Neutron-rich hypernuclei: Lambda-6H and beyond

Recent experimental evidence presented by the FINUDA Collaboration for a particle-stable Lambda-6H has stirred renewed interest in charting domains of particle-stable neutron-rich Lambda hypernuclei, particularly for unbound nuclear cores. We have studied within a shell-model approach several neutron-rich Lambda hypernuclei in the nuclear p shell that could be formed in (pi-,K+) or in (K-,pi+) reactions on stable nuclear targets. Hypernuclear shell-model input is taken from a theoretically inspired successful fit of gamma-ray transitions in p-shell Lambda hypernuclei which includes also Lambda-N<->Sigma-N coupling (Lambda<->Sigma coupling). The particle stability of Lambda-6H is discussed and predictions are made for binding energies of Lambda-9He, Lambda-10Li, Lambda-12Be, Lambda-14B. None of the large effects conjectured by some authors to arise from Lambda<->Sigma coupling is borne out, neither by these realistic p-shell calculations, nor by quantitative estimates outlined for heavier hypernuclei with substantial neutron excess.


Introduction
Dalitz and Levi Setti, fifty years ago [1], discussed the possibility that Λ hyperons could stabilize particle-unstable nuclear cores of Λ hypernuclei and thus allow studies of neutron-rich baryonic systems beyond the nuclear drip line. The Λ's effectiveness to enhance binding is primarily connected with the Pauli principle from which it is exempt, allowing it to occupy the lowest 0s Λ orbital. Several unbound-core Λ hypernuclei have since been identified in emulsion work, the neutron richest of which is 8 Λ He [2]. Of particular interest is the recent FINUDA evidence for a particle-stable 6 Λ H hypernucleus produced in the 6 Li(K − stop , π + ) reaction [3,4]. In distinction from the well-established hyper-hydrogen isotopes 3,4 Λ H, the 5 H nuclear core of 6 Λ H is unbound, and its neutron-proton excess ratio (N − Z)/(N + Z)=0.6 is unsurpassed by any stable or Λ-stabilized core nucleus. The 6 Λ H hypernucleus was highlighted in Ref. [5] as a testground for the significance of ΛΣ coupling in Λ hypernuclei, spurred by the role it plays in s-shell hypernuclei [6,7] and by the far-reaching consequences it might have for dense neutron-star matter with strangeness [8]. In the present work, and as a prelude to its main theme, we discuss the particle stability of 6 Λ H from the point of view of the shell model, focusing on ΛΣ coupling contributions and making comparisons with other calculations and with the FINUDA experimental evidence.
The purpose of this Letter is to provide shell-model predictions for other neutron-rich Λ hypernuclei that could be reached at J-PARC in (π − , K + ) or (K − , π + ) reactions on stable nuclear targets in the p shell. A missing-mass 6 Li(π − , K + ) spectrum from J-PARC in-flight experiment E10 [9] is under study at present, aimed at assessing independently FINUDA's evidence for a particle-stable 6 Λ H hypernucleus. In the present analysis, based on extensive hypernuclear shell-model calculations [10], we use 0p N 0s Λ effective interactions with matrix elements constrained by the most comprehensive set of hypernuclear γ-ray measurements [11]. Included explicitly are also 0p N 0s Λ ↔ 0p N 0s Σ effective interactions based on the 0s N 0s Y G-matrix interactions (Y =Λ, Σ) used in the comprehensive s-shell hypernuclear calculations of Ref. [6]. The methodology of this shell-model analysis is briefly reviewed in Section 2, following which we discuss 6 Λ H in Section 3 and then, in Section 4, the neutron-rich hypernuclei that can be produced on stable nuclear targets in the nuclear p shell. Predictions are made for the corresponding ground-state binding energies . Also outlined in this section is a shell-model evaluation of ΛΣ coupling effects in heavier hypernuclei with larger neutron excess, such as 49 Λ Ca and 209 Λ Pb, demonstrating that the increase in neutron excess is more than compensated by the decrease of the ΛΣ coupling matrix elements with increasing orbital angular momentum ℓ N of the valent nucleon configurations involved in the coherent coupling approximation. We thus conclude that none of the large effects conjectured in Ref. [5] to arise from ΛΣ coupling is borne out in realistic calculations.

Shell-model methodology
The ΛN effective interaction where S N Λ = 3( σ N · r)( σ Λ · r) − σ N · σ Λ , is specified here by its 0p N 0s Λ spin-dependent matrix elements within a 0 ω shell-model space [12]. The same parametrization applies also to the ΛΣ coupling interaction and the ΣN interaction for both isospin 1/2 and 3/2, with an obvious generalization to account for the isospin 1 of the Σ hyperon [10]. The detailed properties of the ΣN interaction parameters hardly matter in view of the large energy denominators of order M Σ − M Λ ≈ 80 MeV with which they appear. To understand the effects of the ΛΣ coupling interaction, it is convenient to introduce an overall isospin factor 4/3 t N · t ΛΣ , where t ΛΣ converts a Λ to Σ in isospace. Matrix elements of the ΛN effective interaction (1) and of the ΛΣ coupling interaction are listed in Table 1. The ΛN matrix elements were fitted to a wealth of hypernuclear γ-ray measurements [13], resulting in values close to those derived from the Y N interaction models NSC97 [14]. The ΛΣ matrix elements are derived from the same NSC97 models used to construct 0s N 0s Y G-matrix interactions in s-shell Λ hypernuclear calculations [6]. By limiting here the ΣN model-space to 0p N 0s Σ , in parallel to the 0 ω 0p N 0s Λ model-space used for ΛN, we maintain the spirit of Akaishi's coherent approximation [5,6] which is designed to pick up the strongest ΛΣ matrix elements. 1 It is clear from Table 1 that the only significant ΛΣ interaction parameters areV ΛΣ and ∆ ΛΣ , in obvious notation. The first one is associated with diagonal matrix elements of the spin-independent part of the ΛΣ interaction, viz. ( This part preserves the nuclear core, specified here by its total angular momentum J N and isospin T , with matrix elements that bear resemblance to 1 By coherent we mean nl N 0s Λ ↔ nl N 0s Σ coupling that preserves hyperon and nucleon orbits, where nl N denotes the nuclear orbits occupied in the core-nucleus wavefunction. For the A = 4 hypernuclei considered by Akaishi et al. [6], this definition reduces to the assumption that the Λ and Σ, both in their 0s orbits, are coupled to the same nuclear core state. The parameters listed in the last line of Table 1 contribute almost 0.6 MeV to the binding of 4 Λ H(0 + g.s. ) and more than 0.5 MeV to the excitation energy of 4 Λ H(1 + exc. ).  the Fermi matrix elements in β decay of the core nucleus. Similarly, the spin-spin part of the ΛΣ interaction associated with the matrix element ∆ ΛΣ involves the operator j s N j t N j for the core, connecting core states with large Gamow-Teller (GT) transition matrix elements as emphasized recently by Umeya and Harada in their study of the 7−10 Λ Li isotopes [15]. Finally, the ΛN spin-independent matrix elementV is not specified in Table 1 because it is not determined by fitting hypernuclear γ-ray transitions. Its value is to be deduced from fitting absolute binding energies. Suffice to say thatV assumes values ofV ≈ −1.0±0.1 MeV [10]. We consider it as part of a mean-field description of Λ hypernuclei, consistent with the observation that on the average throughout the p shell B Λ (A) increases by about 1 MeV upon increasing A by one unit, with a 0.1 MeV uncertainty that might reflect genuine Y NN three-body contributions [16,17].

6 Λ H
The low-lying spectrum of 6 Λ H consists of a 0 + g.s. and 1 + exc. states which, disregarding the two p-shell neutrons known from 6 He to couple dominantly to L = S = 0, are split in 4 Λ H by 1.08±0.02 MeV [18]. These states in 6 Λ H are split by 1.0±0.7 MeV, judging by the systematic difference noted in the FINUDA experiment [3,4] between the mass values M( 6 Λ H) from production and decay, as shown in Fig. 1. The observation of 6 Λ H→ π − + 6 He weak decay in the FINUDA experiment implies that 6 Λ H is particle-stable, with 2n separation energy B 2n ( 6 Λ H g.s. )=0.8±1.2 MeV, independently of the location of the core-nucleus 5 H resonance. , with thresholds marked on its upper-left side and theory predictions beneath [1,19]. The location of the 5 H resonance vs. the 3 H+2n threshold, with the (blue) hatched box denoting its width, is taken from Ref. [20]. The (red) shaded box represents the error on the mean production and decay mass value obtained from the three FINUDA events assigned to 6 Λ H, whereas the positions of the 1 + exc. and 0 + g.s. levels marked on the right-hand side are derived separately from production and decay, respectively. We thank Elena Botta for providing this figure.

Phenomenological shell-model analysis
A phenomenological shell-model estimate for B Λ ( 6 Λ H) cited in Ref. [4], also bypassing 5 H, yields based on the value B Λ ( 7 Λ He) = (5.36 ± 0.09) MeV obtained by extrapolating linearly the known binding energies of the other T = 1 isotriplet members of the A = 7 hypernuclei. 2 The argument underlying this derivation is that the total Λnn interaction, including ΛΣ coupling contributions, is given by the difference of Λ binding energies within the square bracket, assuming that the Λnn configuration for the two p-shell neutrons in 6 Λ H is identical with that in 7 Λ He. Accepting then the 5 H mass determination from Ref. [20] one obtains 2 The value B Λ ( 7 Λ He)=5.68±0.03(stat.)±0.22(syst.) MeV, obtained recently from Jlab E01-011 [21], is irreproducible also in related few-body cluster calculations [22].
the corresponding phenomenological shell-model estimate for B 2n ( 6 Λ H): MeV which, however, is not independent of the resonance location of 5 H. Equation (4) demonstrates the gain in nuclear binding owing to the added Λ hyperon.

Refined shell-model analysis
The phenomenological shell-model estimate outlined above needs to be refined on two counts as follows. First, one notes that the nuclear s shell is not closed in 6 Λ H, in distinction from all other neutron-rich Λ hypernuclei considered in the present work, so that the ΛΣ coupling contribution in 6 Λ H is not the sum of the separate contributions from the s shell through 4 Λ H and from the p shell through 7 Λ He. For an s 3 p 2 core of 6 Λ H, with [32] spatial symmetry and T = 3/2, the coherent ΛΣ coupling diagonal matrix element is is the corresponding A = 4 matrix element (see caption to Table 1), with no contribution from ∆ 0p ΛΣ because the two p-shell neutrons have Pauli spin zero. ΛΣ contributions to the binding energies of Λ hypernuclei involved in the shell model evaluation of the 6 Λ H doublet levels 0 + g.s. and 1 + exc. , calculated for the interaction specified in Table 1, are listed in Table 2.  4 Λ H owing to the ΛΣ coupling, considerably less than the additional 1.4 MeV argued by Akaishi et al. [5,19].
A more important reason for revising the phenomenological shell-model estimate is the reduction of matrix elements that involve spatially extended pshell neutrons in 6 Λ H (B 2n 0.8 MeV) relative to matrix elements that involve a more compact p-shell neutron in 7 Λ He (B n ∼ 3.0 MeV). The average neutron separation energy in 6 Λ H is closer to that in 6 Λ He (B n = 0.26 ± 0.10 MeV). Using our shell-model estimates for the spin-dependent and ΛΣ coupling contributions to B Λ ( 6 Λ He), we deduceV ∼ −0.8 MeV and a similar 20% reduction in other 0p N 0s Λ interaction matrix elements, thereby yielding (5) where the first uncertainty is due to the statistical uncertainty of the bindingenergy input [2] and the second, systematic uncertainty assumes that the 0.8 renormalization factor is uncertain to within 0.1. Accepting, again, the 5 H resonance location from [20], a revised estimate follows: B 2n ( 6 Λ H)≈(0.1±0.4) MeV. Because the s 3 p 2 core for 6 Λ H will be more spatially extended than the α-particle core for the He hypernuclei, a few-body calculation such as that of Hiyama et al. [23], discussed below, is required. 3 We therefore consider the present revised shell-model estimate, listed also in Table 3 below, as an upper bound for B Λ ( 6 Λ H). In Table 3 we compare several theoretical 6 Λ H predictions to each other and to FINUDA's findings. The table makes it clear that FINUDA's results do not support the predictions made in Refs. [5,19] [22] in four-body C-n-n-Λ calculations, where the cluster C stands for 3 H or 4 He respectively, supports our ≈20% reduction of ΛN and ΛΣ matrix elements in going from 7 Λ He to 6 Λ H. We thank Emiko Hiyama for providing us with plots of these densities.  6 Λ H(0 + g.s. ) and a particle-unstable 6 Λ H(1 + exc. ) that decays by emitting a low-energy neutron pair: However, this decay is substantially suppressed both kinematically and dynamically, kinematically since s-wave emission requires a 3 S 1 dineutron configuration which is Pauli-forbidden, and the allowed p-wave emission which is kinematically suppressed at low energy requires that both 6 Λ H constituents, 4. Neutron-rich hypernuclei beyond 6 Λ H In the first part of this section we consider neutron-rich Λ hypernuclei that can be produced in double-charge exchange reactions (π − , K + ) or (K − , π + ) on stable nuclear targets in the p shell. These are 6 Λ H, which was discussed in the previous section, 9 Λ He, 10 Λ Li, 12 Λ Be, and 14 Λ B, with targets 6 Li, 9 Be, 10 B, 12 C, and 14 N, respectively. 4 In distinction from the unbound-core 6 Λ H, 4 We excluded 16 Λ C which can be produced on 16 O target because its analysis involves (1s − 0d) N 0s Λ matrix elements which are not constrained by any hypernuclear datum. all other neutron-rich Λ hypernuclei listed above are based on bound nuclear cores, which ensures their particle stability together with that of a few excited states. In the second half of the present section we outline the evaluation of ΛΣ coupling matrix elements in heavier nuclear cores with substantial neutron excess, 48 Ca and 208 Pb. In Table 4, we focus on ΛΣ contributions to binding energies of normalparity g.s. in neutron-rich p-shell hypernuclei. Column 3 gives the diagonal matrix-element (ME) from the central componentsV 0p ΛΣ and ∆ 0p ΛΣ of the coherent ΛΣ coupling interaction. The contribution fromV 0p ΛΣ is given by Eq. (2), saturating this diagonal matrix element in 9 Λ He and dominating it with a value 3.242 MeV in the other cases. Column 4 gives the corresponding downward energy shift (ME) 2 /80 assuming a constant 80 MeV separation between Λ and Σ states, and column 5 gives the energy shift in the full shell-model calculation, accounting also for the noncentral components of the coherent ΛΣ coupling interaction and including nondiagonal matrix elements as well. Except for 12 Λ Be where the decreased shift in the complete calculation is due to the contribution of the noncentral components to the diagonal matrix element, a very good approximation for the total coherent ΛΣ coupling contribution is obtained using just the central (Fermi and GT) ΛΣ coupling interaction. In these cases, the increased energy shift beyond the diagonal contribution arises from Σ core states connected to the Λ core state by GT transitions. The shift gets smaller with increased fragmentation of the GT strength. Finally, we note that the total ΛΣ contibution listed in column 5 agrees for 10 Λ Li with that computed in Ref. [24] using the same Y N shell-model interactions. More recently, these authors discussed ΛΣ coupling effects on g.s. doublet spacings in 7−10 Λ Li isotopes, obtaining moderate enhancements between 70 to 150 keV [15]. We have reached similar conclusions for all the neutron-rich hypernuclei considered in the present work beyond 6 Λ H.

p-shell neutron-rich hypernuclei beyond 6 Λ H
The total ΛΣ contribution discussed above is only one component of the total beyond-mean-field (BMF) contribution ∆B g.s.
Λ . To obtain the latter, various spin-dependent ΛN contributions generated by V ΛN have to be added to the total ΛΣ contributions listed in column 5 of Table 4. Column 6 lists one such spin-dependent ΛN contribution, arising from the Λ-induced l N · s N nuclear spin-orbit term of Eq. (1). By comparing column 5 with column 6, and both with the total BMF contributions listed in column 7, the last one, it is concluded that the majority of the BMF contributions arise from ΛN spin-dependent terms, dominated by l N · s N . Our Λ binding energy predictions for ground states of neutron-rich hypernuclei A Λ Z in the p shell are summarized in the last column of Table 5. We used the experimentally known g.s. binding energies of same-A normal hypernuclei A Λ Z ′ with Z ′ > Z from [2], averaging statistically when necessary. The BMF contributions ∆B g.s. Λ (normal) to the binding energies of the A Λ Z ′ 'normal' hypernuclei, taken from [10], are listed in column 3 and the BMF contributions ∆B g.s.
Λ (n-rich) to the binding energies of the A Λ Z neutron-rich hypernuclei value, which are reproduced in the last column of Table 4, are listed here in column 4. Finally, the predicted binding energies B g.s.
Λ (n-rich) of the neutron-rich hypernuclei A Λ Z are listed in the last column of Table 5, were evaluated according to The resulting binding energies are lower by about 0.1 MeV than those of the corresponding normal hypernuclei, except for 10 Λ Li with binding energy about 0.5 MeV larger than that of 10 Λ Be-10 Λ B. The relatively small effects induced by the ΛΣ coupling interaction on B g.s.
Λ persist also in the particle-stable portion of neutron-rich Λ hypernuclear spectra. Here we only mention that, except for 12 Λ Be, one anticipates a clear separation about or exceeding 3 MeV between g.s. (in 9 Λ He) or g.s.-doublet (in 10 Λ Li and 14 Λ B) and the first-excited hypernuclear doublet. Thus, the g.s. or its doublet are likely to be observed in experimental searches of neutronrich Λ hypernuclei using 9 Be, 10 B, and 14 N targets, provided a resolution of better than 2 MeV can be reached.

Medium weight and heavy hypernuclei
It was demonstrated in the previous subsection that the ΛΣ coupling contributions to p-shell hypernuclear binding energies are considerably smaller than for the s-shell A = 4 hypernuclei. The underlying ΛΣ matrix elements decrease by roughly factor of two upon going from the s-shell to the p-shell, and the resulting energy contributions roughly by factor of four. This trend persists upon going to heavier hypernuclei and, as shown below, it more than compensates for the larger N − Z neutron excess available in heavier hypernuclei. To demonstrate how it works, we outline schematic shell-model weak-coupling calculations using G-matrix ΛΣ central interactions generated from the NSC97f Y N potential model by Halderson [25], with input and results listed in Table 6 across the periodic table. We note that although the separate contributions ΛΣ(V ) and ΛΣ(∆) in 9 Λ He differ from those using the Akaishi interaction (see Table 4), the summed ΛΣ contribution is the same to within 2%. For 49 Λ Ca, its 48 Ca core consists of 0s, 0p, 1s − 0d closed shells of protons and neutrons, plus a closed 0f 7/2 neutron shell with isospin T = 4. The required 0f N 0s Λ ↔ 0f N 0s Σ radial integrals are computed for HO nℓ N radial wavefunctions with ω = 45A −1/3 − 25A −2/3 MeV and are compared in the table to the 0p N 0s Λ ↔ 0p N 0s Σ radial integrals for 9 Λ He. The decrease in the values ofV ΛΣ and of ∆ ΛΣ upon going from 9 Λ He to 49 Λ Ca is remarkable, reflecting the poorer overlap between the hyperon 0s Y and the nucleon 0ℓ N radial wavefunctions as ℓ N increases. The resulting ΛΣ contributions to the binding energy are given separately for the Fermi spin-independent interaction using the listed values ofV ΛΣ , and for the GT spin-dependent interaction using the listed values of ∆ ΛΣ . The Fermi contribution involves a 0 + → 0 + core transition preserving the value of T , see Eq. (2). The GT contribution consists of three separate 0 + → 1 + core transitions of comparable strengths, transforming a f 7/2 neutron to (i) f 7/2 nucleon with nuclear-core isospin T N = 3, or to f 5/2 nucleon with (ii) T N = 3 or (iii) T N = 4. The overall ΛΣ contribution of 24 keV in 49 Λ Ca is rather weak, one order of magnitude smaller than in 9 Λ He. In 209 Λ Pb, the 44 excess neutrons run over the 2p, 1f , 0h 9/2 and 0i 13/2 orbits. The Fermi 0 + → 0 + core transition is calculated using Eq. (2) where, again,V ΛΣ (with a value listed in Table 6) stands for the (2j+1) average of the separate neutron-excessV nℓ N ΛΣ radial integrals, resulting in a ΛΣ contribution of merely 52 keV to the binding energy of 209 Λ Pb. For the calculation of the GT 0 + → 1 + core transitions, we define ∆ ΛΣ as a (2j + 1) average of ∆ nℓ N ΛΣ radial integrals over the neutron-excess incomplete LS shells 0h 9/2 and 0i 13/2 (with a value listed in the table). These are the neutron orbits that initiate the GT core transitions required in 209 Λ Pb, with structure similar to that encountered in 49 Λ Ca for the 0f 7/2 neutron orbit; details will be given elsewhere, suffice to say here that neither of these transitions results in a ΛΣ contribution larger than 0.2 keV to the binding energy of 209 Λ Pb.

Summary and outlook
In this Letter we have presented detailed shell-model calculations of pshell neutron-rich Λ hypernuclei using (i) ΛN effective interactions derived by fitting to comprehensive γ-ray hypernuclear data, and (ii) theoretically motivated ΛΣ interaction terms. None of the large effects conjectured by Akaishi and Yamazaki [5] to arise from ΛΣ coherent coupling in neutron-rich hypernuclei is borne out by these realistic shell-model calculations. This is evident from the relatively modest ΛΣ component of the total BMF contribution to the Λ hypernuclear g.s. binding-energy, marked ∆B g.s.
Λ in Table 4. It should be emphasized, however, that ΛΣ coupling plays an important role in doublet spacings in p-shell hypernuclei [10], just as it does for 4 Λ H and 4 Λ He [6]. Although the ground-state doublet spacings in 10 Λ Li and 14 Λ B probably can't be measured, ΛΣ coupling contributes 40% and 55% to the predicted doublet spacings of 341 and 173 keV, respectively (similar relative strengths as for 16 Λ O [10]; the 12 Λ Be doublet spacing, however, is predicted to be very small). Forthcoming experiments searching for neutron-rich Λ hypernuclei at J-PARC will shed more light on the N − Z dependence of hypernuclear binding energies.
We have also discussed in detail the shell-model argumentation for a slightly particle-stable 6 Λ H, comparing it with predictions by Akaishi et al. that overbind 6 Λ H as well as with a very recent four-body calculation by Hiyama et al. that finds it particle-unstable. In the former case [5,19], we have argued that the effects of ΛΣ coupling in 6 Λ H cannot be very much larger than they are in 4 Λ H, whereas such effects are missing in the latter case [23]. Genuine Y NN three-body contributions are missing so far in all calculations of 6 Λ H, and need to be implemented. Systematic studies of these contributions, which appear first at NNLO EFT expansions, have not been made to date in hypernuclear physics where the state of the art is just moving from LO to NLO applications [26]. At present, and given the regularity of hypernuclear binding energies of heavy Λ hypernuclei with respect to those in light and medium-weight hypernuclei as manifest in SHF-motivated density-dependent calculations [27], in RMF [28] and in-medium chiral SU(3) dynamics calculations [29] extending over the whole periodic table, no strong phenomenological motivation exists to argue for substantial N − Z effects in Λ hypernuclei. Indeed, this was demonstrated in the calculation outlined in the previous section for 49 Λ Ca and 209 Λ Pb.