Rapidity and Pseudorapidity distributions of the Various Hadron-Species Produced in High Energy Nuclear Collisions : A Systematic Approach

With the help of a phenomenological approach outlined in the text in some detail, we have dealt here with the description of the plots on rapidity and pseudorapidity spectra of some hadron-secondaries produced in various nucleus-nucleus interactions at high energies. The agreement between the measured data and the attempted fits are, on the whole, modestly satisfactory excepting a very narrow central region in the vicinity of y=$\eta$=0. At last, hints to how the steps suggested in the main body of the text to proceed with the description of the measured data given in the plots could lead finally to a somewhat systematic methodology have also been made.


Introduction and Background
In a chain of our previous works we studied extensively the properties of the rapidity (pseudorapidity) spectra of the various secondary particles in some very high energy collisions. The leftovers of the available latest data on rapidity (pseudorapidity) spectra would here be dealt with for some specific secondaries produced in some very high energy nuclear collisions. The secondary species studied herein are mostly clearly identified particles. Our objective here is quite clear. In Amidst the two previous works, this paper is more in line with our second work [1] on understanding the nature of the rapidity spectra for production of some heavy baryons than with the first one. In this paper, an empirical energy-dependence of one of the parameters was introduced and a systematic approach to the study was built up. This work is just a follow-up of that particular methodology. This work is essentially just an exhaustive study on the rapidity-density or pseudorapidity-density of the identified hadronic secondaries produced in some high energy nuclear collisions. This apart, some clues to developing this procedure as a systematic approach have also been highlighted.
The paper is organized as follows. In the next section (Sec.2) we give the basic outlook and the approach to be taken up for this study. The following section (Sec.3) provides description of the data analyses on Au+Au and Pb+Pb interactions mostly in the graphical plots. The last section is reserved for summing up the conclusions with some suggestive remarks to develop the applied procedure into a somewhat complete systematic methodology.

The Phenomenological Setting : Premises and the Pathway
Following Faessler [2], Peitzmann [3], Schmidt and Schukraft [4] and finally Thomé et al [5], we [6,7] had formulated in the past a final working expression for rapidity distributions in proton-proton collisions at ISR (Intersecting Storage Rings) ranges of energy-values by the following three-parameter parametrization, viz, where C 1 is a normalization constant and y 0 , ∆ are two parameters. The choice of the above form made by Thomé et al [5] was intended to describe conveniently the central plateau and the fall-off in the fragmentation region by means of the parameters y 0 and ∆ respectively. Besides, this was based on the concept of both limiting fragmentation and the Feynman Scaling hypothesis. For all five energies in PP collisions the value of ∆ was obtained to be ∼ 0.55 for pions [6] and kaons [7], Now, the fits for the rapidity (pseudorapidity) spectra for non-pion secondaries produced in the PP reactions at various energies are phenomenologically obtained by De and Bhattacharyya [7] through the making of suitable choices of C 1 and y 0 . It is observed that for most of the secondaries the values of y 0 do not remain exactly constant and show up some degree of species-dependence .
However, for Λ, Ξ, Σ, Ω and φ, it gradually increases with energies and the energy-dependence of y 0 is empirically proposed to be expressed by the following relationship [6] : The nature of energy-dependence of y 0 is shown in the adjoining figure (Fig.1). Admittedly, as k is assumed to vary very slowly with c. m. energy, the parameter y 0 is not exactly linearly correlated to ln √ s N N , especially in the relatively low energy region. And this is clearly manifested in Fig.1. This variation with energy in k-values is introduced in order to accommodate and describe the symmetry in the plots on the rapidity spectra around mid-rapidity. This is just phenomenologically observed by us, though we cannot readily provide any physical justification for such perception and/or observation. And the energy-dependence of y 0 is studied here just for gaining insights in their nature and for purposes of extrapolation to the various higher energies (in the centre of mass frame, √ s N N ) for several nucleon-nucleon, nucleon-nucleus and nucleus-nucleus collisions. The specific energy (in the c.m. system, √ s N N ) for every nucleon-nucleus or nucleus-nucleus collision is first worked out by converting the laboratory energy value(s) in the required c.m. frame energy value(s). Thereafter the value of y 0 to be used for computations of inclusive cross-sections of nucleon-nucleon collisions at particular energies of interactions is extracted from Eq. (2) for corresponding obtained energies. This procedural step is followed for calculating the rapidity (pseudorapidity)-spectra for not only the pions produced in nucleon-nucleus and nucleus-nucleus collisions [6]. However, for the studies on the rapidity-spectra of the non-pion secondaries produced in the same reactions one does always neither have the opportunity to take recourse to such a systematic step, nor could they actually resort to this rigorous procedure, due to the lack of necessary and systematic data on them.
Our next step is to explore the nature of f (y) which is envisaged to be given generally by a polynomial form noted below : where α, β and γ are the coefficients to be chosen separately for each AB collisions (and also for AA collisions when the projectile and the target are same). Besides, some other points are to be made here. The suggested choice of form in expression (3) is not altogether fortuitous. In fact, we got the clue from one of the previous work by one of the authors (SB) [8] here pertaining to the studies on the behavior of the EMC effect related to the lepto-nuclear collisions. In the recent past Hwa et al [9] also made use of this sort of relation in a somewhat different context. Now let us revert to our original discussion and to the final working formula for dN dy in various AB (or AA) collisions given by the following relation : where C 2 is the normalization constant and C 3 =C 2 (AB) α is another constant as α is also a constant for a specific collision at a specific energy. The parameter values for different nucleus-nucleus collisions are given in the Tables ( Table2 -Table 11).
However, it is to be noted that the relationship between rapidity and pseudorapidity is given by the following standard relation with the following properties : (a) In the region y>>0, dN dη ≈ dN dy (b) But, in the region y→0, there ia a small depression of the dN dη distribution relative to dN dy due to the above transformation. In experiments at high energies where dN dy has a plateau shape, this transformation gives a small dip in dN dη around η ≈0. (c) In the c.m. frame, the peak of the distribution is located around y≈ η ≈0, and the peak value of dN dη is smaller than the peak value of dN dy ; And this Diminutive Fraction Factor (DFF) is given by 3 Depicting the Results Obtained

A Few Pointed Steps
The procedural steps for arriving at the results could be summed up as follows : (i) We assume that the inclusive cross section (I.C.) of any particle in a nucleus-nucleus (AB) collision can be obtained from the production of the same in nucleon-nucleon collisions by multiplying the inclusive cross-section (I.C.) by a product of the atomic numbers of each of the colliding nuclei raised to a particular function, which is initially unspecified [10].
(ii) Secondly, we accept the property of factorization [1] of that particular function which helps us to perform the integral over p T in a relatively simpler manner.
(iii) Thirdly, we assume a particular 3-parameter form for the pp cross section with the parameters C 1 , y 0 and ∆.
(iv) Finally, we accept the ansatz that the function f(y) can be modeled by a quadratic function with the parameters α, β and γ.

Final Results Delivered
The results are shown here by the graphical plots with the accompanying tables for the parameter values. Here we draw the rapidity-density of pion(π), kaon(K), proton-antiproton(N ), φ, Ω, Σ, Λ, Ξ for symmetric Pb+Pb and Au+Au collisions and pseudorapidity-density of charged-particle (mainly π + ) for symmetric Au+Au collisions at several energies which have been appropriately labeled at the top right corner. In this context some comments are in order. Though the figures represents the case for production of pion(π), kaon(K), proton-antiproton(N ), φ, Ω, Σ, Λ, Ξ, we do not anticipate and/or expect any strong charge-dependence of the results. Besides, the solid curves in all cases-almost without any exception-demonstrate our GCM-based results. Secondly, the data on rapidity(pseudorapidty)-spectra for some high-energy collisions are, at times, available for both positive and negative y(η)-values. This gives rise to a problem in our method. It is evident here in this work that we are concerned with only symmetric collisions wherein the colliding nuclei must be identical. But in our expression (4) the coefficient β multiplies a term which is proportional to y and so is not symmetric under y→(-y). In order to overcome this difficulty we would introduce here β=0 for all the graphical plots (except Fig.12). These plots are represented by Fig.2 to Fig.11 for π, K, φ, N, Σ, Ξ, Λ, Ω in Pb+Pb and Au+Au collision under different conditions. The parameter values in this particular case are presented in tables ( Table 2 -Table 10

Concluding Remarks and Some Comments
On an overall basis, our model-based results are in fair agreement with the most of the datasets, excepting y≈0 or η≈0 region, wherein the data shows flat-plateau structures in almost all the diagrams exhibiting data on both positive and negative rapidities or pseudorapidities. The degree of disagreement in the vicinity of η≈0 region is evidently much stronger for the plots on pseudorapidity-density versus pseudorapidity plots. These discrepancies might probably be ascribed to our simplistic assumption of y=η. Had we been able to compute the diminutive fraction factor (DFF) as given by expression (6), we would have been capable of giving the pseudorapidity-figures much better looks. And this computation is not possible because of the fact that the rapiditydata-sets do not generally offer even the slightest hints on the p T -ranges of the secondaries under observations and/or measurements.
The last figure of this paper carries some special physical significance which we now explain below. This is, by essence, undoubtedly a purely phenomenological model with no or very little predictive capacity. The energy-dependences studied in Fig. 13 for some of the involved parameters,              [11] and the parameter values are taken from Table 2-Table 5. The solid curve provide the GCM-based results.   Table 2-Table 5. The solid curve provide the GCM-based results.  Figure 5: Pseudo-rapidity spectra for π + for nine centrality bins representing 45% of the total cross-section for Au+Au collisions at √ s N N =19.6 GeV for β=0. The different experimental points are taken from [13] and the parameter values are taken from Table 7. The solid curve provide the GCM-based results.  Figure 8: Pseudo-rapidity spectra for π + for five centrality bins representing 45% of the total cross-section for Au+Au collisions at √ s N N =130 GeV for β=0. The different experimental points are taken from [13] and the parameter values are taken from Table 9. The solid curve provide the GCM-based results.  Figure 9: Pseudo-rapidity spectra for π + for six centrality bins representing 45% of the total crosssection for Au+Au collisions at √ s N N =130 GeV for β=0. The different experimental points are taken from [13] and the parameter values are taken from Table 9. The solid curve provide the GCM-based results.  Figure 10: Plot of dN dη vs. η for π + for six centrality bins representing 45% of the total cross-section for Au+Au collisions at √ s N N =200 GeV for β=0. The different experimental points are taken from [13] and the parameter values are taken from Table 10. The solid curve provide the GCM-based results.  Figure 11: Plot of dN dη vs. η for π + for five centrality bins representing 45% of the total cross-section for Au+Au collisions at √ s N N =200 GeV for β=0. The different experimental points are taken from [13] and the parameter values are taken from Table 10. The solid curve provide the GCM-based results.  Table 11.