Calculations of η-nuclear quasi-bound states in few-body systems

We report on our Stochastic Variational Method (SVM) calculations of η-nuclear quasi-bound states in s-shell nuclei as well as the very recent calculation of the p-shell nucleus 6Li. The ηN potentials used were constructed from ηN scattering amplitudes obtained within coupled-channel models that incorporate N(1535) resonance. We found that η6Li is bound in the ηN interaction models that yield ReaηN ≥ 0.67 fm. Additional repulsion caused by the imaginary part of ηN potentials shifts the onset of η-nuclear binding to η4He, yielding very likely no quasi-bound state in η3He.


Introduction
The current status of our theoretical studies of η-nuclear quasi-bound states, including discussion of the self-consistent treatment of the strong energy dependence of ηN scattering amplitudes derived from coupled-channel meson-baryon interaction models have been discussed thoroughly in Refs. [1][2][3]. So far, few-body calculations of η-nuclear quasi-bound states have been restricted to s-shell nuclei up to η 4 He. In this contribution, we present our first SVM calculation of the η-nuclear quasi-bound state in the p-shell nuclear system η 6 Li, taking into account all possible spin-isospin configurations. Moreover, we focus on the effect of the imaginary part of the complex V ηN potential on the η binding energy B η . We show that the effect could be considerable in light η-nuclear systems and must be taken into account in the study of the onset of η-nuclear binding.

Theoretical approach
Properties of η-nuclear quasi-bound states are studied within the SVM with a correlated Gaussian basis [4]. This approach was successfully applied in our previous calculations of s-shell η-nuclei and proved itself as highly accurate method with straightforward extension to systems with the number of particles N ≥ 5.
The wave function of an η-nuclear system with orbital momentum L = 0 is expanded as a linear combination of correlated Gaussians * e-mail: m.schafer@ujf.cas.cz where x stands for Jacobi coordinates and χ k S (ξ k T T z ) are corresponding spin (isospin) parts of a given spin (isospin) configuration. The matrix A k is symmetric positive definite and includes N(N − 1)/2 variational parameters. The SVM optimizes the variational basis step-by-step in a random trial and error procedure (details can be found in Ref. [5]).
SVM calculations of η-nuclear quasi-bound states in p-shell nuclei represent a rather challenging task. First, the computational complexity scales with N!, second, the amount of different spin-isospin configurations starts to increase quite rapidly. Preliminary results [6] showed that taking into account only one configuration underestimated binding of the nuclear core 6 Li by approximately 1.8 MeV. This led to development of a new high-performance SVM code which was used in the very recent fully self-consistent calculation of η 6 Li, taking into account all posible spin-isospin configurations.
In our study of η-nuclear quasi-bound states we use the Minnesota NN central potential [7] which reproduces well properties of the ground states of s-shell and light p-shell nuclei. The interaction of the η meson with nucleons is described by a complex two-body energy dependent effective potential derived from the coupled-channel meson-baryon interaction models GW [8] and CS [9]. The form of ηN potential is taken according to [1] as where µ ηN stands for the ηN reduced mass, δ √ s = √ s − √ s th is the energy shift with respect to the ηN threshold, Λ is a scale parameter which is inversely proportional to the range of V ηN , and b(δ √ s) is an energy dependent complex amplitude. The value of Λ is connected to EFT momentum cut-off; its upper bound corresponds to vector-meson exchange Λ ≤ 3.9fm −1 or more restrictively to Λ ≤ 3.0 fm −1 excluding ρN channel from dynamical generation of the N (1535) resonance [3].
For given Λ, b(δ √ s) is fitted to the phase shifts derived from subthreshold δ √ s < 0 scattering amplitude of the corresponding ηN interaction model. See Ref. [3] for details.
The energy dependence of V ηN is treated self-consistently: we search for a SVM solution that fulfills δ where B is the total binding energy, T N (T η ) denotes the kinetic energy of nucleons (η), and A is the number of nucleons. The energy E η = ψ| H − H N |ψ where H N is Hamiltonian of the nuclear core, ξ N(η) = m N(η) /(m N + m η ), and ξ A = Am N /(Am N + m η ).
The imaginary part of V ηN is significantly smaller than its real part. This allows to calculate the width Γ η perturbatively [1]. The SVM η-nuclear calculations are thus performed only for the real part of the ηN potential and Γ η is evaluated using the expression where Ψ g.s. is the SVM solution for the η-nuclear ground state corresponding to ReV ηN . Another possible way how to calculate Γ η is to solve a generalized eigenvalue problem for complex Hamiltonian (including ImV ηN ) using variationally determined SVM basis states for ReV ηN . This approach, already used in SVM calculations of kaonic nuclei [10], yields complex eigenenergy of the ground state E = Re(E) + iIm(E) and consequently the width as Γ η = −2Im(E). This method takes into account the effect of the non-zero imaginary part of V ηN on the η binding energy. Namely, ImV ηN acts as repulsion and thus makes the η meson less bound in the nucleus.

Results
Results of our SVM calculations of the η binding energies B η and widths Γ η in η 3 He, η 4 He, and η 6 Li are summarized in Fig. 1. The calculations were performed using the GW and CS models and the parameter Λ = 2 and 4 fm −1 . In the GW model, η 6 Li is rather comfortably bound for both values of Λ. On the other hand, the CS model yields η-nuclear quasi-bound state only for Λ = 4 fm −1 , with B η = 0.68 MeV.   In Table 1, we compare two approaches to evaluation of the width Γ η introduced in the previous section: the mean-value approach (Eq. 4) and the complex eigenvalue problem (cmplx) approach. Calculations of η 3 He and η 4 He were performed within the GW model with Λ = 2 and 4 fm −1 . It is apparent that the effect of the imaginary part of V ηN on B η , which is included in the cmplx approach, is quite significant in η 3 He (with δ √ s close to threshold) and decreases in η 4 He with larger energy shift with respect to threshold. For the CS model (not shown in the table) the η 3 He is not bound while in η 4 He the effect of ImV ηN is smaller (few tens of keV) due to the lower value of ImV ηN than in the GW model. Table 1 illustrates that the size of the changes of B η caused by ImV ηN decreases with the magnitude of the subthreshold energy shift δ √ s. Namely, the strength of ImV ηN has for both CS and GW interaction models maximum close to threshold and decreases with √ s, as shown in Figure 2 of Ref. [3]. Moreover, the cmplx method confirms the estimate of Γ η within the mean-value approach (Eq. 4), giving practically the same widths in all considered cases.

Summary
We performed few-body calculations of η-nuclear quasi-bound states in s-shell nuclei as well as in the p-shell nucleus 6 Li within our newly developed high-performance SVM code. We considered the Minnesota NN potential and two ηN interaction models -GW and CS. Calculations of η 6 Li within the GW model yield the binding energy B η and corresponding width consistent with previous RMF calculations [11]. The CS model gives quasi-bound state only for Λ = 4 fm −1 . This suggests that to bind η 6 Li, the real part of the ηN scattering length should be greater than Rea ηN = 0.67 fm, predicted by the CS model.
Next, we repeated our previous study of the onset of η-nuclear binding in He isotopes taking into account the effect of ImV ηN on the binding energy B η . We observed considerable decrease of B η in 3 η He and rather negligible effects in 4 η He as well as in 6 η Li. The η meson is barely bound in 3 η He even for the larger value of the cut-off parameter Λ = 4 fm −1 . This indicates that in order to study the η 3 He system, one has to explore the resonance region as well, e.g., using the complex rotation method [12].
M. Schäfer acknowledges financial support from the CTU-SGS Grant No. SGS16/243/OHK4/3T/14 and from the organizers of the MESON2018 conference.