APPROACH TO MULTIPARTICLE PRODUCTION

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APPROACH TO MULTIPARTICLE PRODUCTION

H. Satz

To cite this version:

H. Satz. APPROACH TO MULTIPARTICLE PRODUCTION. Journal de Physique Colloques, 1973, 34 (C1), pp.C1-149-C1-152. �10.1051/jphyscol:1973115�. �jpa-00215195�

APPROACH TO MULTIPARTICLE PRODUCTION

APPROACH TO MULTIPARTICLE PRODUCTION H. SATZ

Department of T h e o r e t i c a l P h y s i c s , U n i v e r s i t y of B i e l e f e l d , Germany

1. - INTRODUCTION.

-

.

of a n N p a r t i c l e s y s t e m i s g e n e r a l l y known, t h e l e v e l d e n s i t y i s g i v e n by t h e phase s p a c e volume of c a n o n i - c a l momenta and c o o r d i n a t e s , s u b j e c t t o H N = c o n s t . I n m u l t i h a d r o n p h y s i c s , t h e H a m i l t o n i a n i s n o t known, and hence t h e i n t r o d u c t i o n of dynamical f e a t u r e s e i - t h e r r e q u i r e s a d i f f e r e n t method of l e v e l d e n s i t y c a l c u l a t i o n o r a m o d i f i c a t i o n uf t h e p h a s e s p a c e mea- s u r e . Only t h e s t a t i s t i c a l model of Fermi [ I ] r e m a i n s w i t h i n c o n v e n t i o n a l s t a t i s t i c a l mechanics : i t pro- p o s e s a s t h e r e l e v a n t l e v e l d e n s i t y t h a t of a f r e e h a d r o n g a s i n s i d e an i n t e r a c t i o n ' ' volume V of t h e d i m e n s i o n s of t h e r a n g e of h a d r o n i c f o r c e s . Such a n a p p r o a c h , however, e n c o u n t e r s s e r i o u s e x p e r i m e n t a l d i f f i c u l t i e s ( j e t s t r u c t u r e , r e s o n a n c e f o r m a t i o n ) a s w e l l a s t h e o r e t i c a l o b j e c t i o n s [ 2 1, s o t h a t d y n a m i c a l m o d i f i c a t i o n s a p p e a r u n a v o i d a b l e .

Such m o d i f i c a t i o n s c a n be a c h i e v e d i n two d i s t i n c t ways. I n a p u r e l y s t a t i s t i c a l a p p r o a c h , one a t t e m p t s t o r e t a i n a l l dynamics1 f e a t u r e s w i t h i n t h e s t a t i s - t i c a l d e s c r i p t i o n

-

Vle have a l r e a d y e n c o u n t e r e d two methods of l e v e l d e n s i t y c a l c u l a t i o n : t h e c a n o n i c a l p h a s e s p a c e ap- p r o a c h of s t a t i s t i c a l mechanics and t h e momentum spa- c e r e s t r i c t i o n [d3p

-,

j e t model. A t h i r d , r a t h e r d i f f e r e n t approach t o t h e problem i s p r o v i d e d by t h e s t a t i s t i c a l b o o t s t r a p con- d i t i o n of Hagedorn [ 6 ] ; i t w i l l be one of t h e main t o p i c s of t h i s s u r v e y . Knowing t h e mass s p e c t r u m pin(m) o f produced hadrons ( s t a b l e p a r t i c l e s and whatever r e s o n a n c e s a r e f o u n d ) , t h e l e v e l d e n s i t y poUt(M) o f a n e x c i t e d h a d r o n i c s y s t e m of mass M i s o b t a i n e d by c o n f i g u r a t i o n c o u n t i n g

where P2 = ^M 2 and V d e n o t e s t h e common h a d r o n i c i n t e r a c t i o n volume. I f we c o n s i d e r a d i s c r e t e mass s p e c t r u m w i t h one t y p e of p a r t i c l e of mass p

t h e n ( 1 ) simply r e d u c e s t o t h e l e v e l d e n s i t y of a f r e e hadron g a s i n s i d e Vo

.

o r , i n o t h e r words, t h a t e x c i t e d h a d r o n i c s y s t e m s ( f i r e b a l l s " ) be made up of o t h e r s y s t e m s o b e y i n g t h e same c o m p o s i t i o n laws. E q u a t i o n s ( 2 ) and ( 3 ) t o g e - t h e r form t h e s t a t i s t i c a l b o o t s t r a p c o n d i t i o n , de- t e r m i n i n g a t t h e same t i m e t h e l e v e l d e n s i t y a n d t h e mass spectrum.

I n t h e r e m a i n d e r , I s h a l l c o n s i d e r two a r e a s of i n v e s t i g a t i o n : t h e s o l u t i o n of t h e b o o t s t r a p equa- t i o n and t h e q u e s t i o n o f t h e p h y s i c a l c o n t e n t of t h e b o o t s t r a p approach.

2.- THE SOLUTION O F THE BOOTSTRAP EQUATION. - To b e g i n w i t h , we have t o s t a t e ( 3 ) more p r e c i s e l y . I n t h e o r i g i n a l f o r m u l a t i o n [ 6 ] , o n l y t h e l e a d i n g expo- n e n t i a l t e r m s of p. and pout were r e q u i r e d t o con-

~ n

v e r g e a s M -t m ("thermodynamical b o o t s t r a p t ' ) , b u t

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973115

f o r a c t u a l s o l u t i o n s [11 1, one g e n e r a l l y a l l o w e d o n l y power law d i f f e r e n c e s between spectrum and l e v e l den- s i t y ("weak b o o t s t r a p " ) . A t p r e s e n t one a l m o s t always c o n s i d e r s t h e " s t r o n g b o o t s t r a p " of F r a u t s c h i [ 1 2 ]

r e q u i r i n g f u l l a s y m p t o t i c e q u i v a l e n c e . L e t u s w r i t e down t h e c o r r e s p o n d i n g b o o t s t r a p e q u a t i o n i n cova- r i a n t form [12,13]

where B d e n o t e s t h e i n t e r a c t i o n c o n s t a n t ( e . g . B =

2mV 1, and t h e f i r s t t e r m on t h e r . h . s . i n c l u d e s t h e d i s c r e t e p a r t of t h e spectrum. E q u a t i o n ( 5 ) d e t e r m i - n e s t h e l e v e l d e n s i t y and spectrum i n a world of one k i n d of s t a b l e p a r t i c l e s , o b e y i n g Boltzmann s t a t i s - t i c s .

T o s o l v e t h i s b o o t s t r a p e q u a t i o n , we L a p l a c e t r a n s - form 7(p2) t o g e t t h e p a f t i t i o n f u n c t i o n

Z ( b ) r / d 4 P e-@ .(p2) ; & > O , P ~ = P ~ - ~ > o

( 6 ) which t h r o u g h ( 5 ) y i e l d s f o r t h e s i n g l e p a r t i c l e d i s - t r i b u t i o n

t h e r e l a t i o n

I n consequence, doe) a s a f u n c t i o n o f Z(P) h a s a ma- ximum a t BZ( Po) = l n 2

,

where P i s t h e v a l u e of t h e o n l y open p a r a m e t e r P

a t t h e maximum. The p a r t i t i o n f u n c t i o n z ( P ) c a n b e shown [14] t o have a s q u a r e - r o o t branch p o i n t a t

P = Po ^:

z ( P ) N ~ ~ + c o n s t . W , ( 1 0 )

which y i e l d s upon i n v e r s i o n of t h e L a p l a c e t r a n s f o r m 2(>f2)

=

,

0 0

t h r o u g h (7) and ( 9 ) . The r e s u l t i n g l e v e l d e n s i t y t h u s i n c r e a s e s a s a l i n e a r e x p o n e n t i a l i n M , which f o r c e s

~ ( 0 ) t o become s i n g u l a r f o r P < Po

.

of s o l v i n g t h e b o o t s t r a p c o n d i t i o n , now r e p l a c i n g p r e v i o u s i t e r a t i v e methods [ I 1 1.

An a l t e r n a t i v e and, a s we s h a l l s e e , p h y s i c a l l y r a t h e r i l l u m i n a t i n g way [ 1 3 ] t o s o l v e ( 5 ) is by ex- p a n s i o n i n terms of s i m p l e phase s p a c e

Because of ( 5 ) , t h e c o e f f i c i e n t s g~ must obey t h e r e c u r s i o n r e l a t i o n s [ I 5 ]

N -1

p N = ~ [ ? ~ k g k g N - k - ( N - l ) g N - L

1

LC= 1 g1 = 1

which have t h e a s y m p t o t i c s o l u t i o n [13,16]

The c r i t i c a l f e a t u r e t o n o t e h e r e i s t h e r a t h e r weak N dependence of (141, t o be c o n t r a s t e d w i t h t h e 1/N!

p r e s e n t i n t h e c o r r e s p o n d i n g l e v e l - d e n s i t y form of a f r e e g a s ( c f . e q s . l , 2 ) . The l a t t e r t h u s h a s f a r f e - wer l e v e l s t h a n t h e b o o t s t r a p s o l u t i o n , which c a n a l s o be s e e n d i r e c t l y by comparing ( 1 1 ) w i t h t h e a- s.ymptotic l e v e l d e n s i t y of a f r e e g a s [17]

which does n o t i n c r e a s e a s a l i n e a r e x p o n e n t i a l . Be- f o r e a s k i n g f o r t h e p h y s i c a l r e a s o n f o r t h i s i n c r e a - s e i n l e v e l s , l e t u s r e c a l l i t s most i m p o r t a n t con- s e q u e n c e : t h e spectrum i n energy p of a p a r t i c l e e m i t t e d from a system w i t h a l i n e a r e x p o n e n t i a l l e - v e l d e n s i t y eCM i s i n t h e s t a t i s t i c a l approach g i v e n

2 2

w i t h P = M and p d e n o t i n g t h e momentum of t h e secon- d a r y . Hence t h e a v e r a g e e n e r g y p e r s e c o n d a r y becomes a s y m p t o t i c a l l y c o n s t a n t : a h i g h e r i n i t i a l energy M of t h e e x c i t e d system r e s u l t s o n l y i n more seconda- r i e s , n o t f a s t e r ones.

I n c l o s i n g t h i s s e c t i o n , l e t me j u s t n o t e t h a t , i n s p i t e of i n t e r e s t i n g p r e l i m i n a r y work [14,18], t h e problem of f o r m u l a t i n g and a s y m p t o t i c a l l y s o l - v i n g t h e b o o t s t r a p e q u a t i o n f o r p a r t i c l e s w i t h c o r - r e c t s t a t i s t i c s and f u l l i n t e r n a l quantum numbers ( i s o s p i n , h y p e r c h a r g e ) under t h e a b s e n c e of e x o t i c f i r e b a l l s s t i l l remains unsolved. P e r h a p s a good t o o l i n i n v e s t i g a t i o n s o f t h i s q u e s t i o n i s t h e li- n e a r - c h a i n b o o t s t r a p [ 1 9 ] , which r e s u l t s i n a sim- p l e r s i n g u l a r i t y s t r u c t u r e t h a n ( 5 ) , b u t s t i l l con- t a i n s a l l e s s e n t i a l f e a t u r e s of t h e approach.

kPPROACH TO MULTIPARTICLE PROWCTION C1-151

3.

-

T h e q u e s t i o n I want t o s t u d y h e r e i s what dynamical f e a t u r e s l e a d t o t h e l e v e l d e n s i t y o b t a i n e d t h r o u g h t h e b o o t s t r a p e q u a t i o n .

Almost immediately a f t e r t h e p r o p o s a l of t h e F e r m i model

N

6

( - ) ,

N=L

Pomeranchuk [ 2 ] n o t e d t h a t t h e u s e of a volume V

=

Vowl/p 3 of t h e s i z e of t h e n u c l e a r f o r c e r a n g e ~ / J J

t o g e t h e r w i t h a f r e e g a s p i c t u r e i s d i f f i c u l t t o j u s - t i f y . He proposed i n s t e a d a n expanding f i r e b a l l p i c - t u r e : h a d r o n i c m a t t e r , i n i t i a l l y c o n c e n t r a t e d i n a volume Vo , c o n t i n u e s t o expand u n t i l a l l c o n s t i t u - e n t s a r e s e p a r a t e d by a d i s t a n c e l / y and hence no l o n g e r i n t e r a c t hadronicarlly. A t t h i s p o i n t , one may a p p l y a f r e e - g a s p i c t u r e . The Pomeranchuk model t h e - r e f o r e s e t s :

which upon i n s e r t i o n i n t h e c o v a r i a n t v e r s i o n of ( 1 6 ) g i v e s p r e c i s e l y t h e s o l u t i o n of t h e b o o t s t r a p c o n d i - t i o n

M/To

%omeranchuk(M)

=

.

T h e c o n s t a n t a n d u n i v e r s a l a v e r a g e energy p e r secon- d a r y , r e s u l t i n g from t h e l i n e a r l y e x p o n e n t i a l l e v e l d e g e n e r a c y , h e r e i s e a s y t o u n d e r s t a n d : no m a t t e r what t h e i n i t i a l e n e r g y of t h e s y s t e m , i t expands u n t i l a l l i t s c o n s t i t u e n t s a r e a d i s t a n c e of 1 / p a - p a r t . The u n i v e r s a l r a n g e of s t r o n g i n t e r a c t i o n s t h u s l e a d s t o a u n i v e r s a l energy d e n s i t y a t t h e " f r e e g a s t h r e s h o l d " .

An a l t e r n a t i v e dynamical approach t o t h e b o o t s t r a p s o l u t i o n i s o b t a i n e d t h r o u g h t h e i n t r o d u c t i o n of p r e - s e n t motions of r e s o n a n c e s t r u c t u r e [20]. The Fermi model i m p l i e s a n e q u i d i s t r i b u t i o n n o t o n l y i n t h e i n -

d i v i d u a l CMS e n e r g i e s of a l l s e c o n d a r i e s , b u t a l s o i n t e r m s of t h e c l u s t e r e n e r g i e s M. ( c f . F i g . 1 )

Pn

M Mn-1 Mn-2 M 2 m

F i g . l

h he o v e r a l l p i c t u r e p r e s e n t e d h e r e was d e v e l o p e d j o i n t l y by M . I . G o r e n s t e i n , V.A. Miransky, V.P. She- l e s t , G.M. Z i n o v j e v and t h e p r e s e n t a u t h o r ; i t w i l l soon a p p e a r i n more d e t a i l a s a U n i v e r s i t y of B i e l e - f e l d p r e p r i n t .

T h i s i s e a s i l y s e e n by r e w r i t i n g t h e c o v a r i a n t form of f r e e momentum s p a c e i n t e r m s of c l u s t e r v a r i a b l e s

I n c o n t r a s t t o t h e e q u i d i s t r i b u t i o n i n M . a s i m p l i e d by (2?), we p r e s e n t l y s u p p o s e r e s o n a n c e d i s t r i b u - t i o n s t o be governed by Regge t r a j e c t o r i e s l i n e a r i n M . 2 ₁ = S _{i '}

-

L e t u s t h e r e f o r e m o d i f y c o n v e n t i o n a l phase s p a c e a c c o r - d i n g l y and c a l c u l a t e t h e l e v e l d e n s i t y i n l 1 ~ e g g e space" (B = a ' )

A s y m p t o t i c a l l y , t h i s g i v e s [ 2 0 ] w i t h z (M 2 ) cx c o n s t . M

Regge

j u s t t h e b o o t s t r a p r e s u l t . So we c a n now e a s i l y a n s - wer t h e q u e s t i o n of where t h e a d d i t i o n a l s t a t e s i n t h e b o o t s t r a p a p p r o a c h come from : t h e y a r i s e i f we no l o n g e r e q u i d i s t r i b u t e r e s o n a n t s t a t e s i n M , b u t i n s t e a d a c c o r d i n g t o Regge t r a j e c t o r i e s l i n e a r i n M'

.

t o a f r e e g a s i n Regge space. I t s h o u l d be emphasi- z e d t h a t o n l y t r a j e c t o r i e s l i n e a r i n S g i v e t h e de- s i r e d b e h a v i o u r , s o t h a t i n a s t a t i s t i c a l approach t h e b o o t s t r a p c o n d i t i o n c a n be t h o u g h t of a s a way t o e n f o r c e t h e c o n v e n t i o n a l Regge r e s o n a n c e d i s t r i - b u t i o n .

We t h u s s e e t h a t t h e e x p o n e n t i a l l e v e l d e g e n e r a c y c a n be o b t a i n e d e i t h e r by t h e b o o t s t r a p c o n d i t i o n , o r t h r o u g h t h e Pomeranchuk model, o r by a f r e e g a s p i c t u r e i n Regge s p a c e . I t s t i l l a r i s e s even i f one i n t r o d u c e s t h e e x p a n d i n g volume of t h e Pomeranchuk approach i n t h e b o o t s t r a p c o n d i t i o n [ 2 1 ]

.

The f i n a l q u e s t i o n I now want t o c o n s i d e r i s t o what e x t e n t one can e x t r a c t o r d e r i v e a s t a t i s t i c a l p i c t u r e of t h i s k i n d from a g i v e n dynamical model.

T h e d u a l r e s o n a n c e model on t h e one hand r e p r e s e n t s t h e most comprehensive dynamical scheme proposed s o f a r , on t h e o t h e r i t a l s o l e a d s t o a l i n e a r exponen- t i a l i n c r e a s e i n t h e number of s t a t e s b e l o n g i n g t o a g i v e n mass [ 2 2 ] ; t h u s i t a p p e a r s a s t h e i d e a l can- d i d a t e .

C o n s i d e r t h e n i n t h e d u a l r e s o n a n c e model t h e decay of a n e x c i t e d s t a t e M i n t o a s i n g l e p a r t i c l e c p l u s a n y t h i n g [ 2 3 ] ; d e n o t e t h e d e g e n e r a t e s t a t e s of M by

C1-152 H. SATZ

a n d t h e momentum of c by p ( c f . F i g . 2 ) .

F i g . 2

A v e r a g i n g o v e r a l l p o s s i b l e d e g e n e r a t e s t a t e s of M t " s t a t i s t i c a 1 a v e r a g i n g " ) , we h a v e f o r t h e d e c a y s p e c t r u m

F(M,p) =

L

w i t h V ( p ) d e n o t i n g t h e decay v e r t e x . For M + and po/M -' 0 , c a l c u l a t i o n s g i v e [ 2 3 ]

-Po

F(M,p) _N e 9 ( 2 7 )

and h e n c e t h e r e s u l t ( 1 6 ) of t h e b o o t s t r a p approach.

On t h e o t h e r hand, i t i s n o t c l e a r i f a n d where s u c h a s t a t i s t i c a l a v e r a g i n g p r o c e d u r e i s p e r m i s s i b l e . I n t h e f o r m a t i o n o f t h e f i r e b a l l M

,

t i c a l d e s c r i p t i o n of t h e b o o t s t r a p t y p e f r o m t h e d u a l r e s o n a n c e model, i t is n o t a t a l l c l e a r i f t h i s d e s c r i p t i o n p r o v i d e s anywhere a good a p p r o x i m a t i o n t o d u a l dynamics - o r t o n a t u r e .

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[ 4 ] LANDAU (L.D. 1, I z v e s t . Akad. Nauk SSR, S e s . F i z . , 1 7 ( 1 9 5 3 ) 51.

[5] cf.;. hlIESOWICZ (M. ), h he Problem o f F i r e - b a i l s " , Cracow p r e p r i n t 1973, and r e f e r e n c e s t h e r e .

[ 6 ] HAGEWRN (R. ), Nuovo Cimento S u p p l . 3 ( 1 9 6 5 ) 147.

[ 7 ] RANIT (G.) and RAIU'FT ( J . ) , P h y s . L e t t . 32B

(1970) 207.

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[ l o ] JACOB (M. ) and SIANSKY (R. 1, P h y s . Rev. 05 ( 1 9 7 2 ) 1847.

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[ 1 2 ] FRAUTSCHI ( S . ) , Phys. Rev. ( 1 9 7 1 ) 2821.

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