On Production of Hadrons in Proton-Proton Collisions at RHIC and LHC Energies and an Approach

From the very early days of Particle Physics, both experimental and theoretical studies on proton-proton collisions had occupied the center-stage of attention for very simple and obvious reasons. And this intense interest seems now to be at peak value with the onset of the Large Hadron Collider (LHC)-studies at TeV ranges of energies. In this work, we have chosen to analyse the inclusive cross-sections, the rapidity density, the $K/\pi$ and $p/\pi$-ratio behaviours and the $$-values, in the light of the Sequential Chain Model (SCM). And the limited successes of the model encourage us to take up further studies on several other aspects of topmost importance in particle physics with the same approach.


Introduction
Proton-proton collisions are known to be the most elementary interactions and form the very basis of our knowledge about the nature of high energy collisions in general. Physicists, by and large, hold the view quite firmly that the perturbative quantum-chromodynamics (pQCD) provides a general framework for the studies on high energy particle-particle collisions [1]. Obviously, the unprecedented high energies attained at Large Hadron Collider (LHC) offer new windows and opportunities to test the proposed QCD dynamics with its pros and cons. Naturally the normal expectations run high that the bulk properties of the collision system such as all the momentum spectra and correlations of all produced hadrons should follow the strictures of QCD. But this not definite and concretely-shaped knowledge about how this actually happens and to what extent the process could be understood in the perturbative and non-perturative domains. The issues involved here still remain, to a considerable extent, quite open [2], [3]. Thus, having been somewhat repulsed by the so-called standard approach, we try here to explain some crucial aspects of measured data on pp reactions at the LHC range of energies with the help of some alternative approach. Our main thrust would be on the properties of time-tested familiar observables like transverse momenta spectra, rapidity distributions, the ratio-behaviours and average transverse momenta (< p T >) for the charged secondaries in high energy pp interactions.
Comparison with some other model would be made whenever possible.
The organisation of the paper is as follows: In Section 2, we provide a brief outline of the model chosen for study. In Section 3, the results obtained by the model-based study are presented. In Section 4, we end up with a discussion on the results and the observations made in Section 3 and the conclusions.

The Approach: An Outline
This section gives a brief overview of the model-based features for the production mechanism of the secondary hadrons in nucleon-nucleon (p + p) interaction in the context of the Sequential Chain Model (SCM). 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 [4]- [9]. The production of pions in the present scheme occurs as follows: the incident energetic π-mesons in the structure of the projectile proton(nucleon) emits a rho(̺)-meson in the interacting field of the pion lying in the structure of the target proton, the ̺-meson then emits a π-meson and is changed into an omega(ω)-meson, the ω-meson then again emits a π-meson and is transformed once again into a ̺-meson and thus the process of production of pion-secondaries continue in the sequential chain of ̺-ω-π mesons. The two ends of the diagram contain the baryons exclusively [4]- [9].
For K + (K − )or K 0K 0 production the model proposes the following mechanism. One of the interacting π-mesons emits a ̺-mesons; the ̺-mesons in its turn emits a φ 0 -meson and a π-meson.
The π-meson so produced then again emits ̺ and φ 0 mesons and the process continues. The φ 0 mesons so produced now decays into either K + K − or K 0K 0 pairs. The ̺-π chain proceeds in any Fenymann diagram in a line with alternate positions, pushing the φ 0 mesons (as producers of K + K − or K 0K 0 pairs) on the sides. This may appear paradoxical as the φ 0 production crosssection is generally smaller than the KK production cross-section; still the situation arises due to the fact that the φ 0 resonances produced in the collision processes will quickly decay into KK pairs, for which the number of φ 0 will be lower than that of the KK pairs. Besides, as long as φ 0 mesons remain in the virtual state, theoretically there is no problem, is an observed and allowed decay mode, wherein the strangeness conservation is maintained with the strange-antistrange coupled production. Moreover, φ 0 K + K − ( or φ 0 K 0K 0 ) coupling constant is well known and is measured by experiments with a modest degree of reliability. And we have made use of this measured coupling strength for our calculational purposes, whenever necessary. It is assumed that the K + K − and K 0K 0 pairs are produced in equal proportions [4]- [9]. The entire production process of kaon-antikaons is controlled jointly by the coupling constants, involving ̺-π-φ and φ 0 -K + K − or φ 0 -K 0K 0 . Now we describe here the baryon-antibaryon production. According to the SCM mechanism, the decay of the pion secondaries produces baryon-antibaryon pairs in a sequential chain as before. The pions producing baryons-antibaryons pairs are obviously turned into the virtual states. And the proton-antiproton pairs are just a part of these secondary baryon-antibaryon pairs. In the case of baryon-antibaryon pairs it is postulated that protons-antiprotons and neutrons-antineutrons constitute the major bulk, Production of the strange baryons-antibaryons are far less due to the much smaller values of the coupling constants and due to their being much heavier.
The field theoretical calculations for the average multiplicities of the π, K andp-secondaries and for the inclusive cross-sections of those secondary particles deliver some expressions which we would pick up from [4]- [9].
The inclusive cross-section of the π − -meson produced in the p + p collisions given by with 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, for example, |Γ π − | ∼ = 90 for Intersecting Storage Ring(ISR) energy region. The terms p T , x in equation (1) represent the transverse momentum, Feynman Scaling variable respectively. Moreover, by definition, is the longitudinal momentum of the particle. The s in equation (2) is the square of the c.m. energy. (1) 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. In this context, the only additional physical information obtained from the observations made here is: with increase in the peripherality of the collisions the values of θ gradually grow less and less, and vise versa.
Similarly, for kaons of any specific variety ( with |Γ K − | ∼ = 11.22 for ISR energies and with And for the antiproton production in pp scattering at high energies, the derived expression for inclusive cross-section is with |Γp| ∼ = 1.87 × 10 3 and mp is the mass of the antiprotons. For ultrahigh energies 3 The Results Now let us proceed to apply the chosen model to interpret some recent experimental results of charged hadrons production for p + p collisions at different energies. Here, the main observables are the inclusive cross-sections or invariant yields, rapidity distributions, ratio behaviour and the average transverse momenta.

Inclusive Cross-sections
The general form of our SCM-based transverse-momentum distributions for p+p → C − +X-type reactions can be written in the following notation: The value of α π − , for example, can be calculated from the following relation: The values of (α π − ) pp , (N π − R ) pp and (β π − ) pp for different energies are given in Table 1. The experimental data for the inclusive cross-sections versus p T [GeV /c] for π − production in p + p interactions at √ s N N = 62.4 GeV and 200 GeV are taken from Ref. [11] and they are plotted in Figs. 1(a) and 1(b) respectively. The production of π − , K − andp at mid-rapidity in protonproton collisions at √ s N N = 900 GeV has been plotted by lines in Fig. 1(c). Data are taken from [12]. For the data for charged particle distribution Ed 3 N ch /dp 3 = 1/(2πp T )E/p(d 2 N ch /dηdp T ) at energies √ s N N = 546 GeV and √ s N N = 900 GeV we use references [3], [13]. And for LHC data for charged particle distribution for energies √ s N N = 0.9 TeV, 2.36 Tev and 7 Tev we use references [14], [15]. They are plotted in Figure 2 and Figure 3 respectively. The solid lines in those figures depict the SCM-based plots. As the main variety of the charged particles coming out are the pions, we use here eqn.(1) for calculational purposes. The 'NSD'-term, used by the experimentalists, has the meaning of non-single diffractive collisions [16].

The Rapidity Distribution
For the calculation of the rapidity distribution we can make use of a standard relation as given below: In Table 3 we had made a comparison between experimentally found dn/dy for π − , K − andp in Similarly, for LHC-energies, by using eqn.(1) and eqn. (11), the SCM-based dN ch /dη will be given hereunder dN ch dη = 4.75 exp(−0.009 sinh η) f or √ s N N = 2.38 T eV, and dN ch dη = 6.28 exp(−0.011 sinh η) f or √ s N N = 7 T eV.

The Ratio-behaviours for Different Secondaries
The nature of the relation of K/π ratios with the SCM, presented in the previous work [20], would be written in the following form Fig. 6(a) shows the nature of rise of K/π ratio in the light of SCM-based above relation (eqn. (15)). Data are taken from Ref. [21].
Similarly, in Fig. 6(b), we have presented the p/π ratio for the RHIC and LHC-data. [21] The SCM-based calculations are done on the basis of eqn.(1) and Table 1.

< p T > Values
Next we attempt at deriving model-based expression for < p T >.
The definition for average transverse momentum < p T > is given below.

Discussions and Conclusions
Let us make some general observations and specific comments on the results arrived at and shown by the diagrams on a case-to-case basis. d) The agreements between the measured data on K/π and p/π ratio and the theoretical SCM plots (shown in Fig. 6)for different energies are strikingly encouraging.
e) Fig. 7 shows the plots of < p T > vs. √ s N N . The initial indication of the SCM-based theoretical plot shows a modest agreement with the data.