MODELS FOR RISING CROSS-SECTIONS

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MODELS FOR RISING CROSS-SECTIONS

L. Caneschi, M. Ciafaloni

To cite this version:

L. Caneschi, M. Ciafaloni. MODELS FOR RISING CROSS-SECTIONS. Journal de Physique Collo- ques, 1973, 34 (C1), pp.C1-268-C1-273. �10.1051/jphyscol:1973131�. �jpa-00215211�

L. CANESCHI and M. CIAFALONI

MODELS FOR RISING CROSS-SECTIONS

L . CANESCHI and M. CIAFALONI CERN-GENEVA

1.- INTRODUCTION.- A number of i n t e r p r e t a t i o n s of t h e observed r i s e o f CT i n t h e ISR energy range have been proposed. We w i l l b r i e f l y review t h e gene- r a l l i n e s of some of them h e r e , i n d i c a t i n g which f e a - t u r e s of t h e d a t a f i t n a t u r a l l y i n each scheme and which do n o t . I t i s convenient t o c l a s s i f y t h e propo-

sed models i n t h r e e c l a s s e s :

a ) Models t h a t emphasize t h e r o l e o f t-channel an7 g u l a r momentum and u n i t a r i t y .

b) Models t h a t emphasize t h e s-channel a n g u l a r mo- mentum (impact parameter) and u n i t a r i t y .

C ) Models t h a t connect t h e r i s e of a t o t h e v a s t i n c r e a s e observed a t ISR of t h e r a t e of r a r e proces- s e s , l i k e l a r g e p T [ l ] and p production [ 2 1.

The m e r i t s of t h i s t h i r d group a r e e s s e n t i a l l y phe- nomenological, i n t h a t they simply r e l a t e o b s e r v a b l e q u a n t i t i e s and g i v e a j u s t i f i c a t i o n , even i f a poste- r i o r i , of t h e l a r g e energy s c a l e f o r t h e r i s e . I n t h i s t a l k , however, we s h a l l c o n c e n t r a t e on t h e f i r s t two c l a s s e s .

I t i s u s e f u l t o c o n s i d e r , t o g e t h e r w i t h uT , a number of o t h e r q u a n t i t i e s t h a t g i v e a more d e t a i l e d information.

1 ) ar 2 ; 1 I m Aep ( S ,0)

.

2) From baryon number c o n s e r v a t i o n ,

c o n s i d e r t h e b a r y o n i n c l u s i v e s p e c t r a . I n p a r t i c u l a r d o (B)

t h e l i m -7 , connected w i t h t h e p r o p e r t i e s o f x-11 dxdp*

t h e t r i p l e - R e g g e c o u p l i n g s , h a s important b e a r i n g on t h e behaviour of uT . E s p e c i a l l y c r i t i c a l i s t h e i s s u e of whether t h e triple-Pomeron v e r t e x v a n i s h e s a t t=O

.

3) From d i s p e r s i o n r e l a t i o n s t h e v a l u e of Re A e p ( s , t ) a t a given s i s s e n s i t i v e t o 0 7 ( s ) a l s o f o r s > s

.

4) Since u T ( s ) =

1

The p r e d i c t i o n s of t h e v a r i o u s models f o r t h e s e q u a n t i t i e s a r e summarized i n t h e Table I and i l l u s - t r a t e d i n t h e following.

2.- t-CHANNEL ORIENTED MODELS.- The s c a t t e r i n g am- p l i t u d e A ( s , t ) h a s t o s a t i s f y b o t h s a n d t - c h a n n e l u n i t a r i t y i n t h e r e l e v a n t p h y s i c a l r e g i o n s . I n t h e l a c k of a r e a l i s t i c model t h a t e x p l i c i t l y s a t i s f i e s b o t h , what i s u s u a l l y done i s t o t a k e one of t h e two requirements a s t h e s t a r t i n g p o i n t , and make s u r e a p o s t e r i o r i t h a t t h e o t h e r i s n o t obviously v i o l a t e d . I f one s t a r t s from t - c h a n n e l u n i t a r i t y , t h e n a t u r a l v a r i a b l e i s t h e t-channel a n g u l a r momentum J

.

allowed J - p l a n e s i n g u l a r i t i e s a r e known t o be moving p o l e s and " s o f t " c u t s . Adding t h e F r o i s s a r t bound a s a n e x t e r n a l c o n s t r a i n t , and t h e n o t i o n t h a t i f u r i s e s i t i s probably d i f f e r e n t from z e r o asymptoti- c a l l y , t h e l e a d i n g s i n g u l a r i t i e s a t t = O a r e a moving p o l e P a t J = 1 and a s o f t c u t , r e l a t e d t o P i n a n i t e r a t i v e way and t h e r e f o r e a l s o a t J = 1

( i n t h i s s i t u a t i o n " s o f t " means w i t h f i n i t e , n o t ne- c e s s a r i l y v a n i s h i n g d i s c o n t i n u i t y a t t h e t i p ) . Hence t h e asymptotic behavior of o i s g i v e n by

While t h e s i g n of t h e c u t d i s c o n t i n u i t y v a r i e s w i t h t h e model, i t seems agreed t h a t ~ ~ ( 0 ) i s t h e squared r e s i d u e of t h e f i x e d p o l e i n t h e P-p am- p l i t u d e [3 ] [4 ], i .e . t h e p r o p e r a n a l y t i c c o n t i n u a -

t i o n of t h e e x ~ r e s s i o n

S t a r t i n g from t h i s formula one can d i s t i n g u i s h a low- energy c o n t r i b u t i o n y,(o) , and a triple-Pomeron c o n t r i b u t i o n (-gb(o)=

:

t-0

(%) The n e g a t i v e s i g n comes from e v a l u a t i n g t h e d i - v e r g e n t i n t e g r a l (2) u s i n g superconvergence f o r t h e Regge-pole p a r t . T h i s formula, l i k e athers i n . t h e f o l l o w i n g a r e symbolic remainders of more complica- t e d e x p r e s s i o n s .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973131

MODELS FOR RISING CROSS-SECTIONS

TABLE I

COMPARATIVE SUMARY OF THE MODELS CONSIDERED

-

finite only if the triple-Pomeron coupling gp(o,O,O) = 0

.

Model

Physics

Asymptotic a

Rise of u at ISR

% A/Im A

E

L "e e A

S Shall T T

I Large T C

Triple Pomeron Coup 1 ing

do B

- dx

cr

Inclusive

problemsm

2.1.- POSITIVE CUT.- This sign occurs in the lad- complex (non leading) poles in J [ 5 ] . The threshold der multiperipheral model (AFS cut), and generates idea can be applied to the large rapidity gap con- a total cross-section that decreases to a constant text [6] or to other channels, e.g.

;

asymptotically. A temporary rise of aT can be ob- Phenomenologically this scheme faces difficulties tained by the introduction of a high threshold in to cope with the size of kT , and theoretically a the kernel of the integral equation that produces positive cut is definitely unappealing.

P Ladder Multiperipheral

Optical theorem t-Channel iteration

N2 *

CP +Bo

Threshold Effect

(?) rn

-

-

No or two zeros Decreasing

*

up break No dip

zero a D - C - - d

B(s) 1

A

"f 3. . ;

X

Two component Scaling Long range corr.

Absorbed Multiperipheral Gribov

Calculus

a = l a' critical Not self Complete flip

Cheng s Wu Eikonal

Sign of cut Interpretation of Im A consistent Model for amplitude

t-Channel unitarity N2 Cp - Bo

Disappearing cut rn

2 9 2

-9- Y B(s) B(s)

+ ^en-2(s)

One zero

~ecreasing* Parameter Down breakm Dependent

m

Upward B(s) Features

Dip Dip

zero

*

uD a (1-2) (C

-

- 1 1- s

A J ~ + A . , X

Three component Non scaling

- ²

n x f n s

- Non

-

-

of sign Energy dependent potential Generated by J=l Ladder Eikonal absorption 'lane

2 2 % Ro en s

Waves- t limitation 2 % a'g en s Increasing potential (2 MB)

+

Disappearing cut (1 ME)

-

-

+ en (s) One zero

Increasing Up break

Dip Not zero

o -, s- N

Non scaling

'

;;

=

- 3 fn (1-x)

- ¹

A I - 1

g

X

-

-

Scaling Long rang corr.

Hard cuts

C1-270 L. CANESCHI and M. CIAFALONI

2.2

.-

The a p p a r e n t paradox of t h e n e g a t i v e c o n t r i b u t i o n (-ypl/ns) t o 2 I m A , of t h e same magnitude a s t h e p o s i t i v e e l a s t i c c r o s s - s e c t i o n , h a s been s o l v e d I 9 1 w i t h t h e remark t h a t a b s o r p t i v e c o r r e c t i o n s t o t h e m u l t i p a r t i c l e p r o d u c t i o n a m p l i t u d e s can s u b t r a c t from

t h e p o l e a c u t t w i c e a s b i g a s t h e AFS c u t . I f t h e e x t e n s i o n of t h i s r e a s o n i n g t o a l l d i f f r a c t i v e chan- n e l s i s allowed, one o b t a i n s t h e complete change of s i g n of t h e N term, and t h u s (1) w i t h n e g a t i v e s i g n 2

[ l o I .

The observed r i s e of u i s i n t e r p r e t e d a s t h e d i - sappearance of t h e n e g a t i v e c u t , which changes w i t h s a s l / B ( s ) (B(s) = d Pn A ( s , t ) ) . Since i n t h e ISR range t h e s h r i n k a g e i s r a t h e r s m a l l , t h e c o e f f i c i e n t o f l / B ( s ) must be q u i t e l a r g e t o g i v e a 10 % e f f e c t .

[ ~ f taken s e r o u s l y , t h i s produces ow 80 mb and t h e u n p l e a s a n t need of v e r y l a r g e non-leading terms t o f i l l up t h e low-energy r e g i o n (s<lOO G ~ V - ~ ) w i t h t h e consequent breakdown of d u a l i t y , exchange degene- r a c y , e t c . ] . T h i s l a r g e c o e f f i c i e n t

2

N (0) = ( y (0)-g1(o)? can be obtained by means of a P P

l a r g e yP and s m a l l g' o r v i c e v e r s a . P

The f i r s t a l t e r n a t i v e [11 ]

-

-

a ) o e p ( s ) t u r n s o u t t o be d e c r e a s i n g i n t h e ISR r m g e

.

b) B(s) has p o s i t i v e 2 nd d e r i v a t i v e . T h i s i s r e l a - t e d t o t h e p r e v i o u s p o i n t , because t h i s behaviour of B(s) i s needed i f set h a s t o d e c r e a s e i n s p i t e of t h e i n c r e a s e of t h e o p t i c a l p o i n t .

nd d o

c ) The 2 d e r i v a t i c w i t h r e s p e c t t o t of en i s n e g a t i v e from t=O a l l t h e way up t o t h e d i p . These embarassing f e a t u r e s can be somehow circum- vented i f t h e f u n c t i o n N ( t ) h a s a n important 2 t-dependence f o r s m a l l t ( e . g . , a z e r o ) a s i t hap- pens i n t h e f i t of P a j a r e s and S h i f f [12], i n which g;(o) >> y ( 0 ) , but g' ( t ) h a s a s t e e p e r t-depen-

P P

dence than y p ( t )

.

parameters i n t h e theory i n terms of a c t u a l f i n a l s t a t e s i n p-p c o l l i s i o n .

I n a n a l t e r n a t i v e approach t o t h e Gribov c a l c u l u s one could t r y t o s e p a r a t e n o n - d i f f r a c t i v e and d i f - f r a c t i v e c o n t r i b u t i o n s t o t h e p o l e a s w e l l a s t h e c u t . I n such a c a s e one h a s , t o o r d e r gp 2 , ^{a nega-}

t i v e r e n o r m a l i z a t i o n term o f t h e form (-g 2 ens) a s

2 P

w e l l a s t h e c u t c o n t r i b u t i o n (-g l e n s ) both a s s o c i a - P

t e d t o h i g h t k e s h o l d s and n e g l e c t e d a t ISR. I f t h e f i n a l s o l u t i o n i s t o be of t h e p o l e and c u t t y p e a t J=l , one has t o i n t r o d u c e a t ISR e n e r g i e s a "bare"

Pomeron p o l e above 1 ( a s s o c i a t e d w i t h n o - l a r g e gap e v e n t s i n t h e u n i t a r i t y sum) and t r y a parameteriza- t i o n of t h e form

No phenomenology along t h e s e l i n e s h a s been attemp- t e d s o f a r .

3.- S-CHANNEL ORIENTED MODELS.- A l t e r n a t i v e l y i t i s p o s s i b l e t o e x p l i c i t e l y e n f o r c e t h e s-channel uni- t a r i t y bound, and t h e most convenient language f o r t h i s i s t h e impact paramefzer b .

For a p u r e l y a b s o r p t i v e amplitude t h e bound needs

Furthermore t h e F r o i s s a r t bound l i m i t s t h e growth of t h e number of p a r t i a l waves t h a t can c o n t r i b u t e t o A ( s , t ) : bmax < Ro Pns

.

The t - c h a n n e l J s i n g u l a r i t y corresponding t o t h e s a t u r a t i o n of t h e F r o i s s a r t bound i s a t r i p l e p o l e a t t=O 1131, a s i n g u l a r c u t a t t h o

2 2

( - 1 ) - ~ ~ t ] - ~ ' ~ )

.

An e i k o n a l i z a t i o n procedure can be performed a l s o on a Regge Pole p o t e n t i a l w i t h a ( o ) = 1 [15]. I n t h i s c a s e one r e c o v e r s t h e p o l e minus s o f t c u t s t r u c - t u r e of t h e Gribov approach, b u t t h e r e s i d u e of t h e c u t , determined now by ueP/uT , i s i n s u f f i c i e n t t o g i v e t h e desired r i s e of cr

.

MODELS FOR RISING CROSS-SECTIONS C1-271

s-channel u n i t a r i z a t i o n scheme, l i k e t h e e i k o n a l [16]

o r t h e d i s t o r t e d p l a n e wave approach [17].

A convenient dynamical mechanism t o produce a n energy i n c r e a s i n g p o t e n t i a l i s t h e m u l t i p e r i p h e r a l l a d d e r i t e r a t i o r o f a s i n g u l a r i t y a t J=l , y i e l d i n g a Regge P o l e a t some 1+6 , t h e b r e p r e s e n t a t i o n o f which i s given by

The amplitude i s t h e n g i v e n by

1 J2

A(b,s) = 7 + ^V(b,s)-

+

9 4 ,,

A=A& + (1-2A)V) and y i e l d s G = ^{2ue Z= a ' 6 f n L s}

a s y m p t o t i c a l l y .

T h i s asymptotic regime i s however approached very slowly, and h a s n o t h i n g t o do w i t h t h e ISR phenome- nology, f o r which we r a t h e r propose a p e r t u r b a t i v e approach [ 1 8 ] [ 1 9 ] , t h e s m a l l parameters being given by t h e p r e s e n t v a l u e s of uee/u and a D i f f / a 7

.

former measures t h e a v e r a g e importance o f t h e

s-channel i t e r a t i o n of t h e Pomeron, and i t s s m a l l n e s s shows t h a t a b s o r p t i o n i s n o t t o o i m p r r t a n t a t t

=

( t h i s i s n o t t h e c a s e f o r b

--

where d i f f r a c t i v e e f f e c t s l i k e t h e d i p a t t

=

a c t u a l l y show up)

.

For t h e o p t i c a l p o i n t t h e r e f o r e A

--

energy behavior of UT (and of f o r s m a l l t ) i s w e l l reproduced i n t h i s approach by a Regge P o l e w i t h

a zz 1.1 (better 1.082925) [ZO]. The s m a l l v a l u e of a - 1 can be r e l a t e d i n o u r approach t o t h e o t h e r s m a l l parameter uD/ uT , i f one assumes t h a t " p u r e l y i n e l a s t i c " e v e n t s ( i . e . e v e n t s n o t showing l a r g e r a - p i d i t y gaps) sum up t o g i v e a c o n s t a n t c o n t r i b u t i o n t o uT

.

The f i r s t p e r t u r b a t i v e c o n t r i b u t i o n , t h e t r i p l e Pomeron one, i s only s l i g h t l y reduced i n magnitude by t h e e l a s t i c a b s o r p t i o n , and remains p o s i t i v e . Hence one e x p e c t s t o be a b l e t o g i v e a n e s t i m a t e of t h e i n c r e a s e of u by i n t e r p r e t i n g t h e l a r g e x peak of t h e p i n c l u s i v e spectrum, which indeed seems t o y i e l d a c o n t r i b u t i o n t o AS o f about 2 mb i n t h e ISR

range ( t h e r e s t (- 1 mb) comes i n on model by a d i - s a p p e a r i n g c u t e f f e c t r e l a t e d t o s h r i n k a g e , even i f u i n c r e a s e s i t s e l f by i n 1 mb).

e l

Obviously i n t h i s scheme t h e r e i s no need t o have a v a n i s h i n g t r i p l e Pomeron c o u p l i n g a t t=O , ^on

t h e c o n t r a r y t h i s i s a s u f f i c i e n t d e v i c e t o t r i g g e r t h e s a t u r a t i o n mechanism. I n t h e asymptotic s o l u t i o n t h e a b s o r p t i v e c o r r e c t i o n s and t h e consequent non f a c t o r i z a b l e n a t u r e of t h e P s i n g u l a r i t y reduce a

n e a r x=l l i t t l e t h e s i n g u l a r i t y o f - _{u dx}

-

( ( l - x ) l n 3 ( - ) i n s t e a d 'of (1-x)-I en-'(1-x)

1

making i t compatible w i t h t h e c o n s e r v a t i o n sum r u l e s l7-1 I.

The asymptotic scheme i n which t h e F r o i s s a r t bound i s s a t u r a t e d i s indeed only a convenient de- v i c e t h a t a l l o w s t o perform dangerous e x e r c i s e s

( l i k e h a v i n g a > 1 and a non-vanishing t r i p l e Pomeron c o u p l i n g ) s e c u r e i n t h e knowledge t h a t asymp- t o t i c a l l y t h i n g s w i l l work o u t w e l l .

4

.-

-

.

i n t e g r a l over t h e peak i n c r e a s e s t o a c o n s t a n t w i t h s and i n t h e l a r g e gap p e r t u r b a t i o n approach I223 should g i v e an o v e r e s t i m a t e of Au7 , s i n c e t h e b a r e Pomeron i s a t 1-E

.

Also i n t h e Gribov scheme one e x p e c t s a peak a t x = l , b u t i t s s i z e i s n o t r e l a t e d t o t h e i n c r e a s e of u

.

n =

.

I n t h e absorbed m u l t i p e r i p h e r a l i s m t h e peak a t x = l , n o t v a n i s h i n g a t t=O , i s expected t o g i v e t h e main c o n t r i b u t i o n t o AuT

.

C1-272 L. CANESCHI and M. CIAFALONI

bounded s c a l i n g an6 long range c o r r e l a t i o n s ( w i t h o -r c o n s t ) . I n t h e e i k o n a l p i c t u r e [25] t h e momentum t r a n s f e r l i m i t a t i o n c h a r a c t e r i s t i c o f m u l t i p h e r i p h e - r a l i s m does n o t h o l d , and one o b t a i n s ii 2 S P ,

hence b r e a k i n g of s c a l i n g .

5.- FINAL THEORETICAL REMARKS.- The schemes d i s c u s - s e d h e r e p r e s e n t t h e common problem of e x p l a i n i n g t h e l a r g e energy s c a l e ( E 2 100 GeV 2 ) on which t h e r i s e o f a T o c c u r s , o r a l t e r n a t i v e l y t o produce s u i t a b l e non-leading terms t o f i l l up t h e low energy r e g i o n .

kn e x t r a p o l a t i o n t o low energy of t h e term g i v i n g t h e rise o f a a t ISR i s l i k e l y t o g i v e less t h a n 30 mb a t s

--

The t - c h a n n e l o r i e n t e d schemes depend h e a v i l y on gp (0) =O , t h e s-channel ones do n o t : t h i s might p r o v i d e a d e c i s i v e t e s t . The orthodox MPM g i v e s by wide agreement t h e wrong s i g n f o r t h e c u t , and i s

unable t o e x p l a i n n a t u r a l l y t h e d i p a t It 1 ^{= 1 . 4} ,

which i n s t e a d f i t s w e l l i n t h e o t h e r schemes. Fur- thermore, t h e q u e s t i o n of why cf = ¹ i s d i f f i c u l t

P t o answer i n t h e MPM.

The Gribov scheme i s a model f o r t h e a m p l i t u d e , and i t i s n o t s t r a i g h t f o w a r d t o l i n k Im A t o a sum over f i n a l s t a t e s , a s i t would be v e r y d e s i r a b l e f o r phenomenology. Also u n p l e a s a n t i s t h e need of r e l y i n g on t h e ( s m a l l ) v a l u e of a' i n o r d e r t o e x p l a i n t h e

[ I ] CASHER (A.), NUSSINOV (S.) and SUSSKIND ( L . ) , phys. L e t t . 44B (1973) 171.

12 ] SIVERS (D.) and VON HIPPEL (F .) , ANL P r e p r i n t HEP 7323.

[ 3 ] ABARBANEL (H.D.I.), Phys. Rev. (1972) 2788.

[4 ] MUZINICH ( I . J .) , PAIGE (F .) , TRUEMAN (T .L .) and WANG (L.L.), Phys. Rev. (1972) 1048.

[ 5 ] CHEW (G.F.) and SNIDER (D.R.), Phys. L e t t . (1970) 75 ;

CHEW (G.F.), Phys. L e t t . 44B (1973) 169.

[6 1 BISHARI (M.) and KOPLIK ( J . ) , Phys. L e t t . (1973) 175.

[7 1 WHITE (A.R.) , Nuclear P h y s i c s B50 (1912) 130.

181 GRIBOV (V.N.), JETP 6 (1968) 414 ; GRIBOV (V .N

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[12] PAJES (C.) and SCHIFF (D.), Orsay p r e p r i n t LPTHE 12 (1973) 13 ;

NG (J.N.) and SUKHATME (U.P.), S e a t t l e pre- p r i n t RLO 1388-645.

[13]Weaker s i n g u l a r i t i e s , l i k e a double p o l e ( a

--

BAIL ^(J

.

.

p r e p r i n t 68 (1972) 360.

r i s e of a . Very a t t r a c t i v e i s on t h e c o n t r a r y t h e a l t e r n a t i n g s i g n p a t t e r n o f t h e many Pomeron c u t c o n t r i b u t i o n s , which s u g g e s t s a l i n k a g e w i t h a n s-channel e i k o n a l i z a t i o n , and a way of e n f o r c i n g t h e s-channel u n i t a r i t y bound and perhaps u n d e r s t a n d i n g

cf = 1

.

P

The s-channel a b s o r p t i v e models l a c k t - c h a n n e l u n i t a r i t y , y i e l d i n g h a r d c u t s . Whether t h i s i s a s i - g n i f i c a n t drawback i s s t i l l t o be e s t a b l i s h e d 1131.

An a d d i t i o n a l problem i s t h e c h o i c e of t h e n a t u r e o f a b s o r p t i o n . We have l i m i t e d o u r s e l v e s t o e l a s t i c ab- s o r p t i o n , b u t i f one i n c l u d e s low-mass d i f f r a c t i v e s t a t e s a s w e l l , t h e i r c o n t r i b u t i o n t o G f l i p s

7

s i g n a s aeP i s known t o do. I f one goes a l l t h e way i n c l u d i n g a l s o a b s o r p t i o n through d i f f r a c t i v e s t a t e s of l a r g e mass ( a s i f they were quasi-two body s t a t e s ) , t h e t r i p l e Pomeron c o n t r i b u t i o n a l s o changes s i g n , and c a n n o t be used t o e x p l a i n t h e rise of a _[26].

I n t h i s c a s e a v a n i s h e s a s y m p t o t i c a l l y i f gp(o)PO [27 1.

I n c o n c l u s i o n , i f e x p e r i m e n t a l l y g ( 0 ) w i l l t u r n P

o u t t o be nonvanishing, a n s-channel a b s o r p t i v e approach w i l l be t h e most a t t r a c t i v e p o s s i b i l i t y . I f g (o)=O on t h e c o n t r a r y , G r i b o v ' s s o l u t i o n i s p r e -

P

f e r a b l e , and i t should be g r e a t f u n t o r e d e r i v e i t i s s-channel terms u s i n g y e t unknown c a n c e l l a t i o n s and changes o f s i g n which a r e n o t i n c l u d e d i n t h e e l a s t i c a b s o r p t i o n approximation.

[ I 4 1 CREUTZ (M.), PAIGE (F.) and WANG (L.L.), Phys.

Rev. L e t t . 2 (1973) 343 have g i v e n a n exam- p l e

05

.

.

115 1 FRAUTSCHI (S .) and MARGOLIS (B.) , Nuovo Cimento

. . . .

56 (1968) 1155.

[16 ] cHEE(H.) and WU (T.T.) , Phys. Rev. L e t t . 24

(1970) 1456.

[17 1 FINKELSTEIN ( J . ) and ZACHARIASEN (F .), Phys.

L e t t . 34B (1971) 631 ;

CANESCHI (L.) and SCHWIMMER (A.), N a c l e a r P h y s i c s B44 (1972) 31.

[18 ] AMATI (D

.

.

CERN p r e p r i n t TH 1676.

[19] FINKELSTEIN ( J . ) , Columbia U n i v e r s i t y p r e p r i n t (1973).

[20] CHENG (H.), WU (T.T.) and WALKER (J.K.), Phys.

L e t t . & (1973) 97.

[21] CANESCHI (L.) and SCHWIMMER (A.) , Nuclear Phy- s i c s (1972) 519.

[22] FRAZER (W.R.), SNIDER (D.R.) and TAN (C.I.), UCSD p r e p r i n t 10P (1973) 10127.

[23 ] ABRAMOVSKI J (V . A . ) , GRIBOV (V .N .) and KANCHELI (O.V.), Proceedings of t h e XVI I n t e r n a t i o n a l Conference on High Energy P h y s i c s ( B a t a v i a , 1972) .,P. 389

TER MARTIROSYAN (IGA.), Phys . L e t t . 44B (1973) 377.

MODELS FOR RISING CROSS-SECTIONS C1-273

[24] Normal two component models ( a s h o r t r a n g e c o r r e - [25] CHENG (H.) and WU ( T . T . ) , Phys. L e t t . 45B (1973) l a t i o n component p r o d u c i n g a Regge P o l e i n 367.

Im A p l u s a d i f f r a c t i v e c o n t r i b u t i o n ) ob- [26] T h i s p o i n t , a l r e a d y remarked i n § 6 of r e f . 18 v i o u s l y commit t h e m s e l v e s t o a n AFS t y p e h a s been s u b s e q u e n t l y s t r e s s e d by

c u t . To o b t a i n a n e g a t i v e c u t , i t i s n e c e s - BLANKENBECLER (R.), P r e p r i n t SLAC-PUB 1282, s a r y t o keep i n t o a c c o u n t c o r r e c t i o n s t o t h e and i n r e f . 26.

s h o r t r a n g e h y p o t h e s i s on t h e c o n t r i b u t i o n o f [27] BLANKENBECLER ( R . ) , FULL0 ( J . ) and SUGAR (R.), t h e n o - l a r g e r a p i d i t y gap e v e n t s ( l i k e l o n g P r e p r i n t SLAC-PUB 1281 ( 1 9 7 3 ) .

r a n g e a b s o r p t i v e e f f e c t s )

.