Quark model of the NN force and triton

Nuclear Physics A508 (1990)239&246c North-Holland

QUARK

MODEL

OF

THE

239~

NN

Yu.S.Kalashnikova,

FORCE

AND

TRITON

I.M.Narodetskii,

Institute for Theoretical 117259, the USSR

and

V.P.Yurov

Experimental

Physics,

Moscow,

We review recent results on the threeand four-nucleon binding energies and the spectral functions for the twoand three-body obtained within the Quark Compound Bag model taking breakup properly into account the admixture of the six-quark bags in the few-nucleon wave function.

1.

INTRODUCTION The

quark

following

effects

three

inclusion

The

i)

in

of

important

consequences

structure

functions1'2.

constituent effect

quark

on

3He

statistic

can

be

separated

at

at

least

into

the

show

that

quark also

effects

extremely

a to

factors

sizeable

sensitive

fraction

the

in

the

details

naive

of due

the to

requirement

forces

to

elastic

the

effects

The

three-body

has

and

using

coherent

levela.

to

antisymmetrization

form

calculations

attributed

the leads

quark

nuclear

Recent

be

antisymmetry

full

the for

model

may

are

nuclei

categories:

of triton

of

EMC

Fermi quark

4

. Both

quark-quark

interaction; ii)

Effects

of

alternatively iii)

described

A

complete

considered The some

as

selection

topic

has

discussion. 5

been

on

the

the

point

energies

NN

what

the

The

the

which

can

nucleon-nucleon

shall

of

consider

not

be

correlation;

few-nucleon

we

literature. 2,4

shall

main

momentum

In

this

result

the In

systems

Quark

on

here

distributions

0375-9474/ 90 / $3.500 Elsevier Science Publishers B.V. (North-Holland)

will

it

the and

we be are

to

which

are

since

the

from

Bag

momentum

(QCB)

briefly that

consistent

present

theoretical

nucleon

shall

simply

our

recent

Compound

connection

influenced

particular,

energies

the

be

subjects

, we omit

concentrate

binding within

will

with

reviewed

and4He

interaction. ii).

we

in

derived

and

range

calculation

orthogonal

recently

3H

distributions

short

states

ones.

remain

Instead,

results

for

of

to discussed

i)

as

confining

microscopic

multiquark

extent

currently

multiquark

the with

model' discuss binding the

QCB

240~

Yu.S. Kakzshnikovaet al. I Quark model of the NN force

picture

which,

definite

improvement

2.

concerning

THE QCB MODEL The QCB

the

most

triton

role hadron

counterpart

of

that

multiquark

interaction.

the

NN

non-local

parameters

are

determined

parameters

quark

meson-meson, case

could

models.

The

in

theoretical The

employed

written

QCB

that

states

model,

the

be

incorporates play

the

in

short

scattering

calculated

already

and

were

predictions

from

has

meson-baryon

the QCB parameters

this

principle

model

provides

the

range

V is given by fr,r';2), QCB potential which contains

interaction,

and

model

confined

In

energy-dependent which

energy,

NN force models.

SYSTEMS

is a semiphenomenological

short-range

These

binding

of "realistic"

FOR FEW-NUCLEON

model

important

the

over

few

data,

in

microscopic

been

tested 7 channels ,

baryon-baryon

found

the

to be in good agreement

for

some

in

which

with

the

of the MIT bag model'.

potentials

in the momentum

have

representation _ ..,

the 9

most

simple

form

when

f,(k)f,(k’) j,(k,k';z)

= c,

cl"(Z)

l

v

(1)

ext(k,k')

where f,,(k) = t-c,+ xw(zw-

(2b)"'

ka/mN)

} sin

(kb)/k

(2)

(s2-k2ba) and 2 vv(z)

(1 - xi)

=

For

ease

of

(s - z,) + x:

presentation,

2

scv= =+ mb N for es.(l)

(3)

(Z"_ Zc"), write

we

uncoupled

and channels, but in some actual calculations the mixing of the 3S1 3 In eqs.(l)-(3) z is the energy D1 pn channels has been included. parameter, momenta, that

mN

is

nucleon

radius

mass,

of the

k

and

potential

k'

are

in the

the

off-shelf

coordinate

space

coincides

method

of

long-range

with the equivalent bag radius used in the P-matrix 10 is the Fourier transform of the ref. , and V ext(k,k') part

of

most

economically

to6

for

the

parameters that

the

b is the

in

discussion ens.(l)-(3)

zy is expressed

is also

traced

back

of the NN content two-level

QCB

the

in

and

in terms

terms the

of

meson

physical

with

exchange.

meaning

bibliography.

of the bag mass

to the P-matrix

interaction

of

further

in the six-quark

The

V ext(r)e(r-b),

interaction

described

approach

and

We

is

refer

the

Recall

QCB only

E, by zv= E,- 2mN and while

z2+ m,

xv 6

is a measure

we consider

the

cl-t m provided

the

bag. As in ref.

x2=0

of

latter

Yu.S. Kalashnibva

ratio

c

=

is

mFlcl

drop

below

the

what

follows

index

we

held

V

adopt

=

that

will

be

used

respectively. preserve at

even

with

c and

zo' has

seven

the

We

(4)

use

we

the

in

with

the

(b,

use

of

sign

effective

6

the

is

operators

the

with

the

with

k

and

p

three-nucleon an

In argued

this 12

for

way

no

that

the

additional

For

U =

the

case

with

the

12,

that

As

long

31,

23 are

of

force

going

from

has

(6)

been

in

the

verified

by

cluster

multiquark It

forces the

integral

standard

arises.

QCB

the

momenta

the as

the

expressed

- p')

relative

considered

however, since

(5)

Va

and

help

difference

directly L of Vo

terms

N = 3 this

pair-wise

when

to

properties.

essential

be

for

eqs.(l)-(3)

problem,

in

1 one

that

practical

is

Jacobi

particle

modification

that

(b,

.j f

equivalent

- 3p2/4mN)6(p

usual

modified

Note

little

operator

trinucleons

three

Tjktl).

few-nucleon

may

S = V(k,k';z

calculation

applied

is

NN

parameters 1 =

phase

all,

uncoupled

with

off-shell

Sq-bags)

&I&,

the

system.

explicit

methods

being

of

energy

V(k,k';p,p';z)

QCB

stress

is

interaction

Hot

kernels

four

to

which

Hamiltonian

q

the

channels

order

system there _ V. For the

kinetic

Thus

to

at

energy-independent.

or

In

sufficient

and

different

contribution II

Ho

.

be

interaction

made

in

modified

to

QCB

the

NN

is

we

zl.

channels,

x~,~

has

V

the 11

NN

c~+~,

but

few-nucleon

(neglecting

where

_

coupled

(1)

potential

NN

of

the

zc,

zo=

(4)

and

by

coupled

QCB

the

use

described the

notation

constraints

shown

of

the

x = x1 and

1

were

energies

expression

the

ref.

additional 2 + x.)+1= J-1 the uncoupled

are for

modified

deal

between

j

the

also

the

introduced as

=

simplify ci,

x?

large

while

=

the

properties

parameters

choice

define

1

Sj)

c

constraints

causality

asymptotically

channels

letting

1 or

for

These

the

To

constant.

1

also

xj=

241~

et al. I Quark model of the NN force

systems'.

has

do

not

three-

to

been

also

require

any

four-nucleon

system.

so, the

within

six-quark

specific in

the

standard

freedom.

systems

One

in

QCR

method

confined

form

complexity

the

of

the

states

finds the

the

QCB

it

NN

twofold.

encouraging under can

be

of

using to

only

realize

on

it

that

the

effects defines

can

nucleon

that,

consideration, treated

the

First,

interaction

technique

problem method

manifestation

is

short-range

few-nucleon

of

the

be

same

the

treated

degrees

despite the

of

of

of the

few-nucleon footing

as

in

242c

Yu.S. Kalashnikova et al. I Quark model of the NN force

the

ordinary

the

six-quark

way

to

bag

account

similar

to

deuteron

3.

potential

problem.

admixture

for

that

Secondly,

in

the

presence

shows

a

of

systematic

the

quark corrections to trinucleon observables 13 ref. fu;, the six-quark corrections to the

of

RESULTS discussing

summarize the

in

can

be

binding quite

some

central

seen

from

energies

Gedn,

dimensionless value

fm

Faddeev

GA,,

the

three-

BT a

4a 2

Gtdn

four-body

fm

0.28

fm

6.35

fm

1.29

is

not

long-range found bt=

in 6.7

within

of

GeV-' 6.0-7.5

parameters data

the

meson 6 ref. .

are

corresponding

The

while

The

S(3Di) the

model"

is

vertex

commonly used 14 . The quoted

that

somewhat

NN

the

QCB

lower

obtained the

by

results

potentials.

1 for

central

interactions

MeV

29.0*

G2 atp

fm

5.5

G:dd

fm

8.6

S,-3D,

QCB

potentials,

of

the of

the of

Ei and

of

the

the

triplet

quality and

of

is

shown

SP-86 Paris

fit in

several

was

potential to

the

Figs.1.

phase-shift potential

the

including

interactions

potential

singlet the

for

out

carried

parameters

VPI-SW

predictions

Co

"realistic"

were

radius The

.

included

and

the

1.744

the

Co=

B,

radius

GeV-I.

S(3S,), from

'So

tail.

to

use of 11,12

nuclear

1.785t0.015 15 and cross-section

3

combinations

Co=

QCB

the

we

experimental

toy

constants

to

calculations

realistic

more

related

value

results

8.48

correction

of

the mixing

the

values

the D(D,P)~H 16 for various

MeV

Coulomb

The

and

between "the

closely

corresponds

with S-D

within

normalization

Table Some

obtained

agreement

also

calculations

neglecting

calculated

"experimental"

calculations

nucleon

results

Table,

quote

of

three

interactions

those

G2 and atp asymptotic

extrapolation

of

few-body QCB

We

G2 1.29 tdn= the newest

than

realistic

the

and

impressive.

constants

the

the

Table

simplified

As

*

explicit

properties.

Before

2

the

trinucleons

may

be

fixed

to

was

varied

phase

shift

Experimental analyses. model

are

The also

243~

Yu.S. Kalashnikova et al. I Quark model of the NN force

4

80

single energy

b

_

RCB

--

Paris

-

I

I

I

I

j

I

I

600

400

200

TL(MeV)

-Ltu

800

600

400

200

TL

‘y ( MeV)

(Me@

Figs.1 The

shown. T lab? The

QCS

300

MeV

the

given QCB

values

the

B,=

8.02,

of

8.10

leads

due

'So

expected analogous can

of

x

of

8.1

the

MeV

pattern

data

with

the

for

corresponding

models.

MeV,

the is

basis

not

of

very

sensitive

interaction.

=

6.1,

independence

6.7

of

BT

the

One

and on

7.3

the

reduction 11 .Found

modify

the for

S'

inclusion

5 channel a

number of

and

of

BTby

D-wave

of of

result of

BT

relative

underbinding

contribution

The

increase

mixed

in

the only

the

0.3

components

NN

NN

slightly. NN

MeV.

The of

the

the triton

kernels

higher

"realistic" 5

to

to

GeV-I,

radius

of

5

obtains

=

neglected

for

similar.

probability

between

on

bs

the

practically

D-state

singlet

for

considerable

3S,-3Di.

an S,

8.14

in

predicted BT=

bs

coincide the

interaction

considered

results

expect

principle

a

to

and to

Bonn

of

description

1 agrees with the previous findings 11 interactions . The inclusion of the tensor

force

partially than

to

descriprion energies

lies

4.9%,

energy

with

QCB

central

and

and The

central

force of

PD=

radius

potential

better

smaller

calculations, the

respectively. QCB

the

binding

Faddeev

choice

the

deuteron parameters 6 . In particular,

Paris

triton

the

the

ref.

interaction,

channel

the

for

the

in

for

The

gives

while

of

values

those

force

Judging

case is

is from

interactions

three-body

QCB

others

kernels

weights

of for

one of

the wave

244c

Xu.S. Kalashnikova et al. I Quark model of the NN force

function triton that

are D-wave

results

91.7,

1.0

probability from

and

7,3

%,

3/2 T = 0 pairs

calculations

binding for

energy the

Note

satisfies 19 in the triton .

Table The triton

respectively.

approximately

(in MeV) QCD

the

that "3/2

the rule"

2 for 5, 18 and 34 channel

interaction

and

other

Faddeev

"realistic"

potentials Number

of

5

QCB5

8.1

RSC'?

SSC17

v1417

Paris16

7.02

7.46

7.44

7.30 7.38

18

7.23

7.49

7.57

34

7.35

7.53

7.67

As an additional Fig.2

the

two-body

test

of the trinucleon

breakup

amplitudes

wave along

A Jansetal, kin1 ---

Fig.2

A ~archaRd etal QCB RSC

function with

we show the

in

recent

24%

Yu.S. Kalashnikova et al. I Quark model of the NN force

functions those

.

data20s21

3He(e,e'p)

can

be

3

three-body k*a/mN

of

3He(p,pn)

of

pp

the

and

those

extracted

3He(p,pp)

reactions

Paris

theoretical

and

calculated

protons

and

and

the

A

experimental theoretical

which

the

in

over

PWIA

to

models.

neutrons

systems

in

similar

the

distributions

pn 22

very

and

also

the

are

the

energy almost

agree

analysis

the

of

.

CONCLUSIONS Our

is

and

Reid the

spectral

theoretical

interaction

have

--tppn

residual

with

perfectly

4.

We

He

breakup

QCB

between

observed.

distributions

momentum

the

"realistic"

agreement

satisfactory

using

results

to

from

derived

results

E,=

The

corresponding

primary

observation

introduced,

trinucleon

of

function

agrees

reaction

at

Clearly, better,

many

the

model

main

simple

and

QCB

by

of

the

potential

here

need

to

the

is

that

short-range

NN

low-energy

in

theoretical

analysis

at

the

of

improvement

nucleon-spectator

succeeds

universal

an

the

description

conclusion the

QCB

description

while from

already

quantitatively

reproduces revealed

of

the

realistic

provides

energy

momenta

a

reasonable model

determined

to

aspects

The

a

that

up

however,

a

QCB

binding

with

when

that,

to

The

3H

the

least

calculation. provides

leads

it

properties.

prediction

is

wave

of

(e,e'p)

600

MeV/c.

be

understood

first

level

the

of

QCB

model

interaction

that

observables

trinucleon

experiment.

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