Multiperipheral Model for the Nonforward N Absorptive Part in the Double-Regge Region

The forward absorptive part in M2 of a+5 +c elastic scattering has been related by Mueller to the inclusive single-particle spectrum for a+& c+X. The nonforward a+& +c 3-3 M absorptive part in the double-Regge region also has several physical applications. The M absorptive part of this amplitude is calculated in the region of double-Regge ex- change using a multiperipheral model with exponential damping in momentum transfer and Regge behavior in inclusive subenergies. The absorptive part is then extended to an analyt- ic function in M ~. A statistical theory of particle production is formulated in analogy with the generalized Ginzburg-Landau theory of phase transitions in superconductors and intensity fluctuations in lasers.


I. INTRODUCTION
Recent investigations of inclusive spectra have given importance to the three-particle to threeparticle (3-3) amplitude. This is due to the optical theorem of Mueller' which relates the M' =(P, +P, +P;)' absorptive part of the forward 3-3 amplitude for a+b+c-a+b+c to the single-particle inclusive cross section for a+5-@+X. The nonfonoard 3-3 amplitude also plays an important role in the analysis of inclusive reactions. In order to fully study the analytic properties of the forward 3-3 absorptive part it is necessary to examine analytic continuations in two-particle subenergies and in the trajectories by using the nonforward 3-3 amplitude. This was first done by Halliday and Parry in y' theory to demonstrate the factorization of the M discontinuity. ' The nonforward multiperipheral amplitude that we calculate here is used to study the relation Another application of the nonforward 3-3 absorptive part is to inclusive experiments on nuclei as studied by Bander' using the eikonal approach. In the usual eikonal computation of a total cross section on a nucleus from the optical theorem, one uses the amplitude S(b) for nucleonnucleon elastic scattering at an impact parameter b, which requires a full knowledge of its nonfor-ward transform S(q'). In the eikonal treatment of the inclusive 3-3 amplitude' it is similarly necessary to know it at nonforward momentum transfers and energies in order to compute the scatterings with the nucleons located at various impact parameters and parallel displacements. The 3-3 absorptive part calculated in this paper is also evaluated at the forward point to give the pionization spectrum. The method given here is a simpler and more direct way to get this pionization spectrum for the exponentially damped multiperipheral models. This result has recently been used in generalizing to arbitrary forms of damping in momentum transfer, thereby allowing fits to be obtained for pionization spectra over all q, '. ' In this paper we compute the M' absorptive part of the nonforward 3-3 amplitude in the double-Regge region for a simple multiperipheral model with Regge behavior and exponential damping in momentum transfer. After computing the M' absorptive part we then extend it to an analytic function.
The model for the absorptive part in the inclusive central plateau region consists of the production of the observed particle c with a summation over numbers of particles emitted faster or slower than c, Fig. 1. The summation over both of these sets of particles is similar to the summation in a total cross section and is assumed to produce a Regge behavior, s,~i , s, 2 as in the multiperipheral model. ' In order to get the observed exponential falloff in (p~)' in the central region, we include factors for exponential damping in the internal momentum transfers e "l'~, e "~'~. This model has been formulated by Caneschi and Pignotti' and analytically computed by Silverman and Tan' and others. ' By squaring this amplitude and integrating over the inclusive phase space, we get a form for the 2279 single-particle spectrum or equivalently the M2 absorptive part of the forward 3-3 amplitude in the central inclusive region. ' In this paper we use this model to compute the nonforward 3-3 M' absorptive part in the double-Regge region, Fig. 2.
By the double-Regge region we mean that subenergies s"s'"s"', and s"are large enough to be in the Regge region, Fig. 3. The calculations are carried out in the nonforward region adjacent to the forward central plateau region so that p;=p, has negative energy P'; &m, . The absorptive part is then used to construct the analytic nonforward 3-3 amplitude in the double-Regge region.
As a special point we will present the forward 3-3 M absorptive part for double Pomeranchukon exchange n, =1, z2=1, which agrees with the previously calculated single-particle spectrum in this model. ' The method employed in this paper gives a simpler and more direct calculation, however, as well as extending the calculation to tra-  Fig. 2 can be computed by first perf orming the phasespace integrations over all of the particles in the sets labeled by momenta p, and p, . We assume this is done (Ref. 8,Appendix) and produces Regge behavior in the momenta squared s, = p, ', s, = p2'. The remaining integrations over p, and p2 can be performed by considering these sets as intermediate particles of masses squared s"s"and then doing the simple two-body phase-space integral to get a function A . Finally, the intermediate particle masses squared s"s"are integrated over their allowed values to get the absorptive part A.
The dependence on s, and s, enters explicitly and through I', p'"p, ' in Eq. (2.5). Defining We then split up sinh[QM(1 -x-y)] into its two exponentials. In the second exponential we substitute x=1y', y =1 -x', 1xy = -(1 -x'-y'). (3.7) Then both exponentials are identical and can be combined to give ( 3 8) We have taken the limit I'-~h ere and s"s", s'"s"'-~in order to make the thresholds reduce to the limits 0 to 1.
The result (4.11) also holds in the forward direction, where large in magnitude, the M' cut from threshold to infinity is mapped into a cut in z for 0 & a &~.
Therefore we make the analytic continuation in the variable a. The function of v in (4.11) can be written in terms of entire functions'0 M (also called e). " Imf(z)-= e '~"~U (n, +1. ,a, +n, +l, z) The absorptive part in M' calculated in (4.11) can be extended to an analytic function by a procedure equivalent to using a Cauchy integral or dispersion relation in M'. Of course this does not give the entire 3-3 amplitude but only that part arising from the M' cut, or equivalently, from diagrams like Fig. 2.
We note that the only occurrence of M' in the result (4.11) is in the variable z~@ (4.10). Keeping (5.6) The same result is obtained by performing the Cauchy integral, '2 and is unique up to an arbitrary polynomial in g.
Including (5.6) with the remainder of (4.11) gives the analytic extension in M' or tc. The signature structure of the Hegge exchanges n, and e, can be found by adding diagrams like Fig. 2 but with ex-changes (aa'), (bb'), and (tttt', bb') Care must be taken to establish that the diagram of Fig. 2 and the exchanged diagrams from (aa') and (bb') are evaluated above the tt cut, but the diagram from the exchange (aa', bb') is evaluated below the~cut. This has been carried out in Ref. 3.

I. INTRODUCTION
At present accelerator energies the number of secondaries produced in hadron-hadron collisions is large enough so that one can consider statistical theories of particle production. In this paper we shall present such a theory. ' Our basic approach is as follows: We do not attempt to treat in detail the fundamental dynamics underlying particle production. Instead, we imagine integrating out the microscopic degrees of freedom and representing them in terms of a small number of phenomenological parameters. We are then left with a statistical theory of the macroscopic observables. A mell-known example of this approach is the Qinzburg-L andau theory of superconductivity. ' In its generalized form' it provides a statistical theory of the superconducting order parameter which depends upon a small set of experimentally determined parameters. Once a microscopic theory of superconductivity was formulated by BCS, it was possible to derive the Ginzburg-Landau theory from it and to obtain expressions for the phe-