Volume 48B, number 3 PHYSICS LETTERS 4 February 1974

A S I M P L E M O D E L F O R T H E D I F F R A C T I V E O V E R L A P I N T E G R A L

M. TEPER Physics Department, Westfield College, London NW3 7St, UK

Received 30 October 1973

A model is introduced for the diffractive overlap integral in which the overlap is represented by a set of non- forward triple Regge diagrams. The formalism provides a clear framework within which to discuss questions con- cerning the peripherality of diffractive processes.

Recently [1 -5 ] there has been much interest in determining the contribution of inelastic diffractive processes to s-channel unitarity. Following calculations [ 1,2] within the context of the diffractive excitation model [6] attempts [3 -5 ] have been made to incorpo- rate the spins of the excited clusters into the calcula- tions. All these attempts have involved making assump- tions about the spin structure of the diffractive ex- change, e.g. t-channel helicity conservation [3, 5], which, while they lead to a compact description of the quasi two-body excitation amplitude, also lead to a complex representation for the diffractive overlap integral. In this letter we outline a simple prescription for the overlap integral itself, which makes maximal use of two body Regge ideas, and which is the simplest extrapolation of a standard representation [6] for the diffractive cross-section.

Our model for the diffractive overlap integral, "derived" diagrammatically in fig. 1, consists of a pair of non-forward triple Regge diagrams, which we refer to as PPR and PPP, and which are, by naive duality arguments, the overlaps of resonant and non-resonant diffractive excitation processes respectively. We shall only consider, here, single excitation processes: at present energies these are, ] resumably, more probable than double excitation processes. At t=0 the model reduces to the standard [6] representation for the dif- fractive cross-section, and may be considered to draw some of its motivation from that representation. In this note we shall illustrate the use of this model in a discussion of the peripherality o f diffractive processes, and of the consequent implications for the forward structure of diffractive differential cross-sections. We shall only consider, here, the (M 2 channel) non-flip contributions to the Pomeron particle amplitude in

Fig. 1. The contribution of diffractive excitation processes to Im Tel (t) in the s-channel unitatity equation. The terms (a) and (b) we label PPR and PPP respectively. There is an implicit integration over t', t" and the missing mass, M. ~ stands for Pomeron exchange, R for secondary Reggeon exchange.

fig. 1, both because these terms should dominate near t = 0, and because their normalisations are measurable - perhaps [7] 4 - 5 mb for the PPR term and 2 - 3 mb for the PPP term at s -~- 1000 * -whe reas those of the flip terms are not.

Consider first the larger PPR term, in which we have a non-flip secondary exchange. In two-body theory one generally assumes [8] that absorption will suppress such an amplitude at small impact parameters so that the exchange becomes peripheral in the sense that the t dependence has the structure of the Bessel function

The sizes of the individual terms are not well determined. Units are GeV with h = c = 1 unless stated otherwise.

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Volume 48B, number 3 PHYSICS LETTERS 4 February 1974

Jo(RVCi -) with R ~ 1 fermi. If we take over such a prescription for the non-flip secondary exchange in the two-body Pomeron particle amplitude, we obtain a t dependence for the PPR term of the form

exp (at)Jo(RX/q-). (1)

The exponential factor in (1) contains a number of contributions; in particular, the loop integration in fig. 1 a will typically contribute something like exp (2.5 t) if the average inelastic diffractive process has a t dependence like exp (5 t). We expect then that the t dependence of the PPR term should have the form, say,

exp (2.5t) Jo(RVCi - ). (2)

Consider now the PPP term. For small t we expect the diffractive contribution to the Pomeron particle amplitude to give a factor like exp (5 t), while as above the loop integration gives a factor like exp (2.5 t) so that the overall t dependence has the form

exp (7.5 t). (3)

Hence, the sum of the PPR and PPP terms will ap- pear in the s-channel unitarity equation

21m Tel(t) = Tn(ki) Tn(ki) dq~n, (4)

(normalised to the total cross-section) as

o 1 exp (2.5t)Jo(Rx/-i-) + o 2 exp (7.5 t), (5)

where o 1 ~ 4 - 5 mb and o 2 ~ 2 - 3 mb. We now ob- serve [9] that a contribution of the form and magnitude of (5) is entirely capable of accounting for the forward break in the pp differential cross-section at ISR ener- gies.

A distinctive feature of the above scheme lies in the contrasting peripherality properties of resonant and non-resonant excitation processes: the former are peripheral in impact parameter, while the latter are not, (see (2) and (3)). This contrasts with the lack of such a distinction in conventional "intuitive" impact pic- tures [8], in which diffractive excitation is assumed to occur in the grey fringe surrounding a black disc.

An advantage of our scheme is that it is applicable at any energy, while black disc pictures, necessarily asymptotic in character, are not. In particular, while the forward pp slope change was first clearly observed at the ISR, it has more recently been shown to persist to very low energies [10]. In our scheme such a slope

change should indeed persist insofar as the low mass resonant excitations prove to be peripheral as they appear to be. The reason why the effect is noticeable, despite the drastic truncation of the excitation mass spectrum at such energies, is presumably connected with the observed enhancement at low energies o f some low mass resonance excitations.

While our numerical estimates in (5) apply specifi- cally to pp scattering, it is clear that our general argu- ments apply to all elastic processes, and hence so do our predictions concerning a forward slope change. Moreover, one may argue that the forward slopes will not differ greatly in magnetide, The scheme may be further extended to all diffractive excitation processes - we simply consider the contribution of diffractive processes to the unitarity equation for a particular dif- fractive process - although in this case the asymmetric character o f the terms in the unitarity equations makes cancellations between terms possible even at t = 0.

We conclude by pointing out that while we have concentrated on the most straightforward aspects of such a triple Regge scheme, perhaps more interesting is the fact that in such diagrams all the exchanges involve energies lower than that of the amplitude whose dis- continuity is being considered. A more detailed analysis of the questions touched upon in this letter be present- ed elsewhere.

When this work was completed we learnt of some other work [11] in which triple Regge type diagrams were being employed as terms in a unitafity equation, although otherwise the context is somewhat different.

The author wishes to thank Professor E. Leader for useful discussions.

[1] R.C. Hwa, Phys. Rev. D8 (1973) 1331. [2] M. Teper, Nucl. Phys. B59 (1973) 166. [3] Z. Ajduk, Nuovo Cim. 16A (1973) 111. [4] E. Gordon and R. Hwa, Relationship between diffractive

peaks of elastic and excitation processes, Oregon preprint, April 1973.

[5] N. Sakai and J.N.J. White, Nucl. Phys. B59 (1973) 511. [6] M. Jacob and R. Slansky, Phys. Rev. D5 (1972) 1847. [7] P. Slattery, Proton-proton collisions at 102 GeV/c.

Rochester preprint, April 1973. [8] H. Harari, Phenomenological duahty, Tel-Aviv Conf, 1971. [9] E.H. De Groot and H.I. Miettinen, Shadow approach to

diffraction scattering, Rutherford Laboratory preprint, March 1973.

[10] I. Ambats et al., Systematic study of n±p, K±P, PP and pp forward elastic scattering from 3 to 6 GeV/c, Argonne preprint, July 1973.

[ 11 ]Chan Hong-Mo and J.E. Paton, Phys. Lett.B46 (1973) 228.

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