Hydrodynamic attractors, initial state energy and particle production in relativistic nuclear collisions

TL;DR

This paper uses hydrodynamic attractors to connect initial energy states with particle production in nuclear collisions, providing a universal description of particle yields and insights into quark-gluon plasma properties.

Contribution

It introduces a general relation between initial energy and particle multiplicities using hydrodynamic attractors, enhancing understanding of pre-equilibrium dynamics in nuclear collisions.

Findings

Universal centrality dependence of charged particle yields explained

Estimated initial energy density of far-from-equilibrium state

Constraints on non-equilibrium properties of quark-gluon plasma

Abstract

We exploit the concept of hydrodynamic attractors to establish a general relation between the initial state energy and the produced particle multiplicities in high-energy nuclear collisions. When combined with an ab initio model of energy deposition, the entropy production during the pre-equilibrium phase naturally explains the universal centrality dependence of the measured charged particle yields in nucleus-nucleus collisions. We further estimate the energy density of the far-from-equilibrium initial state and discuss how our results can be used to constrain non-equilibrium properties of the quark-gluon plasma.

Equations

$${\partial }_{\tau }e=-\frac{e+P_L}{\tau },$$

$$e({\tau }_{\text{hydro}})=e_0\exp \left(-{\int }_{{\tilde{w}}_0}^{{\tilde{w}}_{\text{hydro}}}\frac{d\tilde{w}}{\tilde{w}}\frac{1+f(\tilde{w})}{\frac{3}{4}-\frac{1}{4}f(\tilde{w})}\right).$$

$$\frac{e(\tau ){\tau }^{4/3}}{e_{\mathrm{hydro}}{\tau }_{\text{hydro}}^{4/3}}=\mathcal{E}\left(\tilde{w}=\frac{T_{\mathrm{eff}}(\tau )\tau }{4\pi \eta /s}\right).$$

$$\begin{array}{rl}\mathcal{E}(\tilde{w}\ll 1)& =C_{\infty }^{-1}{\tilde{w}}^{4/9}(\text{free streaming}),\\ \mathcal{E}(\tilde{w}\gg 1)& =1-\frac{2}{3\pi \tilde{w}}(\text{viscous hydro}),\end{array}$$

$$\displaystyle(s\tau)_{\rm hydro}=\frac{4}{3}\left(\frac{\pi^{2}}{30}\nu_{\rm eff}\right)^{1/4}\Bigg{(}\lim_{\tau\to 0}\frac{e(\tau)\tau^{4/3}}{\mathcal{E}\big{(}\frac{T_{\rm eff}(\tau)\tau}{4\pi\eta/s}\big{)}}\Bigg{)}^{3/4}\!\!.$$

$$(s\tau {)}_{\mathrm{hydro}}$$

$$\frac{d{N}_{\mathrm{ch}}}{d\eta }\approx \frac{1}{J}{A}_{\perp }(s\tau {)}_{\mathrm{hydro}}\frac{N_{\mathrm{ch}}}{S}.$$

$$(e\tau {)}_0({\mathbf{x}}_{\perp })$$

$$(n\tau {)}_0({\mathbf{x}}_{\perp })$$

$$Q_s^2({\mathbf{x}}_{\perp })\propto T({\mathbf{x}}_{\perp }),$$

$$\frac{d{N}_{\mathrm{ch}}}{d\eta }\propto \int d^2{\mathbf{x}}_{\perp }{T}^{<}({\mathbf{x}}_{\perp }),$$

$$\frac{d{N}_{\mathrm{ch}}}{d\eta }\propto \int d^2{\mathbf{x}}_{\perp }{\left(T^{<}({\mathbf{x}}_{\perp })\sqrt{T^{>}({\mathbf{x}}_{\perp })}\right)}^{2/3}.$$

$$\frac{\delta s_{\mathrm{hydro}}}{s_{\mathrm{hydro}}}=\frac{2}{3}\frac{\delta e_0}{e_0},$$

$$e_0\approx 270GeV/f{\mathrm{m}}^3{\left(\frac{{\tau }_0}{0.1\mathrm{fm}/c}\right)}^{-1}{\left(\frac{C_{\infty }}{0.87}\right)}^{-9/8}{\left(\frac{\eta /s}{2/4\pi }\right)}^{-1/2}$$

$${\left(\frac{A_{\perp }}{138{\mathrm{fm}}^2}\right)}^{-3/2}{\left(\frac{d{N}_{\mathrm{ch}}/d\eta }{1600}\right)}^{3/2}{\left(\frac{{\nu }_{\mathrm{eff}}}{40}\right)}^{-1/2}{\left(\frac{S/N_{\mathrm{ch}}}{7.5}\right)}^{3/2},$$

$$\rho(r)=\rho_{0}\biggl{[}1~{}+~{}\exp\biggl{(}\frac{1}{a}\bigl{[}r-R\bigr{]}\biggr{)}\biggr{]}^{-1}\;,$$

$$Q_s^2({\mathbf{x}}_{\perp })=Q_{sp}^2(\sqrt{s})A_{p}T({\mathbf{x}}_{\perp }),$$

$$(e\tau {)}_0({\mathbf{x}}_{\perp })={\mathcal{N}}_e{\left(Q_{sp}^2(\sqrt{s})A_p\right)}^{3/2}{T}^{<}({\mathbf{x}}_{\perp })\sqrt{T^{>}({\mathbf{x}}_{\perp })},$$

$$\frac{d{N}_{\text{ch}}}{d\eta }$$

$${\mathcal{N}}_e^{2/3}{Q}_{sp}^2(\sqrt{s})A_p\int d{\mathbf{x}}_{\perp }{\left(T^{<}({\mathbf{x}}_{\perp })\sqrt{T^{>}({\mathbf{x}}_{\perp })}\right)}^{2/3},$$

$$\frac{d{E}_0}{d{\eta }_s}={\mathcal{N}}_e{\left(Q_{sp}^2(\sqrt{s})A_p\right)}^{3/2}\int d{\mathbf{x}}_{\perp }{T}^{<}({\mathbf{x}}_{\perp })\sqrt{T^{>}({\mathbf{x}}_{\perp })}.$$

$$\frac{4}{3J}{C}_{\infty }^{3/4}{\left(4\pi \frac{\eta }{s}\right)}^{1/3}{\left(\frac{{\pi }^2}{30}{\nu }_{\mathrm{eff}}\right)}^{1/3}\frac{N_{\mathrm{ch}}}{S}{\mathcal{N}}_e^{2/3}{Q}_{sp}^2(\sqrt{s})A_p$$

$$\frac{d{E}_0}{d{\eta }_s}$$

$$A_{\perp }^{-1/2}{\left(\frac{S}{N_{\mathrm{ch}}}\right)}^{3/2}{\left(\frac{d{N}_{\text{ch}}}{d\eta }\right)}^{3/2},$$

$$\sqrt{A_{\perp }}=\frac{{\left(\int d{\mathbf{x}}_{\perp }{\left(T^{<}({\mathbf{x}}_{\perp })\sqrt{T^{>}({\mathbf{x}}_{\perp })}\right)}^{2/3}\right)}^{3/2}}{\int d{\mathbf{x}}_{\perp }{T}^{<}({\mathbf{x}}_{\perp })\sqrt{T^{>}({\mathbf{x}}_{\perp })}}.$$

Full text

††preprint: INT-PUB-19-036

Introduction.— Understanding the equilibration of isolated quantum systems is a fundamental question that touches physical phenomena across vastly different energy scales, from micro kelvin temperatures in cold atom experiments to trillion kelvin temperatures in the dense strong-interaction matter produced in ultra-relativistic nuclear collisions Busza et al. (2018); Schäfer and Teaney (2009); Adams et al. (2012); Schlichting and Teaney (2019). One outstanding discovery made in the field of heavy-ion collisions is that the system created about $1\,\rm{fm}/c$ ($\approx 3\cdot 10^{-24}\,\text{s})$ after the impact of two relativistic nuclei can be described as a deconfined plasma of quarks and gluons (QGP) with macroscopic properties of temperature and velocity Adams et al. (2005); Adcox et al. (2005); Back et al. (2005); Arsene et al. (2005). Such “unreasonable effectiveness of hydrodynamics” in describing the violent expansion of the QGP droplets triggered a new research area in mathematical physics devoted to the study of hydrodynamic attractors, that emerge in out-of-equilibrium systems experiencing very fast memory loss of initial conditions and exhibiting a universal approach towards thermal equilibrium Florkowski et al. (2018); Romatschke and Romatschke (2019).

In this article we show that hydrodynamic attractors can be used to describe entropy production in relativistic nuclear collisions and to make robust estimates of initial-state energy before the onset of equilibration. We derive a simple formula, Eq. (6), that relates the energy density of the initial state to the measured charged particle multiplicity, $dN_{\rm ch}/d\eta$, and point out two important phenomenological consequences of this result. We show that the universal centrality dependence of $dN_{\rm ch}/d\eta$ across a wide range of collision systems can be naturally reproduced by combining the initial-state energy deposition in high-energy quantum chromodynamics (QCD) with the non-linear entropy production during the equilibration process. Secondly, we determine the initial energy per unit space-time rapidity, $dE_{0}/d\eta_{s}$, for different collision centralities at the Relativstic Heavy-Ion Collider (RHIC) and the Large Hadron Collider (LHC). By comparing our results to the experimentally measured $dE_{\text{final}}/dy$ in the final state, we estimate the work performed during the expansion of the system Gyulassy and Matsui (1984) and discuss how such an analysis constrains non-equilibrium and transport properties of the QGP.

Hydrodynamic attractors & Entropy production.— We describe the early time dynamics ($\tau\lesssim 1\,\text{fm}/c$) of the high-temperature QCD plasma created in high-energy heavy-ion collisions by the out-of-equilibrium evolution of a boost-invariant and transversely homogeneous conformal system  Bjorken (1983). Energy-momentum conservation dictates that the energy density, $e=T^{\tau\tau}$, evolves according to

[TABLE]

where $P_{L}\equiv\tau^{2}T^{\eta_{s}\eta_{s}}$ is the longitudinal pressure and we use proper time $\tau\equiv\sqrt{t^{2}-z^{2}}$ and space-time rapidity $\eta_{s}\equiv\text{atanh}\,z/t$ coordinates. In (local) thermal equilibrium, the longitudinal pressure is directly related to the energy density via an equation of state, e.g. $P_{L}=e/3$ for a conformal system. While for small deviations around equilibrium the longitudinal pressure is determined by hydrodynamic constitutive relations in terms of the gradient expansion $P_{L}/e=\frac{1}{3}-\frac{16}{9}\frac{\eta/s}{\tau T}$, where $\eta/s$ is the specific shear viscosity Florkowski et al. (2018); Romatschke and Romatschke (2019), this is generally not the case far from equilibrium, where, for instance, at early times after the collision of heavy nuclei the system is highly anisotropic $P_{L}\ll e$ Schlichting and Teaney (2019). Nevertheless, new insights from microscopic equilibration studies Heller et al. (2012); Heller and Spalinski (2015); Romatschke (2018); Heller et al. (2018); Strickland et al. (2018); Blaizot and Yan (2018); Strickland (2018); Behtash et al. (2019a, 2018, b); Kurkela and Zhu (2015); Keegan et al. (2016a, b); Kurkela et al. (2019a, b); Kurkela and Mazeliauskas (2019a, b); Kurkela et al. (2019c) point to the existence of a hydrodynamic attractor Heller and Spalinski (2015), where the far-from-equilibrium system displays an effective constitutive equation $P_{L}/e=f(\tilde{w})$ well before reaching local thermal equilibrium. Such attractor behavior has been established for a number of different microscopic theories (QCD Kinetic Theory Kurkela et al. (2019a, b); Kurkela and Mazeliauskas (2019a, b), Boltzmann RTA Heller et al. (2018); Strickland et al. (2018); Blaizot and Yan (2018); Strickland (2018); Behtash et al. (2019a) and AdS/CFT Heller et al. (2012); Heller and Spalinski (2015); Romatschke (2018)), where the time evolution on the attractor is controlled by a single scaling variable, $\tilde{w}={\tau T_{\rm eff}(\tau)}/{(4\pi\eta/s)}$, where $T_{\rm eff}(\tau)$ is an effective temperature such that $e(\tau)\equiv\frac{\pi^{2}}{30}\nu_{\rm eff}T_{\rm eff}^{4}(\tau)\;$ ($\nu_{\text{eff}}$ is the number of effective degrees of freedom, e.g., $\nu_{\text{eff}}=16$ for ideal gluonic gas).

Based on these insights, the conservation law in Eq. (1) can be integrated, yielding a universal relation between the initial state energy density $e_{0}$ at very early times $\tilde{w}(\tau_{0})\ll 1$, and the energy density $e(\tau_{\text{hydro}})$ of the near thermal system at later times $\tilde{w}(\tau_{\text{hydro}})\gg 1$

[TABLE]

Close to equilibrium $f(\tilde{w}_{\text{hydro}})\approx 1/3$ and the energy density of the longitudinally expanding plasma follows the Bjorken scaling $e(\tau)=e_{\rm hydro}\left(\tau/\tau_{\text{hydro}}\right)^{-4/3}$, while the entropy density per unit rapidity, $s\tau$, remains constant Bjorken (1983). Eventually, for $\tau\gtrsim R/c$, where $2R$ denotes the transverse extent of the system, the QGP fireball starts expanding in the transverse plane and ultimately freezes out in color neutral hadrons Teaney (2010). During the transverse expansion the QGP remains close to equilibrium and the total entropy per unit rapidity ${dS}/{d\eta_{s}}=A_{\bot}~{}(s\tau)_{\rm hydro}$ (where $A_{\bot}=\pi R^{2}$) is approximately conserved onwards from the time $\tau_{\text{hydro}}$ when the QGP can be described as an almost ideal fluid. Ultimately, on the freeze-out surface ${dS}/{d\eta_{s}}$ becomes proportional to the produced charged hadron multiplicity, $dN_{\text{ch}}/d\eta$. The multiplicity of final-state particles emitted from the QGP is therefore a sensitive probe of the entropy production during the pre-equilibrium phase.

Strikingly, the correspondence between initial-state energy density and charged hadron multiplicity can be quantified further using the theory of hydrodynamic attractors. By factoring out the late time Bjorken scaling from Eq. (2) the evolution of the energy density during the pre-equilibrium phase can be characterized by an attractor curve $\mathcal{E}(\tilde{w})$

[TABLE]

As can be seen from Fig. 1, the function $\mathcal{E}(\tilde{w})$ smoothly interpolates between an early free-streaming and late-stage viscous hydrodynamics Romatschke (2018); Kurkela et al. (2019a)

[TABLE]

where $C_{\infty}$ is a constant of order unity. Even though the evolution at intermediate times can be different for different microscopic theories, the overall similarity between different theories is remarkable. Most importantly for our purpose, all curves have the same universal characteristics, Eq. (4), at early and late times, irrespective of the underlying microscopic theory.

Based on Eq. (3), we can immediately establish a quantitative relation between the energy densities $e(\tau)$ at various stages, which upon use of the thermodynamic relations $Ts=e+p$ and $p=e/3$ once the system is close to equilibrium turns into an estimate of the entropy density per unit rapidity

[TABLE]

Evaluating the limit according to Eq. (4) one arrives at the central result of this paper, namely the relation

[TABLE]

from which one can directly estimate the charged particle multiplicity as discussed above:

[TABLE]

Here $S/N_{\text{ch}}\equiv\left(dS/dy\right)/\left(dN_{\text{ch}}/dy\right)\approx 6.7\text{--}8.5$ is the entropy per charged particle at freeze-out Hanus et al. (2019) and $J\approx 1.1$ is a Jacobian factor between particle rapidity $y$ and pseudo rapidity $\eta$ Adam et al. (2016).

Equations (6) and (7) establish a one-to-one correspondence between the initial-state energy per unit rapidity $dE_{0}/d\eta_{s}\approx A_{\bot}(e\tau)_{0}$ and the charged particle multiplicity ${dN_{\rm ch}}/{d\eta}$. One crucial feature of this result is that it accounts for the entropy production during the pre-equilibrium phase, which gives rise to a nontrivial dependence on the initial-state energy density $(e\tau)^{2/3}_{0}$ as well as on the transport coefficient $(\eta/s)^{1/3}$. Our estimate includes all relevant pre-factors, in particular, the constant $C_{\infty}$, which is the property of the hydrodynamic attractor, Eq. (3), and depends on the microscopic physics of equilibration. However it is striking to observe that for the different theories shown in Fig. 1 the variation of $C_{\infty}$ is only at $\sim 10\%$ level. We emphasize that Eq. (6) is entirely based on the macroscopic evolution described by a generic hydrodynamic attractor, which sets it apart from previous parametric estimates of entropy production based on particular microscopic scenarios Baier et al. (2002, 2011); van der Schee and Schenke (2015); Berges et al. (2017).

Centrality dependence of particle multiplicity.—One important phenomenological consequence of the entropy production in the pre-equilibrium phase concerns the determination of initial conditions for hydrodynamic simulations of heavy ion collisions (see e.g. Heinz and Snellings (2013)). While strictly speaking our estimate of the entropy density in Eq. (6) was derived assuming a one dimensional expansion, the influence of transverse gradients can be neglected over the short pre-equilibrium times and we can directly promote Eq. (6) to an estimate for the local entropy density, $\left.\tau s(\tau,\mathbf{x}_{\bot})\right|_{\tau=\tau_{\text{hydro}}}$. Specifically, the pre-flow $v_{\bot}\sim\tau_{\text{hydro}}\mathbf{\nabla}_{\bot}e/e$ is negligible as long as gradients $\mathbf{\nabla}_{\bot}e/e$ are small on the scale of hydrodynamization time $(c\tau_{\text{hydro}})$ Keegan et al. (2016b); Kurkela et al. (2019a) and the one dimensional constitutive relation for $P_{L}/e$ approximately remains valid, as explicitly confirmed in Kurkela et al. (2020). Effectively, Eq. (6) then provides a non-linear map of the initial-state energy density profile to the entropy density profile at later times $\tau\sim\tau_{\text{hydro}}$.

Now, in order to illustrate the impact of the pre-equilibrium phase, we will study the effects on the centrality dependence of the charged particle multiplicity, within a simple initial state model based on the color-glass condensate effective theory of high-energy QCD Gelis et al. (2010). Within the dilute-dense formulation of this theory Dumitru and McLerran (2002); Lappi (2006); Blaizot et al. (2004); Blaizot and Mehtar-Tani (2009); Blaizot et al. (2010), both the initial energy density per unit rapidity $(e\tau)_{0}$ as well as the initial gluon multiplicity per unit rapidity $(n\tau)_{0}$ can be calculated from $\mathbf{k}_{\bot}$-factorization, and are given in terms of convolutions of unintegrated gluon distributions 111We note that explicit comparisons of dilute-dense and dense-dense calculations in Ref. Blaizot et al. (2010) show that in the momentum range $p_{T}\gtrsim Q_{s}$, which gives the dominant contribution to ${dE}/{dy}$, the dilute-dense $k_{T}$ factorization calculations accurately describe dense-dense Classical Yang-Mills results.. Essentially, one finds that (up to logarithmic corrections) $(e\tau)_{0}$ and $(n\tau)_{0}$ are proportional to the (local) saturation scales $Q_{s}(\mathbf{x}_{\bot})$ of the two colliding nuclei Dumitru and McLerran (2002); Lappi (2006):

[TABLE]

where $Q_{s}^{>/<}$ is the saturation scale of the nucleus representing larger/smaller $Q_{s}$ at position $\mathbf{x}_{\bot}$.

Since the saturation scale locally characterizes the longitudinally integrated density of color charge inside the nucleus, it is generically proportional to the nuclear thickness

[TABLE]

whose definition is recalled in the supplemental material 222The supplemental material provides the description of the Monte Carlo Glauber model of nucleus-nucleus collisions used to determine the nuclear thickness functions, which are needed for the calculation of the initial-state energy profile. In addition, we provide detailed formulas for calculating the centrality dependence of the averaged charged-particle multiplicity and the initial-state energy per rapidity, i.e. the lines and shaded bands in Figs. 2 and 3 of the manuscript. The supplemental material includes additional Refs. De Vries et al. (1987); Albacete and Marquet (2014). Based on these considerations, one can then try to estimate the charged particle multiplicity per unit rapidity from the initial gluon multiplicity $(n\tau)_{0}$ (i.e. w/o pre-equilibrium) according to

[TABLE]

as was done, for example, in Mace et al. (2018); Schenke et al. (2012). However, such an estimate is appropriate only when there is no significant amount of particle production in the final state. Conversely, if the initial state evolves into an almost ideal QGP fluid, one needs to account for the entropy production during the pre-equilibrium phase. By employing Eqs. (6) and (7) the charged particle multiplicity is then estimated from the initial state energy density $(e\tau)_{0}$, (i.e. w/ pre-equilibrium) as

[TABLE]

We illustrate the difference between the two estimates in the upper panel of Fig. 2, where we compare the centrality dependence of the multiplicity ${dN_{\rm ch}}/{d\eta}$ from Eq. (12) (solid line) and Eq. (11) (dashed line), with the nuclear thickness and centrality quantilies determined from the optical Glauber model (see the supplemental material for details Note (2)). Both estimates are normalized to reproduce the experimentally measured value of ${dN_{\rm ch}}/{d\eta}$ in the $10-20\%$ centrality class of Pb–Pb collisions at $\sqrt{s_{\text{NN}}}=2.76\,\text{TeV}$. Different trends in the centrality dependence of ${dN_{\rm ch}}/{d\eta}$ are clearly visible, indicating the importance of the pre-equilibrium phase when comparing observables of this type to experimental data.

Since ${dN_{\rm ch}}/{d\eta}$ is accurately measured in experiment for a wide variety of colliding systems and energies, we can also compare the two estimates directly to experimental data, which are reported as symbols in the upper panel of Fig. 2. It is interesting to note that the average $\left<{dN_{\rm ch}}/{d\eta}\right>$ as a function of centrality possesses a remarkable degree of universality, such that—up to an overall normalization factor for each collision system—data points for different colliding species (Au, Cu, Pb, U, Xe) at RHIC and LHC energies all collapse onto a single curve to high accuracy. Despite the simplicity of our theoretical estimate, we find that the curve including pre-equilibrium effects provides a rather good description of the experimental data, except for the more peripheral bins, where fluctuations play an important role (see below). Due to the non-trivial geometry dependence in Eq. (12), the calculation including pre-equilibrium dynamics provides a much better description of the data than the estimate in Eq. (11), which is based solely on the initial state.

Even though Eq. (12) can be clearly justified from theoretical calculations, our description is by no means unique. Other phenomenological models Drescher and Nara (2007); Alver et al. (2008); Gale et al. (2013); Niemi et al. (2016); Bernhard et al. (2016) successfully reproduce the centrality dependence seen in Fig. 2 by introducing various sources of event-by-event fluctuations such as number and positions of participant nucleons, their interaction strength, etc. However, it is important to emphasize in this context that the pre-equilibrium phase also modifies the statistics of fluctuations, such that for the long wavelength perturbations Keegan et al. (2016b); Kurkela et al. (2019a)

[TABLE]

which follows from the linearization of Eq. (6). While Eq. (12) over-predicts particle production in peripheral collisions, it is therefore not surprising that one can restore agreement with peripheral data by including event-by-event fluctuations. This is demonstrated by the dot-dashed line in Fig. 2, where the average of the nuclear thickness in Eq. (12) has been determined from a Glauber Monte Carlo model Miller et al. (2007) (see the supplemental material for details Note (2)).

Estimating the initial-state energy density.—So far we illustrated the utility of Eq. (6) for describing entropy production in the pre-equilibrium phase of high-energy heavy-ion collisions. However, an equally important application concerns the inverse problem, namely the estimation of the energy density $e_{0}$ of the non-equilibrium state at very early time $\tau_{0}$ from experimental measurements of hadrons in the final state.

By inverting Eqs. (6) and (7) (and inserting typical values of $dN_{\text{ch}}/d\eta\sim 1600$ and $A_{\bot}\approx\pi R_{\text{Pb}}^{2}\approx 138\,\text{fm}^{2}$) we obtain the following estimate for the initial energy density for central Pb-Pb collisions at $\sqrt{s_{NN}}=2.76~{}{\rm TeV}$:

[TABLE]

at a time $\tau_{0}=0.1\,\text{fm}/c$, which should be at least of the order of the formation time $1/Q_{s}\approx 0.1\text{fm}/c$ Schlichting and Teaney (2019), but small compared to the equilibration time $\tau_{\rm hydro}\ \approx 1\,{\rm fm}/c$ Kurkela et al. (2019b, a) for the estimate in Eq. (14) to be valid. One finds that the initial energy density quoted in Eq. (14) is nearly three orders of magnitude higher than the energy density at the QCD cross-over $e_{c}\approx 0.346(41)~{}{\rm GeV}/\text{fm}^{3}$ (for 2+1 flavor QCD) Ding et al. (2015); Bazavov et al. (2019).

We emphasize that, unlike the usual Bjorken estimate based on the measured final-state energy $e_{0}=\frac{1}{\tau_{0}A_{\bot}}{dE_{\text{final}}}/{dy}$ Bjorken (1983), our estimate in Eq. (14) includes the work done during the expansion of the QGP Gyulassy and Matsui (1984). We demonstrate this effect in Fig. 3 where we compare the experimentally measured $dE_{\text{final}}/dy$ in the final state to the initial-state energy per unit rapidity $dE_{0}/d\eta_{s}=\int d{\mathbf{x}}_{\perp}(e\tau)_{0}$ reconstructed from the measured particle multiplicities as in Eq. (14). Note that to better account for the non-trivial transverse geometry, we have estimated the transverse area $A_{\bot}$ from our fluctuating initial state model as described in the supplemental material Note (2).

Based on this analysis, we find that, especially in central collisions at high energies, the initial state $dE_{0}/d\eta_{s}$ can easily exceed the measured ${dE_{\text{final}}}/{dy}$ in the final state by a factor of two to three. Evidently the exact amount of work done during the expansion is subject to uncertainties in the non-equilibrium and transport properties of the QGP, which we quantify by uncertainty bands in Fig. 3, corresponding to variations of $\eta/s$ and $C_{\infty}$ within the anticipated margins ($\eta/s=0.08-0.24$ and $C_{\infty}=0.80-1.15$). Vice versa, the size of the uncertainty bands in Fig. 3 also demonstrates the fact that, if the initial state energy density can be determined precisely, e.g. from theoretical calculations, then the experimentally measured $dN_{\rm ch}/d\eta$ will impose strong constraints on the non-equilibrium evolution of the QGP characterized by $C_{\infty}$ and $\eta/s$. Since, the $dE_{0}/d\eta_{s}$ should always be larger than $dE_{\text{final}}/dy$ due to the work performed against the longitudinal expansion, one can further rule out values in the parameter space, where the initial $dE_{0}/d\eta_{s}$ falls below the experimental data points. Specifically, large values of $\eta/s$ and $C_{\infty}$ can already be ruled out, because the estimated $dE_{0}/d\eta_{s}$ in peripheral collisions turns out to be unphysically small, i.e. below the experimental points of $dE_{\text{final}}/dy$, which provide a lower bound on the initial state energy.

Discussion.—Entropy production in high-energy heavy-ion collisions occurs predominantly during the earliest stages, when the system is significantly out-of-equilibrium; therefore measurements of the charged particle multiplicities—reflecting the total amount of entropy produced in the collision—provide a highly sensitive probe of the pre-equilibrium dynamics. Based on the concept of hydrodynamic attractors, which give a macroscopic description of the early time dynamics of the QGP, we established for the first time a direct relation between the initial-state energy and the final-state entropy. This relation, Eq. (6), is remarkably insensitive to the microscopic details of the approach to equilibrium (see Fig. 1).

By combining the information from $dN_{\text{ch}}/d\eta$ on entropy production and $dE_{\text{final}}/dy$ on the work performed against the longitudinal expansion, we demonstrate that a precise calculation of the initial state energy can impose stringent constraints on the shear-viscosity to entropy density ratio $\eta/s$. Based on our extraction of $dE_{0}/d\eta_{s}$, which assumes a scenario of (nearly-)complete equilibration, we obtain an upper limit for $\eta/s\lesssim 0.4$ for the most favorable choice of all other parameters. Conversely, for $\eta/s\gtrsim 0.4$ we can not expect the QGP to equilibrate in peripheral nucleus-nucleus collisions (see also Bhalerao et al. (2005); Kurkela et al. (2019c)) and our estimates need to be revised. Non-trivial modifications due to incomplete equilibration will arise in this context, which should be investigated further, for instance by means of the KøMPøST pre-equilibrium package Kurkela et al. (2019a, b, 2018). We expect such effects to become particularly important in collisions of smaller nuclei, e.g., p–A or O–O, which may therefore provide even deeper insights into the fascinating out-of-equilibrium dynamics of the QGP.

Acknowledgments.—We thank A. Andronic, J. Berges, N. Borghini, M. Heller, C. Klein-Bösing, A. Kurkela, T. Lappi, M. Martinez, J.Y. Ollitrault, K. Reygers, C. Schmidt, D. Teaney, and R. Venugopalan for stimulating discussions and P. Romatschke for also providing the AdS/CFT attractor curve. This work was supported in part by the German Research Foundation (DFG) through the Collaborative Research Centres “CRC-TR 211: Strong-interaction matter under extreme conditions” (S.S.) and “SFB 1225: Isolated quantum systems and universality in extreme conditions (ISOQUANT)” (A.M.). We thank the Institute for Nuclear Theory at the University of Washington for its kind hospitality and stimulating research environment.

Appendix A Supplemental material

Below we explain the details of the implementation of the initial state model described in the main text. When referring to the optical Glauber calculations, which yield the dashed and solid lines in Fig. 2, we determine the nuclear thickness $T(\mathbf{x}_{\bot})$ of each nucleus according to the longitudinal integral of the (average) nuclear matter density distribution $T(\mathbf{x}_{\bot})=\int_{z}dz~{}\rho(\sqrt{\mathbf{x}_{\bot}^{2}-z^{2}})$. We follow previous works and parametrize $\rho(r)$ by a two-parameter Fermi distribution

[TABLE]

with $a=0.55$ fm and $R=6.62$ fm for 208Pb nuclei, and $a=0.53$ fm, $R=6.40$ fm for 197Au nuclei De Vries et al. (1987), while $\rho_{0}$ is such that $\rho(r)$ is normalized to the total number of nucleons. The thickness functions of two nuclei (A and B) colliding at a finite impact parameter, $\mathbf{b}$, are given by $T^{\text{A}/\text{B}}(\mathbf{x}_{\bot})=T(\mathbf{x}_{\bot}\pm\mathbf{b}/2)$. The centrality of a given collision is obtained from the geometric relation $centrality=\pi|\mathbf{b}|^{2}/\sigma_{\text{tot}}$ Teaney (2010), where $\sigma_{\text{tot}}$ is the total inelastic nucleus-nucleus cross section, and we use $\sigma_{\text{tot}}=767$ fm2 for Pb-Pb collisions, and $\sigma_{\text{tot}}=685$ fm2 in Au-Au collisions.

When referring to fluctuating initial conditions, which yield the dot-dashed line in Fig. 2, we introduce nucleonic degrees of freedom following the Monte Carlo version of the Glauber model Miller et al. (2007). In both colliding nuclei we sample the positions of the nucleons according to $\rho(r)$, which we then shift by $\pm\mathbf{b}/2$ to take into account the finite impact parameter. Subsequently, we determine the participants in a black-disk approximation as those nucleons of nucleus A that are located within a distance $\sqrt{\sigma_{\rm nn}/\pi}$ from at least one nucleon in nucleus B, and vice versa. Here $\sigma_{\rm nn}$ is the nucleon-nucleon cross-section, which is equal to $4.2$ fm2 for $\sqrt{s_{\rm NN}}=200$ GeV and to $6.4$ fm2 for $\sqrt{s_{\rm NN}}=2.76$ TeV. The thickness $T(\mathbf{x}_{\bot})$ of each nucleus is then determined on an event-by-event basis by summing the density profiles of all its participating nucleons, where the density of each participant nucleon is taken as a Gaussian profile of width $0.5$ fm.

Explicitly restoring the constants in Eqs. (10) and (8), the saturation scale $Q_{s}(\mathbf{x})$ of the nucleus is determined as

[TABLE]

where $Q_{sp}(\sqrt{s})$ and $A_{p}=\pi R_{p}^{2}$ denote the saturation scale and (transverse) size of a proton, and the initial state energy density per unit rapidity is then given by

[TABLE]

where $\mathcal{N}_{e}$ is a dimensionless proportionality coefficient (see e.g. Dumitru and McLerran (2002)). By use of Eqs. (6) and (7), the multiplicity in an event is then given by

[TABLE]

while the initial-state energy per unit rapidity is simply given by the transverse integral of $(e\tau)_{0}$ in Eq. (17) as

[TABLE]

Statistical averages of $\frac{dN_{\text{ch}}}{d\eta}$ and $\frac{dE_{0}}{d\eta_{s}}$ at a given centrality percentile are then obtained by sampling many events at the corresponding impact parameter. By matching the multiplicity in Eq. (18) to experimental data, we find that the dimensionless combination of pre-factors

[TABLE]

in Eq. (18) should be equal to $10.7$ for 2.76 TeV Pb-Pb collisions $4.8$ for 200 GeV Au-Au collisions, which is compatible with the expected increase of the saturation scale $Q_{s}^{2}(\sqrt{s})\propto\sqrt{s}^{\lambda}$ with $\lambda\approx 0.3$ Albacete and Marquet (2014).

Vice versa, to determine the initial state energy per unit rapidity shown in Fig. 3, we can use Eq. (18) for the multiplicity to eliminate the pre-factor $N_{e}~{}\left(Q_{sp}^{2}(\sqrt{s})A_{p}\right)^{3/2}$ from Eq. (19) and express $\frac{dE_{0}}{d\eta_{s}}$ in the form of Eq. (14) as

[TABLE]

where the effective area, $A_{\bot}$, is given by

[TABLE]

Evaluating Eq. (22) within our Monte-Carlo calculations yields $A_{\bot}\approx 120$ fm2 in central Pb-Pb collisions, which is in fair agreement with the value anticipated in Eq. (14). Note that, while we could have used the experimental data for $dN_{\rm ch}/d\eta$ to produce Fig. 3, we conveniently make use of the model estimate in Eq. (18) to interpolate between experimental data points; as shown in Fig. 2 the model is within $5\%$ of all experimental data.