Scaling properties of the mean multiplicity and pseudorapidity density in e − + e + , e ± +p, p( ¯p )+p, p+A and A+A(B) collisions

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Particle production in A+A(B) collisions, is frequently described with thermodynamic and hydrodynamical models which utilize macroscopic variables such as temperature and entropy as model ingredients.This contrasts with the microscopic phenomenology (involving ladders of perturbative gluons, classical random gauge fields or strings, and parton hadronization) often used to characterize the soft collisions which account for the bulk of the particles produced in e − + e + , e ± + p, p(p)+p and p+A collisions [10][11][12][13][14].The associated mechanisms, commonly classified as single-diffractive (SD) dissociation, double-diffractive (DD) dissociation and inelastic non-diffractive (ND) scattering in p(p)+p collisions [1], typically do not emphasize temperature and entropy as model elements.
Despite this predilection to use different theoretical model frameworks for p(p)+p, p+A and A+A(B) collisions, it is well known that similar charged particle multiplicity (N ch ) and pseudorapidity density (dN ch /dη) are obtained in p(p)+p, and peripheral A+A(B) and p+A collisions.Moreover, an azimuthal long-range (pseudorapidity difference |∆η| ≥ 4) two-particle angular correlation, akin to the "ridge" which results from collective anisotropic flow in A+A collisions, has been observed in p+p and p+Pb collisions at the LHC [15][16][17][18], and in d+Au and He+Au collisions at RHIC [19,20].Qualitative consistency with these data has also been achieved in initial attempts to describe the amplitudes of these correlations hydrodynamically [19][20][21].Thus, an important open question is whether equilibrium dynamics, linked to a common underlying particle production mechanism, dominates for these systems?
In this work, we use the available dN ch /dη measurements for p+p, p+A and A+A(B) collisions, as well as the N ch measurements for e − + e + , e ± + p, and p(p)+p collisions to search for scaling patterns which could signal such an underlying particle production mechanism.
Our scaling analysis employs the macroscopic entropy (S) ansatz to capture the underlying physics of particle production, where T is the temperature, R is a characteristic size related to the volume, and dN ch /dη and N ch are both proportional to S. A further simplification, N 1/3 pp ∝ R, can be used to relate the number of participant pairs N pp , to the initial volume.These pairs can be specified as colliding participant pairs (e.g.N pp = 1 for e − + e + , e ± + p and p(p)+p collisions), nucleon participant pairs (N npp ) or quark participant pairs (N qpp ).For p+p, p+A and A+A(B) collisions, Monte Carlo Glauber (MC-Glauber) calculations [31][32][33][34][35][36], were performed for several collision centralities at each beam energy to obtain N npp and N qpp .In each of these calculations, a subset initial inelastic N+N (q+q) interaction.The N+N (q+q) cross sections used in these calculations were obtained from the data systematics reported in Ref. [37].Equation 1 suggests similar characteristic patterns for [(dN ch /dη)/N pp ] 1/3 and [ N ch /N pp ] 1/3 as a function of centrality and √ s for all collision systems.We use this scaling ansatz in conjunction with the wealth of measurements spanning several orders of magnitude in √ s, to search for, and study these predicted patterns.
The N ch measurements for e − + e + , e ± + p, and p(p)+p collisions are shown in Fig. 1 .This similarity is compatible with the notion of an effective energy E eff in p(p)+p and e ± + p collisions, available for particle production [49][50][51][52].The remaining energy is associated with the leading particle/s which emerge at small angles with respect to the beam direction -the so-called leading particle effect [53].In a constituent quark picture [54], only a fraction of the available quarks in p(p)+p and e ± + p collisions, contribute to E eff .Thus, ) would be expected to give similar values for E eff [55] and hence, comparable N ch values in e − + e + , p(p)+p and e ± + p collisions.Here, κ 2,3 are scale factors that are related to the number of quark participants and hence, the fraction of the available c.m energy which contribute to particle production.
give the expressions for the mid-pseudorapidity density for INE and NSD p+p collisions.Here, it is noteworthy that the recent inelastic p+p measurements at √ s NN = 13 TeV by the CMS [38] and ALICE [41] collaborations are in very good agree-ment with the scaling prediction shown in Fig. 2(b).The data trends in Figs.1(c), 2(b) and 3(b) also suggest that the mean transverse momentum ( p T ∝ T ) for the particles emitted in these collisions, increase as log( √ s).The scaling properties for p+A and A+A(B) collisions are summarized in Fig. 4 where illustrative plots of [(dN ch /dη| |η|=0.5 )/N npp ] 1/3 vs. dN ch /dη and N npp ), suggesting that T has a logarithmic (linear) dependence on the pseudorapidity density (size) at a given value of √ s NN ; note the slope increase with beam energy, as well as the lack of sensitivity to system type (Cu+Cu, Cu+Au, Au+Au, U+U), for a fixed value of √ s NN .These results suggest that, in addition to the expected increase with √ s NN , the mean transverse momentum p T or transverse mass m T of the emitted particles, should increase as log(dN ch /dη) at a given value of √ s NN .They also suggest that the pseudorapidity density factorizes into contributions which depend on √ s NN and N 1/3 npp respectively.Indeed, the data sets shown for each √ s NN in Fig. 4(b), can be scaled to a single curve with scaling factors that are proportional to log( √ s NN ).collisions across the full range of beam energies.Eq. 5 provides a basis for robust predictions of the value of dN ch /dη| |η|=0.5 as a function of N qpp and √ s across systems and collision energies.For example, it predicts an ∼ 20% increase in the dN ch /dη| |η|=0.5 values for Pb+Pb collisions (across centralities) at 5.02 TeV, compared to the same measurement at 2.76 TeV.This increase reflects the respective contributions linked to the increase in the value of √ s and the small growth in the magnitudes of N qpp .
In summary, we have performed a systematic study of the scaling properties of dN ch /dη measurements for p+p, p+A and A+A(B) collisions, and N ch measurements for e − + e + , e ± + p, and p(p)+p collisions, to investigate the mechanism for particle production in these collisions.The wealth of the measurements, spanning several orders of magnitude in √ s, indicate characteristic scaling patterns for both dN ch /dη and N ch , suggestive of a common underlying entropy production mechanism for these systems.The scaling patterns for N ch validate the essential role of the leading particle effect in p(p)+p and e ± + p collisions and the importance of quark participants in A+A(B) collisions.The patterns for the scaled values of dN ch /dη and N ch indicate strikingly similar trends for NSD p+p and A+A(B) collisions, and show that the pseudorapidity density and the N ch for e − + e + , e ± + p, p+p, and A+A(B) collisions, factorize into contributions which depend on log( √ s) and N pp respectively.The quantification of these scaling patterns, give expressions which serve to systematize the dN ch /dη and N ch measurements for e − + e + , e ± + p, p(p)+p, p+A and A+A(B) collisions, and to predict their magnitudes as a function of N pp and √ s.These scaling results have important utility in the study of a broad array of observables which are currently being pursued at both RHIC and the LHC.
(a).They indicate a nonlinear increase with log( √ s), with N ch ee > N ch pp > N ch ep at each value of √ s.In contrast, Fig. 1(b) shows a linear increase of [ N ch /N pp ] 1/3 (N pp = 1) with log( √ s), suggesting a linear increase of T with log( √ s).Fig. 1(b) also indicates comparable slopes for [ N ch /N pp ] 1/3 vs. log( √ s) for e − + e + , e ± + p, and p(p)+p collisions, albeit with different magnitudes for [ N ch /N pp ] 1/3

Figure 1 (
Figure 1(c) validates this leading particle effect.It shows that the disparate magnitudes of [ N ch /N pp ] 1/3 vs. √ s for e − + e + , p(p)+p and e ± + p collisions (cf.Fig. 1(b)) scale to a single curve for [ N ch /N pp ] 1/3 vs. κ n √ s where κ 1 = 1, κ 2 ∼ 1/2 and κ 3 ∼ 1/6.The values for κ 2,3 validate the important role of quark participants in p(p)+p and e ± + p collisions.A fit to the data in Fig. 1(c) gives the expression
N ch /Npp] 1/3 vs. κn √ s for the κn values indicated.The curves are drawn to guide the eye.