# Analytic structure of nonhydrodynamic modes in kinetic theory

## Abstract

How physical systems approach hydrodynamic behavior is governed by the decay of nonhydrodynamic modes. Here, we start from a relativistic kinetic theory that encodes relaxation mechanisms governed by different timescales thus sharing essential features of generic weakly coupled nonequilibrium systems. By analytically solving for the retarded correlation functions, we clarify how branch cuts arise generically from noncollective particle excitations, how they interface with poles arising from collective hydrodynamic excitations, and to what extent the appearance of poles remains at best an ambiguous signature for the onset of fluid dynamic behavior. We observe that processes that are slower than the hydrodynamic relaxation timescale can make a system that has already reached fluid dynamic behavior to fall out of hydrodynamics at late times. In addition, the analytical control over this model allows us to explicitly demonstrate how the hydrodynamic gradient expansion of the correlation function can be resummed such that the complete and exact non-analytic form of the correlation function can be recovered.

## 1 Introduction

A broad range of physical phenomena is involved in how relativistic nonequilibrium systems reach thermal equilibrium. For near-equilibrium systems, these mechanisms are expected to leave characteristic traces in the analytic structure of the retarded correlation function of conserved quantities \(G_R(\omega , k)\). On the one hand, the prototypic longtime behavior of the correlation functions that describes collective excitations evolving towards global equilibrium is given by hydrodynamic poles, whose locations and residues are dictated by the fluid dynamical gradient expansion. On the other hand the question at which time scales hydrodynamic behavior emerges, and with which confounding mechanisms it may compete, is related to the existence and properties of other nonanalytic structures in the lower complex half plane of the correlators. These nonhydrodynamic modes have been seen to govern the approach to hydrodynamics – or hydrodynamization – not only in static but also in rapidly evolving backgrounds, used in the phenomenological description of heavy-ion collisions [1, 2, 3, 4, 5, 6]. While much of the recent work on nonhydrodynamic modes has focused on strongly coupled theories [7, 8, 9, 11, 12], the present study will deal with nonhydrodynamic modes in weakly coupled theories.

Additional motivations for studying nonhydrodynamic modes in relativistic equilibrating systems come from the apparent phenomenological need to understand how fluid dynamical behaviour arises in nucleus-nucleus, nucleus-nucleon and possibly proton-proton collisions [13, 14]. Some phenomenologically successful descriptions of these systems interface hydrodynamics with transport models (see e.g. [15, 16]), while others do not invoke hydrodynamics explicitly (see e.g. [17]). This asks for a better understanding of where and how kinetic theory differs from hydrodynamics. The standard way of relating kinetic theory to viscous hydrodynamics is to derive the latter by truncating the former to a finite set of moments of the distribution function [18, 19]. However, this truncation is based on the assumption that hydrodynamics works. To understand whether, when, and how it breaks down necessitates investigating kinetic theory beyond the moment expansion. The purpose of the present manuscript is to do so by studying how small deviations from thermal equilibrium relax in a full kinetic theory framework.

### 1.1 Analytic structure at strong and weak coupling

In known examples of strongly coupled systems at large \(N_c\), the remarkable simplicity of the microscopic structures of nonabelian plasmas is reflected in a remarkably simple analytic structure of the full field theoretic correlation functions. More specifically, in \({\mathcal {N}}=4\) SYM theory in the limit of large number of colors \(N_c \rightarrow \infty \) and strong coupling \(\lambda = g^2 N_c \rightarrow \infty \), the retarded correlation functions are known to exhibit an infinite set of nonhydrodynamical poles located (asymptotically for large *n*) at \(\omega _{n}^{\pm } = \omega ^{\pm }_0 \pm 2\pi n T (1 \mp i)\), with \(n \in [1,2,3,\ldots ]\), and \(\omega ^{\pm }_0/\pi T = \pm 1.2139 - 0.7775 i\) [7, 8, 9]. (This position of quasi-normal poles is the same for the stress energy tensor and for the scalar \({\mathrm{Tr F}}^{2}\) [8, 10].) In addition, in the channels where energy momentum conservation demands, the correlation functions exhibit poles whose locations and residues are dictated for small *k* by the hydrodynamic gradient expansion.

In weakly coupled theories, the analytic structure of retarded correlation functions is much richer. In these theories, there is a scale separation between the typical size of the wave packets 1 / *T* and the mean free path between the individual scatterings \(t_\mathrm{scat}\). Therefore for time separations larger than \(\Delta t \gg 1/T\), when interference effects can be neglected, the correlation function is determined by Boltzmann transport theory, in which the collision kernels are given by in-medium scattering processes in the field theory [20, 21, 22, 23]. The nonanalytic features of the full field theory that are absent in the transport theory are well known (see Sect. 2.1). However, the nonanalytic structures appearing in the transport theory are less well understood, and will be the topic of this contribution. In weakly coupled theories, transport theory has a wider regime of validity than hydrodynamics but encompasses it. Therefore, understanding these non-analytic structures provides a technically controlled in-road to understanding the onset of hydrodynamic behaviour in weakly coupled theories.

### 1.2 Kinetic theory in the relaxation time approximation

While there have been numerous numerical studies of the full collision kernel in nonabelian gauge theories [24, 25, 26, 27, 28, 29, 30, 31], including computations of equilibrium and nonequilibrium retarded correlation functions [32], the question of analytical structures has been addressed only recently [33] in the simplest possible model of the collision kernel – that of simple relaxation time \(\tau _R\). In this relaxation time approximation (RTA), an ostensibly crisp and simple picture of the onset of fluid dynamic behaviour appears by a migration of a hydrodynamic pole through a nonhydrodynamic cut for a specific value of Knudsen number \(K = k\, \tau _R \) where *k* is the wave number of the perturbation [33]. However, this simple model forgoes much of the structures of the collision kernel in favour of a single relaxation time. The question of whether this simple picture survives the inclusion of more realistic collision processes is the starting point of this paper.

The main result of the present paper is to establish the analytic structure of the retarded correlators of the energy-momentum tensor for the model (1). This result is sketched in Fig. 1 for the (analytically continued) shear channel correlation function obtained from the model. Causality and the stability of thermal equilibrium make the correlation function analytic in the upper complex half-plane, while the locality of the collision kernels in the Boltzmann equation allows one to write the correlation function as analytic for \(|{\mathrm{Re}\,}\omega | > k\). Of course, ambiguities in the analytic continuation of the physical correlation function allow one in principle in kinetic theory to deform branch cuts to the region \(|{\mathrm{Re}\,}\omega | > k\). Also, in quantum field theories, non-analytic structures can occur in the region \(|{\mathrm{Re}\,}\omega | > k\). In addition to the hydrodynamic pole, the model exhibits two nonhydrodynamic cuts whose branch points are located at \(\omega = \pm k\). For any *k*, the cuts extend to smaller imaginary parts than the hydrodynamic pole; it is these structures that are responsible for a nontrivial competition between hydrodynamics and nonhydrodynamic modes that we discuss in detail.

The paper is organized as follows: in Sect. 2, we first provide simple qualitative arguments for the physical mechanisms and corresponding analytic structures arising in full gauge theories. For the class of models (1), Sect. 3 derives then explicit expressions for the retarded correlation functions. For the case \(\xi = 1\), these correlation functions can be expressed in terms of one single, analytically known generating function *H* that largely determines the analytic structure of the correlation functions. A detailed discussion of this analytic structure, its physical meaning, and its ambiguities is the focus of Sect. 4, before we turn in Sect. 5 to a discussion of the physical response on pre-hydrodynamic, hydrodynamic and post-hydrodynamic time scales. As our study provides explicit analytic control over a model of significant physical complexity, it is also an interesting scholarly playground for understanding how Borel resummation techniques can be applied to the asymptotic hydrodynamic gradient expansion. This will be discussed in Sect. 6, before we conclude with a short summary of main results and open questions.

## 2 Generic analytic properties of retarded correlators and their physical origin

Before analyzing in detail the model (1) in subsequent sections, we discuss here generic features of the analytic structure of retarded correlation functions of the energy momentum tensor. In particular, we aim at providing physical intuition for the features appearing in kinetic theory.

### 2.1 Analyticity properties of retarded correlation functions in gauge theories at finite temperature

*p*. The vertices combine to a function

*V*which depends on the theory and the particular channel studied and is a function of momenta \(\vec p\) and \(\vec k\). For specific cases, see [9, 38] for gauge theories.

#### 2.1.1 Rapidly oscillating part *D*(*t*, *k*)

*p*. It can be seen that by choosing a suitable analytic continuation of

*D*in the lower complex half-plane, the cuts \(m + \sqrt{k^2 + m^2}< \pm \omega < \infty \) along the real axis can be exchanged into a series of cuts that are positioned deep in the negative imaginary region at (for \(m=0\)) \({\mathrm{Im}}\omega = -4 \pi n\, T\) and \(-k< {\mathrm{Re}} \, \omega < k \) with \(n \in [1,2,\ldots ]\), see figure 3 of Ref. [9]. As the nonanalytic structures in

*D*have a distance \(\mathcal{O}\left( T\right) \) from the real \(\omega \)-axis, the contribution

*D*decays on timescale 1 /

*T*, and it is insignificant at late times when fluid dynamic behaviour is expected to take place.

#### 2.1.2 The slowly oscillating part *C*(*t*, *k*) and kinetic theory

*C*arises from contributions that can be written in terms of expectation values of number operators. This suggests that for small

*k*, the physics contained in

*C*(

*t*,

*k*) can be captured by kinetic theory. In Fourier space,

*k*, this expression can be expanded to give

The free theory calculation recalled here and presented, e.g., in [9] is insufficient for \(\omega \sim 1/t_\mathrm{scat} \sim g^4\, T\), where interactions change the dynamics qualitatively. It therefore does not reveal the hydrodynamic pole which is close to the origin at \(\omega \sim g^4\, T\). To obtain even at leading order complete results in this region, a class of ladder diagrams needs to be resummed [21]. Such resummation can be dressed in the language of an effective kinetic theory [22, 23] of nearly massless quasiparticles, where the resummed diagrams appear in the particular scattering kernels of the kinetic equation. The effective kinetic theory is suitable for the computation of correlation functions of the quantum field theory with external momenta \(\omega , k \lesssim 1/t_{\mathrm{scat}}\), and therefore it is suitable for studying the vicinity of the slowly oscillating cut of *C* in more detail than the unresummed calculation. However, this resummation fails for larger (negative imaginary) values of \(\omega \) and does not capture the physics of cuts of *D*.

### 2.2 Analytic structure of retarded correlation functions in kinetic theory

#### 2.2.1 Massless kinetic theory without interaction

As sketched on the left hand side of Fig. 3, a sound channel perturbation in an equilibrium system may be viewed as embedding alternating sheets of overdense and underdense regions that are separated in the *z*-direction by a distance \(2\pi /k\). Analogous sketches can be given for perturbations in other channels. Computing the retarded response at time *t* amounts then to studying the state of the system at some arbitrary point \(\vec x\) which initially is on the peak of the overdense region at \(t=0\) when the perturbation is introduced.

*t*is then the average over a sphere of radius

*ct*. As the overdense regions are spaced \(2\pi /k\) apart, the particles moving in -

*z*direction will give rise to a signal oscillating with frequency \(\omega =k\). This corresponds to a pole at

*k*in the complex \(\omega \) plane. Particles coming from any other direction with velocity \(\vec v\) will result in an oscillating signal with smaller frequency \(\omega = \vec k \cdot \vec v\), corresponding to a pole at \(\vec k \cdot \vec v\) in the complex \(\omega \) plane. Integrating over all orientations \(\vec v\) from which particles reach the point \(\vec x\), one finds a string of poles between \(-k< \omega < k\) that assemble to a logarithmic cut

#### 2.2.2 Massless kinetic theory in the standard RTA

*k*, the location and residues of these poles are dictated by the hydrodynamic gradient expansion. We will call this pole in the following hydrodynamic pole (Fig. 4). For \(k \ge \pi /2 {t_R}\), the pole crosses the cut and enters the next Riemann sheet, thus disappearing from the physical plane. Therefore, the model has two distinct kinematic regimes: one where the pole is above the cut and the late time behaviour of the system is dictated by the hydrodynamic pole, and the other where the cut dominates the dynamics at all times. This was called the hydrodynamic onset transition in [33].

#### 2.2.3 Massless kinetic theory with scale-dependent RTA

*p*. We shall establish this picture in an explicit calculation in Sect. 4. It implies that the hydrodynamic pole is always embedded in the nonanalytic structure. The existence of a clear onset transition of hydrodynamics is therefore a consequence of assuming a single relaxation time in (13). As we discuss in the next subsection, emersing the hydrodynamic pole in a nonanalytic strip results in a subtle interplay between hydrodynamic and nonhydrodynamic modes that can lead to a qualitatively novel phenomenon in the long-time behavior.

### 2.3 Dehydrodynamization in kinetic theory

*k*, physics of different energy scales enters the transport on different time scales. To illustrate this parametrically, consider a generic small deformation of the thermal equilibrium. As by assumption the deformation does not take the system far from equilibrium, the number of perturbed modes will be, for large

*p*, proportional to \(e^{-\beta p}\). Each of these modes will then evolve toward equilibrium in a timescale \(\tau _R(p)\), such that the overall magnitude of the nonhydrodynamic part of the perturbation can be estimated at time

*t*by

*t*, the integral is dominated by the decay of modes at a characteristic scale

This dehydrodynamization mechanism will be seen at work in the model (1) of scale-dependent relaxation time, where hard particles still decay directly to a thermal bath and hydrodynamic fluctuations of the thermal bath are ignored. In the full QCD collision kernel, however, the same process proceeds via a cascade of intermediate quasi-democratic splittings [29, 30, 37, 42]. Also, due to the fluctuation-dissipation theorem, there are other sources of hydrodynamic perturbations that can give rise to late-time power-law hydrodynamic tails [20, 39, 44, 45]. Therefore, while the mechanism discussed here is expected to be part of full QCD, the resulting sub-exponential decay is faster than the power-law decay of hydrodynamic tails, and it is therefore not expected to dominate the late-time behavior of the full theory.

## 3 The model: momentum dependent relaxation time

### 3.1 Solution for linear perturbations induced by an external source

### 3.2 Retarded correlation functions

### 3.3 The fluid dynamic limit of \(G_R\)

*k*, the form of retarded correlation functions is dictated by second order fluid dynamics, namely

*k*or \(\omega \). These fluid dynamic expressions depend on entropy

*s*, temperature

*T*, sound velocity \(c_s^2\), as well as shear viscosity \(\eta \), the shear viscous relaxation time \(\tau _\pi \) and the second order transport coefficient \(\kappa \). To determine these fluid dynamic parameters for the kinetic theory with scale-dependent relaxation time, we want to compare the gradient expansion of (39), (40) and (41) to the hydrodynamic expressions (42), (43) and (44). To this end, we expand the integrand of the integral moments (31) to arbitrary order

*N*in \(\omega \) and

*k*, and we perform the

*p*-integration for each term in this expansion. This leads to

Hydrodynamic poles arise as a consequence of energy momentum conservation. In the kinetic theory calculation of Sect. 3.2, the structures in the retarded correlators that arise from energy-momentum conservation are related to the term \(\delta f_{\mathrm{eq}}\) on the right hand side of (25). Inserting the disturbance (25) into (35) and performing the functional derivative \(\delta T^{\mu \nu }(\omega ,k)/\delta h_{\alpha \beta }(\omega ,k)\), one finds that it is exactly the nontrivial denominators in (40) and (41) that arise from the terms proportional to \(\delta f_{\mathrm{eq}}\). The hydrodynamic poles in (40) and (41) are therefore given by the zeroes of the nontrivial denominators in these two channels that arise from energy momentum conservation.

## 4 Analytic structure of the retarded correlation function in momentum dependent relaxation time approximation

The full retarded correlation functions are defined in terms of the integral moments \(I^{a,b,c}(\omega ,k)\) . To study these correlation functions beyond the simple gradient expansion, one needs to evaluate \(I^{a,b,c}(\omega ,k)\) for nonzero \(\omega \) and *k*. A numerical evaluation of \(I^{a,b,c}(\omega ,k)\) in (32) is possible for arbitrary momentum dependencies of the relaxation time approximation (2), i.e., for arbitrary \(\xi \). However, analytical control is advantageous for studying the analytic structure. We therefore focus in the following sections on the case \(\xi = 1\) for which explicit analytical results can be obtained. However, we expect that the qualitative features found for the case \(\xi = 1\) extend to the generic case \(\xi > 0\).

*k*, and in \(R_2\) the derivative \(\partial _\rho \) appears only in the numerator of the rational function. The generating function reads

We note as an aside that we have attempted to derive expressions similar to (53) for other values of \(\xi \). For other rational values, such as \(\xi = 1/2\), \(\xi = 1/3\) etc, one finds typically expressions in terms of more than one generating function, but we were not able to bring all of them into closed analytical form.

### 4.1 Analytic structure of the generating function *H*

*H*. We therefore discuss now the properties of

*H*in detail. To perform the integral in (57), we note that \(\rho G({\bar{\omega }}, \rho )\) is a function of \({\bar{\omega }}/\rho \) only. The derivative with respect to \(\rho \) can therefore be replaced by a derivative with respect to

*x*,

*x*-integration crosses between \(x=\mathrm{Re}\,{\bar{\omega }} - \epsilon \) and \(x=\mathrm{Re}\,{\bar{\omega }} + \epsilon \) the branch cut of \(\Gamma \left[ 0,\textstyle \frac{i\rho }{\bar{\omega }-x}\right] \) for all values \(\bar{\omega }\) with \({\mathrm{Im}\,}(\bar{\omega }) < 0\). The corresponding discrete contribution to the integral is proportional to

*H*in the sense that where the generating function is nonanalytic, so is the full correlation function. The nonanalytic structures seen in Eq. (61) can therefore be related to some of the nonanalytic structures sketched for the retarded correlation function in the introductory Fig. 1. In particular, in the first line of Eq. (61), the two terms \(\propto \left( \bar{\omega }+ {\bar{k}}\right) G({\bar{\omega }} + {\bar{k}},\rho )\) and \(\propto \left( {\bar{\omega }} - {\bar{k}}\right) G({\bar{\omega }} - {\bar{k}},\rho )\) have a logarithmic branch cut for negative imaginary values of \(\bar{\omega }+ {\bar{k}}\) and \({\bar{\omega }} - {\bar{k}}\), respectively. This corresponds to the two nonhydrodynamic cuts depicted in Fig. 1. Moreover, the term in the second line of Eq. (61) is nonanalytic in the entire strip \({\mathrm{Im}\,}{\bar{\omega }} < 0\) and \(-{\bar{k}}< {\mathrm{Re}\,}{\bar{\omega }} < {\bar{k}}\) due to the explicit appearance of \({\mathrm{Im}\,}\,{\bar{\omega }}\). This corresponds to the grey-shaded area of nonanalyticity in Fig. 1. We note that this nonanalytic contribution becomes nonperturbatively small for small \({\bar{\omega }}\) due to the factor \(\sim e^{1/{\mathrm{Im}\,}{\bar{\omega }}}\) in \(G(\bar{\omega },\rho =1)\). Therefore, the analytic region at \({\mathrm{Im}\,}{\bar{\omega }} \ge 0\) is reached very smoothly, whereas the generating function is discontinuous when crossing the \(({\mathrm{Re}\,}{\bar{\omega }})^2 = {{\bar{k}}}^2 \) lines. In contrast to poles and branch-cuts, the analyticity in this strip is also mild in the sense that a contour integral around a region of area

*A*is proportional to

*A*.

*H*. There are singular points, arising from the zeroes of the denominators of Eqs. (40) and (41). These special points are embedded in the strip of mild nonanalyticity, but they give rise to pole-like structures in the sense that they give a finite contribution even when

*A*goes to zero, provided that the special point lies within

*A*(see Fig. 1). Some of these correspond to the hydrodynamical modes in the model. Indeed, the location of such a special point, in shear channel for example, is given for small

*k*by

*H*cannot contribute because of the nonperturbative suppression factor.

### 4.2 Ambiguities in the analytical structure

*H*in Eq. (62) and, a fortiori, the retarded correlation functions are nonanalytic in an entire two-dimensional region as sketched in Fig. 1.

#### 4.2.1 The analytically continued generating function \(H_a\)

A better strategy for calculating (63), that is more practical and more physically revealing is to note that the correlation function is analytic in the upper complex half-plane and along the contour of integration in Eq. (63). Therefore, for the purposes of calculating measurable quantities like (63), we may replace the correlation function in the lower complex half-plane with the analytic continuation of the function from the upper complex half-plane. The nonanalytic structure of the correlation functions \(G^{\alpha \beta ,\gamma \delta }(\omega , k)\) and their generating function *H* in the lower complex half-plane are thus ambiguous to the extent to which the nonanalytic structures arising in *H* can be substituted by an analytic continuation from the upper half-plane.

*H*from the upper half plane is found simply by removing the nonanalytic part form Eq. (61),

*a*stands for analytic continuation. The function \(H_a\) contains incomplete gamma functions with logarithmic branch cuts whose paths are arbitrary as long as their endpoints are fixed to \({\bar{\omega }}= \pm {\bar{k}}\) and to negative complex infinity. Here, we adopt the simplest, but ambiguous choice of continuing the complex gamma function to the full complex plane, resulting in branch cuts at \({\bar{\omega }} = \pm {\bar{k}} + i{\bar{y}}\), for real \(\bar{y} \le 0\). So, \(H_a\) shows the nonhydrodynamic cuts depicted in Fig. 1, but unlike

*H*, these cuts do not bracket a two-dimensional strip of mild nonanalyticity.

To visualize how the analytic structure of \(H_a\) shapes that of retarded correlation functions, we plot in Fig. 5 the real and imaginary part of the shear channel \(G^{0x,0x}_R\), calculated from \(H_a\). This correlation function clearly shares with \(H_a\) the two branch cuts that run in the negative imaginary half plane along \({\mathrm{Re}\,}({\bar{\omega }})=\pm {\bar{k}}\) from zero to complex negative infinity. Closer inspection also reveals that the discontinuity across these branch cuts is exponentially small for small \({\mathrm{Im}\,}({\bar{\omega }})\), as expected from the factor \(\exp \left[ i\rho /{\bar{\omega }} \right] \) in (58). In addition, there is a prominently visible structure of neighbouring peak and trough close to \({\mathrm{Re}\,}({\bar{\omega }})=0\) at negative \({\mathrm{Im}\,}({\bar{\omega }})\), whose orientation is rotated by \(\pi /2\) between the real and imaginary part of \(G^{0x,0x}_R\). This is the tell-tale signature of a simple pole \(\propto 1/({\bar{\omega }} + i\, \mathrm{const})\) in the complex plane. The precise location of this hydrodynamic pole will be discussed in the following. In the gradient expansion, it is given of course by (62).

#### 4.2.2 Deforming the branch cuts

The purpose of this section is to show that in general, the presence or absence of hydrodynamic poles in the lower imaginary half plane of \(G^{\alpha \beta ,\gamma \delta }(\omega , k)\) is not indicative of the onset or disappearance of fluid dynamic behavior.

*H*in Eq. (61) or from \(H_a\) in Eq. (64) yields physically identical responses \( G_R^{\alpha \beta ,\gamma \delta }(t, k)\) while the analytic structure of \(G^{\alpha \beta ,\gamma \delta }(\omega , k)\) is qualitatively different for both cases in the sense that it has a two-dimensional region of mild nonanalyticity if constructed from

*H*, but not if constructed from \(H_a\). In the present section, we consider formulations of the latter kind, for which \(G_R^{\alpha \beta ,\gamma \delta }(\omega , k)\) is given in terms of branch cuts and poles only. In particular, the construction of \( G_R^{\alpha \beta ,\gamma \delta }(t, k)\) from the generating function \(H_a\) is technically advantageous, since the contour of the integration (63) can be closed by encircling the branch cuts going from \(\pm k\) to \(\pm k - i \infty \) and encircling any hydrodynamical poles \(\omega _i\) that may be found in the given channel,

In Fig. 7, we plot the real and imaginary parts of the retarded correlation function in the shear channel for this choice of branch cuts.^{1} Depending on the depth \(-i\sigma \) in the complex \(\bar{\omega }\)-plane at which the two branch cuts are joined, the shear pole is either clearly visible (left hand side of Fig. 7), or it disappears under the branch cut. We emphasize that while both choices of \(\sigma \) lead to qualitatively different features in the analytical structure of \(G_R^{\alpha \beta ,\gamma \delta }(\omega , k)\), they are physically equivalent in the sense that they give rise to identical physical responses \(G_R^{\alpha \beta ,\gamma \delta }(t, k)\) in the time domain. In this sense, the appearance or disappearance of a hydrodynamic-like pole is related to purely technical and physically ambiguous choice of branch cut and it therefore cannot be related to the onset of fluid dynamic behavior.

#### 4.2.3 Differences between the cases \(\xi = 0\) and \(\xi > 0\)

According to the standard definition, the branch cuts of the logarithms in (67) start at \(\omega = - i/{t_R}\pm k\) and they run parallel to the real axis to \(\omega = -i/{t_R}- \infty \). Therefore, they cancel each other outside the range \(-k \le {\mathrm{Re}\,}\omega \le k\), and this gives rise to the nonanalytic segment sketched in Fig. 4. However, the two logarithmic branch cuts of (67) could also be deformed to run parallel to the imaginary axis from \(\omega = \pm k -i/{t_R}\) to negative complex infinity, \(\omega = \pm k -i\infty \).

These two ways of orienting the branch cuts of (67) are reminiscent of the two choices of branch cuts for \(H_a\) depicted in Fig. 7 and discussed for \(\xi =1\) in the previous subsections. However, there are marked physical differences between the cases \(\xi =0\) and \(\xi > 0\).

First, for \(\xi = 0\), the branch cuts can be oriented such that for sufficiently small *k*, hydrodynamic poles are the unique nonanalytic structure closest to the real axis, thus determining the late-time behavior of retarded correlation functions, see Eq. (65). In contrast, for \(\xi = 1\), the branch cuts start always at \({\bar{\omega }} = \pm {\bar{k}}\), and for a gradient expansion around \({\bar{k}}=0\), poles and the starting point of branch cuts are not separated. This observation is related to the finding that the gradient expansion for the position of the pole converges for the case \(\xi = 0\) (for instance, \(\omega _{\mathrm{shear}}(k)\vert _{\xi = 0} = \textstyle \frac{-i}{{t_R}} + \textstyle \frac{ik}{\tan (\bar{k})}\) [33]), while it is an asymptotic series for \(\xi = 1\) (see discussion of Fig. 8 below).

## 5 Retarded correlation functions \(G_R^{\alpha \beta ,\gamma \delta }(t, k)\) in the time domain

In this section, we utilize our understanding of the nonanalytic structures of \(G_R^{\alpha \beta ,\gamma \delta }(\omega , k)\) in the frequency domain for a discussion of the physical response \(G_R^{\alpha \beta ,\gamma \delta }(t, k)\) in the time domain. The connection between both is given by Eq. (65).

In general, with small but increasing *k*, the pole contributions to \(G_R^{\alpha \beta ,\gamma \delta }(t, k)\) in (65) move deeper into the complex plane and they start being cancelled more efficiently by the discontinuities from the branch cuts. While only the sum of these nonanalytic contributions has unambiguous physical meaning, the separate determination of both, the poles and their residues, and the discontinuities along the branch cuts is needed in practice for a discussion of the full physical response in the time domain. In the following, we discuss these nonanalytic contributions separately for the specific choice of the generating function \(H_a\) in (64) with branch cuts taken along \({\bar{\omega }} = \pm {\bar{k}} + i\, y\, {t_R}\), \(y \in [0, -\infty ]\).

### 5.1 The location of the hydrodynamic poles in the shear and sound channel

#### 5.1.1 The pole in the shear channel

*N*coefficients \(b_i\) in a gradient expansion

#### 5.1.2 The sound channel

In close analogy to the discussion of the pole in the shear channel, the poles in the sound channel can be determined in terms of the zeros of the denominator of (41). While the pole in the shear channel is purely imaginary, the pair of sound poles start at finite real values \({\bar{\omega }}_{\mathrm{sound}}({\bar{k}}=0) = \pm c_s = \pm \textstyle \frac{1}{\sqrt{3}}\) before diving into the negative imaginary half plane. The full numerical solution is shown in Fig. 10.

We note that branch cuts can be chosen such that hydrodynamic poles disappear below the cut in one channel while they do not disappear in another channel. Here, this is the case for the choice of branch cuts in \(H_a\) along the imaginary axis. For this choice, the shear pole will remain visible for all \({\bar{k}}\), while the sound pole disappears at \({\bar{k}} = 4\), see Fig. 10. This is yet another illustration of the general statement that there is no unambiguous relation between the existence of hydrodynamic poles in the retarded correlator and the persistence of fluid dynamic behavior.

*k*, see e.g. Ref. [46]. The asymptotic large-

*k*behavior is given by

### 5.2 Contributions of the branch cuts to \(G_R^{\alpha \beta ,\gamma \delta }(t, k)\)

We now combine the information gathered about the nonanalytic structure of \(G_R^{\alpha \beta ,\gamma \delta }(\omega , k)\) to arrive via Eq. (65) at a qualitative understanding of the time-dependence of the physical response \(G_R^{\alpha \beta ,\gamma \delta }(t, k)\). For the shear channel, this time dependence is illustrated with the numerical results in Fig. 11 that display the three characteristic stages of hydrodynamization, hydrodynamic evolution and dehydrodynamization. The following discussion aims at providing an analytic understanding for how these features arise.

#### 5.2.1 The limit \(t\rightarrow 0\) of the retarded correlation functions

*t*,

*t*expansion of \(G_R^{\alpha \beta ,\gamma \delta }(t, k) \) starts with a positive power of

*t*and that Eq. (74) is satisfied.

*k*. In this limit, the residue of the shear pole of \(\bar{G}^{0x,0x}_R\) is

*k*and small

*t*. In the same limit, the cut contribution is sharply peaked around the location of the pole

*k*and small

*t*reads

*t*, they will cancel partially for short times \(t> 0\).

#### 5.2.2 Hydrodynamization

In applications of hydrodynamics, it is often assumed that hydrodynamic behavior dominates the evolution of near-equilibrium perturbations on time scales \(t > \tau _\pi \). In the kinetic model studied here, this hydrodynamic shear relaxation time (49) is \( \tau _\pi = 6 {t_R}\).

*y*in the complex plane where the discontinuity becomes sizeable. The physics is particularly clear in the limit \(k\rightarrow 0\), where one is dealing with one single cut and avoids issues related to the partial cancellation between different cut contributions. In this limit, the shear viscous correlation function takes the form

In summary, simple physics arguments, the numerical inspection of the imaginary part of the cut discontinuity, and the numerical calculation of the retarded correlation function shown in Fig. 11 all indicate that the physical response to perturbations starts being dominated by hydrodynamics on time scales \(t > \tau _\pi = 6 {t_R}\). We emphasize, however, that it is difficult to make this numerical observation analytically precise. The kinetic theory studied here allows for physics on different momentum scales to relax on different time scales.

#### 5.2.3 Late time limit of the correlation function

The late time behaviour of the correlation function is determined by the nonanalytic structures closest to the real axis which are the cuts running to the real axis at \({\bar{\omega }} = \pm {\bar{k}}\). In the physical response \({{\bar{G}}}_R^{\alpha \beta ,\gamma \delta }(t, \bar{k})\) in Eq. (65), the cut discontinuity \(\mathrm{Disc}{\bar{G}}_R^{\alpha \beta ,\gamma \delta }({\bar{k}} + i {\bar{y}}, {\bar{k}})\) at distance \({\bar{y}}= y\, {t_R}\) from the real axis is weighted with an exponential suppression \(e^{{{\bar{y}}} t/{t_R}}\). For the study of the late time behavior \(t \gg 1/k\) and for sufficiently long wavelengths \(1/k \gg {t_R}\), i.e. \({\bar{k}} \ll 1\), it is therefore sufficient to expand this discontinuity around the “on-shell” point \({\bar{\omega }} = {\bar{k}}\).

*k*, we therefore conclude that in the scale-dependent relaxation time approximation investigated here, the kinetic theory dehydrodynamizes for arbitrarily small

*k*at sufficiently late times,

According to Eq. (82), the timescale at which dehydrodynamization occurs varies strongly with the momentum *k*. While Fig. 11 shows a wide window of close-to-hydrodynamic evolution for \({\bar{k}} = 0.4\), this window closes if \({\bar{k}} \) is increased to values larger than unity. As seen in Fig. 13, already for \({{\bar{k}}} = 2\), the oscillatory late-time behavior is visible at all time-scales and a window of close-to-hydrodynamic behavior does not exist.

## 6 Asymptotic nature of gradient expansion and Borel summability

*k*as an explicit example. Its hydrodynamical gradient expansion corresponds to a Taylor expansion in \(\omega \)

*s*.

## 7 Conclusions

Generically, the path to equilibration in relativistic systems described by Boltzmann transport is governed by an interplay of collective hydrodynamic and non-collective particle excitations. The present study allowed us to expose this interplay in detail. Generically, there is no sharp onset of hydrodynamic behavior. On all time and length scales, both hydrodynamic and non-hydrodynamic modes are present. To which extent the one dominates over the other can be at best a quantitative statement that changes gradually with scale. Also, the appearance of poles in the first (physical) Riemann sheet of retarded correlation functions is a matter of choosing a particular analytical continuation and thus cannot be related unambiguously to the onset of fluid dynamic behavior. Still, even if the pole can be made disappear from the physical Riemann sheet by utilizing the ambiguity in analytic continuation, its weight is translated unambiguously to other non-analytic structures in that sheet. In this sense, the relative closeness of hydrodynamic poles to the real axis carries quantitative information about the onset of hydrodynamic behavior irrespective of whether they are visible.

The hydrodynamic behavior is fully characterized by the coefficients of a gradient expansion. As we showed for a generic kinetic theory, this expansion is asymptotic already for retarded correlation functions, since the starting point of the branch cut approaches the origin for small *k*. This is in marked difference to results obtained for strong coupled field theories and in the standard scale-independent relaxation time approximation, where the gradient expansion for retarded correlation functions converges. Remarkably, however, the latter theories if pushed out of equilibrium by longitudinal expansion exhibit a time-dependent energy density whose gradient expansion (in powers of inverse time) is asymptotic. We note that non-linear transport coefficients appear in this expansion, while the above-mentioned gradient expansions of retarded correlation functions involve linear transport coefficients only. It would be interesting to understand the relation between the analytic structures of the higher *n*-point functions that give rise to non-linear transport coefficients, and the qualitatively different convergence properties of the above-mentioned gradient expansions.

Borel summation is employed in attempts to extract physically meaningful information from non-convergent asymptotic series. This technique is often advocated with the seemingly contradictory claim that it can reveal non-perturbative information from analysis of purely perturbative input. By explicitly resumming the Borel series of the gradient expansion of a retarded correlator, we demonstrated in Sect. 6 how this can function.

Our study could be extended on several fronts. The present discussion remained limited to linear response and it could be extended within the present set-up to non-linear response and, in line with the remarks above, to systems undergoing expansion. It may also be interesting to supplement the kinetic theories studied here with thermal fluctuations that via the fluctuation-dissipation theorem are known to give rise to characteristic long-time hydrodynamical tails. Furthermore, it would be interesting to observe, e.g., in numerical simulations, the features identified here in kinetic theories whose collision kernels are derived directly from quantum field theory. Finally, as mentioned in the introduction, a full quantum field theoretical treatment contains interference effects that go beyond simple kinetic theory and become relevant at higher orders in perturbation theory.

## Footnotes

- 1.
We note that our construction of these branch cuts in (66) involves pairs of logarithmic cuts that cancel each other outside a finite segment. For instance, the two terms in the third line of (66) extend both to \({\bar{\omega }} = -i\sigma + \infty \) but they cancel each other for \({\mathrm{Re}\,}({\bar{\omega }}) > {\bar{k}}\). The numerical evaluation shown in Fig. 7 does not attribute values to these lines along which logarithm contributions cancel each other, even though the correlation function is regular there.

## Notes

### Acknowledgements

We thank the organizers and participants of “Micro-workshop on analytic properties of thermal correlators at weak and strong coupling” at Oxford in March 2017 for giving the initial inspiration to this work. We thank Peter Arnold, Harvey Meyer, Krishna Rajagopal, and Andrei Starinets for useful discussions.

## References

- 1.M.P. Heller, R.A. Janik, P. Witaszczyk, Phys. Rev. Lett.
**110**(21), 211602 (2013). https://doi.org/10.1103/PhysRevLett.110.211602. arXiv:1302.0697 [hep-th]CrossRefADSGoogle Scholar - 2.M.P. Heller, M. Spalinski, Phys. Rev. Lett.
**115**(7), 072501 (2015). https://doi.org/10.1103/PhysRevLett.115.072501. arXiv:1503.07514 [hep-th]CrossRefADSGoogle Scholar - 3.G.S. Denicol, J. Noronha, arXiv:1608.07869 [nucl-th]
- 4.M.P. Heller, A. Kurkela, M. Spalinski, arXiv:1609.04803 [nucl-th]
- 5.D.J. Casalderrey-Solana, N.I. Gushterov, B. Meiring, arXiv:1712.02772 [hep-th]
- 6.M. Spalinski, arXiv:1708.01921 [hep-th]
- 7.D.T. Son, A.O. Starinets, JHEP
**0209**, 042 (2002). https://doi.org/10.1088/1126-6708/2002/09/042. arXiv:hep-th/0205051 CrossRefADSGoogle Scholar - 8.A.O. Starinets, Phys. Rev. D
**66**, 124013 (2002). https://doi.org/10.1103/PhysRevD.66.124013. arXiv:hep-th/0207133 CrossRefADSMathSciNetGoogle Scholar - 9.S.A. Hartnoll, S.P. Kumar, JHEP
**0512**, 036 (2005). https://doi.org/10.1088/1126-6708/2005/12/036. arXiv:hep-th/0508092 CrossRefADSGoogle Scholar - 10.A. Nunez, A.O. Starinets, Phys. Rev. D
**67**, 124013 (2003). https://doi.org/10.1103/PhysRevD.67.124013. arXiv:hep-th/0302026 CrossRefADSMathSciNetGoogle Scholar - 11.S. Grozdanov, N. Kaplis, A.O. Starinets, JHEP
**1607**, 151 (2016). https://doi.org/10.1007/JHEP07(2016)151. arXiv:1605.02173 [hep-th]CrossRefADSGoogle Scholar - 12.S. Grozdanov, A.O. Starinets, JHEP
**1703**, 166 (2017). https://doi.org/10.1007/JHEP03(2017)166. arXiv:1611.07053 [hep-th]CrossRefADSGoogle Scholar - 13.V. Khachatryan et al., Phys. Lett. B
**765**, 193 (2017). https://doi.org/10.1016/j.physletb.2016.12.009. arXiv:1606.06198 [nucl-ex]CrossRefADSGoogle Scholar - 14.M. Aaboud et al. [ATLAS Collaboration], Eur. Phys. J. C
**77**(6), 428 (2017). https://doi.org/10.1140/epjc/s10052-017-4988-1. arXiv:1705.04176 [hep-ex] - 15.R.D. Weller, P. Romatschke, Phys. Lett. B
**774**, 351 (2017). https://doi.org/10.1016/j.physletb.2017.09.077. arXiv:1701.07145 [nucl-th]CrossRefADSGoogle Scholar - 16.J.E. Bernhard, J.S. Moreland, S.A. Bass, J. Liu, U. Heinz, Phys. Rev. C
**94**(2), 024907 (2016). https://doi.org/10.1103/PhysRevC.94.024907. arXiv:1605.03954 [nucl-th]CrossRefADSGoogle Scholar - 17.L. He, T. Edmonds, Z.W. Lin, F. Liu, D. Molnar, F. Wang, Phys. Lett. B
**753**, 506 (2016). https://doi.org/10.1016/j.physletb.2015.12.051. arXiv:1502.05572 [nucl-th]CrossRefADSGoogle Scholar - 18.W. Israel, J.M. Stewart, Ann. Phys.
**118**, 341 (1979). https://doi.org/10.1016/0003-4916(79)90130-1 CrossRefADSGoogle Scholar - 19.G.S. Denicol, J. Phys. G
**41**(12), 124004 (2014). https://doi.org/10.1088/0954-3899/41/12/124004 CrossRefADSGoogle Scholar - 20.P.B. Arnold, L.G. Yaffe, Phys. Rev. D
**57**, 1178 (1998). https://doi.org/10.1103/PhysRevD.57.1178. arXiv:hep-ph/9709449 CrossRefADSGoogle Scholar - 21.S. Jeon, Phys. Rev. D
**52**, 3591 (1995). https://doi.org/10.1103/PhysRevD.52.3591. arXiv:hep-ph/9409250 CrossRefADSGoogle Scholar - 22.S. Jeon, L.G. Yaffe, Phys. Rev. D
**53**, 5799 (1996). https://doi.org/10.1103/PhysRevD.53.5799. arXiv:hep-ph/9512263 CrossRefADSGoogle Scholar - 23.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 CrossRefADSGoogle Scholar - 24.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 CrossRefADSGoogle Scholar - 25.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]CrossRefADSGoogle Scholar - 26.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 CrossRefADSGoogle Scholar - 27.J. Hong, D. Teaney, Phys. Rev. C
**82**, 044908 (2010). https://doi.org/10.1103/PhysRevC.82.044908. arXiv:1003.0699 [nucl-th]CrossRefADSGoogle Scholar - 28.M.C. Abraao York, A. Kurkela, E. Lu, G.D. Moore, Phys. Rev. D
**89**(7), 074036 (2014). https://doi.org/10.1103/PhysRevD.89.074036. arXiv:1401.3751 [hep-ph]CrossRefADSGoogle Scholar - 29.A. Kurkela, E. Lu, Phys. Rev. Lett.
**113**(18), 182301 (2014). https://doi.org/10.1103/PhysRevLett.113.182301. arXiv:1405.6318 [hep-ph]CrossRefADSGoogle Scholar - 30.A. Kurkela, Y. Zhu, Phys. Rev. Lett.
**115**(18), 182301 (2015). https://doi.org/10.1103/PhysRevLett.115.182301. arXiv:1506.06647 [hep-ph]CrossRefADSGoogle Scholar - 31.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]CrossRefADSGoogle Scholar - 32.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]CrossRefADSGoogle Scholar - 33.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]CrossRefADSGoogle Scholar - 34.K. Dusling, G.D. Moore, D. Teaney, Phys. Rev. C
**81**, 034907 (2010). https://doi.org/10.1103/PhysRevC.81.034907. arXiv:0909.0754 [nucl-th]CrossRefADSGoogle Scholar - 35.R. Baier, Y.L. Dokshitzer, A.H. Mueller, S. Peigne, D. Schiff, Nucl. Phys. B
**483**, 291 (1997). https://doi.org/10.1016/S0550-3213(96)00553-6. arXiv:hep-ph/9607355 CrossRefADSGoogle Scholar - 36.R. Baier, Y.L. Dokshitzer, A.H. Mueller, S. Peigne, D. Schiff, Nucl. Phys. B
**484**, 265 (1997). https://doi.org/10.1016/S0550-3213(96)00581-0. arXiv:hep-ph/9608322 CrossRefADSGoogle Scholar - 37.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 CrossRefADSGoogle Scholar - 38.H.B. Meyer, JHEP
**0808**, 031 (2008). https://doi.org/10.1088/1126-6708/2008/08/031. arXiv:0806.3914 [hep-lat]CrossRefADSGoogle Scholar - 39.S. Caron-Huot, Phys. Rev. D
**79**, 125009 (2009). https://doi.org/10.1103/PhysRevD.79.125009. arXiv:0903.3958 [hep-ph]CrossRefADSGoogle Scholar - 40.M. Laine, A. Vuorinen, Y. Zhu, JHEP
**1109**, 084 (2011). https://doi.org/10.1007/JHEP09(2011)084. arXiv:1108.1259 [hep-ph]CrossRefADSGoogle Scholar - 41.M. Laine, JHEP
**1305**, 083 (2013). https://doi.org/10.1007/JHEP05(2013)083. arXiv:1304.0202 [hep-ph]CrossRefADSGoogle Scholar - 42.J.P. Blaizot, E. Iancu, Y. Mehtar-Tani, Phys. Rev. Lett.
**111**, 052001 (2013). https://doi.org/10.1103/PhysRevLett. arXiv:1301.6102 [hep-ph]CrossRefADSGoogle Scholar - 43.J.P. Blaizot, E. Iancu, Phys. Rep.
**359**, 355 (2002). https://doi.org/10.1016/S0370-1573(01)00061-8. arXiv:hep-ph/0101103 CrossRefADSGoogle Scholar - 44.Y. Akamatsu, A. Mazeliauskas, D. Teaney, Phys. Rev. C
**95**(1), 014909 (2017). https://doi.org/10.1103/PhysRevC.95.014909. arXiv:1606.07742 [nucl-th]CrossRefADSGoogle Scholar - 45.P. Kovtun, G.D. Moore, P. Romatschke, Phys. Rev. D
**84**, 025006 (2011). https://doi.org/10.1103/PhysRevD.84.025006. arXiv:1104.1586 [hep-ph]CrossRefADSGoogle Scholar - 46.P.M. Chesler, Y.Y. Ho, K. Rajagopal, Phys. Rev. D
**85**, 126006 (2012). https://doi.org/10.1103/PhysRevD.85.126006. arXiv:1111.1691 [hep-th]CrossRefADSGoogle Scholar - 47.Wolfram Research, Inc., Mathematica, Champaign (2017)Google Scholar

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