Jet Transport Coefficient at the Large Hadron Collider Energies in a Color String Percolation Approach

Within the color string percolation model (CSPM), jet transport coefficient, $\hat{q}$, is calculated for various multiplicity classes in proton-proton and centrality classes in nucleus-nucleus collisions at the Large Hadron Collider energies for a better understanding of the matter formed in ultra-relativistic collisions. $\hat{q}$ is studied as a function of final state charged particle multiplicity (pseudorapiditydensity at midrapidity), initial state percolation temperature and energy density. The CSPM results are then compared with different theoretical calculations from the JET Collaboration those incorporate particle energy loss in the medium.


I. INTRODUCTION
The main objective of tera-electron volt energy heavyion collisions is to form a Quark-Gluon Plasma (QGP)the deconfined state of quarks and gluons, by creating extreme conditions of temperature and(or) energy density [1,2], a scenario that might have been the case after a few microseconds of the creation of the Universe. Jets, collimated emission of a multitude of hadrons originating from the hard partonic scatterings, play an important role as hard probes of QGP. These hard jets lose their energy through medium-induced gluon radiation and collisional energy loss, as a consequence of which one observes suppression of high transverse momentum particles and the phenomenon is known as jet quenching [3][4][5][6][7][8][9]. This is a direct signature of a highly dense partonic medium, usually formed in high energy heavyion collisions. The first evidence of the jet quenching phenomenon has been observed at the Relativistic Heavy-Ion Collider (RHIC) [10][11][12][13][14][15][16][17][18][19][20][21][22][23] via the measurement of inclusive hadron and jet production at high p T , γ-hadron correlation, di-hadron angular correlations and the dijet energy imbalance. The jet quenching phenomena are also widely studied in heavy-ion collisions at the Large Hadron Collider (LHC) [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. All the measured observables are found to be strongly modified in central heavy-ion collisions relative to minimum bias proton-proton collisions, when compared to expectations based on treating heavyion collisions as an incoherent superposition of independent nucleon-nucleon collisions.
In the present work, we studyq and its relation with various thermodynamic properties of the QCD matter in the framework of the Color String Percolation Model (CSPM) [59][60][61][62][63][64] which is inspired by QCD. This can be used as an alternative approach to Color Glass Condensate (CGC) [64] and is related to the Glasma approach [65]. In CSPM, it is assumed that color strings are stretched between the projectile and the target, which may decay into new strings via qq pair production and subsequently hadronize to produce observed hadrons [66]. These color strings may be viewed as small discs in the transverse plane filled with color field created by colliding partons. The final state particles are produced by the Schwinger mechanism, emitting qq pairs in this field [67]. With the increasing collision energy and size of the colliding nuclei, the number of strings grows and they start interacting to form clusters in the transverse plane. This process is very much similar to discs in the 2-dimensional percolation theory [60,62,68,69]. At a certain critical density, called critical percolation density (ξ c ≥ 1.2), a macroscopic cluster appears that marks the percola-tion phase transition [60,62,[68][69][70][71]. The combination of the string density dependent cluster formation and the 2-dimensional percolation clustering phase transition are the basic elements of the non-perturbative CSPM. In CSPM, the Schwinger barrier penetration mechanism for particle production and the fluctuations in the associated string tension due to the strong string interactions make it possible to define a temperature. The critical density of percolation is related to the effective critical temperature and thus percolation may provide information on deconfinement in the high-energy collisions [63,64]. The CSPM approach has been successfully used to describe the initial stages in the soft region in high-energy collisions [59,64,68,[72][73][74][75][76][77]. In addition to this, CSPM has also been quite successful in estimating various thermodynamic and transport properties of the matter formed in ultra-relativistic energies [78][79][80][81][82][83][84].
The paper runs as follows: first, we present the formulation and methodology of the CSPM approach in section II. In section III, we present the result obtained and related discussions. Finally, the important findings of this work are summarized in section IV.

II. FORMULATION AND METHODOLOGY
In the color string percolation model, the charged hadron multiplicity, µ n , where n stands for the number of strings in a cluster reduces with the increase of string interactions while the mean transverse momentum squared, p 2 T n , of these charged hadrons increases, to conserve the total transverse momentum. The µ n and p 2 T n of the particles produced by a cluster are proportional to the color charge and color field, respectively [62,64] and can be defined as where S n denotes the transverse overlap area of a cluster of n-strings and the subscript 1 refers to a single string with a transverse overlap area S 1 = πr 2 0 with the string radius, r 0 = 0.2 fm [64], respectively. For the case when strings are just touching each other S n = nS 1 , and µ n = nµ 1 , p 2 T n = p 2 T 1 . When strings fully overlap S n = S 1 and therefore µ n = √ nµ 1 and p 2 T n = √ n p 2 T 1 , so that the multiplicity is maximally suppressed and the p 2 T n is maximally enhanced. This implies a simple relation between the multiplicity and transverse momentum µ n p 2 T n = nµ 1 p 2 T 1 , which denotes the conservation of the total transverse momentum. In the thermodynamic limit, one can obtain the average value of nS 1 /S n for all the clusters [60,62] as Here, F (ξ) is the color suppression factor by which the overlapping strings reduce the net-color charge of the strings. With F (ξ) → 1 as ξ → 0 and F (ξ) → 0 as ξ → ∞, where ξ = NsS1 S N is the percolation density parameter. Eq. (1) can be written as µ n = nF (ξ)µ 0 and p 2 T n = p 2 T 1 /F (ξ). It is worth noting that CSPM is a saturation model similar to the Color Glass Condensate (CGC), where p 2 T 1 /F (ξ) plays the same role as the saturation momentum scale Q 2 s in the CGC model [65,85]. In the present work we have extracted F (ξ) in pp collisions at √ s = 5.02 and 13 TeV for various multiplicity classes using ALICE published results of transverse momentum spectra of charged particles [86]. In case of Pb-Pb collisions at √ s NN = 2.76 and 5.02 TeV, [87] and Xe-Xe collisions at √ s NN = 5.44 TeV [88], F (ξ) values have been obtained from the published centralitydependent transverse momentum spectra of charged particles by ALICE. To evaluate the initial value of F (ξ) from data, a parameterization of the experimental data of p T distribution in low-energy pp collisions at √ s = 200 GeV (minimum bias), where strings have very low overlap probability, was used [68]. The p T spectrum of charged particles can be described by a power-law [64]: where a is the normalisation factor and p 0 , α are fitting parameters given as, p 0 = 1.98 and α = 12.87 [64]. This parameterization is used in high-multiplicity pp and centrality-dependent heavy-ion (AA) collisions to take into account the interactions of the strings [64]. The parameter p 0 in Eq. (3) is for independent strings and gets modified to Using Eq.(4) and Eq.(2) in Eq.(3), we get where F (ξ) mod is the modified color suppression factor and is used in extracting F (ξ) both in pp and AA collisions. The spectra were fitted using Eq.(5) in the softer sector with p T in the range 0.15 -1.0 GeV/c. In pp collisions at low-energies only two strings are exchanged with low probability of interactions, so that nS 1 /S n pp ≈ 1, which transforms Eq.(5) into In the thermodynamic limit, the color suppression factor F (ξ) is related to the percolation density parameter ξ as

III. RESULTS AND DISCUSSIONS
In the present work, we have extracted F (ξ) in the multiplicity-dependent pp collisions at √ s = 5.02 and 13 TeV [86] and centrality-dependent Pb-Pb collisions at √ s NN = 2.76 and 5.02 TeV [87], and Xe-Xe collisions at √ s NN = 5.44 TeV [88] from the charged particles p T spectra measured in ALICE at the LHC. For pp collisions multiplicity-dependent S ⊥ is obtained from IP-Glasma model [89]. In case of Xe-Xe and Pb-Pb collisions we use S ⊥ values obtained using the Glauber model [90].
We show ξ and F (ξ) as functions of final charged particle multiplicity for pp, Xe-Xe and Pb-Pb collisions in Fig. 1 upper and lower panel, respectively. The error in F (ξ) is obtained by changing the fitting ranges of the transverse momentum spectra and is found within ∼ 3%. For a better comparison of proton-proton and nucleusnucleus collisions, dN ch /dη is scaled by the transverse overlap area S ⊥ for both pp and heavy-ion collisions. For pp collisions, multiplicity-dependent S ⊥ is calculated using the IP-Glasma model [89]. In the case of heavy-ion collisions, the transverse overlap area was obtained using the Glauber model calculations [90]. It is observed that F (ξ) falls onto a universal scaling curve for protonproton and nucleus-nucleus collisions. Particularly, in the most central heavy-ion collisions (high N tracks ) and highmultiplicity pp collisions, F (ξ) values fall in a line. This suggests that the color suppression factor is independent of collision energies and collision systems in the domain of high final state multiplicity. Further, what decides the color suppression factor is the final state multiplicity density of the system, which turns out to be the initial parton density in a system for the case of an isentropic expansion.

A. Temperature
The connection between F (ξ) and the initial percolation temperature T (ξ) involves the Schwinger mechanism for particle production [63,64,67] and can be expressed as [63,68] We adopt the point of view that the universal hadronization temperature is a good measure of the upper end of the cross-over phase transition temperature T h [91]. The single string average transverse momentum p 2 T 1 is calculated at the critical percolation density parameter ξ c = 1.2 with the universal hadronization temperature T h = 167.7 ± 2.6 MeV [91]. This gives p 2 T 1 = 207.2 ± 3.3 MeV. In this way at ξ c = 1.2 the connectivity percolation transition at T (ξ c ) models the thermal deconfinement transition. The temperature obtained for most central Pb-Pb collisions at √ s NN = 2.76 TeV in this work is ∼ 223 MeV, whereas the direct photon measurement up to p T < 10 GeV/c gives the initial temperature T i = 297 ± 12 stat ± 41 sys MeV for 0-20% central Pb-Pb collisions at √ s NN = 2.76 TeV measured by the ALICE Collaboration [92]. The measured temperature shows that the temperature obtained using Eq. (8) can be termed as the temperature of the percolation cluster. Figure 2 shows a plot of initial temperature from CSPM as a function of dN ch /dη scaled by S ⊥ . Temperatures from both pp and AA collisions fall on a universal curve when multiplicity is scaled by the transverse overlap area. The horizontal line at ∼ 167.7 MeV is the universal hadronization temperature obtained from the systematic comparison of the statistical thermal model parametrization of hadron abundances measured in high energy e + e − , pp and AA collisions [91]. One can see that temperature for higher multiplicity classes in pp collisions at √ s = 5.02 and 13 TeV, are higher than the hadronization temperature and similar to those observed in Xe-Xe at √ s NN = 5.44 TeV and Pb-Pb collisions at √ s NN = 2.76 and 5.02 TeV.

B. Energy density
The calculation of the bulk properties of hot QCD matter and characterization of the nature of the QCD phase transition is one of the most important and fundamental problems in finite-temperature QCD. The QGP, according to CSPM, is born in local thermal equilibrium because the temperature is determined at the string level. Beyond the initial temperature, T > T c the CSPM perfect fluid may expand according to Bjorken boost invariant 1-dimension hydrodynamics [93]. In this framework, the initial energy density is given by: where ε is the energy density, S N is the transverse overlap area and τ pro , the production time for a boson (gluon), is described by [94] τ pro = 2.405 m T .
Here, m T = m 2 + p 2 T is the transverse mass. For evaluating ε, we use the charged particle multiplicity dN ch /dy at mid-rapidity and m is taken as the pion mass (pions being the most abundant particles in a multiparticle production process like that is discussed here), which gives the lower bound of the energy density. For the estimation of m T , we use the p T spectra of pions at different collision energies and collision species in the p T range: 0.15 GeV/c < p T < 1 GeV/c.
The purpose of estimating the initial percolation temperature and the initial energy density in the framework of CSPM is to study the jet transport coefficient as a function of these global observables for different collision species and collision energies at the LHC. Let us now proceed to estimateq in the CSPM framework.

C. Jet transport coefficient
The final state hadrons, produced in ultra-relativistic collisions at large transverse momenta, are strongly suppressed in central collisions compared to peripheral collisions. This suppression of hadrons at high p T , which is usually referred to as jet quenching, is believed to be the result of the parton energy loss induced by multiple collisions in the strongly interacting medium. Thus, we are encouraged to study the jet transport coefficient,q, which encodes the parton energy loss in the medium. It is also related to the p T broadening of the energetic partons propagating inside the medium. In kinetic theory framework,q can be estimated by the formula [95], where ρ is the number density of the constituents of the medium, q ⊥ is the transverse momentum exchange between the jet and the medium, and dσ d 2 q ⊥ denotes the differential scattering cross-section of the particles inside the medium.
The transport parameter of jet quenching,q, and the shear viscosity-to-entropy density ratio (η/s), transport parameters describing the exchange of energy and momentum between fast partons and medium, are directly related to each other as [57,[96][97][98] Within the CSPM approach, the shear viscosity-toentropy density ratio, η/s, can be expressed as [64,83] here L is the longitudinal extension of the string ∼ 1 fermi [63]. One can get final expression for jet transport coefficient from Eq. 12 as: The jet quenching parameterq is plotted as a function of initial percolation temperature in Fig. 3. Interestingly, we observe a linear increase inq, with the increase in temperature for both pp and AA collisions. At low temperatures, the value of jet quenching parameter is around 0.02 GeV 3 . This value increases gradually and at high temperatures, it reaches the value around 0.08 Pb-Pb collisions at √ s = 2.76 TeV at LHC, respectively. The variation ofq values between different hydrodynamic models is considered as theoretical uncertainties. The scaled jet quenching parameterq/T 3 at the highest temperatures reached in the most central Au-Au and Pb-Pb collisions are [58] q T 3 ≈ 4.6 ± 1.2 at RHIC 3.7 ± 1.4 at LHC.
The corresponding absolute values forq for a 10 GeV at an initial time τ 0 = 0.6 fm/c. In this work, we use charged particle spectra to calculateq within the CSPM approach, so we can't reach the initial temperature published by the JET Collaboration. Therefore, ourq is significantly smaller than the value published by the JET Collaboration for the most central Pb-Pb collisions at √ s = 2.76 TeV at the LHC.
In Fig. 4, we have plottedq as a function of charged particle multiplicity scaled with transverse overlap area for pp collisions at √ s = 5.02 and 13 TeV, Xe-Xe collisions at √ s NN = 5.44 TeV and Pb-Pb collisions at √ s NN = 2.76 and 5.02 TeV. One can see thatq shows a steep increase at lower charged particle multiplicities in pp collisions and gets saturated at very high-multiplicity for all studied energies. This behaviour suggests that at lower multiplicities, the system is not dense enough to highly quench the partonic jets, whereas with the increase of multiplicity, the quenching of jets becomes more prominent. The dimensionless parameter, T 3 -scaledq is shown in Fig. 5 as a function of charged particle multiplicity scaled with transverse overlap area. In the low multiplicity regime, we observe a steep increase inq/T 3 , and after reaching a maximum at dN ch /dη /S ⊥ ∼ 2 it starts decreasing regardless of the collision system or collision energy. The decrease inq/T 3 is faster in Pb-Pb and Xe-Xe as compared to the pp collisions.
The variation ofq as a function of initial energy density is shown in Fig. 6. To have a better understanding, we have compared our results with that of cold nuclear matter, massless hot pion gas and ideal QGP calculations [99]. We observe that our CSPM result is closer to the massless hot pion gas at low energy density. As initial line is for massless pion gas, the solid red curve is for ideal QGP and the black square is for cold nuclear matter [99].
energy density increases,q values increase and then show a saturation towards heavy-ion collisions, which produce a denser medium. The saturation behaviour observed at high energy densities suggests thatq remains unaffected after a certain energy density. Similar behaviour is observed whenq is studied as a function of multiplicity (shown in Fig. 4). The jet energy loss inside a denser QCD medium goes towards saturation after a threshold in the final state multiplicity is reached. If we compare the behaviour of η/s as a function of T /T c for T > T c (the domain of validity of CSPM), we observe an increasing trend, which in principle should be reflected in a reverse way in the observableq/T 3 . However, the interplay of higher temperature and lower η/s decides the high temperature behavior ofq as shown in Fig. 6. Further, one observes the CSPM based estimations ofq showing a deviation from the ideal QGP behaviour for energy densities higher than 1 GeV/fm 3 . This is because the ideal QGP calculations of Ref. [99], assumes /T 4 a constant value, whereas the CSPM-based estimations show an increasing trend of /T 4 towards high temperature(energy density or final state multiplicity) [84]. In Fig. 7, we have plottedq/T 3 as a function of initial temperature. For the comparison, we have also plotted the results obtained by the JET Collaboration using five different theoretical models that incorporate particle energy loss in the medium. The GLV model [41][42][43] predicted the general form of the evolution of center-ofmass energy of the high transverse momentum pion nuclear modification factor from Super Proton Synchrotron (SPS) and RHIC to LHC energies. CUJET 1.0 explained the similarity between R AA at RHIC and LHC, despite the fact that the initial QGP density in LHC almost doubles that of RHIC, by taking the effects due to multi-scale running of the QCD coupling α(Q 2 ) into account [44]. In   [45][46][47][48][49]. The nuclear initial parton scatterings for jet production are carried out by using PYTHIA8 in the MARTINI model [54]. This model describes the suppression of hadron spectra in heavy-ion collisions at RHIC very well with a fixed value of the strong coupling constant. The MARTINI model calculation is represented by the red line with black crosses. In MCGILL-AMY model [52,53], the scattering and radiation processes are described by thermal QCD and hard thermal loop (HTL) effects [100] and Landau-Pomeranchuck-Migdal (LPM) interference [101]. In this approach, a set of rate equations for their momentum distributions are solved to obtain the evolution of hard jets (quarks and gluons) in the hot QCD medium. The result obtained from this model is represented by the magenta line. We observe thatq/T 3 obtained from the CSPM approach has a similar kind of behaviour as observed by JET Collaboration.

IV. CONCLUSION
We study the thermodynamic and transport properties of the matter formed in pp and AA collisions at LHC energies within the framework of the CSPM. We extract percolation density parameter by fitting transverse momentum spectra within Color String Percolation Model and then calculate initial percolation temperature (T ), energy density (ε) and the jet transport coefficient (q). In the present work, for the first time, we study the jet transport coefficient of produced hot QCD matter within the color string percolation approach as a function of final state charged particle multiplicity at the LHC energies. We show thatq increases linearly with initial temperature regardless of the collision system or collision energy.
At very low multiplicity,q shows a sharp increase and this dependence becomes weak at high-multiplicity (energy density). This behaviour suggests that at lower multiplicity, the system is not dense enough to highly quench the partonic jets, whereas with the increase of multiplicity the quenching of jets becomes more prominent. At very high-multiplicity (energy density),q saturates with multiplicity (energy density). This allows us to conclude that at very high-multiplicity (high energy density),q becomes independent of final state multiplicity when scaled by the transverse overlap area of the produced fireball. Interestingly, we found thatq in the low energy density regime, the system behaves almost like a massless hot pion gas. Theq/T 3 obtained from the CSPM approach as a function of temperature has a similar kind of behaviour as observed by the JET Collaboration using five different theoretical models that incorporate particle energy loss in the medium.
In view of heavy-ion-like signatures seen in TeV highmultiplicity pp collisions at the LHC energies, it would be really exciting to see the jet quenching results in such collisions to infer about the possible QGP-droplet formation. The present study of jet transport coefficient as a function of final state multiplicity, initial temperature and energy density will pave the way for such an experimental exploration making LHC pp collisions unique.