Elastic 4He + 4He scattering in the hybrid quark-hadron model

Nuclear Physics North-Holland

A518 (1990) 717-722

ELASTIC

4He +4He SCA’ITERING QUARK-HADRON HUANG

Chao-hui

IN THE HYBRID

MODEL*

and

WANG

De-an’

Departmeni of Physics, Guangxi University, Nanning, P.R. China MA Wei-hsing Center of 7Xeoretical Physics, CCAST

(World Lab.), Beijing, P.R. China

and Institute of High Energy Physics, Academic Sinica, Beijing, P.R. China Received 13 November 1989 (Revised 26 April 1990) Abstract: A calculation of alpha-alpha scattering at higher energy quark-hadron model. The results show that the quark degrees in explaining available experimental data.

has been performed in the hybrid of freedom play an important role

Only recently, we proposed a hybrid quark-hadron (HQH) model to describe high-energy composite particle scattering I). We have applied the model to protonalpha elastic scattering. The model provides a favorable improvement over various conventional calculations; the results are in good agreement with existing available data and also show that the quark degrees of freedom in NN interaction play an important

role in improving

the theoretical

description

of experimental

data. In this

work we apply the model to high-energy elastic alpha-alpha scattering. The purpose of the present paper is testing our model again in a different case. Firstly, we apply the HQH model to high-energy pp elastic scattering and determine the model parameters

by fitting experimental

data on pp scattering.

to the hybrid model we divide the configuration space of distinct regions: the interior quark region and the exterior inner quark region each nucleon behaves like a three-quark two three-quark bags is given by a quark-quark force which amplitude as parametrized

f&(q) =z

iKNN

~(1 -i&

’ This work was supported in part by National Science ’ Permanent address: Fundamental Science Department, 037%9474/90/$03.50

0

1990 - Elsevier

Science

Publishers

According

NN interaction into two hadronic region. In the bag. Interaction between is related to a scattering

exp [-t&4*1. Foundation of China. Guilin College of Geology, B.V. (North-Holland)

(14 Guangxi,

China.

C.-H. Huang et al. / Elastic ‘He+

718

In the outer which,

hadronic

we assume,

amplitude scattering

region,

NN interaction

also corresponds

as eq. (la)

4He scattering

is produced

by exchange

to the same type of parametrized

by replacing

ao,

po and Pa by their

counterparts

~PN)~~P(-%%I~).

(lb)

So the scattering amplitude for the NN system in the hybrid is decomposed into two sectors,

quark-hadron

FVN(9) = F&;bO(q) + FL’IbO(q) , bO is the boundary

separating

the two regions,

scattering amplitude of two three-quark which can be written as ‘)

theory,

with

r,,(b)

=

systems

jKNN =yJ

F&bo(d

in Glauber

of NN

crN, pN and &,

~NN(~)=~~N(~--

where

mesons scattering

(2) Ftzbo(q)

in the inner

quark

stands

for the

region

(b s b,)

(34

d*b,

eiTbrNN(b)

(1 -rNo(b-Sj))

model

1

4N(l, 2,3)

(3b)

fIdxi

(3c)

9

i=l

where

ro,(

b) is given by eq. (la) 1 Too(b)

The parameters is the quark-structure @Q,

= pQ

e-i’bfoo(a)

dz6 =

uQfrjcQ)

e-b2/WG.

(3d)

2rriKQQ

and can be adjusted to fit the experimental data. tJN( 1,2,3) wave function of a nucleon and can be written as ‘)

$IN(l, 2~ 3, =4

pQ

&ijklqiqjqk)colorXS~( 2~ 3)

x [A

r&2,] P3’2 exp

-E,

(m -&J/2&]

,

(3e)

C.-H. Huang et al. / Elastic ‘He-t 4He scattering

719

with RN = f( r, + r, + r3) and size parameter b, = 0.6 fm [ref. ‘)I. The Fk;“,( q) in eq. (2) is the traditional NN scattering amplitude in the outer hadronic region and can be parameterized as “) _F;b$$qq)

=

!!%$k!Cl- iPN)ew [-Kdl

.

where rrN, pN and PN are given by NN the available scattering data ‘). Using eqs. (2), (3) and (4), differential cross section of proton-proton elastic scattering at the energy 6 = 30.7 GeV can be calculated. The parameters (+N, pN and fi& are taken to be 40.14 mb, 0.037 and 12.2 (GeV/c)-2 [ref. 6)], respectively. In order to compare between the HQH model and the quark model of ref. 6), which differ in ideas on the physics behind the processes, we adjusted and PO to fit pp elastic scattering da/dt over the range 0 s --t s 0.1 (GeV/ c)’ as was done in ref. “). As shown in fig. 1, the resulting du/dt is in excellent agreement with the measurements of ref. ‘) when 0.5336 fm, 0.0506, 0.4056 fm and b, = 0.8 fm. We use our hybrid NN amplitude and the parameters ao, and given by pp scattering data at G = 30.7 GeV to analyse elastic alpha-alpha scattering at the corresponding energy of G = 123 GeV. In the framework of Glauber theory, the amplitude of nucleus-nucleus scattering is then uQ,

@Q

=

pQ

=

@Q

pQ

=

pQ

@Q

pp elastic scattering E,=31

U

GeV

0.10

0.05 -t(GeV/c)*

Fig. 1. Fit of pp elastic amplitude at the energy of G= 30.7 GeV in the hybrid quark-hadron model to the corresponding data of ref.‘) (crosses). The solid curve represents the prediction of the HQH model with crQ = 0.5336 fm’, pQ = 0.0506, pQ = 0.4056 fm and b. = 0.8 fm.

C.-f-i. &hang

720

et At.

/

ElAStiC

4&+

4f#t? SCAttWitlg

with F,4,4(6)r($~11-i~E

E1-~(:(6-Sj+si>lltCIA)I~~>.15b)

ll-(llhil-j~l

O2

where

I,!J~is conventional

nuclear

taken to be a Slater determinant

wave function

which in the present

calculation

is

as was done in ref. I), q = k’- k is the momentum

transfer and KAA is the incident momentum, J&,.,(q) is the factor arising from the c.m. correction. Using eq. (2), the profile function T,(b) in eq. (5b) can be written as follows

with

I I

r3b)=j-&--e--iqb d2b F&C{ NN

where

FkGbo and FNN “bo can be obtained

q) ,

(54

emfrbd2b FkGbo( q) .

from eqs. (3) and (4).

Using these formulas we calculate differential cross section of elastic alpha-alpha scattering at the energy of c= 123 GeV. The results are shown in fig. 2, where the dotted curve represents results predicted by the conventional nucleon model 6), the dot-dashed curve stands for the quark model predictions of ref. “) and the dashed curve is our present traditional calculation. Finally, the solid curve demonstrates our HQH model result for 4He-4He scattering. As is seen from the figure, the traditional calculations of ref. “) fit the existing available data marked in the lower part of fig. 2. Their quark model yielded no improvements over conventional calculations, which is the conclusion of ref. 6), but our HQH model predictions are in a good agreement with the data marked in the upper part of fig. 2 and reproduce good locations of dips of theoretical angular distribution, and the flat experimental third minimum can be predicted theoretically. Indeed, the HQH model yields improvement over the conventional nuclear physics calculations, comparisons the interior

though

discrepancies

between

different

sets of measurements

make the

difficult. Our results also show that the quark degrees of freedom in quark region really play an important role in improving the theoretical

description of experimental data. In particular, at large momentum transfers -t, the quark effect is of essential importance and must be considered in any quantitative assessment of the cross section. In order to elucidate further the influence of quark degrees of freedom, the matrix elements (real and imaginary parts) of the quark (Tl?,(b)) and the hadron (r:=(b)> in the whole range of b are drawn in fig. 3. It can be seen from the figure that I% b,, r% and rza are almost equal, when b is about 0.8 fm, which we call b,. As b ==c in a quantitative is a little larger than rEa’,, so the effect must be considered calculation.

C.-H. Huang et al. / Elastic 4He + 4He scattering lo2

t

1

1

’

I

I

1

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1

721

I

I

1

’

’

4He+4He+4He+4He E = 125 GeV 10’

A

I 0”

IO-’

Akesson

et a/.

c

10-Z

1o-3

The HQH lo+

model

--.-

Our nucleon France’s

quark

model

.‘._......

France’s

nucleon

model model

1O-s I

0.2

I

I

I 0.4

I

I

I 0.6

I

0.8

-r(GeV/c)’ Fig. 2. Predictions of our hybrid quark-hadron model for alpha-alpha scattering at the energy of a= 123 GeV and comparison with experimental data as well as results of various models. The data are from refs. *-lo). The dotted curve represents the traditional nucleon model results given by ref. 6)_ The dot-dashed curve shows the result of the quark model in ref. “). The dashed curve stands for our conventional calculations usingf,,(q) (eq. (lb)) with oN = 40.14 mb, pN = 0.37, and p’, = 12.2 (GeV/c))’ in the whole space. Finally, the solid curve demonstrates our hybrid quark-hadron model predictions with oo, po and PO determined by pp scattering data in the text.

Although our HQH model description of high-energy composite particle scattering is quite attractive, there are, of course, a number of improvements that can be made in principle in our hybrid model. For example, a sophisticated description of the quark-quark interaction at b =s6, region and NN scattering amplitude at b > b,., region, These improvements are in progress.

1

I

I

1

2

3

I 4

J 5

tmpaa parameter & {fm) Fig. 3. Matrix etements of the quark part in the inner region and hadronic part in the outer region. References 1) W.H. Ma et d, Nuci. Phys. A496 (1989) 729 2) W. Czyz et al., Ann. of Phys. 52 (1969) 53; W.H. Ma et ai, Nucf. Whys. A477 fl988) 713 3) S. Brodsky, Proc. Int. School on nuclear physjcs, Erice, 1984 4) RD. Viafiior et ML,preprint, CERN, Ref. IX. 2529-CERN, I8 July 1978 5) U. Amaldi ef al., Nucl. Phys. Bld5 (1978) 367; A, Am& ef aL, Nucl. Phys. 8262 (198% 689 6) I’. France, Phys. Rev, C35 (1987) 1328 7) K.R. Schubert, Talks on nucleon-nucleon scattering, ed. M. Schapper, L~ndo~t-~ornstein ~umerj~ai data and functional relationship in science and technology, New series, vol. 9 JSprjnger, Berlin, 19SO) pT273 8) M. Ambrosia et al., Phys. Lett. B113 (1982) 347 9) W. Bell et al., Phys. Lett. B117 (1982) 131 RI) T. Akesson er aL, Phys. l&t, B152 (1985) 140