$\Lambda NN$ content of $\Lambda$-nucleus potential

A minimally constructed $\Lambda$-nucleus density-dependent optical potential is used to calculate binding energies of observed $1s_{\Lambda}$, $1p_{\Lambda}$ states across the periodic table, leading to a repulsive $\Lambda NN$ contribution $D_{\Lambda}^{(3)}\approx 14$ MeV to the phenomenological $\Lambda$-nucleus potential depth $D_{\Lambda}\approx -30$ MeV. This value is significant in connection with the so-called 'hyperon puzzle'.


Introduction
The Λ-nucleus potential depth provides an important constraint in ongoing attempts to resolve the 'hyperon puzzle', i.e., whether or not dense neutron-star matter contains hyperons, primarily Λs besides nucleons [1]. Figure 1 presents compilation of most of the known Λ hypernuclear binding energies (B Λ ) across the periodic table, fitted by a three-parameter Woods-Saxon (WS) attractive potential. As A → ∞, a limiting value of B Λ (A) → 30 MeV is obtained. Interestingly, studies of density dependent Λ-nuclear optical potentials V Λ (ρ) in Ref. [2], with ρ the nuclear density normalized to the number of nucleons A, conclude that a ρ 2 term motivated by three-body ΛNN interactions provides a large repulsive (positive) contribution to the Λ-nuclear potential depth D Λ at nuclear-matter density ρ 0 : D (3) Λ ≈ 30 MeV. This repulsive component of D Λ is more than just compensated at ρ 0 by a roughly twice larger attractive depth value D (2) Λ ≈ −60 MeV, motivated by a two-body ΛN interaction. Note that D Λ is defined as V Λ (ρ 0 ) in the limit A → ∞ at a given nuclear-matter density ρ 0 , with a value 0.17 fm −3 assumed here.
Most hyperon-nucleon potential models overbind Λ hypernuclei, yielding values of D (2) Λ deeper than −30 MeV. Whereas such overbinding amounts to only few MeV in the often used Nijmegen soft-core model versions NSC97e,f [4] it is considerably stronger, by more than 10 MeV, in the recent Nijmegen extended soft-core model ESC16 [5]. A similar overbinding arises at leading order in chiral effective field theory (χEFT) [6]. The situation at next-to leading order (NLO) is less clear owing to a strong dependence of D (2) Λ on the momentum cutoff scale λ [7]. At λ=500 MeV/c, however, it is found in Ref. [8] that both versions NLO13 [9] and NLO19 [10] overbind by a few MeV. Finally, recent Quantum Monte Carlo (QMC) calculations [11,12] 208  139  89  51  40 32  28  16  13 12 11 10   8  7 Woods-Saxon V = 30.05 MeV, r = 1.165 fm, a = 0.6 fm Figure 1. Compilation of Λ binding energies in 7 Λ Li to 208 Λ Pb from various sources, and as calculated using a three-parameter WS potential [2]. Figure adapted from Ref. [3] how large it is [13]. Repulsive three-body ΛNN interactions go beyond just providing solution of the overbinding problem: as nuclear density is increased beyond nuclear matter density ρ 0 , the balance between attractive D (2) Λ and repulsive D (3) Λ tilts towards the latter. This results in nearly total expulsion of Λ hyperons from neutron-star matter, suggesting an equation of state (EoS) sufficiently stiff to support two solar-mass neutron stars, thereby providing a possible solution to the 'hyperon puzzle'. The larger D (3) Λ is, the more likely it is a solution [14,15]. However, there is no guarantee that threebody ΛNN interactions are universally repulsive. For a recent discussion of this problem within an SU(3) 'decuplet dominance' approach practised in modern χEFT studies at NLO, see Ref. [8].
In this Contribution we adopt the optical potential approach as applied by Dover-Hüfner-Lemmer to pions in nuclear matter [16]. For the Λ-nucleus system, it provides expansion in powers of the nuclear density ρ(r), consisting of a linear term induced by a two-body ΛN interaction plus two higherpower density terms: (i) a long-range Pauli correlations term starting at ρ 4/3 , and (ii) a short-range ΛNN interaction term dominated in the present context by three-body ΛNN interactions, starting at ρ 2 . As demonstrated below, the contribution of the Pauli correlations term is non negligible, propagating to higher powers of density terms than just ρ 4/3 , such as the ρ 2 ΛNN interaction term. This explains why the value derived here, D (3) Λ = (13.9 ± 1.4) MeV, differs from any of those suggested earlier in Ref. [2] and in Skyrme Hartree Fock studies [17] where Pauli correlations are usually disregarded. Our value of D (3) Λ strongly disagrees with the much larger value inferred in QMC calculations [12]. We comment on these discrepancies below.

Nuclear densities
In optical model applications aimed at establishing relations between components with different powers of density ρ = ρ p + ρ n , it is crucial to ensure that the radial extent of the densities, e.g., their r.m.s. radii, follows closely values derived from experiment. For proton densities we used charge densities, with proton finite-size and recoil effects included. Harmonic-oscillator type densities [18] were used for the lightest elements, assuming the same radial parameters for protons and neutrons. A variation of 1% in the r.m.s. neutron radius was found to affect calculated Λ binding energies considerably less than given by most of the experimental uncertainties listed in Table 1 below. For a detailed discussion in the analogous case of light Ξ − hypernuclei, see Ref. [19]. For species beyond the nuclear 1p shell we used two-parameter Fermi distributions normalized to Z for protons and N = A − Z for neutrons, derived from assembled nuclear charge distributions [20]. For medium-weight and heavy nuclei, the r.m.s. radii of our neutron density distributions assume larger values than those for proton density distributions, as practiced in analyses of exotic atoms [21]. Furthermore, once neutron orbits extend beyond proton orbits, it is useful to represent the nuclear density ρ(r) as where ρ core refers to the Z protons plus the charge symmetric Z neutrons occupying the same nuclear 'core' orbits, and ρ excess refers to the (N − Z) 'excess' neutrons associated with the nuclear periphery.

Optical potential
The optical potential employed in this work, consists of terms representing two-body ΛN and three-body ΛNN interactions, respectively: with b 0 and B 0 strength parameters in units of fm ( = c = 1). In these expressions, ρ(r) is a nuclear density distribution normalized to the number of nucleons A, ρ 0 = 0.17 fm −3 stands for nuclear-matter density, µ Λ is the Λ-nucleus reduced mass and f A is a kinematical factor transforming b 0 from the ΛN c.m. system to the Λ-nucleus c.m. system: This form of f A coincides with the way it is used for V (2) Λ in atomic/nuclear hadron-nucleus boundstate problems [21] and its A dependence provides good approximation for V (3) Λ . Next is the density dependent factor C Pauli (ρ) in Eq. (2), standing for a Pauli correlation function: with Fermi momentum k F = (3π 2 ρ/2) 1/3 . The parameter α P in Eq. (5) switches off (α P =0) or on (α P =1) Pauli correlations in a form suggested in Ref. [22] and practised in K − atoms studies [23]. To estimate 1/A correction terms, we also approximated C Pauli (ρ) by [19]: As shown below, including C Pauli (ρ) in V (2) Λ affects strongly the balance between the derived potential depths D (2) Λ and D (3) Λ . However, introducing it also in V (3) Λ is found to make little difference, which is why it is skipped in Eq. (3). Finally we note that the low-density limit of V opt Λ requires according to Ref. [16] that b 0 is identified with the c.m. ΛN spin-averaged scattering length (positive here). The present work does not attempt to reproduce the full range of B Λ data shown in Fig. 1. It is limited to 1s Λ and 1p Λ states listed in Table 1. We fit to such states in one of the nuclear 1p-shell hypernuclei listed in the table, where the 1s Λ state is bound by over 10 MeV while the 1p Λ state has just become bound. This helps resolve the density dependence of V opt Λ by setting a good balance between its two components, V (2) Λ (ρ) and V (3) Λ (ρ), following it all the way to 208 Λ Pb the heaviest hypernucleus marked in Fig. 1. We chose to fit the 16 Λ N precise B  Table 1.

Results
The two strength parameters b 0 , B 0 of the optical potential terms Eqs. (2,3) were obtained by fitting to the 16 Λ N B exp Λ (1s, 1p) values listed in Table 1. Suppressing Pauli correlations by setting α P = 0 in Eqs. (5,6), the resulting Λ potential depth D Λ = −27.4 MeV reflects a sizable cancellation between a strongly attractive two-body potential depth D (2) Λ and a strongly repulsive three-body potential depth D (3) Λ . The overall agreement between calculations and experiment is acceptable, but some underbinding appears to develop for increasing mass numbers A, noticed clearly in the three heaviest 1s Λ and two heaviest 1p Λ states. The resulting b 0 is about half of the known Λp scattering length of (1.7 ± 0.1) fm [26,27].
When the full potential Eqs. (2)(3)(4)(5)(6) is used (marked here as model X, including Pauli correlations through α P = 1) the overall picture remains unchanged regarding underbinding for the heavier elements, see Fig. 3. However, the fit parameter b 0 =1.85 fm agrees now with the Λp scattering length. The other parameter, B 0 = 0.170 fm, is about twice smaller than for α P = 0. The phenomenon of underbinding associated with the optical potential Eqs. (2-6) is likely to be a result of the use of ρ 2 in nuclei where excess neutrons occupy shell-model orbits higher than those occupied by protons. This situation occurs in Fig. 3 for the four hypernuclei with A 50. Expecting that direct three-body ΛNN contributions involving one 'core' nucleon and one 'excess' nucleon vanish upon summing on the T =0 'core' closed-shell nucleons, we modify ρ 2 = (ρ core + ρ excess ) 2 by discarding the bilinear term ρ core ρ excess , thereby replacing ρ 2 in V (3) Λ , Eq. (3), by ρ 2 core + ρ 2 excess = (2ρ p ) 2 + (ρ n − ρ p ) 2 (7) in terms of the input densities ρ p and ρ n . This ansatz is consistent with an overall isospin factor τ 1 · τ 2 in two-pion exchange ΛNN forces, as first realized back in 1958 [28]. Results of applying this ansatz are shown in the lower part of Fig. 4 as model Y, where the underbinding of calculated 1s Λ and

Discussion
The D (2) Λ and D (3) Λ values in Eq. (8) are considerably smaller than those deduced in QMC calculations [11,12]. Note that the QMC nuclear densities ρ QMC (r) are much too compact with respect to our realistic densities, with nuclear r.m.s. radii r N (QMC) about 0.8 of the known r.m.s. charge radii in 16 O and 40 Ca [30]. Since ρ scales as r −3 N , applying it to the density dependence of our V   [2,17]. Apart from small nonlocal potential terms and effective mass corrections, the SHF Λ-nuclear mean-field potential V Λ (ρ) consists of two terms: A large-scale SHF fit [17] of the corresponding Λ potential depths to 35 B Λ data points is listed in the middle row of Table 2. We note that the overall D Λ = −35 MeV value becomes −31 MeV upon including a Λ effective-mass correction, a bit closer to the other D Λ values listed in the table. Similar results, particularly for D (2) Λ , can be obtained in fact by choosing a considerably smaller number of fitted data points, as shown by the fits listed in the other two rows of the table. The 11 MeV difference between the D (3) Λ values derived in these two SHF calculations arises mostly from nonlocal lower-power density terms, like ρ 5/3 , present in [17] but absent in [2]. Interestingly, the last row lists a fit to the two B 1s,1p

Summary
In summary, we have presented a straightforward optical-potential analysis of 1s Λ and 1p Λ binding energies across the periodic table, 12 ≤ A ≤ 208, based on nuclear densities constrained by charge r.m.s. radii. The potential is parameterized by constants b 0 and B 0 in front of two-body ΛN and threebody ΛNN interaction terms. These parameters were fitted to precise B The potential depth D (3) Λ derived here, Eq. (8), suggests that in symmetric nuclear matter the Λnucleus potential becomes repulsive near three times ρ 0 . Our derived depth D (3) Λ is larger by a few MeV than the one yielding µ(Λ) > µ(n) for Λ and neutron chemical potentials in purely neutron matter, respectively, under a 'decuplet dominance' construction for the underlying ΛNN interaction terms within a χEFT(NLO) model [8]. This suggests that the strength of the corresponding repulsive V (3) Λ optical potential component, as constrained in the present work by data, is sufficient to prevent Λ hyperons from playing active role in neutron-star matter, thereby enabling a stiff EoS that supports two solar-mass neutron stars.