Pion, Kaon and Antiproton Production in $Pb+Pb$ Collisions at LHC Energy $\sqrt{s_{NN}}$ = 2.76 TeV : A Model-based Analysis

Large Hadron Collider (LHC) had produced a vast amount of high precision data for high energy heavy ion collision. We attempt here to study (i) transverse momenta spectra, (ii) $K/\pi$, $p/\pi$ ratio behaviours, (iii)rapidity distribution, and (iv) the nuclear modification factors of the pion, kaon and antiproton produced in $p+p$ and $Pb+Pb$ collisions at energy $\sqrt{s_{NN}}$ = 2.76 TeV, on the basis of Sequential Chain Model (SCM). Comparisons of the model-based results with the measured data on these observables are generally found to be modestly satisfactory.


Introduction
Heavy ion collisions at ultra relativistic energies might produce a new form of QCD matter characterized by the deconfined state of quarks and gluons (partons) [1]. Measurements of the production of identified particles provide information about the dynamics of this dense matter. The yield of identified hadrons, their multiplicity distributions, as well as the rapidity and transverse momentum spectra are the basic observables in proton-proton and heavy ion collisions at any energy regime, from a few GeV per nucleon to the new ultra-relativistic LHC regime, spanning c.m. energies of a few TeV. [2] Recently, experimental results in P b + P b collisions at √ s N N = 2.76 TeV in the Large Hadron Collider (LHC) are also reported by the different groups. These results might provide another opportunity to investigate the bulk properties of exotic QCD matter, the so-called QGP-hypothesis. But the exact nature of QGP-hadron phase transition is still plagued by uncertainties. [3] Our objective in this work is to use a model, known as 'Sequential Chain Model (SCM)', which is different from 'standard' framework, in interpreting the transverse momenta spectra, some ratio-behaviours, rapidity spectra and the nuclear modification factor of the pions, kaons and antiprotons for p + p and P b + P b collisions at LHC energy √ s N N = 2.76 TeV. Very recently, a question has been raised about the quark-gluon composition of proton. [4] So, in order to explain the huge amount of heavy ion collision data, we put forward this model which has no QGP-tag and is different from the quark-hypothesis. The organization of this work is as follows. In section 2 we give a brief outline of our approach, the SCM. In the next section (section 3) the results arrived at have been presented with table and figures. And in the last section (section 4) we offer the final remarks and conclusions.

Outline of the Model
This section is divided by two subsections (1) the basic model in p+p collision and (2) subsequent transformation to the A + B collisions.

The Basic Model: An Outline
According to this Sequential Chain Model (SCM), high energy hadronic interactions boil down, essentially, to the pion-pion interactions; as the protons are conceived in this model as p = (π + π 0 ϑ), where ϑ is a spectator particle needed for the dynamical generation of quantum numbers of the nucleons. [5]- [11] The inclusive cross-section of the π − -meson produced in the p + p collisions at high energies has been calculated by using field theoretical calculations and Feynman diagram techniques with the infinite momentum frame approximation method. The inclusive cross-section is given by the undernoted relation [5]- [11] with < n π + > pp ∼ = < n π − > pp ∼ = < n π 0 > pp ∼ = 1.1s 1/5 , where Γ π − is the normalisation factor which will increase as the inelastic cross-section increases and it is different for different energy region and for various collisions. The terms p T , x (1) represent the transverse momentum, Feynman Scaling variable respectively. The s in equation (2) is the square of the c.m. energy.
of the expression (1) is the 'constituent rearrangement term'. It arises out of the partons inside the proton. At high energy interaction processes the partons undergo some dissipation losses due to impact and impulses of the projectile on the target. This term essentially provides a damping term in terms of a power law. The exponent of p T , i.e. N R , varies on both the collision process and the specific p T -range. We have to parametrize this term with the view of two physical points, viz., the amount of momentum transfer and the contributions from a phase factor arising out of the rearrangement of the constituent partons. The relation for N R is to be given by [12] N where < N part > denotes the average number of participating nucleons and θ values are to be obtained phenomenologically from the fits to the data-points.
Similarly, for kaons of any specific variety ( with And for the antiproton production in pp collision at high energies, the derived expression for inclusive cross-section is with

The Path from pp to AB Collisions
In order to transform the inclusive cross-section from pp → C − + X to AB → C − + X collisions ( here, C − stands for π − , K − andp, as the case may be), we use the undernoted relation; [13] Here, in the above equation [eqn. (8)], the first factor gives a measure of the number of wounded nucleons i.e. of the probable number of participants, wherein Aσ B gives the probability crosssection of collision with 'B' nucleus (target), had all the nucleons of A suffered collisions with Btarget. And Bσ A has just the same physical meaning, with A and B replaced. Furthermore, σ A is the nucleon(proton)-nucleus(A) interaction cross-section, σ B is the inelastic nucleon(proton)nucleus(B) reaction cross-section and σ AB is the inelastic AB cross-section for the collision of nucleus A and nucleus B. The values of σ AB , σ A , σ B have been worked here out by the following formula [14] σ inel AB = σ 0 (A 1/3 with σ 0 = 68.8 mb, δ = 1.32. The second term in expression (8) is a physical factor related with energy degradation of the secondaries due to multiple collision effects. The parameter a ′ occurring in this term is a measure of the fraction of the nucleons that suffer energy loss. The maximum value of a ′ is unity, while all the nucleons suffer energy loss. This a ′ parameter is usually to be chosen [13], depending on the centrality of the collisions and the nature of the secondaries.

The Results
This section will be divided in the following sub-sections: (i) the p T -spectra of pion, kaon and antiproton in both p + p and P b + P b collisions at √ s N N = 2.76 TeV; (ii) K/π and p/π ratio behaviour at P b + P b collisions at √ s N N = 2.76 TeV; (iii) rapidity distribution of pion for the most central collisions of P b+P b in the above-mentioned energy and (iv) the nuclear modification factor R AA in the same energy range.

TeV
We can write from expression (8), the transformed SCM-based transverse-momentum distributions for A + B → C − + X-type reactions in the following generalized notation: Where, for example, the parameter α π − can be written in the following form: In a similar way, the values of β π − of the equation (10) have been calculated with the help of eqn.(1), eqn. (2). The values of α K − , αp, β π − and βp have been calculated accordingly by using eqns. (4)- (8). Moreover, for calculation for transverse momenta distribution of antiproton production, the the exponential part will be exp(−βp(p 2 T + mp 2 )) (mp ∼ 938 M eV /c 2 ). N C − R of the expression (10) have been calculated by using eqn. (3).

Production of
In table 1, the calculated values of α, N R and β of eqn.(10) for π − , K − andp have been given.
In figure (1), we have drawn the invariant yields against p T for π − , K − andp. By using equation (10) and Table 1 we have plotted the solid lines against the experimental background [15]. The dotted lines in the Fig. show PYTHIA-based results [15].  Table 2.

The K/π and p/π ratios
The model-based K/π and p/π ratios as a function of p T at energies √ s N N = 2.76 TeV have been obtained from the expression (10), Table 1 and Table 2. Data in Figs. 3(a) and 3(b), for different centralities, viz., for 0-5%, 20-30% and 70-80%, are taken from Ref. [16]. Lines in the Figure show the theoretical plots.

The Rapidity Distribution
For the calculation of rapidity distribution, we can make use of the following standard relation [17], By using eqn. (1), eqn. (8), Table 2 and eqn. (12), we arrive at the SCM-based rapidity distribution, which is given hereunder; dN dy = 1895 exp(−0.025 sinh y cm ).
In Fig.4, we have plotted theoretical dN/dy versus y with the help of above equation [eqn. (13)] against experimental background [18]. The dotted line in this Fig. shows the Gaussian fit [18].

The Nuclear Modification Factor
The nuclear modification factor (NMF) R AA is defined as ratio of charged particle yield in P b + P b to that in p + p, scaled by the number of binary nuclear collisions < N coll > [19] and is given hereunder where < N coll > is related with the average nuclei thickness function(< T AA >) by the following relation [19] < N coll >=< T AA > σ incl pp .
Here, σ incl pp is the total inelastic cross section of p + p interactions. The d 2 N/dp T dη is related to d 2 N/dp T dy by the following relation: [13] In the region, y >> 0, d 2 N dp T dη ∼ d 2 N dp T dy . The SCM-based results on NMF's for π − , K − andp in central P b + P b collisions at energies √ s N N = 2.76 TeV are deduced on the basis of Eqn.(10), Table 1 and Table 2. The equations in connection with (R AA ) π − , (R AA ) K − and (R AA )p are give by the following relations and they are plotted in Fig.5 against p T . The solid lines in the figure show the theoretical results, while the points show the experimentally measured results [20];

Discussions and Conclusions
Let us now make some comments on the results arrived at and shown by the diagrams on the case-to-case basis.
(1) The invariant yields against transverse momenta (p T ) for π − , K − andp in proton-proton collisions obtained on the basis of the SCM are shown in Figure 1. Except for very low-p T region, there is a bit degree of success. The model disagrees in the low-p T region. This is due to the fact that the model has turned essentially into a mixed one with the inclusion of power law due to the inclusion of partonic rearrangement factor. However, the power-law part of the equation might not be the only factor for this type of discrepancy. The initial condition and dynamical evolution in heavy-ion collisions are more complicated than we expect. Till now, we do not know the exact nature of reaction mechanism. One might take into account some other factors like radial flow or thermal equilibrium.
(2) Similarly, in calculating the yields for different transverse momenta and for different centralities for π − , K − andp in lead-lead collisions, we use eqns. (8), (9) and (10) alongwith eqns. (1)- (7). The results are given in Table 2 and are depicted in Figs. 2(a), 2(b) and 2(c) respectively. The top-most curves are for central collisions (0-5%) and the lowest curves are peripheral ones (80-90%). In between these two curves, other centralities (5-10%, 10-20%, 20-30%, 30-40%, 40-50%, 50-60%, 60-70% and 70-80%) have been plotted. For the production of pions, the SCM-based results show good fits from central to peripheral collision. Slight disagreements observed at very low-p T regions for kaons and protons at central collisions. These are due to the power law part of the model. Here, we see that the constituent rearrangement terms are clearly centrality dependent. This explanation is also true for low-p T region data in p + p collision.
(3) The K/π and p/π ratio behaviours for different centralities are calculated in the light if the SCM and they are plotted in Figs. 3(a) and 3(b) respectively. The theoretical K/π ratio behaviours are in good agreement with experimental values. Some disagreement are observed in central p/π-ratio in low-p T regions.
(4) In explaining the rapidity distribution for production of pions (Figure 4), the majority of the produced secondaries, the model works agreeably with data. The comparison with Gaussian fit is satisfactory.
(5) The nuclear modification factors for pion, kaon and proton for central P b + P b-collisions for different transverse momenta have been calculated and they are plotted in Figure 5. While the theoretical plots are agreeable in low-p T regions, they disagree in high-p T . Now, let us sum up our observations in the following points; (1) the model under consideration here explains the data modesly well on P b + P b collisions at √ s N N = 2.76 TeV. (2) The particle production in heavy ion collisions can be viewed alternatively by this Sequential Chain Model. Table 1: Values of α, N R and β for pions, kaons, antiproton and proton productions in p + p collisions at √ s N N = 2.76 TeV   [15]. Table 2: Values of (α) P bP b , (N R ) P bP b and (β) P bP b for different centralities for the π − , K − and p productions in P b + P b collisions at √ s N N =2.76 TeV     Figure 3: Ratios of (a) K/π and (b) p/π as a function p T for 0-5%, 20-30% and 70-80% P b + P b reactions at √ s N N =2.76 TeV. Data in these Figures are taken from [16]. The solid lines show the SCM-based results.