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Three-body Faddeev calculations for 6ΛΛHe and 6ΩΩHe hypernuclei

  • We study the ground-state properties of the6YYHe double hyperon for 6ΛΛHe and 6ΩΩHenuclei in a three-body model (Y+Y+α). We solve two coupled Faddeev equations corresponding to the three-body configurations (αY,Y) and (YY,α) in configuration space with the hyperspherical harmonics expansion method by employing the most recent hyperon-hyperon interactions obtained from lattice QCD simulations. Our numerical analysis for 6ΛΛHe, using three ΛΛ lattice interaction models, leads to a ground state binding energy in the (7.468,7.804)MeV domain and the separations rΛΛand rαΛ in the domains of(3.555,3.629) fm and (2.867,2.902) fm, respectively. The binding energy of the double-Ω hypernucleus 6ΩΩHe leads to 67.21 MeV and consequently to smaller separations rΩΩ=1.521 fm and rαΩ=1.293 fm. In addition to geometrical properties, we study the structure of ground-state wave functions and show that the main contributions are from the swave channels. Our results are consistent with the existing theoretical and experimental data.
  • 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, and NΞ interactions [24], close to the physical pion masses mπ146 MeV and Kaon masses mk525 MeV, by the HAL QCD Collaboration [5, 6], where their physical values are mπ135 MeV and mk497 MeV. The NΩ 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 spacing a=0.0846 fm. While there are sophisticated calculations to study 6ΛΛHehypernucleus [918], in this work, we examine the HAL QCD ΛΛ interactions, which are the most consistent potential with the LHC ALICE data [19, 20], to study the ground state properties of the 6ΛΛHe hypernucleus. Similarly, we explore the ΩΩα system with lattice QCD-based interactions.

    The following motivated our exploration for possible implications of the attractive nature of the ΩN and ΩΩ interactions on few-body ΩΩα 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), hypertriton 3ΛH, consisting of a neutron, a proton, and a Λ-particle, is bound with a separation energy of 0.41±0.12 MeV [21, 22]. In the case of strangeness =2, in systems containing Ξ-particles, an enhancement in the binding energy per baryon is observed by increasing the number of nucleons [2325]. The Extended-Soft-Core (ESC08c) model of Nijmegen interaction [26] supports the bound states of ΞNand ΞNN (T=1/2, Jπ=3/2+) with energies 1.56 and 17.2 MeV, respectively [23, 24, 27]. Recently Garcilazo et al. have implemented ΩN and ΩΩ interactions derived by the HAL QCD Collaboration [2, 3, 28] to study three-body systems containing more strangeness number, i.e., ΩNN and ΩΩN with strangeness =3and 6 [26]. As a result, they obtained Ωd (0, 5/2+) binding energy of about 20 MeV and two resonance states Ωnn (1, 3/2+)and ΩΩN (1/2, 1/2+), with resonance energies of 1 and 4.6 MeV, correspondingly. Besides the Ω-deuteron bound state, theΩΩα bound state would be an interesting system to benchmark the Ωα and ΩΩ 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 as ΛΛ [31, 32] and pΩ [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 [3537], whereas the Kd case is currently in progress [38, 39], and Λd correlation function is in the pipeline [40, 41]. A method to probe the momentum correlation functions of ΩΩ is proposed in Ref. [42]. Very recently, the production of ΩNN and ΩΩN in ultra-relativistic heavy-ion collisions using the Lattice QCD ΩN, ΩΩ potentials has been studied in Ref. [43]. Since the di-Omega appears with the binding energy approximately 1.6 MeV in 1S0 channel [2], there is a possibility that our results could help the future study of ΩΩα (like dα[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 and 6ΩΩHehypernuclei 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 QCD ΛΛ and ΩΩ interactions, Isle-type Gaussian potential for αΛ interactions and a Woods-Saxon type potential for αΩ interaction.

    In 2001, the KEK emulsion/scintillating-fiber hybrid experiment [45], known as the Nagara event, reported a uniquely identified double-Λ hypernucleus 6ΛΛHe with binding energy of 7.25±0.19 MeV. The re-analysis of the Nagara event using the new Ξmass of 1321.71 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, the Be double Λ hypernucleus has been identified as an event called the “MINO event” [50], and a new Ξ-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 tuned ΛΛ Nijmegen interactions to reproduce the Nagara event data [9, 10]. Nemura et al. studied ΛΛ hypernuclei with the stochastic variational method using effective phenomenological ΛN and ΛΛpotentials [11, 12]. Moreover, Filikhin et al. have studied 6ΛΛHe by solving the differential Faddeev equations (DFE) in configuration space using different models of Nijmegen YY interactions [1316]. 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 [6065]. 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 and 11Li 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 as Y+Y+α.

    In Sec. II, we briefly review the HH formalism for Y+Y+α three-body bound state. In Sec. III, we introduce YY and αY two-body potentials used to study the structure of double-hyperon nuclei. Our numerical results for the ground state binding energies and geometrical properties of 6ΛΛHe and 6ΩΩHe 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 momentum j by projection μ, composed of two Y particles and one α particle, is given as a sum of three Faddeev components ψjμi

    Ψjμ=3i=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

    (H0E)ψjμi+Vjk(ψjμi+ψjμj+ψjμk)=0,

    (2)

    where H0 is the free Hamiltonian, E is 3B binding energy, and Vjk is the two body interaction (both the Coulomb and nuclear interactions) between the corresponding pair. The indexes i,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 by

    Figure 1

    Figure 1.  (color online) Two sets of Jacobi coordinates (Yα,Y) and (YY,α) for a description of the YYα three-body system.

    xi=Ajk rjk=Ajk (rjrk),yi=A(jk)i r(jk)i=A(jk)i(riAjrj+AkrkAj+Ak),

    (3)

    where ri is the position vector of particle i, rjk is the relative distance between the pair particles (jk), and r(jk)i is the distance between the spectator particle i and the center of mass of pair (jk). The reduced masses are Ajk=AjAkAj+Ak and A(jk)i=(Aj+Ak)AiAi+Aj+Ak, where Ai=mim, m=1 a.m.u., and mi is the mass of particle i 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 by ρ2i=x2i+y2i; the angular part Ωi{θi,ˆxi,ˆyi} denotes a set of hyperspherical angles, with the hyperangle θi=arctan(xi/yi)and other angles associated with the unit vectors ˆxi and ˆyi. The projection of Faddeev components ψY and ψα, hereafter shown as ψQ, for a given total angular momentum jand its projection μ onto the spherical coordinates is given by [72, 73],

    ψjμQ(ρi,Ωi)=ρ5/2iβRjβ(ρi)Yjμβ(Ωi),

    (4)

    Yjμβ(Ωi) is written in terms of hyperspherical harmonics ΥlxilyiKl(Ωi), which are eigenstates of the hypermomentum harmonics operator ˆK as

    Yjμβ(Ω)={[ΥlxlyKl(Ω)ϕSx]jabκI}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 order n=(Klxly)/2, and NlxlyK is a normalizing coefficient. The parameter β{K,lx,ly,l,Sx,jab} represents a set of quantum numbers of a specific channel coupled to j. K is the hyperangular quantum number, lx and ly are the orbital angular momenta of the Jacobi coordinates x and y, l=lx+ly is total orbital angular momentum, Sx is the total spin of the pair particles associated with the coordinate x, and jab=l+Sx. I denotes the spin of the third particle, and the total angular momentum j is j=jab+I. In Eq. (5), ϕSx is the spin wave function of two-body subsystem, and κI 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 functions Rjβ(ρ) of Eq. (4) as

    [22m(d2dρ2(K+3/2)(K+5/2)ρ2)E]Rjβ(ρ)+βVjμββ(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 particles i and j, 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 of imax hyperradial excitations as

    Rjβ(ρ)=imaxi=0CjiβRiβ(ρ),

    (9)

    where the coefficients Cjiβ can be obtained by diagonalizing the three-body Hamiltonian for i=0,...,imax basis functions. The hyperradial functions Riβ(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 of YYα systems by calculating the matter radius rmat=r2and 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+1μdˆxidˆyi|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 and ΩΩα three-body systems.

    For ΛΛ interaction, we use HAL QCD potentials in 1S0 channel with isospin T=0 [4]

    VΛΛ(r)=2i=1αiexp(r2/β2i)+λ2(1exp(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 of t/a=11,12,13, where the potential parameters are fitted to χ2/d.o.f=1.30(40), 0.76(18), and 0.74(30), respectively. The t-dependence is insignificant within the statistical errors. The fitted potential parameters to Eq. (11) are given in Table 1, where the pion mass is mπ=146 MeV.

    Table 1

    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.
    t/aα1/MeV β1/fm α2/MeV β2/fm λ2/MeVfm2 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
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    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 length a(ΛΛ)0=0.81±0.23+0.000.13 fm and an effective range r(ΛΛ)eff=5.47±0.78+0.090.55 fm. The central values and the statistical errors are extracted from phase shifts at t/a=12, whereas the systematic errors are estimated from the central values at t/a=11 and 13[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 with τ=3.02+0.100.09×1010 s and EB=3.1 MeV [75].

    The HAL QCD ΩΩ potential in the 1S0 channel is fitted to an analytical function as [2]

    VΩΩ(r)=3j=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, an Ωα interaction has been recently extracted from a separable HAL QCD ΩN potential. This potential supports an Ωαbound 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(r1.70.47))1.

    (14)

    All two-body interactions for YY (ΛΛ and ΩΩ) and also Yα (Λα and Ωα) subsystems are shown in Fig. 2.

    Figure 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 distances t/a=11,12,13 (shown as model i, ii, iii) and the HAL QCD ΩΩ potential of Eq. (13). Right panel:Yα potentials for the Isle-type Λα potential given in Eq. (12) and the Woods-Saxon type Ωα 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(32r22r2Coul),rrCoul1r,r>rCoul

    (15)

    with a Coulomb radius rCoul=1.47 fm.

    To calculate the ground state binding energy and the geometrical properties of ΛΛαand ΩΩα, 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 coordinate ρi, 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 are mN= 939 MeV, mΛ= 1127.42 MeV, mΩ=1672.45, and mα= 3727.38 MeV. Table 2 shows the convergence of the three-body ground state binding energy E3 and nuclear matter radius rmat as a function of the maximum values of hyperangular quantum number Kmax and hyperradial excitations imax. The spin and isospin of ΛΛα and ΩΩα systems are equal to zero. The number of strange quark content for ΛΛα and ΩΩα is equal to 2 and 6, respectively. The Coulomb interaction of Eq. (15) is considered in ΩΩα systems leading to an increase of approximately 6 MeV in 3B binding energy.

    Table 2

    Table 2.  Convergence of three-body ground state binding energy E3 and nuclear matter radius rmat calculated for ΛΛα and ΩΩα systems as a function of maximum hyperradial excitations imax (with Kmax=80) and hyperangular quantum number Kmax (with imax=25).
    ΛΛα ΩΩα
    imax E3/MeV rmat/fm E3/MeV rmat/fm
    5 −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/MeV rmat/fm E3/MeV rmat/fm
    5 −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
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    Since Ωα interaction is completely attractive, faster convergence is reached in the calculation of ground state binding energy and nuclear matter radius of ΩΩα to 67.21 MeV and 1.326 fm. For the ΛΛα system, a convergence can be reached at larger values of Kmax and imax. For ΛΛα, the ground state binding energy and nuclear matter radius converge to 7.468 MeV and 1.955 fm. The listed results for ΛΛα are obtained for the ΛΛ interaction with imaginary-time distance t/a=12, whereas for t/a=11 and t/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 and ΩΩα systems, we list the binding energy of two-body YY and Yα 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 between 4 to 6%. As one can see in Table 3, the ΩΩα has a deeper bound state by having two bound subsystems. Our numerical analysis shows that the uncertainties in the HAL QCD ΩΩ (ΛΛ) potential parameters impact the 6ΩΩHe (6ΛΛHe) ground state binding energy for approximately4 (0.1) MeV and the rmat for less than 0.01 (0.007) fm. While the employed Ωα potential in our calculations is derived based on the dominant 5S2 channel of NΩ interactions [3, 79], the contribution of the 3S1 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 the Ω particles to the α as possible contributions of repulsive channels, which are not taken into account, even though their contributions appear to be small [79].

    Table 3

    Table 3.  Three-body ground state binding energies E3 in MeV for YYα systems. The last column shows our results for 3B binding energies with zero interaction in YY subsystems. Two-body binding energies E2 for YY and Yα subsystems are also shown in MeV.
    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
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    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-flavor SU6 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).

    Table 4

    Table 4.  Comparison between our results for three-body ground state binding energy E3 of ΛΛα system and other theoretical and experimental data.
    Ref. YY Model E3/MeV
    present 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
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    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 the α matter radius of 1.47 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

    Table 5

    Table 5.  Expectation values of Jacobi coordinates in ΛΛα and ΩΩα systems. rYY is the separation between identical hyperons, rYα is the separation between Yα pairs, and r(YY)α is the separation between the center of mass of YY pair and the spectator α particle. ρ21/2 is the r.m.s. matter radius of the three-body system containing only point particles, and rmat 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).
    (YYα) systemrYY/fm rYα/fm r(YY)α/fm |Δ|/fm2 ρ21/2/fm rmat/fm
    ΛΛα(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
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    Δ=r(YY)α2+14rYY2rYα2=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 of ΛΛα and ΩΩα systems in both (YαY) and (YYα) Jacobi coordinates. As one can see, the main contributions in 3B wave functions come from the s-wave channels, whereas the higher partial wave channels substantially have an insignificant contribution.

    Table 6

    Table 6.  Contributions of different partial wave channels W to the total norm of 3B ground state wave functions of ΛΛα and ΩΩα systems. For each system, the upper panel shows the contributions in (YαY) Jacobi coordinates, and the lower panel shows the contributions in (YYα) Jacobi coordinates. Channels with a contribution greater than 0.001% are listed.
    K lxi lyi l Sxi jab W
    (ΛαΛ) Jacobi
    0 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
    (ΛΛα) Jacobi
    0 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
    (ΩαΩ) Jacobi
    0 0 0 0 0.0 0.0 0.993
    2 0 0 0 0.0 0.0 0.005
    (ΩΩα) Jacobi
    0 0 0 0 1.5 1.5 0.993
    2 1 1 0 1.5 1.5 0.005
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    In Fig. 3, we show the first three dominant hyperradial components ρ5/2Rβ(ρ) for the ground state wave functions of ΛΛα and ΩΩα systems, obtained by imax=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 of ΛΛα and ΩΩα systems as a function of rYY, the distance between the two identical Y hyperons, and r(YY)α, the distance between the α particle and the center of mass of two Y particles. As we can see for both systems, the distributions along the rYY direction are broader than those along the r(YY)α direction,

    Figure 3

    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), and β {4, 0, 0, 0, 0, 0} (red solid line), of the ground state wave functions of ΛΛα and ΩΩα systems.

    Figure 4

    Figure 4.  (color online) 3B ground state probability density of ΛΛα and ΩΩα systems as a function of rYY, the distance between YY pair, and r(YY)α, the distance between α particle and the center of mass of the YY pair.

    confirming that the distance between identical Y hyperons is greater than the distance between the spectator α particle and the center of mass of the YY pair.

    In this study, we invetigated the ground-state properties of multi-strangeness hypernuclei 6ΛΛHe and 6ΩΩHe 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 excitations imax and hyperangular quantum number Kmax. Our numerical results show that the ground state binding energy of 6ΛΛHe using three models of ΛΛlattice interactions changes in the domain of (7.468, 7.804) MeV, while 6ΩΩHe has a deep binding energy of 67.21 MeV. We should indicate that the implemented Ωαpotential 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 Ωα and consequently the 6ΩΩHe system.

    We studied the geometrical properties of the aforementioned 6He 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 the s-wave channels. Our numerical results for 6ΛΛHe are in agreement with the results of other theoretical studies.

    Considering the contributions of the coupled channels in ΛΛΞN (ΩNΛΞΣΞ) interactions is a complementary task to be implemented in the FaCE toolkit to include the coupled components in the wave function of 6ΛΛHe (6ΩΩHe). As shown in Refs. [81, 84], the contribution of the coupled channels leads to an increase of approximately 0.10.4 MeV in the ΛΛα 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 α cluster binding, which inhibits the effectiveness of the ΛΛΞN coupling. We assume that this should also be valid for the ΩΩα system. Moreover, while the contribution of transition potentials to the inelastic channels [85], i.e., ΩNΛΞΣΞ, 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.

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Faisal Etminan and M. R. Hadizadeh. Three-body Faddeev calculations for 6ΛΛHe and 6ΩΩHe hypernuclei[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac7a22
Faisal Etminan and M. R. Hadizadeh. Three-body Faddeev calculations for 6ΛΛHe and 6ΩΩHe hypernuclei[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac7a22 shu
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Three-body Faddeev calculations for 6ΛΛHe and 6ΩΩHe hypernuclei

  • 1. Department of Physics, Faculty of Sciences, University of Birjand, Birjand 97175-615, Iran
  • 2. College of Engineering, Science, Technology and Agriculture, Central State University, Wilberforce, OH, 45384, USA
  • 3. Department of Physics and Astronomy, Ohio University, Athens, OH, 45701, USA

Abstract: We study the ground-state properties of the6YYHe double hyperon for 6ΛΛHe and 6ΩΩHenuclei in a three-body model (Y+Y+α). We solve two coupled Faddeev equations corresponding to the three-body configurations (αY,Y) and (YY,α) in configuration space with the hyperspherical harmonics expansion method by employing the most recent hyperon-hyperon interactions obtained from lattice QCD simulations. Our numerical analysis for 6ΛΛHe, using three ΛΛ lattice interaction models, leads to a ground state binding energy in the (7.468,7.804)MeV domain and the separations rΛΛand rαΛ in the domains of(3.555,3.629) fm and (2.867,2.902) fm, respectively. The binding energy of the double-Ω hypernucleus 6ΩΩHe leads to 67.21 MeV and consequently to smaller separations rΩΩ=1.521 fm and rαΩ=1.293 fm. In addition to geometrical properties, we study the structure of ground-state wave functions and show that the main contributions are from the swave channels. Our results are consistent with the existing theoretical and experimental data.

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    I.   INTRODUCTION
    • 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, and NΞ interactions [24], close to the physical pion masses mπ146 MeV and Kaon masses mk525 MeV, by the HAL QCD Collaboration [5, 6], where their physical values are mπ135 MeV and mk497 MeV. The NΩ 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 spacing a=0.0846 fm. While there are sophisticated calculations to study 6ΛΛHehypernucleus [918], in this work, we examine the HAL QCD ΛΛ interactions, which are the most consistent potential with the LHC ALICE data [19, 20], to study the ground state properties of the 6ΛΛHe hypernucleus. Similarly, we explore the ΩΩα system with lattice QCD-based interactions.

      The following motivated our exploration for possible implications of the attractive nature of the ΩN and ΩΩ interactions on few-body ΩΩα 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), hypertriton 3ΛH, consisting of a neutron, a proton, and a Λ-particle, is bound with a separation energy of 0.41±0.12 MeV [21, 22]. In the case of strangeness =2, in systems containing Ξ-particles, an enhancement in the binding energy per baryon is observed by increasing the number of nucleons [2325]. The Extended-Soft-Core (ESC08c) model of Nijmegen interaction [26] supports the bound states of ΞNand ΞNN (T=1/2, Jπ=3/2+) with energies 1.56 and 17.2 MeV, respectively [23, 24, 27]. Recently Garcilazo et al. have implemented ΩN and ΩΩ interactions derived by the HAL QCD Collaboration [2, 3, 28] to study three-body systems containing more strangeness number, i.e., ΩNN and ΩΩN with strangeness =3and 6 [26]. As a result, they obtained Ωd (0, 5/2+) binding energy of about 20 MeV and two resonance states Ωnn (1, 3/2+)and ΩΩN (1/2, 1/2+), with resonance energies of 1 and 4.6 MeV, correspondingly. Besides the Ω-deuteron bound state, theΩΩα bound state would be an interesting system to benchmark the Ωα and ΩΩ 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 as ΛΛ [31, 32] and pΩ [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 [3537], whereas the Kd case is currently in progress [38, 39], and Λd correlation function is in the pipeline [40, 41]. A method to probe the momentum correlation functions of ΩΩ is proposed in Ref. [42]. Very recently, the production of ΩNN and ΩΩN in ultra-relativistic heavy-ion collisions using the Lattice QCD ΩN, ΩΩ potentials has been studied in Ref. [43]. Since the di-Omega appears with the binding energy approximately 1.6 MeV in 1S0 channel [2], there is a possibility that our results could help the future study of ΩΩα (like dα[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 and 6ΩΩHehypernuclei 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 QCD ΛΛ and ΩΩ interactions, Isle-type Gaussian potential for αΛ interactions and a Woods-Saxon type potential for αΩ interaction.

      In 2001, the KEK emulsion/scintillating-fiber hybrid experiment [45], known as the Nagara event, reported a uniquely identified double-Λ hypernucleus 6ΛΛHe with binding energy of 7.25±0.19 MeV. The re-analysis of the Nagara event using the new Ξmass of 1321.71 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, the Be double Λ hypernucleus has been identified as an event called the “MINO event” [50], and a new Ξ-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 tuned ΛΛ Nijmegen interactions to reproduce the Nagara event data [9, 10]. Nemura et al. studied ΛΛ hypernuclei with the stochastic variational method using effective phenomenological ΛN and ΛΛpotentials [11, 12]. Moreover, Filikhin et al. have studied 6ΛΛHe by solving the differential Faddeev equations (DFE) in configuration space using different models of Nijmegen YY interactions [1316]. 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 [6065]. 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 and 11Li 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 as Y+Y+α.

      In Sec. II, we briefly review the HH formalism for Y+Y+α three-body bound state. In Sec. III, we introduce YY and αY two-body potentials used to study the structure of double-hyperon nuclei. Our numerical results for the ground state binding energies and geometrical properties of 6ΛΛHe and 6ΩΩHe hypernuclei are presented and discussed in Sec. IV. A summary and outlook is provided in Sec. V.

    II.   THREE-BODY FADDEEV EQUATIONS IN HYPERSPHERICAL COORDINATES
    • The total wave function Ψjμ of the three-body system (YYα) for a given total angular momentum j by projection μ, composed of two Y particles and one α particle, is given as a sum of three Faddeev components ψjμi

      Ψjμ=3i=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

      (H0E)ψjμi+Vjk(ψjμi+ψjμj+ψjμk)=0,

      (2)

      where H0 is the free Hamiltonian, E is 3B binding energy, and Vjk is the two body interaction (both the Coulomb and nuclear interactions) between the corresponding pair. The indexes i,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 by

      Figure 1.  (color online) Two sets of Jacobi coordinates (Yα,Y) and (YY,α) for a description of the YYα three-body system.

      xi=Ajk rjk=Ajk (rjrk),yi=A(jk)i r(jk)i=A(jk)i(riAjrj+AkrkAj+Ak),

      (3)

      where ri is the position vector of particle i, rjk is the relative distance between the pair particles (jk), and r(jk)i is the distance between the spectator particle i and the center of mass of pair (jk). The reduced masses are Ajk=AjAkAj+Ak and A(jk)i=(Aj+Ak)AiAi+Aj+Ak, where Ai=mim, m=1 a.m.u., and mi is the mass of particle i 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 by ρ2i=x2i+y2i; the angular part Ωi{θi,ˆxi,ˆyi} denotes a set of hyperspherical angles, with the hyperangle θi=arctan(xi/yi)and other angles associated with the unit vectors ˆxi and ˆyi. The projection of Faddeev components ψY and ψα, hereafter shown as ψQ, for a given total angular momentum jand its projection μ onto the spherical coordinates is given by [72, 73],

      ψjμQ(ρi,Ωi)=ρ5/2iβRjβ(ρi)Yjμβ(Ωi),

      (4)

      Yjμβ(Ωi) is written in terms of hyperspherical harmonics ΥlxilyiKl(Ωi), which are eigenstates of the hypermomentum harmonics operator ˆK as

      Yjμβ(Ω)={[ΥlxlyKl(Ω)ϕSx]jabκI}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 order n=(Klxly)/2, and NlxlyK is a normalizing coefficient. The parameter β{K,lx,ly,l,Sx,jab} represents a set of quantum numbers of a specific channel coupled to j. K is the hyperangular quantum number, lx and ly are the orbital angular momenta of the Jacobi coordinates x and y, l=lx+ly is total orbital angular momentum, Sx is the total spin of the pair particles associated with the coordinate x, and jab=l+Sx. I denotes the spin of the third particle, and the total angular momentum j is j=jab+I. In Eq. (5), ϕSx is the spin wave function of two-body subsystem, and κI 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 functions Rjβ(ρ) of Eq. (4) as

      [22m(d2dρ2(K+3/2)(K+5/2)ρ2)E]Rjβ(ρ)+βVjμββ(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 particles i and j, 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 of imax hyperradial excitations as

      Rjβ(ρ)=imaxi=0CjiβRiβ(ρ),

      (9)

      where the coefficients Cjiβ can be obtained by diagonalizing the three-body Hamiltonian for i=0,...,imax basis functions. The hyperradial functions Riβ(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 of YYα systems by calculating the matter radius rmat=r2and 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+1μdˆxidˆyi|ψjμ(xi,yi)|2.

      (10)
    III.   TWO-BODY POTENTIALS
    • In this section, we present the two-body interactions, which we use in our calculations for the bound state of ΛΛα and ΩΩα three-body systems.

    • A.   ΛΛα system

    • For ΛΛ interaction, we use HAL QCD potentials in 1S0 channel with isospin T=0 [4]

      VΛΛ(r)=2i=1αiexp(r2/β2i)+λ2(1exp(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 of t/a=11,12,13, where the potential parameters are fitted to χ2/d.o.f=1.30(40), 0.76(18), and 0.74(30), respectively. The t-dependence is insignificant within the statistical errors. The fitted potential parameters to Eq. (11) are given in Table 1, where the pion mass is mπ=146 MeV.

      t/aα1/MeV β1/fm α2/MeV β2/fm λ2/MeVfm2 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 length a(ΛΛ)0=0.81±0.23+0.000.13 fm and an effective range r(ΛΛ)eff=5.47±0.78+0.090.55 fm. The central values and the statistical errors are extracted from phase shifts at t/a=12, whereas the systematic errors are estimated from the central values at t/a=11 and 13[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 with τ=3.02+0.100.09×1010 s and EB=3.1 MeV [75].

    • B.   ΩΩα System

    • The HAL QCD ΩΩ potential in the 1S0 channel is fitted to an analytical function as [2]

      VΩΩ(r)=3j=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, an Ωα interaction has been recently extracted from a separable HAL QCD ΩN potential. This potential supports an Ωαbound 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(r1.70.47))1.

      (14)

      All two-body interactions for YY (ΛΛ and ΩΩ) and also Yα (Λα and Ωα) 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 distances t/a=11,12,13 (shown as model i, ii, iii) and the HAL QCD ΩΩ potential of Eq. (13). Right panel:Yα potentials for the Isle-type Λα potential given in Eq. (12) and the Woods-Saxon type Ωα 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(32r22r2Coul),rrCoul1r,r>rCoul

      (15)

      with a Coulomb radius rCoul=1.47 fm.

    IV.   RESULTS AND DISCUSSION
    • To calculate the ground state binding energy and the geometrical properties of ΛΛαand ΩΩα, 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 coordinate ρi, 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 are mN= 939 MeV, mΛ= 1127.42 MeV, mΩ=1672.45, and mα= 3727.38 MeV. Table 2 shows the convergence of the three-body ground state binding energy E3 and nuclear matter radius rmat as a function of the maximum values of hyperangular quantum number Kmax and hyperradial excitations imax. The spin and isospin of ΛΛα and ΩΩα systems are equal to zero. The number of strange quark content for ΛΛα and ΩΩα is equal to 2 and 6, respectively. The Coulomb interaction of Eq. (15) is considered in ΩΩα systems leading to an increase of approximately 6 MeV in 3B binding energy.

      ΛΛα ΩΩα
      imax E3/MeV rmat/fm E3/MeV rmat/fm
      5 −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/MeV rmat/fm E3/MeV rmat/fm
      5 −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 radius rmat calculated for ΛΛα and ΩΩα systems as a function of maximum hyperradial excitations imax (with Kmax=80) and hyperangular quantum number Kmax (with imax=25).

      Since Ωα interaction is completely attractive, faster convergence is reached in the calculation of ground state binding energy and nuclear matter radius of ΩΩα to 67.21 MeV and 1.326 fm. For the ΛΛα system, a convergence can be reached at larger values of Kmax and imax. For ΛΛα, the ground state binding energy and nuclear matter radius converge to 7.468 MeV and 1.955 fm. The listed results for ΛΛα are obtained for the ΛΛ interaction with imaginary-time distance t/a=12, whereas for t/a=11 and t/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 and ΩΩα systems, we list the binding energy of two-body YY and Yα 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 between 4 to 6%. As one can see in Table 3, the ΩΩα has a deeper bound state by having two bound subsystems. Our numerical analysis shows that the uncertainties in the HAL QCD ΩΩ (ΛΛ) potential parameters impact the 6ΩΩHe (6ΛΛHe) ground state binding energy for approximately4 (0.1) MeV and the rmat for less than 0.01 (0.007) fm. While the employed Ωα potential in our calculations is derived based on the dominant 5S2 channel of NΩ interactions [3, 79], the contribution of the 3S1 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 the Ω particles to the α 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 for YYα systems. The last column shows our results for 3B binding energies with zero interaction in YY subsystems. Two-body binding energies E2 for YY and Yα 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-flavor SU6 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 Model E3/MeV
      present 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 of ΛΛα 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 the α matter radius of 1.47 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) systemrYY/fm rYα/fm r(YY)α/fm |Δ|/fm2 ρ21/2/fm rmat/fm
      ΛΛα(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 and ΩΩα systems. rYY is the separation between identical hyperons, rYα is the separation between Yα pairs, and r(YY)α is the separation between the center of mass of YY pair and the spectator α particle. ρ21/2 is the r.m.s. matter radius of the three-body system containing only point particles, and rmat 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)α2+14rYY2rYα2=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 of ΛΛα and ΩΩα systems in both (YαY) and (YYα) Jacobi coordinates. As one can see, the main contributions in 3B wave functions come from the s-wave channels, whereas the higher partial wave channels substantially have an insignificant contribution.

      K lxi lyi l Sxi jab W
      (ΛαΛ) Jacobi
      0 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
      (ΛΛα) Jacobi
      0 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
      (ΩαΩ) Jacobi
      0 0 0 0 0.0 0.0 0.993
      2 0 0 0 0.0 0.0 0.005
      (ΩΩα) Jacobi
      0 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 of ΛΛα and ΩΩα systems. For each system, the upper panel shows the contributions in (YαY) Jacobi coordinates, and the lower panel shows the contributions in (YYα) 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 of ΛΛα and ΩΩα systems, obtained by imax=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 of ΛΛα and ΩΩα systems as a function of rYY, the distance between the two identical Y hyperons, and r(YY)α, the distance between the α particle and the center of mass of two Y particles. As we can see for both systems, the distributions along the rYY direction are broader than those along the r(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), and β {4, 0, 0, 0, 0, 0} (red solid line), of the ground state wave functions of ΛΛα and ΩΩα systems.

      Figure 4.  (color online) 3B ground state probability density of ΛΛα and ΩΩα systems as a function of rYY, the distance between YY pair, and r(YY)α, the distance between α particle and the center of mass of the YY pair.

      confirming that the distance between identical Y hyperons is greater than the distance between the spectator α particle and the center of mass of the YY pair.

    V.   SUMMARY AND OUTLOOK
    • In this study, we invetigated the ground-state properties of multi-strangeness hypernuclei 6ΛΛHe and 6ΩΩHe 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 excitations imax and hyperangular quantum number Kmax. Our numerical results show that the ground state binding energy of 6ΛΛHe using three models of ΛΛlattice interactions changes in the domain of (7.468, 7.804) MeV, while 6ΩΩHe 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 the s -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 approximately 0.1 \sim 0.4 MeV in the \Lambda\Lambda\alpha binding energy, while using an effective single-channel interaction leads to a reduction of approximately 0.3 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.

    ACKNOWLEDGEMENT
    • F.E. thanks T. Hatsuda and J. Casal for helpful discussions and suggestions.

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