HIGH MULTIPLICITY REACTIONS (THEORY)

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HIGH MULTIPLICITY REACTIONS (THEORY)

A. Mueller

To cite this version:

A. Mueller. HIGH MULTIPLICITY REACTIONS (THEORY). Journal de Physique Colloques, 1973, 34 (C1), pp.C1-307-C1-315. �10.1051/jphyscol:1973141�. �jpa-00215221�

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JOURNAL DE PHYSIQUE Colloque C1, suppl6ment au nO1O, Tome 34, Octobre 1973, page C1-307

HIGH MULTIPLICITY REACTIONS (THEORY) A. rnLLER

Department of Physics, Columbia University, NEW YORK

1.- INTRODUCTION. - Two years ago, and perhaps even a year ago, theorists seemed farahead of experimenta- lists in the area of multiple particle production.

At that time very little was firmly established experimentally at high enough energies to be of real use to phenomenologists, and wildly contradictory theories could easily satisfy the few experimental constraints which did exist. All this has now changed.

An enormous amountof data have come out in the past year, a number of possible theories have been elimi- nated and those that survive are not of a detailed enough nature even to seriously attempt to fit all the experimentally known facts. Probably the major piece of information for theorists in the past year has been the correlation data which indicate that at

some level of approximation a short range correlation model is necessary [I].

Once the idea of approximate short range correlation has been accepted it is natural to attempt to understand the corrections necessary for a realistic picture of hadronic production. The motivation for including corrections to short range correlations is not only forced on theorists by the experimental facts. It has been known for a long time that a purely short range correlation model with constant or rising cross sections is internally inconsistent

[2-61. Most theoretical work in the past year has centered on ways of reconciling the ideas of short range correlation and diffraction scattering, ideas which are phenomenologically rather well represented at present energies by the two component model 161.

These attempts have taken two theoretically distinct directions. On the one hand there is a class of models in which the corrections to factorization, represented by the Pomeranchuk pole at J = 1 , decrease

as 1/Pn s at extremely high energies,making the idea of short range correlations closer to the truth at ultra-high energies than at present energies. On the other hand there is the class of models in which short range correlations are a transient phenomenon at present energies and are completely lost at ultra-

high energies where cross sections increase as en s. 2 Before going on to detailed considerations let me list the experimental basis on which I shall judge theoretical models to be correct or incorrect. Thus, the assumed facts are 171 :

1) Limiting fragmentation (hadronic scaling) has occurred or is being approached in all reactions at present NAL and ISR energies. Whether hadronic scaling will be exactly true at ultra-high energies is left as an open question. Only that an approximate, up to say 15%, scaling is true at present energies is assumed.

2) Cross sections rise throughout the ISR energy range.

3) There is a central plateau to some reasonable extent at ISR.

4 ) Short range correlations are very dominant over

long range correlations in the central region at ISR.

5) Factorization is correct in the fragmentation region.

11.- TWO (AND MORE) COMPONENT MODELS. - ^Before

going into details of two component type models let me give a motivation for the in which I shall discuss such models. If the central region at ISR shows short range correlations, which seems to. be the case, and if most events have particles popula- ting the central region, which also seems to be the case, then a purely short range correlation model may be a good zeroth approximation to reality. This point of view is, I believe, very widely accepted.

If a short range correlation model is a good zeroth approximation then a Regge pole model should be an equally good zeroth approximation for elastic ampli- tudes,at least near forward directions. This is seen simply by the following argument. Consider an idealized event, A+B -, 1 + 2 + ...+ ^{n, in}^thezeroth^appro-

ximation, on a rapidity plot as shown in figure 1.

The particles have been numbered according to an increasing laboratory rapidity. In impact space an idealized event looks like the picture in figure 2,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973141

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A . MUELLER

FIG 'la _{F I G} Ib

F I G 2

where t h e numbering of p a r t i c l e p o s i t i o n s i n f i g u r e 2 i s i n correspondence w i t h t h a t of f i g u r e 1. I n im- p a c t space a random walk w i t h a c o n s t a n t s t e p l e n g t h t a k e s place. The o v e r a l l s i z e i n impact space f o r t h e event i s &( x ) - ^{f y .}Thus t h e whole c o l l i s i o n t a - k e s p l a c e w i t h i n a s i z e a + ^bf i i n impact parame- t e r space. This s i z e must r e f l e c t i t s e l f i n a s h r i n - k i n g d i f f r a c t i o n peak i n t h e e l a s t i c A + B ^-1A + B r e a c t i o n shown i n f i g u r e 3 .

A

5 FIG 3

T h i s s h r i n k i n g d i f f r a c t i o n peak i n t u r n i s t h e s i - gnal t h a t a Regge p o l e i s dominating n e a r t = O . Un- f o r t u n a t e l y , t h e slow i n c r e a s e i n t h e s l o p e parame- t e r w i t h energy means t h a t t h e average s t e p l e n g t h i n impact parameter space i s q u i t e small. L a t e r , we s h a l l s e e t h e t r o u b l e s t h a t t h i s l e a d s t o . Neverthe- l e s s , t h e z e r o t h component of h i g h energy, h i g h mul- t i p l i c i t y r e a c t i o n s should be a f a c t o r i z a b l e Regge p o l e w i t h a ( 0 ) n e a r 1.

C l e a r l y t h i n g s have been l e f t o u t of t h e z e r o t h o r - d e r approximation. I n terms of experimental e v e n t s

t h e r e i s no mechanism i n z e r o t h o r d e r which can g i - v e r i s e t o r a p i d i t y c o n f i g u r a t i o n s having a l a r g e gap a s shown i n f i g u r e 4 .

w A' B'

LARGE GAP -

1 1 1 1 1 1 * - 1 1 1 1 1 1

0 Y

F I G 4

Events w i t h a l a r g e gap a r e c a l l e d d i f f r a c t i v e e v e n t s . I n f a c t such e v e n t s should be given by t h e exchange of a Pomeron a s shown i n f i g u r e 5.

FIG 5

T h i s i s n o t p r e s e n t i n t h e z e r o t h approximation.

(For t h e moment we assume t h e masses of t h e clumps A ' and B ' a r e n o t t o o l a r g e ) . U n i t a r i t y r e q u i r e s a c o n t r i b u t i o n t o t h e e l a s t i c amplitude a s shown i n f i g u r e 6.

F I G 6

Following Gribov [8,9] we s h a l l assume t h a t t h e Regge p o l e s i n f i g u r e 6 e i t h e r correspond t o a de- f i n i t e s e t of Feynman graphs o r a t l e a s t t h a t they behave a s i f t h e r e were such a correspondence. Such a n assumption i s a t t h e h e a r t of t h e Gribov c a l c u - l u s . I f now we t a k e t h e imaginary p a r t of t h e ampli- t u d e from which f i g u r e 5 comes, t h e r e s u l t shown i n f i g u r e 7 i s o b t a i n e d [9,10].

FIG 7

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HIGH MULTIPLICITY

T h i s i s p a r t of t h e two Reggeon c u t c o n t r i b u t i o n . The c o n t r i b u t i o n s a r e shown i n f i g u r e 8 according t o r e g i o n s of r a p i d i t y space which a r e populated.

)

$ + - J z

( ^;I1 - ⁿ^rv^const ^{1111 Y}

, F I G 8 a

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l l l l l l o - n = 2 a y + b 1 Y F I G 8 c

The c o n t r i b u t i o n s (a), (b), ( c ) , (d) of f i g u r e 7 have r e l a t i v e weights 1, -2, -2, 2 S O t h a t t h e ma- g n i t u d e of t h e c o n t r i b u t i o n of f i g u r e 7 i s t h e same a s t h a t of f i g u r e 5, except t h a t t h e s i g n h a s changed. Since t h e c o n t r i b u t i o n of f i g u r e 5 i s obviously a p o s i t i v e c o n t r i b u t i o n t o t h e c r o s s s e c t i o n we may w r i t e t h e c o n t r i b u t i o n s t o t h e c r o s s s e c t i o n of t h e z e r o t h and f i r s t o r d e r terms a s

u ( s ) = C - ALL- a + b P n s ₃

where we assume f o r s i m p l i c i t y t h a t t h e z e r o t h o r d e r term i s c o n s t a n t , i.e. t h e Pomeron h a s u n i t i n t e r - cept. These a r e t h e f i r s t two terms i n a n expansion about a s h o r t range c o r r e l a t i o n l i m i t . The n e g a t i v e s i z e of t h e c u t , f o r which we have argued h e u r i s t i c a l - l y h e r e , h a s been derived by White u s i n g t-channel u n i t a r i t y [ I l l .

F i g u r e 7 and 8 i l l u s t r a t e what may w e l l be a gene- r a l p r o p e r t y of c o r r e c t i o n s t o t h e z e r o t h o r d e r approximation, t h a t i s : c o n t r i b u t i o n s t o d i f f e r e n t r e g i o n s of phase space a r e s t r o n g l y c o r r e l a t e d and should not be t r e a t e d independently. For example, t h e "naive" two component model which c o n s i d e r s t h e z e r o t h o r d e r term along w i t h t h e c o n t r i b u t i o n of f i - gure 7a would s a y t h a t t h e breaking of s h o r t range c o r r e l a t i o n s i s due t o d i f f r a c t i v e l y produced e v e n t s and can be t a k e n i n t o account by f i r s t s u b t r a c t i n g t h e c o n t r i b u t i o n of f i g u r e 7a, t h e e v e n t s w i t h a l a r - ge r a p i d i t y gap, from t h e d a t a . However, t h e c o n t r i - b u t i o n s of f i g u r e 7b, f i g u r e 7c and f i g u r e 7d a l s o c o n t r i b u t e t o long range c o r r e l a t i o n s and cannot be e a s i l y s e p a r a t e d phenomenologically.

One f i n a l comment on e q u a t i o n (l).At l a r g e s , equa-

t i o n (1) shows t h e t o e a 1 c r o s s s e c t i o n r i s i n g t o a c o n s t a n t l i m i t . I t might be tempting t o say t h a t t h i s accounts f o r t h e r i s e i n t o t a l c r o s s s e c t i o n s i n t h e ISR energy range. T h i s i s a dangerous leap.

I n p a r t i c u l a r , a s was mentioned before, t h e s l o p e parameter of t h e d i f f r a c t i o n peak i s weakly depen- d e n t on energy,which means t h a t b < a i n e q u a t i o n (1) so t h a t t h e c o n s t a n t c ' would have t o be q u i t e l a r g e t o g i v e a r e a s o n a b l e e f f e c t . But more t o t h e p o i n t , t h e d e c r e a s e i n e q u a t i o n (1) i s due t o a s h r i n k i n g d i f f r a c t i o n peak. I f such a shrinkage e f - f e c t were causing t h e r i s e of t h e c r o s s s e c t i o n , t h e n t h e e l a s t i c c r o s s s e c t i o n should continue t o decrease, which i s n o t what seems t o be happening.

The conclusion t o be drawn i s t h a t t h e i n c r e a s e i n t h e t o t a l c r o s s s e c t i o n i n t h e ISR range cannot be o b t a i n e d from lowest o r d e r c u t c o n t r i b u t i o n t o t h e e l a s t i c amplitude. We t u r n now t o t h e next term i n t h e component expansion.

Consider t h e e v e n t s which can be represented by t h e schematic i l l u s t r a t i o n i n f i g u r e 9.

A' LARGE GAP

-, ^B'

I I I I I I I I 0

'-

Y FIG 9

Here t h e mass of A ' i s l a r g e w h i l e t h e mass of B ' i s n o t l a r g e . I n p a r t i c u l a r i f B ' c o n s i s t s of a s i n - g l e p a r t i c l e , f i g u r e 9 i s an event corresponding t o t h e t r i p l e Pomeron l i m i t shown i n f i g u r e 10.

FIG 10

The e q u a t i o n corresponding t o f i g u r e 10 i s w e l l known and need n o t be used e x p l i c i t l y f o r o u r purpo- ses. J u s t a s f i g u r e 5 l e d t o f i g u r e 7 so f i g u r e 10 l e a d s t o f i g u r e 11.

FIG 11

Events i n f i g u r e 9 correspond t o term (a) of f i g u r e 11 w h i l e f i g u r e 12 shows t h e e v e n t s corresponding

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C1-310 A . MIJELLER

t o (b) o r (c) and (d) of f i g u r e 11. I n f i g u r e 12b, p o i n t t o be made h e r e i s t h a t t h e r e i s l i t t l e r e a - t h e r i g h t hand p a r t of t h e r a p i d i t y a x i s i s popula- son t o p i c k a p a r t i c u l a r region of phase space a s t e d w i t h e v e n t s w i t h twice t h e d e n s i t y of t h e r e s t a l i k e l y p l a c e f o r a r i s e of t h e c r o s s s e c t i o n w i t h - of t h e r a p i d i t y p l o t . T h i s double d e n s i t y corresponds o u t f i r s t a r g u i n g t h a t o t h e r r e g i o n s of phase spa- t o t h e two Pomeron being c u t j u s t a s i n f i g u r e 8c. c e do n o t g i v e d e c r e a s i n g c o n t r i b u t i o n s t o t h e c r o s s

s e c t i o n .

Beyond t h e s e f i r s t t h r e e components t h e expansion

1 1 1 1 1 1 1 1 1 1 1 1 continues. I n f i g u r e 13 some t y p i c a l terms of t h e Y component expansion (Gribov c a l c u l u s ) a r e l i s t e d FIG 1 2 0 w i t h t h e e n e r g i e s a t which t h e i r a s v m ~ t o t i c - - . beha-

I I I I I I ~ I I I I I I I I I I I I ~ I v i o r s e t s i n . N Jill

0 Y

\ I

FIG 1 2 b

Again t h e c o n t r i b u t i o n s of t h e terms (a), (b), (c) and (d) of f i g u r e I1 c o n t r i b u t e i n t h e r a t i o s 1, -2, -2, 2 b u t now, assuming a z e r o i n t h e t r i p l e Pomeron coupling, t h e t o t a l c o n t r i b u t i o n of f i g u r e 11 t o t h e amplitude i s

The c o n s t a n t c" i s i n t e r p r e t e d a s a r e n o r m a l i z a t i o n of t h e c o n s t a n t c i n e q u a t i o n ( l ) , w h i l e c"' g i v e s t h e s t r e n g t h of t h e t r i p l e Pomeron c o n t r i b u t i o n t o t h e cut.

At p r e s e n t e n e r g i e s i t i s , phenomenologically, probably b e t t e r t o n e g l e c t t h e z e r o of t h e t r i p l e Pome- ron coupling, i f such a zero e x i s t s , and a l s o t o ne- g l e c t t h e Pomeron s l o p e so t h a t t h e c o n t r i b u t i o n of f i g u r e l l a t o (I i s I n s + D . (This, a g a i n , i s a consequence of t h e small s t e p l e n g t h i n impact parameter space mentioned e a r l i e r ) . It i s t h i s term which h a s been suggested by numerous a u t h o r s [12-151 a s t h e source of t h e r i s i n g c r o s s s e c t i o n s a t ISR. There i s no unanimity on t h e s i z e of 1: 1, b u t i t seems possi- b l e t h a t i s s u f f i c i e n t l y l a r g e t h a t a term l i k e

- 1: I t n s could g i v e a 2-3 mb i n c r e a s e i n i n e l a s t i c c r o s s s e c t i o n s i n t h e ISR energy range. The phenome- n o l o g i c a l s i t u a t i o n i s s t i l l t o o confused f o r a d e f i - n i t e answer on t h i s p o i n t . There remains a t h e o r e t i - c a l o b j e c t i o n t o such a r i s i n g c r o s s s e c t i o n mecha- aism. T h i s o b j e c t i o n i s t h a t t h e terms (a), (b), (c) and (d) of f i g u r e 11 a r e n o t r e a l l y independent. At u l t r a h i g h e n e r g i e s , when added t o g e t h e r , t h e y g i v e a d e c r e a s i n g term t o t h e c r o s s s e c t i o n . ( ( a ) a l o n e g i v e a n i n c r e a s i n g term t o t h e c r o s s s e c t i o n . ) It does n o t seem r e a s o n a b l e t o n e g l e c t c o n t r i b u t i o n s (b), (c) and (d) and c l a i m t h a t (a) i s r e s p o n s i b l e f o r t h e i n c r e a s i n g c r o s s s e c t i o n . I t h i n k t h e main

w ^s ^large

M

_s^s ^largel a r g e s % l a r g e

s % ^large

FIG 13

Proponents of t h e component expansion c l a i m t h a t a t p r e s e n t e n e r g i e s one need keep only t h o s e terms which a r e a l r e a d y asymptotic when JY i s l a r g e .

The p r e s e n t a t i o n which I have given h e r e supposes t h a t t h e s i g n of t h e two Pomeron c u t i s n e g a t i v e , a s argued by White. There a r e o t h e r component e x p a n s i o n s somewhat l i k e t h e one d e s c r i b e d h e r e which have a po- s i t i v e s i g n f o r t h e c u t [ 1 4 , 1 6 ] . At t h i s p o i n t 1 t h i n k w e must c o n s i d e r t h e n e g a t i v e s i g n f o r t h e c u t a s e s -

t a b l i s h e d . The above a u t h o r s r14,16] do make an i m - p o r t a n t d i s t i n c t i o n which a s s o c i a t e s a d i f f e r e n t Regge p o l e w i t h a l a r g e r a p i d i t y gap, a s i n f i g u r e 4 , t h a n w i t h t h e z e r o t h approximation i n f i g u r e l a . The Regge p o l e i n f i g u r e 5, o r any o t h e r e x c l u s i v e Regge pole, i s t h e t r u e Pomeranchuk t r a j e c t o r y w h i l e t h a t Regge p o l e corresponding t o t h e z e r o t h a p p r o x i - mation, which i s b u i l t from e v e n t s having no l a r g e r a p i d i t y gaps, i s an e f f e c t i v e Pomeron a t low e n e r - g i e s . The expansion which I have d e s c r i b e d i n d e t a i l i s what Gribov c a l l s a "bare Pomeron" expansion and does not d i s t i n g u i s h between e x c l u s i v e and i n c l u s i v e Pomerons. As a m a t t e r of p r i n c i p l e I b e l i e v e t h a t

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H I G H MULTIPLICITY REACTIONS (THEORY)

such a distinction might be useful in the Gribov calculus, but as a matter of practice there is no essential difference for low energy applications where only a few components are used.

Some more general aspects of cuts will be discussed by Amati*. Let me just remark here that if one follows the philosophy of Bronzan [17] and of Cardy and White [IS], then the component expansion described above will either converge badly or perhaps not at all. These authors feel that the iteration of high order cuts is very important. In particular, White feels that the break in the slope parameter at ISR cannot be obtained from a component expansion.

CardyandWhite claim that for a consistent Pomeron the total cross section factorizes up to terms in 1/Pn s

Clearly such an equation cannot be obtained from a component expansion. Whether the component expansion is useful or whether something more along the lJnes of White is correct cannot yet be decided with any certainty. An objection-free model exhibiting the features which White suggests would be very helpful.

3. - RISING CROSS SECTIONS. - Mechanisms for pro- ducing rising cross sections invariably involve high multiplicity production. We have already discussed

one such mechanism in the previous section, that of diffractive production of high mass states. As mentioned before this is a phenomenological possibility, but there is no direct verification or contradiction from ISR data as yet. In this section some other possible mechanisms will be discussed.

3 1.- ANTI-BARYON PRODUCTION [19,20,21]. - ^The

cross section for anti-baryon production presumably is rising at ISR. There is direct data on p ^produc-

tion which shows that the p cross section rises by about 1/2mb in the ISR energy range. If other anti- baryon (heavy particle) production cross sections were to rise by similar amounts, it would be possible to obtain the 3 m b rise of the inelastic cross section in the ISR range. One nice feature of using this mechanism to obtain increasing cross sections is that a high energy threshold naturally occurs.

We can see how this happens by a glance at figure 14,

F I G 14 ^-\

where solid lines refer to heavy particles and da- shed lines refer to light particles. Assume the mass of the heavy particle is M, then

(kl - k212 = a .

2 2 2 2 2 2

call kl = - KI, k2 = - ^K~^where^K~^z^K~= ^(t) ^>^0.

Then,

-2kl.k2 s M? f 2(t) and

where ~y is the difference of rapidities of the momenta kl and k2, and k and k are transverse mo- e t a I t), ilcl 1 2 ^{and ilc2}l2 2are small compared

to 2, ^then

A z &-ne .

If z 1 G~V' and (t) EJ 0.1 GeV then A y 2 = ^2.

Thus it could easily take almost 4 units of rapidity to produce a baryon anti-baryon pair comforta- bly. This argument is reminiscent of the Chew- Snider [22] argument for an oscillating approach to a constant cross section. In this approach there would be an oscillation for each baryon-anti-baryon pair produced, and the oscillation would occupy - 4 units of rapidity. The amplitude of each successive oscillation is smaller than the previous oscillation and ultimately the cross section approaches a constant limit. At ISR energies of course, one need only be concerned with the increase of the first oscillation.

At present there is certainly no evidence that heavy particle production is responsible for the increasing cross section. The attractive features of this mechanism. are i) the possibili.ty of a very high mass threshold effect, and ii) that it is a mechanism, unlike diffractive production of large mass states, which is presumably not strongly cou- pled to the production of other particles, and can really be treated as a first order perturbation.

* D. AMATI, these proceedings.

3.2.- ELECTROMAGNETIC CORRECTIONS. - There is a possibility, although not a likelihood, that electromagnetic corrections give a sizable portion of

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t h e t o t a l proton-proton c r o s s s e c t i o n a t ISR [23].

For a n e s t i m a t e i t should s u f f i c e t o c o n s i d e r o n l y e 2 c o r r e c t i o n s t o t h e s t r o n g i n t e r a c t i o n s . Then one can say t h a t

Anern = - + uo e n s y .

There must be a P n s i n c r e a s e of t h e e l e c t r o m a g n e t i c c o r r e c t i o n s f o r a v a r i e t y of reasons. An obvious rea- son i s t h a t f o r t h o s e photons which a r e a c t u a l l y produced t h e r e i s a f u l l range of r a p i d i t y p r o p o r t i o n a l t o P n s i n which they can be produced. I f we choose a t o be 40mb, t h e approximate s t r o n g i n t e r a c t i o n c r o s s s e c t i o n , then, t h e d e t e r m i n a t i o n of aoem de- pends on y, t h e b a s i c s t r e n g t h of t h e photon coupling t o a l o c a l r e g i o n of r a p i d i t y i n t h e p-p c o l l i s i o n . Model c a l c u l a t i o n s of y must be considered u n r e l i a - b l e and s o one can probably say t h a t y i s somewhere between a/tr = 11500 and Amn /mr, 31100 , t h e e l e c t r o m a g n e t i c c o r r e c t i o n t o t h e pion mass. T h i s g i v e s aaem i n t h e c e n t e r of t h e ISR r e g i o n somewhere between 1 / 2 and 9mb. Common s e n s e says t h a t t h i s upper e s t i m a t e i s t o o l a r g e . T h i s i s t h e same a s s a y i n g t h a t t h e p i o n mass d i f f e r e n c e , t h e l a r g e s t e l e c t r o m a g n e t i c c o r r e c t i o n known, does n o t f e e d back s t r o n g l y i n t o t h e Pomeranchulc t r a j e c t o r y f u n c t i o n o r o t h e r p l a c e s which c o n t r i b u t e t o a P n s i n c r e a s e i n

0. For example a T ( 0 ) i n u n i t s of m-2 i s only about 31500. I n a l l l i k e l i h o o d t h e i n c r e a s e through t h e ISR energy range of e l e c t r o m a g n e t i c c o r r e c t i o n s i s f a r t o o small t o account f o r a c o n s i d e r a b l e p a r t of t h e r i s i n g c r o s s s e c t i o n .

Another, although o n l y a n a e s t h e t i c d i f f i c u l t y , i s t h e f a c t t h a t t h e r e i s no a p p a r e n t h i g h energy t h r e s - h o l d f o r e l e c t r o m a g n e t i c c o r r e c t i o n s t o t h e t o t a l c r o s s s e c t i o n . T h i s means t h a t any r i s e , o r f a l l , i n c r o s s s e c t i o n due t o e l e c t r o m a g n e t i c c o r r e c t i o n s , o r any o t h e r mechanism which does not have a h i g h energy t h r e s h o l d , should occur continuously from oelow AGS-PS e n e r g i e s through t h e h i g h e s t ISR energy. Thus, any mechanismwhich g i v e s a P n s i n c r e a s e t o t h e

c r o s s s e c t i o n , b u t which does n o t have a h i g h energy t h r e s h o l d e f f e c t , must c o n t r i b u t e about 12mb t o t h e t o t a l c r o s s s e c t i o n of t h e h i g h e s t ISR energy i n o r - d e r t o g i v e a 3mb i n c r e a s e t o t h e i n e l a s t i c c r o s s s e c t i o n i n t h e ISR energy region.

3.3.- NON-SOFT TERMS I N THE INTERACTION. - ^Bjorken

s c a l i n g and hadronic s c a l i n g , although fundamentally u n r e l a t e d , both p o i n t t o t h e f a c t t h a t t h e underlying f i e l d t h e o r y must be s o f t , i . e . super-renormalizable.

Regge c u t e f f e c t s , of t h e Mandelstam v a r i e t y , a r e

a l s o e f f e c t s of a s o f t theory. I f t h e underlying theo- r y a l s o h a s a h a r d component, i . e . t h e r e i s a renor- m a l i z a b l e b u t n o t super-renormalizable term i n t h e Lagrangian which i s on t h e o r d e r of 10% o r l e s s of t h e s o f t component, t h e n v i o l a t i o n s o f B j o r k e n sca- l i n g w i l l o c c u r and presumably a l s o P n s terms might occur i n t h e t o t a l c r o s s s e c t i o n and i n c l u s i v e c r o s s s e c t i o n s [23]. Such terms would be p a r t i c u l a r l y pro- minent a t l a r g e t r a n s v e r s e momenta and could be r e s - p o n s i b l e f o r p a r t of t h e c r o s s s e c t i o n a t l a r g e pT

[24]. As y e t , it i s i m p o s s i b l e t o do c a l c u l a t i o n s i n t h i s context. Furthermore, a s i n t h e p r e v i o u s sec- t i o n , t h e r e i s no obvious large-mass t h r e s h o l d . At t h e moment I t h i n k one must j u s t regard "hard i n t e r - a c t i o n s " a s a p o s s i b i l i t y f o r i n c r e a s i n g c r o s s s e c t i o n s w i t h no compelling argument i n t h e i r favor.

I n t h e n e x t s e c t i o n a whole c l a s s of models w i l l be considered which have a b u i l t - i n mechanism f o r o b t a i - ning i n c r e a s i n g c r o s s s e c t i o n s , t h a t of 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. We s h a l l s e e t h a t t h e s e models have both t h e o r e t i c a l and phenomenological weaknesses but, n e v e r t h e l e s s , they must be considered

s e r i o u s contenders f o r " t h e mechanism'' r e s p o n s i b l e f o r r i s i n g c r o s s s e c t i o n . However, none of t h e mechanisms considered h e r e i s r e a l l y v e r y convincing a t t h e p r e s e n t moment. Perhaps t h e combination of more d a t a

( r e a l p a r t s and t h e energy dependence i n t h e NAL energy range) and more t h e o r e t i c a l i n s i g h t w i l l soon l e a d t o t h e one o r more mechanisms r e s p o n s i b l e f o r t h e i n c r e a s e i n t h e c r o s s s e c t i o n .

4. EIKONAL AND ABSORPTION. - I n h i s o r i g i n a l paper, where t h e Pn s bound f o r t h e i n c r e a s e of any t o t a l 2

c r o s s s e c t i o n was derived from t h e Mandelstam r e p r e - s e n t a t i o n , F r o i s s a r t r251 a l s o gave a h e u r i s t i c a r - gument a s t o how t h i s upper bound could be achieved.

The argument i s based on p o t e n t i a l theory. Suppose

i s a n energy dependent p o t e n t i a l w i t h s t h e energy and = b a n impact parameter v a r i a b l e . E may be a s l a r g e a s d e s i r e d . The ebb' term r e f l e c t s t h e f i - n i t e range of t h e p o t e n t i a l . The p r e c i s e d e t a i l s of t h e p o t e n t i a l a r e not r e l e v a n t . The p o t e n t i a l i s ve- r y s t r o n g so long a s b < E / p Pn s and goes t o z e r o r a p i d l y f o r b > (c/p)Pn s . When u n i t a r i t y i s imposed, s a y by means of t h e Schrodinger e q u a t i o n , t h e s c a t t e - r i n g p r o c e s s should e x h i b i t s t r o n g i n t e r a c t i o n w i t h - i n t h e range b - < Ro = ~ / p P n s . The c r o s s s e c t i o n should be t h a t of a b l a c k d i s c of r a d i u s Ro, so t h a t

2 2

c7 2 n R and t h e energy growth i s l i k e Pn s .

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HIGH MULTIPLICITY REACTIONS (THEORY) C1-313

A cross section growing like Pn s would certainly 2 give a rising cross section in the ISR energy range.

If one writes

u = uo l ~ + c ~ n ~ s / s ~ j .

We should be able to assume that no log s/so term occurs for a reasonable choice of so, say on the order of a GeV. Then uo - 40mb and Pn s/so changes 2 by about 25 over the ISR range so that an increase of 4 m b requires that c be about 11250 (asymptotic energies then are s 2 el6 GeV 2 ). Thus, in order to have a reasonable energy scale, one needs a constant which is very small on any strong interaction scale.

This I feel, is an aesthetic difficulty of the theory. Despite their phenomenological difficulties, models which saturate the Froissart bound have an in- trinsic interest for theorists and may ultimately be useful phenomenologically if a small parameter or large energy scale can be found. I shall consider three classes of such models, including one in which the Pn s term is 2 not the mechanism for rising cross sections at present energies.

4.1.- CHENG-WU AND CHANG-YAN TYPE MODELS. - ^{The mo-}

dels discussed by Cheng-Wu [26] and by Chang-Yan [271 are similar but not identical. I shall briefly des- cribe the Cheng-Wu model as an orientation. In massive quantum electrodynamics, say for electron-electron

scattering, the dominant logarithmic terms, those of the form (e Pn 4 s)~, can be calculated exactly and they give, essentially, the asymptotic form

~(s,t) = i sl+' f (t) , ^E= (11132 )a 2

(3) for the amplitude. The relevant diagrams are shown in figure 15, where the single diagram on the right- hand side is a diagrammuch like a Gribov Reggeon diagram.

FIG 15

The amplitude written above is clearly unacceptable, since the corresponding total cross section grows as s

'

, wich clearly violates the Froissart bound. The graphs shown in figure 15 have correct t-channel analyticity but do not have s-channel unitarity, hence the Froissart bound is not built into this part of the model. All that has been learned is that the

leading logarithmic approximation gives bad results.

The fact that the electrons and photons are massive in this theory means that the f(t) in equation (3) is analytic at t=O. If we go to impact parameter space

which is reminiscent of ~roissart 's singularpotential.

Cheng and Wu suggest that the set of graphs of the type shown in figure 16 gives a field theory realiza- tion of a saturation of the Foissart bound, and that such a saturation is natural in massive quantum electrodynamics.

FIG 16

The graphs of figure 16 are calculated in a particular dominant logarithmic approximation and, indeed, in this approximation an eikonalization occurs which gives the full S matrix as

From equation (4) it is clear that the S matrix is zero when b 5 ~ / 4 @ Pn s and equal to 1 when b 2 € 1 4 ~ ~ en s ; which asymptotically gives a tn s 2 dependence to the cross section. In general, .the graphs of figure 16 are not expected to exhibit an eikonal form, but the hope is that the Froissart bound will be saturated because the potential is sin- gular. Such a saturation is not guaranteed, however, since the graphs of figure 16 do not obey the posf- tivity constraints necessary to derive the Froissart bound.

Abramovskii, Gribov and Kancheli [9] have given a criticism of the above approach. Theee authors argue that since the single particle inclusive distribution comes only from the single tower exchange, as shown in figure 17, one obtains a violation of energy con- servation by using the energy sum rule

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FIG 17

I think that this criticism can be stated in a slightly different way which suggests that field theory models which saturate the Froissart bound are going to be very hard to construct, at least so long as one follows the lines of Froissart's originalpo- tential model example. In equation (4) it is clear that one must keep terms in the sum up to n - ^s2' ^,

for a fixed b. But once n >> ens, the two initial particles do not have sufficient energy to make all the configurations of figure 16 asymptotic. Thus, the detailed cancellations which keep the sum over towers in figure 16 from violating the Froissart bound must come from non-asymptotic terms in the single tower.

In general it would seem a miracle if the graphs of figure 16 actually did respect the Froissart bound, It appears much more likely that these graphs increase without bound at high energy.

4.2.- UNITARY MODELS. - The Santa Barbara School has considered several classes of models in which multi-particle unitarity is a main feature, and main virtue, of their approach. They have a summary paper [28] out on this work so I shall mention only a few points. They have demonstrated that the inclusion of a sufficient number of graphs to insure multi-particle unitarity can create an infinite number of Regge trajectories going out to f w for a fixed t. For example the graphs shown in figure 18 treated in a very approximate sense, have such a feature [291.

FIG 18

In figure 18 the wavy lines may be Regge trajectories or just ordinary particles. The strength of the these graphs, that is, their ability to give a trajectory with a large intercept, is simply due to the fact that there is an enormous number of such graphs. These authors argue that such graphs can well dominate the graphs considered by Cheng-Wu and Chang-Yan. The work of this school is always carried out in the context of an eikonal-type model where unitarity is imposed, in order that the large intercepts of the Regge trajectories do not give violations of the Froissart bound. In fact, the 'cross section may even go to zero despite the high lying trajectories which actually lie on an unphysical sheet of the angular momentum plane.

Although these modelsaremotivated by the structureof field theory graphs it should be mentioned that they are not really field theory models in the sense that asymptotic values of sums of Feynman graphs are never really calculated. Also, since different models which they have considered give widely varying results, there are few direct phenomenological results from this approach. Nevertheless, to a theorist these models remain of interest because of multiparticle unitarity and the question of what happens when

such an enormous number of graphs is taken into account.

4.3.- FINKELSTEIN-ZACHARIASEN MODEL AND REFINE- MENTS. - The Finkelstein-Zachariasen r301 mode1 as developed by Caneschi-Schwimner [31], Finkelstein

r321, and Amati-Caneschi-Ciafaloni [33] is a model which has a self-consistent two-body scattering amplitude which grows as Pn s at infinite energies. 2 Asymptotically it is a long range correlation model, even in the central region, although 1/u da/dy has a constant central plateau. Diffraction production of large mass states is also consistently handled and decoupling problems are avoided since there is no factorization asymptotically. The model is schema- tically pictured in figure 19, where each bubble represents scattering on a black disc of radius propor- tional to the logarithm of the invariant mass of the pair of particles coming into the bubble. The bubble connecting the two incoming particles is interpreted as an initial state absorption factor. The model is a consistent model of diffraction scattering although it is rather ad hoc. Its justification is simply that it is consistent.

The model as described so far has no phenomenological implications at present energies.

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HIGH MULTIPLICITY REACTION mEORY) C1-315

FIG 19

However Finkelstein r321 and Amati-Ciafaloni-Caneschi [33] have given a perturbation scheme for construc- ting the final amplitude. The zeroth-order approximation is a pure short range correlation model as described in section 2 of this paper and corresponds to a bare Pomeron with intercept at or near 1. The next step is as in the Gribov calculus and various other component expansion models. Along with the single bare Pomeron exchange the contributions of large mass diffraction scattering, as shown in figure lla, are included along with further t-channel iterations.

The triple Pomeron coupling is assumed not to be zero and the t-channel iteration raises the intercept of the Pomeron above one by a small amount a zz l+g, where g represents the strength of the triple Pomeron coupling. Inconsistency is avoided by imposing the Finkelstein-Zachariasen model, figure 19 as a futher iteration prescription. The rise in the ISR cross

section is presumably due to large mass diffractive production rather than the beginning of a large black disc as in Cheng-Wu. The large black disc presumably only sets in at higher energies.

This model has three stages of development in energy. First an essentially multiperipheral component dominates. Next diffractive excitation is introduced much as in the component models of section 2, but with no necessity of a triple Pomeron zero at zero momentum transfer. Finally, and for present phenome- nology, irrelevantly, absorption is introduced for consistency. The model thus evolves from an essentially short range correlation model at low enegies to a dominantly long range correlation model at ultra- high energies. This model has no field theory basis underlying the consistency impositions ; its virtue is that it obeys the Froissart bound rather than any compelling logic. Overall, this model has built into it many desirable features and could be a useful starting point for understanding both ISR energy phe- nomenology and a consistent ultra-high energy diffraction component.

ACKNOWLEDGEMENT. - I have benefited much from dis- cussion with L. CANESCHI, C. DE TAR, L. FOA, J. WEIS and A. WHITE. Also, I wish to thank J. DONOHUE for much help in the preparation of this manuscript.

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