η N interactions in the nuclear medium. η -nuclear bound states

. We report on our recent study of in-medium η N interactions and η -nuclear quasi-bound states. The η N scattering amplitudes considered in the calculations are constructed within coupled-channel models that incorporate the S 11 N ∗ (1535) resonance. The implications of self-consistent treatment and the role played by subthreshold dynam-ics are discussed.


Introduction
The ηN attraction generated by the N * (1535) resonance near threshold seems to be strong enough to allow binding of the η meson in nuclei. However, in-medium modifications and strong energy dependence of the ηN scattering amplitudes have to be carefully taken into account. This contribution briefly summarizes systematic treatment of energy dependence within self-consistent calculations of η quasi-bound states in selected nuclei (see [1][2][3] for more details).
The in-medium amplitudes which serve as an input in our many-body self-consistent calculations are obtained from the free-space amplitudes GW and M2 by applying the multiple scattering approach [8] (see Ref. [2] for details). In the GR and CS models, the Pauli principle restricts integration domain in the Green's function which enters the underlying Lippmann-Schwinger equations [2].
The strong energy dependence of the ηN scattering amplitudes F ηN ( √ s) has to be treated selfconsistently [1,2]. The argument √ s in the scattering amplitudes is given by where √ s th ≡ m η + m N and B η and B N are η and nucleon binding energies. In the nuclear medium (for A 1 approximated by the lab system) the momentum dependent term causes additional downward energy shift, since ( p η + p N ) 2 0, which can be approximated as [2]  where MeV is the average nucleon binding energy, andρ is the average nuclear density. For attractive scattering amplitudes, all terms in Eq. 2 are negative definite, providing substantial downward energy shift. A variant of Eq. 2 was used in η-nuclear three-and four-body calculations (see Ref. [3] for details).
In few-body ηNN and ηNNN systems, the η-nuclear cluster wave functions were expanded in a hyperspherical basis and the ground-state binding energies were calculated variationally. For the NN interaction, the Minnesota central potential [9] and the Argonne AV4' potential [10] were used. The ηN interaction was described by energy dependent local ηN potentials that reproduce the ηN scattering amplitudes below threshold in considered interaction models [3].
The interaction of the η meson with the nuclear many-body system was described by the Klein-Gordon (KG) equation of the form whereω η = ω η −iΓ η /2 is complex energy of η, ω η = m η −B η , and Γ η is the width of the η-nuclear bound state. The self-energy operator ρ was constructed selfconsistently using the RMF density distributions in a core nucleus.
It is to be stressed that ReF ηN ( √ s) and B η appear as arguments in the expression for δ √ s (Eq. 2), which in turn serves as an argument for F ηN and thus for the self-energy Π η . Therefore, a selfconsistency scheme in terms of both Π η and B η is required in calculations.

Results
Our few-body calculations of the ηNN system found no bound states in the considered coupledchannel models. For ηNNN, a relatively broad and weakly bound state (with η separation energy below 1 MeV) was found for the Minesota NN potential and one particular variant of the ηN potential that reproduced the GW scattering amplitudes (see Ref. [3] for details). No ηNNN bound states were found using more realistic NN interaction models.  Figure 2. Binding energies (left) and widths (right) of the 1s η-nuclear states in selected nuclei calculated using the GR ηN scattering amplitude [7] with different procedures for subthreshold energy shift δ √ s. Figure 2 illustrates the role of the energy dependence of ηN scattering amplitudes in self-consistent evaluations of η nuclear-states in many-body nuclear systems. A comparison is made for the inmedium GR amplitude: our self-consistency scheme based on δ √ s of Eq. 2 (marked δ √ s) reduces considerably the GR binding energies and widths with respect to the original calculations of Ref. [7] that used the δ √ s = −B η procedure (marked −B η ). However, even the reduced GR widths are still quite high, suggesting that η-nuclear states will be extremely difficult to resolve if the GR model is the realistic one.
The model dependence of the ηN scattering amplitudes shown in Fig. 1 manifests itself in the calculations of η-nuclear states. Figure 3 presents binding energies B η and widths Γ η calculated for the 1s η-nuclear states in selected nuclei using the above ηN amplitudes. The left panel of Fig. 3 demonstrates that for each of the ηN amplitude models the binding energy increases with A and tends to saturate for large values of A. The hierarchy of the curves reflects the strength of ReF ηN ( √ s) in the subthreshold region (see Fig. 1). The M2 amplitude is too weak to produce the 1s η bound state in 12 C. In contrast, ReF ηN ( √ s) of the GW model is strong enough to bind η in 12 C and even in lighter nuclei, e.g., it predicts the 1s η bound state in 4 He with B η = 1.2 MeV and Γ η = 2.3 MeV (calculated using a static 4 He density).
The right panel shows substantial differences between the widths Γ η calculated using the above mentioned models. The CS and GW models yield relatively small uniform widths of order 2 and 4 MeV, respectively. On the other hand, the GR and M2 models predict much larger widths which increase with A. This reflects partly the energy dependence of ImF ηN ( √ s) in the subthreshold region and partly the difference in the in-medium renormalization stemming from ReF ηN ( √ s). For instance, the large downward energy shift due to the subthreshold amplitude in the GW model (57 MeV at ρ 0 ) causes a particularly large reduction in the strength of the ImF ηN ( √ s) input. The widths calculated here do not include contributions from two-nucleon processes which are estimated to add a few MeV. We may therefore conclude that η-nuclear states could in principle be observed if the CS and GW models turn out to be realistic ones, provided a suitable production/formation reaction is found. Other models give either too large widths or are too weak to generate η-nuclear bound states in lighter nuclei. This work was supported by the GACR Grant No. P203/15/04301S, as well as by the EU initiative FP7, Hadron-Physics3, under the SPHERE and LEANNIS cooperation programs.