Above threshold s -wave resonances illustrated by the 1/2 + states in $^{9}$Be and $^{9}$B E. Garrido Instituto de Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid, Spain D.V. Fedorov and A.S. Jensen Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark (Dated: July 6, 2018) We solve the persistent problem of the structure of the lowest 1 / 2 + resonance in $^{9}$Be which is important to bridge the A = 8 gap in nucleosynthesis in stars. We show that the state is a genuine three-body resonance even though it decays entirely into neutron- $^{8}$Be relative s -waves. The necessary barrier is created by “dynamical” evolution of the wave function as the short-distance α - $^{5}$He structure is changed into the large-distance n-$^{8}$Be structure. This decay mechanism leads to a width about two times smaller than table values. The previous interpretations as a virtual state or a two-body resonance are incorrect. The isobaric analog 1/2 + state in $^{9}$B is found to have energy and width in the vicinity of 2.0 MeV and 1.5 MeV, respectively. We also predict another 1/2 + resonance in $^{9}$B with similar energy and width. PACS numbers: 21.45.-v, 21.60.Gx, 27.20.+n Introduction Bound states, resonances and other con- tinuum states are well understood and described for two particles interacting through a potential. By weakening the attraction of the potential bound states are pushed upwards into the continuum as resonances or virtual states when a barrier is present or absent, respectively [1]. For neutral particles virtual states arise for s -waves whereas resonances emerge for higher partial waves. De- creasing the attraction further until the resonance energy is above the potential barrier leads to an increase of the resonance width. Correspondingly the related S -matrix pole moves in the complex energy plane as a resonance with non-vanishing imaginary part. For three particles interacting via two- and three-body potentials the continuum structures can be much more complicated due to combinations of the different struc- tures for the three two-body subsystems [2, 3]. One in- triguing possibility arises when one of the two-body sub- systems has a low-lying s -wave resonance produced by a confining Coulomb barrier, and the third neutral parti- cle has dominating s -wave attractions from the first two particles. Even when all higher partial waves are vanish- ingly small, the structure of the three-body continuum state is a priori not easily determined or described. Let us assume that the two-body resonance is very nar- row (long-lived) and the three-body energy is above zero but less than the two-body resonance energy. Then the three-body continuum state resembles a two-body bound state of the third particle and the composite resonance of the two first particles. The lifetime would be determined by the lifetime of the two-body resonance. When the three-body energy is pushed upwards above the two-body threshold by decreasing the attraction, the corresponding structure can be described as a two or three-body virtual state or resonance [1, 2]. The purpose of the present letter is first to determine Formulation Let us consider three composite struc- tures as point-like particles denoted n , α 1 and α 2 . The two mass scaled Jacobi vector coordinates, ( x , y ), can be substituted by hyperspherical coordinates { ρ, α, Ω$_{x}$ , Ω$_{y}$ } , where (Ω$_{x}$ , Ω$_{y}$) describe the directions of ( x , y ), ρ = √ x 2 + y 2 and α = arctan( x/y ), see [3]. We solve this three-body problem by use of adiabatic hyperspherical expansion of the Faddeev equations, i.e. the angular equations are solved for each ρ , providing a set of an- The $^{9}$Be(1 / 2 $^{+}$) structure is most often assumed to be one neutron and the $^{8}$Be ground state [10] but some- times also the α + α + n recombination reaction is as- sumed to proceed by α -capture on the $^{5}$He ground state [6]. It is apparently very difficult to avoid assumptions of two-body sequential structures and processes via subsys- tems of either $^{8}$Be or $^{5}$He. Interestingly a two-center Born-Oppenheimer model based on symmetries alone may combine these structures as in [7] where the 1 / 2 + is lowest at large distance whereas a 3 / 2 − state is low- est at small distance. We shall allow an entirely general three-body structure without a priori assumptions of sub- structures or decay mechanisms. in general which structure arises, and second specifically to solve the long-standing controversy of the $^{9}$Be(1 / 2 $^{+}$) continuum state. This state is important in the nuclear synthesis of light nuclei in stars [4–6], and has therefore received lots of attention both theoretically [6–9] and ex- perimentally [10–15]. In a measurement of photo disinte- gration the cross section is interpreted and parametrized via R -matrix analysis as one neutron and the $^{8}$Be ground state in a two-body s -wave resonance [11]. This seems to be against the two-body quantum mechanical description as such a state cannot survive as a resonance. In another interpretation the same neutron- $^{8}$Be system is described as a virtual state [9] but the resulting cross section does not reproduce the measurement [11]. arXiv:1001.0459v1 [nucl-th] 4 Jan 2010