The Three-Body Force in the Three-Nucleon System: Proceedings of the International Symposium Held at The George Washington University Washington, D.C., April 24–26, 1986

523 Pages·1986·18.51 MB·English

by  B. L. Berman

Executive Summar X
It is clear that no two-body force model adequately describes all of the
fundamental properties of nuclei and of nuclear reactions. In particular, Faddeev
and variational calculations of the bound-state properties of zH and 3He with
realistic models of the two-nucleon force have demonstrated convincingly that the
calculated binding energies and electron-scattering form factors of the A = 3 nuclei
do not agree with the data. This disagreement extends to polarization measurements
in nucleon-deuteron elastic scattering and possibly to breakup reactions as well.
The three-nucleon force has been invoked to explain some or all of these and other
discrepancies.
A number of theoretical approaches have been employed to model the three-body
force (3BF): a model-independent analysis of low-energy ~N amplitudes, a chiral
Lagrangian, and a coupled-channels NN-NA description. All include two-meson
exchange with an intermediate delta as an important component. A more phenomen-
ological approach has also been advocated, because no a priori meson model has yet
Yielded a quantitative description of the basic nucleon-nucleon force. Neither has
any fundamental field-theory model based upon quarks yet proven tractable. Given a
particular 3BF, nuclear-structure properties and reaction cross sections can be
calculated via Faddeev or variational techniques. Including a 3BF leads to improve-
•ents in the calculated three-nucleon binding energies and radii and the neutron-
deuteron spin-doublet scattering length. However, the details of the 3H and 3He
charge form factors are still not understood for values of the momentum transfer
squared of q2 = 10 fm .2- It is in this region, in which the form factor passes
through zero, that nucleonic and nonnucleonic contributions cancel and that one is
Particularly sensitive to the fundamental nature of the nucleon. It is here that one
may be able to extract important information about the contributions of relativistic
effects as well as those of mesons and deltas, or alternatively, quark
distributions.
~ecommendations for future research on bound-state properties include the
addition of rho-meson exchange terms in the two-pion exchange models and a better
Understanding of the vertex cutoff in the pion-nucleon amplitudes, the inclusion of
nucleon-delta interactions in the NN-NA model, and the testing of phenomenological
approaches in nuclei with A > .3 Experimentally, the new electron-scattering data on
elastic form factors below q2 = 30 fm -2 should be evaluated critically and con-
solidated with older data sets.
The low-energy continuum region (below I00 MeV in excitation energy) contains a
wealth of new data with which to test three-nucleon theory in general and the
three-body force in particular. Neutron-deuteron elastic-scattering data are now of
a precision comparable to proton-deuteron data. Recent studies with polarized pro-
jectiles and/or polarized targets have led to a comprehensive set of polarization
parameters. Less extensive, but good quality~ three-nucleon breakup data also are
available. Some aspects of polarized NN scattering (particularly np) need to be
improved because they provide critically important input information for the three-
body calculations. Vector polarization in neutron-deuteron elastic scattering may
be sensitive to the 3BF; more and better data are needed here.
Given the great advances in experimental capabilities already achieved and with
more expected, an equal effort in the development of codes for the description of
the three-body continuum using realistic two- and three-body forces is sorely
needed. Indeed, perhaps the best place to test theories of the 3BY will turn out to
be in the three-nucleon continuum. Separable-potential calculations seem to provide
good qualitative fits to many data and therefore may provide an excellent basis for
continuing work. Additional effort is needed to determine which features of the
nucleon-nucleon force most affect the calculated values of various three-body
observables and what changes are implied by exotic models of the nucleon. Certain
regions of phase space (star and collinear geometries) appear to be more sensitive
to 3BF effects than others and therefore should receive particular attention.
Breakup reactions induced by tensor-polarized deuterons incident on protons at
intermediate energies also show promise of being a sensitive testing area.
Electromagnetic reactions are another potentially rich source of information on
the 3BF. Two examples are the aHe(~,pp)n and H(~,y)3He reactions. In the first
reaction the one- and two-nucleon currents are suppressed so that the three-nucleon
current may dominate certain regions of phase space. In the second reaction the
polarization parameter T20(O) is very sensitive to the D-state component in the
ground state of 3He and therefore can he expected to be sensitive to 3BF effects as
well.
At intermediate excitation energies (above pion threshold) the pion and the
delta resonance, which are the important ingredients of the 3BY, become on-shell
constituents of the nucleus. Continuum calculations become more complex, requiring
the inclusion of many partial waves. Experimental evidence was presented that
elastic pion scattering from 3H and 3He does not follow the equalities expected from
charge symmetry. Nucleon-deuteron breakup reactions are not well understood in this
region. In the electromagnetic sector one needs a consistent picture of the
connection between the two- and three-nucleon potentials and the currents which they
generate in the A = 3 bound-to-continuum transition amplitudes. One- and two-
nucleon knockout measurements in the quasi-free and delta regions are the best route
to solving this problem. Theoretical guidance again si needed in order to choose
the best kinematic arrangements.
The existence of a three-body force is required by any theory of the strong
interaction based upon the exchange of virtual quanta sa( in a proper relativistic
field theory) and by the antisymmetrization of identical quarks in any three-quark
cluster picture. At high momentum transfer, inclusive electron scattering begins to
prohe the quark structure of the A = 3 wave function. Experimental work here has
just begun, hut the results of measurements on 3He for y-scaling and the EHC effect
for heavier nuclei show substantial nuclear-physics modifications to quasifree
Scattering from a bound nucleon. Extension of the y-scaling measurements to aH and
the EMC measurements to both SHe and 3H undoubtedly will facilitate our under-
standing of the isospin dependence. Likewise, the data base for the elastic form
factors should be completed up to q2 = 70 fm -2, where only the charge form factor
for SHe si currently available. Quark-model calculations for the A = 3 nuclei are
in their initial stages, but already they confirm that the 3BF is an integral part
of QCD. The great challenge lies in understanding and quantifying this connection.
The development of a comprehensive understanding of the three-body force is
important for applications in nuclei with A > .3 The 3BF appears to explain some
phenomenological features of many-body theory, e.g., single-particle high-momentum
components in nuclei, the spin-orbit interaction, and nuclear compressibility. A
ANN potential si required to explain certain properties of hypernuc]ei. In nuclear
matter at high density the 3BF might very well dominate the NN force; the
consequences for our understanding of heavy-ion reactions~ neutron stars, and
supernovae are profound.
We have taken giant steps in the last three to five years in our efforts to
Understand three-body forces in nuclear physics. However, a much greater effort
Will be required before we can claim a true understanding of the important role
played by three-body forces. This ~uarantees that much exciting research lies ahead
in this vital sector of experimental and theoretical nuclear physics.
PLENARY SESSION
Chairman: B.L. Berman
JOINT SESSION
Chairmen: B.F. Gibson, E.F. Redish
ehT papers of S.A. Coon, P.U. Sauer, .K Maltman, G.L. Payne, J. Martino,
R.A. Brandenburg, dna H.O. Kiages were given in the Joint Session. Their
papers appear in these Proceedings under their respective Working Groups.

THE THEORY OF THE THREE NUCLEON FORCE
Bruce .H .J McKellar
School of Physics
University of Melbourne
Parkville, Vic., Australia 3052
I review attempts which have been made to construct the three nucleon force from
models of nucleon interactions with mesons and through other nucleon degrees of
freedom, paying particular attention to developments over the last three years.
tn_~_troduction
The 3-body force between nucleons has been around for a long time as a concept
1
"- at least since 1938. However it is only in recent years that most nuclear
Physicists have taken the concept seriously. This delay has been due to the great
difficulties that have been in the way of deciding that the 3-nucleon force is
necessary for the understanding of nuclear properties. Over the last five years or
so it has become apparent that 2-nucleon forces, used in a non-relativistic frame-
Work, do not quantitatively describe the properties of nuclei. The fault could be
in many places -- relativistic effects, many nucleon interactions, renormalization
of the interaction by the nuclear medium~ quark effects, etc. Of course the
distinction between the various items on this list si not clear cut, and we find
people using different labels to describe the same basic effects.
It should be emphasized that the very concept of a potential between two, three
or more nucleons is a nonrelativistic artifact. Were we able to do a fully
relativistic calculation, it would become necessary to describe interactions in the
many nucleon system through a field theory (of mesons or gluons) to avoid the
instantaneous interactions implied by potentials. Were we able to include all of
the relevant field degrees of freedom in our calculations, potentials would be
banished. As it is, however~ we must still use two- and three-body potentials to
Parameterize our knowledge (and perhaps conceal our ignorance) of the underlying
field theory.
In this review I will emphasize the attempts which have been made to deduce the
three nucleon potential from some more fundamental theory.
After a period in which many different forces were tried on a more or less ad
hoc basis, but the basic forces we still used today were introduced, 2 the modern
development of the subject of three body forces began with the realisation by Brown,
Green, and Gerace 3 that the development of current algebra constraints on K-mesonic
amplitudes provided a relatively unambiguous way to fixing the properties of at
least some of the important components of the three body force. This possibility of
model independent calculations of the 3-nucleon potential was developed by Barrett,
Coon and Scadron 4 and exploited by the Tucson-Melbourne group 5'6
New Developments
The development of the theory of 3-nucleon forces up to ]983 has been surveyed
elsewhere 7 so in this review I want to concentrate on developments since the Bochum
workshop and the Karlsruhe meeting. To briefly enumerate them:
.I ~,p and p,p potentials: These potentials were being developed at about the
time of the Karlsruhe meeting and our understanding of them has improved in the
meantime through the work of the Tucson-Melbourne and Brazilian groups 6'8.
.2 Form factors and how to allow for them: One of the contentious issues at
Karlsruhe was the decision on the scale to be used in the form factors, a parameter
to which the resulting force is extremely sensitive. There was also dispute over
how one should smear the contact part of the potential with the form factor. There
have been some developments in understanding the physics of the form factor, but the
major development has been the Los Alamos-lowa work on the triton 9 which shows that
in many practical circumstances the dominant cut-off is provided by the 2-body
correlations, and the different treatment of the form factor does not change the
result too much.
3. Z graphs, o and w contributions: II years ago Barshay and Brown I0 intro-
duced a 3-0 exchange potential. This year o exchange has been revived by Keister
II
and Wiringa , in the context of the Walecka-type of meson-nucleon interaction
Lagrangian. In this context w exchange also enters.
4. Many channel A contribution: The many channel treatment of the A contribu-
12 .
tion to the many-body force was reported by the Bannover group in 1983, and has
13
been further developed since then as we will hear at this symposium
5. Quark contributions: In contrast to the flurry of activity on quark contri-
butions to the 2-nucleon potential there has been rather little work on the
14
analogous contribution to the 3-nucleon potential. This of course reflects the
conceptual and practical difficulties of working with the constituents of
nucleons -- after some better understanding of the 2-body force is developed we can
expect more applications to the 3-body force.
I will discuss the ~,~ exchange potential first -- briefly because there is
little that is new, but it is important as background material and as the inescap-
able long-range interaction in the three body force -- and then turn to survey each
of the above areas of new developments in turn.
Th~Exchange Potential
Basic Remarks
The basic ~,~ exchange potential is illustrated in Fig. .I It is immediately
apparent that the essential ingredient is the ~-N scattering amplitude. The
question is how to best model the ~-N amplitude in the region of interest for the
three-body amplitude. There are two approaches in the literature -- one either
tries to use as much experimental and theoretical data as is available to tie the
amplitude down as much as possible, or one builds dynamical models of the amplitude~
fitting parameters to the data.
It is perhaps instructive to remark that the same dichotomy confronts those who
Work on the the 2K exchange nucleon-nucleon potential, and both methods have been
Used. The Sydney 15, Paris ,61 and Stony Brook 17 groups adopted the approach of using
as much K-N and ~-~ data as possible, together with theoretical constraints on the
18
~-N amplitude to construct the N-N potential. The Bonn group built A intermediate
state models for the K-N amplitude. The advantage of the first method is that the
data build in at least some information about resonant states which are not included
explicitly -- in fact in the analysis of Epstein and McKellar 15 the constraints are
so limiting that even the A contribution to the s-wave amplitude is largely built in
automatically. On the other hand, the second method has the advantage that
explicitly exhibiting the A structure of the amplitude allows an easy application of
the Pauli principle in the intermediate N-A states in many-body calculations -- an
effect which otherwise appears as a many-body force. Note that, if the Pauli
Principle is used explicitly in the intermediate states, care is required to avoid
double counting the 3 body force.
Tr
Fig. 1. The fundamental ~,~ exchange 3-nucleon potential. The blob
represents the KN+rd4 amplitude with the forward propagating Born term sub-
tracted.
Exactly the same advantages apply to the two approaches to the 3-nucleon
potential. To see the nature of the constraints on the amplitude realize that in
the Feynman Diagram of Fig. I the ~ mesons are off their mass shell, so that the
amplitude should be considered as a function of the usual variables v and t and of
the masses q~ and q~. At constant ~ = 0 the constraints may be usefully
represented in the diagram of Fig. 2, due to Sid Coon .91 The Adler and Weinberg
~f/o-
points at which the amplitude vanishes and takes the fixed finite value are,
a
on hadronic scale, quite close together. Satisfying these constraints places
severe demands on any model. While it is not difficult to construct a mode] which
so21, 22
satisfies these constraints, 20 none of the models in present use do
Because the contraint points in Fig. 2 are separated by small distances (of the
order ~) on the hadronic scale (of order 1GeV), the constraints, together with the
value of the amplitude at any point in Fig. 2 not in the plane of the Adler and
Weinberg points, determine the amplitude up to terms quadratic in the momenta 5.
Should it become necessary to know the amplitude to higher order in the momenta,
then recourse to models becomes essential. Models are already necessary for the nur
amplitudes other than the non-spin-flip amplitudes. The point at issue is not
really whether one can do without models -- that is not possible -- but rather
noitamrofni
whether all of the "model independent" is used effectively in the con-
struction of the model. I submit that amplitudes, and thus potentials, which do not
include the experimental and current algebra information are built on sand.
L./2 2
!
/ It
I / I
/
I/ C" =
//[ --'
..... i_A I
I
I //
I / I
I I/ /
^ /
.... X
2
,,/w I/ q2
2 2
Fig. 2. The u = 0 hyperplane in v, t, ql' q2 space.
10
Should it become necessary to know the amplitude to higher order in the momenta,
then recourse to models becomes essential. Models are already necessary for the Ni/
amplitudes other than the non-spin-flip amplitudes. The point at issue is not
really whether one can do without models -- that is not possible -- but rather
whether all of the "model independent" information is used effectively in the con-
struction of the model. I submit that amplitudes, and thus potentials, which do not
include the experimental and current algebra information are built on sand.
One of the most important open questions regarding the ~,R exchange 3-nucleon
potential is the relative importance to be attached to the terms in the potential
which are of higher order in the momenta. This question will recur when we discuss
the form factors, and when we discuss the A-N coupled channels approach to the three
body forces.
For the present we can make some general remarks about the terms of higher order
in the momenta. It is clear that high order terms in the momenta are describing
short distance effects, and one may hope that short range correlations induced by
the 2-nucleon interaction will reduce the magnitude of the wave function in the
region in which these terms are important to the extent that any observables are
insenstivie to the high momentum effects. There is some evidence from the Los
Alamos-lowa results 9 on the triton that this is indeed the case. But there is
evidence from the Hannover work 12 using the A-N coupled channels approach to the
contrary. A definitive analysis of this point is one of the opportunities for
further work which can be identified.
Some comment on the expected order of magnitude of the effects is possible. The
current algebra constraints on the ~-N amplitude demand that there are rapid varia-
tions in the amplitude on the scale q ~ p~, hut that the amplitude of these
variations is small, of order ~ -l, compared to the overall scale of the ~-N
2
amplitude which is set by the Born amplitude at gKNN/mN. Quadratic terms in the
amplitude suffice to satisfy these constraints, so that the quadratic terms have the
dimensionless form (q2/g~). The open question is the natural scale for the higher
Order terms. Both the Tucson-Melbourne and the Brazilian potentials introduce
higher order terms of the type (q2/A2)2, with A ~ m . On the other hand the
~ p
Hannover group in effect have higher order terms of the same type, but with
A2 ~ mN(m ~ - mN). Since the short range correlations in the nucleus introduce a
length scale of order 0.5 fm, the maximum momentum of interest is of order
(0.5 fm) "l ~ 0.4 GeV. This is relatively small on the scale of mp, but not on the
m(Nm{4 ~
Scale of - mN)}, so that one can understand the different results obtained
by the Los Alamos and Hannover groups. The question which needs to be settled is
the size of the "expansion scale"~ A. It is the value of A which will determine
whether we need to obtain the terms in the amplitude of higher order in the momenta.
Futhermore one can observe that, if we ultimately desire to work in con-
figuration space, then an expansion to high powers of the momentum is not physically
reasonable. As a simple example consider the series
11