Few-body calculations of $\eta$-nuclear quasibound states

We report on precise hyperspherical-basis calculations of $\eta NN$ and $\eta NNN$ quasibound states, using energy dependent $\eta N$ interaction potentials derived from coupled-channel models of the $S_{11}$ $N^{\ast}(1535)$ nucleon resonance. The $\eta N$ attraction generated in these models is too weak to generate a two-body bound state. No $\eta NN$ bound-state solution was found in our calculations in models where Re $a_{\eta N}\lesssim 1$ fm, with $a_{\eta N}$ the $\eta N$ scattering length, covering thereby the majority of $N^{\ast}(1535)$ resonance models. A near-threshold $\eta NNN$ bound-state solution, with $\eta$ separation energy of less than 1 MeV and width of about 15 MeV, was obtained in the 2005 Green-Wycech model where Re $a_{\eta N}\approx 1$ fm. The role of handling self consistently the subthreshold $\eta N$ interaction is carefully studied.


Introduction
The ηN interaction has been studied extensively in photon-and hadroninduced production experiments on free and quasi-free nucleons, and on nuclei [1]. These experiments suggest that the near-threshold ηN interaction is attractive, but are unable to quantify this statement in any precise manner. Nevertheless, η production data on nuclei provide some useful hints on possible η quasibound states for very light species where, according to Krusche and Wilkin (KW) "the most straightforward (but not unique) interpretation of the data is that the ηd system is unbound, the η 4 He is bound, but that the η 3 He case is ambiguous" [1]. Indeed, the prevailing theoretical consensus since the beginning of the 2000s, based on ηNN Faddeev calculations, is that ηd quasibound or resonance states are ruled out for acceptable ηN interaction strengths [2,3]. Instead, the ηd system may admit virtual states [4,5,6]. Searching for reliable few-body calculations of the A = 3, 4 ηnuclear systems, we are aware of none for ηNNNN and of only one ηNNN Faddeev-Yakubovsky calculation [7], although not sufficiently realistic, that finds no η 3 H quasibound state. Rigorous few-body calculations substantiating the KW conjecture quoted above are therefore called for. The present work fills some of this gap, reporting precise calculations of ηNN and of ηNNN few-body systems using the hyperspherical basis methodology [8], similarly to the calculations reported in Ref. [9] for theKNN andKNNN systems. Particular attention is given in the present work to the subthreshold energy dependence of the ηN interaction in a way not explored before in η few-body calculations.  [11]; solid: CS [12]; dotted: KSW [13]; long-dashed: M2 [14]; short-dashed: IOV [15]. The thin vertical line denotes the ηN threshold. Figure adapted from [16].
Theoretically, the ηN interaction has been studied in coupled-channel models that seek to fit or, furthermore, generate dynamically the prominent N * (1535) resonance which peaks about 50 MeV above the ηN threshold. Such models result in a wide range of values for the real part of the ηN scattering length a ηN , from 0.2 fm [10] to almost 1.0 fm [11]. Most of these analyses constrain the imaginary part Im a ηN within a considerably narrower range of values, from 0.2 to 0.3 fm. This is demonstrated in Fig. 1 where the real and imaginary parts of the ηN center-of-mass (cm) scattering amplitude F ηN ( √ s) are plotted as a function of the cm energy √ s for several coupled channel models. The ηN threshold, where F ηN ( √ s th ) = a ηN , is denoted by a thin vertical line. We note that both real and imaginary parts of F ηN ( √ s) below threshold decrease monotonically in all of these models upon going deeper into the subthreshold region, displaying however substantial model dependence. This will become important for the η few-body calculations reported here.
Beginning with the pioneering work by Haider and Liu [17], and using input values of a ηN within these specified ranges, several η-nucleus opticalmodel bound-state calculations concluded that η mesons are likely to bind in sufficiently heavy nuclei, certainly in 12 C and beyond [18,19,20,21,22]. In the few-body calculations reported here we find no ηd quasibound states for values of Re a ηN as large as about 1 fm. We do find, however, a very weakly bound and broad η 3 H-η 3 He isodoublet pair for Re a ηN ≈ 1 fm by solving the ηNNN four-body problem.
The paper is organized as follows. In section 2 we construct local energydependent single-channel potentials v ηN that reproduce two of the s-wave scattering amplitudes F ηN ( √ s) shown in Fig. 1, GW [11] and CS [12]. In section 3 we sketch the hyperspherical-basis formulation and solution of the ηNN and ηNNN Schroedinger equations below threshold using these derived ηN potentials and realistic energy-independent NN potentials. Because of the substantial energy dependence of v ηN in the subthreshold region, a self consistency requirement [9] is applied so that the input energy argument of the two-body potential v ηN for convergent few-body solutions is consistently related to some energy expectation values in the resulting quasibound state. Results are presented and discussed in section 4, followed by a brief summary and outlook section 5.

Construction of ηN effective potentials
We seek to construct energy-dependent local ηN potentials v ηN that reproduce the ηN scattering amplitude F ηN ( √ s) below threshold in given models, e.g. from among those shown in Fig. 1. For convenience, the energy argument E introduced in this section is defined with respect to the ηN threshold, E ≡ √ s − √ s th , and should not be confused with the binding energy of the ηNN and ηNNN few-body states studied in subsequent sections.
with µ ηN the reduced ηN mass and where ρ Λ is a Gaussian normalized to 1: Λ is a scale parameter, inversely proportional to the range of v ηN . Its physically admissible values are discussed in subsection 2.2 below. Two representative values are used here, Λ=2 and 4 fm −1 . For a given value of Λ, one needs to determine the energy-dependent strength parameter b(E) of v ηN , as described in the following subsection 2.1.

Solution
Given a specific value of the scale parameter Λ, the two-body s-wave Schroedinger equation is solved for energies above (E > 0) and below (E < 0) threshold. The radial wavefunction u(r) satisfies the boundary conditions u(r = 0) = 0, u(r → ∞) ∝ r(cos δ 0 j 0 (kr) − sin δ 0 n 0 (kr)), where k = 2µ ηN E, j 0 and n 0 are spherical Bessel and Neumann functions, respectively, and δ 0 (E) is the complex s-wave phase shift derived by imposing these boundary conditions on the wave-equation solution. Above threshold, the wave number k is real and taken positive. Below threshold, k = iκ with κ > 0. The scattering amplitude F is then given by This procedure was used in Ref. [23] for constructing effectiveKN potentials below threshold. In the present case, the subthreshold values of the complex strength parameter b(E) in Eq. (1) were fitted to the complex phase shifts δ(E) derived from subthreshold scattering amplitudes F ηN (E) in several of the coupled-channel models of Fig. 1. This is shown for the GW [11] and CS [12] models in Fig. 2, using two values of the scale parameter Λ=2 and 4 fm −1 for GW and just one value Λ = 4 fm −1 for CS. The curves b(E) are seen to decrease monotonically in going deeper below threshold, except for small kinks near threshold that reflect the threshold cusp of Re F ηN (E = 0) in Fig. 1. Comparing models GW and CS for the same scale parameter Λ = 4 fm −1 , one observes larger values of b(E) in model GW than in CS, for both real and imaginary parts below threshold, in line with the larger GW subthreshold amplitudes compared with the corresponding CS amplitudes. We note furthermore that Im b(E)≪Re b(E) in both models by almost an order of magnitude, see Fig. 2, which justifies treating Im v ηN perturbatively in the applications presented below.  (1), for subthreshold energies E < 0, obtained from the scattering amplitudes F GW ηN [11] and F CS ηN [12] shown in Fig. 1. Two choices of the scale parameter Λ are made for GW, both resulting in the same F GW ηN (E), and just one for CS.
To demonstrate the extent to which the energy dependence of b(E) is essential, we compare in Fig. 3 the GW subthreshold amplitude from Fig. 1, which is also generated here using the b(E) potential strength of Fig. 2 for Λ = 4 fm −1 , to the amplitude marked gw which was calculated using a fixed threshold value b(E = 0). This latter amplitude is seen to decrease too slowly beginning about E ≈ −7 MeV. Obviously, an energy-independent single- channel potential v ηN fails to reproduce the subthreshold energy dependence of the GW coupled-channel scattering amplitude F GW ηN (E).

Choice of scale
It is appropriate at this point to address the model dependence introduced in η-nuclear few-body calculations by the choice of the scale parameter Λ made in constructing v ηN , Eqs. (1) and (2). Λ is often identified with the momentum cutoff used to renormalize divergent loop integrals in on-shell EFT N * (1535) models [14,15]. In separable-interaction coupled channel models, however, the momentum cutoff is replaced by fitted Yamaguchi form factors (q 2 + Λ 2 ) −1 with a momentum-space range parameter Λ, the Fourier transform of which is a Yukawa potential exp(−Λr)/r with r.m.s radius identical to that of the Gaussian potential shape (2). Values of Λ from three such N * (1535) models, including the two used in the present work [11,12], are listed in Table 1.
Inspection of Table 1 reveals a broad range of values that Λ may assume, starting with Λ ≈ 3 fm −1 . The relatively high value in the third column is rather exceptional for meson-baryon separable models. Given this broad spectrum of values spanned for Λ, we chose two values Λ = 2 and Λ = 4 fm −1 to study the model dependence of our η-nuclear few-body calculations. The higher value, Λ = 4 fm −1 , corresponds to a Gaussian exp(−r 2 /R 2 ) spatial range R = 2/Λ = 0.5 fm, a value which is very close to R = 0.47 fm taken from the systematic EFT approach in Ref. [23] and used in ourK-nuclear few-body calculations [9]. As argued there, choosing smaller values for R, namely larger values than 4 fm −1 for Λ, would be inconsistent with staying within a purely hadronic basis. 1 In the Introduction section we loosely identified the strength of the ηN interaction with the size of the real part of its threshold scattering amplitude, Re a ηN 1 fm. However, in terms of the interaction potentials v ηN that enter our few-body calculations, a given value of Re a ηN does not rule out a broad spectrum of spatial ranges, or equivalently momentum scale parameters Λ, as demonstrated in Fig. 2. A model dependence is thereby introduced into our few-body calculations, summarized by stating that the larger the ηN scale parameter Λ is, the larger is the η separation energy, provided it is quasibound. This lack of scale invariance hints towards the necessity of including three-body forces, as is expected from an EFT point of view [25]. Such three-body forces amount to adding a new free parameter determined by tuning it to some η few-body experimental data. 1 The effective energy-dependentKN potential vK N constructed by Hyodo and Weise [23] reproduces theKN − πΣ coupled-channel scattering amplitude which is the one essential for generating dynamically the Λ * (1405) resonance. In that case, the choice of Λ must ensure that theK * N channel that couples strongly toKN via normal pion exchange is kept outside of the model space in which vK N is valid. This argument leads to a choice of Λ = p min (KN →K * N ) = 552 MeV/c or 2.8 fm −1 , corresponding to a Gaussian spatial range of R = 0.71 fm. In a somewhat similar reasoning Garzon and Oset [24] recently argued for extending the EFT description of the N * (1535) resonance to include the ρN channel which couples strongly to the already included πN channel, although not to ηN . Identifying Λ with the minimum momentum needed to excite the πN system to ρN , we obtain Λ = p min (πN → ρN ) = 586 MeV/c or 3.0 fm −1 .

η-nuclear hyperspherical-basis formulation and solution
The hyperspherical-basis formulation of meson-nuclear few-body calculations was initiated in Ref. [9] forK mesons. Here we sketch briefly the necessary transformation fromK mesons to η mesons. The N-body wavefunction (N = 3, 4) in our case consists of a sum over products of isospin, spin and spatial components, antisymmetrized with respect to nucleons. In the spatial sector translationally invariant basis functions are constructed in terms of one hyper-radial coordinate ρ and a set of 3N − 4 angular coordinates [Ω N ], substituting for N − 1 Jacobi vectors. The spatial basis functions are of the form Φ n, where R N n (ρ) are hyper-radial basis functions expressible in terms of Laguerre polynomials and Y N [K] (Ω N ) are hyperspherical-harmonics (HH) functions in the angular coordinates Ω N expressible in terms of spherical harmonics and Jacobi polynomials. Here, the symbol [K] stands for a set of angular-momentum quantum numbers, including those ofL 2 ,L z andK 2 , whereK is the total grand angular momentum which reduces to the total orbital angular momentum for N = 2. The HH functions Y N [K] are eigenfunctions ofK 2 with eigenvalues K(K + 3N − 5), and ρ K Y N [K] are harmonic polynomials of degree K [8].
For the NN interaction we used two forms, the (Minnesota) MN central potential [26] and the Argonne AV4' potential [27] derived from the full AV18 potential by suppressing the spin-orbit and tensor interactions and readjusting the central spin and isospin dependent interactions. In s-shell nuclei the AV4' potential provides an excellent approximation to AV18 which pseudoscalar mesons, such as the η meson, are unlikely to spoil, recalling that their nuclear interactions cannot induce S ↔ D mixing beyond that already accounted for by the NN interaction. 2 AV4' and MN differ mostly in their short-range repulsion which is much stronger in AV4' than in MN.
For the ηN interaction we used the energy-dependent local potential Re v ηN introduced in Sect. 2. In order to distinguish the energy E of the few-body system from the energy argument of v ηN , the latter is replaced by δ √ s ≡ √ s − √ s th from now on. Following Eq. (5) in [9], the subthreshold energy argument δ √ s of v ηN , is chosen to agree self-consistently with where ξ N (η) ≡ m N (η) /(m N + m η ), T η is the η kinetic energy operator in the total cm frame, T N :N is the pairwise NN kinetic energy operator in the NN pair cm frame, B is the total binding energy of the η-nuclear few-body system and B η is the η "binding energy", where H N is the Hamiltonian of the purely nuclear part in its own cm frame and the total Hamiltonian H is evaluated in the overall cm frame. In the limit A ≫ 1, Eq. (7) agrees with the nuclear-matter expression given in Refs. [21,22] for use in calculating η-nuclear quasibound states. It provides a self-consistency cycle in η-nuclear few-body calculations by requiring that the expectation value δ √ s derived from the solution of the Schroedinger equation agrees with the input value δ √ s used in v ηN . Since each one of the four terms on the r.h.s. of (7) is negative, the self consistent energy shift δ √ s s.c. is necessarily negative, with size exceeding a minimum nonzero value obtained from the first two terms in the limit of vanishing η binding. The potential and kinetic energy matrix elements for a given η-nuclear state with global quantum numbers I, L, S, J π were evaluated in the HH basis. The NN and ηN interactions specified above conserve I = I N , S = S N and L. Since no L = 0 η-nuclear states are likely to come out particle stable, our calculations are limited to L = 0. The deuteron in this approximation is a purely 3 S 1 state. Suppressing Im v ηN , the g.s. energy E g.s. was calculated in a model space spanned by HH basis functions with eigenvalues K ≤ K max . Self-consistent calculations were done for √ s ranging from the ηN threshold down to 30 MeV below. Self consistency in δ √ s was reached after a few cycles. Good convergence was achieved for values of K max ≈ 20 − 40. Asymptotic values of E g.s. were found by fitting the constants C and α of the parametrization E(K max ) = E g.s. + C exp(−αK max ) (8) to values of E(K max ) calculated for sufficiently high values of K max . The accuracy reached is better than 0.1 MeV in both the three-body and the four-body calculations reported here.
The conversion width Γ was then evaluated through the expression where V ηN sums over all pairwise ηN interactions. Since |Im V ηN | ≪ |Re V ηN |, this is a reasonable approximation for the width.

Results and discussion
Results of ηNN and ηNNN bound-state hyperspherical-based calculations for the GW ηN interaction, with Re a ηN almost 1 fm, are given in this section. The weaker CS ηN interaction is found too weak to generate bound-state solutions.

ηNN calculations
No I = 0, J π = 1 − ηd bound state solution was found for the ηNN threebody system using the MN NN potential [26] and the GW [11]  Given that the ηN interaction is too weak to bind the I = 0, J π = 1 − ηNN state in which the 3 S 1 NN (deuteron) core configuration is bound, the unbound 1 S 0 NN core configuration in the I = 1, J π = 0 − ηNN state certainly cannot support a three-body bound state. This holds so long as the 1 − state is unbound and also for a certain range of larger ηN potential strengths that make the 1 − bound. This situation is reminiscent of the ΛNN system which is known to have one I = 0 bound state in which the Λ hyperon is bound to a deuteron core, but no I = 1 ΛNN bound state, see e.g. Ref. [29].
Our negative results rule out any ηd bound state, practically in all dynamical models of the N * (1535) resonance where the ηN interaction is coupled in, and are consistent with similar conclusions reached in Refs. [2,3,4,5,6].
This holds also upon replacing the MN NN interaction [26] by the AV4' NN interaction [27] in our ηNN calculations. In fact, somewhat larger ηN interaction multiplicative factors are then required to reach the onset of ηNN binding compared to those specified above. Applying the self-consistency requirement discussed in Sect. 3 to the ηNN calculation, and recalling the decreased strength b(δ √ s) in the ηN subthreshold region, see Fig. 2, would only aggravate the failure to generate a three-body ηNN bound state.
g.s. . Also listed in the table are the self consistent values δ √ s s.c. and the self-consistency reduc- were found using v GW ηN self consistently for Λ = 2 fm −1 .  In order to demonstrate how the self consistency procedure works we plotted in Fig. 4  Applying a similar self-consistency procedure to the weaker CS ηN interaction, rather than to the GW ηN interaction used above, no ηNNN bound state solution was found. With AV4' for the NN interaction, this holds even upon using the threshold energy value in v CS ηN . With the MN NN interaction and for the choice Λ = 4 fm −1 , a bound-state solution is found for small values of the input energy δ √ s, disappearing at −δ √ s ≈ 2.8 MeV which is way below the minimum value of −δ √ s required in the limit of E η sep. → 0. We conclude that the CS ηN interaction is too weak to provide self consistently ηNNN bound states. Finally, the ηNNN width Γ s.c. g.s. ∼ 15 MeV listed in the last column of Table 2 was calculated using Im b(δ √ s s.c. ) in forming the integrand Im V ηN in Eq. (9). This width is about three times larger than the widths evaluated self consistently using optical model methods across the periodic table within the ηN GW model [21]. Some explanation of this difference is offered noting that the magnitude of the downward energy shifts δ √ s s.c. effective in those works is considerably larger by factors of two to three than the ≈15 MeV found in the present ηNNN calculations, reflecting the denser nuclear environment encountered by the η meson as it becomes progressively more bound in the calculations of Ref. [21]. Recalling the steady decrease of the ηN absorptivity Im F ηN in Fig. 1 upon moving deeper into subthreshold energies, a factor of two to three difference could be anticipated in favor of relatively small η widths in heavier nuclei.

Summary and outlook
Precise hyperspherical-based few-body calculations were reported in this work to explore computationally whether or not η mesons bind in light nuclei. To this end, complex energy-dependent local effective ηN potentials v ηN were constructed, for subthreshold energies relevant to η mesic nuclei, from coupled channel ηN scattering amplitudes in several models connected dynamically to the N * (1535) resonance. The scale dependence arising from working with an effective v ηN was studied by using two representative values for the momentum scale, Λ = 2, 4 fm −1 . Noting that Im v ηN ≪ Re v ηN , only the real part of v ηN was used in the bound-state calculations, with a related error estimated as less than 0.2 MeV, added to an estimated 0.1 MeV calculational error. The width of the bound state, making it into a quasibound state, was deduced from the expectation value of Im v ηN summed on all nucleons.
No ηNN quasibound states were found for any of the two scale parameters chosen in models where the real part of the ηN threshold interaction satisfies Re a ηN 1 fm, in agreement with deductions made in several past few-body calculations of the ηd scattering length [2,3,4,5,6]. It is unlikely that the ηd system can reach binding upon increasing moderately the momentum scale parameter Λ.
For ηNNN, essentially the η 3 H and η 3 He isodoublet of η mesic nuclei, a relatively broad and weakly bound state was found with η separation energy of less than 1 MeV using the GW ηN interaction model [11] where Re a ηN is almost 1 fm. This holds for the larger of the two values of momentum scale parameter, Λ = 4 fm −1 , studied here, whereas no bound state was obtained upon using the smaller value of Λ = 2 fm −1 . The energy dependence of v GW ηN , treated here within a self consistent procedure [21,22], played an important role by reducing the calculated binding energy by about 2 MeV from that calculated upon using the ηN threshold energy value in v GW ηN . For such halolike η-nuclear quasibound states, the neglect of Im v ηN in the bound-state calculation requires attention. In the case of the GW ηN effective interaction, we estimate the repulsion added by reinstating Im v GW ηN to second order to be roughly 0.2 MeV, eliminating thereby the very weakly bound ηNNN state calculated here using the AV4' NN potential, but not the weakly bound one calculated using the MN NN potential. It is worth noting that the only other few-body ηNNN study known to us [7] deduced from their calculated η 3 H scattering length that no quasibound state was likely. However, the strength of the ηN interaction tested in these calculations was limited to Re a ηN = 0.75 fm, short of our upper value of approximately 1 fm.
In conclusion, recalling the KW conjecture [1] quoted in the Introduction, it is fair to say that the present few-body calculations support the conjecture's first and last items, namely that "the ηd system is unbound" and "that the η 3 He case is ambiguous". Accepting that the strength of the two-body ηN interaction indeed satisfies Re a ηN 1 fm, which is much too weak to bind the ηN system, a persistent theoretical ambiguity connected with choosing a physically admissible range of values for the ηN scale parameter Λ is demonstrated by our few-body calculational results, particularly for the four-body ηNNN system. By choosing a considerably larger value of Λ than done here one could bind solidly this system. To remove this ambiguity, many-body repulsive interactions involving the η meson need to be derived and incorporated within few-body calculations.
In future work we hope to extend our ηNNN calculations also by applying methods of complex scaling that should enable one to follow trajectories of S-matrix quasibound-state poles and look also for other types of poles such as virtual-state poles or resonance poles, all of which affect to some degree the threshold production features of η mesons in association with 3 He. Furthermore we hope to initiate a precise and realistic calculation of the ηNNNN system in order to test the middle item in the KW conjecture, namely that "η 4 He is bound".