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Submitted on 1 Jan 1973
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PARTICLE PRODUCTION WHEN CROSS SECTIONS GROW
F. Zachariasen
To cite this version:
F. Zachariasen. PARTICLE PRODUCTION WHEN CROSS SECTIONS GROW. Journal de
Physique Colloques, 1973, 34 (C1), pp.C1-379-C1-381. �10.1051/jphyscol:1973152�. �jpa-00215232�
PARTICLE PRODUCTION WHEN CROSS-SECTIONS GROW
PARTICLE PRODUCTION WHEN CROSS SECTIONS GROW
F. ZACHARIASEN
C a l i f o r n i a I n s t i t u t e o f T e c h o l o q y
I f t h e t o t a l c r o s s s e c t i o n g r o w s , w h a t can we s a y I f we c o m b i n e t h i s t y p e o f a " m u l t i p e r i p h e r a l "
a b o u t p a r t i c l e n r o d u c i i o n ? O f a model i n d e p e n d e n t component w i t h a n o r m a l d i f f r a c t i v e o n e , it i s e a s i - n a t u r e , n o t h i n g . Rut i f we a r e w i l l i n g t o make some l y s e e n t h a t
g u e s s e s a b o u t t h e n a t u r e o f t h e o a r t i a l c r o s s s e c t i o n s
a ( s l t o p r o d u c e n p a r t i c l e s , we can a t l e a s t i t e m i z e
u
+ bfl % ( l o g s l v t h e v a r i o u s p o s s i b i l i t i e s .V + l I t h a s become f a s h i o n a b l e
-
and t h e f a s h i o n even < n > + <n> M a [ l o ? 91.
h a s some e x p e r i m e n t a l s u p n o r t
-
t o a s s u m e t h a t 0 h a s t w o comoonen t s , a " m u l t i p e r i o h e r a l " o r s h o r tM D
r a n q e c o r r e l a t i o n o n e , an 2nd a d i f f r a c t i v e o n e a n . L e t u s c o n s i d e r t h e t w o a l t e r n a t i v e s t h a t t h e q r o w t h i n a i s d u e t o aM = T an. ly o r t o bD = 7 0.: ( T h e r e i s , o f c o u r s e , t h e t h i r d n o s s i b i l i t y ? h a t b o t h q r o w l . n
i l F i r s t s u p p o s e t h e " n u l t i p w i n h e r a l " p a r t o f t h e a m n l i t u d e i s r e s o o n s i b l e f o r t h e e r o w t h , a n d t h a t a fl a (lee s l V , 2 >_
v
> 0 .Assuming t h e M u e l l e r f o r m a l i s m a p p l i e s t o t h i s comnonent , we s e e t h a t i n t h e f r a g m e n t a t i o n r e q i o n s t h e o n e p a r t i c l e i n c l u s i v e c r o s s s e c t i o n s b e h a v e l i k e
a s w e l l , w h i l e i n t h e c e n t r a l r e g i o n
However, c o r r e l a t i o n s can become l o n g r a n g e . F o r
i f aM + m w h i l e a D + c o n s t a n t . In a p u r e P o i s s o n d i s t r i b u t i o n , f; = 0 , a n d i n g e n e r a l t h e v a r i o u s moments o f t h e d i f f r a c t i v e d i s t r i b u t i o n a r e c c n s t m t . Hen c e
f:, +
- u
<n>D ( l o g s l 0s o t h e r e a r e l o n g r a n g e c o r r e l a t i o n s . I f t h e m u l t i p e - r i p h e r a l comnonen t i s n o t e x a c t l y P o i s s o n d i s t r i b u t e d , t h e n f 2 + f 2 . N
B e f o r e t h e s e p r e d i c t i o n s a r e t a k e n s e r i o u s l y , howe- v e r , m e s h o u l d r e c a l l t h a t m u l t i p e i - i p h e r a l m o d e l s d o n o t l i k e t o p r o d u c e c u t s o r h i g h o r d e r p o l e s a s T h u s ( l / a T l d a / d v a c o n s t . i n t h e f r a q m e n t a t i o n t h e i r l e a d i n g s i n q u l a r i t i e s . I n d e e d , o n e c a n show r e q i o n s , a n d % [ l o g s l v % Y' i n t h e c m t r a l r e g i o n . t h a t m d e r r a t h e r ~ w e r a l c i r c u m s t a n c e s , t h e m u l t i p e - Hence t h e h e i e h t o f t h e c e n t r a l o l a t e a u g r o w s w i t h r i p h e r a l e q u a t i o n w i t h a p o l e a s i n p u t m u s t h a v e a e n e r q y , w h i l e t h e e n d r e ~ i o n s o f t h e d i s t r i b u t i o n r o n - d e q e n e r a t e s i m p l e p o l e a s i t s o u t p u t , so t h a t we
M
s t a y c o n s t a n t i n s h a o e . c m n o t h a v e
a
( s f ( l o g s l V w i t hv
> O . T h i s r e s u l t TEviden t l y , t h e n f a i l s , o f c o u r s e , i n b o o t s t r a p o e d m u i t i p e r i p h e r a l
m o d e l s , b e c a u s e t h e m u l t i p e r i p h e r a l i n t e g r a l e q u a t i o n
s o t h a t t h e m u l t i p l i c i t y becomes
i s t h e n n o l o n g e r l i n e a r , s o t h a t i n p r i n c i p l e a n y r e s u l t c o u l d b e o b t a i n e d . N e v e r t h e l e s s , g r o w i n g c r o s s s e c t i o n s a r e h a r d t o o b t a i n , s o some s k e p t i c i s m a b o u t a l l t h i s i s j u s t i f i e d .
i i l Next s u p o o s e t h e d i f f r a c t i v e p a r t o f t h e am-
0 V
p l i t u d e y i e l d s t h e q r o w t h i n a,.: a s s u m e UT Q ( l o g 5 1 , O < v 1 2 .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973152
C1-380 F
.
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c u t o f f r e l a t i v e l y s h a r p l y a t c a s e we h a v e some v a l u e n"
N . Then t h e r e a r e t w o n a t u r a l s o u r c e sf o r a e r o w t h i n UT: e i t h e r D 0: i s c o n s t a n t w i t h e n e r g y
-?.
d o / d y % I / L [ s l uw h i l e N g r o w s , o r a l l a: grow w i t h e n e r e y w h i l e N i s
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.
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a
1 d a / d y % [ l o g S I ~ / L [ S I .A q a i n , s c h e m a t i c a l l y , we see t h a t i n c a s e o n e 0 -1 dU/dy can b e d e o i c t e d by
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i f L i s c o n s t a n t , w h e r e a s i f L g r o w s , i t l o o k a s f o l l o w s .
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Now, e v i d e n t l y , i n t h e f i r s t c a s e D 0
<n>'
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n c o /a 5 c o n s t a n t , n Tw h i l e i n t h e s e c o n d
<n> D % \ . l o 4 s l v
.
a n d byWe can a l s o e s t i m a t e t h e i n c l u s i v e d i s t r i b u t i o n s d a / d y i n t h e two c a s e s . L e t u s s u o p o s e t h a t , d u e t o t h e d i f f r a c t i v e mechanism, p a r t i c l e s a r e p r o d u c e d w h i t h i n d i s t m c e s L [ s l i n r a p i d i t y f r o m t h e t w o e n d s
a t
+
Y/2 : ( i . e . d n / d y e x i s t s o n l y f o r Y/2 > y > r e s p o c t i v e l yY/2
-
L, a n d -Y/2 < y < -Y/2 + L. Then i q t h e f i r s t We may c o n s t r u c t t h e f o l l o w i n p , s i m p l e t a b l e t o sum- m a r i z e t h e s i t u a t i o n w i t h r e e a r d t o t h e d i f f r a c t i o nn
'n N OT <n > El
2
d a / d vu
c a s e ? (lee s l V c o n s t . [ l o g s l v c o n s t . I / L \ . s l con s t . c a s e 2 c o n s t .
c lo^
s l V [ l o g s l V [ l o g 5)" 1 1 0 4 SI"/LIS) [lee s l Z VWhw s u c h a d i f f r a c t i v e component i s mixed w i t h a
<n> + <n>
n o r m a l m u l t i p e r i o h e r a l o n e , we o f c o u r s e h a v e D +
%
<">MU + OD % ( l o g s l v . The h i g h e r moments c a n , however, s o t h a t i n c a s e o n e , w h e r e <n>D + c o n s t a n t , we h a v e I
-v
b e more c o m p l i c a t e d . T h u s i f + a n d crN -t c o n s t a r i t , <n> + c o n s t a n t + ( l o g s l w h i c h c a n grow i f
v
< 1 .PARTICLE PRODUCTION W E N CROSS-SECTIONS GROW C1-381
In c a s e two, in c o n t r a s t , we have <n> + [ l o ? s l V Hence in c a s e one, t h e succeeding terms behave l i k e I -V
+ ( I c e S I
.
which c e r t a i n l y prows. For t h e twoa
-v 2-2vo a r t i c l e c o r r e l a t i o n s , s i m i l a r f e a t u r e s hold. f + c o n s t . + c l c e S I + ( l o r s l l - " + (102 S I
2
Thus s o we may have f 2 -t
-
i f v < 2, and in c a s e twof, -+ ( l o t ! s l Z V + ( l o g S I = - ~ + [lo!? s l + ( l o g s l 2-2v which c e r t a i n l y qrows.