arXiv:1005.1599v1 [nucl-th] 10 May 2010
Systematics of binding energies and radii based on realistic two-nucleon
plus phenomenological three-nucleon interactions
A. G¨ unther,1, ∗R. Roth,1, †H. Hergert,2and S. Reinhardt1
1Institut f¨ ur Kernphysik, Technische Universit¨ at Darmstadt, 64289 Darmstadt, Germany
2National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA
(Dated: May 11, 2010)
We investigate the influence of phenomenological three-nucleon interactions on the systematics of ground-
state energies and charge radii throughout the whole nuclear mass range from4He to208Pb. The three-nucleon
interactions supplement unitarily transformed two-body interactions constructed within the Unitary Correlation
Operator Method or the Similarity Renormalization Group approach. To be able to address heavy nuclei as
well, we treat the many-body problem in Hartree-Fock plus many-body perturbation theory, which is sufficient
to assess the systematics of energies and radii, and limit ourselves to regularized three-body contact interactions.
We show that even with such a simplistic three-nucleon interaction a simultaneous reproduction of the experi-
mental ground-state energies and charge radii can be achieved, which is not possible with unitarily transformed
two-body interactions alone.
PACS numbers: 21.30.Fe,21.45.Ff,21.60.Jz
many-body calculations using nuclear Hamiltonians based on
Quantum Chromodynamics (QCD). An important step along
these lines is the formulation of nuclear interactions within
chiral effective field theory [1–3], leading to a consistent hier-
archy of two-, three- and many-nucleon interactions starting
from the relevant degrees of freedom and symmetries for the
low-energy nuclear structure regime. The use of these two-,
three- and many-nucleoninteractions in nuclear structure cal-
culations is a formidable task.
clear structure calculations using the chiral two- plus three-
nucleon interaction consistently have been performed in the
no-core shell model (NCSM) for mid p-shell nuclei . An
immense numerical effort is needed to compute and man-
age the three-body matrix elements in these calculations,
which limits the range of applicability of these calculations
at present. Recently, the use of consistent two- plus three-
nucleon interactions resulting from a Similarity Renormaliza-
tion Group evolution of the chiral two- plus three-nucleon in-
teraction was demonstrated also in the context of the NCSM
. This approach, a unitary transformation of the chiral
Hamiltonianaimingat a pre-diagonalizationthat improvesthe
convergence properties of NCSM substantially, holds great
potential also for the use in other many-body schemes and
will play a significant role in the future. However, the com-
putational effort for including those two- plus three-nucleon
interactions into many-body calculations, be it exact or ap-
proximate, is still the limiting factor for many applications.
In this paper we follow a more pragmatic route to explore
the impact of three-body forces in connection with unitarily
transformed two-nucleon interactions. We start from the Ar-
∗Electronic address: email@example.com
†Electronic address: firstname.lastname@example.org
gonne V18 high-precision two-nucleon potential , which
is still widely used although it does not have the same sys-
tematic link to QCD like the chiral effective field theory in-
teractions and is consideredphenomenologicalin this respect.
We then use the Similarity Renormalization Group [7–11] as
well as the Unitary Correlation Operator Method [10, 12–14]
to construct a transformedtwo-nucleoninteraction, which has
a much better convergence behavior and allow us to use sim-
plified many-body schemes. At this level neither genuine nor
induced three-nucleon interactions are included. From vari-
ous applications of these unitarily transformed two-nucleon
interactions we know that there are characteristic deviations
of basic nuclear observables from the experimental systemat-
ics that might be connected to three-body interactions. For
example, unitarily transformed two-body interactions which
yield a realistic systematics for binding energies tend to un-
derestimate the charge radii [10, 15]. Here we study to what
extend these systematic deviations can be cured by including
a three-bodyinteraction. Note that we are not aiming at a pre-
cision description of individual nuclei but rather the complete
systematics from light nuclei,4He, to heavy nuclei,208Pb.
To facilitate calculations for the full mass range from4He
to208Pb we have to simplify the approach compared to the
consistent. The first simplification consists in the use of a
efficient computation of matrix elements but violates the con-
sistency discussed above. The second simplification consists
in the use of Hartree-Fock plus many-body perturbation the-
ory for the approximate solution of the many-body problem.
Despite ofthese simplifications, we will obtainvaluableinfor-
mation on the interplay between realistic two-body and phe-
nomenological three-body interactions and on how well the
experimental systematics of ground-state energies and charge
radii can be reproduced. Furthermore, these studies prepare
plus three-nucleon interactions.
After a brief reminder of the basic concepts of the Unitary
CorrelationOperatorMethodandthe Similarity Renormaliza-
tion Groupwe introducethe phenomenologicalthree-bodyin-
teraction and calculate the matrix elements in the harmonic
oscillator basis in the second section. In the third section, we
discuss the inclusion of the three-body interaction in Hartree-
Fock and many-body perturbation theory and discuss the sys-
tematics of ground-state energies and charge rms-radii across
the whole nuclear mass range and its dependence on the two-
and three-nucleon interaction.
A. Unitary Correlation Operator Method and Similarity
The Unitary Correlation Operator Method and the Similar-
ity RenormalizationGroupprovidetwo conceptuallydifferent
but physically related approaches for the construction of soft
phase-shift equivalent interactions.
The Similarity Renormalization Group (SRG) [7–11] aims
at the pre-diagonalizationof the Hamiltonian for a given basis
by means of a unitary transformation implemented through
the renormalization-groupflow equation:
= [ηα,Hα] ,
where α is the flow parameter and Hαthe evolved Hamilto-
nian, with H0 = H being the initial or ‘bare’ Hamiltonian.
The anti-hermitian generator ηα defines the specifics of the
flow evolution, e.g. the representation with respect to which
the Hamiltonian should become diagonal or block-diagonal.
Various choicesfor this generatorhavebeen investigated,
we restrict ourselves to the simple generator [7, 9]
ηα= [Tint,Hα] (2)
with Tint= T − Tcmbeing the intrinsic kinetic energy, which
leads to a pre-diagonalizationof the Hamiltonian with respect
to the eigenbasis of the kinetic energyor momentumoperator.
Once the generator is fixed, the Hamiltonian and all operators
of interest can be evolved easily using a matrix representation
of the flow equation (1).
In A-bodyspace the evolutiongenerates up to A-bodyoper-
three-body operators. For reasons of practicability one has to
truncate the evolution at some low particle number—typically
this is done by solving the evolution equation in a matrix rep-
resentation in two- or three-body space. For the moment we
restrict ourselves to transformations in two-body space, i.e.,
we will discard any induced three-body interactions.
The aim of the Unitary Correlation Operator Method
(UCOM) [10, 12–14, 16] is to explicitly treat short-range cor-
relations inducedby the nuclearinteractionvia a static unitary
transformation. This transformationcan either be used to cor-
relate the many-body states or to similarity transform opera-
tors of interest, e.g. the Hamiltonian
˜ H = C†HC ,
using the correlation operator C. The dominant short-range
correlations are induced by the strong short-range repulsion
and the tensor part of the nuclear interaction. Therefore the
correlation operator is written as a product of two unitary op-
erators, Crfor the central correlations and CΩfor the tensor
correlations. We choose an explicit form of the correlation
C = CΩCr= exp
with the following ansatz for hermitian generators grand gΩ:
2[qrs(r) + s(r)qr] ,
2[(σ1· r)(σ2· qΩ) + (σ1· qΩ)(σ2· r)] ,
The strengths and radial dependencies of the two transforma-
tions are governed by the correlation functions s(r) and ϑ(r)
for the central and tensor correlations, respectively. One can
obtain these functions via an energy minimization in the two-
body system . Recently, we have also employed the SRG
as tool for the determination of the UCOM correlation func-
tions s(r) and ϑ(r) as discussed in Refs. [9, 10]. Here, we will
use these SRG-optimized UCOM correlation functions only.
Though the SRG- and UCOM-transformations have a dif-
ferent formal background, they address the same physics of
short-range correlations. A first connection becomes clear at
the level of the generators —the SRG generator (2) in two-
body space at α = 0 reveals the same operator structures that
appear in the UCOM-generators (5). At the level of matrix
elements, both the SRG- and UCOM-transformations lead to
a suppression of the off-diagonal momentum-space matrix el-
ements and an enhancement of the low-momentum matrix el-
ements as discussed in detail in Ref. .
In the following, we employ both transformations to gener-
ate one-parameterfamilies of phase-shiftequivalenttwo-body
interactions starting from a specific initial NN-interaction,
the Argonne V18 (AV18) in our case.
transformation the flow parameter α directly spans this fam-
ily of two-body interactions. We will study two versions of
the SRG-transformation, one where the flow equations are
solved for all partial waves and one where only the partial-
waves containing relative S-waves, i.e. the1S0and the cou-
pled3S1−3D1partial waves, are transformed. The latter is
motivated by the fact that short-range correlations affect the
S-wave channels most, because for all higher orbital angular
momenta the relative wave functions are suppressed by the
centrifugal barrier at short distances. We use the label ‘SRG’
for the fully transformed interactions and ‘S-SRG’ for the S-
wave-only transformations. For the UCOM-transformation
we use correlation functions determined from SRG-evolved
two-body wave functions as discussed in Refs. [9, 10], thus
the flow parameter α also spans a family of different UCOM-
transformed interactions. Note that the standard formulation
of UCOM only uses different transformationsfor the different
(S,T)-channels. We thus use the SRG-evolved wave func-
tions for the lowest partial waves for each (S,T)-channel to
define the correlation functions, leading to a transformed in-
teraction labelled ‘UCOM(SRG)’. Analogously to the S-SRG
r), qΩ= q−r
r·qr, and q =1
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