Quark model potential and three nucleon bound state

Nuclear Physics A508 (1990) 247c-252~ North-Holland

Quark Model

Potential

and Three Sachiko

Nucleon

Takeuchi,

Taksu Cheont

Dept. of Physics and Atmospheric tDept.

of Physics

A non-local, cluster model.

and =Istronomy,

energy

are studied in a five-channel than would be expected

State

and Edward

F. Redisht

Science, Drexel Univ., Philadelphia, Univ. of Maryland,

independent

The properties

Bound

nucleon-nucleon

of the three-nucleon Faddeev

calculation.

from conventional

PA 19104, USA

College Park, MD 20742, USA

potential

is derived

bound state produced

from

We find about 0.5 MeV

meson-exchange

a quark

by this potential more binding

models with a similar strength

tensor force.

1. Introduction Among

the many-body

bound state problem

calculations

out [l], but the results appear behavior

than do positive

properties

for testing

to show more sensitivity

energy tests [a]. E ssentially

show a strong correlation

clear, however,

models.

whether

that determines

the proposed

direct

is truly universal irrespective

particular,

essentially The

correlation

between

of the structure

properties.

[3] that It is not

two- and three-nucleon

of the underlying

dynamics.

In

associated

with

this framework

sharply

limit

the variety

of

seen [4]. between

system and the three-nucleon

The tensor forces arising from traditional

meson-exchange

dynamics

off-shell

It has been suggested

all the three-body

It has been known for a long time that there is a strong correlation

constructing

be carried

all of the low energy three-nucleon

mixing of the S-D waves in a two-nucleon and structures.

the three-nucleon

calculations

all of the models studied up to now are based on meson exchange

assumptions

off-shell behaviors

force,

to the nucleon-nucleon

with the binding energy.

there is only one physical parameter observables

the two-nucleon

is perhaps the best. Not only can converged

In a recent attempt

to explicitly

a realistic nucleon potential

potentials

include subnucleonic

the tensor

binding energy [5].

share common ranges degrees of freedom in

model, it has been observed that quark-exchange

may result in a tensor force with a distinct form of nonlocality.

In this paper, we consider whether

the difference in the tensor force structure produced

by this quark model can produce an off-shell effect different from those seen with traditional meson exchange potentials. a non

relativistic

We therefore derive an effective nucleon-nucleon

resonating

cluster model (QCM),

group

equation

and solve the Faddeev

for nucleon-nucleon equations

We only keep the 3Ss and 3S1-3D1 states of relative to focus on tensor force effects.

03759474 / 90 / $3.50 0 Elsevier Science Publishers B.V. (North-Holland)

potential

scattering

from

in a quark

for the triton using this potential.

two-body

angular momentum

in order

248~

S. Takeuchi et al. I Quark model potential

2. Quark Cluster Model and Effective Nucleon-Nucleon The &CM

of Takeuchi

et al. [G] is a hybrid model,

meson degrees of freedom. with the quantum

numbers of the nucleon.

is assumed to be ]q) respect

= d{(]N))*x(R)},

x(R), is obtained

C

explicitly

quark, gluon, and

to be three valence quarks in a cluster

The wave function

of the two-nucleon

where A is th e antisymmetrizing

to all quarks, and IN) = 1q3;[3]s~(S

wave function,

including

A nucleon is considered

= f, T = f); [13]c)$(b).

by solving the RGM

JdR’HcdR, R’)xdR’)

Potential

equation

system

operator

The relative

with

motion

[7]:

= J%JdR’ NJ& R’)x,v(R’).

P

Since we have only one state of the nucleon included, cluster states.

The only coupling

kernels, H and IV, respectively,

H,dR, R’) = N,(R,

R’)

Jc&mm

(N*~(R.m - R), (&)I 3-Id IN26(RAn (N*~(%B

- R),(o)]

and norm

‘FI and 1 (unity),

A ]N*J(RAB

namely

- R’), (P))

(2)

- R’),(o)).

(3)

3-1,is taken to be x =

The kinetic

The hamiltonian

are made from the quark operators

= JdE,d<,dRaB

The Hamiltonian,

the channel indices specify the two-

is due to the tensor force.

xI
+

vOGEP

+

term ‘XKin is the nonrelativistic

and the one-gluon-exchange

potential,

~Conf

kinetic

+

vEMEP.

energy

(4)

of the quarks with mass mp,

V OGEP [81, acts between

the quarks and is written

as

VoGEp = x(X;

’

$ - ${l

Xj)? 1

i
&&CT, x Pij)

For the confinement

potential,

+ g(C;.

Uj)}h(Tij)

*

’ t”i

+

uj)]

(5)

,

we use linear form

VConf = C(Xi

’

Xj)(-U&d)T;j.

i
[6].

mp, 01s and aConf are determined

The meson exchange

nucleons, not between regularized

the individual

as a Guassian.

are essential to a realistic mechanism

responsible

not well described

description

to also treat the intermediate should be reliable.)

(OPEP), These

and an intermediate

provide

features

of the nucleon-nucleon

for the intermediate

by an energy

to single baryon

prop-

between

the

quarks. This allows us to include one-pion-exchange

with the nucleon form factor

which is parameterized

by the fitting

VEMEP, is taken to act directly

potential,

independent

range attraction potential,

range by energy-independent

range attraction

of the interaction

force.

(Although

is dispersive,

all of the potentials potentials,

which

part of the

and therefore we compare

so the comparison

249~

S. Takeuchi et al. I Quark model potential

To make the many-body two-body

Schrodinger

form:

calculation

VNN, from RGM

potential,

feasible,

we construct

kernels as follows.

Eq.(l)

an energy-

independent

can be rewritten

in the

i.e.

with = NltHaiaN;”

z7,,

and

= N&.

K,

(8)

g is nonlocal but independent of energy. Thus we can identify a two-body effective potential VNN = g - IC that can be used in the ordinary Schrijdinger equation for interacting Three-body

nucleons. equation The

two-nucleon

partial

that would arise from starting

observables

waves can be reproduced

by adjusting

are very

MeV.

Our model

ter similar

well

RGM

described

predicts

energy

deuteron

deuteron

low

8

an interesting naturally

force

energy

,:’ _:. _:.

_’: ,:’ .:;._,_._.-‘2 - ,.:;. /

Our

of 2.225

in our potential

The mid-range

sor force

has

feature which does not arise

in conventional

potentials.

arises from

the inter-quark

the tensor

The

OGEP

part between

part

matrix

element

of the

3S1 and 3D1

and the two tensor terms tend

to cancel each other.

3. Three

Body

I_

,.,.:.‘..’

47

200

300

EM, [MeVl 3S1-3D1

Fig.1

nucleon-nucleon prediction

mixing

scattering.

parameter

of Quark Cluster Model

ted line of Reid Soft Core.

for

Solid line is the and dot-

Dot-dashed

line

is given by the Bonn potential.

Calculation

We write the Faddeev

equation

in the form

I& + X(1 + 2Pr)Il$i) = El&). and solve it as an integral Faddeev

component

particles,

I4 is the two body i),

& #

and P,

dex (i, j, Ic). With

equation

in momentum

of the three body

j

#

-

,:’

B_?JL._.-.:

100

of

But in the 3D1 channel, the signs

are opposite

,:. _:.

n*

poten-

channels has the same sign as that of the OPEP.

0

part of our ten-

one-gluon-eschange

’

7

o-

meson-exchange

tial (OGEP). tensor

I

Reid

but has a

probability.

has a binding tensor

1 ”

an ei parame-

(Fig.1)

D-state

r

up to 300

and a 5.58% D-state.

The

I

to that of the Bonn potential

[9] up to that larger

with a three-cluster

and 3D1 phase

the 3Sc, 3Si,

shifts

in

very well

in VEMEP. In

the parameters

particular,

MeV

effects

are ignored.

interaction

is the circular

space.

wave function, potential

permutation

(9) Here

I$i)

(i =

1,2,3)

is the

Ho is the kinetic energy of three

between

particle j and Ic (j, Ic = 1,2,3,

operator

which act on the particle

the use of the multipole-decomposed

Jacobi momenta

(p, q), Eq.(l)

inis

250~

S. Takeuchi et al. I Quark model potential

QCM

-

-

r

8.5

s

8.0

2

0

10

20

30

6.5t”““““.‘.“,.1 4

Q2 [fmw2] Fig.11

Theoretical

of triton.

Charge form factors

Solid line is the calculation

the Quark

Cluster

Model

using

and the dotted

line is Reid Soft Core potential.

reduced

to a integral

quantum numbers.

equation

Fig.111

5 Deuteron

Deuteron

versus triton

D-state

binding

model potentials.

energy

of two continuous

variables

probability with

various

potentials.

and a set of coupled

we use the j-j

7

The line is a least square

fit to the meson exchange

In the actual calculations,

6 [“h]

P,

angular

coupling basis

(10) where

L, S, T signify

lar momentum, with

p, and !,

the calculation

the orbital

angu-

spin and isospin associated s, t with

q.

to 5 channels:

We restrict

and 16 points with an optimized maximum ,

PSI [%]

PO [%]

QCM

8.02

1.5

7.3

The

RSC

7.02

1.8

8.2

values of the mesh points of our

respectively. eigenvalue

and 4.95

With these truncations,

the three body equation, matrix

choice of

points.

choice for p and q are 14.00 fin-’

fm-'

Bt [MeV]

C = 0,2 and

L = 0,2. We discretize the momenta p and q using a Gaussian quadrature of 14 the scaling of the distributed

Model

Eq.(9)

problem

becomes a

with the size of

about 1200 by 1200, which is readily able with standard

codes.

solv-

Table

Triton

I

Properties

from the quark cluster model Soft Core potential.

Bt is the binding en-

ergy, Ps, is the probability amplitude ponent,

calculated and the Reid

of the relative

of mixed symmetry

and PO is the D-state

s-wave comprobability.

S. Takeuchi et al. I Quark model potential

4. Results The

and Discussion

results

Table Ps,

of our three

body

I, we list the three

calculations

nucleon

and PD in the triton

triton

are presented

binding

energy,

wave function.

well with other

calculations.

To get a better model

potentials, III).

The quark

binding

parameter”

linked

more general

hadron

probability

model

of low-energy

more

tensor

models.

three-nucleon

(such as tensor

charge

versus

binding closely

appears

may be associated one quark-based

calculations

with

energy as noted

from the line toward

the three-nucleon

that

more

the “one

with potential

model

results

leads

to a

are sensitive.

particular

in N-d scattering)

observables

meson-exchange

triton

It therefore

results

the

and agree

form fact,or.

of realistic

deviation

At least

to which

also calculated

fall on a line rather

probability.

three-nucleon

structure

detailed

force

of deuteron results

shows a sizable

D-state

to meson-exchange

non-locality

that

two-body

cluster

triton

In

of S’ and D states

we have

we took number

potentials

I, and the figures.

are also shown in the table

II shows the calculated

D-state

for a given deuteron

character

structures suggests

and plot

The results

on our result,

All the meson-exchange

by Kim [lo]. triton

Figure

perspective

in Table

and the probability

As a benchmark,

with the Reid Soft Core potential.

(Fig.

251c

This

sensitivit,y

to the

could test t,he quark-

boundary.

Acknowledgement We acknowledge

the support

FG05-87ER-40322, S520634. Diego

The

numerical

Supercomputer

of University Computer

from U.S. Department

and from

U.S.

National

work was performed Center

of Maryland

grants

fund under

Foundation

under

on the Cray X/MP

and on the FPS

under

of Energy

Science array

from

processor

the SDSC

Grant

Grant

and SCS-40 at the

No. DE-

No.

Computer

and the University

PHY-

as the San Center

of Maryland

Center.

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1987)

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252c

S. Takeuchi et al. I Quark model potential

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