FIREBALL MODEL WITH DOUBLE REGGE POLE EXCHANGE FOR MULTIPLE MESON PRODUCTION IN THE ACCELERATOR ENERGY REGION.

Recently, considerable experimental data of multiple particle production in the accelerator energy region came to be available.> As one of theoretical models Chang Hong-Mo et al. ) have proposed the Reggeized multiperipheral model applying the Regge-pole model to multiple particle production processes, and have obtained a fairly good agreement with the experimental data on the spectra of secondaries. Another model which stands on experimental analysis is the fireball model. This model is obtained first phenomenologically by analysing the jet phenomena in cosmic ray energy region. > Ratner et al. > and other authors> have shown that the model is also applicable to the multiple particle production with large multiplicity in the accelerator energy region. It is an important problem to clarify theoretically the correlation between these two models. As the first step to approach the problem we propose in this paper a model, which combines the Regge-pole hypothesis and fireball description of experiments, and compare this with the experimental data. Our model is that the fireball is produced by exchanging Regge poles and subsequently decay in :flight into secondary mesons. Here, we deal with the pp collision at 30 Ge VIc as an example of the multiple particle production. There may be the case of multi-fireball production or the case of resonance production. However, since the available energy is not so high (30 Ge VI c) compared with the cosmic ray energy region, we assume that the produced fireball is only one. We shall discuss our model and give its formalism in § 2. In § 3, we shall compare the results of our model with experimental data. Some concluding remarks are given in § 4.

Recently, considerable experimental data of multiple particle production in the accelerator energy region came to be available. 1 > As one of theoretical models Chang Hong-Mo et al. 2 ) have proposed the Reggeized multiperipheral model applying the Regge-pole model to multiple particle production processes, and have obtained a fairly good agreement with the experimental data on the spectra of secondaries. Another model which stands on experimental analysis is the fireball model. This model is obtained first phenomenologically by analysing the jet phenomena in cosmic ray energy region. 3 > Ratner et al. 4 > and other authors 5 > have shown that the model is also applicable to the multiple particle production with large multiplicity in the accelerator energy region. It is an important problem to clarify theoretically the correlation between these two models.
As the first step to approach the problem we propose in this paper a model, which combines the Regge-pole hypothesis and fireball description of experiments, and compare this with the experimental data. Our model is that the fireball is produced by exchanging Regge poles and subsequently decay in :flight into secondary mesons.
Here, we deal with the pp collision at 30 Ge VIc as an example of the multiple particle production. There may be the case of multi-fireball production or the case of resonance production. However, since the available energy is not so high (30 Ge VI c) compared with the cosmic ray energy region, we assume that the produced fireball is only one.
We shall discuss our model and give its formalism in § 2. In § 3, we shall compare the results of our model with experimental data. Some concluding remarks are given in § 4. we assume that one fireball is produced, exchanging Regge poles between two incident protons. Since the experimental data show that final protons are produced in the forward or backward directions at high energies, we assume that the dominated diagram is given in Fig. 1.

T. Morii
The following production amplitude is suggested by the formalism of double Regge pole model : 6 M; the fireball mass.
where spm indices of external particles are neglected.
The quantities s and t are defined as follows: Since we have no data on the spin correlation, the square of production amplitude is averaged over spins. If we denote the residue functions with the signature factor averaged over spins as /3's, the matrix element squared is written as follows: (4) For practical calculation, we neglect the dependence on cp and assume that the trajectories are linearly increasing with t. We assume that each vertex part is sharply peaked at t = 0 and has an exponential behavior on t. 6 ) (5) where a and b are free parameters representing exponential t-dependence of each vertex and B 0 is a normalization constant which depends on the mass of the fireball and two Regge trajectories.
The spectrum of the final proton is given by Hereafter let quantities with the asterisk denote those defined in the C.M.
system, (PA + PB = 0). The calculated results with our model are given in Fig. 2.
In order to get the spectrum of the secondary pion, we must first calculate the spectrum of the fire ball. The latter is given by the following expression: (see the Appendix): From Eq. (11), we can know the direction and velocity of the produced fireball (see Fig. 3). In its rest frame, the fire ball decays isotropically into secondary pions, and we assume that in this frame the spectrum 8 ) of the secondary pion is given by where k is the momentum of the produced pion from the fireball and c is a parameter which depends on the mass of the fireball and multiplicity but is independent of k. Equation (12)   3.0 The experimental data are from reference 1). From this spectrum, we get <13u)=0.618 and <ru)=1.27.
Since the fireball is produced into various directions with various velocities, we should perform Lorentz transformation to Eq. (12) from the rest frame of the fire ball to the C.M. system.
Since we do not want to go into detailed discussions, we simplify the problem and assume that, on the average, the fireball is produced in the forward and backward directions with the average longitudinal velocity <f3 11 ) .
In this approximation, the spectrum of secondary pions in the C.M. system is where <f3 11 ) and <ru) are the velocity and the Lorentz factor of transformation from the rest frame of the fireball to the C.M. system. §

Calculated results and comparison with experiment
In this section, we show our calculated results and compare them with the recent experimental data in the accelerator energy region.
There are eight free parameters in our model: the mass M of the fireball, two sets of the parameters a (O) and a' corresponding to two linear Regge trajectories, a and b specifying the exponential t-dependence of each nucleon vertex and B 0 the normalization constant.
As the trajectory parameters, it should be natural to take those for the Pomeranchuk trajectory; ai (0) = 1.0, and we set a/= 0.6 (i =a, b) in order to fit the data. Next, we assume that the parameters a and b are zero. This assumption is justified from the following observation. If we assume that the forward elastic pp cross section is dominated by a single Regge exchange and that the residue function is of the exponential form in t, the amplitude A for the elastic pp scattering reads (except for the signature factor) ( ) Furthermore, we assume that the factorization theorem is valid; the t-dependence of the r.h.s. in Eq. (19) comes from the product of each vertex. Then, the comparison with the experimental foward differential cross section (d6 / d!21"'./e 8 t) for elastic fYP scattering leads to the following equation for a: where the scaling factor s 0 is set equal to 1 (Ge V Note that, from the fireball spectra, the average parallel velocity <f3 11 ) and the Lorentz factor <ru) of the fireball are also calculated; <f3 11 ) = 0.618 and <r 11 ) = 1.27.
For the momentum spectra of the produced fireballs, the characteristic features of our model are summarized as follows; 1. The spectral shape depends strongly on the parameters a and b as well as the Regge slope a/. When the values for these parameters increase, the position of a peak in the spectra shifts to lower momentum side. 2. The Regge intercepts ai (0) do not give rise to any affection on the spectral shape. This fact follows from (5), (6) and (7). 3. As for the effect of the fireball mass M, it is found that the smaller the mass M is, the broader the spectral shape,  1. The proton spectrum is rather independent of P 11 *.
2. The pion spectrum shows a marked dependence on k 11 *.
In our model the above features are qualitatively well reproduced as shown m Fig. 4. In Fig. 5 is shown the proton spectra transformed into P 1' and P 11 * coordinates. Since there are no data on the pp interaction for the pion angular distribution, we give in Fig. 6 those on the 6-prong n-p interaction at 25 Ge VI c. s) As is seen from these figures, we can conclude that our model succeeds, at least qualitatively, in explaining the characteristic features of the multiple particle production in the accelerator energy region. In particular, we emphasize that the proton and secondary-pion spectra are simultaneously reproduced with the same set of parameters. § 4. Concluding remarks In explaining the experimental features of multiple meson production, several authors employ a model, the so-called Reggeized multiperipheral model in which a number of the secondaries are assumed to be produced through exchange of the Regge poles. The number of produced mesons increases with increasing energy. Agreement between the Reggeized multiperipheral model and the experimental data is rather excellent, but the parameters included become a large amount as energy goes higher. Consequently, very complicated calculation should be needed. Main part of this model may be absorbed in the computer work.
Contrary to the Reggeized multiperipheral model, our model assumes only one fireball, since the energy in question is limited in the accelerator region. It may be reasonable to assume the fireball production because of the fact that there are many phenomena which are accounted for with the fireball model in the multiple particle production in the cosmic ray energy region. Thus, in our model the effect of the increase of energy is entirely reflected to the mass of the fireball and the kinematical conditions such as the produced direction and the average velocity of the fireball. Our model would oversimplify the actual physical situation, nevertheless the results obtained are qualitatively satisfactory for us to conclude that our model is a good one of the zeroth order approximations to the multiple particle production processes in the accelerator region.
As for the fireball mass M, we take a rather small value ,.....-' 1 Ge V in contrast to the usual value of 2~3 Ge V deduced from analyses in the cosmic ray energy region. Usually, the fireball in the cosmic ray energy region is considered in the jet phenomena with n 8 :::::::::4 (n 8 = multiplicity). On the other hand, the multiplicity is not assigned in the data employed in this paper. Therefore, it is not astonishing that our fireball mass is rather small.
At the present stage, experimental knowledge Is still poor for us to give a decisive conclusion to a question whether or not the fireball is produced even in the accelerator energy region, but our model seems to show that the fireball is also produced in a rather low energy region for multiple particle production processes. Further experimental data will check our model.

T. Morii
The fireball spectrum is given in the following expressiOn: (A·26)