Ab initio NCSM/RGM for threebody cluster systems and application to He+n+n
Abstract
We introduce an extension of the nocore shell model/resonating group method (NCSM/RGM) in order to describe threebody cluster states. We present results for the He ground state within a He+n+n cluster basis as well as first results for the phase shifts of different channels of the He+n+n system which provide information about lowlying resonances of this nucleus.
Keywords:
ab initio threebody resonances halo nuclei∎
1 Introduction
The NCSM/RGM was presented in [1; 2] as a promising technique that is able to treat both structure and reactions in light nuclear systems. This approach combines a microscopic cluster technique with the use of realistic interactions and a consistent description of the nucleon clusters.
The method has been introduced in detail for twobody cluster bases and has been shown to work efficiently in different systems [1; 2; 3; 4]. However, there are many interesting systems that have a threebody cluster structure and therefore can not be successfully studied with a twobody cluster approach.
The extension of the NCSM/RGM approach to properly describe threebody cluster states is essential for the study of nuclear systems that present such configuration. This type of systems appear, , in structure problems of twonucleon halo nuclei such as He and Li, resonant systems like H or transfer reactions with three fragments in their final states like H(H,2n)He or He(He,2p)He.
Recently, we introduced threebody cluster configurations into the method and presented the first results for the He ground state [5]. Here we present these results as well as first results for the continuum states of He within a He+n+n basis.
2 Formalism
The extension of the NCSM/RGM approach to properly describe threecluster configurations requires to expand the manybody wave function over a basis of threebody cluster channel states built from the NCSM wave function of each of the three clusters,
(1) 
(2)  
where is the relative vector proportional to the displacement between the center of mass (c.m.) of the first cluster and that of the residual two fragments, and is the relative coordinate proportional to the distance between the centers of mass of cluster 2 and 3. In eq. (1), are the relative motion wave functions and represent the unknowns of the problem and is the intercluster antisymmetrizer.
Projecting the microscopic body Schrödinger equation onto the basis states , the manybody problem can be mapped onto the system of coupledchannel integraldifferential equations
(3) 
where is the total energy of the system in the c.m. frame and
(4)  
(5) 
are integration kernels given respectively by the Hamiltonian and overlap (or norm) matrix elements over the antisymmetrized basis states. Finally, is the intrinsic body Hamiltonian.
In order to solve the Schrödinger equations (3) we orthogonalize them and transform to the hyperspherical harmonics (HH) basis to obtain a set of nonlocal integraldifferential equations in the hyperradial coordinate,
(6) 
which is finally solved using the microscopic Rmatrix method on a Lagrange mesh. The details of the procedure can be found in [5].
At present, we have completed the development of the formalism for the treatment of threecluster systems formed by two separate nucleons in relative motion with respect to a nucleus of mass number A.
3 Application to He+n+n.
It is well known that He is the lightest Borromean nucleus [6; 7], formed by an He core and two halo neutrons. It is, therefore, an ideal first candidate to be studied within this approach. In the present calculations, we describe the He core only by its g.s. wave function, ignoring its excited states. This is the only limitation in the model space used.
We used similarityrenormalizationgroup (SRG) [8; 9] evolved potentials obtained from the chiral NLO NN interaction [10] with = 1.5 fm. The set of equations (6) are solved for different channels using both bound and continuum asymptotic conditions. We find only one bound state, which appears in the channel and corresponds to the He ground state.
Ground state
Approach  E(He)  E(He)  



NCSM/RGM  (=12)  MeV  MeV 
NCSM  (=12)  MeV  MeV 
NCSM  (extrapolated)  MeV  MeV 
The results for the g.s. energy of He within a He(g.s.)+n+n cluster basis and = 12, = 14 MeV harmonic oscillator model space are compared to NCSM calculations in table 1. At 12 the binding energy calculations are close to convergence in both NCSM/RGM and NCSM approaches. The observed difference of approximately 1 MeV is due to the excitations of the He core, included only in the NCSM at present. Therefore, it gives a measure of the polarization effects of the core. The inclusion of the excitations of the core will be achieved in a future work through the use of the nocore shell model with continuum approach (NCSMC) [11; 12], which couples the present threecluster wave functions with NCSM eigenstates of the sixbody system.
Contrary to the NCSM, in the NCSM/RGM the He(g.s.)+n+n wave functions present the appropriate asymptotic behavior. The main components of the radial part of the He g.s. wave function can be seen in fig. (1) for different sizes of the model space demostrating large extension of the system. In the left part of the figure, the probability distribution of the main component of the wave function is shown, featuring two characteristic peaks which correspond to the dineutron and cigar configurations.
A thorough study of the converge of the results with respect to different parameters of the calculation was presented in [5], showing good convergence and stability.
Continuum states
The use of threecluster dynamics is essential for describing He states in the continuum. Therefore, this formalism is ideal for such study. Using continuum asymptotic conditions, we solved the set of equations (6) in order to obtain the lowenergy phase shifts for the and channels in the continuum.
In our preliminary results, we obtain the experimentally wellknown resonance as well as a second lowlying resonance recently measured at Ganil [13]. A resonance is also found in the channel while no lowlying resonances are present in the or channels. In fig. 2 some of the preliminary phase shifts for different channels are shown. Results for bigger model spaces and a study of their stability respect to the parameters in the formalism are presently being calculated and will be presented elsewhere.
4 Conclusions
In this work, we present an extension of the NCSM/RGM which includes threebody dynamics in the formalism. This new feature permits us to study a new range of systems that present threebody configurations . In particular, we presented results for both bound and continuum states of He studied within a basis of He+n+n. The obtained wave functions feature an appropriate asymptotic behavior, contrary to boundstate methods such as the NCSM.
Acknowledgements.
Computing support for this work came from the LLNL institutional Computing Grand Challenge program and from an INCITE Award on the Titan supercomputer of the Oak Ridge Leadership Computing Facility (OLCF) at ORNL. Prepared in part by LLNL under Contract DEAC5207NA27344. Support from the U.S. DOE/SC/NP (Work Proposal No. SCW1158) and NSERC Grant No. 4019452011 is acknowledged. TRIUMF receives funding via a contribution through the Canadian National Research Council.References
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