 
3. Recent HRIBF Research  CoupledCluster 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 "abinitio" approach to nuclear structure and
reactions. The "abinitio" 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 manybody quantum system. The
quantum manybody 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], NoCore shellmodel [Nav00] and Green's function MonteCarlo
[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 "abinitio"
program to mediummass nuclei. Coupledcluster theory is a very
promising candidate for this purpose. Recently Coupledcluster theory
has seen a renaissance in nuclear structure. Coupledcluster 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 Coupledcluster theory, and today it defines
the stateoftheart manybody theory in the quantum chemistry
community, and a recent review of Coupledcluster theory can be found
in Ref.[Bar07]. Coupledcluster 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
"abinitio" methods. It has a polynomial scaling with system size,
favoring it over methods with exponential or combinatorial
scaling. Coupledcluster is also capable of systematic improvements
and recovers the exact wave function in the full
limit. Coupledcluster theory maintains the very important feature of
sizeextensivity: 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 "abinitio" nuclear structure effort that moves into heavier
regions of the nuclear chart.
"Abinitio" calculations of light nuclei, starting from Hamiltonians
with twobody forces only have shown consistent failure to meet
experimental mass values. These calculations have therefore revealed
the need for threenucleonforces (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 lowenergy 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 manybody
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
threebody forces will be sufficient to approximately renormalize the
nuclear manybody problem in a range of energy cutoffs. The modern
understanding is that there are no unique 3NF, all nucleonnucleon
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
nucleonnucleon force. A systematic way of relating lowenergy 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 lowenergy 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 twopion 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 coupledcluster calculations of the ground
state energies of ^{4}He, ^{16}O and ^{40}Ca
using a Hamiltonian with a
renormalized twobody force of the lowmomentum type (Vlowk). Our
results were reasonably well converged with respect to the basis size,
and we estimated that the ^{40}Ca groundstate energy were converged
within 1% of the exact result. The calculated groundstate energy of
^{16}O ( 148.2 MeV) and ^{40}Ca (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 twobody interaction we
used.
One of our major aims in the coupledcluster project is to investigate
the role of 3NF's in mediummass nuclei and in isotopic chains with
extreme isospin asymmetry. Recently we developed and implemented
coupledcluster theory for threebody Hamiltonians [Hag1] and
performed a benchmark calculation of the binding energy of ^{4}He
using a renormalized twobody interaction accompanied with a 3NF at
NNLO in the Chiral EFT expansion. Our results were in excellent
agreement with the numerical exact FaddeevYakubovsky calculation
starting with the same Hamiltonian. We further found that the 3NF
could be very well approximated by a density dependent zero, one
and twobody term, see Fig. 31. This finding is very promising, since
we can account for the full 3NF using well developed tools and
machinery for twobody Hamiltonians. It remains to be seen whether
this finding also holds in heavier nuclei.
Figure 31: Relative contributions ΔE/E to
the binding energy of ^{4}He at the CCSD level. The different
points denote the contributions from
(1) lowmomentum NN interactions,
(2) the vacuum expectation value of the 3NF,
(3) the normalordered onebody Hamiltonian due to the 3NF,
(4) the normalordered twobody Hamiltonian due to the 3NF,
and (5) the residual 3NFs. The dotted line estimates the
corrections due to omitted threeparticlethreehole 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
betastability. At the limits of matter (neutron/proton drip lines),
exotic features, which are not seen in the wellbound 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 oneneutron decay threshold is above the twoneutron decay
threshold. Some of these nuclei, like ^{6}He and the cardinal case of
^{11}Li, 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 wellbound 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 singleparticle basis where bound, resonant, and
continuum states are treated on equal footing. A representation of the
manybody wave function in such a basis allows for description of both
halodensities of looselybound 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
Coupledcluster framework for the first time, and calculated masses
and lifetimes of the Helium chain (^{310}He). This was the first
"abinitio" calculation of lifetimes of a whole isotopic chain. The
results are summarized in Fig.32. 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 semiquantitative agreement with experiment. With
this interaction, all helium isotopes lack binding compared to
experiment. However, the evenodd mass pattern is reproduced fairly
well. We see that ^{5}He is unstable with respect to oneneutron
emission, while ^{6}He is stable towards oneneutron emission. However,
^{6}He is not stable towards twoneutron emission. This is mainly due to
the missing threebody forces and inclusion of full triples in our
calculation. ^{8}He is stable towards one, two, and threeneutron
emissions but not stable against the emission of four neutrons to the
continuum and ^{4}He. 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 32: CCSD results (black dotted line) and
experimental values (red dotted line) for the ground state of the
helium chain ^{310}He using a GamowHF basis and a lowmomentum
interaction generated from the N^{3}LO 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 EquationofMotion CCSD method, so that we can study
excited states and properties of closedshell nuclei and their
neighboring nuclei. A particularly interesting project is to perform an
"abinitio" Coupledcluster calculation of halo nuclei, and for the
first time give "abinitio" predictions of the drip lines. We are also
aiming at merging of structure and reaction theory within the
Coupledcluster 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).
