Relation Between the Multi-Regge Model and the Missing-Mass Spectrum

The integral equation approach to the multi-Regge peripheral model is applied to give the missing-mass spectrum with Regge behavior in s and 3II2. A simple factorizable model for the double Regge coupling then gives the magnitude and t dependence of the cross section. This model is found to be in reasonable agreement with the backward m. +p — + p+X data. for terms of Gve-point


q2 -4@2
If we further use the relation valid to first order in r/= e'/4m. , between the photon spectral function and the annihilation cross section p(q') =Or' ' (q')/Sn2n, . where q' is the c.m. energy of the pair, and combining (4.5) and (4.6), we  The integral equation approach to the multi-Regge peripheral model is applied to give the missing-mass spectrum with Regge behavior in s and 3II2. A simple factorizable model for the double Regge coupling then gives the magnitude and t dependence of the cross section. This model is found to be in reasonable agreement with the backward m. +p -+ p+X data. HK integral equation approach to the multi-Regge production model for computing the contribution to the elastic absorptive part has recently been formulated. ' ' The approach has been used to predict total cross sections at high energies with results that are encouraging. ' Recently, this approach has also been applied by Caneschi and Pignotti' to studying the missing-mass spectrum at high energies. We will now derive the missing-mass spectrum for the observed particle emerging from the left end of Fig. 1 and then present the general result for the observed particle emerging from any position in the multi-Regge chain. The contribution of the n-particle intermediate state to the two-body absorptive part is A"(s) =-', dc" IT"I2, region of a multiperipheral chain. This work emphasizes the importance of missing-mass experiments to the search for a "realistic" multiperipheral model.
To obtain the cross section for missing mass M2, observing the 6nal particle of momentum q, we undo the phase-space integral over q in Kqs. (7). Using Eq. (6), we obtain For simplicity, we will use the form (1) for the production amplitude both in the high-energy and the low--= -Gi(t)2$(t)2 energy regions. Finally, the total contribution to the missing-mass spectrum' is given by (see Fig. 2 Viewing the collision in the c.m. system, these three terms describe, respectively, the spectrum of the observed particle for large positive, intermediate, and large negative longitudinal momentum.~9 Through Eq. (10) and the Regge behavior of 8(t; M'), we now have enough information to perform multi-Regge fits to missing-mass data. ' We can also learn much more if we can find the magnitudes of the multi-Regge graphs from our knowledge of the magnitudes of Reggeized two-body elastic scattering and total cross sections. In order to do this we consider the simple model of an co-angle-independent" and factorizable internal Regge residue P(tq, t~). It is normalized for one particle on the mass shell to be the single Regge coupling P(tq, m') = G(t~), P(m', t~) =G(t2), where G(m') = g, so that the factorizable residue is p(t&, t2) = G(t&)G(t2)/g.
For small t's the integral equation is + independent and becomes where we use the approximation that the t dependence from the integration limits is small. ' ' Because of Eq.
(7) for A(s), we have now a model for 8(t; s) in terms of known two-body total cross sections.
For illustration of its usefulness, we apply Eq. (13) to Eq. (8), observing that the elastic differential cross section at s-channel invariant (sh42/M2) is where we have isolated the contribution from elastic scattering. It is interesting to note that the t dependence of the end term contribution is that of elastic scattering.
In most cases, the existence of several communicating Regge trajectories will complicate the situation. To illustrate our approach, therefore, we have chosen a , particular production reaction where only one known trajectory can be exchanged. We consider the reaction + +p~p+X, ' where the outgoing proton has small momentum transfer to the incoming pion. If we further restrict ourselves to those events where the outgoing particle has large longitudinal momentum (for 16 GeV/c m. incident, we detect only lql&, z of proton & 10 GeV/c, or M'(7 GeV') two major simplifications occur: (i) Among the three terms in Eq. (10) trajectory that can be exchanged is the 6 trajectory.
(The~-angle-independent model can be formulated for an exchanged particle with spin by assuming that only one helicity amplitude is dominant. ") Consequen. tly, we obtain, for~+ p -+ p+X in the laboratory frame, This prediction is compared with the experimental result of Anderson et ul. , and the fit to the experiment is shown in Fig. 3. We have parametrized the data for do/du for backward'5 n. p scattering with nq(t) =0.049 +0.76$, and also used g~~' =30 GeV'. The absorptive part is given by a sum of a Pomeranchon term and an "average" meson term of the form M'Lop+oM(N'/