AN ITERATIVE APPROACH TO THE POMERON

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AN ITERATIVE APPROACH TO THE POMERON

C. de Tar

To cite this version:

C. de Tar. AN ITERATIVE APPROACH TO THE POMERON. Journal de Physique Colloques,

1973, 34 (C1), pp.C1-145-C1-148. �10.1051/jphyscol:1973114�. �jpa-00215194�

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AN ITEFIATIVE APPROACH TO THE POMERON

AN ITERATIVE APPROACH TO THE POMERON C. DE TAR

M.I.T.

In my view the essential problem in constructing a model for the Pomeron is this:

Theory: How do we arrange things so that we satisfy both s-channel and t-channel unitarity at once'?

Experiment: How do we account for the phenomenology?

I shall concentrate on the theoretical question.

In Table 1, I summarize the present theoretical status of the various models as discussed by Ian Hallidq in the previous talk (11.

TABLE 1

Present theoretical status of various models

I should like first to concentrate on the Reggeon caloulus [2,3] and speculate on the question of whether it satisfies s-channel unitarity and second to discuss an alternative solution to the Pomeron problem based on an iterative approach.

The version of Gribov's theory which interests us most is the one in which only the Pomeron is treated in a perturbative sense. In figure 1 I show the first few terms of the Gribov series for a four- point amplitude.

FIG. 1.

-

First few terms of Reggeon calculus expans ion

s-channel unitarity

As in conventional perturbation theory, diagrams with loops generate cuts

-

in this case in the J-plane.

The bubbles are elementary Pomeron-particle scatte- ring amplitudes which are themselves "irreducible"

with respect to the "exchange" of

two

or more

Pomerons, i.e. elementary cuts. The basic amplitudes do contain the elementary Pomeron pole, however. The series is contrived in such a way that it automati- cally satisfies J-plane (t-channel) unitarity.

No one has yet succeeded in summing the whole series. Attempts up to now have resorted to specific simplifying assumptions fortheelementary Pomeron- particle amplitudes and/or the neglect of many terms in the series[3]. Even if one were to succeed in summing the whole series one would have to address sign

egg eon OK OK

calculus

t-channel unitarity

the question of satisfying s-channel unitarity.

In order to study this question we must know how to take s-channel discontinuities of the Gribov series. Let us consider the first two terms as an example. To take the discontinuity of the first term, it is necessary to have a model for the elementary Pomeron, since the absorptive part of the first term is built up from those multiple production amplitudes which generate the elementary Pomeron.

The usual assumption is that these are multiperipheral amplitudes. In the language of the two-component multipar-

ticle

can be arranged

J-plane discontinuity formulae

meaning?

elastic:

aints Froissart

t-plane threshold discontinui-

ty formulae (e.g. Bron- zan-Jones condition) not established

s-channel eikonal

,

absorptive OK

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973114

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C1-146 C. DE TAR

models [4,5] the first term contributes to the "small remarkable resemblance to the eikonal series and rapidity gap" component. The procedure for taking

the discontinuity of the second term is now fairly well established [6].

There are four contributions to the absorptive part as shown in figure 2. Diagramatically these correspond to cutting the graph through the upper Pomeron, between the Pomerons, through the lower Pomeron or through both Pomerons. The first, third

FIG.2.

-

Diagrammatic representation of contributions to s-channel discontinuity from the first two terms of the series of Fig. 1.

and fcurth contributions involve cutting through the Pomerons and therefore reveal the same sum over inter- mediate states as in the first diagram - they therefore correspond to an additionalcontribution to the small rapidity gap component. They may also be inter- preted in the language of the absorptive model as an absorptive "correction" to the basic multi-production process [7]. The second contribution corresponds to a diffractive process involving the exchange of an elementary Pomeron. Included in this term, therefore, are the elastic, quasi-elastic, and other diffractive processes leading to any mass small or large.

It is fairly straightfo~ward to generalize this procedure to any term in the series. We may then ask the interesting question, what terms contribute to the discontinuity across the elastic s-channel cut?

The answer is, only those in figure 3. These bear a

suggest, that the Gribov series actually satisfies s-channel elastic unitarity. In particular, one might anticipate that if the elementary Pomeron were chosen with a(0) > 1 the sum of.the series would have a singularity with a (0) 5 1, just as

P in the eikonal calculation.

It seems highly plausible, therefore, that the Gribov series actually satisfies s-channel aad t-channel unitarity (J-plane unitarity) at once:

As such, it reveals what the eikonal and multiperipheral approaches leave out. And, of course, it also contains the answer to the decoupling dilemma.

But since no one knows how to sum the series, where do we go from here? A very attractive possibility, to my mind, is a suggestion made by proponents of the multiperipheral model [a] and more recently by various authors [5,9,10]

.

What if the Pomeron couples so weakly that for all practical purposes we need keep only the first few terms of the series, at least at presently available machine energies? Let us see how this idea can be realized within the Gribov model.

Treating the Pomeron perturbatively is a favourite procedure in the multiperipheral model and seems to have some phenomenological validity [5]

.

^{In fact}

these models bear a very strong resemblance to the Gribov approach. If we use the multiperipheral models to estimate the magnitude of the various elementary diffractive processes, we can then estimate the magnitudes of the terms in the Gribov series and see how well they might converge in a particular model, since the terms agree approximately with the multiperipheral model in magnitude

-

^though

some of them differ in sign.

Let us estimate the magnitudes of the first two terms shown in figure 1 at t = 0, assuming they represent the forward pp elastic amplitude well at moderate energies. The second term is equal in magnitude to the diffraccive component of the total cross section. For pp collisions at 15 GeV/c the elastic cross section is about 10 mb and known N*

cross sections contribute about 2 mb [ll].

So the diffractive component is at least 12 mb. at these energies. At NAL energies the elastic cross section is about 8 mb. and single diffractive dissociation about 7 mb. [12], giving a total diff-

+

⁰ ^. ^.

FIG. 3.

-

Diagrams contributing to elastic cut in ractive component of at least 15 mb. So the magnitude the s-channel.

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AN ITERATIVE APPROACH TO W E POMERON C1-147

of the second term is at least 40% of the total up to NAL energies. Since it is negative, the first two terms add in the proportions 1 = 1.4

-

0.4, so the second term is about 113 the fixst. According to this crude model the Pomeron is a combination of pole and cut, the former factorizable, the latter presu- mably not, the amount of factorization being of the right order of magnitude to agree with experiment.

Recently several authors have used only the first two terms to obtain the dip in the elastic differential cross section and the rise in total cross section at ISR energies [13].

The diffractive amplitudes which contribute to the discontinuity of the second term involve exchanging only an elementary Pomeron. To allow for cuts here as well would lead, crudely speaking, to a correction of about 113 in the diffractive amplitude, thereby modifying its contribution to the total cross section by a smaller amount. This correction is represented by the third and fourth terms in figure 3.

The expansion seems to converge better in the other direction shown in figure 4.

FIG. 4.

-

Terms contributing to multiple diffraction with elementary Pomeron exchange.

The magnitude of the third term in figure 4 is given by the cross sections for producing three "fireballs".

Frazer, Snider and Tan [5] argue that the cross section for producing one additional fireball is suppressed by a factor p: en

($--

^where

third term in figure 4 has a magnitude of about 1 mb .at ISR energies

-

hence it is quite small indeed.

These estimates are very sensitive to the size of the as yet poorly measured triple-Pomeron coupling and so are subject to orders-of-magnitude error. Because of the energy dependence of the terms in the series, one may be forced to include idore and more terms in order

to represent the amplitude at higher energies.

Present efforts to sum the Gribov series exactly [3] start with a bare Pomeron with intercept one and seek a specific solution in the form of a dominant pole at a(0) = 1. Typically the models emphasize the importance of two-Pomeron cuts while neglecting the full effect of three-Pomeron and higher order cuts. The advantage of the perturbative approach is that one need not commit oneself to the exact form of the Pomeron, be it pole or cut, nor to the exact form of the input and the relative importance of various cuts. I see nothing less arbitrary in assuming that the first few terms correspond to a dominant pole plus less important cuts at present energies, than in assuming that the sum of the whole series has this form. And it is certainly simpler from the technical point of view.

The perturbative approach discussed here is in a sense a retreat from confronting the larger question of what the actual Pomeron really is. Any approximate description of J-plane structure precludes real knowledge of the ultimate asymptotic behaviour.

But I think we theorists have been infatuated with asymptopia for too long. We have been trying to find the whole solution to the problem, while con- sidering only part of the necessary theoretical input. The practicality of an approximate perturbative approach is very appealing to me.

ACKNOWLEDGFENT.

-

I want to express my thanks to and $(t) is the elementary triple-Pomeron vertex

factor. Since np

- -

0.0025 according to their estimates based on present experimental knowledge, the

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the Aspen Center for Physics for its hospitality and stimulating atmosphere during the time I prepared this talk.

nces

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,

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,

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