Another root of the present model is the finding that a very special type of statistical thermodynamics is required to fit the actual behavior of a statistical model of the type (19.1): this was found by evaluating numerical calculations performed at CERN at different c.m. And since it is the main point of the model that will be presented below, we need to explain it in more detail. Therefore, if a simplified model of the type given in Eq. 19.7) with particles of the same mass is used to determine the asymptotic behavior, then the factor 1=nŠ should be omitted in order to get as close as possible to reality.

At the first moment of the collision, a certain number of particles – ranging from pions over kaons, nucleons, hyperons and their resonances to highly excited “fireballs” – are produced according to the statistics of distinguishable particles. By the reinterpretation of our model we have of course abandoned the point of view of the canonical ensemble of temperatureT and obtained the description of a single system of given energyE. ACB, but also the transverse momentum distribution of the many particles produced in cosmic ray beams.

Our model would thus explain the hitherto obscure fact that the transverse momentum distribution in high-energy events is experimentally independent not only of the primary energy but even of the number of particles involved. The above values for T0 agree well with our postulateT m. Furthermore, Cocconi, Koester and Perkins [15] and Fowler and Perkins [11] find from high-energy nucleon-nucleon collisions that the pioneer transverse momentum distribution is given by Eq. ppas a function of the transverse momentum. expected if we remember that the pioneers observed experimentally are not the particles produced at the first moment: they are the end products of a chain of decays, each of which is governed by a law such as Eq. 19.36), and the expansion is just a kinematic effect. In other words, despite the flatness of the distribution of multiplicities, we should find a sharp distribution.

We conclude this section by admitting that our reinterpretation of the original model was not always very convincing.

## Speculations on a More Realistic Model

In the rest of the nucleon system, the energy spectrum of these virtual particles is isotropic; for the forward momentum distribution of the incoming nucleon we have In other words, the collision breaks the rigid walls of the volume and virtual particles can become. The longitudinal momentum distribution will be approximately of the form w.pk/exp.pk=˛T0/,˛1, except for very central collisions where it becomes similar to the transverse distribution.

It remains to explain why, when at most one of the two given distributions can hold, nature apparently chooses the transversal one (for evidence, see Figure 19-2). The actual increase in multiplicities observed experimentally should be interpreted as an increase in the excitation of the fireballs. The energy independence of N and T?0 immediately implies [by Eq. 19.47)] that the average kinetic energy stored in the transverse motion is itself independent of the primary energy [though fluctuating widely from event to event, see Eq.

Almost all the energy is contained as kinetic energy in the longitudinal component (beam) and the (strongly fluctuating) transverse energy is on average independent of the primary energy. The abundance of the first generation particles (fireballs) fluctuates strongly, but is not large. The strongly fluctuating multiplicity of the last generation (final pions, nucleons, hyperons) will increase very slowly on average, since most of the total energy is contained in the kinetic energy of the fireballs and only a little in their excitation.

This is consistent with our observation that most of the energy in jet streams must be in the kinetic energy of the longitudinal component and only a little in the excitation (= mass) of fireballs. It would mean that the volume in which the interaction still occurs would have linear dimensions on the order of nine times the pion Compton wavelength. The other way out of the difficulty might lie in introducing a mass spectrum of excited states.

The mass spectrum of highly excited hadrons (fireballs) should increase less than em=T, where Tis is of the order of m. Since the temperature of the system (at high enough energy) becomes T0 D const:, it follows that asymptotically [that is, when.logm/=m!0],. Since experimental evidence shows that T?0 is of the order of m, it is required that Z imposing a condition on the mass spectrum is consistent with the fact that the mass spectrum should be derived from the theory itself.

At present we can at least hope that the value of the integral in Eq. 19.63) is close to one and therefore neither T?0 nor V0 should have unreasonable values. Particles of the same species should then be considered indistinguishable (and a statistic, Bose or Fermi, should be described) and the number of Niof particles of each type as well as the total NDP number.

## Summary and Conclusions

For astrophysics, it is quite obvious: whenever strong interactions and kinetic energies per particle of size m come into play under the influence of gravitational pressure at the center of a star, the relevant statistics for thermodynamic treatment are not Fermi statistics, but statistics of distinguishable particles. T0 and only sufficient energy transfer would raise the temperature (for transverse motion in cm collisions) to T0. Since the differential elastic cross section and the total cross section are related by the optical theorem, and since all hadron total cross sections are of the same order at high energies, we would conclude that they all have the same order of "a priori temperature".

Bohr's quantum condition for the stability of the proton can later be explained by some generalization of quantum mechanics. Clearly, this condition should give us the hadron mass spectrum, regardless of whether the condition itself can be derived from current quantum theory (which is unlikely) or from its future generalization. The "bootstrap" mechanism would then correspond to a new type of perturbation treatment, in which only a few of the lowest masses of the unlimited number of interacting "fireballs" that make up the hadrons are considered.

It is of course possible that the picture being drawn here is wrong and that the fact that our model works even in the diffraction region is merely coincidental. But if this is not wrong, then it would follow that we have basically all the information we could hope for in our hands: the mass spectrum, the selection rules (SU3, etc.), the decay modes of the lower unstable states (ρ ,ω , etc.), and it goes to higher and. None of our arguments exclude their being found, e.g. in an app p collision with several 100 GeV at the center of mass.

Here the first term in round brackets is.logZ/=zand the second tends toxˇZ=z2. We wish to consider here the broadening of the spectrum due to the decay of a fireball. This fireball can launch another particle of mass m`, again only in the ˙x direction, with a "four-momentum".

This now needs to be multiplied by v.u/du and integrated from 1 to 1 (the negative u values account for the cases where the fireball moves in the x direction and emits a 'meson' in the Cx direction with enough energy to overcompensating for the speed in the x-direction and a positive direction of flight). Despite this very rough analysis, we believe that this is sufficient to make it very likely that the apparent increase in temperature is entirely due to kinematics and that our T0 is indeed independent of primary energy. Open Access This book is distributed under the terms of the Creative Commons Attribution Non-commercial License, which permits any non-commercial use, distribution, and reproduction in any medium, provided that the original author(s) and source are acknowledged and credited.

It does not contradict the assumption that nothing serious would happen in reality (the two-dimensional case). If that turned out to be the case, and if the general case gave a similar result, then it would make it possible to conclude from the transverse momentum distribution [namely the deviations from a pure exp."=T0/]. On the other hand. the chain of decay will then contain more members and this can increase the deviations again.

Actually, the T needed to fit the spectra appears to increase somewhat with the primary energy, although not by more than a factor of two, when the primary energy varies by a factor of one million.