Do nuclear collisions create a locally equilibrated quark-gluon plasma?
aa r X i v : . [ nu c l - t h ] J a n Do nuclear collisions create a locally equilibrated quark-gluon plasma?
P. Romatschke
1, 2 Department of Physics, 390 UCB, University of Colorado at Boulder, Boulder, Colorado 80309, USA Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA (Dated: January 19, 2017)Experimental results on azimuthal correlations in high energy nuclear collisions (nucleus-nucleus,proton-nucleus and proton-proton) seem to be well described by viscous hydrodynamics. It is oftenargued that this agreement implies either local thermal equilibrium or at least local isotropy. Inthis note, I present arguments why this is not the case. Neither local near-equilibrium nor near-isotropy are required in order for hydrodynamics to offer a successful and accurate description ofexperimental results. However, I predict the breakdown of hydrodynamics at momenta of orderseven times the temperature, corresponding to a smallest possible QCD liquid drop size of 0.15 fm.
CONTENTS
I. Preface 2II. Introduction 2III. A historical perspective 3IV. Hydrodynamization or The Onset of Hydrodynamic Applicability 4V. Hydrodynamic versus Non-Hydrodynamic Modes 5VI. Breakdown of Hydrodynamics and Tests 9VII. Concluding Remarks 11VIII. Acknowledgments 12References 12
I. PREFACE
The intention of this note is to provide an impulse to the nuclear collisions community which in my opinion is tryingto reconcile the traditional paradigm of hydrodynamic applicability with the unreasonable success of hydrodynamicsat seemingly describing ever smaller systems. In an attempt of thinking “outside-the-box”, my hope is to start adiscussion as regards to the interpretation of this hydrodynamic success that challenges the traditional paradigm oflocal equilibration. Aiming at challenging the traditional, I found it very hard to conform to the standard manuscriptstyle, which led to this informal note.It should be pointed out that many of my main points have been made by others before me. All I have done is collectthe available information, and based on this information, offer my own interpretations, conclusions and predictions.I expect that these conclusions may appear obvious to some readers and contentious to others. I invite readers whodo not agree with my statements to engage in a constructive dialogue, which I hope could help us make progress.
II. INTRODUCTION
If nuclear collision experiments do not probe near-equilibrium matter, then this may have a number of consequenceswhich to my knowledge have not been appreciated before, providing the motivation for this note. Firstly, it wouldimply that the nuclear experiments do not (directly) probe equilibrium QCD properties as those calculated in first-principle lattice QCD calculations. Depending on the degree of non-equilibrium, experiments may be closer orfarther away from the QCD phase diagram plane spanned by temperature and baryon chemical potential. While forhydrodynamics, a projection from non-equilibrium space to the equilibrium plane is provided by e.g. the Landaumatching condition, for other observables such a projection is not explicitly known. For instance, it is possible thatthe phenomenon of critical fluctuations associated with the experimental search for a QCD critical point would getmodified when experiments probe QCD away from equilibrium.The understanding that the matter created in high energy nuclear collisions does not need to equilibrate or isotropizelocally in order for hydrodynamics to be quantitatively applicable would imply that the “early thermalization puzzle”is to some extent not a genuine puzzle (see more details about this below). On the other hand, having experimentalaccess to a non-equilibrium quantum system could lead to new directions in the field such as e.g. observing non-equilibrium entropy production, properties of non-thermal fixed points or off-equilibrium photo-production.Finally, if systems created in nuclear collisions do not equilibrate this could naturally explain why proton-protoncollision data on azimuthal correlations appears to be so similar to data obtained in nucleus-nucleus collisions. Oncesmall gradients or near-equilibrium is no longer a requirement, hydrodynamics will generically convert initial stategeometry and fluctuations into correlations, thus making large and small systems look alike in their azimuthal cor-relation signals. Pushing this idea even further would imply that any lump of sufficiently high energy density couldexpand according to the laws of hydrodynamics (with one important caveat which will be discussed below). A naturalconsequence of this would be the presence of exponentially falling (thermal) spectra as well as potential azimuthalcorrelations in e + + e − collisions. III. A HISTORICAL PERSPECTIVE
In recent years, it has been demonstrated that experimental results obtained in relativistic nuclear collisions arewell described by hydrodynamic simulations. Based on the paradigm that hydrodynamics requires near-equilibriumin order to be applicable, this successful hydrodynamic description has been interpreted as evidence for a locallyequilibrated state of matter (dubbed the quark-gluon plasma) in high energy nuclear collisions.The question on how the system created in high energy nuclear collisions reaches or at least comes close to equilib-rium subsequently has led to a number of developments. In particular, it was realized that because of the expansionof the matter into the vacuum following the nuclear collision, the system would cool and thus freeze into a hadronicgas quickly, at which point the type of correlations observed in experiment could not longer be built up. Thus, itbecame apparent that a fluid dynamic approximation to the system dynamics had to start early, on a time-scale of τ ∼ α s ≪ τ ≥ . α − / s Q − s (1)which leads to τ > ∼ . Q s ∼ α s ∼ .
3. Clearly, this result seemed to be in tension with thestarting time required from the agreement between hydrodynamics and experimental data.Arnold, Lenaghan, Moore and Yaffe [7] pointed out that a possible way out of the dilemma was that full thermaliza-tion was actually not required for a hydrodynamic description, and that local (near-) isotropy of the pressure tensorwas sufficient. Thus, the attention of the field shifted towards finding a mechanism to quickly achieve local isotropy(isotropization) rather than full thermalization in high energy nuclear collisions.One such possible mechanism was that of non-abelian plasma instabilities, specifically the non-abelian Weibelinstability, which had been extensively studied by Mrowczynski since the 1980s [8, 9]. In a series of numerical studiesby a number of groups the growth and saturation of these plasma instabilities was determined for non-expandingsystems [10–14], see Refs. [15, 16] for a review. While corroborating the initial exponential approach towards isotropy,these numerical studies suggested the system to stall at large pressure anisotropies once the plasma instabilitiesreached the non-perturbative non-abelian scale and could no longer grow exponentially. Even worse, later studiesin expanding systems aiming at more realistically describing experimental nuclear collisions indicated that the effectof plasma instabilities was delayed/diminished to an extent that they could not lead to local pressure isotropy in atime-scale relevant for nuclear collisions at RHIC and the LHC [17–20].While full isotropization seemed difficult to achieve within a weak-coupling QCD based framework, it appearedto be reachable much faster in gravitational duals of gauge theories in the limit of infinite coupling. For instance,Chesler and Yaffe [21] report isotropization to occur at τ ∼ . /T for a non-expanding system, roughly translatingto τ ≃ .
35 fm/c when assuming T ∼ . τ < ∼
10 fm/c [24].For completeness, it should be noted that when including inelastic processes in a weak-coupling based descriptions,recent studies [25, 26] have demonstrated the approach to isotropy, albeit at times later than those found for infinitelystrongly coupled gauge theories. (This better be the case). Thus, the approach to isotropization in expanding gaugetheories is now understood both at weak and strong coupling, and indicates long times.Despite the impressive progress made, I believe it is a correct statement to say that at phenomenologically relevanttimes of τ ∼ no theoretical approach (be it weakly coupled or strongly coupled)finds the longitudinal and transverse pressure to agree with each other to better than a factor of two. Obviously, apressure anisotropy of 50 percent is not close to an isotropic system, let alone a system in thermal equilibrium. Bythe criterion of Arnold, Lenaghan, Moore and Yaffe, hydrodynamics should not apply.But it does. P L / P T t T i Pressure anisotropy λ = ∞λ =10 λ = 5NSBRSSS FIG. 1. Pressure anisotropy versus time for matter undergoing Bjorken-like expansion. Shown are exact results for matter withdifferent coupling constant λ (symbols, calculated using AdS/CFT and kinetic theory), as well as hydrodynamics in first andsecond order gradient expansion (’NS’ and ’BRSSS’, respectively). Note that P L /P T = 1 would correspond to isotropy (idealhydrodynamics). Figure adapted from Ref. [24]. IV. HYDRODYNAMIZATION OR THE ONSET OF HYDRODYNAMIC APPLICABILITY • Q: How do you people know hydrodynamics applies for pressure anisotropies of 50 percent or larger? • A: We checked.Let us consider the following numerical experiment. Take matter described by gauge/gravity duality at infinitecoupling or alternatively described by kinetic theory at some finite (constant) value of the coupling. Let the matterbe initially at rest in equilibrium with some temperature T i in flat Minkowski space. Then, at a time t ∼
0, thespacetime suddenly starts to expand in one dimension so that it effectively mimics the effects of so-called Bjorken flow[27]. The symbols in Fig. 1 show the response of the matter (at various values of the coupling λ ) in terms of the ratioof longitudinal to transverse pressure as a function of time. The matter is initially in equilibrium so P T = P L (zeropressure anisotropy) and also tends to equilibrium at late times when the expansion becomes very slow. However, for t ≃ V. HYDRODYNAMIC VERSUS NON-HYDRODYNAMIC MODES
What is hydrodynamics? The equations of hydrodynamics can be derived using a multitude of approaches. Someassume the system to be close to thermal equilibrium, others assume a weakly coupled microscopic particle description(kinetic theory). In my opinion, the most general derivation of hydrodynamics follows the approach of effective fieldtheory (EFT).According to this viewpoint, hydrodynamics is the EFT of long-lived, long-wavelength excitations consistent withthe basic symmetries of the underlying system. The fundamental variables of the EFT are that of a fluid: pressure P,(energy) density ǫ and fluid velocity u a . To lowest (leading) order in the EFT, only algebraic combinations of thesequantities will enter the description . Corrections can then be systematically obtained by considering gradients of thefundamental variables.Applying this EFT approach to the energy-momentum tensor for a relativistic system in three dimensions leads tothe well-known expansion T ab = ( ǫ + P ) u a u b + P g ab − η ∇ h a u b i + . . . , (2)where g ab denotes the space-time metric, η is the shear viscosity coefficient and the symbols <> denote a particularsymmetric projector that dedicated readers can look up e.g. in Ref. [35]. With Eq. (2), conservation of energy andmomentum ∇ a T ab = 0 then are the relativistic Navier-Stokes equations. Many articles have been written aboutnon-causality of the relativistic Navier-Stokes equations; I will simply ignore this issue here because it is somewhattangential to the following discussion.The above EFT derivation does at no point invoke the presence of an underlying particle-based, kinetic descriptionof the matter. However, given the requirement of the small gradients, it does seem to require the system to be closeto isotropy. So what if gradients were not small in a particular situation of interest? Obviously, stopping at first orderin a gradient expansion would not be a good approximation. However, one could try to include higher order gradientcorrections to obtain a good approximation. I will try to elucidate what happens in this case through a particularexample.For pedagogical purposes, let me pick the example of N = 4 SYM at infinite coupling undergoing Bjorken expansionthat has been worked through in a tour-de-force paper by Heller, Janik and Witaszczyk [36]. In this case one has ahigh degree of symmetry, and the only relevant gradient is ∇ · u = τ . The equations of motion then lead to a solutionfor the temperature T as a function of τ which may be systematically calculated for small gradients (or equivalentlylarge τ ). (Note that the actual dimensionless expansion parameter is τT which scales as τ − / in the hydrodynamiclimit). Calculating the temperature T ( τ ) in a hydrodynamic gradient expansion to order 240 leads to a series of theform [36] T ( τ ) = ˆ τ − / X n =1 α n ˆ τ − n/ ! , (3)where ˆ τ = ττ , α n constant, and τ setting the initial time (or equivalently temperature) scale.If the gradient expansion was convergent, then we would have succeeded in a (high order) theory of hydrodynamicsthat was unconditionally applicable also when the gradients are large. Given that for this theory a very large numberof coefficients α n had to be calculated, it would be cumbersome if not impossible to generalize this approach tosituations with a much lower degree of symmetry (e.g. nuclear collisions), but at least in principle, it would work!Unfortunately, there is mounting evidence that the hydrodynamic gradient expansion generally is not a convergentseries. In the cases that have been examined in detail ( N = 4 and N = 2 ∗ SYM at infinite coupling, weakly coupledkinetic theory in the relaxation time approximation and M¨uller-Israel-Stewart (MIS) theory) it was found that α n ∝ n !for large n, thus making the gradient expansion a divergent series [36–40].However, not all is lost. It turns out that when inspecting the analytic structure of gradient expansions at highorders, it is possible to use a generalized Borel resummation to rewrite the above series for the case of N = 4 SYM as T ( τ ) = T hydro ( τ ) + γ exp " − i Z d ˆ τ ˆ ω Borel ˆ τ − / + X n =1 ˆ ω n ˆ τ − (2 n +1) / + . . . , (4)where T hydro ( τ ) is well approximated by the first few orders in (3) as long as ˆ τ is not too small. In the above expression,ˆ ω Borel ≃ ± . − . i , and both γ, ω have been calculated in Ref. [36]. As one important qualifier, let me point out that a necessary condition for this to work is the presence of a local rest frame, cf. thediscussion in Ref. [34]. Without a local rest frame, the local energy density cannot be defined, and a fluid EFT approach is not applicable.
There are two things to note about the resummed result (4). First, the exponential multiplying the coefficient γ in(4) can not be recast in terms of the hydrodynamic gradient expansion. It is a truly non-hydrodynamic mode, andits presence explains why the naive hydrodynamic gradient series is divergent.Second, the numerical value of ˆ ω Borel is not an arbitrary number. It happens to be consistent with the firstnon-hydrodynamic quasi-normal mode frequency of a near-equilibrium 5d Schwarzschild-AdS black holeˆ ω (1)QNM = ± . − . i , (5)calculated by Starinets in Ref. [41]. A quasi-normal mode corresponds to a pole of a two-point function in the complexfrequency plane located at ω = 2 πT ˆ ω QNM . Linear response then implies the presence of a contribution of the form e iωτ = e i π ˆ ωT τ to the one-point function which upon using the leading order expression in (3) for T ( τ ) then leads toa result e ∼ i ˆ ω ˆ τ / consistent with (4), cf. [42]. While only the first non-hydrodynamic quasi-normal mode ˆ ω (1)QNM hasbeen identified in the Borel transform of the gradient expansion, it is likely that all higher non-hydrodynamic modesalso will contribute likewise to T ( τ ), which has been anticipated through the ellipses in (4). In fact, Buchel, Hellerand Noronha were able to show that for the case of N = 2 ∗ SYM the first 10 quasi-normal modes could be obtainedfrom the relevant Borel transform [38].Thus, the following picture emerges: a naive hydrodynamic gradient expansion of the energy-momentum tensor isdivergent because of the presence of other, non-hydrodynamic degrees of freedom. However, the contribution fromthese non-hydrodynamic modes may be either resummed via a generalized Borel transform, or anticipated throughexplicitly calculating the non-hydrodynamic mode structure of the energy-momentum tensor for the theory underconsideration: T ab = T ab hydro + T ab non − hydro . (6)The term T hydro ( τ ) in (4) is a “generalized” hydrodynamic piece has been dubbed “hydrodynamic attractor” or“all order hydrodynamics” by various authors [37, 43]. As remarked above, it is well approximated by a low-ordergradient expansion approximation even in regime when gradients are moderately strong (see e.g. Ref. [37]). Thus,even though different in principle, it is entirely conceivable that for many applications involving gradient terms oforder unity, T ab hydro will quantitatively be well approximated by the naive, low-order gradient expansion (2) becausethe divergence of the gradient approximation only becomes apparent when including higher orders. High-temperatureperturbation theory in QCD exhibits a similar feature where the leading order ( g ) correction to the free energy offersa quantitatively reasonable description of lattice QCD data even for g ≃ T ab non − hydro will in general not havea universal form, but rather be dependent on the particular underlying microscopic description under consideration(“microscopic” in the sense of QCD, not in the sense of quasi-particles) .This is most easily elucidated when considering the small amplitude (but arbitrary gradient) linear response of theenergy-momentum tensor to an initial source S cd ( x ), δT ab ( t, x ) = Z dωd ke − iωt + i k · x G ab,cdR ( ω, k ) S cd ( k ) , (7)where G ab,cdR ( ω, k ) is the retarded two-point function of the energy-momentum tensor (cf. [45]).For instance, in the case of N = 4 SYM at infinite coupling, G , ( ω, k ) possesses only poles in the complexplane. Two of these poles may be uniquely identified as hydrodynamic sound poles, located at ω h = ± c s | k | − iη | k | s when | k | ≪
1. In addition to the hydrodynamic poles, there is an infinite number of (pairs) of non-hydrodynamicquasinormal modes located at ω = ω (1) nh , ω (2) nh , . . . [47]. Performing the frequency integration in (7) will pick upcontributions at all of these poles, immediately leading to δT ( t, x ) = Z d ke i k · x " e − iω h t a h ( k ) + ∞ X n =1 e − iω ( n ) nh t a n ( k ) = δT + δT − hydro , (8)where the coefficient functions a h ( k ) , a n ( k ) depend on the residues from the integration as well as the source function S ( k ). The integral in (8) cleanly separates into a hydrodynamic piece governed by the sound mode dispersion ω h ( k )and an infinite sum over non-hydrodynamic contributions with dispersion ω ( n ) nh ( k ).More important than the realization that the energy-momentum tensor can be split into a hydrodynamic and a non-hydrodynamic piece is the fact that the hydrodynamic poles cease to exist at some value of k when the coupling is finite[48]. This implies that for | k | larger than some critical value of | k | c (dependent on the coupling), the hydrodynamic -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 -8 -6 -4 -2 0 2 4 6 8 0 2 4 6 8k/T Analytic Structure G R00,00 for τ R T=1k c /T Hydro PoleLogarithmic CutIm ω /T Re ω /Tk/T FIG. 2. Analytic structure of the G , R ( ω, k ) correlator in kinetic theory (from Ref. [46]). At low k/T , there are twohydrodynamic poles and a logarithmic branch cut from − k − iτ R to k − iτ R . Increasing k/T , the hydrodynamic poles acquirelarger and larger negative imaginary part until at k c /T ≃ .
53 they merge with the branch cut and disappear from the physicalRiemann sheet. For k > k c , only the branch cut remains. Grey lines are projections of trajectories to the k = 0 plane. component vanishes from the spectrum and is replaced by purely non-hydrodynamic behavior. At least for | k | > | k | c ,hydrodynamics has broken down.One may criticize that N = 4 SYM is a very special microscopic theory, and worry about drawing general conclusionsbased exclusively on N = 4 SYM. However, it turns out that when calculating G ab,cdR ( ω, k ) in kinetic theory inthe relaxation time approximation [46], similar conclusions apply. In kinetic theory, G , R ( ω, k ) generally has twohydrodynamic poles which are located at ω h = ± c s | k |− iη | k | s when | k | ≪
1. In addition to these hydrodynamic poles, G , R ( ω, k ) exhibits a logarithmic branch cut which may be taken to run from −| k | − iτ R to | k | − iτ R where τ R = 5 ηsT is the relaxation time in kinetic theory. It is interesting to note that when increasing | k | beyond some critical value | k | c , the hydrodynamic poles pass through the logarithmic cut onto the next Riemann sheet, and effectively cease toexist (see Fig. 2). Only the non-hydrodynamic branch cut remains, implying that for | k | > | k | c , hydrodynamics hasbroken down.I will summarize the above observations in the form of a Lemma.
Given the existence of a local rest frame, hydrodynamics offers a valid and quantitatively reliable descrip-tion of the energy-momentum tensor even in non-equilibrium situations as long as the contribution from all non-hydrodynamic modes can be neglected.
Proof: Consider matter possessing a local rest frame everywhere in the space-time patch of interest, such that thelocal energy density is non-negative in any frame. Pick a convenient global frame (“laboratory frame”) and considerthe Fourier decomposition of the energy-momentum tensor in this frame. Now consider real time perturbations δT ab ( t, k ) around the Fourier zero mode T ab background in the laboratory frame. At some initial time t , the differencebetween the local energy-momentum tensor and the background can be viewed as an initial perturbation S ab ( t , k ).In the limit of small perturbation amplitude | S ab | →
0, linear response theory applies, cf. Eq. (7). Furthermore,the retarded two-point function G R is known to be given by Navier-Stokes hydrodynamics [45] in the small wave-number limit k →
0. The two-point correlator in Navier-Stokes hydrodynamics possesses hydrodynamic poles (shearand sound poles) in the complex frequency plane. Contour integration as in Eq. (7) will pick up these poles andlead to a hydrodynamic contribution to δT ab . As k is increased, the location of the hydrodynamic poles may shift,and they may even disappear from the spectrum completely at some critical wave-number. In addition to thehydrodynamic poles, new, non-hydrodynamic singularities may appear in the complex frequency plane. These non-hydrodynamic singularities, upon contour integration in Eq. (7) will lead to a non-hydrodynamic contribution that
200 400 600 800 1000 50 100 150 0 0.5 1 1.5 2 2.5 3 3.5 N on - equ ili b r i u m ne ss ( ξ ) T [MeV]Hadron Gas QCD LiquidRHIC TrajectoriesCritical Point? N eu t r on S t a r s "Au+Au 200 GeV""Au+Au 62.4 GeV""Au+Au 39 GeV" µ B [MeV] N on - equ ili b r i u m ne ss ( ξ ) FIG. 3. A non-artist’s calculation of possible RHIC trajectories for Au+Au collisions at various collision energies √ s =39 , ,
200 GeV. Rather than displaying the common projections in the temperature - baryon chemical potential (
T, µ ) plane,trajectories explored by experiment are more likely to explore at least a third non-equilibrium direction (symbols). Projectionsof the trajectories to the
T, µ and
T, ξ planes are indicated as grey dashed lines. A minimum of ξ ≃ . √ s = 200GeV at T ≃ .
17 MeV, which corresponds to P L /P T ≃ .
86. For reference, the deconfinement cross-over transition (blue dottedline) and the liquid-gas first order transition (full black line) have been indicated. Note that as the collision energy is lowered,one moves further away from equilibrium. has to be added to the hydrodynamic part of δT ab as in the example given in Eq. (8). As the amplitude S ab isincreased, other, non-linear structures will contribute to δT ab which can be expressed as a sum over integrals ofn-point functions with the appropriate powers of the source S ab . In the limit of small wave-number, these non-linearcorrections to the hydrodynamic part will, upon resummation, shift the hydrodynamic poles locally, and in additioncontribute new structures familiar from the hydrodynamic gradient expansion, cf. Eq. (2). Away from k = 0 thenon-linear hydrodynamic part of δT ab is analytically connected to this familiar structure, giving rise to the generalizedhydrodynamic attractor form, until some critical wave-number k = k c is reached. In addition to the hydrodynamicpart, there will be non-linear corrections to the non-hydrodynamic contribution of δT ab . Thus, the global energy-momentum tensor can be written as T ab = T ab background + δT ab = T ab hydro + T ab non − hydro . If the non-hydrodynamiccontribution can be neglected, the lemma follows trivially.(Mathematicians probably would want to see a more formal proof than this, so the above lemma should probablybe called a conjecture).The above lemma may seem trivial: once non-hydrodynamic modes are absent, how can the energy-momentumtensor be described by anything else but hydrodynamics? However, when phrased in this fashion, hydrodynamicsneither requires equilibrium, nor isotropy, nor infinitesimally small gradients. (It does require the presence of a localrest frame, though [34]). The applicability of hydrodynamics is exclusively determined by the relative importance ofnon-hydrodynamic modes.As it stands, the above lemma has at least one potentially important consequence, which is phrased as a Dilemma.
The phenomenological success of hydrodynamics in describing experimental data from high energy nuclearcollisions does not imply near-equilibrium behavior of the matter. Experiments do not directly probe the equilibriumQCD phase diagram at finite
T, µ , but explore trajectories in a space with at least one more (non-equilibrium) direction.
Incomplete equilibration in nuclear collisions is a fact well known to all heavy-ion hydro practitioners in the fieldand has been pointed out more than a decade ago by Bhalerao, Blaizot, Borghini and Ollitrault in Ref. [49]. The abovelemma allows me to go one step further since not even near-equilibrium is required for hydrodynamics. The secondpart of the dilemma is a direct consequence of the first, yet it probably is not as widely appreciated. I have tried tovisualize the last point of the dilemma in Fig. 3. To generate the hypothetical trajectories I have matched the pressure k c / T λ Hydrodynamic Breakdown ScaleGauge-Gravity (numeric)Gauge-Gravity (approx)Kinetic Theory
FIG. 4. Results for the hydrodynamic breakdown scale k c from weak coupling (’kinetic theory’, [46]) and strong coupling(’gauge-gravity’, [48]) frameworks (the curve labeled ’approx’ is a fit to the exact numerical values from Ref. [48], see text).The regime of applicability for kinetic theory is λ ≪
1, while for gauge-gravity duality λ ≫ λ ≃ − anisotropy P L /P T to the momentum anisotropy parameter ξ , first defined in Ref. [50]. I use ξ ∈ [0 , ∞ ) to express thedegree of non-equilibrium (“non-equilibriumness”), where ξ = 0 corresponds to the case of local equilibrium. Thereis an extensive literature in anisotropic hydrodynamics which makes the connection between P L /P T and ξ precise(see e.g. the instructive lecture notes by Strickland, Ref. [51], Eq. (3.19)). For the pressure anisotropy itself, I usedNavier-Stokes hydrodynamics in Bjorken expansion (cf. [24]) and a viscosity that is given by ηs = π for T > . ηs = 1 at T = 0 . τ = 1 fm/c with multiplicities representative of experimental measurements [55] convertedto temperature values T ( τ = 1fm / c) as in Ref. [56]. Clearly, none of these choices do justice to the much moreaccurate descriptions that are currently available. However, I expect the sketch in Fig. 3 to be qualitatively reliable. VI. BREAKDOWN OF HYDRODYNAMICS AND TESTS
According to the central lemma in the previous section, hydrodynamics can be used to describe a system if a localrest frame exists and non-hydrodynamic modes are sub-dominant. I have nothing new to say about how to test forthe presence of a local rest frame, so I will simply follow everyone else’s approach and assume that a local rest frameexists. (This is very likely wrong [34] and actually should be thought about more, but I leave this task to dedicatedreaders).Contrary to the existence of local rest frames, quite a bit of knowledge now exists on those non-hydrodynamic modes.For the case of kinetic theory with relaxation time τ R , we know the analytic structure of non-hydrodynamic modesin the two-point function of the energy-momentum tensor, and we know that hydrodynamic modes completely vanishfrom the spectrum at | k | c τ R ≃ . τ R = ηsT and ηs ≃ g for N = 3 color QCD [57], one finds | k | c ≃ . λ T in terms of the ’t Hooft coupling λ ≡ g N . From gauge gravity duals at large but finite λ , non-hydrodynamic modesimply the breakdown of hydrodynamics at | k | c ≃ πT ln (cid:16) . λ / λ − / (cid:17) (see Fig. 4 for exact numerical results fromRef. [48]). These kinetic theory and gauge-gravity results for the hydrodynamic breakdown scale k c are compared inFig. 4. It is curious to note that – despite the difference of the kinetic theory and gauge-gravity duality approaches –0 C ha r ged H ad r on v n p T [GeV]Pb+Pb √ s=5.02 TeVHydro Non-Hydrov , 00-05%, ALICEv , 00-05%, ALICEv , 00-05%, ALICE 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 2 4 6 8 10 C ha r ged H ad r on v n p T [GeV]Pb+Pb √ s=5.02 TeVHydro Non-Hydrov , 30-40%, ALICEv , 30-40%, ALICEv , 30-40%, ALICE FIG. 5. Experimental data for flow coefficients v n as a function of particle p T for Pb+Pb collisions at √ s = 5 .
02 TeV (ALICE,[59]). No hydrodynamic curves are shown, but it is known that hydrodynamics well describes the experimental data in theregime indicated as ’hydro’ in the plot [60], possibly extending up to p T ≃ p T > ∼ the results for k c turns out to be quantitatively similar in both approaches for moderate values of λ ≃ − k c to QCD with α s ≡ g π ≃ . k c ( λ ≃ T = 4 −
7, where the highervalue is actually coming from the kinetic theory result. Let’s be optimistic and take | k | c ≃ T . Thus, I claim that nohydrodynamic description is possible for QCD systems smaller than k − c ≃ .
15 fm at a typical QCD-scale temperatureof T ≃
200 MeV. The prediction that hydrodynamics must break down for | k | − < .
15 fm is most likely somewhatuseless, because it is very hard to falsify experimentally in nuclear collisions given that the mean proton radius is0 .
86 fm. However, the limit of 0 .
15 fm at least constitutes an actual numerical conjecture for the lower bound of thesmallest possible droplet of QCD liquid.It should be pointed out that the results | k | c ≃ . τ R and the numerical gauge-gravity results shown in Fig. 4 arequantitative upper bounds on the domain of hydrodynamic applicability in weak and strong coupling scenarios. How-ever, it is likely that hydrodynamics breaks down before reaching these values of | k | . One reason why hydrodynamicsmay break down earlier would be that while hydrodynamic modes do not vanish for | k | < | k | c , they may become sub-dominant to certain non-hydrodynamic modes, namely those which happen to be closer to the origin of the complex ω plane. For N = 4 SYM at λ → ∞ , this seems to happen at around | k | ≃ πT , which is at around the same value as k c obtained at finite λ . Yet another reason why hydrodynamics may break down at scales below | k | c could be non-lineareffects. For a particular initial condition, numerical studies of N = 4 at λ → ∞ including full non-linear effects byChesler [58] seem to indicate a breakdown of hydrodynamics at a scale | k | ≃ T . These results are fully consistent withthe “most optimistic” result k c ≃ T and the resulting hard upper bound for the hydrodynamic breakdown scale, butclearly leave room for sharpening the prediction for | k | c .A much more direct route to experimentally constrain k c could be provided by high-momentum data on flowcoefficients, see Fig. 5. Experimental data for collective flow harmonics in Pb+Pb collisions suggests a change inbehavior in the regime between p T = 3 GeV to p T = 4 GeV. The low momentum region is well described byhydrodynamics [60]. Assuming that measured particles originated from a constant-temperature freeze-out surface at T = 0 .
17 GeV, this would indicate a breakdown of hydrodynamic behavior at p T T = 18 −
23. In order to relate thisscale to the hydrodynamic breakdown scale k c < ∼ T , quantitative calculations of the location of non-hydrodynamicmodes in an expanding system are needed.In the hadronic phase, kinetic theory would predict hydrodynamic modes to dominate for k < k c ∝ τ R , whilenon-hydrodynamic (particle) modes dominate for k > k c . As the temperature is lowered, τ R ∝ ηsT increases strongly[52] until k c falls below the typical system wave-number. From this point onward, most of the system dynamicsproceeds according to the non-hydrodynamic particle kinetics, providing a qualitative understanding of the transitionfrom hydrodynamic to particle cascade dynamics (“freeze-out”).The above statements involve hard lower bounds on the smallest scales at which hydrodynamics applies, and aqualitative understanding of the freeze-out transition. However, a quantitative test of the applicability of hydrody-namics, e.g. through testing sensitivity of results with respect to non-hydrodynamic modes, is desirable in the case ofnuclear collisions . Fortunately, the workhorse of relativistic viscous hydrodynamics simulations, “causal relativisticviscous hydrodynamics” (which goes by many names and acronyms but is usually associated with the work of M¨uller,Israel and Stewart [61, 62]) does contain a non-hydrodynamic mode buried within, which may be exploited for testing1 v dN/d η Charged Hadron v p+Au, √ s=200 GeVp+Au, √ s=62.4 GeVp+Au, √ s=7.7 GeV 0 0.02 0.04 0.06 0.08 0.1 1 10 v ( p T > . G e V ) dN/d η Charged Hadron v ATLASRND, η /s=0.08, ζ /s=0.00RND, η /s=0.04, ζ /s=0.02FLC, η /s=0.04, ζ /s=0.02 FIG. 6. Sensitivity of charged hadron v on non-hydrodynamic mode for central p+Au collisions (left panel, no p T cut) andp+p collisions at √ s = 7 TeV (right panel, p T cut at 0 . dNdη <
2. Figures based on results from Ref. [65] and from Ref. [66], respectively. purposes. Specifically, besides the usual hydrodynamic modes, the energy-momentum tensor two point function con-tains a pole located at ω nh = − iτ π , where τ π is the “viscous relaxation time” that also controls the size of the secondorder gradient term in the one-point function of T ab . Any current numerical hydrodynamics simulation of the matterproduced after a relativistic nuclear collision needs a specific value for τ π as an input. Simulators choose values of τ π as they see fit, given that the “correct” value for τ π for QCD is not known, and that primary interest is in extractinginformation about ηs , ζs , not some obscure second-order transport coefficient.However, varying τ π around some “fiducial” value does vary the decay-time of the non-hydrodynamic mode inherentto causal relativistic hydrodynamics, thus offering a direct handle on the sensitivity of final results on the non-hydrodynamic mode. This can be implemented in practice in relativistic viscous hydrodynamic simulations by runningsimulations at multiple values of τ π and expressing final results in terms of a mean value and a systematic error barcovering the variations of final results from changing τ π . Examples are shown in Fig. 6 for the case of central p+Aucollisions at various values of √ s and p+p collisions at √ s = 7 TeV. While the sensitivity on non-hydrodynamic modesis not vanishingly small, the error bars do seem to signal the applicability of hydrodynamics to both p+Au and p+pcollisions in general. However, in the case of p+p collisions and dNdY <
2, the error bars become large, signaling strongsensitivity of the result to non-hydrodynamic modes. This empirical result seems to indicate that hydrodynamicsbreaks down in p+p collisions for dNdY <
2. This multiplicity value of the hydrodynamic breakdown corresponds wellto the results derived by Spalinski [63]. It would be interesting to repeat these sensitivity tests for hydrodynamicswith a different non-hydrodynamic mode structure, for instance along the lines suggested in Ref. [64].
VII. CONCLUDING REMARKS
1. I would argue that there is hard experimental evidence, e.g. through the phenomenon of jet modifications, forthe presence of strongly interacting QCD matter created in nuclear collisions. As argued in this note, I amdoubtful about the hard evidence for this matter to be equilibrated.2. The physics of non-hydrodynamic modes is a rich and a barely studied subject. Given that non-hydrodynamicmodes play an important role in the applicability and breakdown of a hydrodynamic descriptions, I believe thosenon-hydro modes should receive more attention, from theorists and experimentalists alike.3. The central lemma in section V also applies to the case of diffusion, not only momentum transport. In particular,this implies that a constitutive equation of the form J = σE could hold in the early-time, out-of-equilibriumregime following a nuclear collision, if non-hydro modes are sub-dominant. This could potentially explain alonger-than-expected life-time of the magnetic field which is critical to experimental detection of the ChiralMagnetic Effect [67] (see also Ref. [68]).4. As outlined in the central dilemma in section V, the experimental search for the QCD critical point will neces-sarily explore trajectories in some non-equilibrium space (cf. Fig. 3). This implies that the standard equilibrium2theory of critical fluctuations strictly speaking does not apply, and one should try to understand non-equilibriumeffects (see e.g. Ref. [69]) in order to correctly interpret the experimental data.5. In view of the ’QGP drop size lower bound’ of 0 .
15 fm, it is maybe not surprising that the matter created in p+pcollisions would behave hydrodynamically. At this scale, however, p+p collisions may not be the ultimate dropsize test. QCD-QED couplings allow fluctuations of electrons to e.g. quark pairs, thus opening up the possibilityof local energy deposition reminiscent of p+p collisions occurring in e + - e − collisions (cf. Refs. [70–72]). Dataon e + - e − collisions taken at e.g. LEP should be re-analyzed with modern tools in order to find (or rule out)hydrodynamic behavior in these systems.6. The fact that experimental data shows a qualitative change in trend from hydrodynamic behavior at low mo-menta to non-hydrodynamic behavior at high momenta suggests a potential experimental handle on the hy-drodynamic breakdown scale k c in QCD. This potential connection should be made quantitative in furtherstudies.7. The entire discussion in this note ignores the presence of hydrodynamic thermal fluctuations, which arise in SU ( N ) gauge theories at any finite number N . The subfield of relativistic hydrodynamics with thermal fluctu-ations is still in its infancy, but potentially can have important phenomenological consequences [45, 73–80].8. A recurring problem of non-standard cosmology (so-called “viscous cosmology”, cf. [81, 82]) seems to be that“interesting” deviations from standard cosmology occur when gradient corrections become order unity. In the“old-fashioned” picture of hydrodynamics, this was not acceptable since order unity corrections heralded thebreakdown of applicability of the theory. In view of the central lemma in section V, it could be interesting todetermine the relevant non-hydrodynamic modes in cosmology and re-evaluate the regime of applicability ofviscous cosmologies. VIII. ACKNOWLEDGMENTS
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