Geometrical scaling for energies available at the BNL Relativistic Heavy Ion Collider to those at the CERN Large Hadron Collider

Based on the recent RHIC and LHC experimental results, the $\langle p_T\rangle$ dependence of identified light flavour charged hadrons on $\sqrt{(\frac{dN}{dy})/S_{\perp}}$, relevant scale in gluon saturation picture, is studied from $\sqrt{s_{NN}}$=7.7 GeV up to 5.02 TeV. This study is extended to the slopes of the $\langle p_T\rangle$ dependence on the particle mass and the $\langle\beta_T\rangle$ parameter from Boltzmann-Gibbs Blast Wave (BGBW) fits of the $p_T$ spectra. A systematic decrease of the slope of the $\langle p_T\rangle$ dependence on $\sqrt{(\frac{dN}{dy})/S_{\perp}}$ from BES to the LHC energies is evidenced. While for the RHIC energies, within the experimental errors, the $\langle p_T\rangle$/$\sqrt{(\frac{dN}{dy})/S_{\perp}}$ does not depend on centrality, at the LHC energies a deviation from a linear behaviour is observed towards the most central collisions. The influence of the corona contribution to the observed trends is discussed. The slopes of the $\langle p_T\rangle$ particle mass dependence and the $\langle\beta_T\rangle$ parameter from BGBW fits scale well with $\sqrt{(\frac{dN}{dy})/S_{\perp}}$. Similar systematic trends for pp at $\sqrt{s}$=7 TeV are in a good agreement with the ones corresponding to Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV and 5.02 TeV pointing to a system size independent behaviour.


I. INTRODUCTION
Parton density evolution as a function of x and Q 2 , addressed more than 35 years ago [1] and its experimental confirmation at HERA [2] have triggered a real interest in the community studying ultra-relativistic heavy ion collisions. The rise of the structure function at low x, still visible at small values of Q 2 [3,4] where the perturbative QCD does not work anymore, requires new approaches for a complete understanding of the log 1 x -logQ 2 QCD landscape. Low x values and moderate Q 2 are characteristic features for the early stage of hadron collisions starting from RHIC up to LHC energies. For average transverse momentum ( p T ) values of the order of 1-2 GeV/c, specific for this range of energies, the x values at mid rapidity are of the order of ∼ 10 −2 and ∼ 10 −4 respectively. Such initial conditions are used by different theoretical approaches for describing especially the most recent results from LHC energies. Colour Glass Condensate (CGC) is one of such approaches based on strong classical colour fields description of the small x degrees of freedom [5][6][7]. Local parton-hadron duality picture (LPHD) [8] and dimensionality argument [9,10] predict a decrease of the ratio between the average transverse momentum and the square root of the hadron multiplicity per unit of rapidity and unit of the colliding hadrons transverse overlapping area ( p T / (dN/dy)/S ⊥ ) towards central collisions and higher energies. With the latest results from the Beam Energy Scan (BES) at RHIC and the highest energies at LHC, it is worth revisiting such a dependence. Recently evidenced similarities between pp, p-Pb and Pb-Pb collisions at LHC energies in terms of long range near side two particle correlations, transverse flow and strangeness enhancement as a function of charged particle multiplicity [11][12][13][14][15][16][17] support the idea that even in small colliding systems, due to increased parton density at such energies, the probability of multiple parton interaction increases, the rescattering processes become important and a thermalised stage could be reached although the interaction time is extremely short. Such a high density deconfined small system could follow a hydrodynamic type expansion. To what extent the hydrodynamics is applicable in small systems is still under debate [18]. The most successful phenomenological models, UrQMD, HIJING, NeXSPheRIO, AMPT, PHSD, EPOS, describing the latest results obtained at LHC in pp, p-Pb and Pb-Pb collisions are based on combinations of different approaches for different stages of the collision [19][20][21][22][23][24] while the classical phenomenological models used in particle physics like PYTHIA [25], HERWIG [26] and PHO-JET [27] had to implement processes like multiparton interaction, rescattering, colour reconnection [28] or shoving mechanism [29] in order to improve the agreement with the LHC results, especially in the soft sector in pp collisions. In this paper we also present a comparison between pp and Pb-Pb at LHC energies in terms of the dependence of different observables on the (dN/dy)/S ⊥ variable. In the second chapter of the paper the estimates of the overlapping area of the colliding hadrons are presented. Details on the hadron density per unit of rapidity are given in the third chapter. The p T dependence on (dN/dy)/S ⊥ is presented in Chapter IV for BES and √ s N N =62. 4 The overlapping area of the two colliding nuclei for a given incident energy and centrality was estimated based on the Glauber Monte Carlo (GMC) approach [30][31][32][33]. For the nuclear density profile of the colliding nuclei, a Wood-Saxon distribution was considered: with a=0.535 fm, r 0 =6.5 fm for the Au nucleus [34] and a=0.546 fm, r 0 =6.62 fm for the Pb nucleus [35]. Within the black disc approach, the nucleons are considered to collide if the relative transverse distance d ≤ σpp π , where σ pp is the nucleon-nucleon interaction cross section. The σ pp values for the corresponding √ s N N energies were taken from [34][35][36][37]. ) have been estimated by averaging the maximum values of the x and y coordinates determined per event, over many events. S var ⊥ has been estimated as being proportional to the quantity S= < σ 2 x >< σ 2 y > − < σ xy > 2 , σ 2 x , σ 2 y are the variances and σ xy is the co-variance of the participant distributions in the transverse plane, per event [39]. They were averaged (< ... >) over many events.  [35,37], are shown: the colliding system, the collision energy, the centrality, the average number of participant nucleons in the collision ( Npart ), the overlapping areas corresponding to the wounded nucleons (S geom ⊥ , S var ⊥ ), estimated by the two recipes explained in the text, the percentage of the wounded nucleons undergoing more than a single collision (fcore), the corresponding areas of the wounded nucleons undergoing more than a single collision ((S geom ⊥ ) core , (S var ⊥ ) core ) and the hadron density (dN/dy).
The centrality dependent values were rescaled by the fac-tor obtained dividing the geometrical area to S in the

III. dN/dy ESTIMATES
The total hadron density per unit of rapidity has been estimated based on the published identified charged hadrons densities [34,35,38,40] and hyperons densities [42][43][44][45][46][47][48]. For √ s N N =19.6 and 27 GeV BES energies or some of the centralities, where the hyperon yields were not reported, the corresponding values were obtained by interpolation using the energy and centrality dependence fits.
As far as Ω − andΩ + yield values for BES were not reported and the extrapolation from higher energies down  Table I. The dashed lines represent the fit results using a power law function. Dark red and dark blue full dots correspond to pp collision at √ s=7 TeV, the values being estimated based on the IP-Glasma initial state model, using two values of the α parameter (see Chapter VII). For better clarity, the blue dots were artificially displaced in √ sNN . to BES energies shows a negligible contribution, they were not considered in the produced hadron density estimates. Therefore, we used the following approximations: for the BES energies dN dy 3 2 dN dy . The values are listed in the last column of Table I.
As it was already mentioned in the Introduction, in the local parton-hadron duality approach [8], n where n is the number of charged hadrons produced via gluon fragmentation [9,10]. Therefore, neglecting other effects like collective hydrodynamic expansion and suppression,  Fig.4b for S var ⊥ . The data points corresponding to each collision energy were fitted with a first order polynomial function. The trends in the two figures are rather similar and the fit quality, in terms of Data/Fit ratios, presented in the bottom plots of Fig.4 is equally good. The fit parameters are listed in Table II and Table III   all the three species at LHC energies. Using S var ⊥ , Fig.4b, the extracted slopes, represented in Fig.5a by open symbols show a marginal variation as a function of collision energy -dashed lines. The corresponding offsets, represented in Fig.5b by open symbols, within the error bars, are the same for pions and kaons and are systematically larger for protons at RHIC energies compared with the ones corresponding to S geom ⊥ . One should remark that at LHC energies, the results using S geom ⊥ or S var ⊥ are the same. At the LHC energies, in the most central collisions, a saturation trend seems to develop. A natural question which comes is how much of the observed trends is due to core-corona interplay [49][50][51][52][53][54][55][56] and how the p T -dN dy /S ⊥ correlation for core looks like. Based on the recipe presented in [56], we estimated the p T core for pions, kaons and protons for √ s N N =200 GeV, 2.76 TeV and 5.02 TeV: for π + , K + , p were obtained based on the MB p T spectra reported in [57,58].
In Fig.6a Table IV. The fit quality can be followed in the bottom plot of Fig.6a where the ratios between the data points and fit results are represented. One can also observe that the last three points at √ s N N =2.76 TeV and 5.02 TeV, corresponding  ) core , the results being presented in Fig.6b.
The quality of the linear fit, represented in the bottom plot of Fig.6b is equally good as for the experimental data but the slope values presented in Table V are systematically smaller and the difference between the highest RHIC energy and the LHC energies is reduced. The saturation towards the most central collisions at LHC energies does not change. The p T dependence on the mass of pions, kaons and protons at different collision centralities, except for the most peripheral ones, is linear. Therefore, linear fits of the p T particle mass dependence, corresponding to each centrality and energy considered in the paper, were performed. The extracted fit parameters as a function of dN dy /S geom ⊥ are shown in Fig.7 (slope) and Fig.8 (offset). In Fig.7 the slopes are fitted with the following expression: The slopes for particles Fig.7a

DEPENDENCE OF BOLTZMANN-GIBBS BLAST WAVE FIT PARAMETERS
The p T spectra for identified charged hadrons were fitted [34,35,38,40,41] using the BGBW expression inspired by hydrodynamic models [59]: kin is the kinetic freeze-out temperature and n defines the expansion profile. A compilation of all results in terms of the β T dependence on dN dy /S geom ⊥ is presented in Fig.10. One should mention that for the BES energies [38] the BGBW fits were performed simultaneously on particles and antiparticles p T spectra, although they do not present the same trends in many respects. Therefore, in Fig.10, the β T for antiparticles for some energies and centralities, where the azimuthal dependent BGBW fits were published [60,61], were represented by open symblols. One could observe that, with increasing collision energy, the values of β T for antiparticles converge towards the values obtained from a simultaneous fit of particles and antiparticles p T spectra [34,38] which the data corresponding to √ s N N =7.7, 11.5 and 62.4 GeV are excluded, can be followed in Fig.11. For the remaining energies, from √ s N N =19.6 GeV to 5.02 TeV a much better scaling is observed. The dynamics in β T as a function of dN dy /S geom ⊥ for different collision energies can be easier followed in Fig.12 where the ratio between β T at a given centrality relative to β T in the most peripheral collisions, 70%-80% (58%-85% for 130 GeV), β T / β P eripheral T , is plotted as a function of dN dy /S geom ⊥ for all energies.
In Fig.13 the T f o kin and n parameters and their dependence on dN dy /S geom ⊥ are presented. A close to linear dependence with a negative slope is observed in Fig.13a, for T f o kin at RHIC energies. Within the error bars, it is rather difficult to conclude on some collision energy de- pendence of T f o kin for a given value of the geometrical variable. On the other hand, a significant shift of about 20 MeV in T f o kin fit parameter towards larger values is evidenced for a given dN dy /S geom ⊥ at LHC energies relative to the RHIC energies. Similar shifts were mentioned in the previous chapters for p T and the offsets of p T as a function of mass. Such a shift is also evidenced in the T f o kin versus β T representation in Fig.14 where the fit parameters reported in Ref. [34,35,38,40,41] are used.
As far as the n dependence on dN dy /S geom ⊥ is concerned, Fig.13b, the values for BES energies are rather scattered and those corresponding to 62.4 and 200 GeV show an opposite trend to what is observed at LHC. Usually, the flow profile changes from a shell type expansion, large n values, towards n=1 (Hubble type) with increasing centrality, even smaller than 1 for very central collisions. It is worth mentioning that for a consistent interpretation, the fits of the p T spectra using the BGBW expression have to be done at all energies on the same p T range for a given species. The range has to be chosen such to reduce as much as possible the influence of processes other than collective expansion on the extracted fit parameters. Therefore, the lower limit of the fit range for pions has to be chosen such that the contribution coming from resonance decays is reduced, while the upper fit ranges for all species have to be optimised in order to be influenced as little as possible by the suppression effects. Last but not least, the influence of the corona contribution on the fit parameters has to be carefully considered.  βT fit parameters were reported in Ref. [34,35,38,40]. correlation as a function of charged particle multiplicity [15] and near-side long range pseudorapidity correlations at large charged particle multiplicities [11], were evidenced at LHC energies. The extent to which the similarity between pp and Pb-Pb is also evidenced in the behaviour of the observables described in the previous chapters as a function of the saturation momentum, i.e. dN dy /S ⊥ , is further investigated. For this comparison we used the results of the ALICE Collaboration for p T spectra of identified light flavour charged hadrons as a function of charged particle multiplicity at mid-rapidity as well as the results of their fits with the BGBW expression given by Eq.5 [15]. The hadron density per unit of  [34,35,38,40,41]. rapidity for the mid-central charged particle multiplicity was estimated by extrapolating the results reported by the ALICE Collaboration in Ref. [17]. The p T values were estimated based on the p T spectra from [15] extrapolated in the unmeasured regions using fits of the measured spectra with the expression from [62]: The interaction area for pp collisions, S pp ⊥ =πR 2 pp , is calculated using the estimates of the maximal radius for which the energy density of the Yang-Mill fields is larger than ε = αΛ 4 QCD (α ∈ [1, 10]) within the IP-Glasma initial state model [63,64]. Within the present knowledge of QCD, α cannot be precisely estimated. The r max values used in Ref. [63] for α=1 were fitted in Ref. [65] with the following expressions: Using the same recipe we fitted the r max values from Ref. [63] for α=10 with the following expression: for kaons in pp relative to Pb-Pb, uncertainty in estimating the value of α, the large inhomogeneity of the initial state with a direct consequence on the S ⊥ estimate and last but not least the build up of collective expansion in the hadronic phase and suppression effects taking place in the Pb-Pb case and not yet evidenced in pp collisions. The comparison between the two systems in terms of the slopes of the p T particle mass dependence as a function of dN dy /S geom ⊥ is presented in Fig.16. A very good scaling is found using α=1 for pp collisions. The same value of α was used in Refs. [65,66]. These results seem to support the assumption that the global properties of the hadron production are determined by the properties of flux tubes of ∼1/ dN dy /S ⊥ size and are very little influenced by the size of the colliding system [18,65,67]. A similar behaviour was evidenced at the baryonic level at much lower energies where the main features of the dynamic evolution of the fireball are determined by the initial baryon density profile and temperature and not too much by its size [68]. As it is well known, the LPHD approach neglects all collective effects. However, a comparison between pp and Pb-Pb collisions in terms of β T , one of the BGBW fit parameters interpreted as the average transverse flow velocity, could be rather interesting. β T values for pp at √ s=7 TeV [15] and Pb-Pb at √ s N N =2.76 and 5.02 TeV [35,40,41]  Qualitatively the trends are similar and there is even a very good quantitative scaling for α=1 used in the estimate of S ⊥ for the pp case. The origin of the remaining differences was discussed above. This similarity shows that the main features of the dynamical evolution of the systems produced in pp or Pb-Pb collisions at LHC energies are determined by the density of produced hadrons per unit of rapidity and overlapping area.

VIII. CONCLUSIONS
Based on the data for the highest three energies measured at RHIC ( √ s N N =62.4, 130, 200 GeV), the most recent results from BES at RHIC ( √ s N N =7.7-39 GeV) and the highest collision energies at LHC ( √ s N N =2.76, 5.02 TeV), we performed a systematic study of the dependence of different observables on the geometrical variable calculated as the square root of the hadron density per unit of rapidity and unit of overlapping area of two colliding ions. The overlapping area has been estimated in the Glauber MC approach. The experimental p T values follow a rather good scaling as a function of this variable for each energy. Linear fits of the experimental data show slopes which increase from pions to protons and decrease from BES to LHC energies. A saturation trend for the most central collisions at LHC is observed. For √ s N N =200 GeV, 2.76 TeV and 5.02 TeV the p T core and dN dy core /(S geom ⊥ ) core were estimated based on the core-corona approach. The corresponding p T core versus dN dy core /(S geom ⊥ ) core show lower slopes and their decrease going from √ s N N =200 GeV to 5.02 TeV is less evident for all three species. This shows the importance of discriminating between the corona and core contributions in such a type of analysis, for a quantitative comparison. The decrease in the slopes from RHIC to LHC for all species and for the most central collisions at LHC energies seems to support the approach presented in Ref. [10]. A much better scaling as a function of dN dy /S geom ⊥ is observed for the slope from the linear fit of the p T dependence on the particle mass and the BGBW fit parameter, β T . The offset of the p T particle mass dependence and the T f o kin parameter show a clear jump towards larger values between RHIC and LHC energies. As it was already mentioned, other phenomena, like suppression and its azimuthal dependence as well as the hydrodynamic expansion in the deconfined and after hadronization stages, also have to be considered. The very similar dependence of the p T , p T particle mass dependence and the BGBW fit parameter, β T , on dN dy /S ⊥ in pp and Pb-Pb collisions at LHC energies support the assumption that the global properties evidenced at LHC energies are determined by the properties of flux tubes of ∼1/ dN dy /S ⊥ size, the system size playing a minor role.