Scaling of transverse energies and multiplicities with atomic number and energy in ultrarelativistic nuclear collisions

TL;DR

The paper addresses how the initial energy density and produced gluons and quarks scale with atomic number $A$ and beam energy $\sqrt{s}$ in ultrarelativistic heavy-ion collisions. It introduces a saturation-scale framework, evaluating quantities at $p_{\text{sat}}$ determined by the condition $N(p_0)=p_0^2 R_A^2$, and computes initial numbers using perturbative QCD with a constant $K$-factor and shadowing, yielding explicit fits $N_i \sim A^{0.922}(\sqrt{s})^{0.383}$, $p_{\text{sat}} \sim A^{0.128}(\sqrt{s})^{0.191}$, and $E_{Ti} \sim A^{1.043}(\sqrt{s})^{0.595}$. These initial values are then propagated to final observables under the assumption of kinetic thermalisation and adiabatic expansion, leading to predictions for charged multiplicities and transverse energy at RHIC and LHC, and showing that even at SPS energies a largely perturbative, gluon-dominated picture can yield reasonable numbers. The key contribution is demonstrating that the saturation scale can effectively encode the influence of all momentum scales, and that the resulting scaling laws provide a coherent link between early-time QCD dynamics and final-state hadron observables. The approach offers testable predictions and insights into the degree of thermalisation and the role of expansion dynamics in heavy-ion collisions.

Abstract

We compute how the initial energy density and produced gluon, quark and antiquark numbers scale with atomic number and beam energy in ultrarelativistic heavy ion collisions. The computation is based on the argument that the effect of all momentum scales can be estimated by performing the computation at one transverse momentum scale, the saturation momentum. The initial numbers are converted to final ones by assuming kinetic thermalisation and adiabatic expansion. The main emphasis of the study is at LHC and RHIC energies but it is observed that even at SPS energies this approach leads to results which are not unreasonable: what is usually described as a completely soft nonperturbative process can also be described in terms of gluons and quarks. The key element is the use of the saturation scale.

Figures (5)

• Figure 1: The average number of initially produced QCD-quanta with $p_T\ge p_0$ and $|y|<0.5$ as a function of the lower limit $p_0$ for central Pb-Pb collisions at $\sqrt{s}= 5500$ (LHC), 200 (RHIC) and 17 GeV (SPS). The saturation scale $p_{\rm sat}$ for $A=208$ is given by the points of intersection of the dashed curve "saturation" ($p_0^2R_A^2$) with the curves $N(p_0)$.
• Figure 2: (a) The initial $E_T$ in $|y|<0.5$ in a central $A+A$ collision with $A=12,32,64,136,208$ as a function of $\sqrt{s}$. (b) Decomposition of the initial $E_T$ into gluon, quark and antiquark components for $A=208$.
• Figure 3: Proper time dependence of energy density during longitudinal expansion for LHC, RHIC and SPS with the initial values given in Table \ref{['parameters']}.
• Figure 4: The number of charged particles per unit rapidity for $A=12,32,64,136,208$ as a function of $\sqrt{s}$, computed from Eq. (\ref{['nchlaw']}) (solid lines) and compared with $2/3*0.9N_i$ (dotted lines).
• Figure 5: (a) The average final $E_T$ computed from Eq. (\ref{['ET_kataja']}) (solid lines) and from Eq. (\ref{['etc']}) with $T_c=0.18$ GeV (dashed lines) for $A=12,32,64,136,208$ as a function of $\sqrt{s}$. (b) Comparison of the initial and the final $E_T$ for $A=208$ as a function of $\sqrt s$.