An empirical model to describe rapidity density and transverse momentum distributions

The distribution of rapidity density and transverse momentum is formulated empirically and analytically. It describes the data quite well over the wide energy range of √ s= 22.4− 7000 GeV.


Introduction
We formulate the transverse momentum and rapidity density distributions of produced particles in multiple particle production (MPP) empirically and analytically.Compared with the current models which use a theoretical idea and the simulation technique, present model has following advantages.
(1) It describes all existing data concerned reasonably.(2) Consequently it is the most suitable one to be used in analyzing high energy cosmic-ray events.
(3) We can see the least necessary ingredients to describe the MPP.

Transverse momentum and rapidity density distributions 2.1 Assumptions
We enumerate the assumptions to formulate the rapidity density distribution, d 2 N ch /dy * d p T , of produced particles in this section, the details of which are found in Ref. [1].
(1) All produced particles are pions.(The assumption affects slightly the rapidity density distribution.)(2) Produced particles are emitted isotropically from several centers.(Fig.1) The emitting centers are distributed on the rapidity axis in the center of mass system (CMS) with the distribution g(y ′ )dy ′ of where the values of the parameters b = d = 0.25 are determined by trial and error by fitting the (pseudo-)rapidity a e-mail: ohsawa@icrr.u-tokyo.ac.jp b e-mail: shibuya@if.unicamp.brc e-mail: tamada@ele.kindai.ac.jp (3) The energy distribution of produced particles f (p)d p is that of Tsallis type on the respective emitting centers.That is, where the values of the parameters T 0 and q are determined so as to reproduce the transverse momentum (abbreviated as p T hereafter) distributions at various energies.Then the rapidity density distribution is where E = µ cosh(y * − y ′ ), p = √ E 2 − m 2 , µ = p 2 T + m 2 (m : pion mass).(The quantity with an asterisk is the one in CMS.)The distribution can be converted to the pseudo-rapidity density distribution and x distribution (x * ≡ 2p * || / √ s) easily.

p T distributions
The p T distribution at the zenith angle θ * = 90 • , expressed in terms of the invariant cross section, is where p = [µ cosh(y ′ )] 2 − m 2 .The distribution includes three parameters T 0 , q and a 1 since the integration is almost independent of the parameter a 2 at high energies, i.e. ln( √ s/M) ≫ 3. We calculate the p T distributions for several pairs of values of the parameters (T 0 , q) to look for the best fitting one to the data.The value of the parameter a 1 can be obtained through the empirical relation of the quantity ρ 0 , the pseudo-rapidity density at η * = 0, obtained by CMS Collaboration [2], since it includes the parameters T 0 , q and a 1 .The distributions describe the p T distribution well over the whole region of the data at any incident energy.(See Fig. 2.) The values of the parameters T 0 and q in the energy distribution are obtained by fitting the p T distributions to the data at the energies of √ s = 63, 200, 500, 900, 1800, 2360, 7000 GeV.The empirical energy dependences of them are q = 0.0295 log 10 √ s 1.0 GeV + 1.020, with ǫ 0 = 3.82 × 10 2 GeV.

Pseudo-rapidity density distributions
The pseudo-rapidity distribution includes four parameters of T 0 , q, a 1 and a 2 .The former two have been determined already and the latter two are determined by fitting the distributions to the data.The distributions describe the data well over the whole region of the data at any incident energy.(See Fig.  [3] and 500 (upper) GeV [4].Experimental data are for a half of the charged hadrons (or π 0 's) and hence the curves are a half of eq.( 4).The solid lines (at 63 GeV) and chain lines (at 500 GeV) are for the values of the parameters in the table below.The values of q are determined so as to reproduce the average p T 's at respective energies for the values of T 0 .
The values of the parameters a 1 and a 2 are obtained by fitting the pseudo-rapidity density distributions to the data at the energies of

Figure 1 .
Figure 1.Distribution of the emitting centers on the rapidity axis in CMS.The parameter y 0 is defined as y 0 = ln( √ s/M) − ln a 2 where M is nucleon mass and a 2 an adjustable parameter.(See 2.1.)The produced particles are emitted isotropically from several emitting centers.The energy distribution of produced particles is that of Tsallis type, eq.(2), on respective emitting centers.

Figure 3 .
Figure 3. Pseudo-rapidity density distribution at √ s = 22.4GeV.The chain lines are for the value of the parameters a 1 = 1.0 and a 2 = 0.1, 0.2, 0.5, 1.0, 2.0 (attached to the curves).The solid curve of a 1 = 1.38 and a 2 = 0.5 is the best fitting one to the data by EHS-NA22 Collaboration.[5]