12 pages, 2 figures. Presented at the International Symposium on Meson Physics, held on 1 -4 October, 2008 in Krakow, Poland. To appear in Acta Physica Polonica B
Formation of eta-mesic nucleus, a bound state of an eta meson in a nucleus, is reviewed in this paper. Three different theoretical approaches are used to calculate the binding energies and widths of such nuclei. The effect of eta-mesic nucleus in pion double-charge-exchange reaction is discussed. Experimental efforts by different groups to detect the nucleus are also discussed. The ramifications of the theoretical and experimental studies of the bound state of eta in a nucleus are pointed out.
We calculate formation spectra of eta-nucleus systems in (pi,N) reactions with nuclear targets, which can be performed at existing and/or forthcoming facilities, including J-PARC, in order to investigate eta-nucleus interactions. Based on the N^*(1535) dominance in the eta N system, eta-mesic nuclei are suitable systems for study of in-medium properties of the N^*(1535) baryon resonance, such as reduction of the mass difference of N and N^* in nuclear medium, which affects level structure of the eta and N^*-hole modes. We find that clear information on the in-medium N^*- and eta-nucleus intera…
Even if the theoretical definition of an unstable state is straightforward, its experimental identification often depends on the method used in the analysis and extraction of data. A good example is the case of eta mesic nuclei where strong hints of their existence led to about three decades of extensive theoretical and experimental searches. Considering the still undecided status of these states and the limitations in the understanding of the eta-nucleon as well as the eta-nucleus interaction, the present article tries to look back at some unresolved problems in the production mechanism and f…
We calculate theoretically the formation spectra of eta'(958)-nucleus systems in the (p,d) reaction for the investigation of the in-medium modification of the eta' mass. We show the comprehensive numerical calculations based on a simple form of the eta' optical potential in nuclei with various potential depths. We conclude that one finds an evidence of possible attractive interaction between eta' and nucleus as peak structure appearing around the eta' threshold in light nuclei such as 11C when the attractive potential is stronger than 100 MeV and the absorption width is of order of 40 MeV or l…
We discuss the role of QCD symmetries and confinement in understanding eta and etaprime mesic nuclei. Eta and etaprime bound states in nuclei are sensitive to the flavour-singlet component in the meson. The bigger the singlet component, the more attraction and the greater the binding. Recent results on the etaprime mass in nuclei from the CBELSA/TAPS collaboration are very similar to the prediction of the Quark Meson Coupling model. In the model eta-etaprime mixing induces a factor of two enhancement of the eta-nucleon scattering length relative to the prediction with a pure octet eta, with re…
The missing-mass spectrum obtained in a recoil-free transfer reaction p($^{27}$Al,$^{3}$He)$π^{-}$p'X is analyzed. We find that the observed peak structure arises from the coherent contributions from two reaction processes in the energy region corresponding to a bound eta ($η$) meson. In one of the processes the intermediate $η$ is captured by the nucleus to form the $η$-mesic nucleus $^{25}$Mg$_η$. In the other process, the $η$ does not form $η$-nucleus bound state. The interference between these two processes has caused the peak of the spectrum to appear at an $η$ binding energy stronger tha…
The nuclear states of \bar K and ηare bound by a similar mechanism - the excitations of nucleons to Λ(1405) and N*(1535) resonant states. The observed large differences in binding energies are understood in terms of separation of the involved energies and the resonance positions. The other experimental findings: broad \bar K-mesic and narrow η-mesic states are more difficult to understand. A phenomenological model for η-N interactions is used to explain the suppression of the ηabsorption in light nuclei.
Using the quark-meson coupling (QMC) model, we investigate whether $ω, η, η'$ and $D^-$ mesons form meson-nucleus bound states. Our results suggest that one should expect to find $η$- and $ω$-nucleus bound states in all the nuclei considered. Furthermore, it is shown that the $D^-$ meson will form quite narrow bound states with $^{208}$Pb.
We extend the method of time delay proposed by Eisenbud and Wigner, to search for unstable states formed by eta mesons and the 3He nucleus. Using few body equations to describe eta-3He elastic scattering, we predict resonances and unstable bound states within different models of the eta-N interaction. The eta-3He states predicted within this novel approach are in agreement with the recent claim of the evidence of eta-mesic 3He made by the TAPS collaboration.
Eta-mesic nucleus or the quasibound nuclear state of an eta ($η$) meson in a nucleus is caused by strong-interaction force alone. This new type of nuclear species, which extends the landscape of nuclear physics, has been extensively studied since its prediction in 1986. In this paper, we review and analyze in great detail the models of the fundamental $η$--nucleon interaction leading to the formation of an $η$--mesic nucleus, the methods used in calculating the properties of a bound $η$, and the approaches employed in the interpretation of the pertinent experimental data. In view of the succes…
Using the Green function method, binding effects on produced eta-mesons in the two-stage reaction γ+ A \to N + η+ (A-1) \to N + (πN) + (A-2) are studied. The energy spectrum of the correlated pi-N pairs which arise from decays of etas inside the nucleus is strongly affected by an attractive eta-nucleus optical potential. Its resonant behavior gives a clear signal of formating intermediate eta-mesic nuclei.
With the eta-nucleon (eta N) interaction Lagrangian deduced from chiral perturbation theory, we study the possible eta-mesic nuclei in the framework of relativistic mean-field theory. The eta single-particle energies are sensitive to the eta N scattering length, and increase monotonically with the nucleon number A. If the scattering length is in the range of a^{eta N}=0.75-1.05 fm and the imaginary potential V_{0}-15 MeV, some discrete states of C, O and Ne eta bound states should be identified in experiments. However, when the scattering length a^{eta N}< 0.5 fm, or the imaginary potential V_…
The structure of eta-nucleus bound systems (eta mesic nuclei) is investigated as one of the tools to study in-medium properties of the N(1535) (N*) resonance. We show that, as a general consequence, the eta-nucleus potential has a repulsive core at the nuclear center with an attractive part at the nuclear surface, if sufficient reduction of the mass difference of N and N* stems from the in-medium effects to N*. The (d,3He) spectra are evaluated for the formation of these bound states to investigate the experimental feasibility.
The structure of eta-nucleus bound systems (eta mesic nuclei) is investigated as one of the tools to study in-medium properties of the N(1535) (N^*) resonance by using the chiral doublet model to incorporate the medium effects of the N^* resonance in a chiral symmetric way. We find that the shape and the depth of the eta-nucleus optical potential are strongly affected by the in-medium properties of the N^* and the nucleon. Especially, as a general feature of the potential, the existence of a repulsive core of the eta-nucleus potential at the nuclear center with an attractive part at the nuclea…
Using the quark-meson coupling (QMC) model, we investigate theoretically whether $η$ and $ω$ mesons form meson-nucleus bound states. We study several nuclei from $^{6}$He to $^{208}$Pb, including those which are the final nuclei in the proposed experiment at GSI. Results for the $ω$ are compared with those of the Walecka model. Our results suggest that one should expect to find $η$- and $ω$-nucleus bound states in all these nuclei. Furthermore, we investigate the possibility of charmed mesic nuclei. It is shown that the $D^-$ meson will inevitably form narrow bound states with $^{208}$Pb.
The binding system of a hadron and a nucleus is a topic of great interest for investigating the hadron properties. In the heavy-flavor region, the attraction between a $P(=\bar{D},B)$ meson and a nucleon $N$ can appear, where the $PN-P^\ast N$ mixing plays an important role in relation to the heavy-quark spin symmetry. The attraction can produce exotic heavy mesic nuclei that are stable against the strong decay. We study an exotic system where the $\bar{D}$ ($B$) meson and nucleus are bound. The meson-nucleus interaction is given by a folding potential with single-channel $PN$ interaction and …
A microscopic treatment of eta-nucleus system is presented. Using this, the poles of the amplitude describing elastic scattering of eta-meson off the light nuclei ^2H, ^3H, ^3He, and ^4He have been located in the second quadrant of the complex plane of the eta-nucleus relative momentum. The results obtained indicate that, within the existing uncertainties in the eta-Nucleon input data, a bound state could exist. By a qualitative consideration it is shown that such bound-states, cannot be narrow. It is further shown that scattering length calculations cannot provide a definite answer on the que…
We study eta meson properties in the infinite nuclear matter and in atomic nuclei with an emphasis on effects of the eta coupling to N*(1535)--nucleon-hole modes. The N*(1535) resonance, which dominates the low-energy eta-nucleon scattering, can be seen as a chiral partner of the nucleon. The change of the chiral mass gap between the N* and the nucleon in a nuclear medium has an impact on the properties of the eta-nucleus system. If the N*-nucleon mass gap decreases with a density increase (chiral symmetry restoration) the calculations show the existence of the resonance state at the energy ab…
A novel method is proposed to measure eta'(958) meson bound states in 11C nuclei by missing mass spectroscopy of the 12C(p,d) reaction near the eta' production threshold. It is shown that peak structures will be observed experimentally in an inclusive measurement in case that the in-medium eta' mass reduction is sufficiently large and that the decay width of eta' mesic states is narrow enough. Such a measurement will be feasible with the intense proton beam supplied by the SIS synchrotron at GSI combined with the good energy resolution of the fragment separator FRS.
The experimental data of the COSY-GEM Collaboration for the recoil-free transfer reaction p (27Al, 3He) π- p' X, leading to the formation of bound state of eta (η) meson in 25Mg nucleus, is reanalyzed in this paper. In particular, predicted values of binding energy and half-width of the η-mesic nucleus 25Mgη, given by different theoretical approaches, are compared with the ones obtained from the experimental missing mass spectrum. It is found that the spectrum can be explained reasonably well if interference effect of another process, where ηis not bound in 25Mg but is scattered by the nucleus…
We demonstrate that the binding energies and widths of eta-mesic nuclei depend strongly on subthreshold eta-N interaction. This strong dependence is made evident from comparing three different eta-nucleus optical potentials: (1) a microscopic optical potential taking into account the full effects of off-shell eta-nucleon interactions; (2) a factorization approximation to the microscopic optical potential where a downward energy shift parameter is introduced to approximate the subthreshold eta-nucleon interaction; and (3) an optical potential using on-shell eta-nucleon scattering length as the …
The anti-D meson self-energy is evaluated self-consistently, using unitarized coupled-channel theory, by computing the in-medium meson-baryon T-matrix in the C=-1,S=0 sector. The heavy pseudo-scalar and heavy vector mesons, anti-D and anti-D^*, are treated on equal footing as required by heavy quark spin symmetry. Results for energy levels and widths of D^- mesic atoms in 12C, 40Ca, 118Sn and 208Pb are presented. The spectrum contains states of atomic and of nuclear types for all nuclei. anti-D^0--nucleus bound states are also obtained. We find that, after electromagnetic and nuclear cascade, …
We investigate the structure and formation of charmed meson--nucleus systems, with the aim of understanding the charmed meson--nucleon interactions and the properties of the charmed mesons in the nuclear medium. The $\bar{D}$ mesic nuclei are of special interest, since they have tiny decay widths due to the absence of strong decays for the $\bar{D} N$ pair. Employing an effective model for the $\bar{D} N$ and $D N$ interactions and solving the Klein--Gordon equation for $\bar{D}$ and $D$ in finite nuclei, we find that the $D^{-}$-${}^{11}\rm{B}$ system has $1 s$ and $2p$ mesic nuclear states a…
We present a theoretical study on formation rates of ηand ωmeson-nucleus systems induced by (γ,p) reactions on nuclear targets at ideal recoilless condition. We find that the smaller distortion effect in the (γ,p) reaction enables us to investigate properties of the mesons created deeply inside nucleus more clearly. We also consider excitation of scalar-isoscalar (σ) mode in nucleus in order to investigate spectral enhancement around two-pion threshold caused by partial restoration of chiral symmetry. We conclude that valuable information of meson-nucleus interactions can be extracted from glo…
In this paper the search for eta-mesic nuclei with particular focus on light $η$-He bound states is reviewed. A brief description of recent theoretical studies and experimental results is presented.
In this contribution we report on theoretical studies of $η$ nuclear quasi-bound states in few- and many-body systems performed recently by the Jerusalem-Prague Collaboration [1-5]. Underlying energy-dependent $ηN$ interactions are derived from coupled-channel models that incorporate the $N^*(1535)$ resonance. The role of self-consistent treatment of the strong energy dependence of subthreshold $ηN$ amplitudes is discussed. Quite large downward energy shift together with rapid decrease of the $ηN$ amplitudes below threshold result in relatively small binding energies and widths of the calculat…
In this paper, we briefly introduce the interests and the recent research activities on the η-mesic nuclei. We also mention the activities on the η' (958)-mesic nucleus formation.
We propose to use the recoilless (d,$^3$He) reaction to produce $η$-mesic nuclei. This reaction has been used to observe deeply bound pionic states and proven to be powerful recently. We calculate $η$-mesic bound states in the nucleus using an optical potential and their formation cross section with the Green function method. Then, we carefully check the experimental feasibility. We find that $η$-mesic nuclei can be observed experimentally using the (d,$^3$He) reaction. We also mention the possibility to study the formation of $ω$-mesic nuclei.
The $η$ mesic nucleus is considered to be one of the interesting exotic many body systems and has been studied since 1980's theoretically and experimentally. Recently, the formation of the $η$ mesic nucleus in the fusion reactions of the light nuclei such as $d + d \rightarrow (η+ α) \rightarrow X$ has been proposed and the experiments have been performed by WASA-at-COSY. We develop a theoretical model to evaluate the formation rate of the $η$ mesic nucleus in the fusion reactions and show the calculated results. We find that the $η$ bound states could be observed in the reactions in cases wit…
The isoscalar $η$ and $η'$ mesons are special in QCD, being linked both to chiral symmetry and to non-perturbative glue associated with the axial anomaly. The properties of these mesons in medium are sensitive to how these dynamics work in the nuclear environment. In contrast to pionic and kaonic atoms which are mainly bound by the Coulomb interaction with some corrections due to the strong force (Coulomb assisted binding), the $η$ and $η'$ as neutral mesons can only be bound by the strong interaction. Is this interaction strong enough? This topic has inspired a vigorous program of experiments…
We study the structure and formation of the $ϕ$ mesic nuclei to investigate the in-medium modification of the $ϕ$-meson spectral function at finite density. We consider (${\bar p},ϕ$), ($γ,p$) and ($π^-,n$) reactions to produce a $ϕ$-meson inside the nucleus and evaluate the effects of its medium modifications to the reaction cross sections. We also estimate the consequences of the uncertainties of the ${\bar K}$ selfenergy in medium to the $ϕ$-nucleus interaction. We find that it may be possible to see a peak structure in the reaction spectra for the strong attractive potential cases. On the …
We have investigated the 6Li(gamma, d) reaction theoretically for the formation of the eta'(958) mesic nucleus. We have reported the numerical results in this article.
We have investigated the 6Li(gamma,d) reaction theoretically for the formation of the eta'(958) mesic nucleus close to the recoilless kinematics. We have developed the theoretical formula and reported the quantitative results of the formation spectra for various cases in this article. We have found that the formation cross sections are reduced by the effects of the fragile deuteron form factor.
The self-energy of an eta in a nuclear medium is calculated in a chiral unitary model, and applied to eta states in nuclei. Our calculation predicts an attractive eta nucleus optical potential which can accommodate many eta bound states in different nuclei.
The energies and widths of several D^0 meson bound states for different nuclei are obtained using a D-meson selfenergy in the nuclear medium, which is evaluated in a selfconsistent manner using techniques of unitarized coupled-channel theory. The kernel of the meson-baryon interaction is based on a model that treats heavy pseudoscalar and heavy vector mesons on equal footing, as required by heavy quark symmetry. We find D^0 bound states in all studied nuclei, from 12C up to 208Pb. The inclusion of vector mesons is the keystone for obtaining an attractive D-nucleus interaction that leads to the…
Binding energy of eta_c with ^4 He has been calculated on the basis of Yukawa type potential. Final Rank Approximation (FRA) has been used for microscopical treatment of the initial 5-body problem. The results are compared with the results of the folding model and variational calculations.
The T-noninvariant transverse polarization of neutrons is considered for muon capture by a 6-Li nucleus with decay into a quasistationary 2+ state of the three particle alpha+n+n continuum. This polarization is orthogonal to the plane spanned by the polarization axis of the initial mesic atom and the neutron momentum. The situation in which neutrons that are emitted in the plane orthogonal to the axis of the mesic-atom polarization and which have oppositely directed momenta equal in magnitude are selected is analyzed. The wave function of the final state is constructed by using the method of h…
The connection of the binding energy and width of possible bound eta-mesic states is given to the complex scattering length for s-states in the hope that, with knowledge of the final state interaction this could be useful in searches of these states. In spite of the strong direct potential dependence of both observables this connection is seen to be very model independent even for various nuclei once the influence of also the effective range is considered. The importance of this term is pointed also for data analyses. Although the nucleus considered here is 12C, extension to other nuclei is im…
The properties of eta-nucleus interaction and their experimental consequences are investigated with eta-nucleus optical potentials obtained by postulating the N*(1535) dominance for eta-N system. The N*(1535) properties in nuclear medium are evaluated by two kinds of chiral effective models based on distinct pictures of N*(1535). We find that these two models provide qualitatively different optical potentials of the eta meson, reflecting the in-medium properties of N*(1535) in these models. In order to compare these models in physical observables, we calculate spectra of (d,3He) reactions for …
We demonstrate that one can use the occurrence of charge symmetry breaking as a tool to explore the eta--nucleus interaction near the eta threshold. Based on indications that the cross section ratio of pi+ and pi0 production on nuclei deviates from the isotopic value in the vicinity of the eta production threshold, due to, e.g., pi0-eta mixing, we argue that a systematic study of this ratio as a function of the energy would allow to pin down the sign of the real part of the eta-nucleus scattering length. This sign plays an important role in the context of the possible existence of eta-nucleus …
Angular distribution of neutrinos (recoil nucleus) in muon capture for an allowed Gamov-Teller transition is considered by taking account of hyperfine effects. This angular distribution is shown to include a correlation of the form P_2(cos A), where A is the angle between the neutrino momentum and the axis specifying the orientation of the initial mesic atom. This correlation, which arises only if the initial mesic atom is aligned, proves highly sensitive to the form factor g_P of induced pseudoscalar interaction. The proposed method for determinating g_P may be realized for the transition 1+ …
The position and movement of poles of the amplitude for elastic eta-meson scattering off the light nuclei d, t, ^3He, and ^4He are studied. It is found that, within the existing uncertainties for the elementary eta-N interaction, all these nuclei can support a quasi-bound state. The values of the eta-nucleus scattering lengths corresponding to the critical eta-N interaction which produces a quasi--bound state are given.
This is the summary of the 'International Workshop on eta--Nucleus Physics' held in Juelich, May 8--12, 2006. Each talk is represented by an extended abstract. At the same time the document provides a progress report of the corresponding working group.
We develop a model to evaluate the formation rate of the heavy mesic nuclei in the two nucleon pick-up reactions, and apply it to the $^6$Li target cases for the formation of heavy meson-$α$ bound states, as examples. The existence of the quasi-deuteron in the target nucleus is assumed in this model. It is found that the mesic nuclei formation in the recoilless kinematics is possible even for heavier mesons than nucleon in the two nucleon pick-up reactions. We find the formation rate of the meson-$α$ bound states can be around half of the elementary cross sections at the recoilless kinematics …
Back in the mid eighties, a new branch of investigation which was related to the interaction of eta mesons with nuclei came into existence. It started with the theoretical prediction of possible exotic states of eta mesons and nuclei bound by the strong interaction and later developed into an extensive experimental program to search for such unstable states as well as understand the underlying interaction via eta meson producing reactions. The vast literature of experimental as well as theoretical works which studied various aspects of eta producing reactions such as the $π^+$ $n$ $\to ηp$, $p…
In two recent analyses of eta meson production a very small imaginary part was obtained for the eta-He3 scattering length, which is in contradiction with most theoretical predictions. In this report a plausible explanation is given to this unexpectedly weak absorption by suppression of the two main inelasticity channels. It is stressed that the real and imaginary parts of the elementary scattering length do not necessarily always have the same isospin algebraic and spatial properties in a nuclear environment and caution should be exercised in their use in multiple scattering calculations and e…
Due to its dominance in the low energy eta-nucleon interaction, the S11 N$^*$(1535) resonance enters as an important ingredient in the analyses of experiments aimed at finding evidence for the existence of eta-mesic nuclei. The static properties of the resonance get modified inside the nucleus and its momentum distribution is used in deciding these properties as well as the kinematics in the analyses. Here we show that given the possibility for the existence of an N$^*$-$^3$He quasibound state, the relative momentum distribution of an N$^*$ and $^3$He inside such a $^4$He is narrower than that…
The η-meson collision with light nuclei, ^2H, ^3H, ^3He, and ^4He, is considered on the basis of a microscopic approach which utilises Faddeev type and Lippmann-Schwinger equations. The nuclear hamiltonian is approximated by the finite rank approximation which amounts to the coherent approximation. The η-nucleus scattering length and the resonance and bound--state poles of the amplitude describing elastic scattering of η-meson by these nuclei are obtained. For each of the nuclei considered, the minimal factor enhancing the ηN--attraction that moves the poles to the quasi--bound state area of t…
The resonance and bound--state poles of the amplitude describing elastic scattering of $η$-meson off the light nuclei $^2H$,\,$^3H$,\, $^3He$, and $^4He$ are calculated in the framework of a microscopic approach based on few--body equations. For each of the nuclei, the two--body parameters that enhance the $ηN$--attraction which generate quasi--bound states, are also determined.
The existence of etamesic nuclei has been speculated for a long time without firm experimental evidence. Much of the effort has taken place in final state interactions on production. One crucial factor in seeing a quasibound state is its width, which should be related to the imaginary part of the scattering length. Comparing two models for eta-N scattering giving the same elementary scattering length, a simple optical potential and an explicit coupling to the pionic channel, it is seen that the latter yields a much smaller imaginary part for eta-nucleus scattering. This decrease of absorption …
We investigate theoretically whether it is feasible to detect $η$- and $ω$-nucleus bound states. As well as the closed shell nuclei, $^{16}$O, $^{40}$Ca, $^{90}$Zr and $^{208}$Pb, we also investigate $^6$He, $^{11}$B and $^{26}$Mg, which are the final nuclei in the proposed experiment involving the (d,$^3$He) reaction at GSI. Potentials for the $η$ and $ω$ mesons in these nuclei are calculated in local density approximation, embedding the mesons in the nucleus described by solving the mean-field equations of motion in the QMC model. Our results suggest that one should expect to find $η$- and $…
The possible existence of $η$-mesic nuclei poses an interesting and still open issue of research. Since the occurence of such $η$-nucleus bound states is reflected in the corresponding $η$-nucleus scattering length, we critically review the present knowledge for the $η$$^3$He system. Specifically, we scrutinize the available experimental information for the reaction $p{+}d \to η{+} ^3{\rm He}$ which is commonly used to extract the $η$$^3$He scattering length. We point out several striking discrepancies between the various measurements. Subject to those inconsistencies we deduce a value of $a{=…
A microscopic treatment of $η$--nucleus scattering is presented. When applying the underlying exact integral equations, the excitation of the target are neglected, and an input $ηN$ amplitude is chosen which reproduces the $S_{11}(1535)$ resonance. It is shown that the $η$--nucleus scattering lengths are quite sensitive to the $ηN$ parameters and the nuclear wave functions. For a special choice of the parameters an $η{^4\!H}e$ quasi--bound state occurs.
We study the nuclear (or A) dependence of the coherent eta photoproduction reaction in a relativistic impulse approximation approach. We use a standard relativistic parameterization of the elementary amplitude, based on a set of four Lorentz- and gauge-invariant amplitudes, to calculate the coherent production cross section from He-4, C-12, and Ca-40. In contrast to nonrelativistic treatments, our approach maintains the full relativistic structure of the process. The nuclear structure affects the process through the ground-state tensor density. This density is sensitive to relativistic effects…
We propose a simple and model-independent procedure to account for the impact of the nuclear modification of the gluon density as encoded in nuclear collinear PDF sets on two-to-two partonic hard processes in proton-nucleus collisions. This applies to a good approximation to quarkonium, D and B meson production, generically referred to H. Our procedure consists in parametrising the square of the parton scattering amplitude, A_{gg -> H X} and constraining it from the proton-proton data. Doing so, we have been able to compute the corresponding nuclear modification factors for J/psi, Upsilon and …
The structure and the energy spectrum of the $η^{\prime}$ mesonic nuclei are investigated in a relativistic mean field theory. One expects a substantial attraction for the $η^{\prime}$ meson in finite nuclei due to the partial restoration of chiral symmetry in the nuclear medium. Such a hadronic scale interaction for the $η^{\prime}$ mesonic nuclei may provide modification of the nuclear structure. The relativistic mean field theory is a self-contained model for finite nuclei which provides the saturation property within the model, and is good to investigate the structure change of the nucleus…
