Three nucleon forces

435c

Nuclear Physics A416 (1984) 435~464~ North-Holland. Amsterdam

THREE NUCLEON FORCES

Bruce H.J. McKELLAR School of Phvsics. Australia -

University

of Melbourne,

Parkville,

Victoria

3052,

and Walter GLOCKLE Institut fijr Theoretische D-4630 Bochum 1, W-Germany

Physik II,

Ruhr-Universitxt,

This report is derived from both the invited review paper on Three Nucleon Potentials given by BHJMcK at Few Body-X, and from the discussions at the international workshop on Three Nucleon Forces organised by WG and held at Ruhr-Universitit Bochum, on 18th and 19th Auqust 1983. It attempts at the same time to provide a record of the content of the review paper and to summarise results presented at the workshop.

1. INTRODUCTION The concept of the three nucleon potential Yukawa's

proposal

that mesons mediated

tion of the a mediated and the realisation

1~71exchange

3 nucleon potential

by Brown, Green and Gerace3

straints could play an important nucleon potentials

was advanced

ushered

that current algebra con-

role in determining

in the present

of calculations changed, devoted

the meson exchange

era of meson-theoretic

generated

workshop

sophis-

sophistication Now that has

on three nucleon forces was of the effects of 3NP in

systems and in nuclear matter.

by the growing fails

-

in the effects of 3NP in nuclear realisation

that nuclear

we cannot reproduce

nor the binding energy and equilibrium librium properties forces.

of increasingly

by a corresponding

to reports of "state of the art" calculations

In part this interest

tentials

3NP was not matched

three

3NP.

of the effects of the 3NP in many body systems.

and much of the international

few nucleon

The construc-

by Fujita and Miyazawa',

For more than a decade this work on the construction ticated meson theoretic

very soon after

the two nucleon force'.

of nuclei in between

than adequate

input, to suggest

technical

density of nuclear matter, these extremes

that what is missing

B.V.

nor any equi-

with just two body

in our numerical

skill.

0375-9474/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

physics with two nucleon po-

the binding energy and radius of 3H,

And we now have enough confidence

two nucleon potential

systems has been

methods,

and in the

is physics rather

436c

B.H.J. McKellar,

W. Gliickle / Three-,Vucleon

Forces

It seems that the KITE-3NP, which has received the most theoretical so far, is not capable of repairing ready generated tentials

work on the meson theoretic

and on the construction

Undoubtedly

all of these deficiencies4. construction

of empirical

the next Few Body conference

attention

This has al-

of shorter range po-

3NP with the desired

properties.

will hear of many more such develop-

ments. In this report we first of all review the meson theoretic three nucleon lem.

potentials,

Then we review applications

attempts

to develop empirical

approaches

to three nucleon

2. MESON THEORETIC 2.1

then discuss 3NP.

theoretic

illustrated

systems and describe

and draw some conclusions.

POTENTIALS Potential

(WE-3NP)

in figure 1, has been the subject of most meson

studies of the potential,

work contributed

of prob-

Finally we discuss a number of alternative

The Two pion Exchange Three Nucleon

The IWE-3NP,

construction

in the three nucleon

to other many nucleon

potentials

THREE NUCLEON

applications

from the work of Fujita and Miyazawa

to this conference5.

The key ingredient

required

to

to construct

4 T-T

,_-----

l-r -

Fig.l(a) the nnE-3NP

is the

nN scattering

for off mass shell pions. philosophy

(b) The underlying

The T~ITE-~NP

amplitude

of figure l(b) which

Two basic approaches

have been used to obtain

The first is the mode2

independent

ITN Scattering

information philosophy

amplitude

is required

which differ significantly about this amplitude. which seeks to obtain the

in

maximum

information

hold independently this approach

about the off shell amplitude of any dynamical

has been exploited

group6, and the resulting The alternative struct a dynamical is the approach

most extensively

potential

approach

model for the 7N scattering

the 60's by Nogami and his collaborators8, from Robilotta

(and advocated

the advantages

ardently)

and disadvantages

The technique

used in calculations.

approach which seeks to conamplitude

of figure l(b).

of including

and it is appropriate

and is represented

et a1.5

in the contribu-

Both approaches

at the workshop,

This

It was used in

were discussed

and we therefore

review

of each.

resonance

has been advocated

wavefunction)

by the Tucson-Melbourne

has been extensively

in the model makirq

results which

For the F-TE-~NP

of Ore11 and Huang7 and Fujita and Miyazawa.

tions to this conference extensively

from general

model of the amplitude.

states in the Hilbert Space

as a way of including

(a's in the

the three nucleon forces,

to compare this method with the TN amplitude

methods

in

this section. Before doing so we should define the kinematic ing amplitude

structure

of the TIN scatter-

T=l-S, of figure l(b) which we write in the usual isospin and

spin decomposition

as

Tij

= T(+) "ij

,(k) =

F(k)

+

+

T(-) icijk 7k

(1)

,(*) [ef,pI']

.. The amplitudes x

TIJ are understood

Pauli isospinors

tudes F('), B(') are functions variables

to be sandwiched

of four variables

-

The invariant

ampli-

the usual kinematic

L' = _L 4m (p + p')*(q + q') and t = (q - q')2 but also q2 and q12 since

the pions of interest are off mass shell. which the nucleons ignored).

between the Dirac Spinors

of the initial and final nucleons.

Brown and Green' have suggested

pions important

(The relatively

small extent to

are off shell because of the nuclear binding

in nuclear processes

ly small on the typical ~,',"$I"~n~~~g~~~~tt~~~)

that typical

are from -p2 to -15u2, which are relative-

hadronic mass scale of 1 GeV. Iv1 5 $-p and 0 2 t 2 -15~~. and

is customarily

"masses" of the virtual

B(') in this kinematic

An extension

of their

We should therefore

ask

region.

First of all, in the nearby on shell (v,t) plane where q2 = q12 = p2 the 10 subthreshold expansion extrapolates the amplitudes from the physical region to the (v,t) region of interest.

However we need the amplitudes

mass shell, so we must use the subthreshold a constraint

expansion

on the off pion mass shell extrapolations

The off mass shell extrapolation

off the pion

in the on shell plane as which are necessary.

is further constrained

by the soft pion

438c

B. H. J. McKellar, W. Cliickle 1 Three-Nuclem

Forces

theorems established These constraints vector nucleon

from PCAC and current algebra by Adler _ apply only to -(+) F the amplitude F(+",::t

pole term subtracted

v = 0 hyperplane13 the interchange

illustrated

-

and are usefully visualised

in figure 2.

F(+) is obviously

q2 c-f q12, which is the operation

WXC in figure 2.

Adler's

consistency

:~~n~:::~~:

condition

of reflection

constrains

the points marked A and A' in the figure, and Weinberg's

on the

symmetric

under

in the plane

F (+) to vanish at

double soft pion limit

fixes the value of F(+) at the point W to be -u/f:, where o is the nN d term and fn is the TI decay constant (f N 93 MeV with this normalisation convention). (U = 0) T If one assumes that -(+) F is a linear function of t, q2 and q12 these -(+) soft pion constraints determine F everwhere on the plane WAA'. In particu-

ffj

2 r’,_-_ mm-

I’ /I

$--_

+-_

-_

/I ;

I

I I IA I p2_-_ --;I----I ,’ :’ I I ,I A,$____-!’ I x 1

/I

---D-B

Fig.2

The v = 0 hyperplane

in v, t, q2, q12 space

lar they fix F(+) to be +o/fG at the Cheng Dashen point C, which is the intersection of the "soft pion plane" WAA' and the on shell line q* = q12 = u2 (XC) This permits the determination possible

correction

of u from the on shell amplitude

From this point on the two possible figure l(b) part company. struction

values of the arguments

potential

(on the hadronic

will suffice.

for this amplitude.

of

used in the con-

opproaciz

the fact that relatively

small

the most impor-

of T up to quadratic

This means that for F(+) the linear and as crossing

of odd powers of v the linear approximation

plane is adequate approximation

6

scale) are apparently

in v, t, q', and q'* suffices

the appearance

up to

to the YN amplitude

is used to suggest that a knowledge

terms in nucleon 3-momenta approximation

approaches

In the mode2 tndependent

of the Tucson-Melbourne

tant in applications

-

terms of order t2.

symmetry prohibits in the v = 0 hyper-

To fix the F(+) amplitude

we need its value at one point outside

in this

the soft pion plane WAA'.

In ref.6 the point X was used, because the amplitude had been directly evaluated at this point using dispersion the on shell line would,

in principle,

Using this information coefficients

the model

constructing

=

included

approach

is able to fix the

-(+), of F

a + B t + y(q2 + q'2)

(3)

It is important

of all of the resonances

to recognise

(in all channels)

of keeping only the terms quadratic

major invariant

flip amplitude quadratic

independent

are

in these coefficients.

In the approximation theother

However any other point on

.

a model for the amplitude.

that to this order the effects already

14

serve just as well.

N, 8, and y in the expansion

++I without

relations

B(-).

amplitude

which contributes

Since the covariant

in the nucleon momenta

term in that expansion

this amplitude

we need only the constant

sion of B(-) which can be taken directly ever, since the subthreshold

multiplying

expansion

in nucleon momenta,

to the TT~E-~NP is the spin is already

term in the expan-

from the subthreshold

How-

expansion.

is on pion mass shell the "constant"

is really of the form a + y(q2 + q12) with q2 = q12 = u2,

one needs to make a model calculation

to verify that the term 2~1-1~is indeed a

small correction.

In this sense one can say that the Tucson-Melbourne of a detailed

dynamical

has the advantage automatically

model of the underlying

that, to the order considered,

included.

The disadvantaqes

model for the TN amplitude wishes

to include effects

potential

nN scattering

is independent

amplitude.

all resonant contributions

of not having a detailed

are that such a model

This

is in fact required

of higher order in the nucleon momenta,

are

dynamical if one

or if one

B.H.J. McKellar, W. Gliickle / Three-Nucleon

44oc

wishes to consider model

independent

three body exchange

currents.

Forces

(In the latter case some

low energy theorem results would presumably

be obtainable,

but one would need explicit models to go beyond them.) There is a systematic the model

independent

way to graft models for the higher order terms onto

amplitude.

To do this one writes

the amplitude

in the

form T

=

T

+

pole

AT

q'*C*q

+

(4)

where T

is the nucleon pole, AT contains the current algebra terms and the pole terms required to satisfy the soft pion theorems, and q' * C . q contains the resonance

If the latter are included using a dispersion

contributions.

tion for C (rather than for q' . C . q) one is guaranteed

rela-

to not disturb the

soft pion results. The alternative

modei, muking

t :)

approach

to Few Body X by the papers of Robilotta

is represented et a1.5

related to the coupled channel calculations the nuclear wavefunction amplitude

in figure 3

which

in the contributions

This approach

is also closely

include L resonant

states in

15 . We represent a typical dynamical model of the nN

, where we also indicate the correspondence

terms of the model and those of the PCAC Current Algebra model

between the

independent

approach.

I’

I’

I’ I’ If

I

/

--

I

If : /

I

I f

I I I’ I -\ \ \\ \\ \\ \\ \ \

+

+

I

t

I

/

I’ : M

pole

:

I’ I’ Current Commutator

q'.C*q

nM

a Term

"soft pion terms" Figure 3:

The model of Robilotta

et al.

for the TN amplitude.

Dynamical

models

disadvantages mechanism models (i)

for the nN scattering

of the model

independent

amplitude

approach

for going beyond terms quadratic

constructed

up to now have suffered

-

do overcome

some of the

in particular

they offer a

in the nucleon momenta,

However the

from two difficulties

they do not satisfy the soft pion constraints

at the points A, A' and W of

figure 2. in that the o term is included

they are not truly dynamical,

(ii)

way and is not represented Furthermore,

the work of Shimizu,

that the A resonance may contribute

of a physical (or fictious) particle. 14 Polls and Miither has raised the possibility

alone may not be sufficient,

significantly.

the possibility

in an ad hoc

by exchange

Once model making

of adding more diagrams,

and that higher resonances is started,

there is always

and one should look for criteria which

decide where to stop. If, instead of expanding

the model

A pole terms, this suggests the calculations

of nuclear physics

states in which some particles function approach,

This method (7)

(ii)

into

if the Hilbert space is extended to include 15 . This idea, the A's in the wave revived and pursued vigourously

channel calculation

by the

of the three nucleon system

16.

has several advantages.

Additional

discussed

to just the nucleon and

forces can be incorporated

are A's

has been recently

tiannovergroup in a coupled

it is contracted

that three nucleon

three body forces, sucF7as

by Fujita, Kawai and Tanifuji

Those terms of higher order

propagation

of the A are readily

contribution

that of figure 4(a), originally

, are automatically

in momenta included,

generated.

which correspond

and significantly

to non static alter the 3NP

to the energy.

,_( l-l _----_

A

A

l-T

a. Figure 4.

b.

Additional

effects

a.

The 3aE - 3NP of Fujita,

b.

The dispersion

included

in A in the wave function calculations.

Kawai and Tanifuji.

effect on the two body potential

contribution

to the energy.

B.H.J. McKellar,

4422

(iii) Dispersion the diagrams

corrections

W. Gliickle 1 Three-Nucleon

Forces

to the two body force contribution

arising

from

of figure 4(b) are readily included and tend to cancel the effects

of three body forces.

In the conventional

nucleon states in the Hilbert

approach with potentials

space, these dispersion

effects

and only

can be included

only by taking the energy dependence of the two pion exchange two nucleon po18 , and finding ways to handle this energy dependence in many

tential seriously nucleon problems. (iv)

n and F exchange

can be readily

Despite these advantages

included.

the A's in the wave function approach

also has its

short comings. (i)

The implied nN amplitude

amplitude

does not provide a good representation

in the subthreshold

region. To convince 19 the tables of Olsson and Osypowski , multiplying

tables by 1.37 to allow for the different Some coefficients

of the IAN

oneself of this one can use the A amplitude

choice of nNA coupling

in those

constant.

are listed in table I.

TABLE Coefficients

I

f\'), b\-) in . the subthreshold

expansion

al + a,t + (a3 + a,t)v2 + (as + a6t)v4 + a7t2 + ..

(ii)

i

1

2

3

4

5

6

F+

model expt

-2.12 -1.45

1.10 1.18

1.00 1.14

0.19 0.15

0.23 0.20

0.030 0.034

B-

model expt

7.08 -0.15 10.33 0.20

1.23 1.04

-0.074 -0.063

0.25 0.27

-0.023 -0.030

The implied nN amplitude

7 -0.017 0.001~0.04 .003 .004

does not satisfy the off pion mass shell con-

straints of current algebra and PCAC. (iii) The TIN amplitude

implied by the A's in the wavefunction

the A is on its mass shell differs amplitude

in which the A propagates

significantly

approach

from the corresponding

off its mass shell.

In particular

in which Feynman off mass

shell ambiguities

exist in the latter amplitude, but appear in the former only 20 terms in the Hamiltonain are included . While it may be .19 argued that Olsson and Osypowskl found that the subthreshold expansion in irN

when the contact

scattering

is well reproduced

tribution,

in their analysis

contribution

when the off mass shell term makes a small con21 they found that that a large

of photoproduction

from the off mass shell term was required.

the p meson exchange

contributions

to the amplitude

This should influence 20 in a significant way .

For these reasons one cannot escape the conclusion

that, even in a A's in

B.H.J. McKellar, W. Gliickle / Three-.Vuclron

the wavefunction forces

calculation

one cannot avoid including

which one would expect

-

to contribute

same strength as the A contributions 2.2

of including

explicit

three nucleon

to the energy with about the

themselves.

The n-p Exchange Three Nucleon

The importance

443c

Forces

Potential

(3 exchange

(T~JE-~NP)

terms as well as % exchange

the NN -f NA transition

potential

shop on this subject.

The pN + nN amplitude

terms in

was recognised quite early in the development 15 of the A's in the wavefunction approach , but P exchange contributions were 22,23 . There included in explicit three body potentials only quite recently 24,25 and a presentation to the workwere two contributions to the conference

photoproduction

amplitude, 26 to this conference _ Once again model construction approaches

independent

of the amplitude is illustrated

has obvious relations

which has been discussed 23324

and model making

are possible.

to the TI

by Ellis in a contribution 22,25

approaches

The relationship

in figure 5, which is to be compared

between

to the these

to figure 3.

P M

M

Figure 5:

AM

P

The pN -f nN scattering

amplitude

In the model independent approach chiral symmetry breaking terms, analogous to the o term in the nN scattering terms appeared Scadron27. breaking

amplitude,

for the IT photoproduction

However,

in contrast

terms do not contribute

is not such a difference for the TRITE-3NP.

between

arise in the same way that these

amplitude

studied by McMullen

and

to the TN case, this time the chiral symmetry in a numerically

significant

the two approaches

way.

Thus there

for the ~ITE-~NP as there is

444c

B.H.J. McKellur, W. Gliickle / Three-Nuclron

Forces

The term of lowest order in k, which one may have expected 3NP, arises from the analogue may be regarded ys coupling

Kroll-Ruderman

as a contribution

to dominate 28

term in IT photoproduction

from the backward

propagating

scheme, which is how Kroll and Ruderman originally

It may alternatively ysyp coupling

This minimal

This

in the theterm.

be regarded as arising from minimal

scheme

photoproduction

nucleon derived

the

.

substitution in the 29 as pointed out e.g. by Domhey and Read , in the

-

case.

substitution

Ellis, Coon and McKellar

is, in the present context,

choose to emphasise

of the term, and to work with y, coupling, the "contact" or seagull The final contribution

interaction

-ss,

=

whereas

arising

to the amplitude,

M ij V

the "backward propagating"

et al. emphasise substitution.

is in the soft p limit (k a 0)

iFijk

4m

Robilotta

from the minimal

nature

iO"

+

'k

O(k)

for space like V. Robilotta

and Isidro Filho, and earlier McKellar,

that this would be the major contribution the case of nuclear matter ressed relative Kroll-Ruderman correlations

to the pnE-3NP.

the contribution

However,

of the Kroll-Ruderman

to that from the A by the spin-flip, 3NP.

Coon and Scadron

23

claimed

at least in

term is sup-

isospin-flip

nature of the

The contribution

in the wave function,

of the A is further enhanced by tensor 24 and Ellis, Coon and McKellar found that

the A gives the dominant contribution,

although

the Kroll-Ruderman

term is not

insignificant. Two rather subtle points and the related 3NP. amplitude. 7171force.

arise when considering

The first is the treatment

the pN -f nN amplitude,

of the 71 pole in the pN * TN

As can be seen in figure 6 this may be regarded as a QV force or a As there are two possibilities

treat this diagram without in constructing

double counting

is to regard it as a w~E-~NP.

a 3NP from the pN -f nN amplitude

forward propagating

The other subtlety to the background

we must subtract

Thus

both the

nuclear poles and the t channel 7~ pole from pN -f nN ampli-

tude before constructing

the potential.

is that one must ensure that the A contributes

term only

-

readily achieved with derivative and disperses

for the former the only simple way to

C rather than M.

explicitly

i.e. to terms of order k2, q2 or q-k. coupling,

This is

as long as one writes MU = k*CV*q

Should one adopt the alternative

of dispersing

Figure 6:

3 Vays to regard the n pole in the pN + rN amolitude ting to the 3NP.

M, then one obtains -

to the Mdj amplitude

a contribution

this is however just what is required

.

as contribu-

proportional

to cancel the backward

to ~w&S'~

propagating

This cancellation is embodied in the Born term in MAJ in the soft pion limit. 30 , which is the photoproduction analogue of Fubini, Furlan, Rosetti relation the Adler zero in the nN scattering The net result of the detailed Coon and McKellar

analysis

is that the pnE-3NP

the sum of the Kroll-Ruderman by Robilotta

amplitude. of the PN + nN amplitude

by Ellis,

is given to quite a good approximationby

term, discussed

by McKellar,

and Isidro Filho, and the n contribution

Coon and Scadronand

introduced

by Martzolff,

Loiseau and Grange. 2.3

The pp Exchange Three Nucleon

The ,-oE-3NP was also introduced dominance

by Martzolff,

Loiseau and Grange,

In the non relativistic

approximation.

is of the form ((gl x ,l$ x k_) this case, as emphasised

Potential

by McKellar,

These give a 3NP proportional

order in the momenta There

potential

However in in momentum space. 23 Coon and Scadron , the backward propaga-

- ((g2 x k_‘) x k’)

ting Born terms do not cancel but give the p meson equivalent tering.

in the n

limit the resulting

to (OJ x k)

* (z2 x k’)

to Thompson

-

scat-

of lower

than the n term.

is also an additional

contribution

from the current commutator

These give rise to spin and isospin flip terms in the pN + pN amplitude are in the non relativistic

limit

terms. which

M

ij UU

=

(1 +

KP)

E:

ijk 'k '~31

cs"

for space like ii, w, X, and were discovered by Beg in his analysis of "isovec31 . If one were to make models for this term, it is

tor photon" scattering given by figure 7

Figure 7:

-

it arises from the c;meson pole in the t channel.

The p meson pole in the pN + pN amplitude Beg term in the amplitude.

However,

Ellis, Coon and McKellar

lower order in the nuclear momenta, energy than the A term. coupling tude

24

which gives rise to the

have found that these terms, although

nevertheless

contribute

This is partly a consequence

of the strong magnetic

of the F meson, both to the nucleon and to the N-A transition

-

the effective

expansion

parameter

of

much less to the

is not k/m but K

o

ampli-

k/m, which for

k/m Q, l/2 is greater than unity. This "breakdown" potential

approach

could be confined tribution

of the k/m expansion

to the opE-3NP, which according

ot the total energy in nuclear matter.

which should receive further 2.4

meson propagators

ofthe it

Clearly

this is a problem

study.

to extend the Yukawa terms

down to very small values of r.

it is necessary

Less dramatically

to ref.24 makes a small con-

The Form Factor Problem

It is clearly unphysical

problem

could even signal the breakdown

to the nuclear many body problem.

to take into account

-mr Cr.._

implied by the

We know in the two nucleon

other exchanges

with the same

quantum numbers as those of the meson, and that it is common in OBEP models take these exchanges vertex3*.

into account

through form factors at the meson nucleon

to

447c

The same solution that the numerical

to the problem has been used for 3NP.

results3;btained

This observation

factors.

to this conference

are very sensitive

has been revived

by Robilotta,

tances

-

certainly

Transform

of

in a new form in the contribution 5. . 'They point out

to moderate

in fact influences

but rather the Transform

this consider

when the potential

v(r)

which

=

Figure 8:

When this is modula-

.

the contribution.

the simple case of a square root form factor

in configuration

F.T.

is plotted

&

ensures that the short range part

with a range ,".-Iwhere p% is the cut off in the form factor,

has a factor :? which greatly enhances To illustrate

is no2t in general the Fourier

of

ted by a form factor, the k* in the numerator of the potential,

be-

at quite large dis-

of the influence of the form factor

is that the radial potential

--L_k2+p2

the short distance

the potential

up to 2 fm.

Part of the reason for this extension to large distances

is

form

Isidro Filho, Coelho and Das

that the form factor, which was introduced haviour of the potential,

The difficulty

to the assumed

space behaves

~

UL-

,-i”r r

-

like

,-nr

A2 -3

r

in figure 8 for A = 611.

Showing the influence of the form factor on the radial dependence of the potential (after A. CassS3).

(8)

B.H.J. McKellur, W. Gliickle / Three-Nucleon

448c

It should be emphasised exchange

channel,

that this is a physical effect

and hence the nNN form factor,

relation34

the expected

calculation,

exchange35

the one pion from3n

A number of attempts

of the form factor;

fitted with a monopole

the average differential

consistent

behaviour

-

receives contributions

exchange which does indeed have quite a long range. been made to calculate

Forces

have

a dispersion

form gives ii = 5.5~;

fits to

cross section for np Charge Exchange and pp charne

suggest a smaller !i = 4.1~ in the monopole fit; and a recent self 36 to fit a number of form factors used the parameterisation

attempt

F(q 1

and obtained

1

a

1 1 + (qZ/n$)" a$1

1 + q/q

+ a q2IAL

QCD

(9)

)

for the ;INN form factor with n = 2, :I1 = 1 GeV, ‘2, = 9.49 GeV.

the latter two cases information factor from the constituent

about the large qL behaviour

interchange

in the other37 was used to constrain All of these analyses, the small q2 behaviour cal and experimental

model

In

of the form

in the first case, and from QCD

the parameterisation.

while they differ

in details,

serve to suggest that

of the form factor may be able to be fixed by theoreti-

work.

12 % 5.6~ (700-800 MeV) would seem to be a reason-

able value to adopt, and for these values the form factor does induce significant changes

in the potential

in the 1-2 fm region.

We believe that these effects are physical not be discarded

in the way proposed

rather than spurious,

by Robilotta

et al.

However

and should

results ob-

tained for the effects of three body forces will depend sensitively

on the be-

haviour of the form factor, and more effort could usefully be devoted to fixino the form factor from independent

3. EFFECTS OF THREE BODY FORCES 3.1

Contribution

studies.

IN THE THREE NUCLEON SYSTEM

to the Binding Energy

There are a number of approaches

to the calculation

the triton, and by now many of these methods effects of a 3NP, including method,

the Faddeev scheme, the variational

the ATMS method and the hyperspherical

a comparison

of results obtained

with a monopole

harmonic method.

in these calculations

results only for those calculations make this comparison

into the Faddeev or equivalent 3NP, W, does not introduce

To facilitate

3NP - we choose to

in a variety of ways

distinct

T~TE-~NP,

A 2, 6~.

the 3NP may in principle

equations

physically

Monte Carlo

done using the Tucson-Melbourne

form factor with a cut off parameter approach,

Energy of

we will present detailed

which use a comparable

with calculations

In the Faddeev equation

of the Binding

have been used to include the

amplitudes,

be incorporated 38 . Because the

it is possible

to

B.H.J. McKdar, W. Gliichlr / Three-~Vicleon Forces split TW, the partial T matrix various ways giving different

in which the last interaction generalisations

method which has been exploited

in practice

lise the fact that this particular

449c

is the 3NP, in One

of the Faddeev Equations. for the ZnE-3NP3'

W may be written,

has been to uti-

in a natural way, as the

sum of three terms w where the subscript then possible

=

w, + w, f w,j

(10) It is

labels the nucleon which is the active scatterer.

to associate

W, with V1 (the two body interaction

3) and define a partial T matrix

T

=

between

2 and

by

(vl + W,)

+

(V, + W,) Go T

(11)

from which the wavefunction Y

=

(1

+

P12P23

+

(12)

P13PjZ)O

where 1! = may be constructed The T matrix generated

Go T(P,,P,,

+

(13)

P,,P,,)$

in the usual way.

including

three body forces can be expressed

in terms of that

from two body forces only, which we call t, by solution of T

In a momentum

=

space approach

are the Jacobi co-ordinates,

t + (1 + t Go)

W1(l + Go T)

.

(14)

to the three nucleon problem the material whereas

change 3NP are the meson momenta

-

the natural coordinates

for a meson ex-

which depend on both of the Jacobi momenta

for the initial and final states, and so the 3NP has a complicated 40,41,42 expansion The (j#

variables

partial wave

) convention for label ing the three body states is adopted, so that

the states are [[a x s'lJ x [A xl 2$ states generate

lJ Lt x ;I'.

The %a,

3S,, 3S,-JD, two body

the 5 channel truncation of the Hilbert Space. 39 of computing the energy including the 3NP used a trunca-

Gl'dckle's method

tion of W, to the first three channels, the Reid potential the Tucson-Melbourne

a solution of (14) to first order in W,,

for V and the solution nnE-3NP

and obtained

of (13) in momentum the result quoted

Glockle found that the results are sensitive form factor. Muslim,

space.

in Table

He used II.

to the choice of cut off in the

A = 5.8~ gives E, = 1.3 MeV, but A = 7.1~ gives E, = 1.9 MeV. 40 41 , and Bomelburg , calculated Ea perturbatively, from

Kim and Ueda

B. H. J. McKellar, W. Gliickle / Three-Nucleon

45oc

Forces

(15) using a 5 channel calculation parable,

but with a residual

They obtain a negligible

of I$. These results, given in table II, are comnumerical

discrepancy

which needsto

net Ea, s and p wave terms cancelling

be understood.

against each

other. It is interesting obtained

to compare these results with the result Et = 0.65 MeV 43 . They solve the Faddeev equations in configuration

by Torre et al.

space truncated

to 12 channels,

using the Tourreil-Sprung

tion, and a square root form factor with :L = m. haps be attributed

two nucleon

interac-

The small result could per-

to the longer range form factor, were it not for the fact

that the convergence question quoted

of the partial wave expansion of (15) has been called into 44,45 by some recent work of Biimelburg et al. . An 18 channel result,

in table II is obtained which is again very small.

quite large individual tuate in sign.

contributions,

However

For example

< #lIWa(l + P)l#l >

=

-0.164 MeV

(16a)

< #lIWa(l + P)l#18 >

=

0.330 MeV

(16b)

Where

liil> =

/lSo, $;4

> has 44$ of the normalisation,

and

1#18> =

13D2, d5,2

; 4> has just +O.l% of the normalisation.

There is thus no reason to expect that the omitted channels tributions.

The calculation

ating matrix

elements

expansion

they find

even up to the last channel, which fluc-

must either be extended,

(or more directly

Ea) without

give small con-

or a way found of evalu-

recourse

to a partial wave

of the P operator.

One method which has the advantage

of not requiring

such partial wave expan-

sion is the direct evaluation of the matrix elements through Monte Carlo inte46 , andcarlson, Pandharipandeandwirinqa 47 have used this tech-

gration. Wiringa

nique in conjunction

with a variational

of the wavefunction

after including

beyond perturbation

theory in W.

calculation,

They found a relatively

the three body force, 1.15 MeV attraction. 48 Wiringa reported more recent calculations Monte Carlo method

of evaluating

revarying

the parameters

the three body force, so their result goes large net effect from

to the workshop,

the expectation

in which the

value of the Hamiltonian

is

used in conjunction with a three body wave function obtained from Faddeev cal49 . Using the two body Reid potential, the binding energy of the

culations

triton is significantly

lowered

of the n~rE-3Np to the potential

by the new wave function, is not changed.

but the contribution

Wiringa

46,48

is altered.

also reported an interesting

Table

III

effect when the two body potential

shows some of his results.

II

TABLE

Results of Triton Binding Energy Calculations. The Reid Potential is used for the two body potential, and the Tucson-Melbourne T~ITE-~NPwith a monopole form Energies are in MeV. factor and !? % 6~ is used for the three body potential. P

EZ

G

-7.02

-

-

-1.28

-8.3G

3 Channel W

MUSLIM, KIM and UEDA ref. 40

-6.98

+1.03

-0.96

+0.07

-6.91

5 Channel

BiiMELBURG ref. 41

-7.02

0.90

-0.70

0.20

-6.82

5 Channel

BbMELBURG GLDCKLE ref. 44

-7.02

-

-

-0.16

-7.18

18 Channel

CARLSON PANDHARIPANDE WIRINGA ref.47 , 48

-6.62

0.38

-1.53

-1.15

-7.77

Variational function

WIRINGA ref.48

-7.08

0.28

-1.43

-1.15

-8.23

Faddeev Wave Function Monte Carlo Integral

GLiiCKLE ref. 42

It will be observed

that as the repulsive

weakened,

increasing

tribution

becomes more repulsive,

energy.

The physical

Wave

core in the two body potential the three body potential

leading to a somewhat

is con-

smaller total binding

reason for this is the larger probability

that two nucle-

from each other, and so feel more of the short range

in the nnE-3NP. TABLE

Dependence

III

of Triton Binding Energy calculations

Two Body Potential

EZ

on the Two Body Potential.

EZ

E!

E E3

tot

Reid V I4

-7.08

0.28

-1.43

-1.15

-8.23

Argonne

-7.29

0.41

-1.35

-0.94

-8.23

-7.38

0.58

-1.25

-0.67

-8.05

V I4

Supersoft

Core

Now we see that we are entering structed

Comment

tot

the two body binding energy,

ons are a short distance repulsion

E

E3

E3

to be a good approximation

terra incognita

-

cal effects which depend on its short range properties by shorter

the nnE-3NP was con-

to the long range part of the 3NP.

Physi-

are likely to be altered

range parts of the 3NP, which must now be included.

One can either

452

B.H.J. McKellar,

W. Gldckle 1 Threehircleotl

utilise the ~roE-3NP and the ppE-3NP discussed approach, which we describe behaviour

of the 3NP to fit the observed

significant

above, or a phenomenological

below, in which one tries to fix the short range

4He and heavier nuclei. 50 Hajduk et al. have applied They find a considerable

Forces

binding energy discrepencies

the A coupled channels approach

dispersion

in 3H,

to the triton.

effect in the two body interaction,

effect from a non static A propagator.

form factors their results are summarised

and a

With h = 71-1and monopole

in Table IV.

TABLE IV Contributions approach50.

to the energy of "H in a A coupled channels

Interaction

Two Body Dispersion Energy

Three Body Energy

Total Energy

2.1 -2.2 0.6 0.7

-1.4 0.5 -0.8 0.1

0.7 -1.7 -0.2 0.8

ml

TP To%

Once again, there are almost complete leading to a small final result. coupled

channel approach

been discussed

above.

ccancellations

The advantages

as a representation

A consequence

result from this approach

of the cancellations

is that it is important

channels

calculation,

terms,

of the A

of three body force effects

such as the s wave ~T~E-~NP, the Kroll-Ruderman terms in ttie A coupled

of individual

and disadvantages

has

and the small final

to include three body forces,

~rpE-SNP, and off A mass shell

and we see this as the desirable

next step. 3.2

The Doublet Scattering Length 43 have calculated the effect of the p wave part of the Tucson-

Torre et al. Melbourne

nnE-3NP on the doublet scattering

Reid soft core value right direction, 0.65 fm. explain

length.

They found that the large

(a2 = 1.5 fm) was reduced to 1.0 fm

-

a step in the

but not far enough to agree with the experimental

Delfino51,

value of

in a report to the Bochum workshop,

showed that one could 52 this result on the basis of the work of Girard and Fuda which related

a2 to the energy EV of the virtual state of the triton, which threshold

in the second sheet.

a2 including

Delfino then exploited

the full Tucson-Melbourne

state with the triton quantum

numbers.

mation to the Malfliet-Tjon

potentials

His results are sumnarised

in Table V.

is below a n-d

this connection

I~TE-~NP, averaged

to obtain

over the spin isospin

He used a separable

Unitary

Pole Approxi-

I and III as the two body interaction.

453c

TABLE V Binding Energy of the bound and virtual states of 3H, and the doublet scattering length. (after Delfino51).

E3

without

3NP

with 3NP

3.3

EV

a2

-8.50 MeV

-0.49

MeV

0.8 fm

-9.05 MeV

-0.47

MeV

0.5 fm

The Charge Form Factor of 3He

One of the intuitive

arguments

in favour of a non trivial 3NP effect in

three nucleon systems

is the observation

sity, or equivalently

a "large" secondary maximum

However calculations

with present three nucleon

observed

charge form factor.

Carlson,

Pandharipande

log

of the dip in the central charge den53 in the charge form factor potentials

do not reproduce

the

As an example we show in figure 9 the results of 47 and Wiringa for the charge form factor of 3He. The

lFl2

With _._._._

3NP

Without

3NP

Figure 9 The Charge Form Factor of 3He, for the Reid Potential with and without Tucson-Melbourne ITTE-3NP (after ref.47).

the

B.H.J. McKellar, W. Gliickle / Three-Nucleon

454c minimum

Forces

is moved to smaller q, and the height of the secondary maximum

creased,

but neither of these trends is sufficient

form factor to be in agreement to look to exchange

with the observations.

current effects,

pancy between calculated

for the calculated Possibly

or quark effects,

and observed

is incharge

it is necessary

to resolve the discre-

form factors.

3.4

Effects of three body forces in break up reactions 54 Slaus, Akaishi and Tanaka noted the discrepancy between length extracted

from different

creases a

=

-20.7 * 0.2 fm


+ nn)

=

-18.6 i 0.48 fm


=

-16.73 + 0.47 fm.

that including

increases

extracted

three body forces in the analysis

arm extracted

from knock out reactions,

from pick up reactions,

thus removing

However ME?er55,

in a paper presented

tions suggesting

that the effect of the 3NP on the extracted

angle dependent,

and that a succinct

impossible Meier55

statement

and Bomelburg,

Glockle and Meier

in star, collinear

final state phase space. will be difficult

to see

56

45

and quasifree

of the and de-

the discrepancy.

at the Bochum workshop,

to make.

have effects

the n-n scattering

In particular

arm (nd + pnn, knock out)

They then suggested break up reaction

reactions.

reported

calcula-

value of arm is

of the effects of 3NP on arm was

give results that the 3NP could scattering

These effects are predicted

regions

in the pnn

to be small (2 13%) and

, but this seems to be a useful way to look for

effects of the 3NP. Two warnings

are in order.

its important attributes, configurations suppressed

Since the angle dependence

one may have expected

to display an enhanced

cross section.

However,

either the star or the collinear

cross section, and the other to show a

in the calculations

cross section for both configurations

of the 3NP is one of

is enhanced

it is found that the

by the n~rE-3NP.

Thus one

must guard against being led astray by intuition. Almost as important present calculations.

is the warning

the three body force (spin-isospin

of Ea.

Whether

are superseded.

This process makes significant

changes

in

similar changes occur in the break up reaction

will only be known when more extensive approximations

and

matrix element of the Tucson-Melbourne

TT~E-~NP) are subject to improvement. the calculation

that one should not stake too much on the

Both the two body force (an UPA to the Malfeit-Tjon)

calculations

are done and the present

B.H.J. McKellar, W. Glijckle 1 Three-Nucleor~ Forces

45SC

4. NUCLEAR MATTER Nuclear Matter has been the traditional Since the last Few Body Conference in calculations nucleon

there have been two significant

of E,/N, the contribution

in symmetric

nuclear matter.

In the first of this Wiringa tional Monte Carlo approach, isospin dependent tive potential

testing ground for three body forces.

et al.

46-48

have calculated

using the FHNC(4)/SOC

correlations

method

to include spin and

used in earlier evaluations

of the 3NP to nuclear matter6

E,/N in the varia-

This relaxes

in the wavefunction.

approximation"

developments

of the 3NP to the binding energy per

, and allows contributions

the "effec-

of the contribution

from all of the various

spin-isospin components of the 3NP. Of the results obtained by Wiringa et a146-48 , we illustrate in figure 10 those obtained using the Tucson-Melbourne nnE-3NP,

again with monopole

a phenomenological

3NP which we discuss

It will firstly nuclear matter

at the observed

density,

~nE-3Np

curves.

1.9 MeV additional

the undersirable

The calculated

sity and greater binding energy. enhances

two body potentials

the short distance

Including the Tucson-

binding energy at the observed

features of the two body potential

saturation Presumably

repulsion

underbind

(kF = 1.36 fm-l), but predict saturation

and too large a binding energy.

produces

but exacerbates

saturation

density

using

in more detail below.

be noted that the "realistic

at two high a density Melbourne

form factors and !l = 61-1,and those obtained

point is shifted to higher denone should look to a force which

to decrease

the equilibrium

density and

binding energy. It is possible have reported calculating

that the npE-3NP will have this effect.

calculations

the effects of tensor correlations

approximation, to estimate

but allowing

the importance

Ellis et a1.24y57

of the effect of this potential

double exhcnage

in nuclear matter,

using the effective

potential

terms in the matrix element of W

of the Kroll-Ruderman

terms.

Table VI is

from their results.

TABLE VI Contributions of nrE, npE and ppE-3NPpio5$he of nuclear matter (after Ellis et al. 3 ). 3N Potential TlTiE

Contribution

binding energy

to E3/N at kF = 1.36 -3.60 MeV

npE, KroJl-Ruderman

+0.56 MeV

APE,

+2.27 MeV

A

PPE

-0.12 MeV

Total

-0.89 MeV

abstracted

456~

-Melbourne 3NP

-30 E/N (MeVI

-*_.-.-._

Argonne

&L

2NP

Urbana

?;L

2NP

Empirical

--I--___

Figure 10 The saturation curves for the binding energy of nuclear matter, using two body potentials with meson theoretic and phenomenological 3NP (after ref. 46-48).

Note firstly that this approximate

method,

applied

more binding energy than the more exact calculations also that much of the attractive pulsive contribution dominates

significant

contribution

It will be interesting the evaluation exchange

energy from IT~Texchange

from up exchange.

the Kroll-Ruderman

et al.

is cancelled

Note by a re-

Finally note that the A contribution

term, but that the Kroll-Ruderman

term makes a

and should be retained. to see how the inclusion of further correlations

of the matrix elements

3NP alters these results.

tion will be available

to the nnE-3NP gives 80% of Wiringa

in

of these shorter range terms in the meson

We hope that the results of such a calcula-

in the near future.

B. H.J. McKetlur, W. Glijckle / Three-Nucleorr Forces

5. THREE NUCLEON The influence

IN THE n PARTICLE

POTENTIALS

of 3NP in the o particle

in the three nucleon system,

influence

457c

is expected

simply because

to be greater

than its

there are now many more

triples which can interact

through the 3NP. Three body forces in the u particle 58 59,60 46-48 have been studied by Coon et al. , Tanaka et al. and Wiringa et al. .

To illustrate

the effects,

Table VII gives the results obtained

al using the Tucson-Melbourne

nnE-3NP,

in the triton and nuclear matter. enough additional which

binding

is more pronounced

comparing

by Wiringa et

the results with those obtained

While the three body force does not provide

for the triton,

there is a slight tendency

to overbid

in nuclear matter.

TABLE VII Results obtained for 3H, 4He and Nuclear matter by Wiringa et al, body forces only, and with three body forces included. Two body interaction

Argonne

46-48

VI4

Urbanna V1,

E2

-7.0 f 0.1

-7.2 * 0.1

E3

-1

-0.9

E

-8.1 i 0.1

-8.1 -e 0.1

0.3

-23.8 ir0.2

for two

3H

tot

-22.1

E2

4He

tot

E,/N

Nuclear Matter

%ot'N Tanaka"

-6.5

-29.8 1 0.5

-29.3 i 0.5

-18.1

-20.0

-9.5

-4.5

-27.6

-24.5

E,/N

particle

i

-1.7

E3

E

.I

has emphasised

the significance

as a proble of the structure

ber of s state triples,

of the excited

of the three nucleon

and the spatial wave functions

from the ground state to the excited

ground state and about 2 MeV for the excited et al.46-48

proposed

have studied

a phenomenological

The num-

of the dis-

-

it is about 6 MeV for the 61 states. Both Sato et al. and of excited

in the determination

states, and both have of the parameters

of

three body potential.

These investigations structure

the spectrum

to use this as an ingredient

potential.

both change in going

states, as does the magnitude

crepency when two body forces alone are used

Wiringa

states of the CL

show promise that this may be a way of investigating

of the three body force provided we can satisfy ourselves

that the

the

B.H.J. McKelIar,

458c

contribution

6.

from four body forces

OTHER APPLICATIONS

.

62

is either negligible

it is perhaps appropriate

tions of 3NP to the traditional recent applications

Forces

or reliably

estimated!

OF THREE BODY FORCES

At a few body conference

different

W. Gliickle / Three-Nucleon

few body problems.

of the 3NP which are interesting

qualitative

features

to emphasise

the applica-

There are however

two other

because they emphasise

of the interaction.

Ando and Bando63 have calculated

the contribution

of the three body force to

the spin orbit splitting

of one particle and one hole states in 160 and '+OCa. 22 Using the 3NP of Martzolf et al. they obtained a three body force contribution

20-30X of that obtained

from two body potentials.

using the Tucson-Melbourne

Similar results were obtained

potential when a large cut off (!124 8.5~ in a mono-

pole form factor) was used, but at A Q 6~ the 3NP contribution reduced.

We believe this work is important because

spin orbit splitting

is sensitive

to the 3NP.

was dramatically

it demonstrates

that the

With further work it could help

us to decide between different models of the 3NP. 64 Coon, McCarthy and Vary investigated the influence

of 3NP on the magnetic

transition

form factors of 170, which have been difficult to understand in the 65 conventional shell model with only two body interactions . They found that the Tucson-Melbourne

nnE-3NP,

able to give a reasonable even more interesting

in conjunction

approximation

with meson exchange

to the observed

currents, was

form factors.

Perhaps

was the fact that they showed that the Tucson-Melbourne

nrrE-3NP gave significantly

better agreement

than the Fujita-Miyazawa

These results hold out the hope that we may have in the magnetic

T~T~E-~NP.

form factors

not only'evidence

for the three body force, but a means of differentiating

between different

three body forces.

Both of these results are indications 3NP by looking at nuclear

properties

that one may find out more about the

other than the bindinq energy.

Further

work along these lines should be encouraged.

7. PHENOMENOLOGICAL

THREE BODY FORCES

It is clear that the short distance determined situation

properties

by existing meson theoretic models. of the two nucleon potential,

of the 3NP are not well

This of course parallels

where the short distance

the

potential

is

not fixed by the meson theory, and is often parameterised. A similar approach by Tanaka

to the 3NP has been suggested by Wiringa et a1.46-48, and 59-61 . In this approach a simple parameterised

and his collaborators

3NP is written

down, and the parameters

are varied to obtain the best fit to

data, usually chosen to be the A = 3 and A = 4 systems, and for Wiringa et al 46-48 these systems with the saturation properties of nuclear matter. To

date a satisfactory may be indicative (if

fit to all of these data has not yet been obtained.

of at least one of the following

That

three difficulties:

The form chosen for the 3NP may not be the appropriate

one.

An exhaus-

tive study of the possible

spin-isospin covariants has not been performed, as 24 found 22 spin covariants to quadratic far as we are aware, but Ellis et al.

order in the nuclear momenta

in their study of the ~lpE-3NP.

tion that there is a rich field of possible

structures

This is an indica-

waiting

to be explored

in constructing

phenomenological potentials. 62 66 (ii) Four Body potentials or even multibody potentials , may be import'62 is that such potentials have negligible ant. The conventional argument effects at normal nuclear matter densities. But the density dependence of 66 such potentials is dramatic , and they could influence the saturation properties of nuclear matter. (iii)

At short distances

the potential

concept may break down completely

and

we must deal directly with the quarks, rather than working with nucleons

and

mesons.

but we

believe

This possibility its relevance

remains to be

potential

clear that there are close connections

and the three nucleon potential,

one would like to see both potentials same underlying

ular approach

in a consistent way from the 67 The work of Fonsca and Pefia provides an attempt to

theory.

to the NNn system.

as realistic,

tional meson theoretic

Orlowski

it may be interesting

body problem.

In a qualitative

a three body potential

However

when imbedded

sense one must anticipate

there are difficulties

tion used by Orlowski potential

to learn more about the inter-

in the two

in the three

this, because

the

of the third particle will alter the energy of the two body sub-

set of three body equations overcome

molec-

to relate it to the more conven-

in an attempt

between the 2NP and the 3NP. 68 and Kim have pointed out that an energy dependence introduces

system.

in the Born-Oppenheimer

At the present stage this model cannot be

potentials

body potential

introduction

between the two

and that in an ideal world

obtained

do this in a model in which the 2NP is obtained

relationship

transfer

TO THE THREE BODY FORCE

It is intuitively

regarded

at this conference,

physics at low momentum

established.

8. OTHER APPROACHES

nucleon

has been argued forcefully

to "soft" nuclear

In general energy dependence

the suppression

interest to Orlowski

in constructing a consistent 69 and the prescrip-

potentials

and Kim, which is to set El, = El, - q2 does not seem to

these difficulties. reflects

involved

for energy dependent

of degrees of freedom

of the two body -

in the case of

and Kim these are the quark degrees of freedom

may be more straightforward

-

and it

to retain these degrees of freedom in the three

460~

B.H.J. McKellur, W. Gliickle / Three-Nucleon Forces

body problem or to project them out at the point, rather than construct sistent set of three body equations

for energy dependent

a con-

potentials.

9. CONCLUSIONS There is an increasing

body of evidence

enough to account for the behaviour potentials

that two nucleon potentials

of many nucleon systems.

are the obvious next step to introduce

are not

Three nucleon

in trying to understand

that

behaviour. At present results obtained which

should dominate

sensitivity 3NP.

using 7r~ exchange

the long range behaviour

three nucleon

of the 3NP show a great deal of

to the short range, or large momentum

The immediate

The construction

transfer

task facing those who construct

way is to improve our understanding

potentials,

properties,

of the

the 3NP in a fundamental

of the short distance

of 3NP with heavier meson exchanges

regime of the 3NP.

is one step which has been

taken, but we would predict that, before Few Body XI, the quark model will be invoked to try to understand

this part of the potential.

For those who use the 3NP we suggest two lines of approach to be capable of providing the 3NP.

useful insights

The first is in the study of break up reactions

system, the second is the incorporation tial approximation

in the construction

sensitive

form factors.

in the three body

of the 3NP beyond the effective of matrix elements

in many body systems, which has applications tromagnetic

that seem to us

into the presence and the nature of

It is possible

poten-

of one body operators

to spin orbit splitting

that these processes

and elec-

will be more

to 3NP than the binding energy of many body systems.

Of course when we are satisfied

we have understood

the 3NP, and even before,

we will need to start to worry about the effects of four body forces! REFERENCES 1) H. Primakoff

and T. Holstein,

Phys. Rev. 55 (1939) 1218.

2) I. Fujita and H. Fliyazawa, Pron. Theor. Phys. 17 (1957) 360. 3) G.E. Brown, A.M. Green and W.J. Gerace, Nucl. Phys. All5 4) R.B. Wiringa,

invited paper, Bochum Workshop

(1968) 435.

(1983).

5) M.R. Robilotta, M.P. Isidro Filho, H.T. Coelho and T.K. Das, Contributions to this conference (pp. 226, 229) and Phys. Rev., to be published. 6) S.A. Coon, M.D. Scadron, P.C. McNamee, B.R. Barrett, D.W.E. Blatt and B.H.J. McKellar, Nucl. Phys. A317 (1979) 242; S.A. Coon and W. Glockle, Phys. Rev. C 23 (1981) 1790 give some corrections to the potential. 7) S. Drell and K. Huang, Phys. Rev. 91 (1953) 1527.

8) R. Bhaduri, Y. Nogami and C.K. ROSS, Phys. Rev. C 2 (1970) 2082; B.A. Loiseau, Y. Nogami and C.K. ROSS, Nucl. Phys. Al65 (1971) 601. 9) G.E. Brown and A.M. Green, Nucl. Phys. Al37 (1969) 1. 10)G. Hohler, H.P. Jakob and R. Straus, Nucl. Phys. B39 (1972) 237; H. Nielsen and G.C. Dades, Nucl. Phys. 872 (1974) 310. 11)s. Adler,

Phys. Rev. 137 (1965) 81022.

12)s. Weinberg,

Phys. Rev. Lett. 17 (1966) 616.

13)S.A. Coon, private communication. 14)K. Shimizu, A. Polls, H. Mijther and A. Faessler,

Nucl. Phys. A364 (1981) 461.

15)See e.g. A.M. Green, Rep. Prog. Phys. 39 (1976) 1109. 16)Ch. Hajduk, Ch. Hajduk,

P.U. Sauer and W. Strueve, Nucl. Phys. A405 (1983) 581; P.U. Sauer and S.N. Yang, Nucl. Phys. A405 (1983) 605.

17)I. Fujita, M. Kawai and M. Tamifuji,

Nucl. Phys. 29 (1962) 252.

18)G.N. Epstein and B.H.J. McKellar, Lett. Nuovo Cim. 5 (1972) 807; Phys. Rev. DlO (1974) 2169; W.N. Cottingham, M. Lacombe, B. Loiseau, J.M. Richard and R. Vinh Mau, Phys. Rev. D8 (1973) 800. 19)M.G. Olsson and E.T. Osypowski,

Nucl. Phys. BlOl (1975) 136.

20)G. Ball and B.H.J. McKellar, University of Melbourne preprint UM-P-77/38 (1977); B.H.J. McKellar, G. Ball and R.G. Ellis, to be published. 21)M.G. Olsson and E.T. Osypowski, 22)M. Martzolff,

Nucl. Phys. 887 (1975) 399.

B. Loiseau and P. Grange,

23)B.H.J. McKellar, (1981).

Phys. Lett. 92B (1980) 46.

S.A. Coon and M.D. Scadron,

University

24)R.G. Ellis, S.A. Coon and B.H.J. McKellar, Contributions and R.G. Ellis, invited paper, Bochum Workshop (1983). 25)M.R. Robilotta and M.P. Isodro Filho, Contributions and Sao Paulo preprint IFUSP/P-380 (1982). 26)R.G. Ellis, contribution 27)J.T.

MacMullen

of Arizona

to this conference

to this conference

to this conference.

and M.D. Scadron,

28)N. Kroll and M.A. Ruderman,

Phys. Rev. D20 (1979) 1069, 1081.

Phys. Rev. 93 (1954) 233.

29)N. Dombey and B.J. Read, Nucl. Phys. 860 (1973) 65. 30)s. Fubini, G. Furlan and C. Rossetti, 31)M.A.B.

Nuovo Cim. 40 (1965) 1161.

Beg, Phys. Rev. 150 (1966) 1276.

32)See e.g. K. Holinde,

Phys. Rep. 68 (1981) 121.

preprint

B.H.J. McKellar, W. Gliickle / Three-Nucleon Forces

462~

33)A. Cass, Ph.D. Thesis, University of Melbourne (1980); D.W.E. Blatt, Ph.D. Thesis, University of Sydney (1974), and references 6 and 8. 34)A. Cass and B.H.J. McKellar,

Nucl. Phys. 8166 (1980) 399.

35)A. Cass and B.H.J. McKellar,

Phys. Rev. D18 (1978) 3269.

36)U.

Kaulfuss and M. Gari, Bochum preprint

37)See S. Brodsky,

(1983).

invited paper at this conference.

38)B.H.J. McKellar and R. Rajaraman, in "Mesons and Nuclei" (Edited by M. Rho and D. Wilkinson, North Holland Pub. Co., Amsterdam 1979) p. 357. 39)W. Gliickle, Nucl. Phys. A381 (1982) 343. 40)s. Muslim,

Y.E. Kim and T. Ueda, Nucl. Phys. A393 (1983) 399.

41)A. Bomelburg,

Phys. Rev. C28 (1983) 403.

42)See the paper by S.A. Coon and W. GlBckle 43)J. Torre, J.J.

in ref.6.

Benayoun and J. Chauvin, Z. fijr Physik Al00 (1981) 319.

44)A. Bbmelburg and W. Glbckle, Bochum Preprint (1983), and A. Bomelburg, invited paper, Bochum Workshop (1983), Phys.Rev.d (in print). 45)A. BBmelburg,

W. Glbckle and W. Meier, contribution

46)R.B. Wirinqa,

Nucl. Phys. A401 (1983) 86.

47)J. Carlson,

V.R. Pandharipande

and R.B. Wiringa,

to this conference.

Nucl. Phys. A401 (1983) 59.

48)See ref.4, and R.B. Wirinqa, paper presented at 3rd International on Recent Progress in Many Body Theories (1983). 49)J.L. Friar, E.L. Tomusiak, 677.

B.F. Gibson and G.L. Payne, Phys. Rev. C24 (1980)

50)Ch. Hajduk, invited paper, Bochum Workshop (1983); paper, Bochum Workshop (1983); See also ref.16. 51)A. Delfino,

Conference

invited paper, Bochum Workshop

S.N. Yang, invited

(1983).

52)B.A. Girad and M.G. Fuda, Phys. Rev. Cl9 (1979) 579. 53)J.L. 51.

Friar, B.F. Gibson and G.L. Payne, Comments

54)I. Slaus, Y. Akaishi 55)W. Meier,

and H. Tanaka,

57)R.G.

Phys. Rev. Lett. 48 (1982) 993.

invited paper, Bochum Workshop

56)M. Karus et al. contribution

(1983).

to this conference.

Ellis, S.A. Coon and B.H.J. McKellar,

58)S.A. Coon, J.G, Zabolitzky

Nucl. Part. Phys. 11 (1983)

TRIUMF preprint

(1983).

and D.W.E. Blatt, Z. Phys. A281 (1977) 137.

B.ff.J. McKellur, W. Gliickle / Three-Nucleon

Forces

59)H. Tanaka (editor), Proq. Theor. Phys. Supp. 56 (1974); T. Katayama, Y. Akaishi and H. Tanaka, Prog. Theor. Phys. 67 (1982) 236. 60)H. Tanaka,

invited paper, Bochum Workshop

61)M. Sato, Y. Akaishi and H. Tanaka, 62)D.W.E.

Blatt and B.H.J. McKellar,

63)K. Ando and H. Band;, Prog. Theor. 64)S.A. Coon, R.J. McCarthy

65)R.J.

McCarthy

and J.P.

and J.P.

(1983).

Prog. Theor.

Phys. 66 (1981) 930.

Phys. Rev. Cl1 (1975) 2040. Phys. 66 (1981) 227. Vary, Phys. Rev. C25 (1982) 756.

Vary, Phys. Rev. C25 (1982) 73.

66)B.H.J. McKellar and R. Rajaraman, Phys. Rev. Lett. 31 (1973) 1063; Phys. Rev. C 10 (1974) 871; D.W.E. Blatt and B.H.J. McKellar Phys. Rev. C 12 (1975) 637. 67)A.C. Fonseca and M.T. Petia, contributions to this conference. and A.C. Fonseca, invited paper, Bochum Workshop. 68)M. Orlowski 69)B.H.J.

and Y.E. Kim, contributions

McKellar

to this conference.

and McKay, Austr. Journal of Physics

(to be published)

463~