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3. Recent HRIBF Research - Coupled-Cluster Approach to Nuclear Structure
(G. Hagen, spokesperson)

One of the major aims in the nuclear structure and reaction community today, is to understand the nuclear properties from the basic interactions among protons and neutrons. This effort has been labeled the "ab-initio" approach to nuclear structure and reactions. The "ab-initio" approach aims for a theory that is capable of not only explaining experimental data, but also making predictions, and therefore providing guidance for future experimental setups. In the nuclear structure/reaction context, this approach involves treating the nucleus as a many-body quantum system. The quantum many-body problem is a difficult undertaking. Today there exist several theoretical methods capable of virtual exact solution of the nuclear Hamiltonian in the lightest region of the nuclear chart (A<=12); these include the Faddeev [Nog00], Hyperspherical Harmonics [Bar99], No-Core shell-model [Nav00] and Green's function Monte-Carlo [Piep01] approaches. Due to the combinatorial or exponential scaling, these methods are limited to the lightest region of the nuclear chart.

Scientists are exploring different methods to extend the "ab-initio" program to medium-mass nuclei. Coupled-cluster theory is a very promising candidate for this purpose. Recently Coupled-cluster theory has seen a renaissance in nuclear structure. Coupled-cluster theory originated in nuclear theory, and was pioneered by Coester and Kummel in the late 50's Ref.[Coe60]. In quantum chemistry there was a parallel development of Coupled-cluster theory, and today it defines the state-of-the-art many-body theory in the quantum chemistry community, and a recent review of Coupled-cluster theory can be found in Ref.[Bar07]. Coupled-cluster theory is an ideal compromise between computational cost on the one hand and accuracy on the other hand. It brings in correlation in a very economical way when compared to other "ab-initio" methods. It has a polynomial scaling with system size, favoring it over methods with exponential or combinatorial scaling. Coupled-cluster is also capable of systematic improvements and recovers the exact wave function in the full limit. Coupled-cluster theory maintains the very important feature of size-extensivity: the energy of the system scales correctly with number of particles in the system regardless of the order of approximation made. This is crucial property must be a component of any "ab-initio" nuclear structure effort that moves into heavier regions of the nuclear chart.

"Ab-initio" calculations of light nuclei, starting from Hamiltonians with two-body forces only have shown consistent failure to meet experimental mass values. These calculations have therefore revealed the need for three-nucleon-forces (3NF'S) in order to account for this systematic discrepancy. The existence of 3NF's is not surprising since nucleons are not elementary point particles. A theory starting from nucleon degrees of freedom is therefore an effective theory where internal degrees of freedom (quark and gluon) are integrated out. The relevant low-energy degrees of freedom are given by a cutoff or resolution scale at which properties of the system are resolved and probed. The higher the resolution scale the more details of the inner structure is revealed. The removal of degrees of freedom by a cutoff at a given energy scale, has to be compensated by additional many-body forces in order to recover the richness of the system where all degrees of freedom are taken into account. The hope is that two- and three-body forces will be sufficient to approximately renormalize the nuclear many-body problem in a range of energy cutoffs. The modern understanding is that there are no unique 3NF, all nucleon-nucleon forces have their associated cutoffs, and therefore have to be accompanied with their own 3NF. A frontier in nuclear structure concerns how one can consistently relate 3NF's to a given realistic nucleon-nucleon force. A systematic way of relating low-energy nuclear physics to QCD through Chiral Effective Field Theory (EFT), was recently developed. Chiral EFT starts from an effective Lagrangian consistent with the symmetries of QCD. The relevant low-energy degrees of freedom of Chiral EFT are the nucleons and pions, all other degrees of freedom are integrated out of the theory. Expanding the nuclear amplitude in powers of a typical nucleon momentum or pion mass over the chiral symmetry break down scale (~1 GeV) a perturbative series is obtained, where NN forces, 3NF's and forces of higher rank appear systematically at a given order. At each order in the theory there are a finite number of diagrams determined by one- and two-pion exchange terms and contact terms. This approach further accounts for the natural hierarchy of forces, i.e. NN > NNN > NNNN ... In Ref.[Hag01] we performed large scale coupled-cluster calculations of the ground state energies of 4He, 16O and 40Ca using a Hamiltonian with a renormalized two-body force of the low-momentum type (V-lowk). Our results were reasonably well converged with respect to the basis size, and we estimated that the 40Ca ground-state energy were converged within 1% of the exact result. The calculated ground-state energy of 16O ( -148.2 MeV) and 40Ca (-502.9 MeV) were largely overbound when compared to the experimental mass values of -127.6 MeV and -342.1 MeV, respectively. This is not surprising since we did not include the corresponding 3NF's which should accompany the two-body interaction we used.

One of our major aims in the coupled-cluster project is to investigate the role of 3NF's in medium-mass nuclei and in isotopic chains with extreme isospin asymmetry. Recently we developed and implemented coupled-cluster theory for three-body Hamiltonians [Hag1] and performed a benchmark calculation of the binding energy of 4He using a renormalized two-body interaction accompanied with a 3NF at NNLO in the Chiral EFT expansion. Our results were in excellent agreement with the numerical exact Faddeev-Yakubovsky calculation starting with the same Hamiltonian. We further found that the 3NF could be very well approximated by a density dependent zero-, one- and two-body term, see Fig. 3-1. This finding is very promising, since we can account for the full 3NF using well developed tools and machinery for two-body Hamiltonians. It remains to be seen whether this finding also holds in heavier nuclei.

Figure 3-1: Relative contributions ΔE/E to the binding energy of 4He at the CCSD level. The different points denote the contributions from (1) low-momentum NN interactions, (2) the vacuum expectation value of the 3NF, (3) the normal-ordered one-body Hamiltonian due to the 3NF, (4) the normal-ordered two-body Hamiltonian due to the 3NF, and (5) the residual 3NFs. The dotted line estimates the corrections due to omitted three-particle--three-hole clusters.

Another frontier in the nuclear structure and reaction community today concerns the theoretical understanding of structure properties and reaction mechanisms of nuclei located far away from the valley of beta-stability. At the limits of matter (neutron/proton drip lines), exotic features, which are not seen in the well-bound and stable nuclei, start to emerge, such as extreme matter clusterizations, melting and reorganizing of shell structure, ground states embedded in the continuum, and extreme dilute and extended matter densities. Another peculiar feature which appear in some of these exotic nuclei, is that the one-neutron decay threshold is above the two-neutron decay threshold. Some of these nuclei, like 6He and the cardinal case of 11Li, have been labeled as Borromean nuclei. It is a great theoretical challenge to account for the properties of nuclei at the drip lines. In standard shell model approaches, the nuclear wave function is expanded in a finite set of harmonic oscillator states. While this approach works well for well-bound and stable nuclei, it is obvious that this description is not the appropriate description when moving towards the drip lines, where the nuclei become loosely bound and even unbound in their ground states. The proximity of the scattering continuum in these systems, directly relates to the exotic properties observed in these nuclei. As the outermost nucleons approach the scattering thresholds, the tail of their wave functions extend far out in radial space and therefore accounts for the spatially dilute matter distributions or halo densities observed in some of these nuclei.

A very promising way to account for these properties is by expanding the wave function in a Berggren basis [Berg68]. The Berggren basis is a generalized single-particle basis where bound-, resonant,- and continuum states are treated on equal footing. A representation of the many-body wave function in such a basis allows for description of both halo-densities of loosely-bound nuclei and calculation of lifetimes and decay widths of unbound nuclear states. This approach has been applied with great success in shell model calcuations here at Oak Ridge in W. Nazarewich group, led by J. Rotureau [Rot06] and N. Michel [Mic02] . In Ref.[Hag3] we applied a Berggren basis within the Coupled-cluster framework for the first time, and calculated masses and lifetimes of the Helium chain (3-10He). This was the first "ab-initio" calculation of lifetimes of a whole isotopic chain. The results are summarized in Fig.3-2. The black dotted line gives our calculated masses, while the red dotted line gives the experimental mass values. The inset gives our calculated widths of the helium isotopes compared with experimental values. This figure shows that our results are in semi-quantitative agreement with experiment. With this interaction, all helium isotopes lack binding compared to experiment. However, the even-odd mass pattern is reproduced fairly well. We see that 5He is unstable with respect to one-neutron emission, while 6He is stable towards one-neutron emission. However, 6He is not stable towards two-neutron emission. This is mainly due to the missing three-body forces and inclusion of full triples in our calculation. 8He is stable towards one-, two-, and three-neutron emissions but not stable against the emission of four neutrons to the continuum and 4He. We believe that the growing discrepancy between theory and experimental mass values as we move along the helium chain is due to the lack of 3NF's. But for larger systems, triples corrections should play a more prominent role as well. By combining both of these missing ingredients, we believe that our results should be closer to the experimental values.

Figure 3-2: CCSD results (black dotted line) and experimental values (red dotted line) for the ground state of the helium chain 3-10He using a Gamow-HF basis and a low-momentum interaction generated from the N3LO interaction model.

In summary, we are now in a position where we can answer questions in nuclear structure and reactions, which could previously not be addressed. In the near future we are going to explore the role of 3NF's in medium size nuclei. We will look at saturation properties of modern realistic forces in medium mass nuclei. We will implement and derive the Equation-of-Motion CCSD method, so that we can study excited states and properties of closed-shell nuclei and their neighboring nuclei. A particularly interesting project is to perform an "ab-initio" Coupled-cluster calculation of halo nuclei, and for the first time give "ab-initio" predictions of the drip lines. We are also aiming at merging of structure and reaction theory within the Coupled-cluster framework.

[Nog00] Phys. Rev. Lett. 85, 944 (2000).
[Bar99] Phys. Rev. C 61, 054001 (2000).
[Piep01] Ann. Rev. Nucl. Part. Sci. 51, 53 (2001).
[Nav00] Phys. Rev. C 62, 054311 (2000).
[Coe60] Nucl. Phys. 17, 477 (1960).
[Bar07] Rev. Mod. Phys. 79, 291 (2007).
[Hag1] Phys. Rev. C 76 044305 (2007).
[Hag2] Phys. Rev. C 76 034302 (2007).
[Berg68] Nucl. Phys. A 109, 265 (1986).
[Rot06] Phys. Rev. Lett. 97, 110603 (2006).
[Mic02] Phys. Rev. Lett. 89, 042502 (2002).
[Hag3] Phys. Lett. B 656 169 (2007).

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