¼ 1 − ð13Þ − ð24Þ þ ð13Þð24Þ: ð53Þ
However, due to the fact that c- and b-quarks are distin-guishable particles, the operator A consists only on twoterms for the scsb system, and read as
A ¼ 1 − ð13Þ: ð54Þ
III. RESULTS
The low-lying S-wave states of QQs s (Q ¼ c, b)tetraquarks are systematically investigated herein. The parityfor different QQs s tetraquarks is positive under ourassumption that the angular momenta l1, l2, l3, whichappear in Eq. (41), are all 0. Accordingly, the total angularmomentum, J, coincides with the total spin, S, and can takevalues 0, 1, and 2. Note, too, the value of isospin can only be0 for the QQss system. For ccss, bbss and cbss systems,all possible meson-meson, diquark-antidiquark, and K-typechannels for each IðJPÞ quantum numbers are listed inTables IV–IX, respectively. The second column shows the
necessary basis combination in spin ðχσiJ Þ, flavor ðχfjI Þ, andcolor ðχckÞ degrees-of-freedom. The physical channels withcolor-singlet (labeled with the superindex 1), hidden-color(labeled with the superindex 8), diquark-antidiquark (labeledwith ðQQÞðssÞ) and K-type (labeled from K1 to K4)configurations are listed in the third column.Tables ranging from X to XIX summarize our calculated
results (mass and width) of the lowest-lying QQss tetra-quark states and possible resonances. In particular, results ofccss tetraquarks with IðJPÞ ¼ 0ð0þÞ, 0ð1þÞ and 0ð2þÞ arelisted in Tables X, XI and XII; those of bbs s tetraquarks areshown in Tables XIII–XV; and Tables XVI–XVIII collect thecbss cases. In these tables, the first column lists the physicalchannel of meson-meson, diquark-antidiquark and K-type (ifit fulfills Pauli principle); the second column indicates theexperimental/theoretical value of the non-interacting meson-meson threshold; the third column signals the discussedchannel, e.g., color-singlet (S), hidden-color (H), etc.; thefourth column shows the theoretical mass (M) of each singlechannel; and the fifth column shows a coupled calculationresult for one certain configuration. Moreover, the completecoupled channels results for each quantum state are shown atthe bottom of each table. Besides, Table XIX summarizes theobtained resonance states of QQs s tetraquarks in thecomplete coupled-channels calculation.
Figures 2 to 10 depict the distribution of complexenergies of theQQss tetraquarks in the complete coupled-channels calculation. The x-axis is the real part of thecomplex energy E, which stands for the mass of tetraquarkstates, and the y-axis is the imaginary part of E, which isrelated to the width through Γ ¼ −2ImðEÞ. In the figures,some orange circles appear surrounding resonance can-didates. They are usually ∼0.6 GeV above their respectivenon-interacting meson-meson thresholds and ∼0.2 GeVaround their first radial excitation states; moreover, look-ing at the details, we shall conclude that most of theseobserved resonances can be identified with a hadronicmolecular nature.Now let us proceed to describe in detail our theoretical
findings for each sector of QQs s tetraquarks.
A. The ccs s tetraquarks
Within the considered energy region, we find onlyresonances in the channels with quantum numbers IðJPÞ ¼0ð0þÞ and 0ð2þÞ. This result is opposite to the one found inour previous study of ccq q tetraquarks [54] and it is relatedwith the ratio between light and heavy quarks that composethe tetraquark system, but also the interplay between thedifferent flavor quarks. We shall proceed to discuss belowthe J ¼ 0, 1, and 2 channels individually.
TABLE IV. All possible channels for IðJPÞ ¼ 0ð0þÞ ccs s andbbs s tetraquark systems. The second column shows the neces-
sary basis combination in spin (χσiJ ), flavor (χfjI ) and color (χck)
degrees of freedom. Particularly, the flavor indices (j) 1 and 2 areof ccs s and bbs s, respectively. The superscript 1 and 8 stands forthe color-singlet and hidden-color configurations of physicalchannels.
IndexχσiJ ; χ
fjI ; χ
ck
½i; j; k� Channel
1 ½1; 1ð2Þ; 1� ðDþs Dþ
s Þ1; ðB0sB0
sÞ12 ½2; 1ð2Þ; 1� ðD�þ
s D�þs Þ1; ðB�
s B�sÞ1
3 ½1; 1ð2Þ; 2� ðDþs Dþ
s Þ8; ðB0sB0
sÞ84 ½2; 1ð2Þ; 2� ðD�þ
s D�þs Þ8; ðB�
s B�sÞ8
5 ½3; 1ð2Þ; 4� ðccÞðs sÞ; ðbbÞðs sÞ6 ½4; 1ð2Þ; 3� ðccÞ�ðs sÞ�; ðbbÞ�ðs sÞ�7 ½5; 1ð2Þ; 5� K1
8 ½5; 1ð2Þ; 6� K1
9 ½6; 1ð2Þ; 5� K1
10 ½6; 1ð2Þ; 6� K1
11 ½7; 1ð2Þ; 7� K2
12 ½7; 1ð2Þ; 8� K2
13 ½8; 1ð2Þ; 7� K2
14 ½8; 1ð2Þ; 8� K2
15 ½9; 1ð2Þ; 10� K3
16 ½10; 1ð2Þ; 9� K3
17 ½11; 1ð2Þ; 12� K4
18 ½12; 1ð2Þ; 11� K4
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The IðJPÞ ¼ 0ð0þÞ state: Two possible meson-mesonchannels, Dþ
s Dþs and D�þ
s D�þs , two diquark-antidiquark
channels, ðccÞðs sÞ and ðccÞ�ðs sÞ�, along with K-typeconfigurations, are studied first in real-range calculationand our results are shown in Table X. The lowest energylevel, ðDþ
s Dþs Þ1, is unbounded and its theoretical mass just
equals to the threshold value of two noninteracting Dþs
mesons.4 This fact is also found in the ðD�þs D�þ
s Þ1 channelwhose theoretical mass is 4232 MeV. As for the otherexotic configurations, the obtained masses are all higherthan the two di-meson channels. In particular, masses of thehidden-color channels are about 4.6 GeV, diquark-antidiquark channels are lower ∼4.4 GeV, and the otherfour K-type configurations are located in the mass interval of4.2 to 4.8 GeV. Note, too, there is a degeneration betweenðccÞð�Þðs sÞð�Þ, K3 and K4 channels around 4.4 GeV.In a further step, we have performed a coupled-channels
calculation on certain configurations, and still no boundstates are found. The coupling is quite weak for the color-singlet channels Dþ
s Dþs and D�þ
s D�þs . Hidden-color,
TABLE VI. All possible channels for IðJPÞ ¼ 0ð2þÞ ccss andbbss tetraquark systems. The second column shows the necessary
basis combination in spin (χσiJ ), flavor (χfjI ), and color (χck)
degrees of freedom. Particularly, the flavor indices (j) 1 and 2 areof ccss and bbss, respectively. The superscript 1 and 8 stands forthe color-singlet and hidden-color configurations of physicalchannels.
IndexχσiJ ; χ
fjI ; χ
ck
½i; j; k� Channel
1 ½1; 1ð2Þ; 1� ðD�þs D�þ
s Þ1; ðB�s B�
sÞ12 ½1; 1ð2Þ; 2� ðD�þ
s D�þs Þ8; ðB�
s B�sÞ8
3 ½1; 1ð2Þ; 3� ðccÞ�ðs sÞ�; ðbbÞ�ðs sÞ�4 ½1; 1ð2Þ; 5� K1
5 ½1; 1ð2Þ; 6� K1
6 ½1; 1ð2Þ; 7� K2
7 ½1; 1ð2Þ; 8� K2
8 ½1; 1ð2Þ; 10� K3
9 ½1; 1ð2Þ; 12� K4
TABLE V. All possible channels for IðJPÞ ¼ 0ð1þÞ ccs s andbbs s tetraquark systems. The second column shows the neces-
sary basis combination in spin (χσiJ ), flavor (χfjI ) and color (χck)
degrees of freedom. Particularly, the flavor indices (j) 1 and 2 areof ccss and bbss, respectively. The superscript 1 and 8 stands forthe color-singlet and hidden-color configurations of physicalchannels.
IndexχσiJ ; χ
fjI ; χ
ck
½i; j; k� Channel
1 ½1; 1ð2Þ; 1� ðDþs D�þ
s Þ1; ðB0sB�
sÞ12 ½3; 1ð2Þ; 1� ðD�þ
s D�þs Þ1; ðB�
s B�sÞ1
3 ½1; 1ð2Þ; 2� ðDþs D�þ
s Þ8; ðB0sB�
sÞ84 ½3; 1ð2Þ; 2� ðD�þ
s D�þs Þ8; ðB�
s B�sÞ8
5 ½6; 1ð2Þ; 3� ðccÞ�ðs sÞ�; ðbbÞ�ðs sÞ�6 ½7; 1ð2Þ; 5� K1
7 ½8; 1ð2Þ; 5� K1
8 ½9; 1ð2Þ; 5� K1
9 ½7; 1ð2Þ; 6� K1
10 ½8; 1ð2Þ; 6� K1
11 ½9; 1ð2Þ; 6� K1
12 ½10; 1ð2Þ; 7� K2
13 ½11; 1ð2Þ; 7� K2
14 ½12; 1ð2Þ; 7� K2
15 ½10; 1ð2Þ; 8� K2
16 ½11; 1ð2Þ; 8� K2
17 ½12; 1ð2Þ; 8� K2
18 ½13; 1ð2Þ; 10� K3
19 ½14; 1ð2Þ; 10� K3
20 ½15; 1ð2Þ; 9� K3
21 ½16; 1ð2Þ; 12� K4
22 ½17; 1ð2Þ; 12� K4
23 ½18; 1ð2Þ; 11� K4
TABLE VII. All possible channels for IðJPÞ ¼ 0ð0þÞ cbs stetraquark systems. The second column shows the necessary
basis combination in spin (χσiJ ), flavor (χfjI ) and color (χ
ck) degrees
of freedom. The superscript 1 and 8 stands for the color-singletand hidden-color configurations of physical channels.
IndexχσiJ ; χ
fjI ; χ
ck
½i; j; k� Channel
1 ½1; 3; 1� ðDþs B0
sÞ12 ½2; 3; 1� ðD�þ
s B�sÞ1
3 ½1; 3; 2� ðDþs B0
sÞ84 ½2; 3; 2� ðD�þ
s B�sÞ8
5 ½3; 3; 4� ðcbÞðs sÞ6 ½4; 3; 3� ðcbÞ�ðs sÞ�7 ½5; 3; 5� K1
8 ½5; 3; 6� K1
9 ½6; 3; 5� K1
10 ½6; 3; 6� K1
11 ½7; 3; 7� K2
12 ½7; 3; 8� K2
13 ½8; 3; 7� K2
14 ½8; 3; 8� K2
15 ½9; 3; 9� K3
16 ½9; 3; 10� K3
17 ½10; 3; 9� K3
18 ½10; 3; 10� K3
19 ½11; 3; 12� K4
20 ½12; 3; 11� K4
4Note that our discussion on tetraquark results shall refer to theself-consistent calculation, i.e., theoretical output in both mesonand tetraquark sectors. This is in order to avoid, on one hand,theoretical uncertainties coming from the quark model predictionof meson spectra and, on the other hand, misleading statementsabout the nature of the tetraquark structures.
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diquark-antidiquark, and K-type structures do not shed anydifferent with respect the color-singlet channel, coupledenergies range from 4.2 to 4.4 GeV. These results confirmthe nature of scattering states for Dþ
s Dþs and D�þ
s D�þs .
Moreover, in a complete coupled-channels study, the low-est energy of 3978 MeV for Dþ
s Dþs state is remained. The
real-scaling results are consistent with those of ccq qtetraquarks; however, it is invalid for resonances.Figure 2 shows the distributions of ccss tetraquarks’
complex energies in the complete coupled-channelscalculation. In the energy gap from 3.9 GeV to5.0 GeV, most of poles are aligned along the thresholdlines. Namely, with the rotated angle θ varied from 0° to
6°, the Dð�Þþs Dð�Þþ
s energy poles always move along withthe same color cut lines. However, in the high energyregion which is about 0.2 GeV above the Dþ
s ð1SÞDþs ð2SÞ
threshold, one resonance pole is found. In the yellowcircle of Fig. 2, the three calculated dots with black, red,and blue almost overlap. This resonance pole has massand width 4902 MeV and 3.54 MeV, respectively, and it
TABLE VIII. All possible channels for IðJPÞ ¼ 0ð1þÞ cbs stetraquark systems. The second column shows the necessary
basis combination in spin (χσiJ ), flavor (χfjI ), and color (χck)
degrees of freedom. The superscript 1 and 8 stands for the color-singlet and hidden-color configurations of physical channels.
IndexχσiJ ; χ
fjI ; χ
ck
½i; j; k� Channel
1 ½1; 3; 1� ðDþs B�
sÞ12 ½2; 3; 1� ðD�þ
s B0sÞ1
3 ½3; 3; 1� ðD�þs B�
sÞ14 ½1; 3; 2� ðDþ
s B�sÞ8
5 ½2; 3; 2� ðD�þs B0
sÞ86 ½3; 3; 2� ðD�þ
s B�sÞ8
7 ½6; 3; 3� ðcbÞðs sÞ�8 ½5; 3; 3� ðcbÞ�ðs sÞ9 ½4; 3; 4� ðcbÞ�ðs sÞ�10 ½7; 3; 5� K1
11 ½8; 3; 5� K1
12 ½9; 3; 5� K1
13 ½7; 3; 6� K1
14 ½8; 3; 6� K1
15 ½9; 3; 6� K1
16 ½10; 3; 7� K2
17 ½11; 3; 7� K2
18 ½12; 3; 7� K2
19 ½10; 3; 8� K2
20 ½11; 3; 8� K2
21 ½12; 3; 8� K2
22 ½13; 3; 10� K3
23 ½14; 3; 10� K3
24 ½15; 3; 10� K3
25 ½13; 3; 9� K3
26 ½14; 3; 9� K3
27 ½15; 3; 9� K3
28 ½16; 3; 12� K4
29 ½17; 3; 12� K4
30 ½18; 3; 11� K4
TABLE IX. All possible channels for IðJPÞ ¼ 0ð2þÞ cbs stetraquark systems. The second column shows the necessary
basis combination in spin (χσiJ ), flavor (χfjI ), and color (χck)
degrees of freedom. The superscript 1 and 8 stands for the color-singlet and hidden-color configurations of physical channels.
IndexχσiJ ; χ
fjI ; χ
ck
½i; j; k� Channel
1 ½1; 3; 1� ðD�þs B�
sÞ12 ½1; 3; 2� ðD�þ
s B�sÞ8
3 ½1; 3; 3� ðcbÞ�ðs sÞ�4 ½1; 3; 5� K1
5 ½1; 3; 6� K1
6 ½1; 3; 7� K2
7 ½1; 3; 8� K2
8 ½1; 3; 9� K3
9 ½1; 3; 10� K3
10 ½1; 3; 12� K4
TABLE X. The lowest-lying eigenenergies of ccs s tetraquarkswith IðJPÞ ¼ 0ð0þÞ in the real range calculation. The firstcolumn shows the allowed channels and the second one thenoninteracting meson-meson threshold value of experiment/theory. Color-singlet (S), hidden-color (H) along with otherconfigurations are indexed in the third column, the fourth andfifth columns refer to the theoretical mass of each channels andtheir couplings. (unit: MeV).
Channel Threshold Index M Mixed
ðDþs Dþ
s Þ1 3938=3978 1(S) 3978ðD�þ
s D�þs Þ1 4224=4232 2(S) 4232 3978
ðDþs Dþ
s Þ8 3(H) 4619ðD�þ
s D�þs Þ8 4(H) 4636 4377
ðccÞðs sÞ 5 4433ðccÞ�ðs sÞ� 6 4413 4379
K1 7 4802K1 8 4369K1 9 4698K1 10 4211 4201
K2 11 4343K2 12 4753K2 13 4166K2 14 4838 4158
K3 15 4414K3 16 4427 4373
K4 17 4413K4 18 4439 4379
All of the above channels: 3978
QQS S TETRAQUARKS IN THE CHIRAL … PHYS. REV. D 102, 054023 (2020)
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could be identified as a resonance of the Dþs Dþ
s molecu-lar system.The IðJPÞ ¼ 0ð1þÞ state: There are two meson-meson
channels, Dþs D�þ
s and D�þs D�þ
s , one diquark-antidiquarkchannel, ðccÞ�ðs sÞ�; but more K-type channels (18 chan-nels) are allowed due to a much richer combination ofcolor, spin and flavor wave functions which fulfills thePauli Principle. Table XI lists the calculated masses of thesechannels and also their couplings.Firstly, the situation is similar to the IðJPÞ ¼ 0ð0þÞ case,
i.e., no bound state is found in the real-range calculation.Secondly, the couplings are extremely weak for both color-singlet and hidden-color channels. In contrast, strongerbinding is found for the K-type structures; the coupled-channels computation considering the K-type configura-tions, along with hidden-color and diquark-antidiquarkones, results in a mass around 4.4 GeV. Then, after mixingall of the channels listed in Table XI, the nature of thelowest energy level Dþ
s D�þs is still unchanged, it is a
TABLE XI. The lowest-lying eigenenergies of ccss tetraquarkswith IðJPÞ ¼ 0ð1þÞ in the real range calculation. The firstcolumn shows the allowed channels and the second one thenoninteracting meson-meson threshold value of experiment/theory. Color-singlet (S), hidden-color (H) along with otherconfigurations are indexed in the third column, the fourth andfifth columns refer to the theoretical mass of each channels andtheir couplings. (unit: MeV).
Channel Threshold Index M Mixed
ðDþs D�þ
s Þ1 4081=4105 1(S) 4105ðD�þ
s D�þs Þ1 4224=4232 2(S) 4232 4105
ðDþs D�þ
s Þ8 3(H) 4401ðD�þ
s D�þs Þ8 4(H) 4607 4400
ðccÞ�ðs sÞ� 5 4424 4424K1 6 4537K1 7 4536K1 8 4528K1 9 4440K1 10 4445K1 11 4371 4305
K2 12 4417K2 13 4419K2 14 4326K2 15 4699K2 16 4787K2 17 4802 4266
K3 18 4442K3 19 4443K3 20 5013 4424
K4 21 4427K4 22 4426K4 23 4953 4423
All of the above channels: 4105
TABLE XII. The lowest-lying eigenenergies of ccss tetra-quarks with IðJPÞ ¼ 0ð2þÞ in the real range calculation. Thefirst column shows the allowed channels and the second one thenoninteracting meson-meson threshold value of experiment/theory. Color-singlet (S), hidden-color (H) along with otherconfigurations are indexed in the third column, the fourth andfifth columns refer to the theoretical mass of each channels andtheir couplings. (unit: MeV).
Channel Threshold Index M Mixed
ðD�þs D�þ
s Þ1 4224=4232 1(S) 4232 4232
ðD�þs D�þ
s Þ8 2(H) 4432 4432
ðccÞ�ðs sÞ� 3 4446 4446
K1 4 4522K1 5 4385 4381
K2 6 4355K2 7 4666 4354
K3 8 4448 4448
K4 9 4446 4446
All of the above channels: 4232
TABLE XIII. The lowest-lying eigenenergies of bbss tetra-quarks with IðJPÞ ¼ 0ð0þÞ in the real range calculation. The firstcolumn shows the allowed channels and the second one thenoninteracting meson-meson threshold value of experiment/theory. Color-singlet (S), hidden-color (H) along with otherconfigurations are indexed in the third column, the fourth andfifth columns refer to the theoretical mass of each channels andtheir couplings. (unit: MeV).
Channel Threshold Index M Mixed
ðB0sB0
sÞ1 10734=10710 1(S) 10710ðB�
sB�sÞ1 10830=10800 2(S) 10800 10710
ðB0sB0
sÞ8 3(H) 11184ðB�
sB�sÞ8 4(H) 11205 10943
ðbbÞðs sÞ 5 10967ðbbÞ�ðs sÞ� 6 10901 10896
K1 7 11445K1 8 10928K1 9 11259K1 10 10863 10843
K2 11 10877K2 12 11445K2 13 10815K2 14 11441 10802
K3 15 10902K3 16 10960 10895
K4 17 10901K4 18 10980 10897
All of the above channels: 10710
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scattering one. Additionally, comparing the results inTable V for ccq q tetraquarks of our previous work [54],one notices that the deeply bound state with ∼200 MeVbinding energy for DþD�0 is invalid in the Dþ
s D�þs sector.
Our results using the complex scaling method appliedto the fully coupled-channels calculation are shownin Fig. 3. The complex energies of Dþ
s ð1SÞD�þs ð1SÞ,
D�þs ð1SÞD�þ
s ð1SÞ along with their first radial excitationstates Dþ
s ð1SÞD�þs ð2SÞ, Dþ
s ð2SÞD�þs ð1SÞ, and D�þ
s ð1SÞ ×D�þ
s ð2SÞ are generally aligned along the cut lines when therotated angle θ goes from 0° to 6°. Although there are threeregions which change slightly in the mass gap 4.55 to4.70 GeV, the calculated poles still come down graduallywhen the value of complex angle increases. Hence, neitherbound states nor resonances are found within the IðJPÞ ¼0ð1þÞ channel of ccs s tetraquarks.The IðJPÞ ¼ 0ð2þÞ state: Only one D�þ
s D�þs meson-
meson configuration, one ðccÞ�ðs sÞ� diquark-antidiquarkstructure and six K-type structures contribute to the highest
TABLE XIV. The lowest-lying eigenenergies of bbs s tetra-quarks with IðJPÞ ¼ 0ð1þÞ in the real range calculation. The firstcolumn shows the allowed channels and the second one thenoninteracting meson-meson threshold value of experiment/theory. Color-singlet (S), hidden-color (H) along with otherconfigurations are indexed in the third column, the fourth andfifth columns refer to the theoretical mass of each channels andtheir couplings. (unit: MeV).
Channel Threshold Index M Mixed
ðB0sB�
sÞ1 10782=10755 1(S) 10755ðB�
sB�sÞ1 10830=10800 2(S) 10800 10755
ðB0sB�
sÞ8 3(H) 10949ðB�
sB�sÞ8 4(H) 11185 10949
ðbbÞ�ðs sÞ� 5 10906 10906
K1 6 11041K1 7 11048K1 8 11038K1 9 10936K1 10 10949K1 11 10917 10870
K2 12 10911K2 13 10914K2 14 10879K2 15 11216K2 16 11483K2 17 11373 10840
K3 18 10928K3 19 10929K3 20 11557 10907
K4 21 10911K4 22 10908K4 23 11458 10906
All of the above channels: 10755
TABLE XV. The lowest-lying eigenenergies of bbs s tetra-quarks with IðJPÞ ¼ 0ð2þÞ in the real range calculation. The firstcolumn shows the allowed channels and the second one thenoninteracting meson-meson threshold value of experiment/theory. Color-singlet (S), hidden-color (H) along with otherconfigurations are indexed in the third column, the fourth andfifth columns refer to the theoretical mass of each channels andtheir couplings. (unit: MeV).
Channel Threshold Index M Mixed
ðB�sB�
sÞ1 10830=10800 1(S) 10800 10800
ðB�sB�
sÞ8 2(H) 10959 10959
ðbbÞ�ðs sÞ� 3 10915 10915
K1 4 11023K1 5 10894 10879
K2 6 10870K2 7 11186 10869
K3 8 10918 10918
K4 9 10916 10916
All of the above channels: 10800
TABLE XVI. The lowest-lying eigenenergies of cbs s tetra-quarks with IðJPÞ ¼ 0ð0þÞ in the real range calculation. The firstcolumn shows the allowed channels and the second one thenoninteracting meson-meson threshold value of experiment/theory. Color-singlet (S), hidden-color (H) along with otherconfigurations are indexed in the third column, the fourth andfifth columns refer to the theoretical mass of each channels andtheir couplings. (unit: MeV).
Channel Threshold Index M Mixed
ðDþs B0
sÞ1 7336=7344 1(S) 7344ðD�þ
s B�sÞ1 7527=7516 2(S) 7516 7344
ðDþs B0
sÞ8 3(H) 7910ðD�þ
s B�sÞ8 4(H) 7927 7678
ðcbÞðs sÞ 5 7726ðcbÞ�ðs sÞ� 6 7675 7662
K1 7 8171K1 8 8274K1 9 8369K1 10 8145 7613
K2 11 7896K2 12 8266K2 13 7758K2 14 8282 7629
K3 15 8647K3 16 8181K3 17 8321K3 18 8675 8010
K4 19 8199
K4 20 8359 8063
All of the above channels: 7344
QQS S TETRAQUARKS IN THE CHIRAL … PHYS. REV. D 102, 054023 (2020)
054023-11
spin channel (see Table XII). In analogy with the twoprevious cases, no bound state is obtained neither in eachsingle channel calculation nor in the coupled-channelscases. The mixed results of K1 and K2 types are botharound 4.35 GeV, which is lower than those of the otherexotic configurations (∼4.45 GeV); however, these do nothelp in realizing a bound state of D�þ
s D�þs .
Nevertheless, thrilling results are found in the completecoupled-channels study by CSM. Figure 4 shows that thereare three almost fixed resonance poles between theD�þ
s ð1SÞD�þs ð1SÞ and D�þ
s ð1SÞD�þs ð2SÞ threshold lines.
Two of them are wide resonances whereas the remainingone is narrow. The three found D�þ
s D�þs resonances have
mass and width (4821 MeV, 5.58 MeV), (4846 MeV,10.68 MeV), and (4775 MeV, 23.26 MeV). In the excitedenergy region, which is located about 0.5 GeV higher thanthe D�þ
s D�þs threshold but 0.1 GeV below its first radial
excitation, it is reasonable to find resonances whose natureis of hadronic molecular type [50,54,67].
TABLE XVII. The lowest-lying eigenenergies of cbs s tetra-quarks with IðJPÞ ¼ 0ð1þÞ in the real range calculation. The firstcolumn shows the allowed channels and the second one thenoninteracting meson-meson threshold value of experiment/theory. Color-singlet (S), hidden-color (H) along with otherconfigurations are indexed in the third column, the fourth andfifth columns refer to the theoretical mass of each channels andtheir couplings. (unit: MeV).
Channel Threshold Index M Mixed
ðDþs B�
sÞ1 7384=7389 1(S) 7389ðD�þ
s B0sÞ1 7479=7471 2(S) 7471
ðD�þs B�
sÞ1 7527=7516 3(S) 7516 7389
ðDþs B�
sÞ8 4(H) 7900ðD�þ
s B0sÞ8 5(H) 7891
ðD�þs B�
sÞ8 6(H) 7920 7684
ðcbÞðs sÞ� 7 7683ðcbÞ�ðs sÞ 8 7680ðcbÞ�ðs sÞ� 9 7725 7671
K1 10 7796K1 11 8172K1 12 8009K1 13 7695K1 14 7760K1 15 7634 7620
K2 16 7607K2 17 7621K2 18 7510K2 19 8137K2 20 8211K2 21 8209 7505
K3 22 7705K3 23 7706K3 24 7682K3 25 7734K3 26 7733K3 27 8298 7666
K4 28 7687
K4 29 7677
K4 30 7771 7670
All of the above channels: 7389
TABLE XVIII. The lowest-lying eigenenergies of cbs s tetra-quarks with IðJPÞ ¼ 0ð2þÞ in the real range calculation. The firstcolumn shows the allowed channels and the second one thenoninteracting meson-meson threshold value of experiment/theory. Color-singlet (S), hidden-color (H) along with otherconfigurations are indexed in the third column, the fourth andfifth columns refer to the theoretical mass of each channels andtheir couplings. (unit: MeV).
Channel Threshold Index M Mixed
ðD�þs B�
sÞ1 7527=7516 1(S) 7516 7516
ðD�þs B�
sÞ8 2(H) 7712 7712
ðcbÞ�ðs sÞ� 3 7698 7698
K1 4 7804K1 5 7705 7704
K2 6 7624K2 7 8205 7622
K3 8 8311K3 9 7701 7696
K4 10 7697 7697
All of the above channels: 7516
TABLE XIX. Possible resonance states of QQss (Q ¼ c, b)tetraquarks. (unit: MeV)
IðJPÞ Resonance Mass Width
0ð0þÞ Dþs Dþ
s 4902 3.54B�sB�
s 11306 1.86B�sB�
s 11333 1.84B0sB0
s 11412 1.54D�þ
s B�s 7919 1.02
D�þs B�
s 7993 3.22
0ð1þÞ D�þs B�
s 7920 1.20D�þ
s B�s 7995 4.96
0ð2þÞ D�þs D�þ
s 4821 5.58D�þ
s D�þs 4846 10.68
D�þs D�þ
s 4775 23.26B�sB�
s 11329 1.48B�sB�
s 11356 4.18B�sB�
s 11410 2.52D�þ
s B�s 8046 1.42
D�þs B�
s 8096 2.90
GANG YANG, JIALUN PING, and JORGE SEGOVIA PHYS. REV. D 102, 054023 (2020)
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B. The bbs s tetraquarks
We proceed here to analyze the bbs s tetraquark systemwith quantum numbers IðJPÞ ¼ 0ð0þÞ, 0ð1þÞ, and 0ð2þÞ.The situation is similar to the ccs s case, i.e. there are onlynarrow resonances in the IðJPÞ ¼ 0ð0þÞ and 0ð2þÞ chan-nels within the studied mass region; meanwhile, thisresult is also in contrast with the one obtained for bbqqtetraquarks [54]. The details are as following.The IðJPÞ ¼ 0ð0þÞ state: Table XIII summarizes all
possible channels for the IðJPÞ ¼ 0ð0þÞ bbs s tetraquark.In particular, there are two meson-meson channels B0
sB0s
and B�sB�
s , where both color-singlet and hidden-colorconfigurations are considered. Moreover, there are two
diquark-antidiquark structures, ðbbÞðssÞ and ðbbÞ�ðssÞ�,and 12 K-type channels. The calculated mass of each singlechannel ranges from 10.7 to 11.5 GeV, and no bound stateis observed. Additionally, after coupling between the samekind of configurations, one can conclude that the couplingis weak in dimeson case and it is quite comparable amongdiquark-antidiquark and K-type structures.The nature of scattering for lowest state B0
sB0s remains in
the complete coupled-channels calculation with rotatedangle θ ¼ 0°. However, three narrow resonance statesare obtained in the complex-scaling analysis. In the massregion from 10.7 to 11.5 GeV, Fig. 5 establishes thecomplex energy distributions of B0
sB0s , B�
sB�s and their first
radial excitation states. There are two orange circles whichsurround the resonance poles whose masses and widths are(11.31 GeV, 1.86 MeV), (11.33 GeV, 1.84 MeV), and(11.41 GeV, 1.54 MeV), respectively. The first two reso-nances can be identified as B�
sB�s molecular states because
they are ∼0.5 GeV higher than its threshold value and thethird one should be interpreted as a B0
sB0s because its
location is just between B0sð1SÞB0
sð2SÞ and B�sð1SÞB�
sð2SÞstates. Finally, after comparing our results of ccs s andbbs s tetraquarks, we conclude that, with much heavierconstituent quark components included, more narrowmolecular resonances will be found around 0.2 GeVinterval near the first radial excitation states.The IðJPÞ ¼ 0ð1þÞ state: The results listed in Table XIV
highlight that tightly bound and narrow resonance statesobtained in bbqq tetraquarks [54] are not found in this case.Firstly, the lowest channel B0
sB�s is of scattering nature both
in single channel calculation and coupled-channels one.Secondly, the mass of diquark-antidiquark configuration ishigher than meson-meson channels and its value of10.91 GeV is very close to the hidden-color channels,
3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000-30
-25
-20
-15
-10
-5
0D*+
sD*+
sD+
sD+
s
0°
-/2
(MeV
)
M(MeV)
D+sD
+s radial excitation
2°
4°
6°
FIG. 2. Complex energies of ccs s tetraquarks with IðJPÞ ¼0ð0þÞ in the complete coupled channels calculation, θ varyingfrom 0° to 6°.
4100 4200 4300 4400 4500 4600 4700 4800 4900 5000-30
-25
-20
-15
-10
-5
0D*+
sD*+
sD+
sD*+
s
0°
-/2
(MeV
)
M(MeV)
D*+
s D*+
sD+
sD*+
sradial excitation
2°
4°
6°
FIG. 3. Complex energies of ccss tetraquarks with IðJPÞ ¼0ð1þÞ in the coupled channels calculation, θ varying from0° to 6°.
4200 4300 4400 4500 4600 4700 4800 4900 5000-30
-25
-20
-15
-10
-5
0D*+
sD*+
sD*+
sD*+
s
0°
-/2
(MeV
)
M(MeV)
radial excitation
2°
4°
6°
FIG. 4. Complex energies of ccs s tetraquarks with IðJPÞ ¼0ð2þÞ in the coupled channels calculation, θ varying from 0°to 6°.
QQS S TETRAQUARKS IN THE CHIRAL … PHYS. REV. D 102, 054023 (2020)
054023-13
10.95 GeV. Furthermore, the other K-type configurationsproduce masses slightly lower (∼10.90 GeV) than theformer case.Figure 6 shows that the scattering nature of B0
sB�s and
B�sB�
s states is even clearer when the CSM is employed.More specifically, in the mass interval from 10.7 to11.3 GeV, the calculated complex energies always movealong with the varied angle θ. There is no fixed pole in theenergy region which is around 0.6 GeV above the B0
sB�s
threshold. This fact is consistent with the ccs s resultsdiscussed above.The IðJPÞ ¼ 0ð2þÞ state: For the highest spin channel of
bbs s tetraquarks, Table XV summarizes our theoreticalfindings in real-range method. Among our results, the
following are of particular interest: (i) only one dimesonchannel B�
sB�s exists and it is unbounded if we consider
either the single channel or multichannels coupling calcu-lation, and (ii) the other exotic configurations whichinclude hidden-color, diquark-antidiquark, and K-typeare all excited states with masses about 10.9 GeV.In a further step, in which the complex analysis is
adopted, three resonances are obtained. It is quite obviousin Fig. 7 that three fixed poles, marked with orange circles,are located at around 11.35 GeVand near the real-axis. Theexact masses and widths of these B�
sB�s resonances are
(11.33 GeV, 1.48 MeV), (11.36 GeV, 4.18 MeV) and(11.41 GeV, 2.52 MeV), respectively. These found narrowresonances are 0.6 GeV above B�
sB�s threshold, meanwhile
they approach its first radial excitation state.
C. The cbs s tetraquarks
Several narrow resonances with quantum numbersIðJPÞ ¼ 0ð0þÞ, 0ð1þÞ and 0ð2þÞ are found in this sector,and within the studied mass region. However, there is nobound states in the case of cbq q tetraquarks [54].The IðJPÞ ¼ 0ð0þÞ channel: There are two meson-
meson channels, Dþs B0
s and D�þs B�
s , two diquark-antidiquark structures, ðcbÞðssÞ and ðcbÞ�ðssÞ�, and14 K-type channels (see Table XVI). The single channelcalculation produces masses which ranges from 7.34 to8.67 GeV, and all states are scattering ones. The coupled-channels study for each kind of structure reveals weakcouplings in dimeson configuration of color-singlet chan-nels and stronger ones for the other structures, with massesabove 7.6 GeV.If we now rotate the angle θ from 0° to 6° in a fully
coupled-channels calculation, Fig. 8 shows the distributionof complex energy points of Dþ
s B0s and D�þ
s B�s . It is
10800 10900 11000 11100 11200 11300 11400 11500-40
-35
-30
-25
-20
-15
-10
-5
0B*
sB*s
0°
-/2
(MeV
)
M(MeV)
B*sB*
s radial excitation
2°
4°
6°
FIG. 7. Complex energies of bbss tetraquarks with IðJPÞ ¼0ð2þÞ in the coupled channels calculation, θ varying from0° to 6°.
10700 10800 10900 11000 11100 11200 11300-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
0°
-/2
(MeV
)
M(MeV)
B*sB*
sB0
sB*
s
2°
4°
6°
FIG. 6. Complex energies of bbss tetraquarks with IðJPÞ ¼0ð1þÞ in the coupled channels calculation, θ varying from0° to 6°.
10700 10800 10900 11000 11100 11200 11300 11400 11500-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0B*
sB*
s
0°
-/2
(MeV
)
M(MeV)
B0sB0
s
B*sB*
sB0sB0
sradial excitation
2°
4°
6°
FIG. 5. Complex energies of bbs s tetraquarks with IðJPÞ ¼0ð0þÞ in the coupled channels calculation, θ varying from0° to 6°.
GANG YANG, JIALUN PING, and JORGE SEGOVIA PHYS. REV. D 102, 054023 (2020)
054023-14
obvious to notice that there are two stable poles in the massregion from 7.3 to 8.0 GeV. Actually, their calcu-lated masses and widths are (7.92 GeV, 1.02 MeV)and (7.99 GeV, 3.22 MeV), respectively. Because theyare much more close to the D�þ
s B�s threshold lines,
the two narrow resonances can be identified as D�þs B�
smolecules.The IðJPÞ ¼ 0ð1þÞ channel: There are 30 possible
channels in this case and they are listed in Table XVII;in particular, one has three meson-meson channels: Dþ
s B�s ,
D�þs B0
s , and D�þs B�
s , the diquark-antidiquark channelsðcbÞðssÞ�, ðcbÞ�ðssÞ and ðcbÞ�ðssÞ�, and the remaining21 channels are of K-type configurations. In our first kindof calculation, the single channel masses are located in theenergy interval 7.39–8.23 GeV. Particularly, the color-singlet channels of dimeson configurations present masseswhich are below 7.52 GeV, and the other exotic structuresare above this level. Then, in the coupled-channels com-putation, the lowest energy of color-singlet channels is stillthe Dþ
s B�s threshold value, 7.39 GeV. Masses of the other
configurations are about 7.67 GeV, except for 7.51 GeVofK2-type channels.Figure 9 mostly depicts the typical distribution of
scattering states of Dþs B�
s , D�þs B0
s , and D�þs B�
s , i.e., thecalculated complex energies are basically aligned alongtheir respective threshold lines. However, two stable polesare located in the top right corner of this figure. Inside thetwo orange circles, one can find that the black, red and bluedots (which are the results of 2°, 4°, and 6° rotated angle,respectively) almost overlap. Together with the fact thatnearD�þ
s B�s threshold lines appear, they can be identified as
D�þs and B�
s resonances whose masses and widths are(7.92 GeV, 1.20 MeV) and (7.99 GeV, 4.96 MeV),respectively.
The IðJPÞ ¼ 0ð2þÞ channel: Only one meson-mesonchannel, D�þ
s B�s , one diquark-antidiquark channel,
ðcbÞ�ðssÞ�, and 7 K-type configurations must be consid-ered in this case. Their calculated masses are listed inTable XVIII. As all other cases discussed above, no boundstates are found neither in the single channel computationnor in the coupled-channels case. The lowest scatteringstate of D�þ
s B�s is located at 7.52 GeV and all other
excited states, in coupled-channels calculation, are below7.72 GeV.In contrast to the cbq q tetraquarks [54], two cbss
resonances are found in the complete coupled-channelscalculation when complex range method is used. Figure 10shows an orange circle, which is near the threshold linesD�þ
s ð1SÞB�sð2SÞ and D�þ
s ð2SÞB�sð1SÞ, surrounding two
fixed resonance poles. The calculated masses and widths
7500 7600 7700 7800 7900 8000 8100 8200 8300 8400-30
-25
-20
-15
-10
-5
0D*+
sB*
sD*+
sB*
s
0°
-/2
(MeV
)
M(MeV)
radial excitation
2°
4°
6°
FIG. 10. Complex energies of cbss tetraquarks with IðJPÞ ¼0ð2þÞ in the coupled channels calculation, θ varying from0° to 6°.
7300 7400 7500 7600 7700 7800 7900 8000-30
-25
-20
-15
-10
-5
0D*+
sB0
s
0°
-/2
(MeV
)
M(MeV)
D*+s
B*sD+
sB*
s
2°
4°
6°
FIG. 9. Complex energies of cbss tetraquarks with IðJPÞ ¼0ð1þÞ in the coupled channels calculation, θ varying from0° to 6°.
7300 7400 7500 7600 7700 7800 7900 8000-30
-25
-20
-15
-10
-5
0D*+
sB*
sD+
sB0
ss s
0°
-/2
(MeV
)
M(MeV)
2°
4°
6°
FIG. 8. Complex energies of cbss tetraquarks with IðJPÞ ¼0ð0þÞ in the coupled channels calculation, θ varying from0° to 6°.
QQS S TETRAQUARKS IN THE CHIRAL … PHYS. REV. D 102, 054023 (2020)
054023-15
are (8.05 GeV, 1.42 MeV) and (8.10 GeV, 2.90 MeV),respectively. Apparently, these two narrow D�þ
s B�s reso-
nances are ∼0.6 GeV higher than their threshold.
IV. EPILOGUE
TheQQs s tetraquarks with spin-parity JP ¼ 0þ, 1þ, and2þ, and in the isoscalar sector I ¼ 0 have been systemicallyinvestigated. This is a natural extension of our previouswork on double-heavy tetraquarks QQq q (q ¼ u, d);however, not only the meson-meson and diquark-antidiquark configurations, with their allowed color struc-tures: color-singlet, hidden-color, color triplet-antitripletand color sextet-antisextet, are considered but also fourK-type configurations are included herein.The chiral quark model contains the perturbative one-
gluon exchange interaction and the nonperturbative linear-screened confinement and Goldstone-boson exchangeinteractions between anti-strange quarks. This model hasbeen successfully applied to the description of hadron,hadron-hadron and multiquark phenomenology. In order todistinguish among bound states, resonances and scatteringpoles the complex scaling method is used. FollowingRef. [69], we employ Gaussian trial functions with rangesin geometric progression. This enables the optimization ofranges employing a small number of free parameters.For the three types of tetraquarks: ccss, bbss, and cbss,
no bound state is found in any quantum-number channelstudied herein, and this is in contrast with theQQqq sector.However, within the studied energy regions, several reso-nances are available with different quantum numbers andnature. Table XIX collects our results showing the mass andwidth of each found resonance. Some details of suchresonances are summarized below.All found resonances are about 0.6 GeV higher than their
corresponding threshold and near the first radial excitationstates, around 0.2 GeV energy region. For the ccss
tetraquark, one narrow Dþs Dþ
s resonance is obtained inIðJPÞ ¼ 0ð0þÞ channel with mass and width 4.9 GeV and3.54 MeV, respectively. Besides, another narrow resonanceof 5.58 MeV width and two wide ones with widths of10.68 MeV and 23.26 MeV are found for D�þ
s D�þs in the
IðJPÞ ¼ 0ð2þÞ channel; their masses are 4.82 GeV,4.85 GeV and 4.78 GeV, respectively.Similarly to ccs s tetraquarks, narrow resonances are
only found in IðJPÞ ¼ 0ð0þÞ and 0ð2þÞ states for bbsssector. However, with much heavier flavor quarks included,more resonances are available. Specifically, there are twoB�sB�
s and one B0sB0
s resonances in IðJPÞ ¼ 0ð0þÞ channel.Their masses and widths are (11.31 GeV, 1.86 MeV),(11.33 GeV, 1.84 GeV), and (11.41 GeV, 1.54 MeV),respectively. Meanwhile, in the IðJPÞ ¼ 0ð2þÞ channel,three B�
sB�s resonances are obtained with masses and widths
(11.33 GeV, 1.48 MeV), (11.36 GeV, 4.18 GeV), and(11.41 GeV, 2.52 MeV), respectively.Furthermore, two D�þ
s B�s narrow resonances have been
found in each IðJPÞ ¼ 0ð0þÞ, 0ð1þÞ, and 0ð2þÞ channel.Their masses and widths can be summarized as follows:D�þ
s B�sð7.92 GeV; 1.02 MeVÞ and D�þ
s B�sð7.99 GeV;
3.22 MeVÞ within the IðJPÞ ¼ 0ð0þÞ channel;D�þ
s B�sð7.92 GeV; 1.20 MeVÞ and D�þ
s B�sð7.99 GeV;
4.96 MeVÞ within the IðJPÞ ¼ 0ð1þÞ channel; andD�þ
s B�sð8.05 GeV; 1.42 MeVÞ and D�þ
s B�sð8.09 GeV;
2.90 MeVÞ in the case of IðJPÞ ¼ 0ð2þÞ.
ACKNOWLEDGMENTS
Work partially financed by: the National Natural ScienceFoundation of China under Grants No. 11535005 andNo. 11775118; the Ministerio Español de Ciencia eInnovación under Grant No. PID2019–107844 GB-C22;and the Junta de Andalucía under Contract No. OperativoFEDER Andalucía 2014-2020 UHU-1264517.
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