# Azimuthal anisotropies at high momentum from purely non-hydrodynamic transport

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## Abstract

In the limit of short mean free path, relativistic kinetic theory gives rise to hydrodynamics through a systematically improvable gradient expansion. In the present work, a systematically improvable expansion in the opposite limit of large mean free path is considered, describing the dynamics of particles which are almost, but not quite, non-interacting. This non-hydrodynamic “eremitic” expansion does not break down for large gradients, and may be useful in situations where a hydrodynamic treatment is not applicable. As applications, azimuthal anisotropies at high transverse momenta in Pb + Pb and p + Pb collisions at \(\sqrt{s}=5.02\) TeV are calculated from the first order eremitic expansion of kinetic theory in the relaxation time approximation.

## 1 Introduction

Is there a simple description for transport when hydrodynamics fails?

At low momenta, relativistic hydrodynamics has been tremendously successful in offering quantitative descriptions and predictions of experimental data from high energy nuclear collisions (see Ref. [1, 2, 3, 4] for recent reviews.) Hydrodynamics breaks down when non-hydrodynamic modes start to dominate over hydrodynamic modes, and it has been suggested that the experimentally observed peak in azimuthal anisotropies at transverse momenta \(p_T\simeq 4\) GeV indicates the transition from hydrodynamic to non-hydrodynamic transport [5]. For conformal field theories at weak (strong) coupling, this transition happens at a well-defined momentum scale \(k_c\) [6, 7]. For momenta above \(k_c\), the lifetimes of hydrodynamic modes are shorter than those from non-hydrodynamic modes, hence at late times bulk transport will be dominated by purely non-hydrodynamic degrees of freedom. While many studies exist that include both hydrodynamic and non-hydrodynamic modes, little is known about the phenomenological implications and observational consequences of *pure* non-hydrodynamic transport where all hydrodynamic mode contributions have been turned off. The present work is meant as a step in this direction by studying non-hydrodynamic transport for the case of relativistic kinetic theory.

The kinetic theory of classical gases has a long history [8], yet active research on its properties is still ongoing. Recent examples include the divergence of the gradient expansion of kinetic theory [9, 10], its non-perturbative resummation leading to hydrodynamic attractors [11, 12, 13, 14, 15, 16, 17], the characterization of the non-hydrodynamic modes in kinetic theory [6, 18] and the “Lattice Boltzmann Approach” which uses kinetic theory as an efficient algorithm to simulate fluid dynamics [19, 20, 21].

For vanishing mean free path, kinetic theory corresponds to ideal (non-viscous) fluid dynamics that is described by the Euler equation [22]. Small, but non-vanishing mean-free path corrections give rise to viscous fluid dynamics, described by the equations of Navier and Stokes [23, 24]. Higher order corrections to the small mean free path regime can be systematically calculated [25]. The opposite limit of large mean free path is known as rarefied gas dynamics or high Knudsen number regime [26, 27, 28], and in the extreme case of infinite mean free path gives rise to non-interacting (or free-streaming) particle dynamics. For infinite mean free path, the classic kinetic equations can be solved analytically using the method of characteristics, leading to ballistic evolution. In a sense, ballistic evolution and ideal fluid dynamics are analogues of each other, corresponding to opposite extreme limits of infinite and zero mean free path, respectively. However, while the systematic small mean free path expansion has been recognized to lead to viscous fluid dynamics, the equivalent systematic expansion at large but finite mean free path seems to have received less attention in the high energy physics literature, except for two works [29, 30]. The present work is meant to consider the phenomenological consequences arising from such a systematic expansion, extending in particular the pioneering work by Borghini and Gombeaud [30] to the case of large momenta. For large mean free path, particles rarely interact, similar to hermit crabs in their natural environment, hence this systematic expansion will be referred to as “eremitic” expansion in the following.

## 2 Setup

*t*. Because the particles are massless, their dynamics will be governed by relativistic kinetic theory, although it should be straightforward to modify the discussion for massive particles with non-relativistic dynamics. The relativistic Boltzmann equation is given by

*f*. In order to give a more hands-on treatment, it will be useful to consider a concrete and simple example for the collision kernel, such as the relaxation time (or BGK [31]) approximation where

*f*. For this work, the mostly plus metric convention \(g_{\mu \nu }=\mathrm{diag}(-,+,+,+)\) will be used such that \(p^\mu u_\mu =-p^0 u^0+\mathbf{p}\cdot \mathbf{u}\).

*T*in a local rest frame given by the four vector \(u^\mu \) in some global coordinate system, then the equilibrium distribution function for classical particles can be taken as

*T*as

*T*) from the energy density as \(T=\epsilon ^{1/4}\) (cf. the discussion in Ref. [2]). This pseudo-temperature, together with the time-like eigenvector \(u^\mu \) of \(T^{\mu \nu }\), are used to define the pseudo-equilibrium distribution function \(f_\mathrm{eq}[f]\) via Eq. (3), where the functional dependence on the non-equilibrium particle distribution

*f*has been denoted explicitly.

### 2.1 Review of small mean free path expansion

Note that in the small mean free path expansion, the relevant expansion parameter is \(\tau _R\) times a typical gradient strength, cf. Eq. (10). This implies that the expansion fails for large gradients (see the discussion in Ref. [6]).

### 2.2 Zeroth order eremitic expansion: ballistic regime

### 2.3 First order eremitic expansion

### 2.4 Collective modes

Because hydrodynamic and eremitic expansions are opposite limits of kinetic theory (1), their collective mode structure can also be expected to be different. The collective modes of Eq. (1) in the relaxation time approximation have been analyzed in Ref. [6] for constant \(\tau _R T\) (see Ref. [18] for the case of momentum dependent relaxation time), and results are summarized here in order to keep this work self-contained.

^{1}. The hydrodynamic poles approach the fluid dynamic results (17) in the limit of small \(\tau _R |\mathbf{k}|\), as they should. The branch points approach the eremitic results (18) in the limit of large \(\tau _R |\mathbf{k}|\), as they should. The situation is summarized in Fig. 1, which depicts the singularity structure of the sound channel two-point function in the complex frequency plane.

It is common to refer to the collective modes corresponding to the sound poles \(\omega _\mathrm{hydro}\) as hydrodynamic modes, and label the modes corresponding to the branch cuts \(\omega _\mathrm{cut}\) as non-hydrodynamic modes. Thus, the fluid dynamic expansion of kinetic theory contains only hydrodynamic modes, the eremitic expansion contains only non-hydrodynamic modes, while kinetic theory without any expansion contains both.

## 3 Analytic examples

*F*and \(\Lambda \) arbitrary functions. Within this class of examples, the zeroth order eremitic expansion leads to

### 3.1 Single Gaussian hot-spot

*T*and flow vector \(u^\mu \) corresponding to this energy momentum tensor can be obtained via eigenvalue decomposition, cf. Eq. (5). One finds

*t*, the eigenvalue decomposition of \(T^{\mu \nu }_\mathrm{hermit, (0)}\) leads to

^{2}of Eq. 32, confirming the accuracy of the approximation for \(t \ll \sigma \). As comparison, I also show the corresponding evolution of

^{3}

### 3.2 Two Gaussian hot-spots

The present example is very similar to the case of a deformed Gaussian in two dimensions considered in Ref. [30], where many analytic results were obtained. The main difference to Ref. [30] is that by superposition, the above calculation can be generalized to the case of multiple Gaussian hot-spots without additional complications.

## 4 Numerical examples with boost invariance

*Y*and used shorthand notation \(v^a=p^a/p_T=\left( \cosh (Y-\xi ),\mathbf{v}_T,\tau ^{-1}\sinh (Y-\xi )\right) \).

### 4.1 Single Gaussian hot-spot with boost invariance

*Z*parametrizing the number of degrees of freedom, and setting \(\Lambda (\mathbf{x}_T)=T_\mathrm{init}e^{-\mathbf{x}_T^2/(8 \sigma ^2)}\) to be a two-dimensional Gaussian leads to simple integral expressions for the non-vanishing components of the energy momentum tensor:

### 4.2 Two Gaussian hot-spots with boost invariance

^{4}

^{5}.

### 4.3 High energy \(\mathrm{Pb}\)+\(\mathrm Pb\) collisions

The same techniques as for the two hot-spot case may be used to model high energy nuclear collisions, such as Pb+Pb collisions at \(\sqrt{s}=5.02\) TeV. This is because the so-called Glauber model [44, 45] provides initial conditions for the matter distribution deposited after the collision as a sum over Gaussian hot-spots corresponding to the locations of collisions of the individual nucleons (see Ref. [2] for a recent review of relativistic nuclear collision modeling). For the purpose of this work, hot-spot locations are generated by first Monte-Carlo sampling nucleon positions for two lead nuclei from a suitably normalized Woods–Saxon probability distribution function \(\rho (\mathbf{x})\propto (1+e^{(|\mathbf{x}|-r_0)/a_0})^{-1}\) with \(r_0=6.62\) fm and \(a_0=0.546\) fm. Random sampling of impact parameters for the collision of two lead nuclei, nucleons are said to undergo a collision if their respective distance in the transverse \(\mathbf{x}_T\) plane is less than \(|\mathbf{x}_T|<\sqrt{\sigma _{NN}/\pi }\), where \(\sigma _{NN}\simeq 60\) mb is the (collision-energy dependent) nucleon–nucleon cross-section at \(\sqrt{s}= 5.02\) TeV. Each location of a collision is taken to correspond to the location of one Gaussian hot-spot. The sum over these Gaussian hot-spots defines the initial energy-density distribution in the transverse plane, which is successfully used in modern hydrodynamic modeling of lead–lead collisions [2]. The number of nucleons participating in a collision is related to the total entropy of the system, which in turn translates to the number of particles observed in experiment (“multiplicity”). In the following, I will consider mid-central lead-lead collisions corresponding to the 30–40 % highest multiplicity class. For this case, the parameter \(T_\mathrm{init}\) was adjusted such that the hydrodynamic evolution with a QCD equation of state [46] gives multiplicities that are consistent with those found in experiment [47]. Unlike the two hot-spot case treated above, for multiple hot-spots found in the Glauber modeling of Pb + Pb collisions the momentum flow coefficients \(v_n\) for \(n=3,4,\ldots \) are also non-vanishing in general.

^{6}at low momenta \(p_T\le 2\) GeV can be connected to the eremitic curves at high momenta \(p_T\ge 15\) GeV by a type of Padé fit, suggesting a peak in \(\langle v_n(\tau _f,p_T)\rangle \) for specific values of \(p_T\) for \(n=2,3,4\). Note that the available information at low and high momenta, respectively, is not sufficient to unambiguously determine the location or height of the peaks in \(\langle v_n(\tau _f,p_T)\rangle \).

Since the results shown for \(\langle v_n(\tau _f,p_T)\rangle \) are for massless partons obtained when the whole system has cooled down below a pre-defined temperature, the results are not directly comparable to experimental data. However, it is tempting to inspect the relevant experimental data on differential flow coefficients for 30–40% Pb + Pb collisions for unidentified hadrons, shown in the rhs panel of Fig. 4. Interestingly, the experimental data seems to exhibit the qualitative features of the above theoretical calculations at low momenta (rise with \(p_T\) as predicted by hydrodynamic expansions) and high momenta (decrease with \(p_T\) as predicted by eremitic expansions). Curiously, also the magnitude of experimentally measured \(v_n\) coefficients at \(p_T\lesssim 2\) GeV and \(p_T \gtrsim 15\) GeV seem to be consistent with theoretical calculations shown in the lhs panel of Fig. 4. Furthermore, note that the ratio \(\frac{\langle v_3(\tau _f,p_\perp )\rangle }{(\langle v_2(\tau _f,p_\perp )\rangle )^{3/2}}\simeq 1\) exhibits near-constant behavior close to unity as a funtion of \(p_T\) at large momenta in eremitic expansion, similar to what has been observed experimentally [55].

## 5 High energy *p*+\(\mathrm Pb\) collisions

One of the unresolved questions in the context of high energy nuclear collision is the mechanism for the measured sizable \(v_2\) coefficient at high transverse momenta \(p_T\gtrsim 10\) GeV, cf. Fig. 4. It has been suggested that the measured \(v_2\) coefficient arises from jet quenching, with highly energetic particles (jets) losing more energy when traveling through a longer path length in a medium [57, 58]. However, jet quenching seems to be absent in proton-lead collisions, yet the experimentally measured \(v_2\) coefficient exhibits the same behavior as in lead-lead collisions [3], cf. Fig. 5. Eremitic expansions offer a potential alternative explanation for the observed \(v_2\) coefficient, namely through non-hydrodynamic transport of the initial geometry. While the momentum anisotropies in eremitic expansions arise from the dynamics of high energy particles, these particles are nevertheless part of, and flowing with, the medium, as opposed to the modeling of jets, which are by definition treated separately from the medium.

For this reason, I have simulated central p + Pb collisions through Monte-Carlo sampling positions of nucleon collisions from a Glauber model, and using these positions as the initial location of Gaussian hot-spots as explained in the preceding sections. The dynamics encountered in p + Pb is not boost-invariant, but hydrodynamic simulations seem to indicate that nevertheless boost-invariance is not a bad quantitative approximation in practice [59, 60, 61, 62]. The results for the momentum anisotropies \(\langle v_n(\tau _f,p_T)\rangle \) averaged over 10 events for zeroth order hydrodynamic and first order eremitic expansion are shown in Fig. 5. One finds that the same qualitative features as in Pb+Pb emerge: rising \(v_n\) coefficients at low \(p_T\) as predicted by hydrodynamics, and falling \(v_n\) coefficients at large \(p_T\) as predicted by eremitic expansions. Unlike the case for Pb + Pb collisions, the magnitude for \(\langle v_2\rangle \) at large \(p_T\) for massless partons from eremitic expansions of p + Pb collisions, while non-vanishing, is systematically below the experimentally measured values for unidentified hadrons (rhs panel of Fig. 5). Future studies involving more realistic equations of state and a confinement prescription will be needed in order to decide if eremitic expansion qualify as explanation for the observed \(v_2\) coefficient at high momenta.

## 6 Summary and conclusions

In this work, a systematic expansion procedure for relativistic kinetic theory in the large mean-free path regime was considered. This eremitic expansion procedure is complementary to the perhaps more familiar hydrodynamic expansion scheme in that it allows controlled calculations at very high particle momenta, while breaking down at low particle momenta. Eremitic expansions allow to probe purely non-hydrodynamic transport phenomena since hydrodynamic modes are absent in this approach. Using kinetic theory in the relaxation time approximation as an example, first order eremitic expansions for Gaussian hot-spots with and without boost invariance were calculated. Applications for these calculations to evaluating the momentum anisotropy coefficients \(v_n(\tau _f,p_T)\) in Pb + Pb and p + Pb collisions at \(\sqrt{s}=5.02\) TeV were presented, and it was found that eremitic expansions qualitatively describe the experimentally measured behavior of flow coefficients at high momenta. Thus, eremitic expansions offer a potential alternative to jet quenching as the source for the measured elliptic anisotropy at high momenta.

Many generalizations and validations of the present work are possible. For instance, the second order correction to eremitic expansions should be straightforward to calculate for many of the examples given in this work. The quantitative reliability of eremitic expansions should be checked by direct comparison to full numerical solutions of the Boltzmann equation. The application of eremitic expansions to relativistic collision systems should be made more realistic by including a QCD equation of state and a hadronization procedure.

Nevertheless, eremitic expansions seem to have the potential to become an interesting new tool in the study of relativistic collision systems and the phenomenology of non-hydrodynamic transport.

## Footnotes

- 1.
For a common choice of the location of the logarithmic branch cut, the hydrodynamic poles move through the cut onto the next Riemann sheet for \(k \tau _R\gtrsim 4.5313912\dots \equiv k_c \tau _R\) and thus are no longer present on the fundamental Riemann sheet [6]. However, this behavior is not generic because other choices of the logarithmic cut location may be employed [18]. What is generic, however, is that for \(k>k_c\) the hydrodynamic poles are farther away from the real axis then the non-hydrodynamic branch cut, implying late-time transport to be dominated by non-hydrodynamic degrees of freedom.

- 2.
Numerical algorithms employed for this work are publicly available for download from [35].

- 3.
- 4.
The exact definition would imply an integration over space-time rapidity \(\int d\xi \) in both numerator and denominator. However, since

*f*is strongly peaked around \(\xi =Y\), the approximation \(\xi =Y\) should be reasonably accurate. - 5.
The hydrodynamic results have been calculated numerically using the same initial conditions and same equation of state using the numerical solver VH2+1 [41] for the hydrodynamic equations.

- 6.

## Notes

### Acknowledgements

This work was supported in part by the Department of Energy, DOE award No DE-SC0017905. I would like to thank Jamie Nagle, Nicolas Borghini, Ulrike Romatschke and Jürgen Schukraft for helpful comments, and Gowri Sundaresan for collaboration in the early stages of this work.

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