Baryon Resonances in Nuclei

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Baryon Resonances in Nuclei

Michael Danos

To cite this version:

Michael Danos. Baryon Resonances in Nuclei. Journal de Physique Colloques, 1972, 33 (C5), pp.C5-

171-C5-182. �10.1051/jphyscol:1972513�. �jpa-00215115�

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JOURNAL DE PHYSIQUE Coll.oque C5, suppl6ment au no 8-9, Tome 33, AoGt-Septembre 1972, page C5-171

Baryon Resonances i n Nuclei Michael Danos

N a t i o n a l Bureau of S t a n d a r d s Washington DC, 20234

Resume

-

Nous discutons les problhmes souleves par l'inclusion des resonances baryoniques et des mdsons dans la fonction d'onde du noyau. Puisqu'il n'existe pas de theorie realisable pour les interactions fortes, ces problemes doivent Etre affront& pas B pas d'une manihre quasi-thborique. Nous discutons l'dtat actuel de notre compr6hension et les ddveloppements possibles de l'avenir.

A b s t r a c t

-

The problems one e n c o u n t e r s when i n c o r p o r a t i n g baryon resonances and mesons i n t o t h e n u c l e a r wave f u n c t i o n a r e d i s c u s s e d . S i n c e no workable s t r o n g i n t e r a c t i o n t h e o r y e x i s t s t h e y must be a t t a c k e d i n a q u a s i - t h e o r e t i c a l s t e p

-

by

-

s t e p manner. The p r e s e n t s t a t u s i n t h e understanding i s d i s c u s s e d and t h e p o s s i b l e f u t u r e development suggested.

I INTRODUCTION

The d i f f i c u l t y and t h e b e a u t y of t h e problem of baryon resonances i n n u c l e i b o t h emanate from t h e circumstance t h a t no v i a b l e s t r o n g i n t e r a c t i o n t h e o r y e x i s t s a t t h i s time. The d i f f i c u l t y mani- f e s t s i t s e l f i n our i n a b i l i t y t o compute r e l i a b l e , a c c u r a t e numbers; t h e b e a u t y a r i s e s from t h e c o r r o l a r y t o i t , namely, t h a t one i s f o r c e d i n t o q u a l i t a t i v e , a t b e s t s e m i q u a n t i t a t i v e , understanding of t h e phenomena. Furthermore, t h i s problem i s particularly a t t r a c t i v e s i n c e we s t a n d a t t h i s time a t a wide-open f i e l d where b o l d s p e c u l a t i o n s t i l l i s a v i r t u e . I n t h i s paper I s h a l l c o n c e n t r a t e mainly on t h e q u e s t i o n s of p r i n c i p l e , i. e. on t h e c o n c e p t u a l problems one e n c o u n t e r s when t r y i n g t o i n c l u d e e x p l i c i t l y mesons and baryon r e s o n a n c e s i n t o t h e d e s c r i p - t i o n of n u c l e i . I hope t o show t h a t a reexamination of e s s e n t i a l l y a l l b a s i c components of n u c l e a r p h y s i c s i s r e q u i r e d and t h a t t h i s i s a t a s k which i n f a c t can be accomplished. I s h a l l d i s c u s s t h e

e f f e c t s of t h e baryon resonance admix- t u r e s o n l y v e r y b r i e f ~ y . They a r e t h e s u b j e c t of s e v e r a l c o n t r i b u t i o n s t o t h i s Conference. They a r e a l s o d e s c r i b e d i n a review a r t i c l e [ l ] w h i c h w i l l be p u b l i - shed i n t h e n e a r e s t f u t u r e ; i t may a l - ready have appeared i n p r i n t .

To d e f i n e t h e problems l e t us c o n s i d e r t h e d e u t e r o n wave f u n c t i o n :

Most of t h e time t h e p r o t o n and t h e neu- t r o n a r e i n a r e l a t i v e s - s t a t e . Some- t i m e s t h e y s c a t t e r i n t o a r e l a t i v e d- s t a t e , b u t t h e y soon s c a t t e r back i n t o a r e l a t i v e s - s t a t e . T h i s i s t h e o l d n u c l e a r p h y s i c s .

I f one t a k e s a c l o s e r look one s e e s t h a t d u r i n g c l o s e approach t h e n u c l e o n s p o l a r i z e each o t h e r i n t h e

same way a s , s a y , two hydrogen atoms

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972513

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polarize each other during a collision.

Staying for a moment with the hydrogen atoms, one may expand the wave function of the colliding atoms in terms of the complete set of states of the free hydrogen atoms. Then the polarization is ex- pressed as an admixture of excited hydrogen states to the ground state. Since the polarizability of hydrogen is large the amplitudes of the admixtures of

excited hydrogen states may be very large.

Returning to the deuteron, one may express the mutual polarization of the nucleons in terms of the excited states of the free nucleons. These states are called baryon r e s o n a n c e s [ i ? J ( a b b r e v i a t e d hence- forth by 311) (see Table 1).

Table I

Baryon

"' YieV

^BTNU2

N ^LL T ^{I t}

938

⁰ 0.08 A('l236)

; t -

¹²³⁴ ¹²⁰ ^0.35

N'(l470)

i C

¹⁴⁷⁰ ¹⁰⁰ ^0.017

Taking account explicitly of this polarization requires the augmentation of (1)

by appropriate terms :

I. DANOS

Table 2

A 'M~V T S Pa

m

⁰ ⁰ ⁴

99

1 0

A(1236) 300 1 2 0

2 1

NN' (1470) 530 0 1 0.2

1 0

A(1236) (1236) 600 0,2 1,3 1.0 193 072

The allowed 2-particle configurations are llsted in Table 2, Only the space

symmetric configurations are given since the odd L configurations are suppressed owing to the short range of the two-body forces. The quantity A G is ehe mass excess over that; of a two-nucleon system.

To estimate the magnitude of the admixture coefficients one must know the strength of the nucleon-Bd force. They can be estimated from a one-pion exchange potential (OPEP). The required coupling constants are known [3](Table I), and thus the OPEF strength can be found

by standard methods. Having done this one can determine the admixture. The results of such a calculation[~~are given in the last column of Table 2, which shows the sum of che admixture probabilities in all configurations. The order of magnitude of the admixture of the 32-s is I

-

²$6.

This should b e about the order of magnitude for all nuclei, in the sense, that each nucleon has a probability of 1

-

²^U/o

of beinp: present in form of a Bii. [As a 2ochlar aside, thus the probability of findin:; 235 nucleons in

23511

is only 0.01. host of she time a few of them are Bd-s. Waturally, the probability of hitting a BH with a fast projecsile is still only 1

-

²^%of the probability of hitting a nucleon.

3

Qualitatively, the reason for this surprisingly large admixture (not too different from the usual d- state admixture) in spite of the required large excitation energy (2M

-

^,2MN

2 600 NeV) lies in the strength of the OPEP; it is about four times stronger than the 33

-

OPEP strength in view of the large NAw coupling constant.

To explain this point a little more fully, this strong interaction implies that the effective shell model potential for a

A

ⁱⁿ^alarge nucleus

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BARYONIC RESONANCES...

would be

-

²⁰⁰MeV r a t h e r t h a n t h e - 5 0 MeV f o r nucleons i f e v e r y t h i n g e l s e were t h e same. It i s n o t f u l l y t h e same;

--

^t I

- -

a l a r g e p a r t of t h e p o t e n t i a l f o r nucle-

--- - -

^>

-

ens a r i s e s from Wigner f o r c e s . There i s

no r e a s o n t o b e l i e v e t h a t t h e s e f o r c e s

- _--

w i l l a l s o s c a l e b; t h e same f a c t o r 4. An

-

e s t i m a t e f o r t h e b i n d i n g of a A i n

n u c l e i can b e o b t a i n e d from t h e observed

-

down-shift of t h e peak i n t h e e l a s t i c

--

n-nucleus s c a t t e r i n g from i t s u n s h i f t e d

v a l u e of 180 MeV. ^Nucleon

-..-

Fig. 2

What d o e s t h i s a l l mean? T h i s we s h a l l do immediately, and t h e n we s h a l l b r i e f - l y d i s c u s s t h e i m p l i c a t i o n s f o r t h e o r y and experiment.

I1 TIIEOHETICAL CONSIDEHATIONS

.

When embarking on a more c a r e - Namely, i n an e l a s t i c s c a t t e r i n g p r o c e s s

t h e incoming n-meson w i l l undergo m u l t i p l e s c a t t e r i n g w i t h i n t h e n u c l e u s b e f o r e re- emerging and Leaving t h e t a r g e t n u c l e u s behind i n i t s zround s t a t e ( F i g . I ) . The i n t e r m e d i a t e h i g h l y e x c i t e d s t a t e l o o k s l i k e a s t a t e containing a bound (Eig.2).

I f t h e average b i n d i n g energy of t h e A i s l a r g e r t h a n t h a t of a nucleon t h e e l a s t i c p i o n s c a t t e r i n g peak w i l l a p p e a r down-shifted by t h i s b i n d i n g energy d i f f e - rence. I n a d d i t i o n , t h e peak of c o u r s e w i l l be broadened b o t h b y t h e d i s t r i b u t i o n of b i n d i n g e n e r g i e s and by t h e i n e l a s t i c c h a n n e l s l e a d i n g t o t h e d i s i n t e g r a t i o n of t h e nucleus. I n view of t h e small Coup- l i n g c o n s t a n t no such e f f e c t i s expected i n t h e Nt(1470)-nucleus s c a t t e r i n g .

f u l d i s c u s s i o n one h a s t o b e g i n by r e - marking t h a t t h e d i f f i c u l t i e s emanating from t h e absence of a c o n s i s t e n t meson t h e o r y a l r e a d y a r e v e r y much p r e s e n t i n c o n v e n t i o n a l n u c l e a r p h y s i c s , i. e.

b e f o r e even c o n s i d e r i n g t h e a d m i x t u r e . o f BR-s. T h i s comes about because t h e o n l y a v a i l a b l e workable t h e o r i e s a r e of t h e e f f e c t i v e Lagrangian t y p e , and t h e y , i n f a c t , have been d e f i n e d s o a s t o p r e c l u d e t h e i r a p p l i c a t i o n t o bound s t a t e problems.

Namely, t h e y a r e supposed t o be used only:

t o lowest non-vanishing o r d e r i n s c a t t e - r i n g problems: t h e c o u p l i n g c o n s t a n t s e t c . a r e chosen such t h a t t h e t r e e d i a - grams ( F i g . 3) reproduce t h e experimental r e s u l t s . T h i s diagram may r e p r e s e n t a pion-nucleon e l a s t i c s c a t t e r i n g , o r a nucleon-nucleus s c a t t e r i n g event. The This is a quick We ^*Ow compound system, R , may be a ,d (1236) i n must a d d r e s s o u r s e l v e s t o t h e q u e s t i o n : t h e former c a s e , o r an e x c i t e d s t a t e of

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C5-174 PI. DANOS

t h e nucleus i n t h e l a t t e r case. I f one wants t o examine t h e "bowels" of t h e s t r u c t u r e of t h e i n t e r m e d i a t e s t a t e , say 1 3 ~

,

one h a s t o r e s o l v e t h e "heavy l i n e " , a s i n d i c a t e d schematically i n Fig. 4. C l e a r l y t h e r e i s no way i n which t h i s f i g u r e can be redrawn a s a t r e e diagram. An a r b i t r a r y number of meson exchanges t a k e p l a c e i n a bound s t a t e .

Fig. 3

Fig. ⁴

A s i s well known t h i s d i f f i - c u l t y i s hidden away i n conventional n u c l e a r physics by going over t o a non- r e l a t i v i s t i c treatment. One f i r s t ex- t r a c t s a n o n - r e l a t i v i s t i c p o t e n t i a l from t h e t r e e diagram Fig. 5 , d e s c r i b i n g t h e s c a t t e r i n g of two f r e e nucleons, and then

-

a f t e r a d j u s t i n g some parameters- one uses t h i s p o t e n t i a l i n a n o n - r e l a t i v i s t i c d e s c r i p t i o n of t h e nucleus. The p a t h t h u s i s : r e l a t i v i s t i c t r e e diagram -+ non- r e l a t i v i s t i c p o t e n t i a l ^-td i a g o n a l i z a t i o n

Fig.

5

of matrices.

The same p a t h can be a l s o taken when considering t h e admixture of baryon resonances t o nuclear bound s t a t e s . However, one h e r e has t o f a c e c e r t a i n new d i f f i c u l t i e s . The r o o t s of t h e d i f f i c u l - t i e s a r e , ( i ) t h e extremely s h o r t l i f e - time of t h e resonances (P=lO MeV), ( i i ) 2 t h e i r unknown off-mass-shell behavior, ( i i i ) t h e ambiguity of t h e choice of

p a r t i c l e s t o be t r e a t e d e x p l i c i t l y . Since t h e r e e x i s t s no t h e o r y , l e t me d i s c u s s t h e s e problems i n terms of analogous problems which we "understand", v i z . , problems of conventional n u c l e a r physics.

( i ) s h o r t l i f e t i m e s .

One has no d i f f i c u l t y t o admit u n s t a b l e p a r t i c l e s t o t h e r o s t e r of l e g i t i m a t e b u i l d i n g blocks of n u c l e i . Namely, t h e f r e e neutron i s u n s t a b l e ; so i s t h e A - p a r t i c l e (and, so a r e

p

-mesons i n

p

mesic atoms). They a r e considered

"long l i v e d " , and one t r e a t s them a s i f they were s t a b l e p a r t i c l e s and c o n s i d e r s t h a t they l l e x i s t " w i t h i n t h e nucleus.

Consider now t h e resonant s c a t t e r i n g

p + q 2 ~ - , 1 3 ~ * + p +

'*c .

The i n t e r m e d i a t e resonance I3N* ' ' e x i s t s 1 ' , but not s t r i c t l y . I t " l i n g e r s " f o r a while b e f o r e i t decays. Thus:

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BARYONIC RESONANCES

. .

^C5-175

l i m

_r+

₀( l i n g e r s ) = ( e x i s t s )

.

On t h e o t h e r hand, t h e same " r e ~ o n a n c e ~ ~ , 131in, e x i s t s i n t h e ground s t a t e of 160:

where t h e C a r e s u i t a b l e expansion

(1.4

c o e f f i c i e n t s ; t h e y i n s u r e t h e r e q u i r e d symmetries and a l s o depend on t h e r e l a - t i v e d i s t a n c e between t h e fragments. One of t h e s t a t e s Y a i s t h e c o n s i d e r e d "resonance". T h i s expansion i s e x a c t ; b o t h t h e

t# a and t h e

Lfrr

a r e computed with bound s t a t e boundary c o n d i t i o n s . The c o n s i d e r e d resonance which o n l y t l l i n g e r s " i n f r e e

s p a c e , r e a l l y " e x i s t s " i n t h e nucleus. By v i r t u e of b e i n g a c o n s t i t u e n t of a s t a b l e n u c l e u s i t h a s a c q u i r e d = 0, and by b e i n g bound i t i s o f f t h e mass s h e l l . F i n a l l y , i f one c o n s i d e r s an e x c i t e d s t a t e of 160 above t h e p emission t h r e s h o l d t h e decay p r o c e s s of Fig. 3 can a g a i n t a k e p l a c e and t h e "resonance1' a g a i n a c q u i r e s a width.

R e t u r n i n g t o t h e ground s t a t e we s e e t h a t we can d e s c r i b e i t i n terms of (3) s i n c e we have a t h e o r y f o r t h e

"resonance1' ; we know i n p r i n c i p l e how t o a c h i e v e p = 0 and how t o c a r r y t h e resonance o f f t h e mass s h e l l , The correspon- d i n g admixture c o e f f i c i e n t Cab t h u s i s w e l l d e f i n e d and i n p r i n c i p l e can be computed unambiguously~ The e s s e n t i a l p o i n t i s t h a t no c o n c e p t u a l d i f f i c u l t i e s a r i s e i n b u i l d i n g a perhaps v e r y s h o r t - l i v e d resonance i n t o t h e ground s t a t e ; i t simply must be t r e a t e d a s s t a b l e , a l b e i t r e q u i r i n g minor s t r u c t u r a l changes.

The aforementioned can be immediately a p p l i e d t o t h e t r e a t m e n t of baryon r e s o n a n c e s i n n u c l e i . The o n l y d i f f i c u l t y i s t h a t one d o e s n o t know what t h e "minor s t r u c t u r a l changes"

amount t o i n t h i s c a s e . One simply h a s t o

make a n assumption, i, e. b u i l d a model.

The s i m p l e s t c h o i c e i s : t a k e t h e graph of t h e k i n d of Fig. 3 s e r i o u s l y , i. e.

assume t h a t t h e p r o p a g a t o r of t h e e f f e c - t i v e L a g r a n g i a n t h e o r y a p p l i e s a l s o i n a n u c l e a r bound s t a t e , However, remember t h a t t h i s i s o n l y a model assumption. It may be a v e r y good assumption, b u t i t must be proven o u t . ( R e c a l l t h a t t h e e f f e c t i v e Lagrangian was d e f i n e d o n l y f o r t r e e diagrams.)

( i i ) Going o f f t h e mass s h e l l . A u s u a l nucleon moving i n s i d e t h e n u c l e u s w i t h t h e Yermi momentum h a s a k i n e t i c energy of about 50 MeV, b u t i t i s bound. T h e r e f o r e i t i s o f f t h e mass s h e l l by about 50 MeV. A bound baryon resonance i s c o r r e s p o n c i n g l y more o f f t h e mass s h e l l ; a A(1236) w i t h a r e a s o n a b l e

momentum may b e o f f t h e mass s h e l l by 300

-

^{400 MeV.} ^{I s}t h a t t o l e r a b l e ? No d i r e c t i n f o r n a t i o n i s a v a i l a b l e a t t h i s moment. One may, however, l e a r n some- t h i n g from deep i n e l a s t i c e l e c t r o n s c a t - t e r i n g . The k i n e m a t i c s of t h a t experiment i s such ( P i g , 6) t h a t one can c o n t r o l b o t h t h e e x c i t a t i o n energy of t h e p a r t i - c l e , i. e. PIY

,

and i t s k i n e t i c energy, i. e . t h e f i n a l s t a t e momentum P. Both b e f o r e t h e c o l l i s i o n t h e n u c l e o n , and a f t e r t h e c o l l i s i o n t h e e x c i t e d p a r t i c l e , a r e on t h e mass s h e l l . However, by

t e s t i n g how b i g a j o l t ( i . e. momentum t r a n s f e r P) t h e p a r t i c l e can s u r v i v e without s h a t t e r i n g one can g a i n i n s i g h t on how f a r o f f t h e mass s h e l l one might

Fig. 6

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C5-176 M. DANOS

be a b l e t o go without d e s t r o y i n g t h e can b e t r e a t e d l i k e n u c l e o n s i n t h a t one l l i d e n t i t g " of t h e p a r t i c l e whatever t h e

mechanism f o r , o r d i r e c t i o n i n t h e complex energy p l a n e o f , going o f f t h e mass s h e l l may be. The d a t a show t h a t t h e p a r t i c l e s c a n s u r v i v e almost up t o q2;,~a w i t h l i t t l e damage[4> So t h e

A -

peak, M~ = 1.5 GeV 2

,

h a s s u f f e r e d o n l y some damage a t q2 6 = 2 G ~ v ' , w h i l e t h e peak a t

flz1.5

GeV i s s t i l l v i s i b l e even a t q2 = 4 G ~ ( F i g . 7 ) . V ~

TWO PlON THRESHOLD W IGeV) CONE PlON THRECH3LD

Fig. 7

Heemphasizing t h a t a momentum t r a n s f e r p u t s a d i f f e r e n t s t r a i n o,.l a p a r t i c l e t h a n going o f f t h e mass s - i e l l one n o n e t h e l e s s may be r e a s s u r e d and assume r a t h e r c o n f i d e n t l y t h a t one may t a k e a p a r t i c l e o f f i t s mass s h e l l by a reasonable

f r a c t i o n of i t s r e s t mass without having t o worry about it. We t h u s make t h e second model assumption: Baryon resonances

may f o r g e t t h a t t h e y a r e o f f t h e i r mass s h e l l when bound i n s i d e a nucleus.

( i i ) ( a ) The c o n t i n u a t i o n of t h e v e r t e x f u n c t i o n , t h e c o u p l i n g term, o f f t h e mass s h e l l i s c o n c e p t u a l l y much l e s s troublesome s i n c e one i s accustomed t o % h a t : i n e v e r y s c a t t e r i n g g r a p h t h e v e r t i c e s i n v o l v i n g t h e exchanged p a r t i c l e a r e o f f t h e mass s h e l l . One i n g e n e r a l i n t r o d u c e s a form f a c t o r , v ( ~ ~ ) , which e q u a l s u n i t y on t h e mass s h e l l and f a l l s

2 2

o f f i n some way f o r l a r g e

f

q

f

-m

.

^{T h i s}

s i m u l a t e s t h e e m p i r i c a l l y r e q u i r e d hard c o r e e f f e c t s .

( i i i ) Which p a r t i c l e s a r e p r e s e n t , and which have t o be c o n s i d e r e d e x p l i c i t l y i n t h e n u c l e a r wave f u n c t i o n ? T h i s i s t h e q u e s t i o n which i s most d i f f i c u l t t o answer i n t h e absence of a t h e o r y . It i s a l s o t h e q u e s t i o n which h a s l e d t o c o n t r o v e r s i e s .

The f o l l o w i n g c h o i c e s have been t r i e d .

( a ) u s e e x p l i c i t l y o n l y n u c l e o n s and p o t e n t i a l s ; use mesons o n l y i n computing t h e exchange c u r r e n t s ^151, ( b ) u s e e x p l i c i t l y o n l y n u c l e o n s , baryon

r e s o n a n c e s , and p o t e n t i a l s , and compute t h e "exchange c u r r e n t s " from t h e c o n t r i b u t i o n s of t h e baryon 11).

resonance admixtures.

( c ) combine ( a ) and ( b ) . T h i s h a s n o t y e t been widely p r a c t i c e d .

Approaches ( a ) and ( b ) a r e s a f e w i t h r e s p e c t t o overcounting b u t w i t h h i g h p r o b a b i l i t y l e a d t o undercounting;

approach ( c ) l e a d s v e r y l i k e l y t o overcounting. A t any r a t e , making a p a r t i c u - l a r c h o i c e a g a i n i s a model assumption, and must be t r e a t e d a s such.

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BARYONIC RESONANCES..

.

Besldes having emotional over- t o n e s , t h e p h y s i c a l b a s i s f o r t h e con- t r o v e r s i e s l i e s i n t h e d i f f i c u l t y of making p r a c i s e s t a t e m e n t s concerning t h e overcoun-;ing o r undercounting, i. e.

whether. o r n o t , o r by how much, over- c o u n t h g o r undercounting o c c u r s w i t h a p a r t s c u l a r choice. These q u e s t i o n s a r e sometimes t r i v i a l , b u t sometimes of a s u b t l e n a t u r e , even i f a t h e o r y e x i s t s ; t h e y a r e much more opaque i n t h e absence of a theory. To e x p l a i n t h i s s i t u a t i o n l e t u s a g a i n borrow from n u c l e a r p h y s i c s and c o n s i d e r a g a i n t h e expansion o f , s a y , t h e 160 ground s t a t e i n terms of i t s fragments, Eq. ( 3 ) . A s a l r e a d y mentioned, t h i s expansion i s e x a c t s i n c e t h e y , and

rgp

each form a complete s e t . One may use an a l t e r n a t i v e expansion:

b u t , o b v i o u s l y , one may n o t use t h e s e two expansions simultaneously. For example, a p a r t i c u l a r component of t h e expansion (4) may d e s c r i b e I2C! i n t h e ground s t a t e and ' ~ e i n a s c a t t e r i n g s t a t e p + 3 ~ . T h i s w i l l be v e r y c l o s e l y r e l a t e d t o some s t a t e i n ( 3 ) , v i z . t h a t d e s c r i b i n g 'H i n t h e ground s t a t e and 1 3 ~ i n a s c a t t e r i n g s t a t e p + 1 2 ~ - I n a time-dependent des- c r i p t i o n t h i s double c o u n t i n g graph might look l i k e Fig, 8. It t h u s would be an obvious c a s e o f double c o u n t i n g i f one

Fig. 8

would add t o t h e e x p w s i o n ( 3 ) a

"resonanceif of t h e p + 3~ system. Natural- l y , nobody would do a t h i n g l i k e t h a t i n n u c l e a r p h y s i c s . Another k i n d of p o s s i b l e

over c o u n t i n g of r e a c t i o n diagrams h a s l e d t o t h e i n t r o d u c t i o n of t h e concept of d u a l i t y i n t o p a r t i c l e p h y s i c s . A n u c l e a r p h y s i c s analogue may a g a i n s e r v e a s a n i l l u s t r a t i o n .

Fig. 9

Assume t h a t t h e graph of Fig.

9

g i v e s a good d e s c r i p t i o n of a p a r t i c u l a r p r o c e s s i n c e r t a i n kinematic r e g i o n s . One a l t e r n a t i v e l y i n p r i n c i p l e can compute t h e p r o c e s s by, s a y , an R-matrix type t r e a t m e n t i n which a complete s e t of s t a t e s of t h e compound system C i s employed (Fig. 9 b ) . N a t u r a l l y , no n u c l e a r p h y s i c i s t would add t h e amplitude of Fig.

9

b t o t h e amplitude of Pig.

9

a.

( I f o n l y an incomplete expansion f o r C h a s been used, one may add t h e a n p l i t u d e s of Fig.

9,

checking f o r o r t h o g o n a l i t y . ) I n p a r t i c l e p h y s i c s t h i s c a v e a t i s c a l l e d d u a l i t y . It s a y s : u s e e i t h e r t h e t r e e graphs a o r t h e t r e e graphs b , and n o t both. U n f o r t u n a t e l y d u a l i t y can n o t be used i n bound s t a t e problems. The f i r s t of t h e above examples i s almost s t u p i d l y t r i v i a l ; t h e second i n v o l v e s t h e

" c o m p e t i t i o n between d i r e c t and compound p r o c e s s e s f ' , a s it i s c a l l e d i n t h e some- what sloppy n u c l e a r p h y s i c s jargon. A t any r a t e , i n n u c l e a r p h y s i c s double c o u n t i n g can always b e r e c o g n i z e d , and,

(9)

n o t t o o much double c o u n t i n g h a s t a k e n

Fie;. 10

a t l e a s t i n p r i n c i p l e , e x a c t l y accounted f o r and e l i m i n a t e d . T h i s i s n o t s o e a s y , o r perhaps n o t even p o s s i b l e , i n p a r t i c l e p h y s i c s . Let me i l l u s t r a t e t h i s w i t h p i o n photoproduction on a nucleon, Fig.10.

Resolving t h e p r o c e s s i n t o elementary graphs one h a s t h e d i r e c t g r a p h s a , b , c , d . Then one h a s r e s c a t t e r i n g g r a p h s , of which e and f a r e examples f o r r e s c a t t e - r i n g based on graph d. Only mesons appear i n t h e s e graphs. One may want t o add a graph c o n t a i n i n g s a y , t h e

A

(1236), and draw g. Now one s e e s t h a t one may have overcounted s i n c e graphs e and f a r e components of t h e Chew-Low model of t h e

a .

(Note t h a t t h e graph e i s f u l l y analogous t o t h e t r i v i a l double c o u n t i n g graph Fig. 8 of n u c l e a r p h y s i c s . ) Simi- l a r l y , one w i l l exclude t h e r e s c a t t e r i n g graph h. I f one now r e c a l l s t h e d u a l i t y c a v e a t one f i n d s t h a t summing o v e r t h e graphs of t y p e d and g c o n s t i t u t e s double c,ounting. However, i f one u s e s o n l y one graph of each k i n d (e. g. o n l y t h e d i n g and t h e n-meson i n d ) perhaps

p l a c e . Once one g o e s t o many-body n u c l e i , s a y t h e d e u t e r o n ground s t a t e , p r e c i s e s t a t e m e n t s on double c o u n t i n g can n o t b e made: i n t h e absence of a t h e o r y one h a s t o r e l y on t r i a l and e r r o r t o l e a r n whether a p a r t i c u l a r c h o i c e of e x p l i c i t l y i n c l u d e d p a r t i c l e s c o n s t i t u t e s double c o u n t i n g o r n o t .

Another problem i s t h e a n t i - symmetrization o r symmetrization of t h e p a r t i c l e s . A s i s w e l l known, t h e s e q u e s t i o n s a r e t a k e n c a r e of by summation o v e r Feyman g r a p h s - I n a t r e a t m e n t c o n s i s t i n g of u s i n g e f f e c t i v e Lagrangians and b r e a k i n g t h e r u l e s of e f f e c t i v e Lagrangian t h e o r y one must e x p e c t t o do

something a t l e a s t sometimes wrong. T h i s problem i s n o t s e r i o u s f o r Fermions. It could be s e r i o u s f o r Bosons: t o some e x t e n t n u c l e i may a c t a s BASER'S (Boson A m p l i f i e r s by S t i m u l a t e d Emission of R a d i a t i o n ) .

The f i n a l v e r d i c t t h u s is:

depending on t h e s t u d i e d e f f e c t , u s e e i t h e r approach ( a ) , i. e. c o n s i d e r e x p l i c i t l y o n l y mesons, o r u s e approach ( b ) , i. e. c o n s i d e r e x p l i c i t l y o n l y baryon resonances, a s c o n s t i t u e n t s of t h e n u c l e u s , As w i t h any c a s e of comple- m e n t a r i t y one w i l l f i n d t h a t it w i l l be v e r y i n s t r u c t i v e t o s w i t c h back and f o r t h between t h e two a l t e r n a t i v e s i n t h i n k i n g about t h e problems, and one w i l l a l s o l e a r n how t o s t e p g i n g e r l y i n t o t h e approach (c), where one c o n s i d e r s s i m u l t a n e o u s l y mesons and baryon r e s o - nances, a l l t h e time checking o n e ' s r e s u l t s a g a i n s t experiment. I n o t h e r words, one r e a l l y must employ t h e

s c i e n t i f i c method. The procedure now is:

g a i n c o n f i d e n c e , and them u s e it.

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BARYONIC RESONANCES

- -

^C5-179

I11 PRESENT STATUS.

From t h e t h e o r e t i c a l

c o n s i d e r a t i o n s (Sect.11) one should f e e l reasonably confident i n a semi-quanti- t a t i v e way, i. e., one should f e e l t h a t not only q u a l i t a t i v e l y baryon resonances a r e present i n n u c l e i , i n c l u d i n g i n n u c l e a r ground s t a t e s , b u t one should f e e l t h a t t h e numerical semi-phenomeno- l o g i c a l p r e d i c t i o n s should be trustworthy t o w i t h i n a f a c t o r of 2

-

^{3 ,}^i.^{e. t o}

h a l f an o r d e r of magnitude. Is it t r u e ? The b e s t information i s a v a i l - a b l e from t h e magnetic moment of t h e deuteron: t h e d i f f e r e n c e between t h e experimental v a l u e and t h e t h e o r e t i c a l value computed n e g l e c t i n g exchange c u r r e n t s i s about 10 times l a r g e r t h a n t h e combined experimental and t h e o r e t i c a l u n c e r t a i n t i e s . One f i n d s : f o r t h e s t a t i c magnetic moment t h e meson exchange

c u r r e n t s do v e r y l i t t l e , v i z a t most

-

^{I 0}^%of r e q u i r e d amount. E s s e n t i a l l y t h e f u l l discrepancy can b e explained by t h e c o n t r i b u t i o n s r e s u l t i n g from t h e baryon resonance admixturel61. The l a r g e s t u n c e r t a i n l y i n t h i s r e s u l t a r i s e s from t h e need t o use a t h e o r e t i c a l e s t i m a t e f o r t h e s t a t i c moment of t h e

A

(1236).

Thus, h e r e approach ( a ) does not work, while approach (b) s u f f i c e s ,

From t h e s e r e s u l t s one may conclude t h a t t h e above t h e o r e t i c a l

confidence e s t i m a t e , i. e. a f a c t o r 2

-

3, i s r a t h e r conservative. Perhaps a confidence w i t h i n a f a c t o r 1.2

-

²could b e j u s t i f i e d .

I V CONCLUSIONS.

The momentum space wave func- t i o n s f o r t h e d i v e r s e A c o n f i g u r a t i o n s i n t h e deuteron computed with p o t e n t i a l s which have no r e p u l s i v e c o r e s a r e shown i n Fig. 11. One s e e s t h a t above

- ⁱ

^GeV

t h e nucleon c o n f i g u r a t i o n s have vanishing amplitude, compared with t h a t of t h e A b configurations. One t h u s may chose such kinematic c o n d i t i o n s i n a s c a t t e r i n g experiment which would allow t o s e p a r a t e t h e s c a t t e r i n g of t h e p r o j e c t i l e off a

a

from t h a t off a nucleon. More p r e c i s e l y , one may t r a c e . o u t t h e momentum d i s t r i - b u t i o n of both t h e NN and t h e 4 A con- f i g u r a t i o n s by s u i t a b l e s c a t t e r i n g experiments. ( N a t u r a l l y , wave f u n c t i o n s derived with b e t t e r p o t e n t i a l s w i l l have t o be computed before t h e high momentum t a i l s of t h e wave f u n c t i o n s can be

employed with confidence.) A n experiment of t h i s kind may seem very d i f f i c u l t . However, an analogous experiment i n which t h e RMS charge r a d i u s of t h e K-meson was measured using t h e " v i r t u a l " n-mesons The transition derived surrounding a nucleon a s t a r g e t s , has from t h e observed p-n c a p t u r e c r o s s

been r e c e n t l y The ex-

s e c t i o n a t thermal neutron e n e r g i e s , can

perimental difficulties thus seem

be t o a l a r g e e x t e n t explained by meson surmountable with present-day techniques.

exchange currents. there remains There e x i s t s , however, a more conventio- an unexplained gap. This a g a i n i s ex- n a l , i f not l e s s s p e c t a c u l a r , consequence p l a i n e d by t h e c o n t r i b u t i o n a r i s i n g mainly

of the -admixture to the deuteron - .--

from t h e off-the-mass-shell photo-produc- wave function. 'his concerns t h e a n a l y s i s t i o n of t h e

a

(1236) ( F i g - 10 g ) * 17] since of ?;he e-d scata;ering experiments under- t h i s v e r t e x can be d e r i v e d from t h e ex- taken t o e x t r a c t t h e form f a c t o r s of t h e perimental on-the-mass-shell process t h i s

free neutron. The analysis has been done r e s u l t i s r a t h e r convincing. Thus here p r e v i o u s l y i n term of t h e conventional approaches ( a ) and ( b ) alone do n o t s u f f i c e - ,,clear physics. A re-analysis including A c a u t i o u s use of approach ( c ) i s required.

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M. DANOS

-16; 0.4 o.a 1 2 q ( G ~ v ) I

Pig. 11

t h e

A b

-admixture h a s been r e c e n t l y a t t e m p t e d i n a v e r y s i m p l i f i e d way [ 9 ).

It t u r n s o u t t h a t perhaps t h e n e u t r o n i s e l e c t r i c a l l y e x a c t l y n e u s r a l f o r a l l q 2 ( o f c o u r s e , except f o r t h e r e l a t i v i s t i c e f f e c t which r e q u i r e s a moving magnetic d i p o l e t o e x h i b i t an e l e c t r i c d i p o l e a s w e l l ) . The e x c e s s c h a r e e form f a c t o r can be f u l l y accounted f o r by t h e admixture of t h e A A c o n f i g u r a t i o n t o t h e deuteron.

One v e r y i m p o r t a n t problem t o t h e b e s t of my knowledge h a s n o t y e t been s a t i s - f a c t o r i l y worked out. T h i s concerns t h e r e n o r m a l i z a t i o n of t h e two-body poten- t i a l s , t a k i n g e x p l i c i t l y i n t o account t h e 8R a&mixtures a s c l o s e d c h a n n e l s i n t h e nucleon-nucleon s c a t t e r i n g s t a t e .

c a l r e s u l t s f o r t h e d i v e r s e e f f e c t s of t h e BH admixtures w i l l , i n f a c t , c o n t a i n some i n c o n s i s t e n c i e s . F u r t h e r a p p l i c a - t i o n s abound. L e t me o n l y r e f e r t o n u c l e a r physics. One ought t o b e g i n t o t a k e t h e BR admixtures f o r g r a n t e d , t h u s i n s o l a t i n g t h e p u r e l y n u c l e a r p h y s i c s a s p e c t s ; e. g. one may remove t h e con- t r i b u t i o n s t o f3 -decay m a t r i x elements, o r magnetic moments, a r i s i n g from t h e BR, and be c o n f i d e n t t h a t t h e remainder must be e x p l a i n e d by c o n v e n t i o n a l n u c l e a r p h y s i c s . I n o t h e r words, "mesonic e f f e c t s "

should n o t be used any more a s excuses f o r b a d l y f i t t i n g t h e o r e t i c a l v a l u e s . I n summary, i t should be p o s s i b l e t o d e s c r i b e r a t h e r a c c u r a t e l y t h e e f f e c t s of t h e p o l a r i s a b i l i t y of t h e n u c l e o n s , and t h e e f f e c t s of t h e d i v e r s e mesons, on n u c l e a r p r o p e r t i e s , and t o t u r n t h e

problem around and t o determine t h e c h a r a c t e r i s t i c s of BR-s using n u c l e i a s BR c o n t a i n e r s . T h i s seems t o o b t a i n i n s p i t e of t h e absence of a f u n c t i o - n i n g s t r o n g i n t e r a c t i o n t h e o r y . A t p r e s e n t i t i s unknown how f a r t h i s approach can be c a r r i e d . One w i l l have t o proceed s t e p by s t e p , c a u t i o u s l y r e f i n i n g t h e o r y and experiment, remembe- r i n g t h a t indeed double counting may o c c u r ; i n f a c t t o some e x t e n t always does occur. Any d i s c r e p a n c y which may t u r n u p w i l l be of g r e a t v a l u e ; a f t e r a l l one may f i n d l i m i t a t i o n s of a p h y s i c a l p i c t u r e e i t h e r by going with c r u d e measurements t o t h e l i m i t s of t h e p i c t u r e o r doing p r e c i s e measurements i n more a c c e s s i b l e r e g i o n s .

The investment i n work needed f o r t h i s

problem i s n o t s m a l l ; t h i s t a s k should, REFERENCES however, be much l e s s formidable t h a n

t h e g e n e r a t i o n of t h e u s u a l p o t e n t i a l s , ¹ Arenh5vel (H.) and Weber

(H.J.)

s i n c e t h e c l o s e d B2. c h a n n e l s should Nuclear I s o b a r C o n f i g u r a t i o n s ; t o be r e s u l t o n l y i n r a t h e r minor m o d i f i c a t i o n s p u b l i s h e d 1972, S p r i n g e r T r a c t s , of t h e c o n v e n t i o n a l p o t e n t i a l s . Neverthe- Vol. 65. T h i s paper c o n t a i n s an exten- l e s s , b e f o r e having such a s e t of e f f e c - s i v e l i s t of r e f e r e n c e s .

t i v e p o t e n t i a l s a v a i l a b l e t h e t h e o r e t i -

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BARYONIC RESONANCES C5-181

2 R i t t e n b e r g (A.) e t . a l .

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Rev. Mod. Phys.

1971,

43.

S 1

3 E b e l (G.) e t , a l .

-

Nucl. Phys. 1970,

E,

L 4 T a y l o r (R.E.)

-

P r o c , 4 t h , I n t . Symp. o n E l e c -

t r o n and Photon I n t e r a c t i o n s a t High E n e r g i e s , L i v e r p o o l , 1969, p. 251

5 E r i c s o n (M.) and Rho (M.)

-

From s o f t p i o n s t o r e a l p i o n s i n n u c l e i , t o be p u b l i s h e d i n P h y s i c s R e p o r t s , 1972

6 Arenhovel (H.), Danos (M.) and Williams ( H . T . )

-

Nucl. Phys. 1971,

31B.

109

7 R i s k a (D.O.) and Brown (G.E.)

-

Phys. L e t t e r s 1972,

388,

¹⁹³

8 S h e p a r d (P.)

-

B u l l , A.P.S. 1972, II,17, 435 ( I n v i t e d p a p e r a t t h e APS Washington Meeting)

9 W l l i a m s (H.T.), Arenh'dvel (H.) and M i l l e r (H.G.) Phys, L e t t e r s 1971,

s,

²⁷⁸

DISCUSSION

W. J, SWIATECKI i ~ e r k e l e y )

I t i s v e r y s t i m u l a t i n g t o t h i n k of t h e v a r i o u s e f f e c t s t h a t n u c l e o n s t r u c t u r e might have on n u c l e a r p h y s i c s . One a s p e c t of n u c l e o n s t r u c - t u r e is t h e mass d i f f e r e n c e of 1.3 MeV between a f r e e n e u t r o n and a f r e e p r o t o n . I would l i k e t o a s k t h e s p e a k e r and o t h e r e x p e r t s on n u c l e a r s t r u c t u r e t h e f o l l o w i n g q u e s t i o n : what d o you t h i n k of t h e h y p o t h e s i s t h a t t h e n-p m a s s d i f - f e r e n c e d i s a p p e a r s f o r n u c l e o n s immersed i n s i d e n u c l e i ? The r e a s o n f o r t h i s q u e s t i o n is t h a t e m p i r i c a l l y B i l l Myers i n B e r k e l e y f i n d s t h a t i n t r y i n g t o f i t n u c l e a r ground s t a t e masses one e n c o u n t e r s a n anomaly t h a t i s v e r y d i f - f i c u l t t o remove, e x c e p t by d r o p p i n g t h e n-p mass d i f f e r e n c e ! Could i t b e t h a t t h e n u c l e a r i n t e r a c t i o n s , which a r e of t h e o r d e r o f 5 0 MeV, d e s t r o y t h e s p e c i a l s t r u c t u r e d i f f e r e n c e s b e t - ween n e u t r o n s and p r o t o n s which a r e r e s p o n s i b l e f o r t h e 1.3 M e V n-p mass d i f f e r e n c e ?

M. DANOS (Washington)

I f t r u e , i t would b e v e r y i n t e r e s t i n g . I n a n y c a s e , i t would b r e a k t h e model a s s u m p t i o n t h a t t h e c h a r a c t e r i s t i c s of t h e p a r t i c l e s do n o t c h a n g e when t h e y are b e i n g i n c o r p o r a t e d i n t o a bound system.

M. DANOS (Washington)

I may have b e e n u n c l e a r i n my p r e s e n t a t i o n . The meaning of t h e l i n e C i n F i g . 9 b i s t h a t i t r e p r e s e n t s a c o m p l e t e set o f s t a t e s of t h e compound s y s t e m d e f i n e d , s a y , i n t h e i n s i d e of a s p h e r e of r a d i u s 100 fm

.

Then a l l p r o c e s - s e s , i n c l u d i n g F i g . 9 a , a r e a l r e a d y c o n t a i n e d i n t h e set C. However, i f o n e a p p r o x i m a t e s t h e set by one of i t s members, a s i n t h e o n e - l e v e l B r e i t - W i g n e r f o r m u l a , o n e c a n , and p e r h a p s must, add e x p l i c i t l y t h e a m p l i t u d e of a p a r t i - c u l a r p r o c e s s o f t h e c l a s s F i g . 9 a

.

M. RHO ( S a c l a y )

1. The e f f e c t s one wants t o l o o k a t depend upon t h e s i t u a t i o n s : w h e t h e r N* o r mesonic d e g r e e s of freedom a r e more i m p o r t a n t s h o u l d depend upon e x p e r i m e n t a l s i t u a t i o n s .

2. D u a l i t y : d u a l i t y , a n o t e v e n p r o v e n f a c t , i s t o make s e n s e i f e v e r y t h i n g is p u t i n o n e way o r o t h e r way. T h i s s o u n d s n o t f e a s i b l e i n c a l c u l a t i n g s o m e t h i n g i n n u c l e i .

3. Magnetic moments : t h e f i r s t p o i n t a p p l i e s h e r e ; ~ * ' s p r o b a b l y p l a y n o r o l e i n m a g n e t i c moments

.

4. I t would b e n i c e i f t h e s p e a k e r c o u l d pro- p o s e a way t o g o a b o u t p u t t i n g i n N*'S i n nu- c l e i . H e f a i l e d t o d o s o i n t h i s t a l k . I h a v e O.M. BILANIUK (Swarthmore C o l l e g e , P e n n . and O r s a y )

a l i t t l e d i f f i c u l t y i n b e i n g o p t i m i s t i c a b o u t What r e a c t i o n times a r e a s s o c i a t e d w i t h y o u r

i t . p r o c e s s e s A and B ? A r e n ' t t h e t i m e s c a l e s f o r

t h e s e two p r o c e s s e s a l t o g e t h e r d i f f e r e n t ? How ^M. DANOS (Washington)

t h e n c a n you t a l k of combining A a n d B ? C o n c e r n i n g t h e proof of d u a l i t y , t h i s would r e - q u i r e t h e e x i s t e n c e o f a t h e o r y . No t h e o r y

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(13)

n o p r o o f . The f a c t s c o n c e r n i n g t h e c o n t r i b u - t i o n s o f t h e d i f f e r e n t mechanisms t o t h e ma- g n e t i c moments a r e , I b e l i e v e , a s s t a t e d abo- v e i n t h e p a p e r .

G. IGO (U.C.L.A.)

With r e g a r d t o your comment a b o u t p e e l i n g t h e d e u t e r o n a p a r t i n o r d e r t o s t u d y t h e v a r i o u s components o f t h e d e u t e r o n , p a r t i c u l a r l y t h e h i g h mcnnentum components of t h e d e u t e r o n s which w i l l b e p a r t i c u l a r l y s e n s i t i v e t o nu- c l e o n i s o b a r s . W e h a v e made measurements i n a momentum t r a n s f e r r e g i o n , u s i n g 433 MeV, 362 MeV and 291 MeV d e u t e r o n s , where t h e n u c l e o n i s o b a r components a r e v e r y s m a l l and t h e D- wave component dominates. The D-wave component i s n o t u n i q u e l y determdned by nucleon-nucleon s c a t t e r i n g and t h e ground s t a t e p r o p e r t i e s of t h e d e u t e r o n .

A.M. GREFN ( H e l s i n k i )

I n s u p p o r t of Danos and h i s l a r g e p e r c e n t a g e s o f baryon r e s o n a n c e c o n f i g u r a t i o n s i n n u c l e a r w a v e f u n c t i o n s , Schucan and myself l o o k e d a t

t h e t r i n u c l e o n s y s t e m where a c o n f i g u r a t i o n w i t h o n l y o n e h ( 1 2 3 6 ) is n o t f o r b i d d e n . W e f i n d t h a t t h i s c o n f i g u r a t i o n h a s w 4-556 proba- b i l i t y . Such Ei p e r c e n t a g e i s i n l i n e w i t h re- c e n t c a l c u l a t i o n s of t h e &decay of t h e t r i t o n . I n g e n e r a l , i f we t a k e i n t o a c c o u n t NN d - s t a t e s g e n e r a t e d by O.P.E. we s h o u l d a l s o t a k e i n t o a c c o u n t N A s t a t e s g e n e r a t e d by O.P.E.

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^{t h e y}

a r e e q u a l l y i m p o r t a n t , i f t h e y a r e n o t comple- t e l y f o r b i d d e n .

K. BLEULER (Bonn)

I d o n o t see a n y problem i n u s i n g t h e w e l l - known e x c i t e d s t a t e s o f n u c l e o n s i n a n a t u r a l e n l a r g e m e n t o f n u c l e a r p h y s i c s . T h e r e i s t h e p o s s i b i l i t y of m e a s u r i n g a l a r g e number of new e f f e c t s (new k i n d of n u c l e a r e x c i t a t i o n s i n h e a v i e r n u c l e i , i

.

^e."exchangev o f a n " e x c i t o n " ; y - x p r o c e s s e s as d i s c u s s e d a l r e a d y y e a r s a g o ; e f f e c t s on magic numbers i n n u c l e a r ground s t a - t e s v i a t h e P a u l i p r i n c i p l e , e t c . ) and t o make i n t e r e s t i n g c a l c u l a t i o n s .