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Physics of few-body a hypernuclei
NuclearPhysics A479 (1988) 11% - 134~ North-Holland,Amsterdam
PHYSICS
OF FEW-BODY
A HYPERNUCLEI
B. F. GIBSON Theoretical 07545
Division,
Los Alamos
School of Physical Sciences, Bedford Park, SA, Australia
National
Flinders 5042
Laboratory*,
University
Los Alamos,
NM, USA
of South Australia,
The energies of the particle-stable states in few-body A hypernuclei are Other topics reviewed include: summarized. the role of the hypertriton in determining the spin dependence of the AN force, the role of the hypertriton in three-body force investigations, the effect of medium modifications upon AN-EN coupling in the A=4 isodoublet and the spin dependence of the AN force, the importance of exact equation formalisms in interpreting precision data, and the need for a renewed effort to identify and measure the masses of Ah hyperfragments.
1. INTRODUCTION Although
the first hypernucleus
was in the >arly state energies the strangeness in nature. exchange
1960's
that one realized
of the s-shell hypernuclei
the A has isospin
tail and does not support (RH) is the lightest
the binding
occurs
a molecular
BA(;H)
the energy deuteron,
BA(iH)
= B(;H)
from the systematics
with
a (deuteron-like)
S = -1 multibaryon the A clings
type state.
The A separation
bound bound
tenuously
ago', it
of the ground the physics
the nonstrange
0, the AN interaction
only because
physics
of found
has no one-pion2 state The A=3 system.
However,
to the deuteron
in
energy
- B(eH),
required
to remove
the A from the hypertriton
leaving behind
the
was only3
= 0.13 ? .05 MeV.
[Here, B(*H) = -E(2H) = 2.225 MeV.] ;1H was used to establish J; Permanent
more than 30 years
just how different
(S) -1 systems was compared
Because
hypertriton
almost
was identified
Nonetheless,
the pionic
weak decay of the
that the spin is l/2 and not 3/Z (the spin of the A
address.
037%9474/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
B. F. Gibson /Physics of few-body A hypernuclei
116~
is 1/2),4 and this implies that the AN spin-singlet force is stronger than the 5 at least in the :H bound state. force, The clear difference
spin-triplet between
the A separation
energies
of the A=4 isodoublet3
BA(iHe) - B(iHe) - B(sHe) = 2.39 f 0.03 MeV and
BA(iH) = B(AH)
indicated
- B(3H) = 2.04 f 0.04 MeV
that there was a distinct
AN interaction.
That is, the Ap and An interactions
the A=4 A separation symmetry
breaking
Furthermore,
BA(iHe)
energy
deduced
difference
= B(iHe)
energy
in the
in such a way that
energy
difference.
in iHe
- B(4He) = 3.12 f 0.02 MeV
was only about half that estimated
from central
force potential
models
that
to the A-3 and 4 hypernuclear
the available progress
differ
component6
was three times as large as any charge
from the sH-sHe binding
the A separation
were fitted
charge-symmetry-breaking
Ap bubble
chamber
in understanding
have developed.
data and were also consistent with scattering data. 5,7 We have since made some
this physics,
In particular,
so, then it places 8
but there remain puzzles
does the double A hypernucleus
and new ones
*iHe exist?
limits on the mass of the S = -2 dibaryon,
If
the "H" particle
of Jaffe.
In this review, The intriguing symmetry
I will look briefly
of sH will be discussed. The question of charge A in the isodoublet will be examined. The l+ + O+ transition
in the A-4 system and its relationship will be explored.
The anomalously
of M-hypernuclei
be discussed.
A-hypernuclei.
aspects
breaking
The bearing
at the data on few-body
Three
to the spin dependence
small binding
of :He will be touched upon.
upon, and the existence
important
aspects
of the AN force
of the physics
of the H dibaryon
will
may be summarized
as
follows:
1)
An improved
measurement
of the AsH binding
the models
of the hyperon-nucleon
scattering
from tagged A beams
are data on An scattering 2)
AN-CN
coupling
coupling
in nuclear
and the B is narrow.
interaction.
energy
is needed
New low-energy
in pp 4 AA production
to constrain
data on Ap
are anxiously
awaited
from K-d + Any.
is more
physics,
important because
This produces
in hypernuclear
physics
the A-X mass difference a complex
spin-dependence
than NN-NA
is only 80 MeV of the AN
as
B.F. Gibson /Physics of few-body
interaction simple
as a function
spin-dependent
data, which nuclei, 3)
provides
will
is confirmed,
approach
of the nuclear
core state.
A
on the free space scattering
to describing
are more
Ah hypernuclei
the NN interaction
important is needed.
for the lightest
in
in hypernuclei. If their existence
such hyperfragments
upon the mass of any possible
would
S = -2 dibaryon.
DATA SUMMARY
The experimental Is-shell
to identify
constraints
(isospin) modeled
corrections
mass measurements
severe
2. S-SHELL
a successful Medium
fail.
A new effort
provide
of the mass
AN interaction
117c
A hypemuclei
information
hypernuclei
available 3,9-10
for particle-stable in Table
are summarized
I, where
states
of the
A separation
energies
BA(AA) = B(AA)
- B(A-1)
and AA separation
energies
BAACAAN = HAAN are given.
The uncertainty
to extract
from emulsion
The value
of BA(iHe)
statistics. studies.
- B(A-2)
experiments,
was determined
It was the most common
The photon
coincidence
energies
measurements.
the mass 4 system. particle-stable
within
is fractionally because
because
light hyperfragment
They provide
the nuclear
that describe
formed
in emulsion
were determined
case, here we have
the system of strongly
Hamiltonian
BA(MeV)3
model
plus excitation
E7(MeV)9
2.04 f .04
1.04
A4He
2.39 f .03
1.15
AsHe
3.12 + .02
M
aHe
10.6 (?)
two solve
framework.
A4H
10
by to model
interacting
0.13 + .05
BM(MeV)
energy.
of the available
for which we can numerically
Table 1. Ground-state A and AA separation energies of particle-stable states for Is-shell hypernuclei.
AaH
It was difficult
a real test of our ability
in the same nucleus
a nonrelativistic,
large.
of the small binding
most reliably
for the A-4 isodoublet
That is, unlike
states
the set of exact equations baryons
in BA(iH)
energies
B.F. Gibson /Physics of few-body A hypemuclei
118c
There was some controversy that was identified corresponds
as the decay of AlHe. The GBe
to SBe.
and seems reasonably that these two M framework
10 of the emulsion event 11 A second event was reported which
about the interpretation
event
(BAA =18 MeV) has been throroughly
well established.
separation
Cluster model
energies
are consistent
based upon A-o and u-a potentials
calculations within
that reproduce
binding
and to confirm
3. THE HYPERTRITON Because
spin-$ or ; Anp states.
all two-body
interactions
+
analysis which
extracted
scattering
to a deuteron).
The J = $ system
(The np interaction
Ap bubble
of the interactions lengths
bound
state.)
An observation
system
chamber
scattering
data
are, in fact, highly
physics
related
together. which
convention
cannot
is the stronger. 13 correlated. However,
to hold either
In fact, even the Ahnn system limit the strength
it was
that there is no
in the A-4 discussion.
that the hypertriton
is that Ann is not bound.
The it
state had Jn =
Correspondingly,
a < 0 implies
to this point
to the statement
is not strong enough
13,14
(singlet or triplet)
We shall return
to the deuteron
calculations
is dominated
is a spin
from the pionic weak decay of RH that the ground
(Recall that in the nuclear
interaction
to
That is, one finds
1+ Thus, V; is stronger than ViN in the sH system. 2. A concluded that for the scattering lengths
a A bound
are needed.
to the spin-l deuteron
must be spin triplet.
of available
determine
two-body
of the :H
1t 3s VAN = 4 'AN + 4 'AN
:
was deduced4
our knowledge
: vm = ViN
+ J"=:
A direct
hypernuclei
approximately
It is clear that in the J = 2 system that
AN interaction. 5'12
corresponding
$
=
of M
model
the binding
AND RELATED
by the spin-singlet
J"
the existence
to improve
ISSUES + the A has spin J* - i , it can couple
form either
triplet,
new experiments
indicate
a potential
energies of 6He and 9Be; that is, the same AA model force agrees A A with the quoted AA separation energies for A=6 and 10. As we shall see below,
checked
corresponds
to
That is, the An
the unbound is unbound
of the AA interaction
np-singlet
or nn
in model to be no more than
B.F. Gibson /Physics of few-body A hypernuclei
Thus, the AN force is a relatively
that of the AN interaction. This is the result
exchange
an isospin-1
in lowest discuss
of their being no one-pion-exchange
(The A has isospin
interaction.
119c
zero, so that the AN system
Because
pion.)
of this the AN tensor
these details
corollary
to the lack of binding
to the
cannot
simply
force, which
is also not large. 15 in his presentation.
Holinde
order from i< and I?* exchange,
An interesting
feeble one.
contribution
comes
will
in the Ann system
is that
The XN interaction is even weaker than the is also unbound. 16 AN interaction. That is unfortunate, because a bound X-nn system would be the E-nn system
unable
to decay
into ANN due to charge
We shall see in the A-4 discussion 4He-iH ground A
ABA(O+)
state binding
is magnified
in comparison energy
of the two protons
breaking
breaking
deduced
in the
difference
to the charge
difference
symmetry
after correcting
for the Coulomb
from the interaction
in 3He
*BCSB = [B(3H) - B(3He)]
Similarly,
coupled-channel
three-body
force
effects
symmetry
= 2.39 - 2.04 = 0.35 MeV
3H-3He binding
magnified
energy
conservation. that charge
- EC = 0.76 - 0.64 = 0.12 MeV.
effects
(ANN) effects
in :H compared
to NN++NA
in the triton, because
(AN+-+W conversion),
when the EN channel coupled-channel
the hyperon
(or NNN three-body
mass difference
language
eliminated,
are
force)
mE-mA = 80 MeV is
much smaller To make
than m AmN' this clear, let us recall
or in another
is formally
that the coupled-channel
interaction
%Nleads to the "box diagram" formally
V
eliminated
by iterating
1 AN - "AN - "EN s-E+Am
Schematically converts
in a one-channel
when
the EN channel
is
equations:
"EN'
this is described
to a C (through
the coupled
formalism,
in Fig. 1, where
the transition
potential
in the second v
EN)
term the A
and then back
into a A.
B.F. Gibson /Physics of few-body A hypernuclei
12oc
A
N
A
N
ri
A
N
A
ri
Schematic picture AN-EN coupling
channel
models
describes Diagram
nucleon
such a coupled-channel
is eliminated.
now includes
which weakens
three-body
is
an effect often neglected
2(c) describes three baryons
between
to the box diagram the kinetic
in Fig. 2.
However,
This is referred energy
of the interaction
involved
Diagram
2(a)
the A and one of the nucleons. of Fig. 1.
a repulsive
the more conventional are directly
of two types, when the
energy of the second,
its contribution.
from modification
in the three-body
forces
This is illustrated
force in the literature,
force arising
interaction
to three-body
the simple vAN interaction
2(b) corresponds
denominator
in one-
calculations.
one is led immediately
EN channel
level of including
the importance
Note also that it is energy dependent,
When one embeds problem,
A
of including the box diagram 15 Note that the box diagram of the AN interaction.
will emphasize
in nuclear
I
“XN
FIGURE 1 of the AN interaction at the box diagram
Holinde
attractive.
A
N
three-body
to as the dispersive
dependence
in the medium.
force,
the energy
or spectator,
in the NN Diagram
that resulting
when all
in the interaction.
-__ .____ --__. I II N
NAN
A
__-
N
A
(A)
N
N
N
z
AN
(B)
N
A
N
x _--
N
A
N
ta
FIGURE 2 Schematic picture of three contributions to the AN interaction in the hypertriton when AN-EN coupling is allowed: (A) direct AN two-body interaction; (B) medium modification of the two-body interaction; (C) conventional three-body interaction
B.F. Gibson /Physics of few-body h hypemuclei
Because
the isospin
121c
of the A (T-O) and the X (T-l) differ,
the h++C
conversion
alters the isospin (and therefore spin) of the interacting nucleon17 Schematically one has two types of terms in the hypertriton pair.
nucleon
wave function:
[A @ dlT-'
+ [C D d*lT='.
(d) has T-O as does the A, so that the spin-singlet
The deuteron
T-l) must couple A++X conversion interaction appearing
has an antibound in the energy
a medium
correction
significantly
force effects
a model
Ap scattering hypernuclear
the (np)
Hence, T-l
energy
than 2 MeV, and such
from diagram
coupling)
2(b)
are clearly evident in the 18 have been
Bodmer
and Usmani
in which AN and ANN forces are parameterized
states.
in reproducing
supports
B degrees
energies
the data.
energies
Bodmer's
involves
disagree
finding
of freedom
force contribution
(In contrast,
which
AN potentials
to be 19 Shinamura reports
only phenomenological
markedly
essential.)
The Bodmer
contains
parameters:
(1) that of the AN interaction
a
AN
with the Ap scattering
that, if one adopts
are eliminated,
analysis
by analyzing
of the A - 3, 4, 4*, and 5
They find a three-body
His extracted
explicit
the attraction
energies.
plus the binding
indispensible
data, which
is only a little more
(or AN++XN
binding
fit to just the binding forces.
to the T=O :H.
Because
state at about 60 keV, the excitation
denominator
does not quench
hypernuclear
developing
of the np pair.
(d*,
in this system.
Three-body few-body
to the C (T-l) in order to contribute forces a recoupling
NN state
a formalism
then ANN three-body
in which
forces are
effectively
four potential strength combination svt 4 AN + :VI, found in
Ap scattering and in iHe; (2) that due to the AN interaction spin dependence t - VI,; (3) that of the dispersive diagram 2(b); (4) that of the long-range vAN attractive ANN force of diagram 2(c). The spin dependence of the AN force is ill determined,
due primarily
Furthermore,
B(;H).
three-body
result
appears
to disagree
of $H, 4He, and nuclear
find the contribution (more attractive)
corresponding
the three-nucleon
contribution
Also,
in our knowledge
to dominate
of
the required
ANN
to some extent with a similar by Pandharipande
to diagram
three-body
and coworkers,
2(c) to be about 1 MeV larger 20 term. We shall return to
Sauer finds in the Hanover
diagram
who
(repulsive)
force in terms of NN++NA
of the dispersive
two-pion-exchange
matter
than the dispersive
this in the next section. models
term appears
force.
The latter analysis
to the lack of precision
the dispersive
coupling,
approach,
which
that the repulsive
is slightly smaller than the attractive 21 force in the triton. The most complete
122c
B. F. Gibson /Physics of few-body A hypernuclei
calculations separable
for :H using a AN *EN
potential
calculations
calculations
were not designed
to provide
they showed that AN*EN
estimates,
coupled-channel by Dabrowski
energy by as much as 200 keV.
neglect
such effects without
Because which
the hypertriton
to study three-body
defined
short-range
two-pion-exchange
in
the
precision
are
effects
component
of
the
before
required
further
are simple 22
binding
could enhance
Although
the
energy
aH A BA(?AH) is so small, one cannot
Because detailed
the model
investigation.
is loosely bound, it is an ideal laboratory in 23 Here one is relatively insensitive to ill forces.
occur by one pion exchange)
data
quantitative
conversion
binding
interaction
and Fedorynski.
such as heavy meson exchange. of the three-body
such
energy
The long-range,
(AN-EN
conversion
HCWevei-, a significant
will dominate.
:H binding
force
as well
an investigation
as
can
AN scattering
be made quantitative.
improved
constraints
potential
models.
A step in that direction using tagged A beams from the CERN 24 In addition, the K-d+Any reaction realizable.
proposed
two-body
appears
as a means
scattering
hyperon-nucleon
One
must have
pp-+Ah reaction
on the realistic
improved
can
improvement
of obtaining 25 is being parameters
information tested
about the low-energy An 26 for feasibility at BNL.
4. THE A-4 ISODOUBLET The 0" ground hypernuclear energies.
states and l+ spin-flip
isodoublet Because
excited
states of the mass 4
are shown in Fig. 3 in terms of their A separation
one defines
1.00
0.24
1.24
2.04
0.35
2.39
Level diagram
FIGURE 3 for the mass 4 isodoublet in terms of A separation
energies
B. F. Gibson / Physics of few-body
123~
A hypernuclei
B,,(;H:H)= B(;H) - B(3H) and
BA(;He) - B(;He) - B(3He) for both
the ground
states
(4) and excited
in 3He and :He or AHe* cancels*'
energy
charge
previously, a difference interaction
symmetry
between
breaking
and one less An interaction
is three times larger
deduced
from the experimental
is a small Coulomb
actually
20 keV,27'28 strong
binding
energy
increases
yielding
interaction
ABESB(O+)
interaction. couples
AB,,.
the two
It has been estimated
energy
in SHe.
in AHe This
to be around
energy
difference
due to the
= 0.37 MeV.
effect
of some type is expected
in A
because of the significant AN*CN coupling in the hyperon-nucleon 6 For example, the C+ and X- masses differ by some 10 MeV, and hp An couples
included
in the commendable meson-theoretical
interaction.16'2g'30
exhibited
the Coulomb
than that occurring
a charge-symmetry-breaking
realistic
calculation
between
of
to C+n whereas
parameters
in the SHe-SH nuclear
interaction
to ABA, because
(more repulsive)
A charge-symmetry-breaking hypernuclei,
difference
Coulomb
symmetry
interactions)
- EC = 120 keV.
correction
to be larger
sign to) the charge
the nn and pp strong
for the repulsive 28 in 3He, one obtains
is expected effect
between
Correcting
*BCSB - [B(sH) - B(3He)]
There
states due to
(AHe has one more hp
than does AH),
than (and of opposite
(due to differences
protons
ground
as noted
- BA(iH) = 350 keV,
breaking
isodoublet.
Coulomb
Therefore,
in the hypernuclear
the hp and An interactions
- BA(AHe) ABA@+)
(4*), the repulsive
state
to first order.
using (a,r,)
to C-p.
effort
of the Nijmegen
potential
In particular,
separable
potentials
of the Nijmegen
by that potential
Effects
models
of this ilk have been group to construct
of the hyperon-nucleon
it has been demonstrated31S3*
in a model
fitted to the low-energy scattering 16 that the charge symmetry breaking
model D
(Vt # Vi,, Vs # VIn) AP AP
is sufficient
to account
B.F. Gibson /Physics of few-body A hypernuclei
124~
for such a value
of ABiSB(O+)
if one uses a true
is, one must solve exact four-body analysis value
that a folding model prescription
of ABESB(O+)
evidenced
four-body
equations.
using
too small by a factor of 2.)
the same potentials
yielded
The charge
breaking
= BA(iHe*)
- BA(iH*)
The fact that there exist two particle-stable us with a unique forces.
opportunity
Generating
is not a trivial
forces
that reproduce
the low-energy
(Such a test of our ability
wave
Yakubovsky-like states
exercise,
properties
to model
amplitudes
exact equations
spin in this four-body
in Fig. 4.
i.e., they correspond
to configurations Amplitude
an N coupled
in which
A describes
a trinucleon
of the
ground
state are depicted having
one baryon
a A coupled
state), while
[3,1] symmetry, is removed
from
to a three-nucleon amplitudes
B and C
that one finds in the
(Cl
(6)
CD)
in terms
the five types of amplitudes
to the two types of amplitudes
(A)
data.
nuclei
of the structure
of either
There are three amplitudes
the remaining
describe
few-body
expect.
system,
schematically
(not necessarily
to utilize
generated by solving the Faddeev31,33 that the spin dependence of the two
the Schrijdinger wave function
core
state
shows
that comprise
three.
of the hyperon-
of the YN scattering
An analysis
is not as simple as one might naively
If one neglects
in the A=4 isodoublet
if one is required
the nonstrange
does not exist.)
function
states
to test our models
the same model
of the NN interaction
equations.
both the O+ ground state and l+ excited
within
four-body
symmetry
a
= 240 keV,
has yet to be analyzed in terms of exact four-body + 4.1. The O+++l Transition
nucleon
That
by AB,(l+),
AB,(l+)
provides
formalism.
(It was also shown in that
(F)
FIGURE 4 Schematic representation of the five amplitudes that determine the A=4 wave function in the spin-independent limit of the separable potential equations
125~
B.E. Gibson /Physics offew-body A hypernuclei
hypertriton coupled
and describe of baryons
spin, the number
there are 15 amplitudes
in the three-body
four-body
and spin-triplet three-body
different
pairs
in the Of state expands
spin-singlet
In a central
force
and spin-triplet
they must couple
pairs
interacting core
to the fourth baryon
in the three-body
to
to
the same spin-singlet
subsystems,
but the
core states can have a spin of 3/2 as well as l/2 and still couple
Approximating
to form the spin-l
either
the case of 4He where
causes
an unacceptable structure
with
of amplitudes
four identical
[A @ 3H]J is inadequate.
reduces
nucleons,
one to miss l/3 of the o-particle approximation.
[d* D d*] form.
0+ or l+ state. amplitudes
state.
are coupled the two other
neglecting
4He is not just composed
Similarly,
the [2,2] or D 34 clearly
energy,
of states with
important
:H is not just
In the model calculations
In
to two (A and D)
binding
[p D 3H] and [n P 3He] but contains
like
[d D d] and
four-body
the O+ or the I+ state by
the number
one is dealing
amplitude
largest,
of amplitudes
The 1+ state involves
state.
interacting
to the fourth baryon
B,C,D,
[2,2] symmetry
in which
but the total spin of the three-body
subsystems,
can be at most l/2 because
form the spin-0
because
states
in the l+ state.
the O+ state involves
approximation,
[A
q
to be discussed,
elements
of the
3H] in either
the
all five types of
to one another.
Although
[3,1] amplitudes
and the two [2,2] amplitudes
the A amplitude
is the (that is,
and F) are each of the order of 10% of A.
In a naive picture
analysis
in which
that approximates
one retains
the [l/2 CxI3/2]l amplitude because
the J-3/2 excited
the J=1/2
WI0
into asymptotic
NN pair and a nucleon
D and F have
interact.
10, while
states
to an interacting
The amplitudes
AN pair.
the decomposition
When one includes
pairs
-- a A coupled
core states
to an interacting
ground
amplitudes
trinucleon) amplitudes, are related
state.
only the A type amplitudes), one might argue that + contributes to the 1 state can be neglected,
which
states of the trinucleon
and spin-triplet
l+ state would
"core" states
require
the remaining
be included.
of the hypertriton
are nearly
contain
two [l/2 GS pairs
in the
degenerate,
a similar
amplitudes,
which
The B and C amplitudes
system.
The hypertriton
the exact four-
contain
spin-l/2
three-body and spin-
and the J-3/2
states cannot be neglected. + of the A=4 hypernuclear 1 states as just a A
any simple model analysis + spin-flip imposed upon the 0 state structure Furthermore,
contain
interacting
two [l/2 m 1/2]l + + which is the origin of the assumption that the 0 and 1 4H states A by a simple spin-flip transition. However, the argument clearly
fails when one cannot neglect
Thus,
"core" lie far enough above
If so, then the O+ state would
(with spin-singlet
and the model
body equations
3/2 states
:H states as [A 1813H]J (that is, a
we shall see that AN-XV
coupling
can be highly is important,
misleading. because
the A
B.F. Gibson /Physics of few-body A hypernuclei
126~
(T=O) and the C (T-l) have different core states.
isospin
and therefore
This was clear in the analysis
of NN states
couple
to different
that contribute
to
3H when A-X conversion is included. Bodmer17 suggested this as an A explanation of the anomalously small value of BA(iHe). That is, the B can couple
only to the highly
some 40 MeV or higher of ;He have
isospin
To illustrate
T-l, even parity states of 4He which lie 35 because both the A and the 4He core
0.
4.2. A 0+-l+ Model
correct
excited
in the spectrum,
Problem
the importance
(exact equation)
of treating
formalism,
this A=4 system
we consider
calculations.
We use the Stepien-Rudza and Wycech 36 approximation to the Nijmegen YN coupled channel F.3O
We include
the two-channel potentials
the h++X conversion potentials
which have
length and effective
effects
identical
low-energy
the full set of 10 and 15 coupled, + that describe the 0 and l+ A=4 isodoublet
- 10.7 B(O+)
model potential
(AN-EN) potential
parameters
energies
effective
AN
(scattering
that result
two-dimensional 33 states are:
model
In other words,
by one-channel
scattering
The A-4 binding
solving
separable
only implicitly.
of ref. 36 are replaced
range).
in terms of a
the following
integral
from equations
MeV
and
B(l+) = 11.7 MeV.
The states are inversely this approximation equations,
to experimental
of using the free AN scattering potentials + than the 0 state.
is understandable.
a bound
For two attractive
state
la1 > Ia'1 + V is more attractive
than V'
and
r. > rb + V is more attractive
If the potential
a
observation.
2
does support
>B'
2
than V'.
a bound
state, then
potentials
In
in the exact A=4
the l+ state is more bound
The reason support
ordered with respect
that do not
B. F. Gibson / Physics of few-body
r.
>
r;, + B2 > B2'.
(As a potential where
127~
A hypemuclei
becomes
it just supports
attractive
nature
to think of a simple
is further
effective
increasing
square well,
effective
are related
range fixed,
a
n
(strength)
strengths
and r' o, then one can demonstrate
r
0 systems
Bn in various
the depth
or the size
However,
for the two-body system. 37,34 where n > 2. systems
V and V' with scattering
potentials ranges
-m
For those who prefer
enhanced.)
it more attractive
same does not hold true for n-body attractive
length a approaches
state, and then a falls from +m as the
of the potential
of the well makes
the scattering
more attractive, a bound
a and a' and
that the binding
to the potentials
the
Given two
energies
Holding
as follows.
the
then one finds that
>B' n, n = 2, 3, 4, . . .
That is, the binding
energy Bn due to potential
V' in a 2-, 3-, 4-,
. . . body system.
not support
state,
a bound
or closer
to supporting
4, . ...)
Because
(For an attractive
Ial > (a'1 means
a bound
state,
the scattering
potential,
this result
scattering
length and varies
and
length
is expected
V is greater
potential
that does
that V is more attractive Ial > la'1
is related
intuitively.
the effective
than that due to
+ Bn > B' n, n=
to a volume
However,
range,
than V', 3,
integral
of the
when one fixes the
then one finds
r. > r;, -+ B2 > B;
but
r. > r;, + Bn < Bn, n = 3, 4, . ..
That is, as the effective in the two-body variational
range
.
is increased,
sense, but less attractive
model
calculation
illustrating
in 1935 as an argument 38,39 __ range otherwise,
the potential
in many-body
this effect was,
by Thomas
for why the nuclear
nonzero
the triton would
nucleus.)
Thus, a mean-field,
effective
body system may lead to an incorrect measurements.
Exact calculations
cannot be obtained one of an effective
two-body
two-body
interpretation
can reveal novel
in any approximate equation.
is more attractive
systems.
theory
(A
in fact, put forth
force had to be of
collapse model
to a point
approximation
of precision aspects
that reduces
to an n-
experimental
of physics
which
the calculation
to
B. F. Gibson / Physics of few-body A hypernuclei
128c
.This is illustrated scattering
by the AN potential
parameters
quoted
in Table
II.
The
and effective ranges are those of the separable potential 36 30 to the Nijmegen interaction. approximation The X and ,5 are the
model
lengths
and range of the rank-one
strength
V(P,P')
- -k
separable
potential
P(P) g(p')
P(P) - (p2 + 8*)-l
that reproduces system.)
a and
differences averages
(Here p is the reduced
rO.
The scattering
lengths
are contained
that correspond
mass of the two-body
are approximately
in the effective
the same.
ranges.
The potential
The effective
spin
to the two states are'
o+: 1s 5t 6'AN + zVAN'
l+.
' Thus,
the l+ state is dominated
by the spin-triplet
AN interaction.
Because
rt < rs o, the l+ state is more bound in the four-body calculation than is the 0 0 state. Based upon the above analysis, it is clear that in an effective two-body triplet
formalism
just the opposite
force is weaker
Although
having
effective
the O+ state more bound
two-body
One important
that describe
coupling.
Because
that is missing
Neglecting
trinucleon
Table II. The potential interaction.
the physics
is the isospin
the A (T-O) and C(T-1)
couple
medium modification
core states
parameters
would be wrong!
related
differently
of three T-1/2 nucleons
there is a significant T-3/2
The spin-
from this model based upon AN
free scattering
that are composites
nucleons,
would be found.
force in a two-body sense. + than the 1 state in a mean field,
model might be pleasing,
effect
potentials
states
ordering
than the spin-singlet
to ANNc*ZZN
to T-1/2 core
than to elementary
T=1/2
of the AN interaction.
(having excitation
energies
of some 80
along with a and r. for the free space AN
S "AN
X(fm-3)
0.0952
B(fm‘l)
1.2011
t "m
0.3262 1.7251
a(fm)
-1.97
-1.95
ro(fm)
3.90
2.43
129~
B. F. Gibson / Physics of few-body A hypemuclei
MeV),
the free space coupled-channel
exhibit
altered
In particular,
spin-isospin
The rank-one quoted
energies
III,
parameters
that reflect
which
II in the few-body
is combined
with
the l+ state, has been weakened
more
terms.
sense.
However,
modifications
(Ial is smaller Hence,
(coefficient
spin-singlet
interaction,
spin-triplet
interaction,
in the O+ state.
which
the modified
spin-triplet interaction
-l/5 compared
is combined
with
The model binding
are
and r. is
the binding
the free space spin-singlet
the modified
W+) -
these medium
are weaker
of both states will be reduced.
interaction,
coupling
in the O+ state
+ in the 1 state
Both interactions
than those in Table
for the off-diagonal
force is modified
force is modified
separable
in Table
larger)
coefficients
the spin-singlet
and the spin-triplet
potentials
to
l/3) than
the free space 33 are
energies
9.6 MeV
B(l+) = 8.2 MeV.
Table III. Potential interactions.
parameters
along with a and r. for the medium
v&w+) x B a rO
0.0739
1.1828
vb(l+) 0.1814 1.6061
-1.33
-0.95
4.68
3.50
modified
in
B.F. Gibson / Physics of few-body A hypernuclei
13oc
Indeed, medium modification However,
of both states.
suffers
interaction, The model
E
of the A-C
conversion
the 1+ state, dominated
process
lowers
the binding
by the spin-triplet
the larger change.
0+-l+ energy difference
now has the correct
sign and is
= 1.4 MeV. 7
This is a model calculation effects
such as tensor
that AN*XN complex
coupling
nature
medium.
plays
However,
more attractive
than the spin-triplet
important
the important
the s-shell A-hypernuclei
of the AN interaction
the spin-singlet
in the two-body
other possibly
it does illustrate
in understanding
of the spin-dependence
Furthermore,
be weaker
which has neglected
forces.
AN interaction interaction
sense in free space.
role
and the
in the nuclear
may turn out to appear
in few-body
Hypernuclear
bound
states but
physics
is most
interesting.
5. Ah HYPERFRAGMENTS Two emulsion
rather
events have been reported
throughly
somewhat
interpreted
as Ah
= 18 MeV was found first and has been M 10 The AiHe event with BAA = 10.6 MeV has been
checked.
controversial.
unquestioned.
The importance
They provide
their existence symmetric
which were
The i:Be event 1' with B
hypernuclei.
of such AA hyperfragments
our only window
to study the M
bears upon that of the "H" dibaryon8
combination
strong magnetic-color
of 6 quarks forces
is
interaction,
-- a uuddss
that could take maximal
in the one-gluon-exchange
and
spatially
advantage
interaction
of the among
quarks. Although accepted,
the interpretation
consistent.40 energy
to indicate
When ha forces are parameterized
in :He and aa forces are parameterized
sBe levels,
the M
quoted value Because dibaryon
force needed
to account
one would expect
a M
to reproduce to describe
for BAA(iiBe)
hyperfragment
if the H has a mass smaller argues against
events
oo scattering
to decay quickly
into an H
of an H bound with respect
events
to Ah
above
in which both As in 'OBe decay weakly must be very AA Many more ;XBe hypernuclei must have been formed and decayed
Therefore,
undetected
and for the
of Ah
Emulsion events are identified by weak decay (A+Nn) of the 4 which is strongly suppressed as the mass is increased
rare indeed.
are
the A separation
also accounts
than 2mA, the observation
the existence
hypernucleus, A=5.
that the two M
of BAA(AiHe).
hyperfragments decay.
of the AAaHe event has not been universally seem
model calculations
by the hN+NN weak process.
B. F. Gibson / Physics of few-body A hypernuclei
Because
of the serious
for this unique consideration signature
consequences
perturbative
of the existence
QCD prediction
hypernuclei
efforts
to confirm
lighter
mass M
of GHe
serious
to exploitation
other than their pionic
the existence
of Ah hyperfragments
of the H dibaryon,
should be given by experimentalists
for M
131c
are called
of a
decay modes.
Renewed
for, as is a search for
From B,.A(iiBe) = 18 meV, one can deduce
hypernuclei.
that
“H > 2m - 20 MeV. A the Ah pair should
Otherwise,
then one can surmize
decay rapidly
into an H.
If GHe
is confirmed,
that
mH z 2mA - 10 MeV.
If Af;He is not confirmed, because
model
calculations
then one can bound mH between these two values, 40,27 that are consistent with BAA(iiBe) being about
18 MeV also yield
an estimate
of BM(AHe)
pair do not decay
into an H.
If AHe
exist? an H.
Because
:H binds,
The 'He(K-,K")AiH
momentum
transfer
favorable
of about 10 MeV
is confirmed,
AAH will also bind, reaction
AiH
pair do not decay into
for the search,
in such double-strangeness-exchange
to ground-state
the Ah
does the hyperfragment
if the M
is a candidate
-- assuming
reactions
although
the
is not
formation.
6. SUMMARY An improved
measurement
of the A3H binding
the spin dependence
of the hyperon-nucleon
to model
forces.
three-body
tagged A beams coupling
Improved
is an important
low-energy
aspect
of hypernuclear
is small and the X is narrow.
dependence
for the AN interaction
state.
normal
Medium
nuclei.
of a repulsive identify
dibaryon.
corrections
This effect three-body
Ah hypernuclei
hyperfragments
is called
Ap scattering
are more
is maximal
(nuclear
is called
would provide
physics,
This produces
that varies
important
because a complex
spin
of the nuclear
in hypernuclei
force.
Mass measurements constraints
AN-EN
the A-C mass
than in
in :He and can be interpreted
core dependent) for.
data from
are needed.
with the isospin
important
for to constrain
and to test our ability
as well as An data from the K‘d-+Any reaction
difference
core
energy
interaction
A renewed
in terms effort to
of the lightest
such
on the mass of any S = -2
132~
B. F. Gibson / Physics of few-body A hypernuclei
ACKNOWLEDGEMENT The work of the author has been supported of Energy.
He gratefully
Scholar by Flinders D. R. Lehman
acknowledges
University
in part by the IJ. S. Department
the appointment
as a Visiting
Research
and a long and fruitful
collaboration
with
in this area of physics.
REFERENCES 1) J. Pniewski, Nukleonika 25 (1980) 341; D. H. Davis and J. Sacton, in: High Energy Physics, Vol. II, ed. E. H. S. Burhop (Academic Press, New York, 1967) pp. 365-455. 2) J. J. de Swart and C. Iddings,
Phys. Rev. 128 (1962) 2810.
3) M. Juric et al., Nucl. Phys. B52 (1973) 1. 4) R. H. Dalitz, in: Nuclear Physics, Les Houches 1968, eds. C. de Witt and V. Gillet (Gordon and Breach, New York, 1969) pp. 703-787. 5) R. C. Herndon and Y. C. Tang, Phys. Rev. 153 (1967) 1091; 159 (1967) 853; 165 (1968) 1093. 6) R. Dalitz
and F. von Hipple,
Phys. Lett. 10 (1964) 153.
7) A. Gal, in: Advances in Nucl. Phys., Vol. 8, eds. M. Baranger (Plenum Press, New York, 1975) pp. l-120. 8) R. L. Jaffe,
Phys. Rev. Lett.
and E. Vogt
38 (1977) 195.
9) A. Bamberger et al., Nucl Phys. B60 (1973) 1; M. Bejidian et al., Phys. Lett. 83B (1979) 252; H. Piekarz, Nucleonika 25 (1980) 1091. 1O)J. Prowse, ll)M. Danysz
Phys. Rev. Lett. 17 (1966) 782. et al., Nucl. Phys. 49 (1963) 121.
12)B. F. Gibson
and D. R. Lehman,
Phys. Rev. C 22 (1980) 2024.
13)G. Alexander et al., Phys. Rev. Lett. 13 (1964) 484; G. Alexander Phys. Rev. 173 (1968) 1452.
et al.,
14)B. Sechi-Zorn et al., Phys. Rev. Lett. 13 (1964) 282; B. Sechi-Zorn Phys. Rev. 175 (1968) 1735. 15)K. Holinde,
Meson Picture
Including
Strangeness,
16)M. M. Nagels, 2547.
T. A. Rijken,
17)A. R. Bodmer,
Phys. Rev. 141 (1966) 1387
et al.,
this volume.
and J. J. de Swart, Phys. Rev. D 15 (1977)
18)A. R. Bodmer an Q. N. Usmani, Phys. Rev. C 31 (1985) 1400; Nucl. Phys. A450 (1986) 257~; A. R. Bodmer, Q. N. Usmani, and J. Carlson, Phys. Rev. C 29 (1984) 684. 19)s. Shinamura, Nucl. Phys. A463 (1987) 215~; S. Shinamura, H. Tanaka, Prog. Theor. Phys. 71 (1984) 546.
Y. Akaishi,
and
133c
B.F. Gibson /Physics of few-body A hypernuclei
20)V. R. Pandharipande, in: Lecture Notes in Physics, Vol. 260, eds. B. L. Berman and B. F. Gibson (Springer-Verlag, Heidelberg, 1986) pp. 59-78. 21)P. U. Sauer, in: Lecture Notes in Physics, Vo. 260, eds. B. L. Berman and B. F. Gibson (Springer-Verlag, Heidelberg, 1986) pp. 107-118; in: Progress in Particle and Nuclear Physics, Vol. 16, ed. A. Faessler (Pergammon Press, Oxford, 1986) pp. 35-102. 22)J. Dabrowski
and E. Fedorynska,
Nucl. Phys. A210
23)B. F, Gibson and B. H. J. McKellar, System, to be published in Few-Body 24)K. Kilian,
Hyperon-Antihyperon
Three-Body Systems.
Interaction
(1973) 509.
Forces
in the Trinucleon
at Low Energies,
this volume.
25)B. F. Gibson, G. J. Stephenson, V. R. Brown, and M. S. Weiss, in: Proceedings of the Summer Study Meeting on Nuclear and Hypernuclear Physics with Kaon Beams (Brookhaven Nat. Lab. Report BNL 18335, 1973) pp. 296-306. 26)B. L. Roberts, this volume 27)B. F. Gibson,
Radiative
Kaon Capture
A. Goldberg,
and Hyperon
and M. S. Weiss,
28)J. L. Friar and B. F. Gibson,
Weak Radiative
Phys. Rev. 181 (1969) 1486.
Phys. Rev. C 18 (1978) 908.
29)M. M. Nagels, 338.
T. A. Rijken,
and J. J. de Swart, Ann. Phys.
30)M. M. Nagels, 1633.
T. A. Rijken,
and J. J. de Swart,
31)B. F. Gibson (1979) 289. 32)B. F. Gibson,
and D. R. Lehman,
Nucl.
Decay,
Nucl.
Phys. A450
Phys. A329
(N.Y.) 79 (1973)
Phys. Rev. D 20 (1979)
(1979) 308; Phys. Lett. 83B
(1986) 243~.
33)B. F. Gibson and D. R. Lehman, Four-Body Calculation of the 0+-l+ Binding Energy Difference in A-4 A Hypernuclei, to be published in Phys. Rev. C. 34)B. F. Gibson
and D. R. Lehman,
Phys. Rev. C 14 (1976) 685; 15 (1977) 2257.
35)B. F. Gibson, A. Goldberg, and M. S. Weiss, in: Few Particle Problems in the Nuclear Interaction, eds. I. Slaus, S. Moszowski, R. P. Haddock, and W. T. van Oers (North Holland, Amsterdam, 1972) pp. 188-190. 36)W. Stepien-Rudza
and S. Wycech,
Nucl.
37)B. F. Gibson and G. J. Stephenson, (1975) 1448. 38)L. H. Thomas,
(1981) 349.
Jr., Phys. Rev. C
8 (1973) 1222; 11
Phys. Rev. 47 (1935) 903.
39)H. A. Bethe and R. F. Bather, 40)A. R. Bodmer
Phys. A362
and Q. N. Usmani,
Rev. Mod. Phys. Nucl.
8 (1936) 82.
Phys. A463
(1987) 221~.
PHYSICS
OF FEW-BODY
A HYPERNUCLEI
B. F. GIBSON Theoretical 07545
Division,
Los Alamos
School of Physical Sciences, Bedford Park, SA, Australia
National
Flinders 5042
Laboratory*,
University
Los Alamos,
NM, USA
of South Australia,
The energies of the particle-stable states in few-body A hypernuclei are Other topics reviewed include: summarized. the role of the hypertriton in determining the spin dependence of the AN force, the role of the hypertriton in three-body force investigations, the effect of medium modifications upon AN-EN coupling in the A=4 isodoublet and the spin dependence of the AN force, the importance of exact equation formalisms in interpreting precision data, and the need for a renewed effort to identify and measure the masses of Ah hyperfragments.
1. INTRODUCTION Although
the first hypernucleus
was in the >arly state energies the strangeness in nature. exchange
1960's
that one realized
of the s-shell hypernuclei
the A has isospin
tail and does not support (RH) is the lightest
the binding
occurs
a molecular
BA(;H)
the energy deuteron,
BA(iH)
= B(;H)
from the systematics
with
a (deuteron-like)
S = -1 multibaryon the A clings
type state.
The A separation
bound bound
tenuously
ago', it
of the ground the physics
the nonstrange
0, the AN interaction
only because
physics
of found
has no one-pion2 state The A=3 system.
However,
to the deuteron
in
energy
- B(eH),
required
to remove
the A from the hypertriton
leaving behind
the
was only3
= 0.13 ? .05 MeV.
[Here, B(*H) = -E(2H) = 2.225 MeV.] ;1H was used to establish J; Permanent
more than 30 years
just how different
(S) -1 systems was compared
Because
hypertriton
almost
was identified
Nonetheless,
the pionic
weak decay of the
that the spin is l/2 and not 3/Z (the spin of the A
address.
037%9474/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
B. F. Gibson /Physics of few-body A hypernuclei
116~
is 1/2),4 and this implies that the AN spin-singlet force is stronger than the 5 at least in the :H bound state. force, The clear difference
spin-triplet between
the A separation
energies
of the A=4 isodoublet3
BA(iHe) - B(iHe) - B(sHe) = 2.39 f 0.03 MeV and
BA(iH) = B(AH)
indicated
- B(3H) = 2.04 f 0.04 MeV
that there was a distinct
AN interaction.
That is, the Ap and An interactions
the A=4 A separation symmetry
breaking
Furthermore,
BA(iHe)
energy
deduced
difference
= B(iHe)
energy
in the
in such a way that
energy
difference.
in iHe
- B(4He) = 3.12 f 0.02 MeV
was only about half that estimated
from central
force potential
models
that
to the A-3 and 4 hypernuclear
the available progress
differ
component6
was three times as large as any charge
from the sH-sHe binding
the A separation
were fitted
charge-symmetry-breaking
Ap bubble
chamber
in understanding
have developed.
data and were also consistent with scattering data. 5,7 We have since made some
this physics,
In particular,
so, then it places 8
but there remain puzzles
does the double A hypernucleus
and new ones
*iHe exist?
limits on the mass of the S = -2 dibaryon,
If
the "H" particle
of Jaffe.
In this review, The intriguing symmetry
I will look briefly
of sH will be discussed. The question of charge A in the isodoublet will be examined. The l+ + O+ transition
in the A-4 system and its relationship will be explored.
The anomalously
of M-hypernuclei
be discussed.
A-hypernuclei.
aspects
breaking
The bearing
at the data on few-body
Three
to the spin dependence
small binding
of :He will be touched upon.
upon, and the existence
important
aspects
of the AN force
of the physics
of the H dibaryon
will
may be summarized
as
follows:
1)
An improved
measurement
of the AsH binding
the models
of the hyperon-nucleon
scattering
from tagged A beams
are data on An scattering 2)
AN-CN
coupling
coupling
in nuclear
and the B is narrow.
interaction.
energy
is needed
New low-energy
in pp 4 AA production
to constrain
data on Ap
are anxiously
awaited
from K-d + Any.
is more
physics,
important because
This produces
in hypernuclear
physics
the A-X mass difference a complex
spin-dependence
than NN-NA
is only 80 MeV of the AN
as
B.F. Gibson /Physics of few-body
interaction simple
as a function
spin-dependent
data, which nuclei, 3)
provides
will
is confirmed,
approach
of the nuclear
core state.
A
on the free space scattering
to describing
are more
Ah hypernuclei
the NN interaction
important is needed.
for the lightest
in
in hypernuclei. If their existence
such hyperfragments
upon the mass of any possible
would
S = -2 dibaryon.
DATA SUMMARY
The experimental Is-shell
to identify
constraints
(isospin) modeled
corrections
mass measurements
severe
2. S-SHELL
a successful Medium
fail.
A new effort
provide
of the mass
AN interaction
117c
A hypemuclei
information
hypernuclei
available 3,9-10
for particle-stable in Table
are summarized
I, where
states
of the
A separation
energies
BA(AA) = B(AA)
- B(A-1)
and AA separation
energies
BAACAAN = HAAN are given.
The uncertainty
to extract
from emulsion
The value
of BA(iHe)
statistics. studies.
- B(A-2)
experiments,
was determined
It was the most common
The photon
coincidence
energies
measurements.
the mass 4 system. particle-stable
within
is fractionally because
because
light hyperfragment
They provide
the nuclear
that describe
formed
in emulsion
were determined
case, here we have
the system of strongly
Hamiltonian
BA(MeV)3
model
plus excitation
E7(MeV)9
2.04 f .04
1.04
A4He
2.39 f .03
1.15
AsHe
3.12 + .02
M
aHe
10.6 (?)
two solve
framework.
A4H
10
by to model
interacting
0.13 + .05
BM(MeV)
energy.
of the available
for which we can numerically
Table 1. Ground-state A and AA separation energies of particle-stable states for Is-shell hypernuclei.
AaH
It was difficult
a real test of our ability
in the same nucleus
a nonrelativistic,
large.
of the small binding
most reliably
for the A-4 isodoublet
That is, unlike
states
the set of exact equations baryons
in BA(iH)
energies
B.F. Gibson /Physics of few-body A hypemuclei
118c
There was some controversy that was identified corresponds
as the decay of AlHe. The GBe
to SBe.
and seems reasonably that these two M framework
10 of the emulsion event 11 A second event was reported which
about the interpretation
event
(BAA =18 MeV) has been throroughly
well established.
separation
Cluster model
energies
are consistent
based upon A-o and u-a potentials
calculations within
that reproduce
binding
and to confirm
3. THE HYPERTRITON Because
spin-$ or ; Anp states.
all two-body
interactions
+
analysis which
extracted
scattering
to a deuteron).
The J = $ system
(The np interaction
Ap bubble
of the interactions lengths
bound
state.)
An observation
system
chamber
scattering
data
are, in fact, highly
physics
related
together. which
convention
cannot
is the stronger. 13 correlated. However,
to hold either
In fact, even the Ahnn system limit the strength
it was
that there is no
in the A-4 discussion.
that the hypertriton
is that Ann is not bound.
The it
state had Jn =
Correspondingly,
a < 0 implies
to this point
to the statement
is not strong enough
13,14
(singlet or triplet)
We shall return
to the deuteron
calculations
is dominated
is a spin
from the pionic weak decay of RH that the ground
(Recall that in the nuclear
interaction
to
That is, one finds
1+ Thus, V; is stronger than ViN in the sH system. 2. A concluded that for the scattering lengths
a A bound
are needed.
to the spin-l deuteron
must be spin triplet.
of available
determine
two-body
of the :H
1t 3s VAN = 4 'AN + 4 'AN
:
was deduced4
our knowledge
: vm = ViN
+ J"=:
A direct
hypernuclei
approximately
It is clear that in the J = 2 system that
AN interaction. 5'12
corresponding
$
=
of M
model
the binding
AND RELATED
by the spin-singlet
J"
the existence
to improve
ISSUES + the A has spin J* - i , it can couple
form either
triplet,
new experiments
indicate
a potential
energies of 6He and 9Be; that is, the same AA model force agrees A A with the quoted AA separation energies for A=6 and 10. As we shall see below,
checked
corresponds
to
That is, the An
the unbound is unbound
of the AA interaction
np-singlet
or nn
in model to be no more than
B.F. Gibson /Physics of few-body A hypernuclei
Thus, the AN force is a relatively
that of the AN interaction. This is the result
exchange
an isospin-1
in lowest discuss
of their being no one-pion-exchange
(The A has isospin
interaction.
119c
zero, so that the AN system
Because
pion.)
of this the AN tensor
these details
corollary
to the lack of binding
to the
cannot
simply
force, which
is also not large. 15 in his presentation.
Holinde
order from i< and I?* exchange,
An interesting
feeble one.
contribution
comes
will
in the Ann system
is that
The XN interaction is even weaker than the is also unbound. 16 AN interaction. That is unfortunate, because a bound X-nn system would be the E-nn system
unable
to decay
into ANN due to charge
We shall see in the A-4 discussion 4He-iH ground A
ABA(O+)
state binding
is magnified
in comparison energy
of the two protons
breaking
breaking
deduced
in the
difference
to the charge
difference
symmetry
after correcting
for the Coulomb
from the interaction
in 3He
*BCSB = [B(3H) - B(3He)]
Similarly,
coupled-channel
three-body
force
effects
symmetry
= 2.39 - 2.04 = 0.35 MeV
3H-3He binding
magnified
energy
conservation. that charge
- EC = 0.76 - 0.64 = 0.12 MeV.
effects
(ANN) effects
in :H compared
to NN++NA
in the triton, because
(AN+-+W conversion),
when the EN channel coupled-channel
the hyperon
(or NNN three-body
mass difference
language
eliminated,
are
force)
mE-mA = 80 MeV is
much smaller To make
than m AmN' this clear, let us recall
or in another
is formally
that the coupled-channel
interaction
%Nleads to the "box diagram" formally
V
eliminated
by iterating
1 AN - "AN - "EN s-E+Am
Schematically converts
in a one-channel
when
the EN channel
is
equations:
"EN'
this is described
to a C (through
the coupled
formalism,
in Fig. 1, where
the transition
potential
in the second v
EN)
term the A
and then back
into a A.
B.F. Gibson /Physics of few-body A hypernuclei
12oc
A
N
A
N
ri
A
N
A
ri
Schematic picture AN-EN coupling
channel
models
describes Diagram
nucleon
such a coupled-channel
is eliminated.
now includes
which weakens
three-body
is
an effect often neglected
2(c) describes three baryons
between
to the box diagram the kinetic
in Fig. 2.
However,
This is referred energy
of the interaction
involved
Diagram
2(a)
the A and one of the nucleons. of Fig. 1.
a repulsive
the more conventional are directly
of two types, when the
energy of the second,
its contribution.
from modification
in the three-body
forces
This is illustrated
force in the literature,
force arising
interaction
to three-body
the simple vAN interaction
2(b) corresponds
denominator
in one-
calculations.
one is led immediately
EN channel
level of including
the importance
Note also that it is energy dependent,
When one embeds problem,
A
of including the box diagram 15 Note that the box diagram of the AN interaction.
will emphasize
in nuclear
I
“XN
FIGURE 1 of the AN interaction at the box diagram
Holinde
attractive.
A
N
three-body
to as the dispersive
dependence
in the medium.
force,
the energy
or spectator,
in the NN Diagram
that resulting
when all
in the interaction.
-__ .____ --__. I II N
NAN
A
__-
N
A
(A)
N
N
N
z
AN
(B)
N
A
N
x _--
N
A
N
ta
FIGURE 2 Schematic picture of three contributions to the AN interaction in the hypertriton when AN-EN coupling is allowed: (A) direct AN two-body interaction; (B) medium modification of the two-body interaction; (C) conventional three-body interaction
B.F. Gibson /Physics of few-body h hypemuclei
Because
the isospin
121c
of the A (T-O) and the X (T-l) differ,
the h++C
conversion
alters the isospin (and therefore spin) of the interacting nucleon17 Schematically one has two types of terms in the hypertriton pair.
nucleon
wave function:
[A @ dlT-'
+ [C D d*lT='.
(d) has T-O as does the A, so that the spin-singlet
The deuteron
T-l) must couple A++X conversion interaction appearing
has an antibound in the energy
a medium
correction
significantly
force effects
a model
Ap scattering hypernuclear
the (np)
Hence, T-l
energy
than 2 MeV, and such
from diagram
coupling)
2(b)
are clearly evident in the 18 have been
Bodmer
and Usmani
in which AN and ANN forces are parameterized
states.
in reproducing
supports
B degrees
energies
the data.
energies
Bodmer's
involves
disagree
finding
of freedom
force contribution
(In contrast,
which
AN potentials
to be 19 Shinamura reports
only phenomenological
markedly
essential.)
The Bodmer
contains
parameters:
(1) that of the AN interaction
a
AN
with the Ap scattering
that, if one adopts
are eliminated,
analysis
by analyzing
of the A - 3, 4, 4*, and 5
They find a three-body
His extracted
explicit
the attraction
energies.
plus the binding
indispensible
data, which
is only a little more
(or AN++XN
binding
fit to just the binding forces.
to the T=O :H.
Because
state at about 60 keV, the excitation
denominator
does not quench
hypernuclear
developing
of the np pair.
(d*,
in this system.
Three-body few-body
to the C (T-l) in order to contribute forces a recoupling
NN state
a formalism
then ANN three-body
in which
forces are
effectively
four potential strength combination svt 4 AN + :VI, found in
Ap scattering and in iHe; (2) that due to the AN interaction spin dependence t - VI,; (3) that of the dispersive diagram 2(b); (4) that of the long-range vAN attractive ANN force of diagram 2(c). The spin dependence of the AN force is ill determined,
due primarily
Furthermore,
B(;H).
three-body
result
appears
to disagree
of $H, 4He, and nuclear
find the contribution (more attractive)
corresponding
the three-nucleon
contribution
Also,
in our knowledge
to dominate
of
the required
ANN
to some extent with a similar by Pandharipande
to diagram
three-body
and coworkers,
2(c) to be about 1 MeV larger 20 term. We shall return to
Sauer finds in the Hanover
diagram
who
(repulsive)
force in terms of NN++NA
of the dispersive
two-pion-exchange
matter
than the dispersive
this in the next section. models
term appears
force.
The latter analysis
to the lack of precision
the dispersive
coupling,
approach,
which
that the repulsive
is slightly smaller than the attractive 21 force in the triton. The most complete
122c
B. F. Gibson /Physics of few-body A hypernuclei
calculations separable
for :H using a AN *EN
potential
calculations
calculations
were not designed
to provide
they showed that AN*EN
estimates,
coupled-channel by Dabrowski
energy by as much as 200 keV.
neglect
such effects without
Because which
the hypertriton
to study three-body
defined
short-range
two-pion-exchange
in
the
precision
are
effects
component
of
the
before
required
further
are simple 22
binding
could enhance
Although
the
energy
aH A BA(?AH) is so small, one cannot
Because detailed
the model
investigation.
is loosely bound, it is an ideal laboratory in 23 Here one is relatively insensitive to ill forces.
occur by one pion exchange)
data
quantitative
conversion
binding
interaction
and Fedorynski.
such as heavy meson exchange. of the three-body
such
energy
The long-range,
(AN-EN
conversion
HCWevei-, a significant
will dominate.
:H binding
force
as well
an investigation
as
can
AN scattering
be made quantitative.
improved
constraints
potential
models.
A step in that direction using tagged A beams from the CERN 24 In addition, the K-d+Any reaction realizable.
proposed
two-body
appears
as a means
scattering
hyperon-nucleon
One
must have
pp-+Ah reaction
on the realistic
improved
can
improvement
of obtaining 25 is being parameters
information tested
about the low-energy An 26 for feasibility at BNL.
4. THE A-4 ISODOUBLET The 0" ground hypernuclear energies.
states and l+ spin-flip
isodoublet Because
excited
states of the mass 4
are shown in Fig. 3 in terms of their A separation
one defines
1.00
0.24
1.24
2.04
0.35
2.39
Level diagram
FIGURE 3 for the mass 4 isodoublet in terms of A separation
energies
B. F. Gibson / Physics of few-body
123~
A hypernuclei
B,,(;H:H)= B(;H) - B(3H) and
BA(;He) - B(;He) - B(3He) for both
the ground
states
(4) and excited
in 3He and :He or AHe* cancels*'
energy
charge
previously, a difference interaction
symmetry
between
breaking
and one less An interaction
is three times larger
deduced
from the experimental
is a small Coulomb
actually
20 keV,27'28 strong
binding
energy
increases
yielding
interaction
ABESB(O+)
interaction. couples
AB,,.
the two
It has been estimated
energy
in SHe.
in AHe This
to be around
energy
difference
due to the
= 0.37 MeV.
effect
of some type is expected
in A
because of the significant AN*CN coupling in the hyperon-nucleon 6 For example, the C+ and X- masses differ by some 10 MeV, and hp An couples
included
in the commendable meson-theoretical
interaction.16'2g'30
exhibited
the Coulomb
than that occurring
a charge-symmetry-breaking
realistic
calculation
between
of
to C+n whereas
parameters
in the SHe-SH nuclear
interaction
to ABA, because
(more repulsive)
A charge-symmetry-breaking hypernuclei,
difference
Coulomb
symmetry
interactions)
- EC = 120 keV.
correction
to be larger
sign to) the charge
the nn and pp strong
for the repulsive 28 in 3He, one obtains
is expected effect
between
Correcting
*BCSB - [B(sH) - B(3He)]
There
states due to
(AHe has one more hp
than does AH),
than (and of opposite
(due to differences
protons
ground
as noted
- BA(iH) = 350 keV,
breaking
isodoublet.
Coulomb
Therefore,
in the hypernuclear
the hp and An interactions
- BA(AHe) ABA@+)
(4*), the repulsive
state
to first order.
using (a,r,)
to C-p.
effort
of the Nijmegen
potential
In particular,
separable
potentials
of the Nijmegen
by that potential
Effects
models
of this ilk have been group to construct
of the hyperon-nucleon
it has been demonstrated31S3*
in a model
fitted to the low-energy scattering 16 that the charge symmetry breaking
model D
(Vt # Vi,, Vs # VIn) AP AP
is sufficient
to account
B.F. Gibson /Physics of few-body A hypernuclei
124~
for such a value
of ABiSB(O+)
if one uses a true
is, one must solve exact four-body analysis value
that a folding model prescription
of ABESB(O+)
evidenced
four-body
equations.
using
too small by a factor of 2.)
the same potentials
yielded
The charge
breaking
= BA(iHe*)
- BA(iH*)
The fact that there exist two particle-stable us with a unique forces.
opportunity
Generating
is not a trivial
forces
that reproduce
the low-energy
(Such a test of our ability
wave
Yakubovsky-like states
exercise,
properties
to model
amplitudes
exact equations
spin in this four-body
in Fig. 4.
i.e., they correspond
to configurations Amplitude
an N coupled
in which
A describes
a trinucleon
of the
ground
state are depicted having
one baryon
a A coupled
state), while
[3,1] symmetry, is removed
from
to a three-nucleon amplitudes
B and C
that one finds in the
(Cl
(6)
CD)
in terms
the five types of amplitudes
to the two types of amplitudes
(A)
data.
nuclei
of the structure
of either
There are three amplitudes
the remaining
describe
few-body
expect.
system,
schematically
(not necessarily
to utilize
generated by solving the Faddeev31,33 that the spin dependence of the two
the Schrijdinger wave function
core
state
shows
that comprise
three.
of the hyperon-
of the YN scattering
An analysis
is not as simple as one might naively
If one neglects
in the A=4 isodoublet
if one is required
the nonstrange
does not exist.)
function
states
to test our models
the same model
of the NN interaction
equations.
both the O+ ground state and l+ excited
within
four-body
symmetry
a
= 240 keV,
has yet to be analyzed in terms of exact four-body + 4.1. The O+++l Transition
nucleon
That
by AB,(l+),
AB,(l+)
provides
formalism.
(It was also shown in that
(F)
FIGURE 4 Schematic representation of the five amplitudes that determine the A=4 wave function in the spin-independent limit of the separable potential equations
125~
B.E. Gibson /Physics offew-body A hypernuclei
hypertriton coupled
and describe of baryons
spin, the number
there are 15 amplitudes
in the three-body
four-body
and spin-triplet three-body
different
pairs
in the Of state expands
spin-singlet
In a central
force
and spin-triplet
they must couple
pairs
interacting core
to the fourth baryon
in the three-body
to
to
the same spin-singlet
subsystems,
but the
core states can have a spin of 3/2 as well as l/2 and still couple
Approximating
to form the spin-l
either
the case of 4He where
causes
an unacceptable structure
with
of amplitudes
four identical
[A @ 3H]J is inadequate.
reduces
nucleons,
one to miss l/3 of the o-particle approximation.
[d* D d*] form.
0+ or l+ state. amplitudes
state.
are coupled the two other
neglecting
4He is not just composed
Similarly,
the [2,2] or D 34 clearly
energy,
of states with
important
:H is not just
In the model calculations
In
to two (A and D)
binding
[p D 3H] and [n P 3He] but contains
like
[d D d] and
four-body
the O+ or the I+ state by
the number
one is dealing
amplitude
largest,
of amplitudes
The 1+ state involves
state.
interacting
to the fourth baryon
B,C,D,
[2,2] symmetry
in which
but the total spin of the three-body
subsystems,
can be at most l/2 because
form the spin-0
because
states
in the l+ state.
the O+ state involves
approximation,
[A
q
to be discussed,
elements
of the
3H] in either
the
all five types of
to one another.
Although
[3,1] amplitudes
and the two [2,2] amplitudes
the A amplitude
is the (that is,
and F) are each of the order of 10% of A.
In a naive picture
analysis
in which
that approximates
one retains
the [l/2 CxI3/2]l amplitude because
the J-3/2 excited
the J=1/2
WI0
into asymptotic
NN pair and a nucleon
D and F have
interact.
10, while
states
to an interacting
The amplitudes
AN pair.
the decomposition
When one includes
pairs
-- a A coupled
core states
to an interacting
ground
amplitudes
trinucleon) amplitudes, are related
state.
only the A type amplitudes), one might argue that + contributes to the 1 state can be neglected,
which
states of the trinucleon
and spin-triplet
l+ state would
"core" states
require
the remaining
be included.
of the hypertriton
are nearly
contain
two [l/2 GS pairs
in the
degenerate,
a similar
amplitudes,
which
The B and C amplitudes
system.
The hypertriton
the exact four-
contain
spin-l/2
three-body and spin-
and the J-3/2
states cannot be neglected. + of the A=4 hypernuclear 1 states as just a A
any simple model analysis + spin-flip imposed upon the 0 state structure Furthermore,
contain
interacting
two [l/2 m 1/2]l + + which is the origin of the assumption that the 0 and 1 4H states A by a simple spin-flip transition. However, the argument clearly
fails when one cannot neglect
Thus,
"core" lie far enough above
If so, then the O+ state would
(with spin-singlet
and the model
body equations
3/2 states
:H states as [A 1813H]J (that is, a
we shall see that AN-XV
coupling
can be highly is important,
misleading. because
the A
B.F. Gibson /Physics of few-body A hypernuclei
126~
(T=O) and the C (T-l) have different core states.
isospin
and therefore
This was clear in the analysis
of NN states
couple
to different
that contribute
to
3H when A-X conversion is included. Bodmer17 suggested this as an A explanation of the anomalously small value of BA(iHe). That is, the B can couple
only to the highly
some 40 MeV or higher of ;He have
isospin
To illustrate
T-l, even parity states of 4He which lie 35 because both the A and the 4He core
0.
4.2. A 0+-l+ Model
correct
excited
in the spectrum,
Problem
the importance
(exact equation)
of treating
formalism,
this A=4 system
we consider
calculations.
We use the Stepien-Rudza and Wycech 36 approximation to the Nijmegen YN coupled channel F.3O
We include
the two-channel potentials
the h++X conversion potentials
which have
length and effective
effects
identical
low-energy
the full set of 10 and 15 coupled, + that describe the 0 and l+ A=4 isodoublet
- 10.7 B(O+)
model potential
(AN-EN) potential
parameters
energies
effective
AN
(scattering
that result
two-dimensional 33 states are:
model
In other words,
by one-channel
scattering
The A-4 binding
solving
separable
only implicitly.
of ref. 36 are replaced
range).
in terms of a
the following
integral
from equations
MeV
and
B(l+) = 11.7 MeV.
The states are inversely this approximation equations,
to experimental
of using the free AN scattering potentials + than the 0 state.
is understandable.
a bound
For two attractive
state
la1 > Ia'1 + V is more attractive
than V'
and
r. > rb + V is more attractive
If the potential
a
observation.
2
does support
>B'
2
than V'.
a bound
state, then
potentials
In
in the exact A=4
the l+ state is more bound
The reason support
ordered with respect
that do not
B. F. Gibson / Physics of few-body
r.
>
r;, + B2 > B2'.
(As a potential where
127~
A hypemuclei
becomes
it just supports
attractive
nature
to think of a simple
is further
effective
increasing
square well,
effective
are related
range fixed,
a
n
(strength)
strengths
and r' o, then one can demonstrate
r
0 systems
Bn in various
the depth
or the size
However,
for the two-body system. 37,34 where n > 2. systems
V and V' with scattering
potentials ranges
-m
For those who prefer
enhanced.)
it more attractive
same does not hold true for n-body attractive
length a approaches
state, and then a falls from +m as the
of the potential
of the well makes
the scattering
more attractive, a bound
a and a' and
that the binding
to the potentials
the
Given two
energies
Holding
as follows.
the
then one finds that
>B' n, n = 2, 3, 4, . . .
That is, the binding
energy Bn due to potential
V' in a 2-, 3-, 4-,
. . . body system.
not support
state,
a bound
or closer
to supporting
4, . ...)
Because
(For an attractive
Ial > (a'1 means
a bound
state,
the scattering
potential,
this result
scattering
length and varies
and
length
is expected
V is greater
potential
that does
that V is more attractive Ial > la'1
is related
intuitively.
the effective
than that due to
+ Bn > B' n, n=
to a volume
However,
range,
than V', 3,
integral
of the
when one fixes the
then one finds
r. > r;, -+ B2 > B;
but
r. > r;, + Bn < Bn, n = 3, 4, . ..
That is, as the effective in the two-body variational
range
.
is increased,
sense, but less attractive
model
calculation
illustrating
in 1935 as an argument 38,39 __ range otherwise,
the potential
in many-body
this effect was,
by Thomas
for why the nuclear
nonzero
the triton would
nucleus.)
Thus, a mean-field,
effective
body system may lead to an incorrect measurements.
Exact calculations
cannot be obtained one of an effective
two-body
two-body
interpretation
can reveal novel
in any approximate equation.
is more attractive
systems.
theory
(A
in fact, put forth
force had to be of
collapse model
to a point
approximation
of precision aspects
that reduces
to an n-
experimental
of physics
which
the calculation
to
B. F. Gibson / Physics of few-body A hypernuclei
128c
.This is illustrated scattering
by the AN potential
parameters
quoted
in Table
II.
The
and effective ranges are those of the separable potential 36 30 to the Nijmegen interaction. approximation The X and ,5 are the
model
lengths
and range of the rank-one
strength
V(P,P')
- -k
separable
potential
P(P) g(p')
P(P) - (p2 + 8*)-l
that reproduces system.)
a and
differences averages
(Here p is the reduced
rO.
The scattering
lengths
are contained
that correspond
mass of the two-body
are approximately
in the effective
the same.
ranges.
The potential
The effective
spin
to the two states are'
o+: 1s 5t 6'AN + zVAN'
l+.
' Thus,
the l+ state is dominated
by the spin-triplet
AN interaction.
Because
rt < rs o, the l+ state is more bound in the four-body calculation than is the 0 0 state. Based upon the above analysis, it is clear that in an effective two-body triplet
formalism
just the opposite
force is weaker
Although
having
effective
the O+ state more bound
two-body
One important
that describe
coupling.
Because
that is missing
Neglecting
trinucleon
Table II. The potential interaction.
the physics
is the isospin
the A (T-O) and C(T-1)
couple
medium modification
core states
parameters
would be wrong!
related
differently
of three T-1/2 nucleons
there is a significant T-3/2
The spin-
from this model based upon AN
free scattering
that are composites
nucleons,
would be found.
force in a two-body sense. + than the 1 state in a mean field,
model might be pleasing,
effect
potentials
states
ordering
than the spin-singlet
to ANNc*ZZN
to T-1/2 core
than to elementary
T=1/2
of the AN interaction.
(having excitation
energies
of some 80
along with a and r. for the free space AN
S "AN
X(fm-3)
0.0952
B(fm‘l)
1.2011
t "m
0.3262 1.7251
a(fm)
-1.97
-1.95
ro(fm)
3.90
2.43
129~
B. F. Gibson / Physics of few-body A hypemuclei
MeV),
the free space coupled-channel
exhibit
altered
In particular,
spin-isospin
The rank-one quoted
energies
III,
parameters
that reflect
which
II in the few-body
is combined
with
the l+ state, has been weakened
more
terms.
sense.
However,
modifications
(Ial is smaller Hence,
(coefficient
spin-singlet
interaction,
spin-triplet
interaction,
in the O+ state.
which
the modified
spin-triplet interaction
-l/5 compared
is combined
with
The model binding
are
and r. is
the binding
the free space spin-singlet
the modified
W+) -
these medium
are weaker
of both states will be reduced.
interaction,
coupling
in the O+ state
+ in the 1 state
Both interactions
than those in Table
for the off-diagonal
force is modified
force is modified
separable
in Table
larger)
coefficients
the spin-singlet
and the spin-triplet
potentials
to
l/3) than
the free space 33 are
energies
9.6 MeV
B(l+) = 8.2 MeV.
Table III. Potential interactions.
parameters
along with a and r. for the medium
v&w+) x B a rO
0.0739
1.1828
vb(l+) 0.1814 1.6061
-1.33
-0.95
4.68
3.50
modified
in
B.F. Gibson / Physics of few-body A hypernuclei
13oc
Indeed, medium modification However,
of both states.
suffers
interaction, The model
E
of the A-C
conversion
the 1+ state, dominated
process
lowers
the binding
by the spin-triplet
the larger change.
0+-l+ energy difference
now has the correct
sign and is
= 1.4 MeV. 7
This is a model calculation effects
such as tensor
that AN*XN complex
coupling
nature
medium.
plays
However,
more attractive
than the spin-triplet
important
the important
the s-shell A-hypernuclei
of the AN interaction
the spin-singlet
in the two-body
other possibly
it does illustrate
in understanding
of the spin-dependence
Furthermore,
be weaker
which has neglected
forces.
AN interaction interaction
sense in free space.
role
and the
in the nuclear
may turn out to appear
in few-body
Hypernuclear
bound
states but
physics
is most
interesting.
5. Ah HYPERFRAGMENTS Two emulsion
rather
events have been reported
throughly
somewhat
interpreted
as Ah
= 18 MeV was found first and has been M 10 The AiHe event with BAA = 10.6 MeV has been
checked.
controversial.
unquestioned.
The importance
They provide
their existence symmetric
which were
The i:Be event 1' with B
hypernuclei.
of such AA hyperfragments
our only window
to study the M
bears upon that of the "H" dibaryon8
combination
strong magnetic-color
of 6 quarks forces
is
interaction,
-- a uuddss
that could take maximal
in the one-gluon-exchange
and
spatially
advantage
interaction
of the among
quarks. Although accepted,
the interpretation
consistent.40 energy
to indicate
When ha forces are parameterized
in :He and aa forces are parameterized
sBe levels,
the M
quoted value Because dibaryon
force needed
to account
one would expect
a M
to reproduce to describe
for BAA(iiBe)
hyperfragment
if the H has a mass smaller argues against
events
oo scattering
to decay quickly
into an H
of an H bound with respect
events
to Ah
above
in which both As in 'OBe decay weakly must be very AA Many more ;XBe hypernuclei must have been formed and decayed
Therefore,
undetected
and for the
of Ah
Emulsion events are identified by weak decay (A+Nn) of the 4 which is strongly suppressed as the mass is increased
rare indeed.
are
the A separation
also accounts
than 2mA, the observation
the existence
hypernucleus, A=5.
that the two M
of BAA(AiHe).
hyperfragments decay.
of the AAaHe event has not been universally seem
model calculations
by the hN+NN weak process.
B. F. Gibson / Physics of few-body A hypernuclei
Because
of the serious
for this unique consideration signature
consequences
perturbative
of the existence
QCD prediction
hypernuclei
efforts
to confirm
lighter
mass M
of GHe
serious
to exploitation
other than their pionic
the existence
of Ah hyperfragments
of the H dibaryon,
should be given by experimentalists
for M
131c
are called
of a
decay modes.
Renewed
for, as is a search for
From B,.A(iiBe) = 18 meV, one can deduce
hypernuclei.
that
“H > 2m - 20 MeV. A the Ah pair should
Otherwise,
then one can surmize
decay rapidly
into an H.
If GHe
is confirmed,
that
mH z 2mA - 10 MeV.
If Af;He is not confirmed, because
model
calculations
then one can bound mH between these two values, 40,27 that are consistent with BAA(iiBe) being about
18 MeV also yield
an estimate
of BM(AHe)
pair do not decay
into an H.
If AHe
exist? an H.
Because
:H binds,
The 'He(K-,K")AiH
momentum
transfer
favorable
of about 10 MeV
is confirmed,
AAH will also bind, reaction
AiH
pair do not decay into
for the search,
in such double-strangeness-exchange
to ground-state
the Ah
does the hyperfragment
if the M
is a candidate
-- assuming
reactions
although
the
is not
formation.
6. SUMMARY An improved
measurement
of the A3H binding
the spin dependence
of the hyperon-nucleon
to model
forces.
three-body
tagged A beams coupling
Improved
is an important
low-energy
aspect
of hypernuclear
is small and the X is narrow.
dependence
for the AN interaction
state.
normal
Medium
nuclei.
of a repulsive identify
dibaryon.
corrections
This effect three-body
Ah hypernuclei
hyperfragments
is called
Ap scattering
are more
is maximal
(nuclear
is called
would provide
physics,
This produces
that varies
important
because a complex
spin
of the nuclear
in hypernuclei
force.
Mass measurements constraints
AN-EN
the A-C mass
than in
in :He and can be interpreted
core dependent) for.
data from
are needed.
with the isospin
important
for to constrain
and to test our ability
as well as An data from the K‘d-+Any reaction
difference
core
energy
interaction
A renewed
in terms effort to
of the lightest
such
on the mass of any S = -2
132~
B. F. Gibson / Physics of few-body A hypernuclei
ACKNOWLEDGEMENT The work of the author has been supported of Energy.
He gratefully
Scholar by Flinders D. R. Lehman
acknowledges
University
in part by the IJ. S. Department
the appointment
as a Visiting
Research
and a long and fruitful
collaboration
with
in this area of physics.
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Three-Body Systems.
Interaction
(1973) 509.
Forces
in the Trinucleon
at Low Energies,
this volume.
25)B. F. Gibson, G. J. Stephenson, V. R. Brown, and M. S. Weiss, in: Proceedings of the Summer Study Meeting on Nuclear and Hypernuclear Physics with Kaon Beams (Brookhaven Nat. Lab. Report BNL 18335, 1973) pp. 296-306. 26)B. L. Roberts, this volume 27)B. F. Gibson,
Radiative
Kaon Capture
A. Goldberg,
and Hyperon
and M. S. Weiss,
28)J. L. Friar and B. F. Gibson,
Weak Radiative
Phys. Rev. 181 (1969) 1486.
Phys. Rev. C 18 (1978) 908.
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and J. J. de Swart, Ann. Phys.
30)M. M. Nagels, 1633.
T. A. Rijken,
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Nucl.
Decay,
Nucl.
Phys. A450
Phys. A329
(N.Y.) 79 (1973)
Phys. Rev. D 20 (1979)
(1979) 308; Phys. Lett. 83B
(1986) 243~.
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and D. R. Lehman,
Phys. Rev. C 14 (1976) 685; 15 (1977) 2257.
35)B. F. Gibson, A. Goldberg, and M. S. Weiss, in: Few Particle Problems in the Nuclear Interaction, eds. I. Slaus, S. Moszowski, R. P. Haddock, and W. T. van Oers (North Holland, Amsterdam, 1972) pp. 188-190. 36)W. Stepien-Rudza
and S. Wycech,
Nucl.
37)B. F. Gibson and G. J. Stephenson, (1975) 1448. 38)L. H. Thomas,
(1981) 349.
Jr., Phys. Rev. C
8 (1973) 1222; 11
Phys. Rev. 47 (1935) 903.
39)H. A. Bethe and R. F. Bather, 40)A. R. Bodmer
Phys. A362
and Q. N. Usmani,
Rev. Mod. Phys. Nucl.
8 (1936) 82.
Phys. A463
(1987) 221~.