# Flow in AA and pA as an interplay of fluid-like and non-fluid like excitations

## Abstract

To study the microscopic structure of quark–gluon plasma, data from hadronic collisions must be confronted with models that go beyond fluid dynamics. Here, we study a simple kinetic theory model that encompasses fluid dynamics but contains also particle-like excitations in a boost invariant setting with no symmetries in the transverse plane and with large initial momentum asymmetries. We determine the relative weight of fluid dynamical and particle like excitations as a function of system size and energy density by comparing kinetic transport to results from the 0th, 1st and 2nd order gradient expansion of viscous fluid dynamics. We then confront this kinetic theory with data on azimuthal flow coefficients over a wide centrality range in PbPb collisions at the LHC, in AuAu collisions at RHIC, and in pPb collisions at the LHC. Evidence is presented that non-hydrodynamic excitations make the dominant contribution to collective flow signals in pPb collisions at the LHC and contribute significantly to flow in peripheral nucleus–nucleus collisions, while fluid-like excitations dominate collectivity in central nucleus–nucleus collisions at collider energies.

## 1 Introduction

Azimuthal asymmetries \(v_n\) in soft transverse multi-particle distributions have been measured in detail in PbPb, pPb and pp collisions at the LHC [1, 2, 3, 4, 5, 6]. At lower energies they have been studied in nucleus–nucleus and deuteron-nucleus collisions at RHIC [7, 8, 9], and at yet lower center-of-mass energies reached in fixed-target experiments, e.g., [10]. For sufficiently central nucleus–nucleus collisions at collider energies, model analysis support a fluid-dynamic interpretation of the measured asymmetries [11, 12]. The recently established persistence of sizeable azimuthal asymmetries in the smaller pPb and pp collision systems [3, 4, 5, 6] now raises the question of whether small transient fluid droplets are also formed in these systems, or whether other physics mechanisms are at the origin of the observed signatures of collectivity. The main purpose of the present work is to address a particular variant of this central question: *To what extent are fluid-dynamic or particle-like excitations at the origin of the flow phenomena observed in proton-proton, proton-nucleus and nucleus–nucleus collisions? And how does the interplay between these two sources of collectivity change as a function of system size and energy density?*

To set the stage for these questions, we recall that any self-interacting matter that does not spontaneously break Lorentz symmetry carries both fluid-dynamic and non-fluid dynamic excitations. Analysis of the retarded two-point correlation functions \(G_R^{\mu \nu ,\alpha \beta }(x;t) = \langle [ T^{\mu \nu }(x,t), \, T^{\alpha \beta }(0,0)]\rangle \) is one way of characterizing the excitations that a system can carry. In particular, the Fourier transform \(G_R^{\mu \nu ,\alpha \beta }(k,\omega )\), understood as a function of complex frequency \(\omega \), displays non-analytic structures that can be related unambiguously to excitable physical degrees of freedom. Fluid dynamic excitations correspond to poles at positions \(\omega _{\mathrm{hyd}}(k)\) close to the real axis in the negative imaginary \(\omega \) half plane. In all known stable and causal dynamics, they are supplemented by other non-analytic structures whose precise nature depends on the microscopic dynamics. Prominent examples are sketched in Fig. 1. For instance, for the non-abelian plasma of \(\mathcal{N}=4\) super–Yang–Mills theory in the large-\(N_c\) limit at strong coupling, \(G_R^{\mu \nu ,\alpha \beta }(k,\omega )\) displays—in addition to fluid dynamic poles—a tower of quasi-normal modes that are located deep in the negative complex plane at depths \(\propto - i 2\pi T n\), \(n\in 1,2,3,...\) [13, 14, 15]; somewhat distorted versions of this non-fluid dynamic pole structure are found in a class of related, strongly coupled quantum field theories with gravity duals [16, 17, 18]. In contrast, kinetic theory in the relaxation time approximation (RTA) supplements fluid-dynamic excitations with a branch-cut of quasi-particle excitations that is located at a depth \(-i/\tau _R\) set by the relaxation time \(\tau _R\) [19]; scale-dependent versions of this RTA allow for more general but related non-analytic structures [20]. Also Israel-Stewart dynamics [21] is more than viscous fluid dynamics. The ad–hoc requirement that the shear-viscous tensor relaxes to the constitutive fluid-dynamic 1st-order relation on a timescale \(\tau _\pi \) results in a non-fluid-dynamic mode in Israel-Stewart dynamics.

The experimental determination of these non-hydrodynamic structures would be tantamount to determining the microscopic structure of quark–gluon plasma. While experimental reality prevents us from directly measuring the response of a static and infinite quark–gluon plasma to a linearized perturbation and directly accessing the non-hydrodynamic structures in Fig. 1, the non-hydrodynamic sector may leave its imprint to the dynamical evolution of the system created in physical collisions. This is especially the case if the hydrodynamic and non-hydrodynamic modes are not well separated from each other, or even more so if the non-hydrodynamic modes are closer to the real axis than the hydrodynamic ones and thus dominate the evolution of the system.

The non-hydrodynamic sector becomes more prominent in smaller systems. Firstly, the smallest wavenumber that excitations in a system with transverse size *R* can have is \(k \ge 1/R\). With decreasing *R*, the spectrum of fluid-dynamic excitations in the system thus corresponds to poles that lie deeper and deeper in the negative imaginary \(\omega \)-plane (at positions \({\mathrm{Im}}\left[ \omega _{\mathrm{hyd}} \right] \lesssim - \textstyle \frac{\eta }{\varepsilon + p} \textstyle \frac{1}{R^2}\) for the shear channel), and thus approach the non-hydrodynamic sector. Secondly, in a system that lives for a lifetime of \(\Delta \tau \), the information of modes with \({\mathrm{Im}}(\omega )> 1/\Delta \tau \) is lost as the decay of the modes is dictated by the imaginary part of frequency \(\exp ({\mathrm{Im}}(\omega ) \tau )\). Therefore with decreasing \(\Delta \tau \sim R\), one may access an increasing amount of information about the non-hydrodynamic structures that lie deeper in the complex plane. For both of these reasons, systems that are small enough are inevitably dominated by physics that is not fluid-like and the specific way of how the hydrodynamic description fails for small systems carries the information of what lies beyond.

A systematic comparison of system-size dependencies between models of different microscopic dynamics and measured azimuthal anisotropies may offer an empirical pathway to discriminate between different possibilities. We note that while disagreement with the data would exclude a given model, the opposite does not hold true. In fact, an example of such an agreement comes from the phenomenological success of Israel–Stewart hydrodynamics in small systems where the dynamics is dominated by the non-hydrodynamic pole governing the relaxation of the shear-stress tensor to its Navier–Stokes value. Apparent agreement with data surely does not imply that the microscopic dynamics of Israel–Stewart theory are at play in Nature but does mean that the non-hydrodynamic sector of Israel-Stewart theory at least somewhat resembles that of the true underlying dynamics.

To what extent we may discriminate whether the underlying dynamics contains quasiparticles, quasi-normal modes of AdS black hole, or perhaps something else depends to what extent we can identify features inconsistent with data with decreasing system size. The most prominent feature exhibited by the small systems is the free-streaming dynamics upon which the standard modelling of pp-collisions is based. Hence, it is a natural starting point to ask how this limit is approached in the only one of the above mentioned models that does support free streaming, namely the kinetic theory.

## 2 Background

Kinetic transport has been considered since the beginning in the study of relativistic nucleus–nucleus collisions [22]. On the one hand, there are parton-cascade formulations [23, 24, 25, 26, 27] that follow the time evolution of the individual partons. On the other hand, direct solutions of the Boltzmann transport equation trace the particle distribution functions.

Parton cascades are nowadays embedded in widely used phenomenological models like the AMPT [28] and BAMPS [29] that aim at a complete dynamical description of all stages of nucleus–nucleus collisions, and that reproduce in particular many aspects of the measured azimuthal anisotropies in nucleus–nucleus collisions, and their system size dependence [30, 31, 32, 33]. Parton cascades have also been used to study the Boltzmann transport equation in isolation with the aim of gaining qualitative insights into how kinetic theory approaches a hydrodynamic regime [24, 26, 34, 35, 36], how it builds up collective flow [37, 38, 39, 40], and how this affects the flow of heavy quarks [41]. In addition to the above mentioned partonic cascades, we mention in passing that hadronic transport codes, such as [42, 43, 44], play an important role in phenomenology.

Aspects of real-time dynamics of QCD can be followed in an effective kinetic theory (EKT) [45] which provides a systematic expansion in the strong coupling constant \(\alpha _s\). Transport coefficients of weak-coupling QCD have been evaluated by directly solving the corresponding Boltzmann transport equation near thermal equilibrium [46, 47, 48, 49, 50, 51]. Solutions to the EKT Boltzmann equation have been studied also in more complex geometries. In particular, solutions to the EKT Boltzmann equation in 1 + 1 dimensional longitudinally expanding systems have shown that the out-of-equilibrium systems of saturation framework (see, e.g., [52])—described by classical gluodynamics at early times [53]—can be smoothly evolved to late times where the system allows for a fluid-dynamic description [54, 55, 56, 57]. In support of its potential phenomenological relevance, this perturbative approach—if supplemented with realistic values of the coupling constant—leads to short hydrodynamization timescales [55] comparable to those obtained from non-perturbative strong coupling techniques [58, 59, 60, 61, 62] and favored in phenomenological models. This approach has been extended to 2 + 1 dimensional solutions [63] which form the foundation of tools that can be used to match the out-of-equilibrium initial stage models to a hydrodynamic stage in nucleus–nucleus collisions [64, 65]. In these solutions the transverse translational symmetry is, however, broken only by small linear perturbations limiting their use in the study of small systems which potentially do not reach the fluid-dynamic regime before disintegrating due to transverse expansion.

In the past years there has been significant interest also in solutions to Boltzmann transport equations with simplified collision kernels, such as relaxation time approximation or diffusion-type kernels [66, 67, 68, 69]. These formulations have the advantage that due to their simpler structure they allow for more detailed analysis. That has allowed the study of, e.g., the role of out-of-equilibrium attractors and their relation to the non-hydrodynamic modes [66, 70]. While this class of models is not directly based on QCD (cf. [51, 71]), they do share characteristics with EKT and may be used as toy model of EKT. In addition, they may reveal qualitative features of systems with quasiparticle descriptions even in a regime where weak coupling approximations of EKT are not appropriate. Besides works that solve the Boltzmann transport equation perturbatively in the collision kernel around the free-streaming solution [40, 72, 73], these studies are still constrained to simple geometry of 1 + 1 dimensions with the notable exception of [74] that breaks the translational symmetry but retains a de Sitter symmetry and remains restricted to azimuthal rotational symmetry.

The first non-perturbative solution to Boltzmann transport equation under longitudinal transverse expansion and broken transverse translational symmetry, including broken azimuthal symmetry was given in [75]. The approach in [75] that we document in detail in Sect. 3 one starts from an essentially analytic formulation of kinetic transport in isotropization time approximation (ITA) and extends it to the calculation of azimuthal flow. The fluid-dynamic properties of the model are analytically known which allows us in Sect. 4 to disentangle unambiguously and quantitatively fluid-dynamic from non-fluid dynamic features in the dynamical evolution of kinetic theory. We thus investigate a system in which the modern conceptual debate about a quantitatively controlled understanding of hydrodynamization that has remained largely limited so far to arguably academic, lower dimensional analytic set-ups, can be extended to a problem of sufficient complexity to provide insight into the current phenomenological practice. We hasten to remark that the kinetic transport theory considered here remains a simple one-parameter model designed mainly to address conceptual questions like those posed at the beginning of this introduction. However, in an effort of pushing the relevance of this starting point as far as possible into the realm of full phenomenological studies explored so far with multi-parameter fluid dynamic and parton cascade codes of significant complexity, we shall compare in Sect. 5 results of this one-parameter kinetic theory to results on the system size dependence of measures of collectivity in hadronic collisions at RHIC and at the LHC.

## 3 The model

We solve the kinetic theory in *isotropization-time approximation* (ITA) with boost invariant geometry but with strongly broken transverse translational symmetry. Our initial conditions have an approximate azimuthal symmetry that is broken by linear perturbations. In this section, we describe the model that we solve using free-streaming coordinate-system methods introduced in [75] that we fully document in Appendix Appendix A.

### 3.1 Isotropization-time approximation

*p*-integrated moment of the distribution function

*p*, but does depend on the direction of the propagation of particles \(\vec v_\perp \) and \(v_z\) collectively denoted by \(\Omega \), subject to constraint that \(v_z^2+v_\perp ^2=1\). The evolution equation of the integrated distribution function is obtained by taking the first integral moment of Eq. (1)

*C*[

*F*] can be expressed in terms of

*F*. While this does not generically happen, it is the case in the

*isotropization-time approximation*(ITA). The ITA is a simple implementation of the physical picture that interactions drive the system towards isotropy even if the system is far from equilibrium and may be too short-lived to thermalize. The ITA therefore assumes that the distribution

*F*evolves in the local rest frame \(u^{\mu }(\tau ,\vec x_\perp )\) to an isotropic distribution \(F_{iso}\) in timescale \(\tau _{iso}\), and that the collision kernel is taken to be of the form

As discussed already in previous work [73, 75], the isotropization time approximation is found to reproduce the evolution of the QCD weak coupling effective kinetic theory within \(~\sim 15\%\). It shares important commonalities with the well-known relaxation time approximation, but it is free of the assumption that the distribution function *f* evolves directly to a thermal equilibrium. As such it is better suited for the description of non-equilibrium systems such as anisotropic plasmas to which a local equilibrium distribution is difficult to associate to.

### 3.2 Initial conditions

*F*cannot depend on \(\phi \) initially. Moreover, since ultra-relativistic pp, pA and AA collisions are expected to display very large momentum anisotropies between the transverse and longitudinal direction, the initial condition will be taken to be of the form

*cf.*Eq. (30)). The initial conditions are then of the form

- Initial Gaussian profileThe normalization of this isotropic distribution is chosen such that the initial energy density \(\varepsilon (\tau _0, r=0)\) calculated from \(F(\tau _0, \vec x_\perp ; \phi , v_z)\) in Eq. (9) satisfies \(\varepsilon (\tau _0, r=0) = \varepsilon _0\).$$\begin{aligned} P_{\mathrm{iso}}\left( \frac{r}{R}\right) = 2 \varepsilon _0\, \exp \left[ -\frac{r^2}{R^2} \right] . \end{aligned}$$(10)
- Initial Woods–Saxon (WS) profile For a Pb nucleus, the standard Woods–Saxon parametrization of the density profile is \(\rho _{\mathrm{WS}} (r,z)= 1/\left( \exp \left[ \textstyle \frac{\sqrt{r^2 + z^2}-R_{\mathrm{WS}}}{\delta _{\mathrm{WS}}} \right] + 1 \right) \) where \(R_{\mathrm{WS}} = 1.12 A^{1/3} - 0.86 A^{-1/3} \vert _{A=208} = 6.49\)fm, and \(\delta _{\mathrm{WS}} = 0.54 \)fm. The corresponding nuclear profile function \(T_{\mathrm{WS}}(r) \equiv \int _{-\infty }^\infty \rho _{\mathrm{WS}}(r,z)\, dz\) is the projection of this density onto the transverse plane. Simple models of the local energy density in the transverse plane take \(\varepsilon (\tau _0, r)\) proportional to the nuclear overlap of two Pb nuclei, i.e., for vanishing impact parameter, proportional to \(T^2_{\mathrm{WS}}(r)\) [76]. This motivates us to make the ansatzwhere we choose the number \(c_{\mathrm{rms}}\approx 4.42\) such that the radius mean square of \(P_{\mathrm{iso}}\left( \frac{r}{R}\right) \) is properly normalized to unity.$$\begin{aligned} P_{\mathrm{iso}}\left( \frac{r}{R}\right) = \frac{2\varepsilon _0\, T_{\mathrm{WS}}^2\left( \frac{r\, c_{\mathrm{rms}}}{R} \right) }{ T_{\mathrm{WS}}^2\left( 0\right) }, \end{aligned}$$(11)

*the opacity*\({{\hat{\gamma }}}\)

## 4 Numerical results

We analyze now in detail how the time evolution of the energy-momentum tensor varies with transverse system size between the extreme limiting cases of free streaming and ideal fluid dynamics by varying the only available dimensionless parameter, the opacity \({{\hat{\gamma }}}\). In the following, at \(z=0\) time *t* and radial *r* coordinates are plotted in units of the system size *R*.

### 4.1 Time evolution of \(T^{\mu \nu }_{\mathrm{kin}}\)

We consider first the time evolution of the energy-momentum tensor \(T^{\mu \nu }_{\mathrm{kin}}\) for azimuthally symmetric initial conditions. This corresponds to solving the Boltzmann transport equation (A21) for the zeroth harmonic \(F_0\) with initial conditions (11) brought into the form similar to (A56). The solution defines the energy momentum tensor (A13), which we shall denote with the subscript “kin” to make clear that this is the result of kinetic theory.

We start by characterizing in Fig. 2 the time evolution of the local energy density \(\varepsilon (t,r) = u_\mu (t,r)\, T^{\mu \nu }_{\mathrm{kin}}(t,r)\, u_\nu (t,r)\) of a system initialized with the radial Woods–Saxon distribution (11). To explain the main features of this figure, we note the following: For a free-streaming gas of massless particles, all components propagate unperturbed along the light cone. As initially \(v_z =0\), the particles can reach positions close to \(r= 0\) at late times only if they originate at large radius and propagate inwards. The energy density \(\varepsilon (t,r=0)\) at the center can therefore take non-negligible values only at times \(t<r_{\mathrm{tail}} \simeq R\) limited by the extent of the tail of the initial radial distribution. These particles will propagate in the (*r*, *t*) diagram on a light-like trajectory below \((r, r+r_{\mathrm{tail}})\). On the other hand, particles that start initially at the same radial position \(r_{\mathrm{tail}}\) but propagate exactly outwards will show up in Fig. 2 along \((r,t)=(r+r_{\mathrm{tail}},r)\) and all particles produced initially at smaller *r* or emitted not exactly inwards will stay above that line. A free-streaming density distribution is therefore a density distribution that takes non-vanishing values only in the strip limited by \((r,t)=(r, r+r_{\mathrm{tail}})\) and \((r,t)=(r+r_{\mathrm{tail}},r)\) in the (*r*, *t*)-diagram. This is clearly seen for the free-streaming limit \({{\hat{\gamma }}} = 0\) in Fig. 2.

As one endows the system with a small interaction rate, large-angle scatterings can deflect freely propagating particles towards the neighborhood of \(r=0\). As a consequence, a non-vanishing energy-density around \(r=0\) persists for a longer time. This effect is sizeable even if scattering is infrequent: the case \({{\hat{\gamma }}} = 0.25\) in Fig. 2 corresponds to a system size that extends only over one quarter of the in-medium path length, so most particles will escape unscattered from the collision region. Yet, compared to the free-streaming case, the system maintains at its origin a non-vanishing energy density for a significantly longer time.

As one increases \({{\hat{\gamma }}}\) and thus the collision propability per particle per system size, one finds that the general features described above evolve gradually, and an increasing fraction of the total energy density remains for an increasing time duration in the neighborhood of the origin of the system, see Fig. 2. In this sense, the evolution of the energy density moves with increasing \({{\hat{\gamma }}}\) further away from the light cone, although particles located in the tails of the system and propagating outwards will always escape along the light cone.

### 4.2 Quantifying deviations of \(T^{\mu \nu }_{\mathrm{kin}}\) from fluid dynamics

*t*and

*r*. Since \(\left( T_{\mathrm{kin}}- T_{\mathrm{hyd}} \right) ^{\mu \nu } = \Pi ^{\mu \nu }_{\mathrm{kin}} - \Pi ^{\mu \nu }_{\mathrm{hyd}}\), and since the shear viscous tensor is transverse, \(u_\mu \Pi ^{\mu \nu } = 0\), only the spatial components of \(\left( T_{\mathrm{kin}}- T_{\mathrm{hyd}} \right) ^{\mu \nu }\) can be non-zero in the local rest frame. Therefore, the contractions under the square root in (21) yields a sum over squares with positive coefficients. Thus, \(Q_2(t,r)\) is a positive definite measure of the difference between \(T_{\mathrm{kin}}^{\mu \nu }\) and \(T_{\mathrm{hyd}}^{\mu \nu }\). At the space-time point (

*t*,

*r*), it measures the differences between the shear viscous tensors of kinetic theory and hydrodynamics in units of the local energy density, \(T_{\mathrm{id}}^{\mu \nu } {T_{\mathrm{id}}}_{\mu \nu } = \textstyle \frac{4}{3} \varepsilon (t,r)\).

*t*,

*r*) in its local rest frame. The results shown for \(Q_0\) in Fig. 3 indicate that for any non-transparent system with \({{\hat{\gamma }}}>0\), there is always a region of space-time in which the system is almost isotropic. For \(r=0\), the corresponding blue regions in which \(Q_0\) is small are centered around \(t/R = 2\). The time \(t/R=2\) is special in that it is the maximum time up to which particles produced initially at large radius can reach the center \(r=0\) along free-streaming inward-going trajectories. For small \({{\hat{\gamma }}}\), these particles will cross each other with negligible isotropizing interactions. In the local rest frame defined by Landau matching, such crossing particle streams lead to transient, locally isotropic systems. This effect is known from other fields in which one uses fluid dynamics to describe collective phenomena that is inherently non-fluid dynamical. For instance, this feature is referred to as

*shell-crossing*in the literature on cosmological Large Scale Structure [78, 79]. There, one refers to a velocity field as

*multi-valued*if the velocity of individual particles in the neighborhood of a space-time point (

*x*,

*t*) is not characterized by the (thermal) dispersion around the unique rest frame velocity \(u^\mu (t,x)\), but where it is determined instead by the (multiple) directions in which particles were located in the distant past and from which they were streaming towards (

*t*,

*x*). Kinetic theory realizes this scenario for almost transparent systems with small opacity \({{\hat{\gamma }}}\).

Non-interacting or weakly interacting streams of particles that cross each other can yield transient approximate isotropization in a neighborhood of (*t*, *x*). The tell-tale sign of this phenomenon is a negligible \(Q_0(t,x)\) combined with a sizeable \(Q_1(t,x)\) or \(Q_2(t,x)\) that indicate that fluid dynamics predicts *larger* shear viscous components in the energy momentum tensor that is realized in the kinetic theory.

Fluid dynamics becomes a quantitatively controlled description of the collective dynamics in space-time regions where increasing the order of the fluid dynamic constitutive relation leads to a better description of the full energy-momentum tensor. We therefore refer to a system evolved with kinetic equations as behaving fluid-dynamical in a space-time region around (*t*, *r*) if \(Q_1(t,r)\) is sufficiently small and if this smallness persists to higher order in gradients, i.e., for \(Q_2(r,t)\). To be specific, we use the criterion \(Q_i(t,r)<0.1\) which corresponds to a situation in which the summed deviations of the kinetic shear viscous tensor from its constitutive hydrodynamical form is less than 10% of the total enthalpy \(\varepsilon + p\) of the system. This condition corresponds to the blue regions in Fig. 3.

For \({{\hat{\gamma }}}=1\), even though a small band with \(Q_0(r,t) < 0.1\) indicates that the system passes locally through approximately isotropic distributions, the region in which \(Q_1(r,t) < 0.1\) is vastly different from the region in which \(Q_0(r,t) < 0.1\) or \(Q_2(r,t) < 0.1\). This indicates that the gradient expansion does not converge and that the system displays characteristics that are very different from those of a fluid. We draw a similar conclusion for systems characterized by \({{\hat{\gamma }}} = 2\), since there is no extended region in (*t*, *r*) for which the difference \(Q_1(t,r)\) is small *and* for which this smallness persists in \(Q_2(t,r)\). For \({{\hat{\gamma }}} > 4\), however, our calculations indicate that the fluid dynamic gradient expansion is reliable, since the differences between \(T^{\mu \nu }_{\mathrm{kin}}\) and the fluid dynamic ansatz for the energy-momentum tensor improves order by order in the gradient expansion. Further increasing \({{\hat{\gamma }}} \) to 8, we find that the regions of small \(Q_1(r,t)\) and \(Q_2(r,t)\) widen, and that they extend in particular to earlier time. In the phenomenological practice, such behavior would allow one to switch at an earlier time from a full kinetic theory description to a fluid-dynamic one. For a discussion in the present framework, see, [75].

*k*is expected to propagate more fluid-like or more particle-like, depending on whether \(\omega _{\mathrm{cut}}\) or \(\omega _{\mathrm{pole}}\) is closer to the real axis. Since cut- and pole-terms depend inversely on \(\gamma \), it is clear that for decreasing \(\gamma \), particle-like excitations dominate. Also, since we consider the evolution of the components of the energy-momentum tensor which receives contributions from all wavelengths 1 /

*k*, it is clear that the transition from a system dominated by particle-like excitations to a system dominated by fluid dynamic excitations proceeds gradually with increasing opacity \({{\hat{\gamma }}}\). Quoting a precise value of \({{\hat{\gamma }}}\) for this transition would rely on agreeing on a criterion (such as \(Q_i < 0.1\)) of what is meant by

*small*. Qualitatively, however, we observe from Fig. 3 that systems with \({{\hat{\gamma }}} < 2\) display characteristics in the evolution of their energy-momentum tensor that show marked differences from a fluid dynamic evolution, while systems with \({{\hat{\gamma }}} > 4\) evolve fluid-like from early times \(t /R\ll 1\) onwards. In the first case, the fluid-dynamic gradient expansion does not converge in any extended space-time region relevant for building up elliptic flow, while it does in the second case. This indicates that as the system becomes more opaque, particle-like excitations cease to dominate the collective dynamics in the range \(2 \lesssim {{\hat{\gamma }}} \lesssim 4\).

### 4.3 Work in kinetic theory

The time-evolution of the transverse energy per unit rapidity and its azimuthal distribution \(\frac{dE_\perp (t)}{d\eta _s d\phi }\) can be calculated from kinetic theory according to (A18). The late time limit of these quantities corresponds to the experimentally accessible transverse energy distribution. In the phenomenological discussion in Sect. 5, we need to know to what extent the azimuthal average \(\frac{dE_\perp (t\rightarrow \infty )}{d\eta _s}\) at late time differs from that at early times which is tantamount to asking how much work the system does as a function of opacity \({{\hat{\gamma }}}\).

Finally, we note the remarkable insensitivity of \(f_{\mathrm{work}}\) to the initial profile illustrated by the numerical similarity between the results in obtained using either gaussian (black line) or Woods–Saxon (green dashed line) initial profile in Fig. 4.

### 4.4 Results for the linear response \(v_n/\epsilon _n\)

*energy*flow. Here we have used the fact that particles with momentum rapidity

*y*after the last interaction will end up in a spacetime rapidity \(\eta _s = y\). See Appendix A2 for details.

Spatial deformations \(\delta _{n,m}\) are introduced in the initial conditions according to (8). For the kinetic theory formulation explored in this work, the first results for the linear response of the elliptic energy flow \(v_2\) to initial elliptic perturbations \(\epsilon _2\) were reported in Ref. [75]. The purpose of the present subsection is to complete this information with results on higher harmonics, and to demonstrate that the results depend negligibly on the transverse profile function. These results are complete to all orders in \({{\hat{\gamma }}}\), but they are first order in \(\epsilon _n\). The formulation in Sect. 3 one allows for an extension to all orders in \(\epsilon _n\) in future work.

The black and the purple lines in Fig. 5 correspond to Gaussian background initial conditions of (10) and (8) with \(\delta _{2,2}\ne 0\) (black line) or \(\delta _{2,3}\ne 0\) (purple line). This illustrates the insensitivity of the response to details of the radial profile of the perturbation. Moreover, there is a remarkable insensitivity to the background profile, since Wood-Saxon (yellow line) and Gaussian (black line) initial conditions yield numerically similar linear responses. Furthermore, the green line in Fig. 5 corresponds to an isotropic initial momentum distribution \(F(r,\theta ,v_z) = \varepsilon _0 e^{-r^2/R^2}\) with a \(\delta _{2,2}\) perturbation. Again no significant sensitivity to the initial condition is visible. Similar observations can be made for \(v_3/\epsilon _3\) from the bottom panel of Fig. 5.

Fig. 5 illustrates for an important class of observables how kinetic theory interpolates between the limits of free-streaming and perfect fluidity. In the free streaming limit (\({{\hat{\gamma }}} = 0\)) momentum asymmetries \(v_n\) are not built up from initial spatial asymmetries \(\epsilon _n\). As a consequence, \(v_n/\epsilon _n\) starts from zero, and its \({{\hat{\gamma }}}\)-slope (red lines in Fig. 5) close to \({{\hat{\gamma }}} = 0\) can be understood as the consequence of perturbatively rare final state interactions in a small and/or dilute system. In the opposite limit of ideal fluid dynamics (\({{\hat{\gamma }}} \rightarrow \infty \)), momentum asymmetries are built up from spatial eccentricities with vanishing dissipation and thus most efficiently. As a consequence, \(v_n/\epsilon _n\) increases monotonously with increasing opacity \({{\hat{\gamma }}}\) and it approaches at very large \({{\hat{\gamma }}}\) the ideal fluid limit (dashed blue lines in Fig. 5) from below. For \({{\hat{\gamma }}} < 16\), the difference between this ideal fluid limit and the result of full kinetic theory indicate dissipative effects. As discussed in the context of Fig. 3, these dissipative effects are accounted for by viscous fluid dynamics or, for smaller \({{\hat{\gamma }}}\), by particle-like excitations outside the fluid dynamic description.

## 5 Data comparison

The kinetic theory defined in Sect. 3 one and analyzed in Sect. 4 is a one-parameter model. By comparing it to experimental data, we do not intend to claim that a complete phenomenological understanding of data could be based on this one-parameter model. It is obvious that it cannot, and by stating at the beginning of this section the obvious, we hope to preempt any possible misunderstanding of this point. Rather, by comparing to data, we want to understand to what extent the one generic physics effect implemented in this model could account for one central qualitative feature in the data, namely for the remarkable system size dependence of measures of collectivity. Is the centrality dependence of flow measurements consistent with a dynamics that interpolates between free-streaming and viscous fluid dynamics, or is it more consistent with a collective dynamics that is always fluid-like in the sense that it depends negligibly on particle-like excitations even for the smallest collision systems studied at the LHC?

### 5.1 The centrality dependence of opacity \({{\hat{\gamma }}}\)

*R*of the initial transverse energy profile. We note that for LHC energies, the transverse energy \(dE_\perp /d\eta \) varies by a factor 80 between the most central (1%) and most peripheral (80%) centrality, while the rms radius R changes by less than a factor 2. As a consequence, the centrality dependence of \({{\hat{\gamma }}}\) is largely determined by the centrality dependence of the transverse energy, and modeling uncertainties of

*R*play a minor effect in determining \({{\hat{\gamma }}}\) (e.g., varying

*R*by 10% amounts only to doubling the line width in Fig. 6). This allows us to determine for simplicity the rms radius R from an optical Glauber model [82].

#### 5.1.1 Opacity \({{\hat{\gamma }}}\) in PbPb at the LHC

The centrality dependence of \({{\hat{\gamma }}}\) shown in Fig. 6 for \(\sqrt{s_{\mathrm{NN}}} = 5.02\) TeV PbPb collisions at the LHC was obtained with \(\frac{dE_\perp }{d\eta _s}\) values determined from the \(p_\perp \)-spectrum \(\textstyle \frac{dN}{dp_\perp d\eta _s}\) published by ALICE in Ref. [83] for nine centrality bins between 0–5% and 70–80%, corrected for the presence of neutrals and interpolated to finer intermediate centrality bins. Combined with the conclusions drawn from Fig. 3, Fig. 6 informs us for which values of \(\eta /s\) fluid dynamic behavior can be expected as a function of system size.

For instance, for \(\eta /s=2/4\pi \), the value \({{\hat{\gamma }}} \approx 8\) reached in the most central PbPb collisions (see, Fig. 6) signals according to Fig 3 that for matter of this viscosity the collective dynamics realized in central PbPb collisions is fluid-like. But for semi-peripheral PbPb collisions with centrality \(> 50\%\), the \(\eta /s=2\)-curve in Fig. 6 shows values \({{\hat{\gamma }}} < 4\) for which a fluid dynamic picture of the evolution becomes questionable, and for \(> 80\%\) centrality, values \({{\hat{\gamma }}} < 2\) signal a collective dynamics in peripheral collisions which is qualitatively different from that of a fluid. As seen from Fig. 6, the centrality range for which a fluid dynamic gradient expansion is expected to be quantitatively reliable (\({{\hat{\gamma }}} > 4\)) or qualitatively indicative (\({{\hat{\gamma }}} > 2\)) changes rapidly with \(\eta /s\).

#### 5.1.2 Opacity \({{\hat{\gamma }}}\) in AuAu at RHIC

In close analogy to the calculation of \({{\hat{\gamma }}}\) for PbPb collisions at the LHC, we calculate \({{\hat{\gamma }}}\) for \(\sqrt{s_{\mathrm{NN}}} = 200\) GeV AuAu collisions based on STAR data of the \(N_{\mathrm{part}}\)-dependence of \(dE_\perp /d\eta _s\) [84]. For the same centrality and the same assumed value of \(\eta /s\), \({{\hat{\gamma }}}\) is approximately \(20\%\) smaller at RHIC than at the LHC, and the parameter regions in which fluid-like and particle-like excitations dominate collectivity shift accordingly, see, Figs. 6 and 7. For nucleus–nucleus collisions at RHIC to show a more fluid-like behavior (i.e. correspond to a larger \({{\hat{\gamma }}}\)) than at the LHC, the matter produced at RHIC would thus have to display a significantly smaller \(\eta /s\), since at comparable values of \(\eta /s\), the higher center of mass energy and the higher transverse energy density attained at LHC facilitates fluid-like behavior.

#### 5.1.3 Opacity \({{\hat{\gamma }}}\) in pPb at the LHC

To calculate the opacity \({{\hat{\gamma }}}\) for pPb collisions at the LHC, we make the simplifying assymption that the rms-width *R* of the transverse area initially active in pPb collisions does *not* change with event multiplicity. This assumption has a physical and a pragmatic motivation. Physically, it amounts to stating that the dispersion in the multiplicity distribution at fixed impact parameter is so broad that the correlation between event multiplicity and event centrality (i.e., impact parameter) is weak. From a pragmatic point, choosing *R* to be multiplicity-independent is reasonable, since we do not have firm knowledge about how the initial geometry of pPb collisions changes with multiplicity. Moreover, for the purpose of calculating \({{\hat{\gamma }}}\), this lack of knowledge is partially alleviated by the fact that information about the multiplicity-dependence of *R* enters \({{\hat{\gamma }}}\) only as a fourth root. We therefore choose in the following \(R = 1\) fm as default value for pPb collisions, noting that even extreme values of \(R= 2\) fm would affect results for \({{\hat{\gamma }}}\) only by 20%.

### 5.2 Comparing kinetic theory to \(v_n/\epsilon _n ({{\hat{\gamma }}})\): assumptions and uncertainties

Before presenting in the following subsections comparisons of kinetic theory to data on \(v_n/\epsilon _n\), we list here the main elements of this comparison and we discuss the underlying assumptions and resulting uncertainties.

#### 5.2.1 Transverse energy \( \frac{dE_\perp }{d\eta _s} \) and energy flow \(v_2\) and \(v_3\)

Particle flow coefficients \(v_{m}^{\mathrm{particle}}\) are measured with different methods that are designed to include different sources of particle correlations and that therefore yield numerically different results. In particular, the 2nd and higher order cumulant flows \(v_{m}^{\mathrm{particle}}\lbrace 2 \rbrace \), \(v_{m}^{\mathrm{particle}}\lbrace 4\rbrace \), \(v_{m}^{\mathrm{particle}}\lbrace 6\rbrace \), ..., are designed such that they include only particle correlations that persist in connected 2-, 4-, 6-, ... particle correlation functions. Moreover, to eliminate contributions from jet-like structures, data for \(v_{m}^{\mathrm{particle}}\) are often designed to include only the correlation of particles separated by a sizeable rapidity gap \(\vert \Delta \eta _s\vert \).

To choose the variant of particle flow measurements from which we infer \(v_m^{\mathrm{energy}}\) via (33), we start from the following two conisiderations. First, one of the simplest perturbative corrections to free-streaming that is accounted for by kinetic theory may be thought of as a single rare scattering that leads to correlations amongst very few of the final state particles and that leads, a fortiori, to an azimuthal correlation of \(\textstyle \frac{dE_\perp }{d\eta _s\, d\phi } \) carried by a small fraction of the entire \( \frac{dE_\perp }{d\eta _s} \). As such a correlation is not shared by all particles in the event, it would not fully persist in higher order cumulants, while it is fully accounted for in kinetic theory. Second, the formulation of the kinetic theory presented here is not sufficiently detailed to include the characteristic dynamics of jet-like fragmentation patterns, and a meaningful comparison of kinetic theory to data should thus start from an energy flow \(v_m^{\mathrm{energy}} \) from which jet-like correlations are eliminated to the extent possible. The combination of both considerations prompts us to calculate \(v_m^{\mathrm{energy}} \) from 2-particle cumulant measurements with rapidity gap. Consistent with this choice, we use throughout this work second order cumulants to characterize spatial eccentricities, \(\epsilon _m = \epsilon _m\lbrace 2\rbrace \).

- 1.
*Data for PbPb collisions at the LHC*\(p_\perp \)-differential particle flow \(v_{2,3}\lbrace 2, \vert \Delta \eta _s\vert \rbrace (p_\perp )\) [2] published in nine centrality classes between 0–5% and 70–80%, and single inclusive charged hadron spectra [83] in the same centrality classes and the same mid-rapidity range are used to determine \(c_{\mathrm{e-corr,2}} = 1.34 (1.33)\) and \(c_{\mathrm{e-corr,3}} = 1.52 (1.51)\) for \(\sqrt{s_{NN}} = 5.02\) \((\sqrt{s_{NN}} = 2.76)\) TeV PbPb collisions. Fluctuations of these numbers with centrality are small, do not reveal any systematic trend, and may be attributed to uncertainties of \(p_\perp \)-differential flow data at higher \(p_\perp \). The \(p_\perp \)-integrated particle flow calculated from these data according to (34) is checked to be consistent with the \(p_\perp \)-integrated \(v_2 \lbrace {2, \vert \Delta \eta _s\vert > 1} \rbrace \) [2] restricted to the experimentally chosen range \(0.2 {\mathrm{GeV}}< p_\perp < 3 {\mathrm{GeV}} \). The factors \(c_{\mathrm{e-corr,2}}\) and \(c_{\mathrm{e-corr,3}}\) are then used to infer the energy flow via (35) for the 80 different \(1\%\)-wide centrality bins published in [2] for \(\sqrt{s_{NN}} = 5.02\) TeV PbPb, and for the somewhat fewer bins between 0% and 80% centrality published for \(\sqrt{s_{NN}} = 2.76\) TeV PbPb.

The data on single inclusive charged hadron spectra [83] are further used to determine \(dE_\perp /d\eta _s\) for the nine bins between 0% and 80% centrality, and are then interpolated to obtain centrality dependent values for \(dE_\perp /d\eta _s\) in 80 1%-wide centrality bins. This is checked to be consistent with published results on the centrality dependence of \(dE_\perp /d\eta _s\).

In addition to these data of the ALICE collaboration, there are CMS-data on \(v_2\lbrace {2, \vert \Delta \eta _s\vert > 2}\rbrace \) in \(\sqrt{s_{NN}} = 5.02\) PbPb [4] that could allow one to extend particle flow measurements to \(\approx 93\% \) centrality (Ref. [4] reports data on pp and pPb, but its HEPDATA-repository documents the latest CMS data on PbPb collisions). As these data are presented as a function of off-line tracks \(N^{\mathrm{offline}}_{\mathrm{trk}}\), as they are integrated over a slightly different \(p_\perp \)-range, and as they are taken in a much wider rapidity range, there are subtelties in consistently infering energy flow as a function of centrality with CMS and ALICE-data on the same plot. We are in contact with both collaborations to arrive at such a combination, but this lies outside the scope of the present manuscript.

- 2.
*Data on pPb collisions at the LHC*CMS data on the pseudorapidity dependence of the transverse energy \(dE_\perp /d\eta _s\) in pPb collisions at \(\sqrt{s_{\mathrm{NN}}} = 5.02\) were presented in [85] as a function of \(N_{\mathrm{part}}\), and pPb particle flow measurements \(v_2\lbrace {2, \vert \Delta \eta _s\vert > 2}\rbrace \) were presented in [4] against the experiment-specific number of offline tracks \(N^{\mathrm{offline}}_{\mathrm{trk}}\). Information tabulated in Ref. [86] relates \(N^{\mathrm{offline}}_{\mathrm{trk}}\) to the fraction of the total pPb cross section (i.e. centrality), and this average centrality can be converted into an average \(N_{\mathrm{part}}\), as done for instance in Ref. [87]. However, CMS data points for \(N_{\mathrm{offline}}^{\mathrm{track}} \gtrsim 150\) that correspond to the \(< 1\%\) most active events of the total cross section cannot be related meaningfully to \(N_{\mathrm{part}}\). As \(v_2\lbrace {2, \vert \Delta \eta _s\vert > 2}\rbrace \) for \(N_{\mathrm{offline}}^ {\mathrm{track}} \gtrsim 150\) is approximately constant [85] information from all data points at larger \(N^{\mathrm{offline}}_{\mathrm{trk}}\) can be regarded as grouped into the data point of largest \(N_{\mathrm{part}}\). In this way, both particle flow and transverse energy in pPb at the LHC is known as a function of \(N_{\mathrm{part}}\).

The following discussion uses the \(N_{\mathrm{part}}\)-dependence of \(v_2\lbrace {2, \vert \Delta \eta _s\vert > 2}\rbrace \) and \(dE_\perp /d\eta _s\) thus obtained

*without*relating \(N_{\mathrm{part}}\) to a geometrical picture of the collision. The same factor \(c_{\mathrm{e-corr,2}}=1.34\) determined for \(\sqrt{s_{\mathrm{NN}}} = 5.02\) PbPb collisions is used to determine energy flow, since the absence of sufficiently fine-binned \(p_\perp \)-differential \(v_2\) data does not allow one to perform the same determination of \(c_{\mathrm{e-corr,2}}=1.34\) as for PbPb. We anticipate that none of our conclusions will depend on the precise numerical value of \(c_{\mathrm{e-corr,2}}=1.34\) used in pPb. - 3.
*Data for AuAu collisions at RHIC*The centrality dependence of \(dE_\perp /d\eta _s\) is taken from Ref. [84]. We infer energy flow \(v_2\) for \(\sqrt{s_{\mathrm{NN}}} = 200\) GeV Au+Au collisions at RHIC by paralleling closely the procedure described for PbPb collisions at the LHC. STAR \(p_\perp \)-differential charged hadron \(v_2(p_\perp )\) data [88] in the wide 10–40% centrality bin is weighted with charged single inclusive hadron spectra from [89] according to Eqs. (33) and (34) to infer a correction factor \(c_{e-corr,2} = 1.5\) for \(p_\perp ^{\mathrm{min}} = 0.1\) GeV. This factor is applied to particle flow data of \(p_\perp \)-integrated \(v_2(\vert \Delta \eta _s\vert )\) presented in 11 \(N_{\mathrm{part}}\)-bins by PHOBOS [7]. To allow for a better comparison with other calculations presented in this paper, the \(N_{\mathrm{part}}\)-dependence is then converted into the impact parameter dependence obtained from an optical Glauber calculation [82].

#### 5.2.2 Non-ideal equation of state

#### 5.2.3 Non-linear response to spatial eccentricities

#### 5.2.4 Centrality dependence of initial eccentricities \(\epsilon _2\) and \(\epsilon _3\)

In comparing data to the kinetic theory prediction of \(v_n/\epsilon _n ({{\hat{\gamma }}})\), the model-dependence of the initial eccentricities \(\epsilon _n\) needs to be accounted for. Here, we consider two recent initial state models that were studied in Ref. [92] (referred to as GGMLO in the following) and that are based on different physics assumptions. They compare one of the current phenomenological standards, the so-called Trento model for \(p=0\) [93], to a model based on saturation physics for which the variation of the saturation scale \(Q_s\) translates into a band in the centrality dependence of initial eccentricities. Ref. [92] shows the corresponding eccentricities for impact parameter \(b < 10\) fm, since saturation physics is thought to lose predictive power for more peripheral collisions. As we study in the following systems up to 80% centrality, we require the impact parameter dependence of \(\epsilon _n\) for \(b \lesssim 13.9\) fm. We thank the authors of Ref. [92] for providing their data for this extended regime which we replot in Fig. 9. We have further obtained corresponding data sets for 2.76 TeV PbPb collisions that take for the saturation model slightly larger values of the eccentricity (data not shown) and that we shall use. The gray band in these figures will enter as model-dependent uncertainty in the following study.

### 5.3 Confronting kinetic transport to data on \(v_n/\epsilon _n({{\hat{\gamma }}})\) in PbPb at the LHC

- i.
the energy flow \(v_n\) from data,

- ii.
the transverse energy per unit rapidity \(\frac{dE_\perp }{d\eta _s}\) from data,

- iii.
the rms radius

*R*of the initial distribution as a function of centrality from a Glauber model, - iv.
the eccentricity \(\epsilon _n\).

We use data for 80 1%-wide centrality bins in 5.02 TeV PbPb collisions to evaluate (39) and to plot the results against the 80 \({{\hat{\gamma }}}\)-values of the corresponding centrality bins (see Eq. (32) and Fig. 6). Figures. 10 and 11 show the comparison of these data to kinetic theory results for \(\textstyle \left( \frac{v_2}{\epsilon _2} \right) _{\mathrm{corr}}({{\hat{\gamma }}})\) and \(\textstyle \left( \frac{v_3}{\epsilon _3} \right) _{\mathrm{corr}} ({{\hat{\gamma }}})\).

As seen from Fig. 6, the 80 bins between 1% and 80% centrality correspond to ranges \( 4\lesssim {{\hat{\gamma }}} \lesssim 16\) ( \(2\lesssim {{\hat{\gamma }}} \lesssim 7.5\), \(1.3\lesssim {{\hat{\gamma }}} \lesssim 5.0\)) for \(\eta /s = 1/4\pi \) (\(\eta /s = 2/4\pi \), \(\eta /s = 3/4\pi \)), respectively. Accordingly, the blue, red and brown bands for \(\eta /s = 1/4\pi \) (\(\eta /s = 2/4\pi \), \(\eta /s = 3/4\pi \)) in Fig. 10 scan over the corresponding \({{\hat{\gamma }}}\)-ranges, with the most peripheral centrality bin determining the left-most data of the corresponding band, and the most central bin determining the right-most data point. For fixed value of \({{\hat{\gamma }}}\), the sizes of the plotted error bars reflect only the modelling uncertainties in the initial eccentricities which are depicted by the grey band in Fig. 9. These uncertainties get larger for more peripheral centralities, when the uncertainties in the models (see, Fig. 9) are larger. The uncertainties are also somewhat larger in the most central bins, when \(\textstyle \left( \frac{v_2}{\epsilon _2} \right) _{\mathrm{corr}}\) is obtained by dividing \(v_2\) with a small eccentricity \(\epsilon _2\) of non-negligible uncertainty.

To illustrate the size of the correction terms in Eqs. (39), (40), we display the same data in the lower panel of Fig. 10 without correcting for nonlinearities, that is, by setting \(f_{\mathrm{corr}} = 1\). Similarly we expose the theoretical uncertainty arising from the equation-of-state correction factor \(c_{\mathrm{eos}}\) by displaying the kinetic theory result of Fig. 5 corrected with \(c_{\mathrm{eos}} = 0.8\) and 0.9 in both panels.

This analysis is repeated for triangular flow in Fig. 11. Since the maximal \(\epsilon _3\) values are significantly smaller than those for \(\epsilon _2\) (see, Fig. 9), a correction for a non-linear dependence on eccentricity \(f_{\mathrm{corr}}(\epsilon _3)\) turns out to be negligible (data not shown). Also, since the gray uncertainty band shown for the centrality dependence of eccentricity in Fig. 9 is in semi-central collisions much broader for \(\epsilon _3\) than for \(\epsilon _2\), the corresponding uncertainties in plotting \(\textstyle \left( \frac{v_3}{\epsilon _3} \right) _{\mathrm{corr}} \) in Fig. 11 are more pronounced for data from semi-central events. We see that kinetic theory accounts for the magnitude and centrality-dependence of \(\textstyle \left( \frac{v_2}{\epsilon _2} \right) _{\mathrm{corr}} \) and \(\textstyle \left( \frac{v_3}{\epsilon _3} \right) _{\mathrm{corr}} \) with values of \(\eta /s\) that lie within the same ball-park, although the triangular flow data prefer slightly lower \(\eta /s\) or—alternatively—prefer a slightly smaller \(\epsilon _3\) than the one displayed in Fig. 9.

Given the simplicity of the kinetic theory studied here and the uncertainties enumerated in Sect. 5.2, we do not draw too tight numerical conclusions about \(\eta /s\) from the present study. We solely note that the parameter range \(2/4\pi< \eta /s < 4/4\pi \) is favored by this analysis. We further note that this parameter range is consistent with the values favored in recent kinetic theory studies (\(0.2<\eta /s < 0.4\), [31]), and it is consistent with a full phenomenological analysis that includes a hydrodynamic evolution stage and that favors \(\eta /s = 2.5/(4\pi )\) [94]. Other full phenomenological analyses [95] favor smaller values for \(\eta /s\).

The constraint \(\eta /s < 4/4\pi \) is significantly lower than the value favored in our earlier analysis [73]. We note that this earlier analysis was based on the single hit line in Fig. 5, and that the difference compared to the present study arises from a combination of several improvements. In particular, the present analysis includes non-perturbative control over the \({{\hat{\gamma }}}\)-dependence beyond the linear approximation, it takes into account the correction factors in (39), (40), and it exploits the full experimentally available centrality dependence.

In combination with the \(\eta /s\)-dependence of \({{\hat{\gamma }}}\), the loose constraint \(2/4\pi< \eta /s < 4/4\pi \) implies an interesting qualitative statement. Irrespective of where the favored value of \(\eta /s\) is located in the range \(2/4\pi< \eta /s < 4/4\pi \), semi-peripheral or peripheral PbPb collisions reach down to values \({{\hat{\gamma }}} < 2\) for which a fluid dynamic picture does not correspond to the physical reality, while the most central PbPb collisions reach up to values \({{\hat{\gamma }}} > 4\) for which a fluid dynamic picture is a suitable interpretation of the physical reality.

### 5.4 The centrality dependence of eccentricities \(\epsilon _2\) and \(\epsilon _3\) preferred by kinetic theory

We start this discussion with the 5.02 TeV PbPb data, for which Fig.14 shows the centrality dependence of eccentricities \(\epsilon _2\), \(\epsilon _3\) reconstructed from (41) and overlaid with the eccentricity models depicted in Fig. 9. The conclusion is obviously the same as the one reached in Sect. 5.3: combining LHC data and the eccentricity model of Fig. 9 favors a value of \(\eta /s\) close to \(3/4\pi \). In addition, (41) informs us by how much one would have to distort the model of initial eccentricity to favor a value outside the range \(2/4\pi< \eta /s < 4/4\pi \).

We have reconstructed the eccentricity \(\epsilon _2\) for 200 GeV AuAu collisions from data sets discussed in Sect. 5.2. The initial geometrical distribution in AuAu collisions at RHIC and in PbPb collisions at the LHC can differ not only because of the slighly different nuclear size, but also because particle production processes and the ensuing event-wise fluctuations are expected to evolve over an order of magnitude in \(\sqrt{s_{\mathrm{NN}}}\). Because of the uncertainties discussed in Sect. 5.2, the reconstructed eccentricities displayed in Fig. 15 do not exclude small difference in the eccentricity of both collision systems. Such differences could arise within the present analysis for instance if the value of \(\eta /s\) varies significantly with \(\sqrt{s_{\mathrm{NN}}}\), or if the factor \(c_{\mathrm{eos}}\) in (41) takes a slighly different value for both collision systems. Also, the slightly different size of the Au and Pb nucleus implies that larger differences would certainly become visible if RHIC data were extended to more peripheral collisions. Despite these caveats, Fig. 15 seems to supports an approximately \(\sqrt{s_{\mathrm{NN}}}\)-independent elliptic eccentricity.

### 5.5 Reconstructing eccentricity for pPb collisions

The eccentricity in pPb collisions is particularly difficult to infer from multiplicity distributions close to mid-rapidity, since the correlation between event multiplicity and transverse geometry weakens when the dispersion of the multiplicity produced at fixed impact parameter approaches the width of the entire multiplicity distribution. In plotting \({{\hat{\gamma }}}\) for pPb collisions in Fig. 8, we had assumed already the limiting case that this correlation is negligible so that events of different multiplicity (classified experimentally as a function of \(N_{\mathrm{part}}\)) correspond to systems of initially similar transverse width *R*.

In reconstructing eccentricity according to (41) from the 5.02 TeV pPb data discussed in Sect. 5.2, Fig. 16 gives independent support to this assumption of a multiplicity-independent initial geometry of the active area in pPb collisions. While the value of the measured eccentricity \(\epsilon _{2}^{\mathrm{pPb}}\) depends on the assumed value of \(\eta /s\), the qualitative feature that it does not depend on the final state multiplicity remains unchanged for all values of \(\eta /s\). This suggests that the multiplicity in a pPb collision is not determined by geometric overlap but rather by fluctuations which are independent of the geometrical fluctuations leading to the eccentricity.

### 5.6 Comment on measures of system size in small systems

^{1}

*R*and on \(\frac{dN}{d\eta _s}\) are characteristically different from the ones obtained for the kinetic theory studied here, see Eq. (47), where we find

It is important to understand whether conformal scaling (51) of anisotropic flow with multiplicity can be established experimentally in the smallest systems. Whether or not corrections to this scaling could be established, would inform us about important elements of the microscopic dynamics underlying collectivity.

## 6 Conclusions

To ask what is the microscopic structure of quark–gluon plasma, is to ask how the plasma behaves away from the hydrodynamic limit. The study of the onset of signs of collectivity as a function of system size offers one setup where this question can be addressed. The effective description of the plasma in terms of fluid dynamics must eventually break, giving way to non-hydrodynamic modes to start dominating the dynamics, and the precise way how this happens reflects the microscopic structure of the plasma. Here, we have studied the consequences of assuming that the plasma has an underlying quasiparticle description. We have chosen this as our starting point because it is consistent with free-streaming particles in small systems.

Many microscopic models may appear to be consistent with the data on small systems and given the large unknowns in the initial condition of the collision it may be difficult to discriminate amongst them. One qualitative feature that quasiparticle models have in common is that the location of the hydrodynamic pole (the value of \(\eta /s\)) determines also the location of the non-hydrodynamic sector in the complex frequency plane. While the details of this correspondence may vary between different specific quasiparticle models, the scale at which non-hydrodynamic structures become dominant \(\Delta R\) is given by the mean free path \(\Delta R \sim 1/(\gamma \varepsilon ^{1/4})\) which also determines the specific shear viscosity \(\eta /s \sim 1/\gamma \). Therefore the knowledge of transport properties of the plasma also uniquely determines how the free-streaming behaviour is reached in systems that have transverse sizes smaller than the mean free path. In Sect. 5 we have seen that this prediction is not in contradiction with the available data.

Based on this exercise, to what extent can we then conclude that the plasma does have a quasiparticle description? In quantum field theory at weak coupling, there are additional structures in the complex plane to the hydrodynamic poles and the quasiparticle cut. These appear at the scale of the first Matsubara mode \({\mathrm{Im}}\left[ {\omega }\right] \approx - 4 \pi T\). They reflect the quantum mechanical nature and the de Broglie wavelength of the quasiparticles. When the mean free path becomes \(\sim 1/T \sim 1/\varepsilon ^{1/4}\), the quantum field theory becomes strongly coupled and there is no clear separation between the quasiparticle cut and the quantum mechanical Matsubara modes. We have no firm knowledge about what happens in quantum field theory at these couplings, but we do know that in the limit of infinite coupling the nonhydrodynamics structures still lie at the scale of \({\mathrm{Im}}\left[ {\omega }\right] \sim - T\). Therefore one could expect that in a model that goes beyond quasi-particles, the scale at which non-hydrodynamical modes appear saturates such that \(\Delta R \sim \min (\frac{1}{\gamma \varepsilon ^{1/4}}, \frac{1}{\varepsilon ^{1/4}})\). This is qualitatively in contrast to the quasiparticle models for which \(\Delta R \sim \frac{1}{\gamma \varepsilon ^{1/4}}\). It may be that such models also describe the data well, but given the good agreement with the data to the quasiparticle model, we must acknowledge that we do not have evidence that the quark–gluon plasma does not have quasiparticle excitations.

## Footnotes

## References

- 1.K. Aamodt et al., [ALICE Collaboration], Phys. Rev. Lett.
**107**, 032301 (2011). arXiv:1105.3865 [nucl-ex] - 2.S. Acharya et al., [ALICE Collaboration], JHEP
**1807**, 103 (2018). https://doi.org/10.1007/JHEP07(2018)103. arXiv:1804.02944 [nucl-ex] - 3.V. Khachatryan et al. [CMS Collaboration], Phys. Rev. Lett.
**115**(2015) no.1, 012301. arXiv:1502.05382 [nucl-ex] - 4.A. M. Sirunyan et al. [CMS Collaboration], Phys. Rev. Lett.
**120**(2018) no.9, 092301 https://doi.org/10.1103/PhysRevLett.120.092301. arXiv:1709.09189 [nucl-ex] - 5.B. B. Abelev et al. [ALICE Collaboration], Phys. Rev. C
**90**(2014) no.5, 054901. arXiv:1406.2474 [nucl-ex] - 6.M. Aaboud et al. [ATLAS Collaboration], Eur. Phys. J. C
**77**(2017) no.6, 428. arXiv:1705.04176 [hep-ex] - 7.B. B. Back et al. [PHOBOS Collaboration], Phys. Rev. C
**72**(2005) 051901 https://doi.org/10.1103/PhysRevC.72.051901. arXiv:nucl-ex/0407012 - 8.A. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett.
**114**(2015) no.19, 192301. arXiv:1404.7461 [nucl-ex] - 9.L. Adamczyk et al., [STAR Collaboration], Phys. Lett. B
**747**, 265 (2015). arXiv:1502.07652 [nucl-ex] - 10.C. Alt et al. [NA49 Collaboration], Phys. Rev. C
**68**(2003) 034903 https://doi.org/10.1103/PhysRevC.68.034903. arXiv:nucl-ex/0303001 - 11.D. A. Teaney. arXiv:0905.2433 [nucl-th]
- 12.U. Heinz, R. Snellings, Ann. Rev. Nucl. Part. Sci.
**63**, 123 (2013). https://doi.org/10.1146/annurev-nucl-102212-170540. arXiv:1301.2826 [nucl-th]ADSCrossRefGoogle Scholar - 13.D.T. Son, A.O. Starinets, JHEP
**0209**, 042 (2002). https://doi.org/10.1088/1126-6708/2002/09/042. arXiv:hep-th/0205051 ADSCrossRefGoogle Scholar - 14.A.O. Starinets, Phys. Rev. D
**66**, 124013 (2002). https://doi.org/10.1103/PhysRevD.66.124013. arXiv:hep-th/0207133 ADSMathSciNetCrossRefGoogle Scholar - 15.S.A. Hartnoll, S.P. Kumar, JHEP
**0512**, 036 (2005). https://doi.org/10.1088/1126-6708/2005/12/036. arXiv:hep-th/0508092 ADSCrossRefGoogle Scholar - 16.S. Grozdanov, N. Kaplis, A.O. Starinets, JHEP
**1607**, 151 (2016). https://doi.org/10.1007/JHEP07(2016)151. arXiv:1605.02173 [hep-th]ADSCrossRefGoogle Scholar - 17.S. Grozdanov, A.O. Starinets, JHEP
**1904**, 080 (2019). https://doi.org/10.1007/JHEP04(2019)080. arXiv:1812.09288 [hep-th]ADSCrossRefGoogle Scholar - 18.S. Grozdanov, A.O. Starinets, JHEP
**1703**, 166 (2017). https://doi.org/10.1007/JHEP03(2017)166. arXiv:1611.07053 [hep-th]ADSCrossRefGoogle Scholar - 19.P. Romatschke, Eur. Phys. J. C
**76**(6), 352 (2016). https://doi.org/10.1140/epjc/s10052-016-4169-7. arXiv:1512.02641 [hep-th]ADSCrossRefGoogle Scholar - 20.A. Kurkela, U.A. Wiedemann. arXiv:1712.04376 [hep-ph]
- 21.W. Israel, J.M. Stewart, Ann. Phys.
**118**, 341 (1979). https://doi.org/10.1016/0003-4916(79)90130-1 ADSCrossRefGoogle Scholar - 22.G. Baym, Phys. Lett.
**138B**, 18 (1984). https://doi.org/10.1016/0370-2693(84)91863-X ADSCrossRefGoogle Scholar - 23.M. Gyulassy, Y. Pang, B. Zhang, Nucl. Phys. A
**626**, 999 (1997). https://doi.org/10.1016/S0375-9474(97)00604-0. arXiv:nucl-th/9709025 ADSCrossRefGoogle Scholar - 24.B. Zhang, M. Gyulassy, Y. Pang, Phys. Rev. C
**58**, 1175 (1998). https://doi.org/10.1103/PhysRevC.58.1175. arXiv:nucl-th/9801037 ADSCrossRefGoogle Scholar - 25.B. Zhang, M. Gyulassy, C.M. Ko, Phys. Lett. B
**455**, 45 (1999)ADSCrossRefGoogle Scholar - 26.D. Molnar and M. Gyulassy, Nucl. Phys. A
**697**(2002) 495 Erratum: [Nucl. Phys. A**703**(2002) 893] https://doi.org/10.1016/S0375-9474(01)01224-6, https://doi.org/10.1016/S0375-9474(02)00859-X. arXiv:nucl-th/0104073 - 27.D. Molnar, M. Gyulassy, Phys. Rev. C
**62**, 054907 (2000). https://doi.org/10.1103/PhysRevC.62.054907. arXiv:nucl-th/0005051 ADSCrossRefGoogle Scholar - 28.
- 29.Z. Xu, C. Greiner, Phys. Rev. C
**71**, 064901 (2005). https://doi.org/10.1103/PhysRevC.71.064901. arXiv:hep-ph/0406278 ADSCrossRefGoogle Scholar - 30.J .D. Orjuela Koop, A. Adare, D. McGlinchey, J.L. Nagle, Phys. Rev. C
**92**(5), 054903 (2015). arXiv:1501.06880 [nucl-ex]ADSCrossRefGoogle Scholar - 31.J. Uphoff, F. Senzel, O. Fochler, C. Wesp, Z. Xu, C. Greiner, Phys. Rev. Lett.
**114**(11), 112301 (2015). arXiv:1401.1364 [hep-ph]ADSCrossRefGoogle Scholar - 32.L. He, T. Edmonds, Z.W. Lin, F. Liu, D. Molnar, F. Wang, Phys. Lett. B
**753**, 506 (2016). arXiv:1502.05572 [nucl-th]ADSCrossRefGoogle Scholar - 33.M. Greif, C. Greiner, B. Schenke, S. Schlichting, Z. Xu, Phys. Rev. D
**96**(9), 091504 (2017). arXiv:1708.02076 [hep-ph]ADSCrossRefGoogle Scholar - 34.P. Huovinen, D. Molnar, Phys. Rev. C
**79**, 014906 (2009). https://doi.org/10.1103/PhysRevC.79.014906. arXiv:0808.0953 [nucl-th]ADSCrossRefGoogle Scholar - 35.A. El, Z. Xu, C. Greiner, Phys. Rev. C
**81**, 041901 (2010). https://doi.org/10.1103/PhysRevC.81.041901. arXiv:0907.4500 [hep-ph]ADSCrossRefGoogle Scholar - 36.K. Gallmeister, H. Niemi, C. Greiner, D.H. Rischke, Phys. Rev. C
**98**(2), 024912 (2018). https://doi.org/10.1103/PhysRevC.98.024912. arXiv:1804.09512 [nucl-th]ADSCrossRefGoogle Scholar - 37.H.J. Drescher, A. Dumitru, C. Gombeaud, J.Y. Ollitrault, Phys. Rev. C
**76**, 024905 (2007). https://doi.org/10.1103/PhysRevC.76.024905. arXiv:0704.3553 [nucl-th]ADSCrossRefGoogle Scholar - 38.G. Ferini, M. Colonna, M. Di Toro, V. Greco, Phys. Lett. B
**670**, 325 (2009). https://doi.org/10.1016/j.physletb.2008.10.062. arXiv:0805.4814 [nucl-th]ADSCrossRefGoogle Scholar - 39.C. Gombeaud, J.Y. Ollitrault, Phys. Rev. C
**77**, 054904 (2008). https://doi.org/10.1103/PhysRevC.77.054904. arXiv:nucl-th/0702075 ADSCrossRefGoogle Scholar - 40.N. Borghini, C. Gombeaud, Eur. Phys. J. C
**71**, 1612 (2011). https://doi.org/10.1140/epjc/s10052-011-1612-7. arXiv:1012.0899 [nucl-th]ADSCrossRefGoogle Scholar - 41.P.P. Bhaduri, N. Borghini, A. Jaiswal, M. Strickland. arXiv:1809.06235 [hep-ph]
- 42.H. Sorge, H. Stoecker, W. Greiner, Nucl. Phys. A
**498**, 567C (1989). https://doi.org/10.1016/0375-9474(89)90641-6 ADSCrossRefGoogle Scholar - 43.M. Bleicher et al., J. Phys. G
**25**, 1859 (1999). https://doi.org/10.1088/0954-3899/25/9/308. arXiv:hep-ph/9909407 ADSCrossRefGoogle Scholar - 44.H. Petersen, D. Oliinychenko, M. Mayer, J. Staudenmaier, S. Ryu, Nucl. Phys. A
**982**, 399 (2019). https://doi.org/10.1016/j.nuclphysa.2018.08.008. arXiv:1808.06832 [nucl-th]ADSCrossRefGoogle Scholar - 45.P.B. Arnold, G.D. Moore, L.G. Yaffe, JHEP
**0301**, 030 (2003). https://doi.org/10.1088/1126-6708/2003/01/030. arXiv:hep-ph/0209353 ADSCrossRefGoogle Scholar - 46.P.B. Arnold, G.D. Moore, L.G. Yaffe, JHEP
**0305**, 051 (2003). https://doi.org/10.1088/1126-6708/2003/05/051. arXiv:hep-ph/0302165 ADSCrossRefGoogle Scholar - 47.S.C. Huot, S. Jeon, G.D. Moore, Phys. Rev. Lett.
**98**, 172303 (2007). https://doi.org/10.1103/PhysRevLett.98.172303. arXiv:hep-ph/0608062 ADSCrossRefGoogle Scholar - 48.M.A. York, G.D. Moore, Phys. Rev. D
**79**, 054011 (2009). https://doi.org/10.1103/PhysRevD.79.054011. arXiv:0811.0729 [hep-ph]ADSCrossRefGoogle Scholar - 49.J. Ghiglieri, G.D. Moore, D. Teaney, JHEP
**1803**, 179 (2018). https://doi.org/10.1007/JHEP03(2018)179. arXiv:1802.09535 [hep-ph]CrossRefGoogle Scholar - 50.J. Ghiglieri, G.D. Moore, D. Teaney, Phys. Rev. Lett.
**121**(5), 052302 (2018). https://doi.org/10.1103/PhysRevLett.121.052302. arXiv:1805.02663 [hep-ph]ADSCrossRefGoogle Scholar - 51.P.B. Arnold, C. Dogan, G.D. Moore, Phys. Rev. D
**74**, 085021 (2006). https://doi.org/10.1103/PhysRevD.74.085021. arXiv:hep-ph/0608012 ADSCrossRefGoogle Scholar - 52.F. Gelis, E. Iancu, J. Jalilian-Marian, R. Venugopalan, Ann. Rev. Nucl. Part. Sci.
**60**, 463 (2010). https://doi.org/10.1146/annurev.nucl.010909.083629. arXiv:1002.0333 [hep-ph]ADSCrossRefGoogle Scholar - 53.T. Lappi, Phys. Lett. B
**703**, 325 (2011). https://doi.org/10.1016/j.physletb.2011.08.011. arXiv:1105.5511 [hep-ph]ADSCrossRefGoogle Scholar - 54.R. Baier, A.H. Mueller, D. Schiff, D.T. Son, Phys. Lett. B
**502**, 51 (2001). https://doi.org/10.1016/S0370-2693(01)00191-5. arXiv:hep-ph/0009237 ADSCrossRefGoogle Scholar - 55.A. Kurkela, Y. Zhu, Phys. Rev. Lett.
**115**(18), 182301 (2015). arXiv:1506.06647 [hep-ph]ADSCrossRefGoogle Scholar - 56.A. Kurkela, Nucl. Phys. A
**956**, 136 (2016). https://doi.org/10.1016/j.nuclphysa.2016.01.069. arXiv:1601.03283 [hep-ph]ADSCrossRefGoogle Scholar - 57.A. Kurkela, A. Mazeliauskas, Phys. Rev. Lett.
**122**, 142301 (2019). https://doi.org/10.1103/PhysRevLett.122.191801. arXiv:1811.03040 [hep-ph]ADSCrossRefGoogle Scholar - 58.G. Beuf, M.P. Heller, R.A. Janik, R. Peschanski, JHEP
**0910**, 043 (2009). https://doi.org/10.1088/1126-6708/2009/10/043. arXiv:0906.4423 [hep-th]ADSCrossRefGoogle Scholar - 59.P.M. Chesler, L.G. Yaffe, Phys. Rev. Lett.
**106**, 021601 (2011). https://doi.org/10.1103/PhysRevLett.106.021601. arXiv:1011.3562 [hep-th]ADSCrossRefGoogle Scholar - 60.B. Wu, P. Romatschke, Int. J. Mod. Phys. C
**22**, 1317 (2011). https://doi.org/10.1142/S0129183111016920. arXiv:1108.3715 [hep-th]ADSCrossRefGoogle Scholar - 61.M.P. Heller, D. Mateos, W. van der Schee, M. Triana, JHEP
**1309**, 026 (2013). https://doi.org/10.1007/JHEP09(2013)026. arXiv:1304.5172 [hep-th]ADSCrossRefGoogle Scholar - 62.L. Keegan, A. Kurkela, P. Romatschke, W. van der Schee, Y. Zhu, JHEP
**1604**, 031 (2016). https://doi.org/10.1007/JHEP04(2016)031. arXiv:1512.05347 [hep-th]ADSCrossRefGoogle Scholar - 63.L. Keegan, A. Kurkela, A. Mazeliauskas, D. Teaney, JHEP
**1608**, 171 (2016). https://doi.org/10.1007/JHEP08(2016)171. arXiv:1605.04287 [hep-ph]ADSCrossRefGoogle Scholar - 64.A. Kurkela, A. Mazeliauskas, J.F. Paquet, S. Schlichting, D. Teaney, Phys. Rev. Lett.
**122**(12), 122302 (2019). https://doi.org/10.1103/PhysRevLett.122.122302. arXiv:1805.01604 [hep-ph]ADSCrossRefGoogle Scholar - 65.A. Kurkela, A. Mazeliauskas, J.F. Paquet, S. Schlichting, D. Teaney, Phys. Rev. C
**99**(3), 034910 (2019). https://doi.org/10.1103/PhysRevC.99.034910. arXiv:1805.00961 [hep-ph]ADSCrossRefGoogle Scholar - 66.M.P. Heller, A. Kurkela, M. Spaliński, V. Svensson, Phys. Rev. D
**97**(9), 091503 (2018). https://doi.org/10.1103/PhysRevD.97.091503. arXiv:1609.04803 [nucl-th]ADSCrossRefGoogle Scholar - 67.W. Florkowski, E. Maksymiuk, R. Ryblewski, Phys. Rev. C
**97**(2), 024915 (2018). https://doi.org/10.1103/PhysRevC.97.024915. arXiv:1710.07095 [hep-ph]ADSCrossRefGoogle Scholar - 68.M. Strickland, J. Noronha, G. Denicol, Phys. Rev. D
**97**(3), 036020 (2018). https://doi.org/10.1103/PhysRevD.97.036020. arXiv:1709.06644 [nucl-th]ADSMathSciNetCrossRefGoogle Scholar - 69.J.P. Blaizot, L. Yan, JHEP
**1711**, 161 (2017). https://doi.org/10.1007/JHEP11(2017)161. arXiv:1703.10694 [nucl-th]ADSCrossRefGoogle Scholar - 70.M.P. Heller, V. Svensson, Phys. Rev. D
**98**(5), 054016 (2018). https://doi.org/10.1103/PhysRevD.98.054016. arXiv:1802.08225 [nucl-th]ADSMathSciNetCrossRefGoogle Scholar - 71.J. Hong, D. Teaney, Phys. Rev. C
**82**, 044908 (2010). https://doi.org/10.1103/PhysRevC.82.044908. arXiv:1003.0699 [nucl-th]ADSCrossRefGoogle Scholar - 72.P. Romatschke, Eur. Phys. J. C
**78**(8), 636 (2018). https://doi.org/10.1140/epjc/s10052-018-6112-6. arXiv:1802.06804 [nucl-th]ADSCrossRefGoogle Scholar - 73.A. Kurkela, U.A. Wiedemann, B. Wu, Phys. Lett. B
**783**, 274 (2018). https://doi.org/10.1016/j.physletb.2018.06.064. arXiv:1803.02072 [hep-ph]ADSCrossRefGoogle Scholar - 74.A. Behtash, C.N. Cruz-Camacho, M. Martinez, Phys. Rev. D
**97**(4), 044041 (2018). https://doi.org/10.1103/PhysRevD.97.044041. arXiv:1711.01745 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 75.A. Kurkela, U.A. Wiedemann, B. Wu. arXiv:1805.04081 [hep-ph]
- 76.P.F. Kolb, U.W. Heinz, P. Huovinen, K.J. Eskola, K. Tuominen, Nucl. Phys. A
**696**, 197 (2001). https://doi.org/10.1016/S0375-9474(01)01114-9. arXiv:hep-ph/0103234 ADSCrossRefGoogle Scholar - 77.R. Baier, P. Romatschke, D.T. Son, A.O. Starinets, M.A. Stephanov, JHEP
**0804**, 100 (2008). https://doi.org/10.1088/1126-6708/2008/04/100. arXiv:0712.2451 [hep-th]ADSCrossRefGoogle Scholar - 78.C. Rampf, U. Frisch, Mon. Not. Roy. Astron. Soc.
**471**(1), 671 (2017). https://doi.org/10.1093/mnras/stx1613. arXiv:1705.08456 [astro-ph.CO]ADSCrossRefGoogle Scholar - 79.M. Pietroni, JCAP
**1806**(06), 028 (2018). https://doi.org/10.1088/1475-7516/2018/06/028. arXiv:1804.09140 [astro-ph.CO]ADSMathSciNetCrossRefGoogle Scholar - 80.A. Bazavov et al., [HotQCD Collaboration], Phys. Rev. D
**90**, 094503 (2014). https://doi.org/10.1103/PhysRevD.90.094503. arXiv:1407.6387 [hep-lat] - 81.S. Borsanyi, Z. Fodor, C. Hoelbling, S.D. Katz, S. Krieg, K.K. Szabo, Phys. Lett. B
**730**, 99 (2014). https://doi.org/10.1016/j.physletb.2014.01.007. arXiv:1309.5258 [hep-lat]ADSCrossRefGoogle Scholar - 82.D. Kharzeev, M. Nardi, Phys. Lett. B
**507**, 121 (2001). https://doi.org/10.1016/S0370-2693(01)00457-9. arXiv:nucl-th/0012025 ADSCrossRefGoogle Scholar - 83.S. Acharya et al., [ALICE Collaboration], JHEP
**1811**, 013 (2018). https://doi.org/10.1007/JHEP11(2018)013. arXiv:1802.09145 [nucl-ex] - 84.J. Adams et al. [STAR Collaboration], Phys. Rev. C
**70**(2004) 054907 https://doi.org/10.1103/PhysRevC.70.054907. arXiv:nucl-ex/0407003 - 85.A. M. Sirunyan et al. [CMS Collaboration]. arXiv:1810.05745 [hep-ex]
- 86.S. Chatrchyan et al., [CMS Collaboration], Phys. Lett. B
**724**, 213 (2013). https://doi.org/10.1016/j.physletb.2013.06.028. arXiv:1305.0609 [nucl-ex]ADSCrossRefGoogle Scholar - 87.J. Adam et al. [ALICE Collaboration], Phys. Rev. C
**91**(2015) no.6, 064905 https://doi.org/10.1103/PhysRevC.91.064905. arXiv:1412.6828 [nucl-ex] - 88.B.I. Abelev et al., [STAR Collaboration], Phys. Rev. C
**77**, 054901 (2008). https://doi.org/10.1103/PhysRevC.77.054901. arXiv:0801.3466 [nucl-ex] - 89.J. Adams et al. [STAR Collaboration], Phys. Rev. Lett.
**91**(2003) 172302 https://doi.org/10.1103/PhysRevLett.91.172302. arXiv:nucl-ex/0305015 - 90.H. Niemi, K.J. Eskola, R. Paatelainen, Phys. Rev. C
**93**(2), 024907 (2016) https://doi.org/10.1103/PhysRevC.93.024907. arXiv:1505.02677 [hep-ph] - 91.J. Noronha-Hostler, L. Yan, F.G. Gardim, J.Y. Ollitrault, Phys. Rev. C
**93**(1), 014909 (2016). https://doi.org/10.1103/PhysRevC.93.014909. arXiv:1511.03896 [nucl-th]ADSCrossRefGoogle Scholar - 92.G. Giacalone, P. Guerrero-Rodríguez, M. Luzum, C. Marquet, J. Y. Ollitrault. arXiv:1902.07168 [nucl-th]
- 93.J.S. Moreland, J.E. Bernhard, S.A. Bass, Phys. Rev. C
**92**(1), 011901 (2015). https://doi.org/10.1103/PhysRevC.92.011901. arXiv:1412.4708 [nucl-th]ADSCrossRefGoogle Scholar - 94.H. Niemi, K.J. Eskola, R. Paatelainen, K. Tuominen, Phys. Rev. C
**93**(1), 014912 (2016). https://doi.org/10.1103/PhysRevC.93.014912. arXiv:1511.04296 [hep-ph]ADSCrossRefGoogle Scholar - 95.J. E. Bernhard, J. S. Moreland, S. A. Bass, J. Liu and U. Heinz, Phys. Rev. C
**94**(2016) no.2, 024907 https://doi.org/10.1103/PhysRevC.94.024907. arXiv:1605.03954 [nucl-th] - 96.P. Bozek, Phys. Rev. C
**85**, 014911 (2012). https://doi.org/10.1103/PhysRevC.85.014911. arXiv:1112.0915 [hep-ph]ADSCrossRefGoogle Scholar - 97.P. Bozek, W. Broniowski, Phys. Rev. C
**88**(1), 014903 (2013). https://doi.org/10.1103/PhysRevC.88.014903. arXiv:1304.3044 [nucl-th]ADSCrossRefGoogle Scholar - 98.M. Gyulassy, L. McLerran, Nucl. Phys. A
**750**, 30 (2005). https://doi.org/10.1016/j.nuclphysa.2004.10.034. arXiv:nucl-th/0405013 ADSCrossRefGoogle Scholar - 99.H. Heiselberg, A.M. Levy, Phys. Rev. C
**59**, 2716 (1999). https://doi.org/10.1103/PhysRevC.59.2716. arXiv:nucl-th/9812034 ADSCrossRefGoogle Scholar - 100.S.A. Voloshin, A.M. Poskanzer, Phys. Lett. B
**474**, 27 (2000). https://doi.org/10.1016/S0370-2693(00)00017-4. arXiv:nucl-th/9906075 ADSCrossRefGoogle Scholar

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