-
While hyperons, i.e., baryons with a strangeness content, play an important role in compact star mergers and core-collapse events [1], there are limited experimental data on doubly strange hypernuclear systems, and the systems containing higher strangeness are almost unknown. Different phenomenological models have been developed for the nucleon-hyperon (NY) and hyperon-hyperon (YY) interactions. Nonetheless, recent developments in computational technologies and theoretical progress in Lattice QCD methods facilitated the derivation of
ΩN ,ΩΩ ,ΛΛ , andNΞ interactions [2–4], close to the physical pion massesmπ≃146 MeV and Kaon massesmk≃525 MeV, by the HAL QCD Collaboration [5, 6], where their physical values aremπ≃135 MeV andmk≃497 MeV. TheNΩ and di-Ω interactions were suggested and predicted before the lattice QCD simulation in [7, 8].The potentials are obtained on a large space-time volume
L4=(8.1 fm)4 with a lattice spacinga=0.0846 fm. While there are sophisticated calculations to study6ΛΛHe hypernucleus [9–18], in this work, we examine the HAL QCDΛΛ interactions, which are the most consistent potential with the LHC ALICE data [19, 20], to study the ground state properties of the6ΛΛHe hypernucleus. Similarly, we explore theΩΩα system with lattice QCD-based interactions.The following motivated our exploration for possible implications of the attractive nature of the
ΩN andΩΩ interactions on few-bodyΩΩα systems on the basis of first-principle lattice QCD-based interactions. In few-body systems the presence of additional nucleons may increase the binding, as demonstrated in many straightforward examples in nature. Although there are no dibaryon bound states with strangeness=−1 (Λ -nucleon system), hypertriton3Λ H, consisting of a neutron, a proton, and aΛ -particle, is bound with a separation energy of0.41±0.12 MeV [21, 22]. In the case of strangeness=−2 , in systems containingΞ -particles, an enhancement in the binding energy per baryon is observed by increasing the number of nucleons [23–25]. The Extended-Soft-Core (ESC08c) model of Nijmegen interaction [26] supports the bound states ofΞN andΞNN (T=1/2, Jπ=3/2+) with energies−1.56 and−17.2 MeV, respectively [23, 24, 27]. Recently Garcilazo et al. have implementedΩN andΩΩ interactions derived by the HAL QCD Collaboration [2, 3, 28] to study three-body systems containing more strangeness number, i.e.,ΩNN andΩΩN with strangeness=−3 and−6 [26]. As a result, they obtainedΩd (0, 5/2+) binding energy of about−20 MeV and two resonance statesΩnn (1, 3/2+) andΩΩN (1/2, 1/2+) , with resonance energies of1 and4.6 MeV, correspondingly. Besides theΩ -deuteron bound state, theΩΩα bound state would be an interesting system to benchmark theΩα andΩΩ interactions in a three-body system.As the femtoscopic analysis of two-particle correlation functions in heavy-ion collisions provides information on hadron-hadron interactions at low energies [29, 30], investigating the
6ΩΩHe system can also be interesting for this purpose. The correlation function in multistrange systems such asΛΛ [31, 32] andpΩ [20, 33, 34] have been measured recently in high-energy nuclear collisions. Furthermore, as a next step in femtoscopic analyses, the hadron-deuteron correlation functions would be promising. So far, experimental investigations of correlations for pd, dd and even for light nuclei have been already performed [35–37], whereas theK−d case is currently in progress [38, 39], andΛd correlation function is in the pipeline [40, 41]. A method to probe the momentum correlation functions ofΩΩ is proposed in Ref. [42]. Very recently, the production ofΩNN andΩΩN in ultra-relativistic heavy-ion collisions using the Lattice QCDΩN ,ΩΩ potentials has been studied in Ref. [43]. Since the di-Omega appears with the binding energy approximately1.6 MeV in1S0 channel [2], there is a possibility that our results could help the future study ofΩΩ−α (liked−α [44]) two-particle momentum correlation functions, and can be measured in high energy heavy-ion collisions. We explore this hypothetical system for the first time, and to the best of our knowledge, there is no other study performed on this system thus far.In the present work, we study the ground state properties of
6ΛΛHe and6ΩΩHe hypernuclei as a three-body(α+Y+Y) bound state. For this, we solve two coupled Faddeev equations in configuration space with the hyperspherical harmonics expansion method to calculate the ground state binding energy and the geometrical structures of these hypernuclei. In our study, we use the HAL QCDΛΛ andΩΩ interactions, Isle-type Gaussian potential forαΛ interactions and a Woods-Saxon type potential forαΩ interaction.In 2001, the KEK emulsion/scintillating-fiber hybrid experiment [45], known as the Nagara event, reported a uniquely identified double-
Λ hypernucleus6ΛΛHe with binding energy of−7.25±0.19 MeV. The re-analysis of the Nagara event using the newΞ mass of1321.71 MeV [46], revised the binding energy to−6.91±0.16 MeV [47, 48], considerably shallower than the earlier measured value−10.9±0.8 MeV [49]. In the recent J-PARC E07 experiment emulsion analysis, several hypernuclear events have been observed. For example, theBe doubleΛ hypernucleus has been identified as an event called the “MINO event” [50], and a newΞ -nuclear deeply bound state has been reported [51]. Furthermore, the high precision spectra for light to heavy multi-strange hypernuclei are planned to be measured in the future at JLab and with the new high-intensity high-resolution line at [51, 52].Hiyama et al. performed a three-body calculation for
Λ+Λ+4He , with the Gaussian expansion method, using properly tunedΛΛ Nijmegen interactions to reproduce the Nagara event data [9, 10]. Nemura et al. studiedΛΛ hypernuclei with the stochastic variational method using effective phenomenologicalΛN andΛΛ potentials [11, 12]. Moreover, Filikhin et al. have studied6ΛΛHe by solving the differential Faddeev equations (DFE) in configuration space using different models of NijmegenYY interactions [13–16]. Recently, double-strangeness hypernuclei were studied in an effective field theory approach using the stochastic variational method at leading order [17] and with the Jacobi no-core shell model at the next-to-leading order [18]. The cluster structure of light hypernuclei [53, 54] has been studied with different methods, including the generator coordinate method [55], orthogonality condition model [56], Gaussian expansion method [57, 58], and Tohsaki-Horiuchi-Schuck-Röpke wave function approach [59].The Faddeev-Yakubovsky (FY) equations are extensively used to study the structure of three- and four-body bound states, with identical and non-identical particles, in different sectors of physics [60–65]. FY equations are solved with different techniques such as direct projection in momentum space [66], hyperspherical harmonics (HH) [67], adiabatic hyperspherical [68], and variational methods [69, 70]. The HH method has been implemented to study the complex structure of
6He and11Li halo nuclei in a three-body picture [67, 71]. In this work, we apply this method to study the ground state properties of double hyperon nuclei in a three-body picture asY+Y+α .In Sec. II, we briefly review the HH formalism for
Y+Y+α three-body bound state. In Sec. III, we introduceYY andαY two-body potentials used to study the structure of double-hyperon nuclei. Our numerical results for the ground state binding energies and geometrical properties of6ΛΛHe and6ΩΩHe hypernuclei are presented and discussed in Sec. IV. A summary and outlook is provided in Sec. V. -
The total wave function
Ψjμ of the three-body system(YYα) for a given total angular momentumj by projectionμ , composed of twoY particles and oneα particle, is given as a sum of three Faddeev componentsψjμi Ψjμ=3∑i=1ψjμi(xi,yi).
(1) Each component
ψjμi is expressed in terms of two Jacobi coordinates(xi,yi) and can be obtained from the solution of coupled Faddeev equations(H0−E)ψjμi+Vjk(ψjμi+ψjμj+ψjμk)=0,
(2) where
H0 is the free Hamiltonian,E is 3B binding energy, andVjk is the two body interaction (both the Coulomb and nuclear interactions) between the corresponding pair. The indexesi,j,k run through(1,2,3) in circular order. To solve the coupled Faddeev equations (2) in configuration space, one needs two different sets of Jacobi coordinates(x1,y1) and(x3,y3) , as shown in Fig. 1, defined byFigure 1. (color online) Two sets of Jacobi coordinates
(Yα,Y) and(YY,α) for a description of theYYα three-body system.xi=√Ajk rjk=√Ajk (rj−rk),yi=√A(jk)i r(jk)i=√A(jk)i(ri−Ajrj+AkrkAj+Ak),
(3) where
ri is the position vector of particlei ,rjk is the relative distance between the pair particles(jk) , andr(jk)i is the distance between the spectator particlei and the center of mass of pair(jk) . The reduced masses areAjk=AjAkAj+Ak andA(jk)i=(Aj+Ak)AiAi+Aj+Ak , whereAi=mim ,m=1 a.m.u., andmi is the mass of particlei in a.m.u.The projection of coupled Faddeev equations onto the Jacobi coordinates
(xi,yi) leads to two-dimensional partial differential equations that can be transformed into two sets of coupled one-dimensional equations using the hyperspherical coordinates(ρi,Ωi) . The hyperradius is defined byρ2i=x2i+y2i ; the angular partΩi≡{θi,ˆxi,ˆyi} denotes a set of hyperspherical angles, with the hyperangleθi=arctan(xi/yi) and other angles associated with the unit vectorsˆxi andˆyi . The projection of Faddeev componentsψY andψα , hereafter shown asψQ , for a given total angular momentumj and its projectionμ onto the spherical coordinates is given by [72, 73],ψjμQ(ρi,Ωi)=ρ−5/2i∑βRjβ(ρi)Yjμβ(Ωi),
(4) Yjμβ(Ωi) is written in terms of hyperspherical harmonicsΥlxilyiKl(Ωi) , which are eigenstates of the hypermomentum harmonics operatorˆK asYjμβ(Ω)={[ΥlxlyKl(Ω)⊗ϕSx]jab⊗κI}jμ,
(5) ΥlxlyKlml(Ω)=φlxlyK(θ)[Ylx(ˆx)⊗Yly(ˆy)]lml,
(6) φlxlyK(θ)=NlxlyK(sinθ)lx(cosθ)lyPlx+1/2,ly+1/2n(cos2θ),
(7) where
Pa,bn is the Jacobi polynomial of ordern=(K−lx−ly)/2 , andNlxlyK is a normalizing coefficient. The parameterβ≡{K,lx,ly,l,Sx,jab} represents a set of quantum numbers of a specific channel coupled toj .K is the hyperangular quantum number,lx andly are the orbital angular momenta of the Jacobi coordinatesx andy ,l=lx+ly is total orbital angular momentum,Sx is the total spin of the pair particles associated with the coordinatex , andjab=l+Sx .I denotes the spin of the third particle, and the total angular momentumj isj=jab+I . In Eq. (5),ϕSx is the spin wave function of two-body subsystem, andκI is the spin function of the third particle. Applying this expansion in the Faddeev equations and performing the hyperangular integration, one obtains a set of coupled differential equations for the wave functionsRjβ(ρ) of Eq. (4) as[−ℏ22m(d2dρ2−(K+3/2)(K+5/2)ρ2)−E]Rjβ(ρ)+∑β′Vjμββ′(ρ)Rjβ′(ρ)=0,
(8) The coupling potentials are the hyperangular integrations of the two-body interaction
Vjμββ′(ρi)=⟨Yjμβ(Ωi)|ˆVij|Yjμβ′(Ωj)⟩ , andˆVij are the two-body potentials between particlesi andj , which will be introduced in Section III.In order to solve these coupled equations, the hyperradial wave functions
Rjβ(ρ) are expanded in a finite basis set ofimax hyperradial excitations asRjβ(ρ)=imax∑i=0CjiβRiβ(ρ),
(9) where the coefficients
Cjiβ can be obtained by diagonalizing the three-body Hamiltonian fori=0,...,imax basis functions. The hyperradial functionsRiβ(ρ) can be written in terms of Laguerre polynomials.By having the three-body wave function
Ψjμ in the hyperspherical coordinates, one can study the geometrical structure ofYYα systems by calculating the matter radiusrmat=√⟨r2⟩ and the correlation densities, giving the probability to have definite distances between the particles in the three-body system:P(rjk,r(jk)i)=r2jkr2(jk)i2j+1∑μ∫dˆxidˆyi|ψjμ(xi,yi)|2.
(10) -
In this section, we present the two-body interactions, which we use in our calculations for the bound state of
ΛΛα andΩΩα three-body systems. -
For
ΛΛ interaction, we use HAL QCD potentials in1S0 channel with isospinT=0 [4]VΛΛ(r)=2∑i=1αiexp(−r2/β2i)+λ2(1−exp(−r2/ρ22))2(exp(−mπr)r)2.
(11) The effective
ΛΛ interaction is handled by the coupled-channel formalism [74] at three imaginary-time distances oft/a=11,12,13 , where the potential parameters are fitted toχ2/d.o.f=1.30(40), 0.76(18) , and0.74(30) , respectively. Thet -dependence is insignificant within the statistical errors. The fitted potential parameters to Eq. (11) are given in Table 1, where the pion mass ismπ=146 MeV.t/a α1/MeV β1/fm α2/MeV β2/fm λ2/MeV⋅fm2 ρ2/fm 11 1466.4 0.160 407.1 0.366 −170.3 0.918 12 1486.7 0.156 418.2 0.367 −160.0 0.929 13 1338.0 0.143 560.7 0.322 −176.2 1.033 Table 1. Fitted parameters of
VΛΛ(r) potential, shown in Eq. (11), taken from Ref. [4]. The statistical errors in fitted parameters are not taken into account in our calculations.The low-energy data derived from this interaction indicate no bound or resonant di-hyperon around the
ΛΛ threshold in (2+1)-flavor QCD at nearly physical quark masses. These data predict a scattering lengtha(ΛΛ)0=−0.81±0.23+0.00−0.13 fm and an effective ranger(ΛΛ)eff=5.47±0.78+0.09−0.55 fm. The central values and the statistical errors are extracted from phase shifts att/a=12 , whereas the systematic errors are estimated from the central values att/a=11 and13 [4]. The systematic errors are estimated by the difference between the results obtained by the fit range, and the statistical errors are estimated by the jackknife sampling of the lattice QCD configurations. The source of systematic error is the contamination from inelastic states.For the
Λα interaction we use the Isle-type Gaussian potential [14]VΛα(r)=450.4exp(−(r/1.25)2)−404.9exp(−(r/1.41)2).
(12) This potential reproduces the experimental data for the lifetime and binding energy of the
5ΛHe hypernucleus withτ=3.02+0.10−0.09×10−10 s andEB=−3.1 MeV [75]. -
The HAL QCD
ΩΩ potential in the1S0 channel is fitted to an analytical function as [2]VΩΩ(r)=3∑j=1cje−(r/dj)2,
(13) where the potential parameters, without considering the statistical errors, are
(c1, c2, c3)=(914, 305, −112) MeV and(d1, d2, d3)=(0.143, 0.305, 0.949) fm. Using a single-folding potential method, anΩα interaction has been recently extracted from a separable HAL QCDΩN potential. This potential supports anΩα bound state with a binding energy of approximately−22 MeV and is parameterized in the form of the Woods-Saxon type potential [76]VΩα(r)=−61(1+exp(r−1.70.47))−1.
(14) All two-body interactions for
YY (ΛΛ andΩΩ ) and alsoYα (Λα andΩα ) subsystems are shown in Fig. 2.Figure 2. (color online) Left panel: YY potentials for three models of the HAL QCD
ΛΛ potential of Eq. (11) with the parameters given in Table 1 at the imaginary-time distancest/a=11,12,13 (shown as model i, ii, iii) and the HAL QCDΩΩ potential of Eq. (13). Right panel:Yα potentials for the Isle-typeΛα potential given in Eq. (12) and the Woods-Saxon typeΩα potential given in Eq. (14).In our calculations, we consider the Coulomb interaction in the
ΩΩα system using a hard-sphere model as [77]VCoul(r)=Z2e2×{1rCoul(32−r22r2Coul),r≤rCoul1r,r>rCoul
(15) with a Coulomb radius
rCoul=1.47 fm. -
To calculate the ground state binding energy and the geometrical properties of
ΛΛα andΩΩα , we solve the coupled Faddeev equations (2) by implementing the FaCE computational toolkit [78] using the two-body interactions discussed in Sec. III. To discretize the continuous hyperradius coordinateρi , we use the Gauss-Laguerre quadrature with 100 grid points, while the hyperangular integrations are performed using the Gauss-Jacobi quadrature with 60 grid points. The hyperradius cutoffs are selected high enough to achieve the cutoff-independent binding energies, converging with four significant figures. In our calculations, the masses of particles aremN = 939 MeV,mΛ = 1127.42 MeV,mΩ=1672.45 , andmα = 3727.38 MeV. Table 2 shows the convergence of the three-body ground state binding energyE3 and nuclear matter radiusrmat as a function of the maximum values of hyperangular quantum numberKmax and hyperradial excitationsimax . The spin and isospin ofΛΛα andΩΩα systems are equal to zero. The number of strange quark content forΛΛα andΩΩα is equal to 2 and 6, respectively. The Coulomb interaction of Eq. (15) is considered inΩΩα systems leading to an increase of approximately6 MeV in 3B binding energy.ΛΛα ΩΩα imax E3 /MeVrmat /fmE3 /MeVrmat /fm5 −7.442 1.930 −67.19 1.327 10 −7.467 1.954 −67.21 1.326 15 −7.468 1.955 −67.21 1.326 20 −7.468 1.955 −67.21 1.326 25 −7.468 1.955 −67.21 1.326 Kmax E3 /MeVrmat /fmE3 /MeVrmat /fm5 −6.897 1.953 −66.58 1.327 10 −7.321 1.948 −67.04 1.326 15 −7.404 1.951 −67.14 1.326 20 −7.446 1.953 −67.19 1.326 25 −7.456 1.954 −67.20 1.326 30 −7.463 1.955 −67.21 1.326 35 −7.465 1.955 −67.21 1.326 40 −7.466 1.955 −67.21 1.326 45 −7.467 1.955 −67.21 1.326 50 −7.467 1.955 −67.21 1.326 55 −7.467 1.955 −67.21 1.326 60 −7.468 1.955 −67.21 1.326 65 −7.468 1.955 −67.21 1.326 70 −7.468 1.955 −67.21 1.326 75 −7.468 1.955 −67.21 1.326 80 −7.468 1.955 −67.21 1.326 Table 2. Convergence of three-body ground state binding energy
E3 and nuclear matter radiusrmat calculated forΛΛα andΩΩα systems as a function of maximum hyperradial excitationsimax (withKmax=80 ) and hyperangular quantum numberKmax (withimax=25 ).Since
Ωα interaction is completely attractive, faster convergence is reached in the calculation of ground state binding energy and nuclear matter radius ofΩΩα to−67.21 MeV and1.326 fm. For theΛΛα system, a convergence can be reached at larger values ofKmax andimax . ForΛΛα , the ground state binding energy and nuclear matter radius converge to−7.468 MeV and1.955 fm. The listed results forΛΛα are obtained for theΛΛ interaction with imaginary-time distancet/a=12 , whereas fort/a=11 andt/a=13 , the calculated ground state binding energies are−7.605 MeV and−7.804 MeV, respectively.In Table 3, in addition to the converged 3B binding energies for
ΛΛα andΩΩα systems, we list the binding energy of two-bodyYY andYα subsystems. Furthermore, the 3B binding energies in which the 2B interactions between identical hyperons are set to zero, i.e.,VYY=0 , are listed. Our numerical results show that the relative percentage difference(B3(VYY=0)−2B2(VYα))/B3(VYY=0)×100 varies between4 to 6%. As one can see in Table 3, theΩΩα has a deeper bound state by having two bound subsystems. Our numerical analysis shows that the uncertainties in the HAL QCDΩΩ (ΛΛ ) potential parameters impact the6ΩΩHe (6ΛΛHe ) ground state binding energy for approximately4 (0.1 ) MeV and thermat for less than0.01 (0.007 ) fm. While the employedΩα potential in our calculations is derived based on the dominant5S2 channel ofNΩ interactions [3, 79], the contribution of the3S1 channel can be reasonably ignored. Although the implemented two-body interactions in our calculations are restricted to only one angular momentum channel, we should point out that, to the best of our knowledge, no lattice two-body interactions developed to higher channels. This restriction in the interactions should explain the deep binding of theΩ particles to theα as possible contributions of repulsive channels, which are not taken into account, even though their contributions appear to be small [79].YYα SystemE2(YY) E2(Yα) E3 E3 (VYY=0 )ΛΛα(t/a=12) Not Bound −3.146 −7.468 −6.463 ΩΩα −1.408 −22.01 −67.21 −48.96 Table 3. Three-body ground state binding energies
E3 in MeV forYYα systems. The last column shows our results for 3B binding energies with zero interaction inYY subsystems. Two-body binding energiesE2 forYY andYα subsystems are also shown in MeV.In Table 4, we compare our numerical results with other theoretical results such as those of the Gaussian expansion method (GEM), stochastic variational method (SVM), differential Faddeev equations (DFE), quark-cluster-model (QCM) by different
YY interaction models like the Nijmegen model D (ND), simulating Nijmegen hard-core model F(NFs) , modified simulating Nijmegen hard-core model D(mNDs) , simulating Nijmegen hard-core model D(NDs) , spin-flavorSU6 quark-model (fss2), Nijmegen soft-core model (NSC97e), G-matrix interaction based on the bare ND interaction (ND(G)), and Nijmegen extended soft-Core (ESC00) model as well as the experimental data (Exp).Ref. YY ModelE3 /MeVpresent HAL QCD (t/a=11) −7.605 present HAL QCD (t/a=12) −7.468 present HAL QCD (t/a=13) −7.804 [10] (GEM) ND −7.25 [12] (SVM) NFs −7.52 [12] (SVM) mNDs −7.53 [12] (SVM) NDs −7.93 [80] (DFE) fss2 −7.653 [14] (DFE) NSC97e −6.82 [14] (DFE) ND −9.10 [14] (DFE) ND(G) −10.1 [14] (DFE) ESC00 −10.7 [81] (QCM) ND −9.7 [81] (QCM) ND(G) −9.4 [82] (GEM) ND −9.34 [83](G-matrix) ND(G) −9.23 [47, 48] (Exp.) – −6.91±0.16 [45] (Exp.) – −7.25±0.19 [49] (Exp.) – −10.9±0.8 Table 4. Comparison between our results for three-body ground state binding energy
E3 ofΛΛα system and other theoretical and experimental data.By having 3B wave functions of
YYα systems in terms of the HH basis, we calculate the geometrical quantities, i.e., the r.m.s. distances between the particles and the r.m.s. matter radius, presented in Table 5. In our calculations, we consider theα matter radius of1.47 fm. For comparison, we also present the DFE results in Ref. [14]. Since the studied 3B systems consist of two identical hyperons and one alpha particle, interacting by scalar potentials, three particles form a ground state, where the most probable positions of the particles have the shape of an isosceles triangle. As shown in Table 5, our numerical results for the expectation values of the Jacobi coordinates in(YY,α) and(Yα,Y) configurations satisfy the following Pythagorean theorem with high accuracy(YYα) system⟨rYY⟩ /fm⟨rYα⟩ /fm⟨r(YY)α⟩ /fm|Δ| /fm2 ⟨ρ2⟩1/2 /fmrmat /fmΛΛα(t/a=11) 3.598 (3.36) 2.902 (2.70) 2.276 (2.11) 0.005 (0.015) 3.943 1.944 ΛΛα(t/a=12) 3.629 2.926 2.295 0.002 3.976 1.955 ΛΛα(t/a=13) 3.555 2.867 2.248 0.007 3.895 1.929 ΩΩα 1.521 1.293 1.047 0.003 2.037 1.326 Table 5. Expectation values of Jacobi coordinates in
ΛΛα andΩΩα systems.⟨rYY⟩ is the separation between identical hyperons,⟨rYα⟩ is the separation betweenYα pairs, and⟨r(YY)α⟩ is the separation between the center of mass ofYY pair and the spectatorα particle.⟨ρ2⟩1/2 is the r.m.s. matter radius of the three-body system containing only point particles, andrmat is the r.m.s. matter radius. The numbers in parentheses are from the DFE calculations using the Nijmegen model D (ND)YY interaction [14].Δ shows the accuracy of satisfaction of the Pythagorean theorem in Eq. (16).Δ=⟨r(YY)α⟩2+14⟨rYY⟩2−⟨rYα⟩2=0.
(16) In Table 6, we present the contribution of different channels, indicated by the quantum numbers
{K,lxi,lyi,l,Sxi,jab} , to the total norm of 3B ground state wave functions ofΛΛα andΩΩα systems in both(Yα−Y) and(YY−α) Jacobi coordinates. As one can see, the main contributions in 3B wave functions come from thes -wave channels, whereas the higher partial wave channels substantially have an insignificant contribution.K lxi lyi l Sxi jab W (Λα−Λ) Jacobi0 0 0 0 0.0 0.0 0.980 2 0 0 0 0.0 0.0 0.004 4 2 2 0 0.0 0.0 0.012 (ΛΛ−α) Jacobi0 0 0 0 0.5 0.5 0.980 2 1 1 0 0.5 0.5 0.004 4 1 1 0 0.5 0.5 0.001 4 0 0 0 0.5 0.5 0.010 (Ωα−Ω) Jacobi0 0 0 0 0.0 0.0 0.993 2 0 0 0 0.0 0.0 0.005 (ΩΩ−α) Jacobi0 0 0 0 1.5 1.5 0.993 2 1 1 0 1.5 1.5 0.005 Table 6. Contributions of different partial wave channels
W to the total norm of 3B ground state wave functions ofΛΛα andΩΩα systems. For each system, the upper panel shows the contributions in(Yα−Y) Jacobi coordinates, and the lower panel shows the contributions in(YY−α) Jacobi coordinates. Channels with a contribution greater than 0.001% are listed.In Fig. 3, we show the first three dominant hyperradial components
ρ5/2Rβ(ρ) for the ground state wave functions ofΛΛα andΩΩα systems, obtained byimax=25 . As the binding energy of the 3B system increases from the left to right panel, the system becomes more compact in the configuration space. In Fig. 4, we illustrate the probability density of 3B ground states ofΛΛα andΩΩα systems as a function of⟨rYY⟩ , the distance between the two identicalY hyperons, andr(YY)α , the distance between theα particle and the center of mass of twoY particles. As we can see for both systems, the distributions along the⟨rYY⟩ direction are broader than those along ther(YY)α direction,Figure 3. (color online) Hyperradial wave function
ρ5/2Rβ(ρ) for the first three dominant channels,β≡ {0, 0, 0, 0, 0, 0} (green dash-dotted line),β≡ {2, 0, 0, 0, 0, 0} (blue dashed line), andβ≡ {4, 0, 0, 0, 0, 0} (red solid line), of the ground state wave functions ofΛΛα andΩΩα systems.Figure 4. (color online) 3B ground state probability density of
ΛΛα andΩΩα systems as a function of⟨rYY⟩ , the distance betweenYY pair, andr(YY)α , the distance betweenα particle and the center of mass of theYY pair.confirming that the distance between identical
Y hyperons is greater than the distance between the spectatorα particle and the center of mass of theYY pair. -
In this study, we invetigated the ground-state properties of multi-strangeness hypernuclei
6ΛΛHe and6ΩΩHe in a hyperharmonic three-body model of(YYα) . For this, we solved two coupled Faddeev equations in configuration space with the hyperspherical harmonics expansion method using the most modern two-body interactions, including the recent lattice QCD potentials, to calculate the ground state binding energies and geometrical properties. In our numerical analysis, we checked the convergence of 3B ground state binding energies and nuclear matter radii as a function of the maximum value of hyperradial excitationsimax and hyperangular quantum numberKmax . Our numerical results show that the ground state binding energy of6ΛΛHe using three models ofΛΛ lattice interactions changes in the domain of(−7.468, −7.804) MeV, while6ΩΩHe has a deep binding energy of-67.21 MeV. We should indicate that the implemented\Omega \alpha potential in our calculations is restricted to one angular momentum channel only, whereas the contribution of repulsive channels is not considered. Thus, this could explain the deep binding of the\Omega \alpha and consequently the{}_{\Omega\Omega }^{\;\;\;6}{{\rm{He}}} system.We studied the geometrical properties of the aforementioned
^6 He double hyperon by calculating the expectation values of the Jacobi coordinates and the r.m.s. matter radius and correlation density. Our numerical results confirm that the studied 3B systems, composed of two identical hyperons and one alpha particle, form isosceles triangles, where the most probable positions of the particles perfectly satisfy the Pythagorean theorem. Our numerical analysis on the structure of 3B ground state wave functions shows that the main contributions of over 99% are from thes -wave channels. Our numerical results for{}_{\Lambda\Lambda }^{\;\;\;6}{{\rm{He}}} are in agreement with the results of other theoretical studies.Considering the contributions of the coupled channels in
\Lambda \Lambda- \Xi N (\Omega N-\Lambda\Xi - \Sigma \Xi ) interactions is a complementary task to be implemented in the FaCE toolkit to include the coupled components in the wave function of{}_{\Lambda\Lambda }^{\;\;\;6}{{\rm{He}}} ({}_{\Omega\Omega }^{\;\;\;6}{{\rm{He}}} ). As shown in Refs. [81, 84], the contribution of the coupled channels leads to an increase of approximately0.1 \sim 0.4 MeV in the\Lambda\Lambda\alpha binding energy, while using an effective single-channel interaction leads to a reduction of approximately0.3 MeV [14]. This reduction is due to the tight\alpha cluster binding, which inhibits the effectiveness of the\Lambda \Lambda-\Xi N coupling. We assume that this should also be valid for the\Omega \Omega \alpha system. Moreover, while the contribution of transition potentials to the inelastic channels [85], i.e.,\Omega N-\Lambda\Xi - \Sigma \Xi , are expected to be small [79], they have not yet been derived from the lattice QCD calculations and can be considered in a future study when they are developed. -
F.E. thanks T. Hatsuda and J. Casal for helpful discussions and suggestions.
