PHENOMENOLOGICAL MODELS OF MULTIPLE PRODUCTION

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PHENOMENOLOGICAL MODELS OF MULTIPLE PRODUCTION

A. Bassetto

To cite this version:

A. Bassetto. PHENOMENOLOGICAL MODELS OF MULTIPLE PRODUCTION. Journal de

Physique Colloques, 1973, 34 (C1), pp.C1-341-C1-345. �10.1051/jphyscol:1973144�. �jpa-00215224�

PHENOMENOLOGICAL MODELS OF MULTIPLE PRODUCTION

PHENOMENOLOGICAL MODSLS OF MULTIPLE PRODUCTION

I s t i t u t o d i F i s i c a d e l l ' U n i v e r s i t a , Padova

The o u t s t a n d i n g f e a t u r e o f m u l t i p l e p r o d u c t i o n pro- c e s s e s i s t h a t s e c o n d a r i e s a r e g e n e r a t e d w i t h s m a l l t r a n s v e r s e momenta. I n r e c e n t y e a r s two d i f f e r e n t p i c t u r e s have been advocated t o d e s c r i b e t h i s phe- nomenon ; t h e i r extreme r e a l i z a t i o n s a r e fragmenta- t i o n o r Nova models 11-41 on one s i d e , and simple m u l t i p e r i p h e r a l models [ 5 , 6 1 on t h e o t h e r . Indeed, d u r i n g t h e p a r t y e a r t h e r e h a s been a c o n t r o v e r s y between people s u p p o r t i n g one o r t h e o t h e r p i c t u r e . A number of t e s t s t o d i s t i n g u i s h between t h e models have been proposed.

The d a t a show t h a t n e i t h e r of t h e two extreme pic- t u r e s i s q u i t e c o r r e c t ( a s one might have e x p e c t e d ) , b u t t h a t c e r t a i n l y a m u l t i p e r i p h e r a l p i c t u r e i s nea- r e r t o t h e t r u t h , while c e r t a i n f e a t u r e s , t y p i c a l

N a t u r a l consequences o f t h i s model a r e c o n s t a n t c r o s s - s e c t i o n s s i n c e t h e p r o c e s s e s a r e d i f f r a c t i v e , and l i m i t i n g f r a g m e n t a t i o n f o r t h e one p a r t i c l e in- c l u s i v e d i s t r i b u t i o n s . A c e n t r a l p l a t e a u i n t h e one p a r t i c l e i n c l u s i v e d i s t ' r i b u t i o n c a n a l s o be accomo- d a t e d by choosing a s u i t a b l e e x c i t a t i o n spectrum f o r t h e l a r g e mass f i r e b a l l s .

A c r u c i a l c o n s t r a i n t i n t h i s model is t h e i s o t y o - p i c decaz. From i t , and from t h e observed l i m i t a t i o n on t h e t r a n s v e r s e momenta, i t f o l l o w s t h a t t h e num- b e r of p a r t i c l e s coming from one f i r e b a l l i s pro- p o r t i o n a l t o t h e f i r e b a l l mass M :

and then, i f one wants a l o g a r i t h m i c i n c r e a s e w i t h of f r a g m e n t a t i o n models, a r e n o t a c c e p t a b l e . So now

energy o f t h e t o t a l average m u l t i p l i c i t y , one must i t appears t h a t models of t h i s kind have t o be aban-

impose t h e e x c i t a t i o n spectrum : doned o r a t l e a s t d r a s t i c a l l y modified. On t h e o t h e r

haad, t h e concept of m u l t i p e r i p h e r a l dynamics, i f p(M)

w

M - ~ .

^{( 2 )}

want t o a p p l y i t t o phenomenology, h a s t o be en-

I n t h e Mueller aanguage [ 7 ] t h i s would c o r r e s a o n d l a r g e d i n o r d e r t o t a k e i n t o account e f f e c t s which

t o a PPf c o u p l i n g . The PPP term i s missing i n a r e not p r e s e n t i n t h e o r i g i n a l f o r m u l a t i o n : e f f e c t s

t h i s scheme. There i s no s c a l i n g of t h e l e a d i n g par- of c l u s t e r i n g , d i f f r a c t i o n , and a b s o r p t i o n .

t i c l e peak : Following t h i s l i n e of t h o u g h t , l e t me now go i n t o

more d e t a i l s . F i r s t , I s h a l l b r i e f l y d e s c r i b e t h e f r a g m e n t a t i o n model i n i t s most i n t u i t i v e v e r s i o n ,

E - do- - ^{I 1} _3/2.

_'

^{( 3 )}

d3p

X1l

&- (1-X) t h e Nova model [3] . We have h e r e a d i f f r a c t i v e e x c i -

t a t i o n of o n e cor b o t h ) incoming particles into fire- T h i s i s a s t r o n g l e a d i n g p a r t i c l e e f f e c t . F i n a l l y , balls of rather large mass, which subsequently decay from t h e behaviour of t h e i n t e g r a t e d p a r t i a l c r o s s -

s e c t i o n s : i s o t r o p i c a l l y i n t h e i r r e s t system, a s shown i n f i -

c - n -2

( 4 )

gure 1.

one g e t s very l a r g e f l u c t u a t i o n s i n t h e m u l t i p l i c i t y d i s t r i b u t i o n :

+ ^{+ possibly f, =} ( n ( n - 1 ) ) - ( n ) c < 2 v'Z . ^{( 5 )}

A q u i t e d i f f e r e n t p i c t u r e i s p r e s e n t i n t h e m u l t i -

Fig.1

p e r i p h e r a l model (MPM), f i r s t proposed by Amati, B e r t o c c h i , F u b i n i , S t a n g h e l l i n i and Tonin. I n t h i s model t h e l i m i t a t i o n on t h e t r a n s v e r s e momenta comes from a l i m i t a t i o n on t h e momentum t r a n s f e r s between

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973144

t h e produced p a r t i c l e s . T h i s l i m i t a t i o n e n t a i l s a n o r d e r i n g i n t h e l o n g i t u d i n a l momenta, o r i n t h e r a - p i d i t i e s , s o t h a t t h e most probable c o n f i g u r a t i o n o f s e c o n d a r i e s i s o b t a i n e d when they a r e e q u a l l y spa- ced i n r a p i d i t y space. The model c a n accomodate a c o n s t a n t t o t a l c r o s s - s e c t i o n when a s u i t a b l e v a l u e f o r t h e "coupling c o n s t a n t " a t t h e v e r t i c e s i n cho- sen. N a t u r a l consequences a r e :

- L i m i t i n g fragmentation.

- C e n t r a l p l a t e a u .

- Short-range o r d e r coming from a f a c t o r i z e d in- c l u s i v e Pomeron.

- Weak c l u s t e r i n g ( f 2 w l o g s ) .

- P o i s s o n t a i l o f t h e prong c r o s s - s e c t i o n f o r v e r y high m u l t i p l i c i t y .

The model h a s good "crossed-channel'' p r o p e r t i e s , and t h e r e f o r e i s t h e n a t u r a l framework f o r a Mueller d e s c r i p t i o n of i n c l u s i v e d i s t r i b u t i o n s . However i t h a s poor u n i t a r i t y p r o p e r t i e s .

S e v e r a l t e s t s have been proposed t o d i s c r i m i n a t e between t h e s e two p i c t u r e s : t e s t s on m u l t i p l i c i t y d i s t r i b u t i o n s [8], behaviour w i t h t o t a l energy o f t h e two p a r t i c l e c o r r e l a t i o n f u n c t i o n , average mul- t i p l i c i t y f l u c t u a t i o n s i n a momentum space r e g i o n 0 e x c l u d i n g t h e forward and backward cones [9] .

The main c o n c l u s i o n s a r e :

- ^{CnO( n-2} i s excluded. However, t h e m u l t i p l i c i t y d i s t r i b u t i o n i s much broader t h a n Poisson.

- I n t h e p l a t e a u t h e two p a r t i c l e c o r r e l a t i o n f u n c t i o n , normalized t o t h e product of t h e one par- t i c l e i n c l u s i v e d i s t r i b u t i o n s , i s n e a r l y energy i n - dependent, i n agreement with a

?@

p i c t u r e , and i t depends mainly on t h e d i f f e r e n c e of t h e r a p i d i - t i e s Ay [101[111 .

- F i n a l l y , t h e r e a r e i n d i c a t i o n s i n f a v o u r o f s c a l i n g o f t h e l e a d i n g p a r t i c l e peak [12] . ^There

i s t h e r e f o r e a r e v i v a l o f t h e t r i p l e Pomeron v e r t e x .

However, t h i s mechanism produces a t t h e same t i m e a d i p i n t h e c e n t r a l p l a t e a u [141 , s i n c e t h i s r e - g i o n i n t h e FM i s mainly populated by s e c o n d a r i e s coming from heavy f i r e b a l l decays. T h e r e f o r e even t h i s a t t e m p t must be r e j e c t e d ; and t h e only way o u t i s t o abandon i s o t r o p i c decay of t h e b i g f i r e - b a l l s . With a n i s o t r o p i c f i r e b a l l decays t h e model c o u l d s t i l l d e s c r i b e d i f f r a c t i v e p r o d u c t i o n , b u t n o t t h e e n t i r e phenomenon. Once a n i s o t r o p i c f i r e b a l l s a r e c o n s i d e r e d , t h e r e a r e s t r o n g c o n n e c t i o n s w i t h m u l t i p e r i p h e r a l dynamics :

I s h a l l t h e r e f o r e postpone t h e d i s c u s s i o n of t h e s e m o d i f i c a t i o n s ; I s h a l l d i s c u s s them i n t h e frame- work of a g e n e r a l i z e d MP dynamics,

I n o r d e r t o e n l a r g e t h e concept o f multiperiphe- r i s m beyond t h e l a d d e r models, one must f i r s t con- s i d e r c a r e f u l l y what t h e c h a r a c t e r i s t i c f e a t u r e s of such a p i c t u r e a r e . I n my. o p i n i o n they a r e :

- ^A damping o f t h e momentum t r a n s f e r s between t h e produced s e c o n d a r i e s .

- The p o s s i b i l i t y o f g e t t i n g a r e c u r s i v e r e l a t i o n between e x c l u s i v e p r o d u c t i o n a m p l i t u d e s w i t h d i f f e - r e n t numbers o f f i n a l p a r t i c l e s .

Both t h e s e requirements c a n be met i f one imposes a s u i t a b l e upper bound o n t h e e x c l u s i v e p r o d u c t i o n amplitudes w i t h i s f a c t o r i z e d w i t h r e s p e c t t o sub- e n e r g i e s and momentum t r a n s f e r s [IS]. I n t h i s way -2 n o n - t r i v i a l p r o p e r t i e s l i k e l o n g r a n g e c o r r e l a t i o n s I n o r d e r t o modify t h e p r e d i c t i o n on

^N

n of 2

( f & ( l o g s )

)

and a b s o r p t i v e e f f e c t s c a n be acco- FM, t h e r e h a s been a n a t t e m p t t o i n t r o d u c e a damping 2

modated i n t o a g e n e r a l m u l t i p e r i p h e r a l e q u a t i o n o f t h e h i g h m u l t i p l i c i t i e s by i n t r o d u c i n g a f a c t o r

whose k e r n e l i s a p r i o r i unknown, b u t independent coming from t h e minimum ptomentum t r a n s f e r i n double

of t h e t o t a l energy and of t h e number of produced f i r e b a l l e x c i t a t i o n [13] . T h i s f a c t o r h a s t h e e f -

s e c o n d a r i e s . However, t h i s i s not a model, b u t r a - f e c t o f d e p r e s s i n g h i g h mass f i r e b a l l s , t h e r e b y

t h e r a d e f i n i t i o n of a c l a s s of models. T h e r e f o r e improving t h e m u l t i p l i c i t y curve :

l e t u s s e e i n more d e t a i l how t h e e f f e c t s we men- t i o n e d c a n be r e a l i z e d i n phenomenological models.

F i r s t o f a l l , one can i n t r o d u c e f u r t h e r c l u s t e r i n g e f f e c t s . The MPM a l r e a d y p r o v i d e s c l u s t e r i n g e f - f e c t s through secondary t r a j e c t o r i e s i n a Mueller Fip.2 Fm.3 d e s c r i p t i o n ; however, t h i s c a n o n l y be a p p l i e d when

t h e d i f f e r e n c e i n r a p i d i t y of t h e observed p a r t i c l e s

PHENOMENOLOGICAL MODELS OF MULTIPLE PRODUCTION C1-343

i s r a t h e r l a r g e . One c a n i n s t e a d i n t r o d u c e c l u s t e r s

"by hand", t r y i n g t h e r e b y t o f i t t h e observed c o r - r e l a t i o n s a t s m a l l v a l u e s of hy . T h i s approach h a s been pursued by s e v e r a l a u t h o r s [161[171 , t a k i n g more o r l e s s independent emission models f o r c l u s t e r p r o d u c t i o n , and assuming a n i s o t r o p i c Gaussian decay.

The c l u s t e r s we a r e h e r e c o n s i d e r i n g a r e q u i t e d i f - f e r e n t from Novas s i n c e t h e i r average mass i s n o t v e r y l a r g e , and t h e number of s e c o n d a r i e s they pro- duce i s l i m i t e d ( n

4

3 o r 4 ) . I n t h e s e approaches two f i n a l p a r t i c l e s a r e s h o r t range c o r r e l a t e d only i f they belong t o t h e same c l u s t e r . T h e r e f o r e a s t u - dy of c o r r e l a t i o n s g i v e s a n i d e a of t h e average c l u s - t e r s i z e . Although remarkable f i t s t o t h e d a t a , c a n be produced, I f i n d t h e s e models s a t h e r c r u d e , s i n c e t h e y n e g l e c t c o r r e l a t i o n e f f e c t s between d i f f e r e n t c l u s t e r s , a s w e l l a s " i n t e r f e r e n c e " e f f e c t s t h a t c a n be important i n a s i t u a t i o n where t h e m u l t i p l i c i t y of t h e produced p a r t i c l e s i s l a r g e .

More s p e c u l a t i v e m u l t i p e r i p h e r a l c l u s t e r s models have a l s o been c o n s i d e r e d [181 . The new f e a t u r e o f t h e s e approaches i s t h a t t h e c l u s t e r masses a r e a l - lowed t o i n c r e a s e with t o t a l energy. However, t h e damping of t h e momentum t r a n s f e r s , t o g e t h e r with t h e requirement of a n approximately c o n s t a n t t o t a l c r o s s - s e c t i o n , d e p r e s s e s s t r o n g l y t h e high mass c o n t r i b u - t i o n s . One c a n g e t a m u l t i p l i c i t y t a i l f a l l i n g a s a power of n , b u t t h e long range c o r r e l a t i o n s one i n t r o d u c e s "by hand" (each v e r t e x of t h e c h a i n "re- members" t h e t o t a l energy of t h e p r o c e s s ) a r e not i m p o r t a n t a s y m p t o t i c a l l y . I t i s not c l e a r a t t h e moment how such a c o n c l u s i o n could be modified by a l l o w i n g cT t o i n c r e a s e .

L e t u s now come t o a d i f f e r e n t Sype of c l u s t e r i n g e f f e c t , namely t h e d i f f r a c t i v e e x c i t a t i o n of a c l u s - t e r , followed by a n i s o t r o p i c decay. T h i s l e a d s us t o c o n s i d e r two components models [191, a p i o n i z a t i o n component d e s c r i b i n g p r o d u c t i o n without l a r g e r a p i - d i t y gaps between s e c o n d a r i e s , and a d i f f r a c t i v e component where a t l e a s t one l a r g e gap i s p r e s e n t . I n t h e s i m p l e s t v e r s i o n s of t h i s model, two d i f f e r e n t mechanisms a r e c o n s i d e r e d , w i t h s e p a r a t e average mul- t i p l i c i t i e s having d i f f e r e n t energy behaviours. I n p a r t i c u l a r t h e average m u l t i p l i c i t y coming from t h e d i f f r a c t i v e c o m p 0 n e n t . i ~ assumed t o be c o n s t a n t . I n t h i s way one can g e t good f i t s t o t h e m u l t i p l i c i t y d i s t r i b u t i o n [201, a s w e l l a s long range c o r r e l a t i o n s coming from t h e i n t e r p l a y of t h e two components :

b u t a d i p should e v e n t u a l l y develop i n t h e m u l t i p l i - c i t y d i s t r i b u t i o n , a s shown i n f i g u r e 5.

Fig.5

One c a n a l s o f i t t h e two p a r t i c l e c o r r e l a t i o n func- t i o n :

0- 0-

D D

7C

c 2 ( y 1 ' Y ~ ) = - ^{c2 (y1,y2) + 7 c;} ( Y , ' Y ~ ) +

"in i n

w i t h a mixture of s h o r t range and l o n g range c o n t r i - b u t i o n s [21] . Making a n e x p l i c i t model f o r t h e two components [ l l ] , one c a n e x t r a c t from t h e d a t a

dED(T)

and e s t i m a t e - ^, which t u r n s o u t t o be a n dCX(7)

i n c r e a s i n g f u n c t i o n of t h e c.m.s. r a p i d i t y , a s nay- v e l y expected. However, t h e s e p a r a t i o n between t h e two components i s somewhat model dependent, Moreover, i n o r d e r t o g e t long range c o r r e l a t i o n s , i t i s s u f - f i c i e n t t h a t thetwo m u l t i p l i c i t y d i s t r i b u t i o n s be- have d i f f e r e n t l y w i t h energy, b u t (n)D i s allowed a l s o t o i n c r e a s e [22 I. T h i s seems t o one a more na- t u r a l behaviour.

We s h a l l s e e now t h a t one c a n indeed c o n s t r u c t mo- d e l s i n which t h e d i f f r a c t i v e component by i t s e l f l e a d s t o a n average m u l t i p l i c i t y l o g a r i t h m i c a l l y i n c r e a s i n g with energy, and t o important long range c o r r e l a t i o n s . T h i s h a s been done on a r a t h e r phe- nomenological b a s i s i n r e f e r e n c e s [17 1, [23 1, where t h e d i f f r a c t i v e e x c i t a t i o n of a f i r e b a l l a c c o r d i n g t o t h e mass spectrum p(M)

N

- 1 i s assumed ; t h e

M

f i r e b a l l t h e n decays a n i s o t r o p i c a l l y i n t o a number of s e c o n d a r i e s i n c r e a s i n g l o g a r i t h m i c a l l y w i t h M :

T h i s new p o s s i b i l i t y becomes r a t h e r i n t e r e s t i n g i f

one r e l a t e s p r o d u c t i o n p r o c e s s e s w i t h a l a r g e r a p i -

d i t y gap t o a t r i p l e Pomeron graph, a c c o r d i n g t o t h e

Cl-344 A. BASSETTO

scheme :

I f one assumes a f a c t o r i z e d i n p u t Pomeron, one pre- d i c t s t h a t t h e average m u l t i p l i c i t y of hadrons pro- duced i n d i f f r a c t i o n d i s s s o c i a t i o n i n t o a s t a t e of mass M r i s e s l i n e a r l y with log(M 2

)

:

t h e c o e f f i c i e n t A being t h e same a s t h a t found i n t h e average m u l t i p l i c i t y of normal hadronic r e a c t i o n s [24 I.

The d i f f r a c t i v e component of t h e m u l t i p l i c i t y d i s - t r i b u t i o n i s f l a t f o r low n and c u t s o f f a t v a l u e n s A l o g s , a s shown i n f i g u r e 7 :

T h i s should be c o n t r a s t e d w i t h f i g u r e 5. T h i s mecha- nism, w i t h o u t a b s o r p t i v e c o r r e c t i o n s , c a n c o n t r i b u t e a l o g a r i t h m i c a l l y i n c r e a s i n g term t o t h e t o t a l c r o s s

s e c t i o n , provided t h e t r i p l e Pomeron v e r t e x does n o t v a n i s h a t t = 0 . T h i s w i l l be more e x t e n s i v e l y d i s - cussed i n M u e l l e r ' s r e p o r t . I s h a l l only mention h e r e t h a t i f t h e q u a n t i t y 2a'10 i s small (where a ' i s t h e Pomeron s l o p e and b i s t h e s l o p e of t h e to- t a l v e r t e x ) , t h e t o t a l c r o s s - s e c t i o n c o n t a i n s a term behaving l i k e l o g s , G b e i n g p r o p o r t i o n a l t o t h e

b

t r i p l e Pomeron coupling. T h i s term could perhaps e x p l a i n t h e observed r i s e i n t h e t o t a l c r o s s - s e c t i o n

h 5 1 . Of c o u r s e , t h i s has t o be c o n s i d e r e d a s a p e r t u r b a t i v e t r e a t m e n t , s i n c e t h e "input" Pomeron

d i f f e r s from t h e "output". E v e n t u a l l y a "renormaliaed"

Porneron t r a j e c t o r y i s o b t a i n e d w i t h a n i n t e r c e p t above one. Then one must invoke a b s o r p t i o n tn t h e d i r e c t channel i n o r d e r t o r e c o n c i l e t h i s c o n t r i b u - t i o n w i t h t h e F r o i s s a r t bound. A d e f i n i t e example which c o u l d a p p l y a t extremely high energy i s pro- vided by t h e F i n k e l s t e i n - Z a c h a r i a s e n model

[26

]. I n t h i s model t h e F r o i s s a r t bound i s s a t u r a t e d ; one g e t s s c a l i n g f o r t h e normalized one p a r t i c l e i n c l u - s i v e d i s t r i b u t i o n :

L E , , , c o n s t . ,

a dY

l a r g e f l u c t u a t i o n s :

a

f2 6 ( l o g s ) (12) and s c a l i n g f o r t h e l e a d i n g p a r t i c l e peak :

However, t h e model has problems w i t h "crossed chan- n e l " u n i t a r i t y . Moreover, i t c e r t a i n l y does not ap- p l y a t p r e s e n t e n e r g i e s , where t h e F r o i s s a r t bound i s f a r from being s a t u r a t e d . Also, no s i m p l e Mueller d e s c r i p t i o n i s p o s s i b l e , s i n c e f a c t o r i z a t i o n i s com- p l e t e l y l o s t .

F i n a l l y , I should j u s t mention a d i f f e r e n t possi- b i l i t y , t h a t i n which t h e t r i p l e Pomeron decouples a t t = 0 . One can apply Gribov's Reggeon c a l c u l u s t o e v a l u a t e t h e t r i p l e Pomeron c o n t r i b u t i o n t o t h e t o t a l c r o s s - s e c t i o n i n t h i s case. One T:?;~; , c o r ~ r - t i o n term

[ 2 7 ]

which behaves l i k e -

l o g

_-

s very h i g h energy.

E v e n t u a l l y t h e t o t a l c r o s s - s e c t i o n i s c o n s t a n t . Even i n t h i s c a s e a simple Mueller d e s c r i p t i q n i f n o t p o s s i b l e , due t o t h e complicated s i n g u l a r i t i e s which a r e generated.

I n c o n c l u s i o n , t h e t r i p l e Pomeron graph seems t o p l a y a n i m p o r t a n t r o l e b o t h f a r t h e t o t a l c r o s s -

s e c t i o n s and f o r i n c l u s i v e d i s t r i b u t i o n s . However,

no simple s e l f - c o n s i s t e n t p i c t u r e seems t o emerge

a t t h e moment, n e i t h e r from t h e o r y nor from t h e d a t a .

We a r e s t i l l i n a n i n t e r m e d i a t e energy r e g i o n where

new t r e n d s , l i k e t h e i n c r e a s e of t h e t o t a l c r o s s -

s e c t i o n , a r e appearing. I t i s hard t o s a y whether

t h e y w i l l p e r s i s t o r change a t h i g h e r e n e r g i e s , t h u s

s u p p o r t i n g one o r a n o t h e r t h e o r e t i c a l p i c t u r e .

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