Three-nucleon forces and trinucleon observables

Nuclear Physics A463 (1987) 315c - 326c North.Holland, Amsterdam

THREE-NUCLEON

315 c

FORCES AND TRINUCLEON

OBSERVABLES

J. L. FRIAR Theoretical USA

Division,

Los Alamos National Laboratory,

Los Alamos,

NM

87545

Three-body forces in classical, atomic, solid-state, and nuclear physics are reviewed. The basic ingredients used in constructing three-nucleon forces are discussed. The experimental evidence for three-nucleon forces is presented, as well as the results of calculations using these forces.

i.

INTRODUCTION i.i

Examples

Three-body physics,

and definitions

forces

including

In classical actions

physics

between

interactions basis

for

atoms

is

particles

atomic a

fields

only

effects

are

phenomena ability these

the elementary

large composite and

description

interactions

calculations.

ranges

sophistication

interpreted Are

and

this

usually

no,

in

hinders

requires

duced bits

three-body by of

Additional integral tween

forces

matter

but, yes,

individual

in all fields.

(atoms)

particles calculus

efforts

them.

are ideal places

necessary

relativity,

so

is

in

all

in practice.

those

radially

two

systems masks

This statement, this conference. The answer is

physics,

which is the archetype intro-

interaction between two tiny

directed force assumed by Newton.

composite

0 3 7 5 - 9 4 7 4 / 8 7 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

in-

where computa-

If we ignore the nonlinearities

that he could linearly superimpose in

pinpoint

the general

The basis for this obscure

interact by means of the same mechanism.

particles

force

to

to look.

In

using

Three-body

Indeed,

circumstances?

the gravitational the

used in

potential

from calculations

is one of the reasons we are attending forces

principle,

general

nuclear

Simple systems,

answer is provided by an example from classical for

potentials

Familiar

equation for quantum many-body

is greatest,

generally,

three-body

form the

interaction between

from good to excellent.

description

solve the Schrodinger

inter-

the interaction between two nucleons.

effects except in special circumstances.

tional

of

The two-body Coulomb

atom-atom

of nature which results

small

accurate

objects.

gravitational

nuclei and electrons,

phenomenological

solid-state

generally

between

pairwise

The long-range van der Waals

all

phenomenologically

the

whose to

of

or

two-body

or

physics.

small role in most branches

atomic I, solid-state 2, and nuclear physics 3'4.

electrons,

feature

describe

these

play a relatively

one considers

between

condensed-matter models

(3BF)

classical,

systems.

Newton invented

the interactions

be-

Such a superposition

~ L. Friar / Three-nucleon forces and trinucleon observables

316c

rules

out

elementary

three-body

forces which would arise if the interaction

between any pair of particles depended on the position of a third particle. Let and

us

assume for the moment that the two composite systems are the earth

moon.

What

is the force between these two objects and a small satellite

orbiting the earth? sum of

of

composite tides

This of

the two radially directed forces between the satellite and the centers

the

ocean

is

by

The earth, however,

is deformable and the earth's

the moon will affect the earth's gravitational

earth, moon, and satellite must be simultaneously known. bulge

easily

produces

a small change in the satelliters motion,

detectable and must be taken into account.

that, the

objects.

caused

force.

a three-body force, because the relative coordinates and orientations

the

tidal

Within an excellent approximation this force is simply the

although (tens

of)

composite Several

the

objects

(orientation

of

magnitude

requires

points of

The lesson to be learned is

elementary interaction is purely two-body between each of

orders

other

Although the the effect is

three-body

which

the

of atoms, forces

treating the system as only 3 for

bear repetition are:

3 objects)

an adequate description. (i) the angular dependence

is important in this example,

as it is in the

nuclear problem;

(2) distortion of an originally spherical object is the physi-

cal

many three-body forces;

origin

tidal

of

potential

inside

the earth,

far from the earthrs center,

The long-range nuclear three-body force

The

pion is the lightest of all known hadrons,

of

both

nuclear

force

tensor

is exceptionally

force

dominates

shown

dividual

physical

arising

from

shell)

in

the

scattering

nucleon.

The

many

the

two

pions

the

nuclear

perturbation force

emission

the

pions

exchanged

between

3BF (2~3BF).

potential

three

Many in-

(la), which depicts the force the (off-

and absorption of the pion by a third

possible charge states of a pion and three nucleons, lead to a rich isotopic structure.

in (ib) corresponds

theory

Two

of a pion (dashed line) by one nucleon,

a second nucleon,

to the iteration of OPEP;

only

The exchange of that is, writing

in the Schrodinger equation as VI2 + VI3 + V23 leads in

to graphs such as (ib).

as we have defined it. only

binding.

mechanisms contribute to Fig.

by

the long-range

The OPEP part of the

important in the few-body systems, because the

Fig. i, generate the longest-range

one of which is shown here, the

and mediates

the two-nucleon and three-nucleon forces.

nucleons,

play

is important.

1.2

part

has

(3) because we have no interest in the

it is clear that only the long-range part,

Clearly this is not a three-body

Subtracting out the iterated OPEP is tricky, and

recently been performed in a general way 5.

Other processes which do

a significant role are the nucleon-antinucleon pair contribution in (id), isobar

(A)-mediated force 6 in (le), and the p- and a-mediated portions in

J.L. Friar

(le)

and

(if).

three-electron example

The

"pair"

force I,

force is the analogue

while

the

and of the Axilrod-Teller

Fortunately sively

for

nuclear

Much

is

phenomenological

known

Several

of the Primakoff-Holstein

the analogue

of our classical

force 2 between three atoms.

the pion-nucleon

groups

shorter-range

Graphically,

in Fig.

317c

interaction was exten-

two decades ago in the context of (broken)

Recently,

introducedl0'll.

virtual pions

long-range

is

about this interaction,

context.

this knowledge 7-9.

A-force

physics,

studied approximately

symmetry.

been

/ Three-nucleon forces and trinucleon observables

both in principle

have designed components

they correspond

chiral

and in a

3BF models based on

based on p-exchange have to replacing

one or both

(i) by p mesons.

(o)

(b)

(c)

I.fl tfl (~)

(e)

Physical processes 1.3

(two-body)

OPEP

relativistic

of the 2~3BF has the schematic and

M

the

is

the

corrections.

obtain a rough estimate

nucleon

Using

mass.

energy

FOR THREE-NUCLEON

2.1

Development

The

Faddeev

(-50 MeV).

to

the

technique 12

traditionally over

latter write

three-nucleon

These

terms

are therefore

which is 2 percent is commensurate

with

corrections.

force solutions

is a method for solving the Schrodinger of boundary

(e.g.,

conditions;

Calculations

nucleon-nucleon

partial waves

in a single NN partial wave

is the

value of = 30 MeV, we

This estimate

two-nucleon

equation. the

where V

FORCES

of accurate

which allows easier implementation alent

V2/Mc 2,-

of i MeV for the 2~3BF contribution,

total potential

EVIDENCE

form:

a representative

other estimates 3 of the size of relativistic

2.

forces.

Size scales

One component

of

(f)

FIGURE i contributing to three-nucleon

the

(NN) potential

(or "channels"), IS0).

using

equation

it is exactly equivFaddeev

approach

as an (infinite)

sum

each term of which acts only

The Faddeev equations

are then solved

J.L. Friar / Three-nucleon forces and trinucleon observables

318c

exactly

(in a numerical sense) for a finite number of these terms.

major

technical

solve

the

waves.

Faddeev

The

Hajduk

has

of

equations

standard

positive-parity by

developments

One of thu

the previous decade has been the ability to

for

a large number of such (potential) partial

calculation of a decade ago involved 5 channels,

NN waves with total angular momentum j ~ I.

or all

This was extended

and Sauer 13 to 18 channels,

or all waves with j ~ 2.

Recently this

been extended 14'15 to 34 channels,

or all waves with j ~ 4.

The potential

energy contribution decreases rapidly with increasing j, because of the angular momentum barrier, 58

MeV

and amounts only to about 20 keV for j = 4, out of a total of

for the RSC potential model.

various

groups

potential

models.

VI4,

Super

and

Moreover,

there is a consensus among the

performing these calculations for most of the commonly used NN One finds E B = [7.35, Soft Core (C) models,

7.67, 7.53] MeV for the RSC, Argonne

respectively.

The 34-channel result is

320 keV more bound than the 5-channel case for the RSC potential. 2.2

Development of three-nucleon force solutions

These

two-body

force

results,

together

Tourreil-Rouben-Sprung potential results, and

suggest

Faddeev) used:

that

models.

For

Incorporating

the

gives

not

and

behavior

determine

of

these

models

does indeed lead to additional

1.5

of pion-nucleon form factor 16, each of these models MeV

additional

binding,

which

would overbind the

the argument in favor of this choice of form factor is

one finds a disturbingly strong sensitivity to the shortAll of the powerful arguments invoked

the long-range behavior are of no help in explicating the short-

addition

The binding results are suggestive but problematical. to

these

calculations

the

explicit-A

model of Hajduk and

Sauer 17 incorporates A degrees of freedom in the wave function. implicit

3-nucleon

relationship

(or

(TM), Brazilian 8 (BR), and Urbana-Argonne 9 (UA)

assumed for this force.

range behavior. In

any

choice

Unfortunately,

compelling

and de

is roughly independent of the concomitant two-body force.

approximately

range to

which

standard

triton.

Paris

Several different three-nucleon forces have been commonly

Tucson-Melbourne 7

binding 14'15,

comparable

three-nucleon forces should be added to the Schrodinger

equation. the

with

are roughly I MeV too low in binding,

between

This model has

forces and generates only 0.3 MeV additional binding. these

The

results for the explicit and implicit A-models is

not thoroughly understood. The ing

many different combinations of two- and three-body forces,

varying numbers of channels,

spite

of

In

the fact that we can not make a compelling quantitative case for the

additional norance

incorporat-

lead to a wide range of binding energies.

binding

accrual

due

to

three-nucleon forces, we can use our ig-

as a tool to investigate other trinucleon observables.

Many (or most)

J.L. Friar / Three-nucleon forces and trinucleon observables

of

the

latter

will

depend primarily 2.3

vary

with

the triton binding energy,

on that parameter

(i.e.,

An

example of this scaling behavior 18 is shown in Fig. 2, in which the rms nucleon)

charge

radii

for

3He

binding energy for the many different models

solved

at

Los Alamos.

square

radius

~, or E B.

In fact,

3H

component state

of

in

that fashion,

charge the

to the outer, ~1/2

or asymptotic,

force

The inthe mean-

portion of the

]' it is correspondingly

sensitive

to

data

the radius which is determined by the scales more nearly as EE 1 .

The latter

is largely determined by the overlap of S- and S'-

components,

The

although

density difference density

wavefunction

increases 18

of two- and three-body

force was included in 3He.

if one assumes the asymptotic form for the entire wavefuncshould scale as m~ I/2" The isoscalar, or mass, radius does

radius

scale

and

is highly sensitive

[-exp(-~p)/p 5/2, where ~-mB

the

indeed

combinations

the triton

and the lines are simple fits. Because •

wavefunction

and 3H are plotted versus

No Coulomb

dividual points are calculated

3He

EB, and some will

scale with EB).

Scaling

(point

tion,

319c

from

and

the

latter decreases

rapidly as binding

a Saclay analysis 19 shown on this figure are in

good agreement with the fits. 2.2

~ ,

,

,

i

. . . .

1

. . . .

i

. . . .

sH radius fit ...... 2

"~

a%.a

1.6

7

Calculated second

demonstrated 652 ± 2 keV. spherical input.

example

inverse

trinucleon

and

g

10

FIGURE 2 rms radii together with data. Ec, which is roughly proportional

should increase with increased binding.

This is

in

Fig. 3, where the fit at the triton binding energy gives E ¢ = This is slightly higher than the 638 keV obtained from the hyper-

approximation 20, This

8 El} (MeV)

is the Coulomb energy,

radius

-IP..

SHe detum

6

the

,,%

SH ClIIIIJI~" `~+ o }/,

A

fff

%.

.

to

3 1 ~ rsdkJs

""'"

approximation

which uses experimental

charge form factor data as

is known to work at the one percent level 18.

The

/ Three-nucleon forces and

Z L. Friar

320c

higher

former

proximation momentum Both

number

is

calculations

transfers,

numbers

are

due to the inability of the theoretical

to reproduce

to

which

amount 20 ground states

the experimental

the

significantly

nontrivial

trinucleon observables

Coulomb energy is relatively

lower than the 764 keV datum,

of non-Coulombic

charge-symmetry

impulse ap-

form factor data at high

breaking

insensitive.

and indicate a

in the trinucleon

7OO

550

.

.

.

.

i

6

.

.

.

.

i

7

.

.

.

.

i

8

.

.

.

.

9

EB (MoV) FIGURE 3 trinucleon Coulomb energies.

Calculated 2.4

Charge densities

The

3He

sistent

charge form factor and associated

problem

for theorists

dramatic presentation factor

data

nucleons' the in

and form factors

into a (point nucleon)

charge density,

charge form factor contribution

quasi-experimental

assumptions

are

and

extrapolations

is nevertheless

order

realize

to

place

that

a manifestation

with

was

resolved primarily

by

by

forces the

of certain problems

the graphic

in the form factors. it is important to

and

to

of relativistic

fill

corrections.

Fabre de la Ripelle 22 that both the binding problem

inclusion

sensitive

calcula-

a variety of theoretical

1 percent of the total 3He charge would completely

suggested

two-body

Although

the

The large "hole"

theoretical

this result in the proper perspective,

roughly

the form

This leads to

needed to obtain the "data",

the hole, which has a size characteristic It

way.

charge density data shown in Fig. 4.

discussed.

The most

after first extracting

in an approximate

the density is not reproduced by any of the associated

"hole"

taken.

of the problem was made by Sick 21, who converted

tions, which will be subsequently

In

charge density have been a per-

since the data were originally

the of

charge density problem at the origin could be three-body

forces.

isosceles configurations

The

binding

energy is

of the three nucleons,

where

J.L. Friar / Three-nucleon forces and trinucleon observables

each

nucleon

charge

feels

density,

finding

of

configuration.

tions of Pch(0), a

angular

attractive

of both of its neighbors.

r, from the trinucleon

Moreover,

is the probability center-of-mass.

force

in order to reduce

which is repulsive.

dependence,

in

in general,

an isosceles

the theoretical

Because

of

If the

it is possible

calcula-

must be reduced,

three-body

and

forces have a

for such a force to be

configuration while being repulsive

This could simultaneously

The

it is clear that the nucleons must lie in a

the collinear wave function configuration

requires

strong

attraction

(in the impulse approximation)

r, is chosen to be zero,

collinear

one.

force

a proton at a distance,

distance,

this

the

Peh(r),

321 c

in a collinear

solve the lack of binding from two-body forces

and the charge density problem. The Fig.

effect of three-body

4.

The

without from

a

line

three-nucleon

the corresponding

nucleon

forces

depression, shown the

solid

in

but Fig.

two-body

forces 23 on the charge density of 3He is shown in is a RSC 34-channel

force.

5-channel

case.

reduces

the

central

clearly

the

amount

"complete")

calculation different

Inclusion of the TM, BR, and UA threehump

and

produces

is inadequate.

5 display similar features. case

(i.e.,

This density is not significantly

a

small

central

The 3H charge densities

The small dip near the origin for

is caused by the D-waves.

This is accentuated when three-

nucleon forces are added, but the overall effect is rather small. The

form

nitudes

are

factors are quantities which are directly measured, and the mag2 versus q (momentum transfer squared) in Figs. 6 and 7.

plotted

The main features

illustrated

1.25

,

,

there are easily understood.

, ~ - ~ - ~ - T

. . . .

7~-TT-'--{

T~-T-'

i

'

'"

The most important '~

NO 3-body fome 1

.-. ~

'

,

~

"~'~<

- .....

TM 3-body force

-

BR 3 - b o d y f o r c e ---

,, \ \

UA 3-body force

.

0.25

~ _ L _ ~

0

0

0.5

,

,~_~._l

1

~_~.~_l

,5

, ~

2

~

2.5

3

r fire)

Calculated

FIGURE 4 3He charge densities and quasi-experimental

data.

J.L. Friar / Three-nucleon forces and trinucleon observables

322c

1.25

. . . .

i ~

'

'

I

. . . .

I

. . . .

•

f~,. , ~.~\--"

~\

A

~

. . . .

I

'~

'

'

NO 3 - ~ v fo,c. TM3-bodV,o,~.

.....

BR 3-bOdy force

0.75

c~0.60 0.26

0

0

0.6

1

1.5

2

2.6

r (fro) ~IGURE 5 Calculated -H charge densities. global change due to the 3BF is the change in binding energy. obvious

change

following is

added:

ratio

in the asymptotic wave function,

(approximate)

of

binding

Fourier

argument form

and the multiplicative

produces

the

charge

which is a simple "stretching" of

the

factor

for

mechanism ima,

energies

where the parameter

This crude model simply contracts

transformed,

Fch(q/A),

by

which

given earlier,

an

and motives the

ansatz for the change in the charge density when a 3BF

Pch(r) ~ %3pch(~r),

normalization.

This produces

A(>I)

is determined by the

factor assures the correct the charge density and, when

form

factor

of the momentum

change:

Fch(q )

transfer scale in the

form

factor. The effect of this stretching is to raise the 2 small q and shift the diffraction minimum outward. This

itself is incapable of changing the height of the secondary maxis

observed

in

both Figs.

6 and 7, and is caused instead by the

angular

dependence of the 3BF. The data, including the recent tritium data 19 from Saclay , are in obvious disagreement with the calculations, being too 2 large in the secondary maxima and having a minimum at too small a value of q . Some

theoretical

currents

in

be relativistic calculations making

is

the

necessary

have included the long-range

pseudoscalar

an

in

These two-body contributions

and are both model dependent

current

unimportant

trinucleons. which

corrections,

used

the

relatively

calculations

the charge operator 24.

the

pion-nucleon

algebra/PCAC

theory and experiment.

model,

are known to

and ambiguous.

and

Unfortunately,

Early

coupling Born term, without 25 . This correction is

correction

deuteron case, but is quite important

Recent attempts have preferred acceptable

pion-exchange

the pseudovector

dramatically

improves

for the

coupling version, agreement between

the ambiguity problem prevents

any

J.L. Friar / Three-nucleon forces and trinucleon observables

lo °

. . . .

I

. . . .

I

'

'

'

-

'

' ~

10''

"=~x "x\

=

"•10.

~

I

. . . .

-

-

-

I

. . . .

I

'

'

323c

'

'--.

NO 3-body force -

-

.........

TM 3-body force -

-

BR 3-body force ~_

-

-

-

-

-

UA 3-body force !

-

2

u..

10 3

10'

,

,

,

,

I

,

,

5

0

,

,

I

10

11,1,1111

I

,

,

15

q 2 (fm -2)

,

,

J

. . . .

20

I

,

,

,

25

,

30

FIGURE 6 Calculated 3He charge form factors together with data.

10 ° ~,_, , , , ....

, ....

, ....

, ....

--

-

,,,o ~-boOy,orc,

E ' %

........

",,,,, 3-,:,~,,,orc,

- -

BR 3-body force

10" ~

~ 1 0

, ....

F %

~

2

1°" I 10''

.... , .... . . . .

0

'

5

. . . .

' ' ' " " -

10

15

20

25

3O

q2 (fm-2) FIGURE 7 Calculated 3H charge form factors together with data.

Z L. Friar / Three-nucleon forces and trinucleon observables

324c

definite conclusions. has

shown

that

A decade of calculations of relativistic corrections

in order to obtain an unambiguous result for observables,

wave

functions

used

same

level

relativistic

of

operator. Because

Moreover,

the

corrections

same

as

formalism

used

obtaining

the charge

the most commonly used procedure

is ad hoc.

Scattering of nucleons from deuterons

In

addition to the bound-state evidence for three-nucleon forces discussed

above,

there is some evidence from scattering states.

regime

should

In principle

the latter

be much more useful for unraveling 3BF effects, but the lack of

continuum

calculations

and kinematically complete experiments of suffi-

cient accuracy has hindered the search. 4 very

in

must be used in both calculations.

corrections are not included in the Hamiltonians used to

calculate wave functions,

good

'

the

to calculate matrix elements must be generated using the

relativistic

2.5

24

low-energy

scattering

(below

Most of the attention has centered on

breakup

threshold), which determines the

scattering lengths. The

nucleon-deuteron

quartet. the

has

two

spin

configurations:

doublet and

The latter is primarily sensitive to the deuteron binding energy, and

quartet

body

system

scattering

lengths are nearly identical for all "realistic" two-

force models and virtually unchanged 26 by the inclusion of a 3BF.

other

hand,

trinucleon energies

the

doublet

binding energy, were.

scattering

length,

a2,

is

On the

very sensitive to the

in much the same way that the rms radii and Coulomb

Both nd and pd scattering lengths are plotted below for a wide

variety of two-body and three-body force models together with the corresponding 3 H and 3He binding energies 27

Simple fits to these calculated points are also

indicated

two data.

together

with

the

agreement with experiment.

Unfortunately,

reason

known.

for

theoretical

this

is

not

consequences

The nd calculations are in excellent the pd calculations are not, and the

Moreover,

a controversy has arisen over the

of adding the Coulomb interaction in the pd case.

recent

theoretical analysis,

latter

problem,

A

described here by C. Chandler 28, has resolved the 29 . is

while a recent theoretical calculation by the Graz group

in qualitative agreement with the features displayed in Fig. 8.

3.

CONCLUSIONS Most throe-body forces arise from the mutual distortions of three composite

systems.

The longest-range nuclear 3BF involves the exchange of two pions.

Powerful

theoretical arguments based on current algebra and PCAC constrain the

off-shell

pion-nucleon

scattering amplitudes used in constructing this force.

Two-body potentials underbind the triton,

in some cases by more than 1 MeV.

J.L. Friar / Three-nucleon forces and trinucleon observables

"~ 4

g

"""

t

nd Phillips line fit

t

ixl Philps line fit

325c

o

-4

//

nd datum

+

Ixl datum

6

7

8

9

10

11

12

EB (MeV) FIGURE 8 Calculated Nd doublet scattering lengths Inclusion

of

calculations assumed.

three-body

forces can dramatically

show a disturbing

Many

of

the

sensitivity

calculated

to

these

radii

model calculations the

so

Coulomb

best

the binding,

trinucleon observables

extrapolated

theoretical

increase

to the short-range

behavior when plotted versus the corresponding

provide

together with data.

estimates

have a very simple

trinucleon binding energy.

energy

is

roughly

length

is

contradistinction,

the

in

110-120 excellent

The rms charge

keV

too low.

although the

The calculated nd doublet

agreement with the experimental

pd value is not.

charge form factors and charge densities 2 values of q .

Fits

to the physical binding energies may of these observables.

obtained are in good agreement with recent experiments,

scattering

but the

behavior which is

The impulse approximation

datum;

in

trinucleon

are not in good agreement with experi-

ment for substantial

ACKNOWLEDGEMENT This I

work

would

was performed under the auspices like

to

recently deceased,

dedicate

this

of the U. S. Department

review to Henry Primakoff

who first investigated

three-body

of Energy.

and Ted Holstein,

forces.

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