Onset of $\eta$ nuclear binding

Recent studies of $\eta$ nuclear quasibound states by the Jerusalem-Prague Collaboration are reviewed, focusing on stochastic variational method self consistent calculations of $\eta$ few-nucleon systems. These calculations suggest that a minimum value Re$\,a_{\eta N} \approx 1$ fm (0.7 fm) is needed to bind $\eta\,^3$He ($\eta\,^4$He).


Introduction
The ηN near-threshold interaction is attractive, owing to the N * (1535) resonance to which the s-wave ηN system is coupled strongly [1]. This has been confirmed in chiral meson-baryon coupled channel models that generate the N * (1535) dynamically, e.g. [2]. Hence η nuclear quasibound states may exist [3] as also suggested experimentally by the near-threshold strong energy dependence of the η 3 He production cross sections shown in Fig. 1. However, the η 3 He scattering length deduced in Ref. [4], a η 3 He = [−(2.23 ± 1.29) + i(4.89 ± 0.57)] fm, although of the right sign of its real part, does not satisfy the other necessary condition for a quasibound state pole: −Re a > Im a.  Figure 1. Near-threshold η 3 He production cross sections. Left: dp → η 3 He [4]. Right: γ 3 He → η 3 He [5].
⋆ Presented by A. Gal (avragal@savion.huji.ac.il) at EXA2017, Vienna, Sept. 2017 Quite generally, experimental searches for η nuclear quasibound states in proton, pion or photon induced η production reactions are inconclusive. Regarding the onset of η nuclear binding, Krusche and Wilkin [6] state: "The most straightforward (but not unique) interpretation of the data is that ηd is unbound, η 4 He is bound, but that the η 3 He case is ambiguous." Indeed, with η 3 He almost bound, one might expect that the denser 4 He nucleus should help forming a bound η 4 He. Nevertheless, a recent Faddeev-Yakubovsky evaluation [7] of the scattering lengths a η A He for both He isotopes, A = 3, 4, finds this not to be the case, with the denser 4 He apparently leading to a stronger reduction of the subthreshold ηN scattering amplitude than in 3 He.
The present overview reports and discusses recent few-body stochastic variational method (SVM) calculations of ηNNN and ηNNNN using several semi-realistic NN interaction models together with two ηN interaction models that, perhaps, provide sufficient attraction to bind η in the 3 He and 4 He isotopes [8][9][10]. Real and imaginary parts of the ηN cm scattering amplitude near threshold in two meson-baryon coupled channel models: GW [11] and CS [12]. Figure 2 shows ηN s-wave scattering amplitudes F ηN (E) calculated in two meson-baryon coupledchannel models across the ηN threshold where Re F ηN has a cusp. These amplitudes exhibit a resonance about 50 MeV above threshold, the N * (1535). The sign of Re F ηN below the resonance indicates attraction which is far too weak to bind the ηN two-body system. The threshold values F ηN (E th ) are given by the scattering lengths

ηN and NN interaction model input
with lower values below threshold (E th = 1487 MeV). These free-space energy dependent subthreshold amplitudes are transformed to in-medium density dependent amplitudes, in terms of which optical potentials V opt η (ρ) are constructed and used to calculate self consistently η nuclear quasibound states. This procedure was applied in Refs. [13,14] to several ηN amplitude models, with results for 1s η quasibound states in models GW and CS shown in Fig. 3 from 12 C to 208 Pb. Figure 3 demonstrates that in both of these ηN amplitude models the 1s η binding energy increases with A, saturating in heavy nuclei. Model GW, with larger ηN real and imaginary subthreshold amplitudes than in model CS, gives correspondingly larger values of B η and Γ η . While model GW binds  η also in nuclei lighter than 12 C (not shown in the figure) this needs to be confirmed in few-body calculations.  Few-body calculations, in distinction from optical model calculations, require the use of effective ηN potentials v ηN which reproduce the free-space ηN amplitudes below threshold. Fig. 4 shows subthreshold values of the energy dependent strength function derived from the scattering amplitude F GW ηN (E) of Fig. 2 for several choices of inverse range Λ. The normalized Gaussian function δ Λ (r) is perceived in / πEFT (pionless EFT) as a single ηN zero-range Dirac δ (3) (r) contact term (CT), regulated by using a momentum-space scale parameter Λ. Regarding the choice of Λ, substituting the underlying short range vector-meson exchange dynamics by a single regulated CT suggests that the scale Λ is limited to values Λ m ρ (∼4 fm −1 ). Similarly, a / πEFT energy independent v NN (r) is derived at leading order (LO) by fitting a single regulated CT ∼ δ Λ (r) in each spin-isospin s-wave channel to the respective NN scattering length. A pp Coulomb interaction is included. To avoid NNN and ηNN Thomas collapse in the limit Λ → ∞, one introduces a three-body regulated CT for each of these three-body systems [9]: where With no further contact terms, B calc ( 4 He) is found in this / πEFT version [15] to vary moderately with Λ and to exhibit renormalization scale invariance by approaching a finite value B Λ→∞ ( 4 He)=27.8±0.2 MeV that compares well with B exp ( 4 He)=28.3 MeV. In contrast, no η-related experimental datum is available for the ηNN CT d Λ ηNN to be fitted to. Two versions for choosing this CT were tested: , one finds that each of these versions prevents a potential collapse of ηd, with calculated values of B(η A He) that for Λ ≥ 4 fm −1 are nearly independent of the adopted version, as shown in Fig. 7 below.  Having derived energy dependent ηN potentials v ηN (E; r), see Eq. (2) and Fig. 4, a two-body subthreshold input energy δ √ s ≡ E − E th needs to be chosen. However, δ √ s is not conserved in the η nuclear few-body problem, so the best one can do is to require that this choice agrees with the expectation value � δ √ s � generated in solving the few-body problem, as given by [10] �δ Here , T A and T η denote the nuclear and η kinetic energy operators in appropriate Jacobi coordinates, B is the total binding energy, and E η = �H − H N � with each Hamiltonian defined in its own cm frame. Self consistency (SC), � δ √ s � = δ √ s, is imposed in our calculations, as demonstrated graphically in Fig. 6 (left). Applications of SC to meson-nuclear systems are reviewed in Ref. [16]. For recent K − -atom and nuclear applications see Refs. [17,18]. More recently, Hoshino et al. [19] argued in a K − d study that by applying this procedure one violates the requirement of total momentum conservation. In Appendix B here we show specifically for A = 2 that our choice of SC Eq. (4) is not in conflict with any conservation law.
Finally, we note that Eq. (4) in the limit A >> 1 coincides with the optical model downward energy shift (supplemented by a Coulomb term) used in recent K − atom and nuclear studies [17,18]: where T N = �T A �/A = 23.0 MeV at the average nuclear densityρ, B N = B nuc /A ≈ 8.5 MeV is an average nucleon binding energy and B η denotes the calculated η separation energy. All terms here are negative, thereby leading to a downward energy shift.  The SC procedure is demonstrated in Fig. 6 (left) for η 4 He binding energy calculated using the AV4' NN potential and GW ηN potential with Λ=4 fm −1 . The η 4 He bound state energy E (excluding rest masses) and the output expectation value �δ √ s�, where δ √ s stands for the ηN cm energy with respect to its threshold value E th , are plotted as a function of the subthreshold input energy argument δ √ s of the potential v GW ηN . The SC condition requires δ √ s = �δ √ s� which is satisfied at −32.4 MeV. The corresponding value of E(�δ √ s�) then represents the SC energy of η 4 He, with B SC η = 3.5 MeV, considerably less than the value B th η = 13 MeV obtained by disregarding the energy dependence of v GW ηN and using its threshold value corresponding to δ √ s = 0. In Fig. 6 (right) we present the ηN downward energy shift δ √ s = E − E th as a function of the relative nuclear density ρ/ρ 0 in Ca, evaluated self consistently via Eq. (5) in the CS and GW models. The energy shift at ρ 0 is −55±10 MeV, about twice larger than the SC condition δ √ s = −B η applied in some other works, e.g. [20]. The GW shift exceeds the CS shift owing to the stronger GW amplitude of Fig. 2 and both were incorporated in the calculation of 1s η quasibound nuclear states, Fig. 3.

Results of η nuclear few-body calculations
Our fully self consistent ηNN, ηNNN and ηNNNN bound-state calculations [8][9][10] use the following nuclear core models: (i) / πEFT including a three-body contact term [15], (ii) AV4p, a Gaussian basis adaptation of the Argonne AV4' NN potential [21], and (iii) MNC, the Minnesota soft core NN potential [22]. Models GW [11] and CS [12] were used to generate energy dependent ηN potentials which prove too weak to bind any ηNN system when using AV4p or MNC for the nuclear core model.  Fig. 7 demonstrates in / πEFT the moderating effect that imposing SC (red, squares) by using v GW ηN (E sc ), rather than using threshold values v GW ηN (E th ) (blue, circles), bears on the calculated B η values and their Λ scale dependence [9]. Near Λ=4 fm −1 , imposing sc lowers B η (η 3 He) by close to 5 MeV and B η (η 4 He) by close to 10 MeV. The figure demonstrates that B η (η 4 He) is always larger than B η (η 3 He). Focusing on scale parameters near Λ=4 fm −1 one observes that η 3 He is hardly bound by a fraction of MeV, whereas η 4 He is bound by a few MeV. The choice of three-body CT d Λ ηNN hardly matters for Λ > 4 fm −1 , becoming substantial at Λ < 4 fm −1 . Fig. 8 demonstrates in non-EFT calculations the dependence of B η , calculated self consistently, on the choice of NN and ηN interaction models. Using the more realistic AV4' NN interaction results in less η binding than using the soft-core MNC NN interaction. For v GW ηN near Λ=4 fm −1 the difference amounts to about 0.3 MeV for η 3 He and about 1.5 MeV for η 4 He; η 3 He appears then barely bound whereas η 4 He is bound by a few MeV. The weaker v CS ηN does not bind η 3 He and barely binds η 4 He using the MNC NN interaction, implying that η 4 He is unlikely to bind for the more realistic AV4' NN interaction. For smaller, but still physically acceptable values of Λ down to Λ = 2 fm −1 , η 3 He becomes unbound and η 4 He is barely bound using the AV4' NN and GW ηN interactions. Figure 8. B η (η 3 He) (left) and B η (η 4 He) (right) as a function of 1/Λ from few-body calculations [10] using NN and ηN interactions, as marked, and imposing self consistency.
The B η values calculated in Refs. [8][9][10] were calculated assuming real Hamiltonians, justified by Im v ηN ≪Re v ηN from Fig. 4. This approximation is estimated to add near threshold less than 0.3 MeV to B η . Perturbatively-calculated widths Γ η of weakly bound states amount to only few MeV, outdating those reported in Ref. [8].  Figure 9. Preliminary SVM results for binding energies B η (left) and widths Γ η (right) of 1s η quasibound states in 3 He, 4 He and 6 Li, calcualted using the Minnesota NN potential and the GW ηN potential for Λ = 2 and 4 fm −1 .
In future work it will be interesting to extend the present SVM few-body calculations to heavier nuclei, beginning with light p-shell nuclei. This represents highly non-trivial task. In Fig. 9 6 Li nuclear core consisted of a single S = 1, T = 0 spin-isospin configuration, yielding B( 6 Li)=34.66 MeV which is short by almost 2 MeV with respect to a calculation reported in Ref. [24] that used the same NN interaction while including more spin-isospin configurations. The figure suggests that η 6 Li is comfortably bound, even for as low value of scale parameter as Λ = 2 fm −1 .

Summary
Based mostly on the AV4' results in Fig. 8, which are close to the / πEFT results in Fig. 7, we conclude that η 3 He becomes bound for Re a ηN ∼ 1 fm, as in model GW, while η 4 He binding requires a lower value of Re a ηN ∼ 0.7 fm, almost reached in model CS. These Re a ηN onset values, obtained by incorporating the requirements of ηN subthreshold kinematics, are obviously larger than those estimated in Sect. 3 upon calculating with v ηN (E = E th ; r) threshold input. Finally, Re a ηN < 0.7 fm if η 4 He is unbound, as might be deduced from the recent WASA-at-COSY search [23].
intrinsic kinetic energies, T N:N for the internal motion of the deuteron core and T η for that of the η meson with respect to the NN cm, one gets for this A = 2 special case which agrees with Eq. (4) for A = 2 upon realizing that T N:N here coincides with T A=2 there. To get idea of the relative importance of the various terms in this expression, we assume a near-threshold ηd bound state for which both E η and �T η � are negligible (fraction of MeV each) and B → B d ≈ 2.2 MeV. With �T N:N � → �T d �, and with a deuteron kinetic energy �T d � in the range of 10 to 20 MeV, this term provides the largest contribution to the downward energy shift which is then of order −5 MeV for the diffuse deuteron nuclear core.