Hidden-charm and bottom tetra- and pentaquarks with strangeness in the hadro-quarkonium and compact tetraquark models

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Hidden-charm and bottom tetraand pentaquarks with strangeness in the hadroquarkonium and compact tetraquark models © The Authors. Article funded by SCOAP3. Published version Ferretti, J.; Santopinto, E. Ferretti, J., & Santopinto, E. (2020). Hidden-charm and bottom tetraand pentaquarks with strangeness in the hadro-quarkonium and compact tetraquark models. Journal of High Energy Physics, 2020(4), Article 119. https://doi.org/10.1007/JHEP04(2020)119 2020 JHEP04(2020)119 Published for SISSA by Springer Received:January 9, 2020 Revised:February 19, 2020 Accepted:April 2, 2020 Published:April 20, 2020 Hidden-charm and bottom tetraand pentaquarks with strangeness in the hadro-quarkonium and compact tetraquark models J. Ferrettia,b and E. Santopintoc aCenter for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, Connecticut 06520-8120, U.S.A. bDepartment of Physics, University of Jyv¨askyl¨a, P.O. Box 35 (YFL), 40014 Jyv¨askyl¨a, Finland cIstituto Nazionale di Fisica Nucleare (INFN), Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy E-mail: [email protected],[email protected] Abstract: In two recent papers, we used the hadro-quarkonium model to study the properties of hidden-charm and bottom tetraquarks and pentaquarks. Here, we extend the previous results and calculate the masses of heavy-quarkonium-kaon/hyperon systems. We also compute the spectrum of hidden-charm and bottom tetraquarks with strangeness in the compact tetraquark (diquark-antidiquark) model. If heavy-light exotic systems with non-null strangeness content were to be observed experimentally, it might be possible to distinguish among the large variety of available theoretical pictures for tetraand pentaquark states and, possibly, rule out those which are not compatible with the data. Keywords: Phenomenological Models, QCD Phenomenology ArXiv ePrint: 2001.01067 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP04(2020)119 JHEP04(2020)119 Contents 1 Introduction 1 2 Hadro-quarkonium model 3 3 Spectra of strange hidden-charm and bottom tetraand pentaquarks in the hadro-quarkonium model 4 3.1 Hidden-charm and hidden-bottom pentaquarks with strangeness in the hadro-quarkonium model 5 3.2 Hidden-charm and hidden-bottom tetraquarks with strangeness in the hadro-quarkonium model 7 4 Relativized diquark model 8 5 Masses of cs¯c¯nand bs¯ b¯nstates in the compact tetraquark model 10 5.1 Ground-state energies of cs¯c¯nand bs¯ b¯ntetraquarks 10 5.2 Spectra of cs¯c¯nand bs¯ b¯ntetraquarks 11 6 Conclusion 13 1 Introduction Multiquark states are baryons/mesons which cannot be described in terms of qqq/q¯qdegrees of freedom only. They include XY Z suspected tetraquarks, like the X(3872) [now χc1(3872)] [1–3] and X(4274) [also known as χc1(4274)] [4–6], and pentaquark states. The latter were recently discovered by LHCb in Λb→J/ψΛ∗and Λb→P+ cK−→(J/ψp)K− decays [7,8]. The structure of XY Z tetraquarks and Pcpentaquarks is still unclear. This is why there are several alternative models to explain their properties. For a review, see refs. [9–12]. To distinguish among the different pictures (molecular model, diquark model, unquenched quark model, . . .) one should compare their theoretical predictions for the spectrum, decay amplitudes, production cross-sections, and so on, with the experimental data. A clean way to discriminate among the previous theoretical interpretations for suspected XY Z tetraquarks was suggested in ref. [13]. There, Voloshin pointed out that if Zcresonances exist then, because of the SU(3)fsymmetry, one may also expect the emergence of their strange partners, Zcs [13]. The author also argued that the one-pionexchange interaction of the meson-meson molecular model is impossible between strange and nonstrange heavy mesons, like Band Bs[13]. Hidden-charm and bottom mesons with strangeness are also forbidden in the context of the Unquenched Quark Model (UQM) formalism. Indeed, one cannot dress heavy quarkonium Q¯ Qstates with Q¯s−n¯ Qor Q¯n−s¯ Q higher Fock components (where n=uor d) by creating a light n¯nor s¯spair with vacuum quantum numbers. Therefore, hidden-charm and bottom tetraquark states with non-null – 1 – JHEP04(2020)119 Figure 1. Schematic representation of heavy-light hadro-quarkonium (right) and compact tetraquark (left) states. strangeness content cannot take place neither in the UQM [14–27] nor in the molecular model [28–34] interpretations. On the contrary, these exotic configurations are expected (if above threshold) both in the compact tetraquark [35–50] and hadro-quarkonium [13,51–66] models. See figure 1. In light of this, the experimental observation of XY Z states with non-null strangeness content would make it possible to rule out a few possible theoretical interpretations for tetraquarks. Voloshin did not compute the spectrum of Zcs states, but only discussed phenomenological indications for the emergence of those states [13]. The study of their spectrum and that of their pentaquark counterparts is thus the subject of the present manuscript. Here, we extend the hadro-quarkonium model findings of refs. [64,65] and calculate the spectrum of hidden-charm and bottom tetraquarks and pentaquarks with strangeness. The hadro-quarkonium picture was developed to explain the experimental observation of heavy-light tetraquark candidates characterized by peculiar properties [53,67]. Firstly, these exotics are supposed not to be particularly close to a specific heavy-light mesonmeson threshold, unlike D0¯ D∗0in the X(3872) case. Secondly, such states may decay into heavy quarkonia plus one or more light mesons, like ηc+η. Even though it was meant for the description of tetraquarks, the hadro-quarkonium model can be easily extended to the baryon sector to study pentaquarks [59–61,65]. We also compute the masses of heavy-light tetraquarks with non-null strangeness content in the compact tetraquark model of refs. [46,48,50]. In the compact tetraquark model, heavy-light qQ¯q¯ Qstates are modeled as the bound states of a diquark, qQ, antidiquark, ¯q¯ Q, pair. The diquark constituents are treated as inert against internal spatial excitations. Their binding is the consequence of one-gluon-exchange forces and their relative dynamics can be described in terms of a relative coordinate rrel. The calculation of the spectrum of compact pentaquark configurations in the diquark model is more difficult than that of compact tetraquarks because one has to deal with a three-body problem instead of a two-body one; moreover, one also has to consider both diquark-diquark and diquark-antiquark interactions. This is why here we do not provide results for compact (diquark-diquark-antiquark) pentaquarks, which will be the subject of a subsequent paper. Our predictions for strange hidden-charm and bottom tetraquarks and, especially, those for Pcand Pbpentaquarks with non-null strangeness content may soon be tested by LHCb. – 2 – JHEP04(2020)119 H MGE h 1 2 Figure 2. Hidden-flavor transition ψ1→ψ2+hin the QCD multipole expansion. Here, ψ1and ψ2are the initial and final charmonium states, hlight hadron(s). The two vertices are those of the multipole gluon emission, MGE, and hadronization, H. Picture from ref. [64]; Elsevier Copyright. 2 Hadro-quarkonium model The possible existence of binding mechanisms of charmonium states in light-quark matter was discussed long ago [68–70] in terms of the interaction of charmonium inside nuclei. The idea of hadro-charmonium (hadro-quarkonium) bound states resembles the previous one. Hadro-quarkonia are heavy-light tetraor pentaquark configurations, where a compact Q¯ Qstate (with Q=cor b), labelled as ψin the following, is embedded in light hadronic matter, H=qqq or q¯q(where q=u, d or s) [13,51–66]. The heavy and light constituents, ψand H, develop an attractive force, which is the result of multiple-gluon exchange between them. Such interaction, Heff , can be written in terms of the multipole expansion in QCD [71–73]. In particular, if one considers as leading term the E1 interaction with chromo-electric fields Eand E0[53,69], one gets the effective Hamiltonian Heff =−1 2αψψ0E·E0,(2.1) where αψψ0is the so-called heavy quarkonium chromo-electric polarizability. By making use of additional approximations, Heff can be further reduced to a simple square-well potential [53,64,65], Vhq(r) = (−2παψψMH 3R3 H for r < RH 0 for r > RH ,(2.2) where RH=RBor RMis the light baryon/meson radius. Eq. (2.2) can be plugged into a Schr¨odinger equation and solved for light hadron-heavy quarkonium systems. There are four quantities to be given as input in the calculation. They are the masses Mψand MH, the radius RH, and the diagonal chromo-electric polarizability, αψψ. See table 1. The values of Mψand MHare extracted from the PDG [74]. In principle, non-diagonal quarkonium chromo-electric polarizabilities, αψψ0, can be fitted to the data by considering ψ→ψ0+hhadronic transitions [67,75]; see figure 2. However, no experimental information can be used to estimate the αψψ’s. Therefore, the – 3 – JHEP04(2020)119 Parameter Value Parameter Value αψψ(1P)c¯c11 GeV−3αψψ(2S)c¯c18 GeV−3 α(1) ψψ(1P)b¯ b14 GeV−3α(1) ψψ(2S)b¯ b23 GeV−3 α(2) ψψ(1P)b¯ b21 GeV−3α(2) ψψ(2S)b¯ b33 GeV−3 RΣ0.863 fm RΞ0.841 fm RK0.560 fm RK∗0.729 fm Table 1. Hadro-quarkonium model. Input values and parameters. diagonal chromo-electric polarizabilities, αψψ(n`), where nand `are the radial quantum number and orbital angular momentum of ψ, respectively, have to be extracted from the phenomenology. In the case of charmonia, we consider [65]: αψψ(1P)c¯c= 11 GeV−3and αψψ(2S)c¯c= 18 GeV−3. In the case of bottomonia, we make use of two sets of values for the chromo-electric polarizabilities. They are [65]: α(1) ψψ(1P)b¯ b= 14 GeV−3and α(1) ψψ(2S)b¯ b= 23 GeV−3;α(2) ψψ(1P)b¯ b= 21 GeV−3and α(2) ψψ(2S)b¯ b= 33 GeV−3. We also need the strange mesons’ and hyperons’ radii. While for the kaon we can use the well-established value of the K±charge radius reported on the PDG [74], RK= 0.560±0.031 fm, in the Σ, Ξ and K∗ cases the situation is different.1Indeed, due to the lack of well-established experimental data, we are forced to extract RΣ,RΞand RK∗from phenomenological estimates. For example, see refs. [76–80]. Following ref. [79], we have: RΣ=1 2(RΣ++RΣ−)=0.863 fm; RΞ= 0.841 fm. The K∗(892)’s radius is calculated in the relativized quark model for mesons of ref. [76]: RK∗= 0.729 fm. Finally, the hadro-quarkonium quantum numbers are obtained by combining those of the hadrons ψand H, |Φhqi=(Lψ, Sψ)Jψ; (LH, SH)JH; (Jhq, `hq)JP tot.(2.3) Here, Jhq =Jψ+JH, the hadro-quarkonium parity is P= (−1)`hq PψPH, and `hq is the relative angular momentum between ψand H. From now on, unless explicitly indicated, we assume that `hq = 0. 3 Spectra of strange hidden-charm and bottom tetraand pentaquarks in the hadro-quarkonium model In this section, we discuss our results for the spectrum of heavy quarkonium-strange hadron bound states. The binding energies are computed in the hadro-quarkonium model of section 2and refs. [53,64,65] by solving the two-body eigenvalue problem of eq. (2.2) via a finite differences algorithm [81, Vol. 3, section 16-6]. As a check, the same results are also obtained 1The values of the proton and kaon radii reported by the PDG [74] can be regarded as reliable, because they are the result of the average over several measurements. On the contrary, the value of the Σ−radius from the PDG is the outcome of a single experiment; moreover, there is no available data for the charge radius of the Σ+. This is why here we do not extract RΣfrom the PDG. – 4 – JHEP04(2020)119 Composition Quark content αψψ(n`) [GeV−3]JP tot Mass (Binding) [MeV] χc0(1P)⊗Σnnsc¯c11 1 2 +4440 (−166) ηc(2S)⊗Σnnsc¯c18 1 2 −4474 (−355) ψ(2S)⊗Σnnsc¯c18 1 2 −or 3 2 −4522 (−355) χc1(1P)⊗Σnnsc¯c11 1 2 +or 3 2 +4535 (−166) hc(1P)⊗Σnnsc¯c11 1 2 +or 3 2 +4550 (−167) χc2(1P)⊗Σnnsc¯c11 3 2 +or 5 2 +4580 (−167) [ηc(2S)⊗Σ]`hq=1 nnsc¯c18 1 2 +or 3 2 +4653 (−175) [ψ(2S)⊗Σ]`hq=1 nnsc¯c18 1 2 +,3 2 +or 5 2 +4701 (−176) ηc(2S)⊗Ξnssc¯c18 1 2 −4500 (−459); 4955 (−5) χc0(1P)⊗Ξnssc¯c11 1 2 +4510 (−226) ψ(2S)⊗Ξnssc¯c18 1 2 −or 3 2 −4548 (−460); 5002 (−5) χc1(1P)⊗Ξnssc¯c11 1 2 +or 3 2 +4605 (−227) hc(1P)⊗Ξnssc¯c11 1 2 +or 3 2 +4620 (−227) χc2(1P)⊗Ξnssc¯c11 3 2 +or 5 2 +4650 (−228) [ηc(2S)⊗Ξ]`hq=1 nssc¯c18 1 2 +or 3 2 +4685 (−274) [ψ(2S)⊗Ξ]`hq=1 nssc¯c18 1 2 +,3 2 +or 5 2 +4733 (−275) Table 2. Hadro-quarkonium model predictions for charmonium-Σ and Ξ bound states. The pentaquark binding energies and masses (5th column) are calculated with the values of the chromoelectric polarizabilities αψψ(n`) (3rd column). Here, n=uor d. The bound states are S-wave configurations (i.e. `hq = 0), except where explicitly indicated. In some cases, the Vhq potential well is deep enough to give rise to a heavy-quarkonium−baryon bound state and its radial excitation. In this instance, the masses of both the ground-state and excited hadro-quarkonium configurations are reported in the fifth column. by means of a numerical code based on the Multhopp method; see [82, section 2.4]. The values of the heavy quarkonium chromo-electric polarizabilities and light hadron radii used here are given in table 1. 3.1 Hidden-charm and hidden-bottom pentaquarks with strangeness in the hadro-quarkonium model The first step of our investigation is the study of heavy quarkonium-hyperon bound states. Our findings are enlisted in tables 2and 3. It is worth noting that: I) According to our predictions, heavy-quarkonium-hyperon states may be deeply bound; II) In some cases, the Vhq potential well is deep enough to give rise to a heavy-quarkonium-baryon bound state and its radial excitation; III) Our results show a strong dependence on the hyperon’s radius, RB. See eq. (2.2). The theoretical predictions for RB’s are highly model dependent and span a relatively wide range [77–80]. – 5 – JHEP04(2020)119 Composition Quark content αψψ(n`) [GeV−3]JP tot Mass (Binding) [MeV] ηb(2S)⊗Σnnsb¯ b23 1 2 −10671 (−519); 11118 (−72) Υ(2S)⊗Σnnsb¯ b23 1 2 −or 3 2 −10695 (−519); 11142 (−72) χb0(1P)⊗Σnnsb¯ b14 1 2 +10784 (−267) χb1(1P)⊗Σnnsb¯ b14 1 2 +or 3 2 +10817 (−267) hb(1P)⊗Σnnsb¯ b14 1 2 +or 3 2 +10824 (−267) χb2(1P)⊗Σnnsb¯ b14 3 2 +or 5 2 +10836 (−267) [ηb(2S)⊗Σ]`hq=1 nnsb¯ b23 1 2 +or 3 2 +10840 (−350) [Υ(2S)⊗Σ]`hq=1 nnsb¯ b23 1 2 +,3 2 +or 5 2 +10864 (−350) ηb(2S)⊗Σnnsb¯ b33 1 2 −10383 (−807); 10885 (−305) Υ(2S)⊗Σnnsb¯ b33 1 2 −or 3 2 −10407 (−808); 10909 (−306) [ηb(2S)⊗Σ]`hq=1 nnsb¯ b33 1 2 +or 3 2 +10564 (−626); 11175 (−15) [Υ(2S)⊗Σ]`hq=1 nnsb¯ b33 1 2 +,3 2 +or 5 2 +10588 (−626); 11199 (−15) χb0(1P)⊗Σnnsb¯ b21 1 2 +10588 (−462); 11016 (−34) χb1(1P)⊗Σnnsb¯ b21 1 2 +or 3 2 +10622 (−462); 11049 (−34) hb(1P)⊗Σnnsb¯ b21 1 2 +or 3 2 +10628 (−462); 11056 (−34) χb2(1P)⊗Σnnsb¯ b21 3 2 +or 5 2 +10641 (−462); 11069 (−34) ηb(2S)⊗Ξnssb¯ b23 1 2 −10664 (−657); 11126 (−194) Υ(2S)⊗Ξnssb¯ b23 1 2 −or 3 2 −10688 (−657); 11150 (−195) χb0(1P)⊗Ξnssb¯ b14 1 2 +10832 (−349) [ηb(2S)⊗Ξ]`hq=1 nssb¯ b23 1 2 +or 3 2 +10833 (−488) [Υ(2S)⊗Ξ]`hq=1 nssb¯ b23 1 2 +,3 2 +or 5 2 +10857 (−488) χb1(1P)⊗Ξnssb¯ b14 1 2 +or 3 2 +10865 (−349) hb(1P)⊗Ξnssb¯ b14 1 2 +or 3 2 +10872 (−349) χb2(1P)⊗Ξnssb¯ b14 3 2 +or 5 2 +10885 (−349) ηb(2S)⊗Ξnssb¯ b33 1 2 −10315 (−1006); 10818 (−502) Υ(2S)⊗Ξnssb¯ b33 1 2 −or 3 2 −10339 (−1006); 10842 (−503) [ηb(2S)⊗Ξ]`hq=1 nssb¯ b33 1 2 +or 3 2 +10495 (−826); 11137 (−183) [Υ(2S)⊗Ξ]`hq=1 nssb¯ b33 1 2 +,3 2 +or 5 2 +10519 (−826); 11161 (−184) χb0(1P)⊗Ξnssb¯ b21 1 2 +10593 (−588); 11044 (−138) χb1(1P)⊗Ξnssb¯ b21 1 2 +or 3 2 +10627 (−588); 11077 (−138) hb(1P)⊗Ξnssb¯ b21 1 2 +or 3 2 +10633 (−588); 11083 (−138) χb2(1P)⊗Ξnssb¯ b21 3 2 +or 5 2 +10646 (−588); 11096 (−138) Table 3. As table 2, but for bottomonium-Σ and Ξ bound states. – 6 – JHEP04(2020)119 However, the use of different values of the hyperon’s radius does not change our first conclusion qualitatively. As an example, we consider the ηc(2S)⊗Σ state. If we extract the value of the Σ radius from ref. [80], RΣ=1 2(RΣ++RΣ−)=0.91 fm, we get a binding energy Bηc(2S)⊗Σ=−294 MeV; if we use the experimental value [74], RΣ=RΣ−= 0.780 fm, we obtain Bηc(2S)⊗Σ=−492 MeV. The previous results can be compared to our prediction from table 2,Bηc(2S)⊗Σ=−355 MeV, calculated with RΣ=1 2(RΣ++RΣ−)=0.863 fm [79]; IV) In bottomonium-hyperon configurations, the presence of a heavier (nonrelativistic) b¯ bpair is expected to make the hadro-bottomonium system more stable than the hadrocharmonium one due to kinetic energy suppression. This is why the strange hidden-bottom pentaquarks are more tightly bound than their hidden-charm counterparts; V) If we consider the second set of values for the bottomonium chromo-electric polarizabilities of table 1, we get bottomonium-Σ bound states characterized by very large binding energies. The hadro-quarkonium picture may break down in these specific cases. Thus, one may have to consider the possibility of a mixing between hadro-quarkonium and compact five-quark components: H= Hhq Vmixing Vmixing Hcompact !.(3.1) Here, Hhq =Vhq+Thq is the hadro-quarkonium Hamiltonian, with Thq being the ψHrelative kinetic energy and Vhq the potential of eq. (2.2); Hcompact is an effective Hamiltonian, which describes a compact five-quark system; Vmixing is an off-diagonal interaction, which mixes hadro-quarkonium and compact five-quark components. 3.2 Hidden-charm and hidden-bottom tetraquarks with strangeness in the hadro-quarkonium model As a second step, we study heavy quarkonium-kaon and K∗configurations. Our findings are enlisted in tables 4and 5. Heavy quarkonium-kaon bound states show similar features as the heavy-light pentaquarks of section 3.1. In particular, one can notice that: I) The hadro-quarkonium interaction, eq. (2.2), may determine the emergence of deeply-bound charmonium-kaon tetraquark configurations; II) Even more stable configurations are the bottomonium-kaon ones; III) In both previous cases, if one substitutes the kaon with the K∗, one obtains extremely stable systems. As discussed in section 3.1, a more realistic description of ψK∗ systems may be accomplished by making use of the Hamiltonian (3.1), where one also takes mixing effects between hadro-quarkonium and compact tetraquark components into account. Compact heavy-light tetraquarks have been extensively studied. For example, see the potential model calculations of refs. [41,42,45,48–50] and sections 4and 5. The quality of the approximation of neglecting mixing effects between the heavy, ψ, and the light, H, hadron components in the hadro-charmonium states of table 4can be evaluated by calculating the wave function overlap of the previous components at the hadro-quarkonium center Poverlap =ZRH 0 d3rΨψ(r)ΨH(r).(3.2) – 7 – JHEP04(2020)119 ψ(2S)-, Υ(2S)-, and χb,c(1P)-hyperon/kaon bound states and the possible formation of cs¯c¯nand bs¯ b¯ntetraquarks as diquark-antidiquark bound states. Our results suggest that: I) strange hadro-quarkonium systems may be strongly bound. On the other hand, if the heavy quarkonium- (ψ) light hadron (H) binding energies become too large, the hadro-quarkonium picture may break down. As a consequence, the ψand H components may overlap, and a compact four/five-quark system could be realized rather than a ψ-Hbound state; II) both cs¯c¯nand bs¯ b¯ncompact tetraquarks may be bound, even though bs¯ b¯nconfigurations are more likely to manifest; III) in the case of cs¯c¯nconfigurations, the compact tetraquark ground-state is around 200 MeV below the lowest energy hadro-charmonium state, ηc⊗K. On the contrary, in the bs¯ b¯ncase the compact tetraquark ground-state is above the lowest energy hadro-bottomonium configuration, ηb⊗K; IV) by combining the conclusions discussed at points I) and II), we suggest the experimentalists to look for strange tetraand pentaquark configurations with hidden-bottom. They should be more stable than their hidden-charm counterparts due to kinetic energy suppression; thus, there is a higher probability of observing them. Finally, as pointed out in ref. [13], the meson-meson molecular model cannot be used to describe heavy-light tetraquarks with non-null strangeness content. The reason is that one-pion-exchange cannot take place between strange and nonstrange heavy mesons, like Band Bs. Hidden-charm and bottom mesons with strangeness are also forbidden in the context of the Unquenched Quark Model (UQM) formalism. Indeed, one cannot dress heavy quarkonium Q¯ Qstates with Q¯s−n¯ Qor Q¯n−s¯ Qhigher Fock components (where n=uor d) by creating a light n¯nor s¯spair with vacuum quantum numbers. Tetraquarks with non-null strangeness content can only take place either in the compact tetraquark or hadroquarkonium models. Therefore, a possible way to discriminate between the compact tetraquark and hadro-quarkonium models on one side and the molecular model and UQM interpretations on the other is the experimental search for strange hidden-charm and bottom four-quark states. Our predictions for Pcand Pbpentaquarks with non-null strangeness content may be soon be tested by LHCb. Acknowledgments This work was supported by the U.S. Department of Energy (Grant No. DE-FG-02-91ER40608) and the Academy of Finland, Project No. 320062. – 14 – JHEP04(2020)119 Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. References [1] Belle collaboration, Observation of a narrow charmonium-like state in exclusive B±→K±π+π−J/ψ decays,Phys. Rev. Lett. 91 (2003) 262001 [hep-ex/0309032] [INSPIRE]. 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