Scaling Behavior in Multiperipheral Dynamics*f

We demonstrate the scaling of the single-particle momentum distribution as a general property of all the multiperipheral models which have been proposed. We also show that in these models, pionization is ap- proached as a smooth limit from scaling. The proof is based only on the most general multiperipheral assumption and on Pomeranchuk-pole dominance at high energies. Thus the experimental observation of scaling is required for the validity of any multiperipheral model.


I. INTRODUCTION
I SlN G a great amount of physical insight, Feynman' has recently proposed that the longitudinalmomentum distributions in hadron collisions should exhibit certain simple scaling and limiting features. %e consider the simplest possible inclusive experiment where only one 6nal particle is detected, and the invariant momentum distribution all multiperipheral models which have been proposed' 4 and to show that in these models pionization is approached as a smooth limit from scaling lim P(x,k,) =F(k~) .
(4) Analysis of existing experimental data supports this scaling behavior. ' The phenomenon of pionization has been demonstrated with the original ABFST multiperipheral model' and the proof can be directly extended to the more general multiperipheral models. The original pionization analysis, however, was performed only for x~0, and was not applicable to scaling in what we term the production region (forward production for x)0, backward for x&0). The present authors' have recently demonstrated the scaling phenomena in an analysis of the inclusive single-particle experiment for special multiperipheral models with exponential damping in momentum transfer. %e present here a proof applicable to all multiperipheral models so far studied by making use of the CGL model, which includes the ABFST model as a special case. Ke find that the important requirement for scaling is that the output auxiliary forward amplitude 8 is dominated, above some 6nite subenergy, by a Pomeranchon of intercept unity, i.e. , that the solution leads to a constant total cross section.

II. INCLUSIVE SINGLE-PARTICLE SPECTRUM
In the CGI. multiperipheral model, the singleparticle momentum distribution~' is given by

SCALING OF JACOBIAN
We compute the Jacobian in a useful form by 6rst using the b'(q"+q'+k) to do the q' integration so that the Jacobian is a(s4', s"', t,t") J -'=det~( qo pqz pqu yq» )where q»" and q»" are the components of q&" parallel and perpendicular to k&, respectively. This may be rearranged to give (pp)» (g) lf q» + d'q'd'q" 04(q'+q"+k)B( q', q"; p-') (2s)' x IO(q"', , q") I'B(-q", q'; p) +I G. ((Pk)') I 'B(P, P k; P'), - (5) where B(q', q"; p') is the auxiliary function which satisfies the CGI equation, 6 is a Reggeon-particleparticle vertex function, and P is a Reggeon-Reggeonparticle vertex function which may include dependence on the Toiler angle or. The first and last terms are the contribution of the left and right end diagrams, as illustrated in Fig. 1. The middle term is the contribution of the central diagram, which dominates for IxI not close to 1.It can be easily seen that the end diagrams directly exhibit scaling, ' so we shall concentrate on the central diagram. Our proof proceeds by Grst transforming integrations to the invariant subenergy and momentumtransfer variables and showing how the Jacobian of this transformation has scaling behavior because of the multiperipheral hypothesis that the momentum transfers are small. We are then able to show that in the important regions of integration, the auxiliary forward amplitudes 8 exhibit scaling and produce and s && ) behavior to cancel the 1/s from the flux. This will complete the proof of scaling for the inclusive singleparticle spectrum.
Pro. 2. Kinematics of the central diagram.
The invariants in the Jacobian become 2p p'=sm'm", 2p g =s"-t"m- %e now change from the integration variables s~', s, ' to the scaled integration variables y, 2', which will be shown to be of 0(1) due to multiperipheralism, In the last formula and later on in computing terms of In terms of these we have anally the exact expression 0(1), we observe that in to go into the Jacobian Eq. (10): In terms of these invariants the phase-space regions are as follows.
The inclusive integration at large s is now expressed. in terms of scaled variables with the use of Eqs. (10) and (13): ( d'q'd'q" 6'(q'+q" +k) We have completed calculating the Jacobian at large s using the multiperipheral hypothesis and 6nd from Eq. (18) that it depends only on x and k1' and integration variables, which proves that the Jacobian scales.
In the pionization region x(0(1/gs), the terms in x' may be dropped and the Jacobia, n then agrees with that used by ABFST to show independence of x in the pionization region. The terms in x' are an essential reason why the production regions have a nontrivial x dependence.

A. Pionization Region
Since the subenergies become asymptotic in this region, we use the asymptotic relations that follows from the invariance properties of the CGL equation4 and the hypothesis of Pomeranchuk-pole dominance:
Pionization: Both B» and B"arePomeranchuk-pole dominated, so that (s)')~y(0)(s~)~n (0) The assumption of Pomeranchuk-pole dominance for the auxiliary forward amplitudes B above a finite energy will now be shown to provide an asymptotic behavior s )'") to cancel the 1/s flux factor, and the remaining dependence in the B's will exhibit scaling by being a function only of x, k», and the integration variables. %e may express the B's in terms of invariants including the energies s) -= (p'q') ' = (s1't)m") s"-= (pq")'= (s"' -t"m') +( u, +m'+t), ')+t)+-m', the rest of the dynamical input is seen to exhibit not only scahng but also independence of x. The~-angle dependence of the coupling P(t),~,t,) also exhibits scaling, since for large s»', s"' it is related to the scaled variables by h(u2, t t")) t1' -t1t,+2(ttt")'"cosa& s +m"x+0(1/s) = (k,'+ty, ')/x+ m"x, which is independent of s. From Eqs. (15) and (20) we may then convert the subenergies to scaled variables without introducing any dependence on s: Consequently B(s&,,sI", t&,t") is a function only of x, ky2, y, s, t&, and t", and it is independent of s. The co angle in the production region can also be shown to depend only on these variables, but we omit showing this since the calculation is lengthy, though straightforward. In concert with Eq. (24), we have proved scaling in the forward production region. The proof for the backward region follows by similarity.

C. Transition from Production to Pionization
Q'e now show how the scaled momentum distribution in the production region F(x,ky) approaches as x y 0 the pionization result F(ki), which was obtained by the limiting procedure in Eq. (3). To do this we consider x to be very small and~ed. Then the terms in the Jacobian in x', Eqs. (18) and (19), are negligible and the Jacobian smoothly approaches the pionization limit.
Considering again the analysis of the 8 functions in the forward production region, for very small x, the limit on s&' given by y& 0(1) can become very large:
V. CONCLUSION We have presented a proof of scaling in the inclusive single-particle spectrum based only on the most general multiperipheral assumption and on Pomeranchuk-pole dominance at high energies. The experimental observation of scaling is thus a crucial necessity for the validity of any multiperipheral model. However, in order to differentiate between specific multiperipheral models, it is necessary to calculate the detailed dependence of the spectrum on x and k~' in each of these models. To this end, we have completed an analytical study of the predictions of a simple multiperipheral model which assumes exponential damping in momentum transfers. 6 Pote added iN proof. After the submission of our paper we received reports of two studies of single particle distributions in the multiperipheral models from Bali, Pignotti, and Steele, University of Washington report (unpublished), and DeTar, Lawrence Radiation Laboratory report (unpublished  For suSciently small x, the majority of the range of s&' One of the authors (D. S. ) wishes to thank the Aspen Center for Physics for their hospitality