Scaling Properties of the Mean Multiplicity and Pseudorapidity Density in e⁻+e⁺, e^±+p, p( p ¯ )+p, p+A and A+A(B) Collisions

Abstract

The charged-particle pseudorapidity density ($\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta$) for p($\overline{\mathrm{p}}$)+p, p+A and A+A(B) collisions and the mean multiplicity $⟨N_{\mathrm{ch}}⟩$ for e${}^{-}$+e${}^{+}$, ${\mathrm{e}}^{\pm }+\mathrm{p}$, and p($\overline{\mathrm{p}}$)+p collisions are studied for a wide range of beam energies ($\sqrt{s}$). Characteristic scaling patterns are observed for both $\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta$ and $⟨N_{\mathrm{ch}}⟩$, consistent with a thermal particle production mechanism for the bulk of the soft particles created in all of these systems. The scaling patterns found also validate an essential role for quark participants in these collisions. The measured values for $\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta$ and $⟨N_{\mathrm{ch}}⟩$ are observed to factorize into contributions that depend on $log\left(\sqrt{s}\right)$ and the number of nucleon or quark participant pairs $N_{\mathrm{pp}}$. The quantification of these contributions gives expressions that serve to systematize $\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta$ and $⟨N_{\mathrm{ch}}⟩$ measurements spanning nearly 4 orders of magnitude in $\sqrt{s}$ and to predict their values as a function of $\sqrt{s}$ and $N_{\mathrm{pp}}$.

1. Introduction

Measurements of particle yields and kinematic distributions in electron–positron (e${}^{-}$+e${}^{+}$), electron–proton (${\mathrm{e}}^{\pm }+\mathrm{p}$), proton–proton (p($\overline{\mathrm{p}}$)+p), proton–nucleus (p+A) and nucleus–nucleus (A+A(B)) collisions are essential for characterizing their global properties and to develop a good understanding of the mechanism(s) for particle production [1,2,3,4,5,6,7,8,9]. The p+p measurements also provide crucial reference data for studies of nuclear-medium effects in A+A(B) and p+A collisions, as well as improved constraints to differentiate between particle production models and to fine-tune event generators.

Particle production in A+A(B) collisions is frequently (but of course not always) described with thermodynamic and hydrodynamical models that utilize macroscopic variables such as the 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 that account for the bulk of the particles produced in e${}^{-}$+e${}^{+}$, ${\mathrm{e}}^{\pm }+\mathrm{p}$, p($\overline{\mathrm{p}}$)+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 (INEL) non-diffractive (ND) scattering in p($\overline{\mathrm{p}}$)+p collisions [1], typically do not emphasize the temperature and entropy as model elements.

Despite this predilection to use different theoretical model frameworks for p($\overline{\mathrm{p}}$)+p, p+A and A+A(B) collisions, it is well known that similar charged-particle multiplicity ($N_{\mathrm{ch}}$) and pseudorapidity density ($\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta$) results are obtained in p($\overline{\mathrm{p}}$)+p and peripheral A+A(B) and p+A collisions, as is seen, for example, in measurement results utilized further on in this paper. Moreover, an azimuthal long-range (pseudorapidity difference $|\Delta \eta |\ge 4$) two-particle angular correlation, akin to the “ridge” that results from collective anisotropic flow in A+A collisions, has been observed in $\mathrm{p}+\mathrm{p}$ and p+Pb collisions at the Large Hadron Collider (LHC) [15,16,17,18] and in d+Au and He+Au collisions at Relativistic Heavy Ion Collider (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 a large part of the available $\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta$ measurements for $\mathrm{p}+\mathrm{p}$, p+A and A+A(B) collisions, as well as the $\langle N_{\mathrm{ch}}\rangle$ measurements for e${}^{-}$+e${}^{+}$, ${\mathrm{e}}^{\pm }+\mathrm{p}$, and p($\overline{\mathrm{p}}$)+p collisions, to search for scaling patterns that could signal such an underlying particle production mechanism.

2. Analysis

Our scaling analysis employs the macroscopic entropy (S) ansatz:

$$\begin{array}{c}\hfill S\sim {\left(TR\right)}^3\sim \mathrm{const}.\end{array}$$

(1)

to capture the underlying physics of particle production, where T is the temperature, and R is a characteristic length related to the volume. This equation is based on the basic thermodynamic relation between the entropy density and temperature. We furthermore assume that $\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta$ and $\langle N_{\mathrm{ch}}\rangle$ are both proportional to S. A further simplification, $N_{\mathrm{pp}}^{1/3}\propto R$, can be used to relate the number of participant pairs $N_{\mathrm{pp}}$ to the initial volume (as is seen in measurements of Bose–Einstein correlations [22]). These pairs can be specified as colliding participant pairs (e.g., $N_{\mathrm{pp}}=1$ for e${}^{-}$+e${}^{+}$, ${\mathrm{e}}^{\pm }+\mathrm{p}$ and p($\overline{\mathrm{p}}$)+p collisions), nucleon participant pairs ($N_{\mathrm{pp}}=N_{\mathrm{npp}}$) or quark participant pairs ($N_{\mathrm{pp}}=N_{\mathrm{qpp}}$). In heavy-ion physics, $N_{\mathrm{npp}}$ is denoted as $N_{\mathrm{part}}$, but this is not to be mixed with our more general notation of $N_{\mathrm{pp}}$. For $\mathrm{p}+\mathrm{p}$, p+A and A+A(B) collisions, Monte Carlo Glauber calculations [23,24,25,26,27,28] were performed for several collision centralities at each beam energy to obtain $N_{\mathrm{npp}}$ and $N_{\mathrm{qpp}}$. In each of these calculations, a subset $N_{\mathrm{np}}=2N_{\mathrm{npp}}$ ($N_{\mathrm{qp}}=2N_{\mathrm{qpp}}$) of the nucleons (quarks) became participants in each collision by undergoing an initial INEL N+N (q+q) interaction.

The N+N (q+q) cross-sections used in these calculations were obtained from the data systematics reported in [29].

Equation (1) suggests similar characteristic patterns for ${[(\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta )/N_{\mathrm{pp}}]}^{1/3}$ and ${[\langle N_{\mathrm{ch}}\rangle /N_{\mathrm{pp}}]}^{1/3}$ as a function of centrality and collision energy for all the collision systems. We use this scaling ansatz in conjunction with the wealth of measurements spanning several orders of magnitude in $\sqrt{s}$ to search for and study these predicted patterns.

The $\langle N_{\mathrm{ch}}\rangle$ measurements (describing multiplicities extrapolated to the full phase-space) for e${}^{-}$+e${}^{+}$, ${\mathrm{e}}^{\pm }+\mathrm{p}$, and p($\overline{\mathrm{p}}$)+p collisions are shown in Figure 1a. These indicate a nonlinear increase with $log\left(\sqrt{s}\right)$, with ${\langle N_{\mathrm{ch}}\rangle }_{\mathrm{ee}}>{\langle N_{\mathrm{ch}}\rangle }_{\mathrm{pp}}>{\langle N_{\mathrm{ch}}\rangle }_{\mathrm{ep}}$ at each value of $\sqrt{s}$. In contrast, Figure 1b shows a linear increase of ${[\langle N_{\mathrm{ch}}\rangle /N_{\mathrm{pp}}]}^{1/3}$ ($N_{\mathrm{pp}}=1$) with $log\left(\sqrt{s}\right)$, suggesting a linear increase of T with $log\left(\sqrt{s}\right)$. Figure 1b also indicates comparable slopes for ${[\langle N_{\mathrm{ch}}\rangle /N_{\mathrm{pp}}]}^{1/3}$ versus $log\left(\sqrt{s}\right)$ for e${}^{-}$+e${}^{+}$, ${\mathrm{e}}^{\pm }+\mathrm{p}$, and p($\overline{\mathrm{p}}$)+p collisions, albeit with different magnitudes for ${[\langle N_{\mathrm{ch}}\rangle /N_{\mathrm{pp}}]}^{1/3}$. This similarity is compatible with the notion of an effective energy ($E_{\mathrm{eff}}$) in p($\overline{\mathrm{p}}$)+p and ${\mathrm{e}}^{\pm }+\mathrm{p}$ collisions, available for particle production [30,31,32,33]. The remaining energy is associated with the leading particle(s), which emerge at small angles with respect to the beam direction—the “leading particle effect” [34]. In a constituent quark picture [35], only a fraction of the available quarks in p($\overline{\mathrm{p}}$)+p and ${\mathrm{e}}^{\pm }+\mathrm{p}$ collisions contribute to $E_{\mathrm{eff}}$. Thus, ${\kappa }_1{\sqrt{s}}_{\mathrm{ee}}\approx {\kappa }_2{\sqrt{s}}_{\mathrm{pp}}\approx {\kappa }_3{\sqrt{s}}_{\mathrm{ep}}$ (with ${\kappa }_1\equiv 1$) would be expected to give similar values for $E_{\mathrm{eff}}$ [36] and hence comparable $\langle N_{\mathrm{ch}}\rangle$ values in e${}^{-}$+e${}^{+}$, p($\overline{\mathrm{p}}$)+p and ${\mathrm{e}}^{\pm }+\mathrm{p}$ collisions. Here, ${\kappa }_{2,3}$ are scale factors that are related to the number of quark participants and hence the fraction of the available c.m energy that contributes to particle production.

Figure 1c validates this leading particle effect through the center of mass collision energy relation stated above. It shows that the disparate magnitudes of ${[\langle N_{\mathrm{ch}}\rangle /N_{\mathrm{pp}}]}^{1/3}$ versus $\sqrt{s}$ for e${}^{-}$+e${}^{+}$, p($\overline{\mathrm{p}}$)+p and ${\mathrm{e}}^{\pm }+\mathrm{p}$ collisions (cf. Figure 1b) scale to a single curve for ${[\langle N_{\mathrm{ch}}\rangle /N_{\mathrm{pp}}]}^{1/3}$ versus ${\kappa }_n\sqrt{s}$, where ${\kappa }_1=1$, ${\kappa }_2\sim 1/2$ and ${\kappa }_3\sim 1/6$. A fit to the data in Figure 1c gives the expression

$$\langle N_{\mathrm{ch}}\rangle ={\left[b_{\langle {\mathrm{N}}_{\mathrm{ch}}\rangle }+m_{\langle {\mathrm{N}}_{\mathrm{ch}}\rangle }log\left({\kappa }_n\sqrt{s}\right)\right]}^3,b_{\langle {\mathrm{N}}_{\mathrm{ch}}\rangle }=1.22\pm 0.01,m_{\langle {\mathrm{N}}_{\mathrm{ch}}\rangle }=0.775\pm 0.006,$$

(2)

which can be used to predict $\langle N_{\mathrm{ch}}\rangle$ as a function of $\sqrt{s}$ for e${}^{-}$+e${}^{+}$, ${\mathrm{e}}^{\pm }+\mathrm{p}$, and p($\overline{\mathrm{p}}$)+p collisions.

Figure 2a and Figure 3a show $\mathrm{d}{N}_{\mathrm{ch}}{/\mathrm{d}\eta |}_{\eta \approx 0}$ measurements for INEL and non-single-diffractive (NSD) $\mathrm{p}+\mathrm{p}$ collisions, respectively, for beam energies spanning the range $\sqrt{s_{\mathrm{NN}}}\sim 15$ GeV − $13$ TeV (we note that for $\mathrm{p}+\mathrm{p}$ collisions, $\sqrt{s}=\sqrt{s_{\mathrm{NN}}}$, we however use the latter term to be comparable to p+A and A+A(B) collisions; we note furthermore that for some of the cases, $\mathrm{d}{N}_{\mathrm{ch}}{/\mathrm{d}\eta |}_{\left|\eta \right|<0.5}$; hence here we use the more general $\mathrm{d}{N}_{\mathrm{ch}}{/\mathrm{d}\eta |}_{\eta \approx 0}$ term). These plots indicate a monotonic increase of $\mathrm{d}{N}_{\mathrm{ch}}{/\mathrm{d}\eta |}_{\eta \approx 0}$ with $\sqrt{s_{\mathrm{NN}}}$, similar to that observed for $\langle N_{\mathrm{ch}}\rangle$ in Figure 1a. Figure 2b and Figure 3b, by contrast, confirm the expected linear growth of ${[(\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta {|}_{\eta \approx 0})/N_{\mathrm{npp}}]}^{1/3}$ with $log\left(\sqrt{s_{\mathrm{NN}}}\right)$ (where again we make use of the $\sqrt{s}=\sqrt{s_{\mathrm{NN}}}$ relation for $\mathrm{p}+\mathrm{p}$ collisions). Here the expectation for the linear trend is based on the mean transverse momentum ($〈p_T〉\propto T$) for the particles emitted in these collisions, increasing as $log\left(\sqrt{s_{\mathrm{NN}}}\right)$. The open points and dotted curves in these figures affirm the expected trend for the $\sqrt{s_{\mathrm{NN}}}$ dependence of ${[(\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta {|}_{\eta \approx 0})/N_{\mathrm{qpp}}]}^{1/3}$. Here, the change in magnitude largely reflects the difference in the proportionality constants for $N_{\mathrm{npp}}$ and $N_{\mathrm{qpp}}$ (as in $a_{\mathrm{qpp}}{N}_{\mathrm{qpp}}^{1/3}\propto R$ and $a_{\mathrm{npp}}{N}_{\mathrm{npp}}^{1/3}\propto R$). The fits, indicated by the dashed curves in Figure 2b and Figure 3b, give the expressions

$$\begin{array}{cc}\hfill \mathrm{d}{N}_{\mathrm{ch}}{/\mathrm{d}\eta |}_{\mathrm{INEL}}& ={\left[b_{\mathrm{INEL}}+m_{\mathrm{INEL}}log\left(\sqrt{s_{\mathrm{NN}}}\right)\right]}^3,b_{\mathrm{INEL}}=0.826\pm 0.008,m_{\mathrm{INEL}}=0.220\pm 0.004\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \mathrm{d}{N}_{\mathrm{ch}}{/\mathrm{d}\eta |}_{\mathrm{NSD}}& ={\left[b_{\mathrm{NSD}}+m_{\mathrm{NSD}}log\left(\sqrt{s_{\mathrm{NN}}}\right)\right]}^3,b_{\mathrm{NSD}}=0.747\pm 0.022,m_{\mathrm{NSD}}=0.267\pm 0.007\hfill \end{array}$$

(4)

for the mid-pseudorapidity density for INEL and NSD $\mathrm{p}+\mathrm{p}$ collisions. Here, it is noteworthy that the recent INEL $\mathrm{p}+\mathrm{p}$ measurements at $\sqrt{s_{\mathrm{NN}}}=13$ TeV by the CMS [46] and ALICE [47] collaborations are in very good agreement with the scaling prediction shown in Figure 2b.

The scaling properties for p+A and A+A(B) collisions are summarized in Figure 4, where illustrative plots of ${[(\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta {|}_{\left|\eta \right|<0.5})/N_{\mathrm{npp}}]}^{1/3}$ versus $\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta$ and $N_{\mathrm{npp}}^{1/3}$ and ${[(\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta {|}_{\left|\eta \right|<0.5})/N_{\mathrm{qpp}}]}^{1/3}$ versus $N_{\mathrm{qpp}}^{1/3}$ are shown. Analogous plots have been obtained for other collision systems and beam energies. We note that in these cases, $\left|\eta \right|<0.5$ was used to determine the pseudorapidity density at midrapidity—unlike the case of the previous figures, where a subscript $\eta \approx 0$ was given. Figure 4a,b shows that, irrespective of the collision system, ${[(\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta {|}_{\left|\eta \right|<0.5})/N_{\mathrm{npp}}]}^{1/3}$ increases as $log(\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta )$ ($N_{\mathrm{npp}}^{1/3}$), suggesting that T has a logarithmic (linear) dependence on the pseudorapidity density (size) at a given value of $\sqrt{s_{\mathrm{NN}}}$; we note the slope increase with the beam energy, as well as the lack of sensitivity to the system type (Cu+Cu, Cu+Au, Au+Au or U+U), for a fixed value of $\sqrt{s_{\mathrm{NN}}}$. These results suggest that, in addition to the increase with $\sqrt{s_{\mathrm{NN}}}$, the mean transverse momentum $〈p_T〉$ or transverse mass $〈m_T〉$ of the emitted particles should increase as $log(\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta )$ at a given value of $\sqrt{s_{\mathrm{NN}}}$ (in line with the ansatz of our paper). These results also suggest that the pseudorapidity density factorizes into contributions that separately depend on $\sqrt{s_{\mathrm{NN}}}$ and $N_{\mathrm{npp}}^{1/3}$.

Figure 4c contrasts with Figure 4a,b. It shows that, when $N_{\mathrm{qpp}}$ is used instead of $N_{\mathrm{npp}}$, the size dependence of ${[(\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta {|}_{\left|\eta \right|<0.5})/N_{\mathrm{npp}}]}^{1/3}$ (but not its $\sqrt{s_{\mathrm{NN}}}$ dependence), apparent in Figure 4b, is suppressed. We attribute the flat dependence of ${[(\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta {|}_{\left|\eta \right|<0.5})/N_{\mathrm{qpp}}]}^{1/3}$ on the size ($N_{\mathrm{npp}}^{1/3}$ or $N_{\mathrm{qpp}}^{1/3}$) to the linear dependence of $N_{\mathrm{qp}}/N_{\mathrm{npp}}$ on $N_{\mathrm{npp}}^{1/3}$ (related to the initial size), as illustrated in Figure 5a for Pb+Pb and Au+Au collisions. We note that for central and mid-central p+Pb collisions, $N_{\mathrm{qp}}/N_{\mathrm{npp}}$ decreases with $N_{\mathrm{npp}}^{1/3}$; this results in a reduction of the energy released in these collisions, as well as large multiplicity fluctuations.

The $\sqrt{s_{\mathrm{NN}}}$ dependence of ${[(\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta {|}_{\left|\eta \right|<0.5})/N_{\mathrm{qpp}}]}^{1/3}$ for A+A(B) and NSD $\mathrm{p}+\mathrm{p}$ collisions are compared in Figure 5b. The comparison indicates strikingly similar trends for NSD $\mathrm{p}+\mathrm{p}$ and A+A(B) collisions, as would be expected for a common underlying particle production mechanism in these collisions. We note that for $\sqrt{s_{\mathrm{NN}}}\lesssim 2$ TeV, higher temperatures (and larger $〈p_T〉$) are implied for the smaller $\mathrm{p}+\mathrm{p}$ collision systems. Figure 4c and Figure 5b also indicate that the centrality and $\sqrt{s_{\mathrm{NN}}}$-dependent values of $\mathrm{d}{N}_{\mathrm{ch}}{/\mathrm{d}\eta |}_{\left|\eta \right|<0.5}$, obtained for different collision systems, scaled as $N_{\mathrm{qpp}}$ and $log\left(\sqrt{s_{\mathrm{NN}}}\right)$. A fit to the A+A(B) data in Figure 5b gives the expression

$$\begin{array}{c}\hfill \mathrm{d}{N}_{\mathrm{ch}}{/\mathrm{d}\eta |}_{\left|\eta \right|<0.5}=N_{\mathrm{qpp}}{\left[b_{\mathrm{AA}}+m_{\mathrm{AA}}ln\left(\sqrt{s_{\mathrm{NN}}}\right)\right]}^3,b_{\mathrm{AA}}=0.530\pm 0.008,m_{\mathrm{AA}}=0.113\pm 0.002\end{array}$$

(5)

which systematizes the collision energy and centrality dependencies of the pseudorapidity density in A+A(B) collisions across the full range of beam energies. Equation (5) provides a basis for robust predictions of the value of $\mathrm{d}{N}_{\mathrm{ch}}{/\mathrm{d}\eta |}_{\left|\eta \right|<0.5}$ as a function of $N_{\mathrm{qpp}}$ and $\sqrt{s}$ across systems and collision energies. For example, it predicts a ∼20% increase in the $\mathrm{d}{N}_{\mathrm{ch}}{/\mathrm{d}\eta |}_{\left|\eta \right|<0.5}$ 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 $\sqrt{s_{\mathrm{NN}}}$ and the small growth in the magnitudes of $N_{\mathrm{qpp}}$.

3. Summary

In summary, we have performed a systematic study of the scaling properties of $\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta$ measurements for $\mathrm{p}+\mathrm{p}$, p+A and A+A(B) collisions and $\langle N_{\mathrm{ch}}\rangle$ measurements for e${}^{-}$+e${}^{+}$, ${\mathrm{e}}^{\pm }+\mathrm{p}$, and p($\overline{\mathrm{p}}$)+p collisions, to investigate the mechanisms for particle production in these collisions. The wealth of the measurements, spanning several orders of magnitude in collision energy, indicates characteristic scaling patterns for both $\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta$ and $\langle N_{\mathrm{ch}}\rangle$, suggestive of a common underlying entropy production mechanism for these systems. The scaling patterns for $\langle N_{\mathrm{ch}}\rangle$ validate the essential role of the leading particle effect in p($\overline{\mathrm{p}}$)+p and ${\mathrm{e}}^{\pm }+\mathrm{p}$ collisions and the importance of quark participants in A+A(B) collisions. The patterns for the scaled values of $\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta$ and $\langle N_{\mathrm{ch}}\rangle$ indicate fairly similar trends for NSD $\mathrm{p}+\mathrm{p}$ and A+A(B) collisions and show that the pseudorapidity density and $\langle N_{\mathrm{ch}}\rangle$ for e${}^{-}$+e${}^{+}$, ${\mathrm{e}}^{\pm }+\mathrm{p}$, $\mathrm{p}+\mathrm{p}$, and A+A(B) collisions factorize into contributions that depend on $log\left(\sqrt{s}\right)$ and $N_{\mathrm{pp}}$, respectively. The quantification of these scaling patterns gives expressions that serve to systematize the $\mathrm{d}{N}_{\mathrm{ch}}/\mathrm{d}\eta$ and $\langle N_{\mathrm{ch}}\rangle$ measurements for e${}^{-}$+e${}^{+}$, ${\mathrm{e}}^{\pm }+\mathrm{p}$, p($\overline{\mathrm{p}}$)+p, p+A and A+A(B) collisions and to predict their magnitudes as a function of $N_{\mathrm{pp}}$ and $\sqrt{s}$. These scaling results have an important utility in the study of a broad array of observables that are currently being pursued at both RHIC and the LHC.