S-CHANNEL PHENOMENOLOGY OF DIFFRACTION SCATTERING

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S-CHANNEL PHENOMENOLOGY OF DIFFRACTION SCATTERING

Hannu Miettinen

To cite this version:

Hannu Miettinen. S-CHANNEL PHENOMENOLOGY OF DIFFRACTION SCATTERING. Journal

de Physique Colloques, 1973, 34 (C1), pp.C1-263-C1-267. �10.1051/jphyscol:1973130�. �jpa-00215210�

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S-CHANNEL PHENOMENOLOGY OF DIFFRACTION SCATTERING

Hannu I. MIETTINEN +)

CERN

A fruitful way to look at diffraction scattering is to consider it as a shadow of inelastic processes S-channel unitarity gives :

large extent different regions of the phase space.

This suggests that it might be useful to consider their contributions to elastic scattering separa- tely. Hence we split Ginel(t) into two parts :

where Tfi is elastic amplitude and Tni (Tnf) the amplitude of the production process i + n (f + n).

The q-.:,ntities Gi, i = el, inel, are called the elastic and inelastic overlap functions, respecti- vely. C1I

Diffraction scattering has been studied for a long time with the help of the Eq.(l) but the pro- gress has not been very rapid

-

partially due to lack of knowledge of the particle production am- plitudeswhichare the input in the approach, partially because Eq.(l) is in spite of its simple appearance actually a very complicated relation. In the last year a new wave of enthusiasm in the above

"shadow approach" has grown. This enthusiasm has been triggered by the results of the experiments at the NAL and the ISR : rising total cross-sections, detailed measurements of the s- and t- dependewe of the elastic differential cross-section in proton- proton scattering and

-

^{of course}

-

by the enormous flow of data on multiparticle production which has greatly improved the understanding of the dynamics of multi-body processes.

I .

-

TheThree-Component Pomeron : The measurements of the inclusive proton spectra at the NAL and the ISR show that at high energies inelastic diffraction and non-diffractive production populates to a

---

+) Herman Rosenberg Foundation Fellow. On leave of absence from the Research Institute for Theoreti- cal Physics, University of Helsinki, Finland.

Here, GT(t) is the shadow of the non-diffractive production (s= "pionization") and G (t) that of

D

diffractive production. Combining Eqs.(l) and (2) one has

This "three-component structure" of the Pomeron is illustrated below :

In the forward direction Eq.(l) reduces to the Optical Theorem. Thus at t=O the size of the indi- vidual components are known : G (0) =ue1=7 mb,

el

GT (0) = UT

=

^{25 mb and}^G⁽⁰⁾⁼U D 3 7 mb at D

200 GeV/c. A question of great interest is : How do these three components build up the t-dependence of the elastic amplitude, especially the interes- ting structures observed experimentally (the break

2 2

at t = -0.1 GeV and the dip at t = -1.3 GeV ) ?

2. Ginel (t) from Experiment : The inelastic com-

-

ponent of the Pomer-on can be directly solved from experimental data..

p'

3j0ne rewrites Eq. (I) :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973130

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C1-264 H. I. MIETTINEN

s o l v e s Im Tfi and Re T from e l a s t i c s c a t t e r i n g f i

d a t a w i t h a r e a s o n a b l e assumption on t h e e l a s t i c phase and performes t h e loop i n t e g r a t i o n . The r e - s u l t of t h i s e x e r c i s e i s shown i n Fig. I .

Fig. 1. I n e l a s t i c o v e r l a p f u n c t i o n c a l c u l a t e d from t h e 1500 GeV/c proton-proton d a t a . From Ref. ( 3 ) .

A s t r i k i n g o b s e r v a t i o n i s t h a t G ( t ) changes s i g n

2 i n e l

a t t = -0.6 GeV

.

This r e s u l t has s e v e r a l important consequences. I f Ginel(t) would be a p o l e a s i s assumed e.g. i n t h e o l d AFS approach, t h i s zero would be a p r o p e r t y of t h e r e s i d u e of t h e p o l e . Hence i t should propagate over t o o t h e r p r o c e s s e s s t a y i n g a t a f i x e d v a l u e of t . Such a r e s u l t i s i n a p p a r e n t disagreement w i t h t h e experimental d a t a

( r e p e a t i n g t h e above e x e r c i s e i n ~ + p s c a t t e r i n g g i v e s t h e z e r o a t much l a r g e r t-values t ~ - 1 . 3 GeV 2 ) Another important p o i n t i s t h a t t h e change of s i g n of Ginel(t) p r o v i d e s s t r o n g c o n s t r a i n t s on theore- t i c a l m u l t i - p a r t i c l e models; i t t u r n s o u t n o t t o be e a s y t o g e n e r a t e n e g a t i v e shadows.

3. Models f o r G 7T&: L e t u s s e e how some popu- l a r p a r t i c l e p r o d u c t i o n models b u i l d up t h e t-dependence of G ( t ) . I n g e n e r a l t h e s l o p e r e c e i v e s con- t r i b u t i o n s from t h r e e d i f f e r e n t s o u r c e s : t h e magni- tudes of t h e p r o d u c t i o n a m p l i t u d e s , t h e phases of t h e p r o d u c t i o n amplitudes and t h e s p i n s of t h e produced p a r t i c l e s . I n d i f f e r e n t models t h e s e e f f e c t s combine i n d i f f e r e n t ways.

I n t h e u n c o r r e l a t e d j e t model (UJM) t h e produced p a r t i c l e s d o n o t have any mutual c o r r e l a t i o n s i n t h e momentum space. I n p o s i t i o n s p a c e , however, t h e y a r e s t r o n g l y c o r r e l a t e d c l u s t e r i n g around t h e posi-

t i o n of t h e i n i t i a l s t a t e p a r t i c l e s . The model g i v e s

-

i g n o r i n g phases and s p i n s

-

an o v e r l a p f u n c t i o n of roughly Gaussian shape i n b-space. [I]

The RMS r a d i u s of t h i s o v e r l a p f u n c t i o n can be measured by measuring t h e t r a n s v e r s e momentum dependence of t h e s p e c t r a of t h e produced p a r t i c l e s . The e x p e r i m e n t a l l y observed v a l u e of t h e average t r a n s v e r s e momentum of t h e p a r t i c l e s , <p,> N 350MeV corresponds t o a r a d i u s of t h e p r o d u c t i o n volume of l e s s t h a n 112 f e r m i . This v a l u e i s much t o o small t o f i t t h e d a t a , r41it corresponds t o a s l o p e of t h e e l a s t i c d i f f e r e n t i a l c r o s s - s e c t i o n of t h e o r d e r of 2 G ~ v - ~ , whereas e x p e r i m e n t a l l y b z 1 0 G ~ v - ~ Hence t h e model i s i n v i o l e n t disagreement w i t h experiment.

This unhappy s i t u a t i o n can be avoided by i n c l u - ding momentum dependent phases i n t h e model. 14]Such phases correspond t o t r a n s l a t i o n s i n b-space.

I3.l

Thus t h e phases can s p r e a d o u t t h e b-space volume occupied by t h e produced p a r t i c l e s ( s e e t h e f i g u r e below). It i s obvious t h a t by i n c l u d i n g phases t h e model can be brought i n t o agreement w i t h t h e d a t a . h u t t h e r e s u l t s t h u s o b t a i n e d depend completely on how t h e phases a r e handled. This means t h a t t h e mo- d e l has e s s e n t i a l l y no p r e d i c t i v e power concerning e l a s t i c s c a t t e r i n g . I t can accommodate, f o r example, an a r b i t r a r i l y f a s t s h r i n k a g e a s w e l l a s no s h r i n - kage a t a l l .

NO PHASES : PHASES INCLUDED : absorption region

in the UJM.

absorption region from elastic \

scattering doto

I n t h e m u l t i p e r i p h e r a l model (MPM) t h e r e l e v a n t v a r i a b l e s a r e n o t t h e t r a n s v e r s e momenta of t h e produced p a r t i c l e s b u t t h e momentum t r a n s f e r s of t h e v i r t u a l p a r t i c l e s ( o r Reggeons) exchanged. I n simple v e r s i o n s of t h e f.ITM t h e m p l i t u d e i s assumed

;I

t o f a c t o r i z e i n t o a product of ;he momentum t r a n s - f e r s t (short-range o r d e r ) :

j

n - I

T (ab-4 ... n ) =

^I^I

= TT

^{e a t i}

i = l

b n

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S-C?IANNEL PHENOMENOLOGY O F DIFFRACTION SCA'ITERING Cl-265

The impact parameter s t r u c t u r e o f t h e MPM i s ve- r y d i f f e r e n t from t h a t of t h e UJM. I n t h e MPM t h e n a t u r a l b-space v a r i a b l e s a r e t h e impact parameter s t e p s Abj, i. e . t h e d i f f e r e n c e s between t h e im- p a c t parameters of t h e p a r t i c l e s produced n e x t t o each o t h e r i n t h e chain. The assumption t h a t t h e t . ' s f a c t o r i z e i m p l i e s t h a t t h e impact parameter

3

s t e p s a r e independent. T h i s means t h a t t h e MPM i s

7 1

e q u i v a l e n t t o a random walk i n impact parameter.

The parameters c h a r a c t e r i z i n g t h e b-space walk of t h e MPM, t h e s t e p l e n g t h <Ab

'

and t h e t o t a l

j-

number of s t e p s , can be e s t i m a t e d from p a r t i c l e p r o d u c t i o n d a t a . Numerically t h e above n a i v e v e r - s i o n of t h e MPM t u r n s o u t t o be i n s t r o n g d i s a g r e e ment w i t h experiment : t h e d i f f e r e n t i a l cross-sec-

t i o n g i v e n by t h e model has much too s t e e p t-depen dence ( b ~ 3 0 G ~ v - ' a t 100 GeV/c) and s h r i n k s much too f a s t (a' N 3 G ~ V - ' ) .

C6'

]These t r o u b l e s can

P

b e cured a t l e a s t p a r t i a l l y by i n t r o d u c i n g c l u s t e r i n g e f f e c t s i n t o t h e model. Q u a l i t a t i v e l y i t i s obvious t h a t c l u s t e r i n g works i n t h e r i g h t d i r e c - t i o n : i t reduces t h e number of s t e p s i n t h e c h a i n . Thus i t reduces b o t h t h e t - s l o p e and t h e speed of t h e shrinkage. Q u a n t i t a t i v e l y , however, i t i s n o t c l e a r i f t h e c l u s t e r i n g e f f e c t s a r e enough t o b r i n g t h e model i n t o agreement w i t h d a t a ( n o t i c e t h a t t h e s i z e of t h e c l u s t e r s i s s t r o n g l y c o n s t r a i n e d by t h e m u l t i p a r t i c l e c o r r e l a t i o n d a t a I). C a r e f u l nu- m e r i c a l s t u d i e s of t h e c l u s t e r i n g e f f e c t s i n t h e WM would c l e a r l y be of g r e a t i n t e r e s t .

Another p o s s i b l e way t o modify t h e simple MPM i s t o i n t r o d u c e u n i t a r i t y c o r r e c t i o n s . They cause long-range c o r r e l a t i o n s among t h e produced p a r t i - c l e s . Hence t h e walk i n t h e u n i t a r i z e d MF'M i s n o t

random b u t c o r r e l a t e d . Q u a n t i t a t i v e s t u d i e s o f t h e b-space s t r u c t u r e of t h e u n i t a r i z e d v e r s i o n s of

t h e m u l t i p e r i p h e r a l model would be v e r y i n t e r e s - t i n g .

The problem of how t o b u i l d up t h e Pomeron (and t h e meson t r a j e c t o r i e s ) i n t h e d u a l models has been a c t i v e l y s t u d i e d i n t h e l a s t y e a r . S e v e r a l v e r y i n t e r e s t i n g new i d e a s h a s been p r e s e n t e d . They w i l l be reviewed i n t h e t h e o r e t i c a l s e s s i o n s so we s h a l l n o t d i s c u s s them h e r e .

4 . The D i f f r a c t i v e Component G ( t ) : The shadow D-

of t h e i n e l a s t i c d i f f r a c t i o n G ( t ) c a n be c a l c u - D

l a t e d assuming i ) t h e e x c i t a t i o n spectrum, i i ) t h e phase of t h e e x c i t a t i o n and i i i ) t h e decay of t h e d i f f r a c t i v e l y produced systems. GD(t) i s t h e n ob- t a i n e d by c a l c u l a t i n g t h e loop i n t e g r a l .

T h i s problem h a s been s t u d i e d i n d e t a i l by Sa- k a i and White

p?

These a u t h o r s show t h a t t h e b- space s t r u c t u r e of t h e d i f f r a c t i v e o v e r l a p func- t i o n GD(b) depends c r i t i c a l l y on t h e assumed s p i n and h e l i c i t y s t r u c t u r e s of t h e e x c i t e d systems.

They d e t e r m i n e t h e e x c i t a t i o n v e r t i c e s from t h e ISR p r o t o n s p e c t r a . A c a l c u l a t i o n w i t h t h e assump- t i o n s of a l i n e a r mass-spin r e l a t i o n and of s-chan- n e l h e l i c i t y c o n s e r v a t i o n l e a d s t o a v e r y c e n t r a l p r o f i l e f o r G ( b ) . A s i m i l a r c a l c u l a t i o n w i t h t h e

D

assumption of t-channel h e l i c i t y c o n s e r v a t i o n , on t h e o t h e r hand, l e a d s t o a p e r i p h e r a l p r o f i l e : t h e d i f f r a c t i v e p r o d u c t i o n i s happening a t t h e edge of t h e a b s o r p t i o n r e g i o n around b z l f e r m i ( s e e f i g . 2 )

The experimental d a t a of low mass d i f f r a c t i v e p r o d u c t i o n i s known t o be i n rough agreement w i t h t-channel h e l i c i t y c o n s e r v a t i o n (and t o completely d i s a g r e e w i t h s-channel h e l i t i c i t y c o n s e r v a t i o n . ) . The assumption of e x a c t t-channel h e l i c i t y conser- v a t i o n i s a c t u a l l y n o t a t a l l c r u c i a l f o r o b t a i - ning a p e r i p h e r a l G ( b ) . Any combination of ampli-

D

tudes w i t h a "reasonable" amount of s-channel he- l i c i t y f l i p amplitudes would g i v e a p e r i p h e r a l r e s u - l t f o r t h e d i f f r a c t i v e shadow.

19

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C1-266 H . I . MIET

I

b 0.5 1.0 1.5 fermi

-Fig. 2 : The decomposition of t h e imaginary p a r t of t h e e l a s t i c amplitude i n t o i t s components assuming t-channel h e l i c i t y c o n s e r v a t i o n i n i n e l a s t i c d i f f r a c t i o n . From Ref. 191.

5.- Pomeron Phenomenology : The measurements

bOl

of e l a s t i c d i f f e r e n t i a l c r o s s - s e c t i o n s i n proton- p r o t o n s c a t t e r i n g a t t h e ISR have a t t r a c t e d much p ~ ~ e n o m e n o l o g i c a l i n t e r e s t . The d a t a show c l e a r s t r u c t u r e s a t small t-values : a r a p i d change of t h e s l o p e of t h e o r d e r of Ab 2 t a k e s p l a c e a t t " -0.1 Gev 2

.

Another i n t e r e s t i n g f e a t u r e of t h e d a t a i s t h a t t h e break observed a t lower e n e r g i e s a t t = -1.3 GeV 2 h a s developed t o a b e a u t i f u l d i f - f r a c t i o n minimum a t t h e ISR.

The observed s m a l l - t s t r u c t u r e can be simply ac- comodated by a two-component "disk"

+

"ring" para- m e t r i z a t i o n of t h e i n e l a s t i c o v e r l a p f u n c t i o n Ginel(t) ( o r , a l t e r n a t i v e l y , of t h e e l a s t i c s c a t t e - r i n g a m p l i t i d e i t s e l f ) . S e v e r a l a u t h o r s have p e r f o r - med such two-component f i t s

Cl?

T h e i r r e s u l t s show

t h a t i ) i t i s easy t o f i t t h e d a t a w e l l and i i ) t h e r e l a t i v e s i z e of t h e two components depends c r i t i c a l - l y on t h e i r assumed shapes. The s m a l l - t break c o r r e s - ponds t o a long t a i l i n G (b) extending o u t t o

i n e l

b-values much l a r g e r than one f e r m i . I n t h e two-component f i t s t h i s large-b t a i l can be d e s c r i b e d e i t h e r by t h e "disk" component ( s e e e.g t h e f i t s of Sakai

t l J

and White

lgl

and Henzi and V a l i n o r by t h e "ring"

1 1

1

component (Henyey e t al.[

).

I t i s v e r y tempting t o i d e n t i f y t h e "ring" compo- n e n t w i t h i n e l a s t i c d i f f r a c t i o n and t h e "disk" component w i t h n o n - d i f f r a c t i v e production[31. I n such

r e s u l t s of Sakai and White a l s o s u p p o r t t h i s i d e a . I n t h e i r model c a l c u l a t i o n shown i n f i g . 2 t h e d i f - f r a c t i v e component i s e s s e n t i a l l y a r i n g c e n t e r e d a t I f e r m i and nail-diffractive component dominates b o t h a t s m a l l and a t l a r g e impact parameters. Henyey e t a l . have a d i f f e r e n t i n t e r p r e t a t i o n . They a s s o c i a t e t h e "ring" component t o " d i s s o c i a t i o n p r o c e s s e s " , t h a t i s t o p r o c e s s e s i n which t h e beam p a r t i c l e s d i s - s o c i a t e i n t o t h e i r v i r t u a l c o n s t i t u e n t s b e f o r e s c a t - t e r i n g and then o n l y some of t h e s e c o n s t i t u e n t s i n t e r - a c t w i t h r h e t a r g e t .

L e t u s f i n a l l y comment on t h e observed energy dependence of t h e d a t a . A c a r e f u l s t u d y of G inel@) a s a f u n c t i o n of t h e incoming energy r e v e a l s two v e r y important r e s u l t s ( s e e Ugo Amaldi's review i n t h e s e p r o c e e d i n g s ) : i ) t h e v a l u e of G (b=O) i s e s s e n t i a l -

i n e l

l y c o n s t a n t through t h e whole ISR energy r a n g e a t a l e v e l which i s c l e a r l y below t h e u n i t a r i t y l i m i t (roughly 92-94% of t h e u n i t a r i t y l i m i t ) and i i ) t h e observed r i s e of t h e proton-proton t o t a l c r o s s - s e c t i o n comes from t h e r e g i o n around 6 = 1 f e r m i . The f i r s t r e s u l t i n d i c a t e s t h a t t h e r i s e of tot is n o t caused by a "naive" s a t u r a t i o n of t h e u n i t a r i t y l i n i t ( i n which t h e a b s o r p t i o n f i r s t r e a c h e s t h e 100% c e i l i n g a t b = 0 and then s t a r t s t o expand o u t i n b) b u t i s a a more complicated e f f e c t . A f t e r t h i s o b s e r v a t i o n i t

$$ n a t u r a l tFiat t h e r i s l n g c o n t r i B u t i o n i s c e n t e r e d a t I f e r m i . An expanding "disk" component w i t h a r a d i u s of about 1 f e r m i would obviously g i v e such a r i s e . A l t e r n a t i v e l y , t h e r i s e could be caused by a growing "ring" c o n t r i b u t i o n c e n t e r e d a t I f e r m i . The q u e s t i o n of t h e p h y s i c a l o r i g i n of t h i s r i s e i s one of t h e most e x c i t i n g problems of t o d a y ' s hadron phy- s i c s and i n s o l v i n g i t one w i l l probably l e a r n q u i t e a l o t about t h e s t r u c t u r e of elementary p a r t i c l e s .

ACKNOWLEDGEMENTS :

I thank Chan Hong-Mo, E.H. De Groot, F.S. Henyey, G.L. Kane, R.J.N. P h i l l i p s , R.G. R o b e r t s , DP. Roy e t R. Worden f o r many u s e f u l d i s c u s s i o n s on d i f f r a c - t i o n s c a t t e r i n g . I am g r a t e f u l t o D r . C . Michael f o r c r i t i c a l r e a d i n g of t h e manuscript.

an i n t e r p r e t a . t i o n one should probably blame t h e non- d i f f r a c t i v e component f o r t h e large-b t a i l . The

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S-CHANNEL PHENOMENOLOGY OF DIFFHKCTION SCATTERING

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@]

~e GROOT (E

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