Production of J/$\Psi$-Particles at RHIC and LHC energies: An Alternative `Psi'-chology

We attempt here to understand successfully some crucial aspects of $J/\Psi$-production in some high energy nuclear collisions in the light of a non-standard framework outlined in the text. It is found that the results arrived at with this main working approach here is fairly in good agreement with both the measured data and the results obtained on the basis of some other models of the `standard' variety. Impact and implications of this comparative study have also been precisely highlighted in the end.


Introduction
The study of the J/Ψ-mesons in ultra-relativistic heavy ion collisions has consistently been considered to be a potentially powerful tool for studying the properties of the hypothetical 'hot and dense matter' created in these collisions. Predictions about the suppression of J/Ψ-meson production in the nuclear collision, at certain stages [1] and an anomalous enhancement at certain other stage as well [2] are always treated as very powerful diagnostics. The deciding factor, in terms of the Standard Model (SM), in both the cases, is the number of charm-anticharm (cc) pairs (N cc ) created in the early stage of hard parton collisions in A + A (or in A + B) reactions.
On the basis of it, one might arrive at the assumptions of the J/Ψ-suppression (at N cc < 1) and the J/Ψ-enhancement (at N cc > 1) effects.
In the past [3]- [5] we had constantly refrained from giving such undue importance to any controversy about either suppression/enhanced suppression or an enhancement of J/Ψ mesons in reality. Rather our primary intent would centre around understanding and interpreting the nature of data of inclusive cross-sections and some other important observables of J/Ψ mesons in BNL-RHIC and CERN-LHC experiments. In fact, we have an altogether different and heretic view (which will be outlined very briefly in the next section) about the mechanism of J/Ψ production in high energy particle and nuclear collisions.
We define our objectives here as: 1) to explain the main and major features of the latest data on J/Ψ-production in BNL-RHIC and CERN-LHC experiments from the proposed alternative approach built up by us in a set of previous works done in the both the remote and recent past [6]- [7] and 2) to compare our model-based calculations with some other competing models.
Our plan of work presented here is as follows. In section 2, we provide brief outlines of the models chosen for this study. The section 3 provides very brief outlines of the models which are founded on the gantlets of assumptions of the 'Standard'-model and which have been reckoned with for the sake of comparison. In section 4 we give the results and general discussion. And in the last section (section 5) we offer the final remarks-cum-conclusions.

The Main Approach: An Outline
In the present work, we will make use of a theoretical model to interpret some of the latest and topical observables of the J/Ψ production which were measured and reported by the different groups in the Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) exper-iments in the recent past. The model has a modest degree of dynamical basis and some prior check-ups with data [3]- [5]. It is called here as the sequential chain model (SCM).
The outline and features of the model we use here are obtained, in the main, from our previous works [3]- [5]. According to this model, 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.
The multiple production of J/Ψ-mesons in a high energy proton-proton collisions is described in the following way. The secondary π-meson or the exchanged ̺-meson emit a free ω-meson and pi-meson; the pions so produced at high energies could liberate another pair of free ̺ and trapped ω-mesons (in the multiple production chain). These so-called free ̺ and ω-mesons decay quite a fast into photons and these photons decay into Ψ or Ψ ′ particles, which, according to this alternative approach is a bound state of ΩΩ or Ω ′Ω′ particles.
The inclusive cross-section of the Ψ-meson produced in the pp collisions given by where the expression for for average multiplicity for Ψ-particles in pp scattering would be given by < n J/Ψ > pp = 4 × 10 −6 s 1/4 .
In the above expression, the term |C J/Ψ | is a normalisation parameter and is assumed here to have a value ∼ = 0.09 for Intersecting Storage Ring(ISR) energy, and it is different for different energy and for various collisions. The terms p T , x and m J/Ψ represent the transverse momentum, Feynman Scaling variable and the rest mass of the J/Ψ particle respectively. Moreover, by definition, x = 2p L / √ s where p L is the longitudinal momentum of the particle. The s in equation (2) is the square of the c.m. energy.
The second term in the right hand side of the equation (1), the constituent rearrangement term arises out of the partonic rearrangements inside the proton. It is established that hadrons (baryons and mesons) are composed of few partons. These rearrangements mean undesirable loss of energy , in so far as the production mechanism is concerned. The choice of N R would depend on the following factors: (i) the specificities of the interacting projectile and target, (ii) the particularities of the secondaries emitted from a specific hadronic or nuclear interaction and (iii) the magnitudes of the momentum transfers and of a phase factor (with a maximum value of unity) in the rearrangement process in any collision. The parametrisation is to be done for 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. Collecting and combining all these, we propose the relation to be given by [8] 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 [9].
In order to study a nuclear interaction of the type A + B → Q + x, where A and B are projectile and target nucleus respectively, and Q is the detected particle which, in the present case, would be J/Ψ-mesons, the SCM has been adapted, on the basis of the suggested Wong [10] work to the Glauber techniques. The inclusive cross-sections for J/Ψ production in different nuclear interactions of the types A + B → J/Ψ + X in the light of this modified Sequential Chain Model (SCM) can then be written in the following generalised form as: where a J/Ψ , N R and c are the factors to be calculated under certain physical constraints. The set of relations to be used for evaluating the parameters a J/Ψ is given below.
Here, in the above set of equations, the third 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 B-target. And Bσ A has just the same physical meaning, with A and B replaced.
Besides, in expression (5), the fourth term is a physical factor related with energy degradation of the secondaries due to multiple collision effects. The parameter a ′ occurring in eqn.(5) above 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 [10], depending on the centrality of the collisions and the nature of the secondaries.
The "a" factor in the expression (5)  3 Some Competing Models: The Brief Outlines

The Gluon Saturation Approach
In this approach [11], it is assumed that the nuclear wave functions in very high-energy nuclear collisions can be described by the Color Glass Condensate (CGC). The primary effect is the suppression of J/Ψ production and narrowing of the rapidity distribution due to saturation of the gluon fields in heavy ion collisions relative to p + p collisions. In addition, the production mechanism is modified from p + p such that the multigluon exchange diagrams are enhanced in heavy ion reactions. It should be noted that this model does not include any hot medium effects, but does have a free parameter for the overall normalization factor.

The Quark Coalescence Model (QCM)
The Quark Coalescence Model (QCM) [12], is a two-stage simulation. In stage I the incoming energies comover suppression of directly produced charmonia is expected to be large, due to the increased particle numbers and densities.

Double Color Filter-Oriented Approach
A mechanism called double color filtering for cc dipoles [13], makes nuclei significantly more

J/Ψ (total) Crosssections and Rapidity Distribution in p + p Interactions
As the psi-productions are generically treated rightly as the resonance particles, the standard practice is to express the measured J/Ψ (total) crosssections times branching ratio to muon or electrons, i.e.for lepton pairs , that is by B ll ′ σ J/Ψ p+p . By using expression (4) we arrive at the expressions for the differential cross-sections for the production of J/Ψ-mesons in the mid and forward-rapidities (i.e. |y| < 0.35 and 1.2 < |y| < 2.2 respectively) in p + p collisions at √ s N N =200 GeV at RHIC. and For deriving the expressions (6) and (7) we have used the relation where m T , y cm are the transverse mass of the produced particles and the rapidity distributions.
For the calculation of the rapidity distribution from the equation (4) we can make use of a standard relation as given below: For p + p collisions, the calculated rapidity distribution equation at RHIC-energy √ s N N = 200 GeV is dN dy | p+p→J/Ψ+X = 1.215 × 10 −6 exp(−0.23 sinh y cm ), In Fig. 2 we have plotted the rapidity distributions for J/Ψ-production in p + p collisions at √ s N N = 200 GeV. Data in the figure are taken from Ref. [17] and the line shows the SCM-based output.
In a similar fashion, the SCM-based rapidity distribution equation at LHC-energy √ s N N = 7 TeV in p + p collisions has been given hereunder dN dy | p+p→J/Ψ+X = 8.025 exp(−0.043 sinh y cm ), In Figure 3, we have drawn the solid lines depicting the SCM-based results with the help of above equation (12)  In Figure 4, we have drawn the solid lines depicting the SCM-based results with the help of above two equations (13) and (14) against the experimental measurements [19].
For calculating the values of N R , in general, we have used the values of < N part > from [11], [19].
In the Fig. 6, the solid lines are the plots of SCM-based invariant yields vs. p T as described by equations (17) and (18) at forward and mid-rapidities for Au + Au collisions, while the dotted curve in the Fig. shows results of coalescence model [12]. The experimental data points in the

The Nuclear Modification Factor R AB
There is yet another very important observable called nuclear modification factor (NMF), denoted here by R AA which for the production of J/Ψ is defined by [20] R AA = d 2 N AA J/Ψ /dp T dy < N coll (b) > d 2 N pp J/Ψ /dp T dy .
the SCM-based results on NMFs for Cu + Cu and Au + Au collisions for forward rapidities are deduced on the basis of Eqn.(6), Eqn. (15), Eqn. (17) and Eqn. (19) and they are given by the undernoted relations and herein the value of < N coll (b) > to be used is ≈ 170.5 ± 11 [21] for Cu + Cu collisions and for Au + Au collisions it is taken as ≈ 955.4 ± 93.6 [22].
In Fig. 7(a), we plot R AA vs. p T for 0−20% central region in Cu+Cu and Au+Au collisions.
The solid lines in the figure show the SCM-based results against the experimentally measured results [19], [20]. The dotted lines in the fig. represent the double Color Filtering approach [13].
And in Fig. 7

Summary and Outlook
Let us first concentrate on what we have achieved here: (i) The features related to p T -spectra for J/Ψ production in some particle-particle and nuclear collisions at various high energies have been In the past such were the recurrent observations made by us valid for many other obsevables measured in the various high statistics high energy particle and nuclear experiments.
Thus, summing up our past experiences and considering the weightage of the results reported here, we are forced to comment finally that this work essentially represents a case of paradigm shift in the domain of particle theory, as we have eschewed the conventional views of cc approach to J/Ψ production in the 'standard' framework. And this is just the reflection of our radical views about the particle structure and the nature of particle collisions. Obviously we obtain the fair agreement with data on some observables without inductions of (i) any QGP concept, (ii) any prognosis of suppression or enhancement of J/Ψ-production. The production characteristics resemble all other hadrons.   The data points are from [19], [17]. The solid curves show the SCM-based results. 200GeV as function of p T . The data points are from [17]. The solid curve shows the SCM-based results while the dotted curve depicts the Coalescence Model [12]. 200GeV as function of p T . The data points are from [19], [20]. The solid and dotted curves show respectively the SCM and the Double Color Filter-oriented [13] results. (b) Plot of the R AA vs. N part for J/Ψ production in Au + Au collisions at √ s N N = 200GeV and P b + P b collisions at √ s N N = 2.76T eV . The red circles depict Au + Au collisions [11] while the black squares represent P b + P b collisions [23] . The solid curves show the SCM-based results while the dotted curve depicts the Gluon Saturation Approach [11].