MUELLER-REGGE PHENOMENOLOGY

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MUELLER-REGGE PHENOMENOLOGY

Chan Hong-Mo

To cite this version:

Chan Hong-Mo. MUELLER-REGGE PHENOMENOLOGY. Journal de Physique Colloques, 1973,

34 (C1), pp.C1-352-C1-355. �10.1051/jphyscol:1973146�. �jpa-00215226�

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CHAN HONG -MO

MUELLER-RWE PHENOMENOMGY CHAN HONG-MO Rutherford Laboratory

1.- SINGLE PARTICLE DISTRIBUTIONS I N THE FRAGMEN- TATION REGION.

-

The g e n e r a l i z e d o p t i c a l theorem may b e r e p r e s e n t e d a s i n Fig.1, where, i n t h e fragmen- t a t i o n r e g i o n of b

,

t i s s m a l l and M 2 /s i s h e l d f i x e d . I n terms of t h e diagram on t h e r i g h t hand s i d e of f i g u r e 1 , t may be i n t e r p r e t e d a s t h e mass of t h e system and M 2 /s i s some v a r i a b l e which d e s c r i b e s i t s o r i e n t a t i o n with r e s p e c t t o a

.

I t i s convenient t o t h i n k of t h e i n c l u s i v e c r o s s s e c t i o n t h e r e f o r e a s t h e t o t a l c r o s s s e c t i o n f o r t h e s c a t - t e r i n g of p a r t i c l e a from a "pseudo-particle" b

: .

The g r e a t freedom

i n choosing t h e quantum numbers of b

:

, a s w e l l a s i t s mass ( t ) and s p i n o r i e n t a t i o n (M 2 / s ) accounts f o r t h e r e l a t i v e r i c h n e s s of i n c l u s i v e phenomenology compared with t h a t i n

T '

( a ) E x o t i c i t y c r i t e r i o n f o r e a r l y s c a l i n g . I n analogy w i t h ITT

,

^{i t}was suggested t h a t a b c e x o t i c * f ( t , M / s , s ) 2 independent of s . ( l ) A t t h e time of t h e Amsterdam Conference (19711, t h e r e was some c o n t r o v e r s y a s t o t h e v a l i d i t y of ( 1 ) . I n

~ e f . [ l ] i t i s shown f o r production t h a t t h e

"exotic" r e a c t i o n s pp + 7;

,

K+p + X- and K+p + X- f a l l a s s-"~ towards a c o n s t a n t v a l u e

a s s *

,

while t h e "non-exotic" r e a c t i o n s X-P -t X-

,

K-p + X- and ?IP * ^X- appear t o be independent of energy. T h i s s i t u a t i o n p r e v a i l s only f o r X-production ; f o r h e a v i e r p a r t i c l e s t h e r e a r e a l s o kinematic r e s t r i c t i o n s a t low e n e r g i e s which make i t hard t o judge t h e v a l i d i t y of t h e c r i t e r i o n .

( b ) F a c t o r i e a t i o n .

Asymptotically, Pomeron f a c t o r i s a t i o n i m p l i e s

T h i s can o n l y be t e s t e d when i n c l u s i v e c r o s s s e c t i o n s w i t h secondary beams a r e m a s u r e d a t NAL. At lower e n e r g i e s presumably both t h e Pomeron and the meson t r a j e c t o r i e s p-f-&A2 a r e important. I f one assumes t h a t o n l y t h e s e t r a j e c t o r i e s c o n t r i b u t e , t h e e x o t i - c i t y c o n d i t i o n (1) and f a c t o r i z a t i o n imply a h o s t of r e l a t i o n s which have been compared f a v o r a b l y with experiment [z].

There a r e two i n t e r e s t i n g e x c e p t i o n s r e p o r t e d by c o n t r i b u t i o n s t o t h i s Conference

The c r o s s s e c t i o n s exceed, (by f a c t o r s ranging from 2 t o 8 ! ) t h o s e p r e d i c t e d by f a c t o r i z a t i o n assuming P and p-f-*A2 dominance. The p o l a r i z a t i o n observed i n (B) i s a l s o i n c o n s i s t e n t w i t h f a c t o r i z a t i o n . These d i s c r e p a n c i e s p o i n t t o a d d i t i o n a l exchanges,

presumably r e l a t e d t o t h e a n n i h i l a t i o n of baryon quantum number i n (A) aad of s t r a n g e n e s s i n (B). Such exchanges a r e n o t unexpected i n t h e d u a l model. In- deed, t h e quark diagrams f o r t h e a n n i h i l a t i o n chan- n e l s i n (A) and (B) a s shown i n f i g u r e 2 suggest r e s p e c t i v e l y e x o t i c and cp excharg es. The observed s t e e p dependence of t h e c r o s s s e c t i o n s on s i s i n accord w i t h the low i n t e r c e p t of t h e s e uncommon t r a j e c t o r i e s .

K' Fig. 2

PW

Fig. 2

-

Quarks diagrams f o r the a n n i h i l a t i o n chan- n e l s i n r e a c t i o n s A and B

.

( c ) Semi-local d u a l i t y .

Semi-local d u a l i t y , f o r t h e Regge phenomeno- l o g i s t s , means t h a t when t h e asymptotic Regge formula

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973146

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MUELLER-REGGE

i s e x t r a p o l a t e d down t o low e n e r g i e s , i t s t i l l re- p r e s e n t s t h e semi-local average of t h e imaginary p a r t o f t h e amplitude, d e s p i t e t h e p o s s i b l e dominance t h e r e of resonance s t r u c t u r e s . When i t i s ap- p l i e d t o t h e R.H.S. of f i g u r e 1, t h e s c a t t e r i n g energy i s r e p l a c e d by the m i s s i n g mass b12

.

One t h e n c l a i m s t h a t t h e asymptotic f o r m l a :

which i s s t r i c t l y v a l i d only f o r M2 l a r g e , should s t i l l r e p r e s e n t t h e semi-local average o v e r M 2 f o r m a l l M~

.

Now s i n c e t h e Regge r e s i d u e s fi(t,M / s ) 2 a r e f u n c t i o n s o n l y of t and M 2 /s ( i . e. , they s c a l e ) , they can b e measured f o r any v a l u e of M2/s by going t o s u f f i c i e n t l y high s

.

T h e r e f o r e , when s c a l e d by t h e a p p r o p r i a t e f a c t o r s of (M2)aj(o), they should r e p r e s e n t t h e semi-local average of t h e low energy d a t a , t h u s a f f o r d i n g a d i r e c t check o f semi- l o c a l d u a l i t y .

Fig. 3

-

Comparison of high and low energy d a t a on K-p -+ p +X , ref.[51 d a t a from r e f . [ 6 ]

A rough a p p l i c a t i o n of t h i s i d e a i s shown i n f i g u r e 3;

where t h e dominance of a s i n g l e t r a j e c t o r y i s assu- med. One does indeed s e e t h a t t h e h i g h energy d a t a f o r x-p * p +X r e p r e s e n t a semi-local average of t h e low energy d a t a .

The concept of semi-local d u a l i t y e n a b l e s one t o a p p l y f a c t o r i z a t i o n a l s o t o t h e resonance r e g i o n s , a t l e a s t i n an average sense. Thus, f o r example, t h e c r o s s s e c t i o n s f o r

and

x - + P -' p + X ,

a r e r e l a t e d , on t h e average, by f a c t o r i z a t i o n , even i n t h e s m a l l M2 r e g i o n , where they r e p r e s e n t pro- d u c t i o n c r o s s s e c t i o n s of N* and meson resonances r e s p e c t i v e l y . An example of t h i s i s shown i n Fig.4,

where t h e p r e d i c t i o n from f a c t o r i z a t i o n and

x-

*

( p -' p ) does indeed average a p p r o x i m a t e l y , K p r o d u c t i o n d a t a .

Fig. 4

-

Comparison o f experimental r e s u l t s o n ( p I(;p) w i t h t h e o r e t i c a l p r e d i c t i o n s ( s o l i d l i n e ) . Data from r e f . [ 8 ] f o r X* p r o d u c t i o n i n ( p

5

P )

.

( d ) T r i p l e Regge Analysis.

I n t h e kinematic r e g i o n where s/M2 i s l a r g e one c a n make a f u r t h e r expansion of t h e R.H.S.

of f i g u r e 1 i n terms of t h e t r a j e c t o r i e s i n t h e b c

-

channel, o b t a i n i n g t h u s f i g u r e 5 , which may be con-

Fig. 5

v e n i e n t l y i n t e r p r e t e d a s a p a r t i c l e - r e g g e o n amplitude. I f M2 i s a l s o l a r g e , one t h e n h a s t h e scr c a l l e d t r i p l e - r e g g e formula,

i , J , k

( 4 ) w i t h e x p l i c i t dependence on b o t h M' and s

.

C o n s i d e r a b l e e f f o r t h a s been s p e n t i n t r y i n g t o e s t a b l i s h t h e v a l i d i t y of ( 4 ) from experiment. One f i n d s , i n g e n e r a l t h a t ( 4 ) i s c o n s i s t e n t w i t h d a t a , but c o n t a i n s t o o much freedom t o be t e s t e d conclu- s i v e l y a s y e t . The d i f f i c u l t y i s t h e l a c k of high energy d a t a which s a t i s f y t h e s t r i n g e n t c r i t e r i a of v a l i d i t y f o r ( 4 ) , namely, M~ and s d l a r g e . One s i g n i f i c a n t r e s u l t which emerges from such ana- l y s e s i s t h e e s t a b l i s h m e n t from NAL and ISR d a t a of t h e e x i s t e n c e of a s i z a b l e component f o r t h e t r i - p l e Pomeron v e r t e x

CQI .

U n f o r t u n a t e l y , t h e e x p e r i - mental evidence f o r o r a g a i n s t t h e v a n i s h i n g o f t h i s v e r t e x a t t = ⁰ i s n o t y e t conclusive.

( e l D u a l i t y f o r p a r t i c l e - r e g g e o n amplitudes.

Semi-local d u a l i t y , when a p p l i e d t o t h e par- t i c l e - r e g g e o n amplitude of f i g u r e 5 f o r i = ^ji m p l i e s

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C1-354 CHAN HONG-MO

t h a t t h e c r o s s s e c t i o n a t low M2 should behave on t h e average a s

F o r r e a c t i o n s dominated i n M2 by resonances t h i s means one h a s a p r e d i c t i o n f o r t h e mss-dependence of resonance p r o d u c t i o n c r o s s s e c t i o n s v i a reggeon i exchange, provided one knows t h e t r a j e c t o r y a

k d u a l t o t h e d i r e c t channel resonances. F o r example, f o r t h e r e a c t i o n s

t h e meson resonances a r e d u a l t o vector-meson exchan- g e w i t h a k ( o ) ^% 1/2

.

One t h u s p r e d i c t s t h a t t h e s e resonance p r o d u c t i o n c r o s s s e c t i o n s have t h e mass depemience

which i s found e x p e r i m e n t a l l y , t o w i t h i n a good approximation.

I n c a s e s where t h e d u a l t r a j e c t o r y k i s unknown, t h e formula (6) s e r v e s a s a u s e f u l means o f probing t h e d u a l p r o p e r t i e s . For example, i n baryon-antibaryon s c a t t e r i n g , t h e quark diagrams of f i g u r e 6 s u g g e s t t h a t d i r e c t channel resonances a r e dual t o e x o t i c e x c b n g e s whereas t h e background may be d u e t o t h e exchange o f o r & nary meson poles.

Fig 6

Fig. 6

-

Quark diagrams f o r baryon-baryon s c a t t e r i n g T h i s i s i n complete c o n t r a s t t o t h e u s u a i c a s e , but i t cannot be t e s t e d i n a c t u a l p a r t i c l e - p a r t i c l e s c a t - t e r i n g , bscause of t h e high p h y s i c a l t h r e s h o l d s , which a r e w e l l above t h e r e g i o n o f prominent d i r e c t channel resonances. T h i s d i f f i c u l t y i s removed i f one of t h e p a r t i c l e s i s r e p l a c e d by a reggeon con- t i n u e d off-mass-shell. Such a p a r t i c l e - r e g g e o n s c a t - t e r i n g i s d i r e c t l y measurable i n t h e i n c l u s i v e reac- t i o n (n- q p ) , which c o r r e s p o n d s t o t h e s c a t t e r i n g

-

of p a g a i n s t t h e reggeon Ai+

.

One then e x p e c t s , according t o ( 5 1 , t h a t t h e resonance c r o s s s e c t i o n w i l l f a l l much f a s t e r than t h e background. T h i s a p p e a r s t o h o l d e x p e r i m e n t a l l y , a s may be s e e n i n f i g u r e 7.

Fig. 7

-

I l l u s t r a t i o n of resonance c r o s s s e c t i o n dependence [ l o ] .

The preceeding example s e r v e s a l s o a s a good il- l u s t r a t i o n of t h e manner i n which i n c l u s i v e r e a c - t i o n s may be e x p l o i t e d i n o r d e r t o s t u d y s c a t t e r i n g p r o c e s s e s w i t h unusual t a r g e t s which a r e n o t auai- l a b l e e x p e r i m e n t a l l y .

2.- SINGLE PARTICLE DISTRIBUTIONS I N THE CENTRAL REGION.

-

Assuming a g a i n t h a t one h a s a s y m p t o t i c a l - l y t h e dominance of Pomeron and t h e meson Regge p o l e s p-f -w-A , t h e l e a d i n g t e r n s i n t h e e x p a m i o n of 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 i n t h e c e n t r a l r e - g i o n a r e

2 2

where 8 = u t / s = p, + ^m A t e* = ^90°

,

one then c D

h a s u = t

u p / z

^{i l l 2}

,

which p r e d i c t s a n approach t o t h e s c a l i n g limit a s s -1/4

I t i s tempting t o a p p l y t h i s d i r e c t l y t o t h e experimental d a t a , and t o quote a s a c o n f i r m a t i o n t h e F e r b e l p l o t [111

.

D e s p i t e t h e a t t r a c t i v e n e s s of t h i s , I wish t o r a i s e a few o b j e c t i o n s which I t h i n k a r e s e r i o u s .

( i ) Except f o r t h e NAL and ISR p o i n t s , a l l d a t a i n t h e F e r b e l p l o t have s _( 60 GeV 2 , correspon- d i n g t o l u l , It1

2

2.7 G ~ v ~ , v a l u e s f o r which one would n o t e x p e c t t h e asymptotic formula (8) t o be v a l i d . I f one t a k e s o n l y NAL and ISR d a t a , t h e r e is

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MUELLER-REGGE PHENOMENOLOGY

no a p p a r e n t s -'I4 behaviour.

( i i ) The F e r b e l p l o t does n o t work b e t t e r f o r t h e d i f f e r e n c e i n c r o s s s e c t i o n between pp

- - -

⁺X+X and pp + X X

,

n o r f o r p and K p r o d u c t i o n o v e r t h e same range of energy, a s m y b e s e e n i n Fig.8 ( s m a l l PI ).

( i i i ) I f t h e r i s e i n c r o s s s e c t i o n i n t h e cen- t r a l r e g i o n i s indeed a t h r e s h o l d e f f e c t , a s i s ge- n e r a l l y b e l i e v e d , t h e a p p l i c a t i o n of ( 8 ) t o i t i s l i k e u s i n g t h e meson t r a j e c t o r i e s t o d e s c r i b e t h e t h r e s h o l d r i s e i n cT , which would n o t be admis- s a b l e .

I b e l i e v e t h a t t h e formula ( 8 ) should be a p p l i e d only t o v a l u e s of lu

1,

It

1 >

^5cev2

.

I b e l i e v e a l s o t h a t ( 8 ) h a s a b e t t e r chance of b e i n g c o r r e c t f o r t h e d i f f e r e n c e s between p a r t i c l e and a n t i - p a r t i c l e c r o s s s e c t i o n s r a t h e r t h a n f o r t h e c r o s s s e c t i o n s themselves, s i n c e i n t a k i n g d i f f e r e n c e s t h r e s h o l d e f f e c t s would be suppressed. Such a c o n j e c t u r e i s borne o u t by t h e high p d a t a i n f i g u r e 8 , and i t

I

a l s o works w e l l f o r t h e y* dependence of OCpp + pX)

-

q p p +

>),

a s d e s c r i b e d by Roy i n

~ e f . [IS].

Fig. 8

-

Energy dependence o f c r o s s s e c t i o n d i f f e - r e n c e s [121

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[TI

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[8

1

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,

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