FORMAL THEORY OF WEAK Nd SCATTERING

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FORMAL THEORY OF WEAK Nd SCATTERING

Y. Avishai

To cite this version:

Y. Avishai. FORMAL THEORY OF WEAK Nd SCATTERING. Journal de Physique Colloques,

1984, 45 (C3), pp.C3-71-C3-73. �10.1051/jphyscol:1984313�. �jpa-00224027�

JOURNAL DE PHYSIOUE

Colloque C 3 , supplement au n ° 3 , Tome 45, m a r s 1984 page C3-71

FORMAL THEORY OF WEAK Nd SCATTERING

Y. Avishai

Centre de Recherches Nucleaires et Universite Louis Pasteur de Strasbourg, Groupe de Physique Nucleaire Theorique, B.P. 20CR, 67037 Strasbourg Cedex, France

Résumé - Nous présentons une théorie de la diffusion pour un système de trois nucléons soumis à une interaction NN qui comprend une partie forte ainsi qu'une partie faible.

Abstract - We develop a theory for three nucleon scattering in which the NN interaction contains strong and weak parts.

1 - INTRODUCTION

There has been a great excitement following the neutron spin rotation experiments at ILL ' ) . The ensuing theoretical works ^) were focused on the possibility of enhance- ments due to nuclear structure effects (most notably neutron resonances). These ef- forts, successful as they are, cannot lead to a clear and transparent relation bet- ween the calculated results and the fundamental weak NN interaction. Indeed, neutron

resonance is a complex many particle phenomena and this turns the interpretation rather obscur. Thus, it is now evident that neutron spin rotation experiments should be performed on the lightest nuclei, for which theoretical interpretation is feasi- ble in terms of the basic weak and strong NN interactions.

A natural starting point would of course be n-p scattering. However, it is clear that hydrogen target is not appropriate for the spin rotation experiment (among others, protons have large cross-section for the capture of cold neutrons). Furthermore, the

Z Z

strong isospin dependence of the weak NN force (e.g, isovector terms with T ±T ) sug- gested that n-n interaction should also be studied, together with the n-p interaction.

Consequently, it appears that an experiments like nd and pd scattering are indispen- sible in the study of the strangeness conserving non-leptonic part of the weak inter- action Hamiltonian which is responsible for parity non conservation (PNC) in nuclei.

On the theoretical part, one should have a theory of the three nucleon system in which the NN interaction contains both strong and weak parts. At present, such

theories are based on non-relativistic potential scattering approach and hence they are limited to low and medium energy scattering. In the present contribution, a prac- tical scheme for calculating weak interaction observables in the three nucleon sys- tem will be briefly reviewed (A more detailed account has recently been published by the author

J

) . Earlier works in this direction include those of Desplanques et^

al

4

) and of Kloet et al

5

) .

In section 2 we will discuss the two nucleon problem while the three nucleon system is treated in section 3.

2 - THE TWO NUCLEON SYSTEM

2A - The Weak and Strong NN Interaction. At present, one cannot derive the weak interaction between hadrons from basic principles. It is suggested by several au- thors -') that at energy less than 300 MeV the form of the weak NN potential arises from the exchange of mesons. This picture is convenient for the assignment of range and spin-isospin structure of the weak NN force. It should be stated that this

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984313

C3-72

JOURNAL DE PHYSIQUE

apprcach i s phenomenological. The physics i s s t i l l hidden i n t h e weak N M (M=~,p,w) v e r t e x and i s taken c a r e of by p a r a m e t r i z i n g t h e form f a c t o r s and weak coupling cons- t a n t s fNNM. ( I n f a c t t h i s i s a l s o t h e s i t u a t i o n i n t h e s t r o n g i n t e r a c t i o n (based on meson exchange) regime w i t h s t r o n g form f a c t o r s and coupling c o n s t a n t s gNNM. This p o i n t w i l l c e r t a i n l y undergo a d r a s t i c change i n veiw of t h e bag models). The non- r e l a t i v i s t i c form of t h e weak NN pote?ti$l d z r i y e d from t h e Teszn exshange model c o n t a i n s i s o s c a l a r terms l i k e e . g C

T~ ( i ~ ~ x ~ ~ . E , v ( r ) ] + ( o , - 0 2 ) . { p , v ( r ) l isovec- t o r terms l i k e e.g ~ ~ i ( ? ~ x ? ~ ) ~ ( ~ ~ + ~ ~ ) . E , v ( r ) ] and i s o t e n s o r terms l i k e e . g

C T ( ; , x ; ~ ~ l ( ~ I - ~ Z ) . { ; , v ( r ) ~ l where t h e C I S a r e c o n s t a n t s p o r p o r t i o n a l t o f x g , and W + "

'

+ "

v ( r ) = e - p i r / r (]-Ii= mT,m ,m )

.

Thus, t h e weak NN p o t e n t i a l V ( T ] , T~

,a,

,02 , p , r ) as- P

w

sumes h e r e a d e f i n i t e a n a l y t i c form (though r a t h e r involved). I n momentum space, t h e W " + ->

' ' +

i n t e r a c t i o n V ( ~ , . ~ ~ , 0 ~ , ~ ~ , k , k ' ) can be decomposed i n t o i t s p a r t i a l wave components!I wj v

i R s ; i ' Q 1 s ' ( k , k t ) where R , s , j and

i

a r e s p i n a n g u l a r momentum and i s o s p i n of t h e NN

+ '

system w h i l e ~ = ( T , + T ~ ) ~ . Here

IQ-Q' I = I

and s i n g l e t t r i p l e t t r a n s i t i o r s a r e p o s s i b l e .

..

- W

I t should b e s t r e s s e d h e r e t h a t V

i s

n o t s e p a r a b l e and cannot be approximated by a s h o r t sum of s e p a r a b l e terms.

S

s s

For t h e s t r o n g NN i n t e r a c t i o n

vS

we adopt h e r e a s e p a r a b l e form V =?Iy ><y

I .

^{I n}

p r i n c i p l e , a small non-separable term can be added b u t t h a t o p t i o n w l l l n o t be d i s - cussed h e r e .

2B

-

Formal S o l u t i o n of t h e Two Body Problem. Consider then a two nucleon sys- S W

tem w i t h t h e i n t e r a c t i o n V=V +V d i s c u s s e d above. Our i n t e n t i o n a t t h i s p o i n t i s t o o b t a i n e x p r e s s i o n s f o r t h e q u a n t i t i e s which s e r v e a s i n p u t t o t h e t h r e e body equa- t i o n s . These i n c l u d e t h e s t r o n g form f a c t o r

1~

S ^> (assumed t o be given) which con- t r o l s t h e s t r o n g d i s s o c i a t i o n of two nucleon i s o b a r s ( f i g . l a ) , t h e weak form f a c t o r

lyW>

^=VWg

1

yS> r e s p o n s i b l e f o r weak decay of two nucleon i s o b a r s ( f i g . Ib ; go

i s

the two nucleon propagator) and t h e two nucleon s c a t t e r i n g amplitude f r e s u l t i n g

S W W W S W S

from V=V +V

.

To f i r s t o r d e r i n V one h a s f = V + ( l y > + l y > ) 6 ( < y /+<yW1) ( f i g . I c ) where 6={~-'-<y Ig lyS>}-l S i s t h e ( s t r o n g ) i s o b a r propagator. Thus, f c o n t a i n s

small non s e p a r a b l e term t o g e t h e r w i t h small and l a r g e s e p a r a b l e terms. The expres- s i o n f o r f t h u s o b t a i n e d seems r a t h e r formal b u t i n f a c t it can e a s i l y be used i n p r a c t i c e once t h e n o n - r e l a t i v i s t i c form of t h e i n t e r a c t i o n i s giv,en.

3

-

^THE THREE BODY EQUATIONS

We now have a t h r e e nucleon system i n which each two nucleon p a i r i n t e r a c t s b o t h s t r o n g l y and weakly a s d e s c r i b e d i n s e c t i o n 2. The form f a c t o r s r e s p o n s i b l e f o r s t r o n g and weak d i s s o c i a t i o n of two nucleon i s o b a r s a r e c a l c u l a b l e t o g e t h e r w i t h t h e two nucleon s c a t t e r i n g amplitude. Since t h i s amplitude c o n t a i n s s e p a r a b l e p l u s very s m a l l non-se a r a b l e term t h e most a p p r o p r i a t e t h r e e body formalism i s t h e one developed by AGS

Y

). However, s i n c e t h e two body t m a t r i x c o n t a i n s a l s o s m a l l separa- b l e terms, (which should b e taken i n f i r s t o r d e r ) t h e AGS a l g o r i t h m must b e s l i g h t l y

-

modified and combined w i t h d i s t o r d e d wave formalism. Let us f i r s t denote by a,B,y...

t h e u s u a l p a i r s p e c t a t o r asymptotic channels and w r i t e t h e t h r e e body amplitude X =X

s +xW

a s a sum of s t r o n g and weak p a r t s . Assume t h a t w e have solved t h e t h r e e

aB af3

aB

body e q u a t i o n s i n which t h e weak i n t e r a c t i o n i s a b s e n t . I n t h e s e p a r a b l e approxima- t i o n we t h e n g e t t h e s t r o n g a m p l i t u d e s

xS

from t h e s e t of i n t e g r a l e q u a t i o n s

S S

aB

X = Z

+ zS

^T

xS

where ( w i t h G b e i n g t h e t h r e e n u c l e o n f r e e p r o p a g a t o r ) t h e

aR

a B

Y$B

UY

Y aB s

O

s t r o n g d r i v i n g terms Z = < y ] G

ly

> a r e d e p i c t e d i n f i g . 2a w h i l e

r

i s t h e p a i r

a B u o B s s s s y

s p e c t a t o r ( s t r o n g ) p r o p a g a t o r . Using m a t r i x n o t a t i o n X =Z +Z TX we now c o n s t r u c t

S S S

t h e s t r o n g M & l l e r o p e r a t o r s Q,=I+X

r,

Q = I + T X

.

^It i s t h e n shown 3, t h a t t h e weak

0

a m p l i t u d e s a r e g i v e n by

W S

w

S W S

ZaB=<ya

I

^Go

1 y

_B > + S

*

W+<Yu

I

GoV Go

1

^{Y B >}

The f i r s t two terms of Z

W

( f i g . 2b) r e s u l t from weak d i s s o c i a t i o n of two n u c l e o n a R

i s o b a r s . I f t h i s i s o b a r 1 s t h e p h y s i c a l d e u t e r o n , t h e n t h e s e terms r e f l e c t t h e oc- c u r e n c e of P wave components i n t h e d e u t e r o n wave-function. The t h i r d t e r m i n 2ZB ( f i g . 2c) r e s u l t s from t h e n o n - s e p a r a b i l i t y of

vW

and r e p r e s e n t s weak NN s c a t - t e r i n g a t t h e i n t e r m e d i a t e s t a t e s . I n c a s e t h e o n l y two n u c l e o n i s o b a r

i s

t h e p h y s i - c a l d e u t e r o n i t i s t h i s t e r m which d i s t i n g u i s h e s nd from pd s c a t t e r i n g . Hence, can- c e l l a t i o n of t h i s t e r m w i t h t h e o t h e r two of f i g . 2b i s improbable.

F i n a l l y , we n o t i c e t h a t a t z e r o e n e r g y ( a s i n t h e n e u t r o n s p i n r o t a t i o n e x p e r i m e n t ) t h e s t r o n g a m p l i t u d e c a n b e approximated by s c a t t e r i n g l e n g t h

,

and hence t h e s t r o n g X d l l e r o p e r a t o r s a r e e a s i l y c a l c u l a t e d ( a d d o p t i n g o n - s h e l l a p p r o x i m a t i o n ) . Hence, t h e e v a l u a t i o n of t h e s p i n r o t a t i o n a n g l e i s f e a s i b l e . The f o r m a l i s m d e s c r i b e d above

i s

a p p l i c a b l e o f c o u r s e a l s o a t h i g h e r e n e r g i e s i n t h e c a l c u l a t i o n of t h e p e r t i n e n t asymmetries.

R e f e r e n c e s

1 ) M. F o r t e

Gal,

Phys. Rev. L e t t .

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(1980) 2088.

2) V.E. Bunakov and V.P. Gudkov, Nucl. Phys. (1983) 93,and r e f e r e n c e s t h e r e i n . 3 ) Y. A v i s h a i , Nucl. Phys.

A399

(1983) 575.

4) B .

Desplanques

et,

Nucl. Phys.

A324

(1979) 221.

5 ) W.M. K l o e t e , Phys. Rev. C27 (1983) 2529.

6 )

K.R. L a s s e y and B.H.J. M c K e l l K Nucl. Phys.

A260

(1976) 413.

7)

E .

A l t

9,

Nucl. Phys. (1967) 167.