## On the phenomenological analysis of the S-matrix in multiple production

##### Abstract

An attempt is made to analyse the structure of S-matrix elements for multiple production by phenomenological considerations based upon the two center model. We start from two assumptions: 1) the momentum distributions of pions emitted in the for- and backward cones are isotropic in each center-of-mass system (the fire ball rest system) and the average values e{open}0 of the pion energies in this system do not depend on the details of collisions; 2) the mean value of the momentum transfer Δ is always of the order of the nucleon mass M. From these assumptions the following results are derived in Sect. 2: 1) the multiplicity is given by n{reversed tilde equals}ξ√2 E0|Δ|/e{open}0, where ξ is the ratio ℳ/ℳmax of the fire ball mass ℳ and its maximum value ℳmax. Experimentally, ξ{reversed tilde equals}0.5 at 1013 eV; 2) the lower limit of the inelasticity K is given by K≥ξ2. At 1013 eV we have K≥ξ2{reversed tilde equals}0.2 which is consistent with experiments. The relation between multiplicity and inelasticity is discussed; 3) in the center-of-mass system of the fire ball and the recoil nucleon (« isobar system » IBS), the ratio |M|/ℳ of the momentum of the nucleon M and the fire ball mass ℳ is also expressed by the parameter ξ as |N|/ℳ=(1-ξ2)/2ξ. At 1013 eV we have |M|/ℳ{reversed tilde equals}1. This means that an equipartition of energy is approximately valid for the three possible degrees of freedom: the motion of the recoil nucleon, the translational and internal motion of the fire ball. |M|≪ℳ is excluded, unless K is unity, contrary to experiments; 4) when the equipartition holds, the mean value of the transverse momentum is constant and its value is of the order of e{open}0. In Sect. 3, the ℳ and n-dependence of the S-matrix is discussed. And by using the corresponding results we estimate the asymmetry in numbers of particles n1 and n2 emitted in the for- and backward cones. The parameter [〈(n1-n2)2〉/〈(n1+n2)2〉]1/2, a measure of the asymmetry, becomes 1/3 due to statistical fluctuation alone. It becomes larger if the energy transfer Δ0 is not zero. Discussion is also made of the condition by which the average value e{open}0 of the pion energy in the fire ball is constant. The properties of the fire ball seem to be independent from the mechanism of its production. Extending this point of view, the similarity between the multiple production in N-N collisions and N-N annihilation is discussed in Sect. 4. Conclusions are given in Sect. 5. © 1961 Società Italiana di Fisica.