# Nearly isentropic flow at sizeable

###### Abstract

Non-linearities in the harmonic spectra of hadron-nucleus and nucleus-nucleus collisions provide evidence for the dynamical response to azimuthal spatial eccentricities. Here, we demonstrate within the framework of transport theory that even the mildest interaction correction to a picture of free-streaming particle distributions, namely the inclusion of one perturbatively weak interaction (“one-hit dynamics”), will generically give rise to all observed linear and non-linear structures. We further argue that transport theory naturally accounts within the range of its validity for realistic signal sizes of the linear and non-linear response coefficients observed in azimuthal momentum anisotropies with a large mean free path of the order of the system size in peripheral ( centrality) PbPb or central pPb collisions. The shear viscosity to entropy density ratio of such a transport theory is approximately an order of magnitude larger than that of an almost perfect fluid. The phenomenological success of transport simulations thus challenges the perfect fluid paradigm of ultra-relativistic nucleus-nucleus and hadron-nucleus collisions.

Introduction. The applicability of relativistic transport theory ranges from the description of free-streaming particle distributions via weakly interacting systems of long mean free path up to systems exhibiting viscous fluid dynamic behavior. In this way, transport theory encompasses within a unified framework a broad range of candidate theories for the collective flow-like behavior observed in ultra-relativistic nucleus-nucleus ALICE:2011ab () and, more recently, proton-nucleus Abelev:2014mda (); Khachatryan:2015waa (); Aaboud:2017acw (), deuterium-nucleus Adare:2014keg (); Adamczyk:2015xjc () and high multiplicity proton-proton collisions Khachatryan:2016txc (); Aad:2015gqa ().

Arguably the best-studied candidate theory for collective dynamics in nuclear collisions is the viscous fluid dynamic limit of kinetic transport. Fluid dynamic simulations of ultra-relativitic heavy ion collisions at RHIC and at the LHC can account for hadronic spectra in nucleus-nucleus collisions, their azimuthal anisotropies , their -differential hadrochemical composition, as well as for correlations between different ’s, see Refs. Heinz:2013th (); Romatschke:2017ejr () for recent reviews. Generically, these fluid dynamic simulations favor an almost perfect fluid with a very small ratio of shear viscosity to entropy density , suggesting that the system is so strongly interacting that the notion of quasi-particles and mean free path becomes doubtful since no particle excitation propagates over distances larger than . However, up to finer differences Adam:2015bka (); Bzdak:2014dia () that may or may not be improvable via tuning of model parameters, the main signal sizes and kinematic dependencies of ’s in nucleus-nucleus collisions have also been accounted for Xu:2011jm () by A-Multiphase-Transport-Model (AMPT) Lin:2004en (), a simulation code of partonic and hadronic transport whose applicability relies on a sufficiently large mean free path. The supposed dichotomy between a strong coupling picture (based on negligible mean free path and almost perfect fluidity) and a weak coupling picture based on spatially well-separated interactions is further challenged by the apparent phenomenologically compatibility of both descriptions with smaller collision systems. In particular, exploratory fluid dynamic simulations of pPb collisions have predicted Bozek:2011if (); Bozek:2013uha () for an almost perfect fluid () the correct signal size and -dependence of anisotropic elliptic and triangular flow in pPb prior to data taking at the LHC Abelev:2014mda (); Khachatryan:2015waa (); Aaboud:2017acw (), while the AMPT code provides a phenomenologically reasonable description of pPb data at LHC Bzdak:2014dia () and smaller collision systems at RHIC Koop:2015wea () with an apparently dilute system from which approximately of all partons escape without rescattering He:2015hfa () (numbers quoted for d+Au collisions at RHIC). The discovery of heavy ion like behavior in smaller (pp and pPb) collision systems at the LHC and its confirmation at RHIC energies thus raises the question of whether the matter produced in these collisions shares properties of an almost perfect fluid, or whether the observed signatures of collectivity can arise in a system of particle-like excitations of significant mean free path.

Increasing in fluid dynamic simulations i) reduces the efficiency of translating spatial gradients into momentum anisotropies and ii) increases entropy production. Within a model parameter space that includes uncertainties in the initial conditions, one may imagine to increase by compensating effect i) with an increased initial density (and thus with increased initial spatial gradients within a fixed initial spatial overlap area). But entropy increases with increasing initial density or , and any such variation of model parameters is therefore tightly constrained by event multiplicity. This illustrates that a phenomenologically valid collective dynamics needs to combine a nearly isentropic dynamics with an efficient mechanism for translating spatial anisotropies into momentum anisotropies. Remarkably, in the opposite limiting cases of a (nearly) perfect liquid and of a (nearly) free-streaming gas of particles, kinetic transport theory gives rise to a (nearly) isentropic dynamics. The first limit is clearly realized in fluid dynamic simulations that are known to translate efficiently spatial into momentum anisotropies. At face value, the fact that transport codes like AMPT and MPC Molnar:2000jh () can build up an efficient collective dynamics with very few spatially separated collisions indicates that they dynamically realize the opposite limit of a nearly isentropic evolution close to free-streaming. This has the potential for a shift away from the perfect fluid paradigm and it thus deserves to be understood in detail.

For any sufficiently complex simulation, the code is the model in the sense that the code is more than a tool for solving an easily stated set of equations of motion. For instance, fluid dynamic simulations do not code only for a solution to the viscous fluid dynamic equations of motion, but they interface those with a model of initial conditions, with a model for hadronization and with a hadronic rescattering phase. This necessary phenomenological complexity of a simulation can obscure the relation between phenomenological success and the physical mechanism at work. For state-of-the-art fluid dynamic codes, however, the fluid dynamic part of the simulation is benchmarked against analytically known results of viscous Israel-Stewart theory, and many kinematic dependencies and measurable properties of ’s and their correlations can be understood at least qualitatively by solving in isolation for the viscous fluid dynamic equations of motion with suitable initial conditions. It is the qualitative and semi-quantitative consistency between full phenomenological simulations and benchmarked solutions to an unambiguously defined set of equations of motion that gives strength to the phenomenological conclusion of fluid dynamic behavior in heavy ion collisions.

For transport theory close to the free streaming limit, a corresponding program of anchoring major dynamical stages of full-fledged simulations on unambiguously and analytically defined benchmark calculations is largely missing (exceptions include early tests of parton cascades between the free-streaming and hydrodynamic limits Gyulassy:1997ib (); Zhang:1998tj (), more recent comparisons of kinetic theory to fluid dynamics Gabbana:2017uvc (); Bouras:2010hm (); Kurkela:2015qoa ()). What is at stake is to understand from simple physical principles and beyond any specific model implementation whether a particular manifestation of collective behavior is a generic property of transport theory. Motivated by the conceivable phenomenological relevance of transport close to the free-streaming limit for ultra-relativistic hadronic collision systems, the aim of this paper is to pursue such a programme of establishing benchmark results, and to determine which characteristic signatures of collectivity will arise generically in such a framework.

The model. We start from massless kinetic transport of a distribution , assuming longitudinal boost invariance and focussing on the slice of central spatial rapidity ,

(1) |

We denote space-time coordinates by , and normalized momenta by with and . We are mainly interested in -integrated distribution functions that satisfy

(2) |

The collision kernel depends on the microscopic details of the interactions of the constituents. However, it is a generic property of interactions that they bring the system towards isotropy, and hence we consider a model — isotropization-time approximation — in which the scattering between particles in a given volume element is assumed to distribute the particles isotropically in the rest frame of that given volume element

(3) |

where is the isotropic distribution in the rest frame . We further assume that the system is conformally symmetric such that the time scale for the interactions is proportional to the energy density scale of the medium, , where is our single model parameter setting the isotropization rate. While in more realistic models, different momentum scales isotropize on different timescales Arnold:2002zm (); Kurkela:2017xis (), the momentum integral in the definition of is dominated by a single scale, whose isotropization time corresponds to our model parameter (see e.g. Heller:2016rtz () for a comparison of this model to QCD effective kinetic theory, finding differences in relaxation of when of the two models are matched). This caveat in mind, we shall restrain ourselves from making -differential statements which would depend on details of the collision kernel that are not captured by a single isotropization timescale.

The local rest frame energy density and the rest frame velocity with normalization are defined by the Landau matching condition, , with the energy-momentum tensor defined by the second velocity moments of . The form of can be found by demanding that the unintegrated isotropic distribution depends only on the Lorentz scalar , . Even though the exact functional dependence is not known for a non-equilibrium system, the integrated isotropic distribution can be determined explicitly from the dimensionality of the integrand and the requirement of local energy conservation,

(4) |

For -integrated distributions, the usual relaxation time approximation with, say, Maxwell-Boltzmann distribution and collision kernel reduces to this model with where we used . The transport coefficients for this system are well known and read , and

(5) |

Propagating eccentricities. We want to investigate how azimuthal anisotropies in the transverse flow of energy density and non-linear correlations between these ’s arise in transport theory (2) for large mean free path (i.e. small ). Our starting point will be an azimuthally isotropic distribution with vanishing longitudinal component distributed in a spatial profile that is longitudinally boost invariant and composed of an azimuthally isotropic background with perturbatively small deviations from azimuthal symmetry

(6) |

The azimuthally symmetric initial background profile is of Gaussian form, with . The perturbations are pure in the -th harmonic, and similarly for the perturbation in with . Here, () denote the azimuthal angles in coordinate (momentum) space. Since the integration measure over the normalized longitudinal momentum is , the factor in front of (6) implies that is the initial energy density at the origin.

We want to determine how transport theory in the limit of large in-medium pathlength translates the small spatial perturbations into momentum anisotropies. To first order in (other analyses of this limit are found in Refs. Heiselberg:1998es (); Kolb:2000fha (); Borghini:2010hy (); Romatschke:2018wgi ()), this amounts to a one-hit dynamics in which the initial distribution (6) is free-streamed up to proper time ,

(7) |

before entering once the collision kernel. The latter depends on , and thus it depends on the velocity of the local rest frame and on the comoving energy density . Both need to be determined from the free-streaming distribution (7). To this end, we make a tensor decomposition of

(8) |

For the azimuthally symmetric background distribution , the eigensystem of the resulting energy momentum tensor can be given analytically in terms of the harmonic moments . In particular, with , and the flow field is purely radial . Although free-streaming of preserves the global coordinate space azimuthal symmetry of the system, the presence of non-zero harmonics in these expressions illustrates that local momentum space anisotropies develop over time. This is so, since the free-streaming distribution at any given position will depend on the direction in which particles propagate. Different positions have different anisotropies that the interactions tend to isotropize with the potential to create imbalance of the particle flow in the final state. In the azimuthally symmetric case the net effect is, however, zero and no anisotropy is generated. In contrast, in the case of an azimuthally anisotropic transverse profile momentum imbalance is generically formed as sketched in Fig. 1.

Small spatial eccentricities in the initial distribution (6) lead to computable perturbations of away from . It is then straightforward to determine the eigensystem of the full energy momentum tensor perturbatively from that of to the needed order in the ’s. For instance, the local energy density of the perturbed free-streaming solution is of the general form

(9) |

where the tilde indicates that the angular dependence has been factored out. Here, and equation (9) is written for a generalization of the initial conditions (6) for which different harmonic perturbations have different azimuthal reaction plane orientations , that means . These coefficient functions can be determined trivially in explicit form although they are generally lengthy. For instance

We have calculated analogously explicit but lengthy harmonic decompositions of the other eigenvalues and eigenvectors of the full perturbed energy momentum tensor that is obtained from free-streaming of (6). Armed with this information, we can determine the

Collision kernel and energy flow. Free-streaming leads to local azimuthal anisotropies of the spatial distribution, but the experimentally accessible momentum distribution remains azimuthally isotropic. To change the latter, collisions are needed to which we turn now. We are particularly interested in the measurable angular dependence of the radial flow of energy density at late time when space-time rapidity can be identified with momentum rapidity

(10) |

Since measures energy-momentum components per unit volume , the explicit -dependence arises here from . After the last scattering at time , the particles will free-stream from to . Since the -integration renders the free propagation trivial, , we can undo this free-streaming and write eq. (10) as

(11) |

Assuming only a single collision, the correction to the integrated distribution due to the interactions that take place in the infinitesimal time interval from to is given by . Then integrating over all possible times when the single interaction takes place, we get for the correction of order

(12) |

The collision kernel that enters this one-hit dynamics is known explicitly from (2) and (4) in terms of the free-streamed solution obtained from the initial condition (6), the energy density calculated as described in (9) and the rest-frame velocity that we have also calculated perturbatively in powers of the eccentricities . Given that the -dependencies of are known explicitly, we can Taylor expand in the eccentricities and perform the integrals in (11), thus obtaining the main result of this work

(13) | |||||

where we have followed the angular dependence analytically and performed the remaining - and -integrals numerically.

Dimensional analysis of the kinetic theory (2) reveals that irrespective of the initial conditions, physical results like (14) depend only on one particular combination of system size , initial energy density and scale of the inverse mean free path.

(14) |

Here, the latter form shows that the expansion parameter is proportional to the mean free path at time when flow is mainly generated; it is obtained with the help of . The one-hit dynamics studied here is a truncation of transport theory to first order in . This expansion is justified for but it breaks down for . This is clearly seen from the correction to free-streaming for an azimuthally symmetric distribution (the term ) that renders (13) unphysical for . This term arises from scatterings that transfer transverse into longitudinal momentum and it indicates that multiple scatterings need to be resummed for of this order.

Within its range of validity (), the main qualitative conclusion of (13) is that both, linear response to spatial eccentricities and non-linear mode-mode coupling are natural consequences of a perturbative one-hit dynamics and thus must not be taken for a tell-tale sign of a hydrodynamic mechanism at work. In fact, we could have easily expanded (13) to higher orders in eccentricities to obtain higher non-linear mode-mode couplings (such as ) or we could have included higher harmonics in the initial condition (6) to find e.g. linear response coefficients and . All the linear and non-linear structures observed in the azimuthal distributions of single inclusive hadron spectra can be obtained from transport models in the limit of long mean free path.

A quantitative comparison of the coefficients in (13) to other model calculations and data is complicated by several issues. For instance, coefficients are typically defined as azimuthal anisotropies of particle distributions . Instead, eq. (13) provides anisotropies of the -integrated transverse energy flow. Due to the different -weighting, , the anisotropies thus obtained can be larger. Also, the prefactors in (13) are model-dependent in the sense that they depend on the shape of the transverse profile of the initial condition. Only to the extent to which the dynamical response to main spatial characteristics such as and is robust against finer details of transverse profile, can the response coefficients calculated here be taken as indicative of the typical signal sizes obtained from transport theory in the one-hit limit. We note in this context that constructions of linear response coefficients by dividing data on with values of obtained from model calculations ALICE:2011ab (); Liu:2018hjh () show variations of depending on the model from which the eccentricities are calculated. While these considerations caution us that any numerical comparison at face value can only be indicative of order of magnitude effects, we nevertheless share in the following numerical observations to address the fundamental question whether a transport mechanism with significant mean free path can be sufficiently efficient to account for the observed signal size within the range of its validity.

We base the following numerical comparison on one particularly compact and recent compilation of linear and non-linear response coefficients, see Fig. 1 of Ref. Liu:2018hjh (). This study shows values for () that drop from () in the most central (0-5%) PbPb collisions at 2.76 TeV to values in the most peripheral 70-80% (in peripheral 40-50% centrality) collisions. At face value, if we assume in peripheral PbPb collisions of centrality, then these findings are consistent with the numbers extracted from (13),

(15) |

Analogously, one can construct linear response coefficients for central pPb collisions at 5.02 TeV by dividing the measured asymmetries (, Sirunyan:2017igb ()) by estimates of average eccentricities in hadron-nucleus collisions (, , Bozek:2013uha ()). Again, this lies in the ballpark of (15) for . Within the above-mentioned uncertainties, this indicates that the range of validity of one-hit dynamics extends to systems as large as those created in peripheral PbPb ( centrality) or central pPb collisions.

For a choice , also the size of the major non-linear mode-mode couplings extracted from (13)

(16) |

seem to be, within the stated uncertainties, consistent with the strength of the couplings in peripheral ( centrality) PbPb collision shown in Ref. Liu:2018hjh (), thus reinforcing the qualitative conclusion drawn from the comparison to linear response coefficients. However, the contribution to in (13) seems to be significantly larger than the value shown in Ref. Liu:2018hjh (). We note in this context that while the coefficients in (15) and (16) are generated within a time comparable to the system size (see Fig. 2), significant contributions to come in our calculation from times as late as . We refrain from speculating whether these or other reasons are at the origin of the observed discrepancy for .

Further qualitative and quantitative features of measured momentum anisotropies may also be consistent with one-hit dynamics. For instance, it has been argued in Borghini:2010hy () that the experimentally observed mass ordering of ’s can also result as a generic feature of one-hit dynamics. This is so, because the dynamics leading to ’s depends essentially on velocities, and this corresponds to a mass ordering in the transverse momenta.

Since the mean free path at the typical time at which scattering occurs in our calculation is , it follows from (14) that corresponds to collision per particle per system size. Since remains almost unchanged with system size while increases almost linearly with system size, the same transport theory that accounts with for peripheral PbPb collisions would automatically apply to more central nucleus-nucleus collisions. It is not the transport theory, but only its one-hit approximation studied here that becomes more questionable for increasing centrality; the intrinsic matter properties of the system remain (almost) unchanged with centrality.

It may be noteworthy that also the AMPT (see Figs. 1 and 2 of Ref. He:2015hfa ()) and BAMPS Greif:2017bnr () transport codes simulate central hadron-nucleus collsions with collision. Of course, the number or the mean free path is not free of model-dependent assumptions about the nature of the collision. In the transport theory (1), the mean free path corresponds to the time scale over which a distribution evolves to the isotropic one . If instead a more sophisticated collision kernel dominated by small-angle scatterings is implemented in a transport code, a larger number of collisions may be needed for isotropization. Thus, in any given transport code, the efficiency to isotropize will depend on the the microscopic dynamics (whose specification introduces some model-dependence); and in the present transport theory (1), this isotropization efficiency is parametrized in terms of one single parameter , irrespective of how it arises microscopically. There is no reason that eq. (1) must arrive at the same number of collisions than a transport code (that may supplement transport mechanisms by other model-dependent features such as string melting, hadronization prescriptions, etc), but the study of (1) teaches us that there is a class of transport models that can generate within the range of their validity a distribution (13) by a microscopic dynamics that realizes a particular value of .

Ultimately, the main motivation for studying soft particle production in hadron-nucleus and nucleus-nucleus collisions is to learn about fundamental properties of the dense QCD matter produced in these ultra-relativistic collisions. The question arises to what extent a system described by the transport theory studied here is qualitatively different or qualitatively similar to one described by an almost perfect liquid. Does a transport theory that is phenomenologically successful for a given system size formulate a dynamics which is consistent with a low viscous fluid dynamic description, or does it support a qualitatively different picture of the microscopic physics? For instance, for BAMPS to reproduce measured elliptic flow values, one requires non-perturbatively small values for the shear viscosity to entropy density ratio ( in the low temperature range in which flow is built up Uphoff:2014cba ()), thus suggesting that BAMPS implements a nearly perfect fluid dynamic picture for nucleus-nucleus collisions. We are unaware of a related statement for the transport code AMPT. We ask more generally whether a sufficiently efficient collective dynamics can be realized in transport codes also for interaction rates for which the medium can be viewed as a collection of quasi-particle excitations (). To shed light on this question, we focus now for the class of transport models (1) that are determined by a single isotropization time scale on the model-independent property that characterizes them in the hydrodynamic limit, namely the specific transport coefficient whose value is given in eq. (5).

We have argued that describes peripheral (50% centrality) PbPb collisions. The transverse energy produced in such collisions is set by measured quantitities, GeV Aamodt:2010cz () and their system size can be estimated from the nuclear overlap function at impact parameter fm Kolb:2001qz (), which is fm. In our set-up, converts these values to an energy density at time that takes the value . This leads to and, according to eq. (5), this translates into a shear viscosity

(17) |

Uncertainties about the centrality class for which enter this expression only via the fourth root of the system size and the transverse energy in that centrality class. This makes the above estimate robust. The first conclusion is that this shear viscosity is an order of magnitude larger than that of a perfect fluid. Second, in physical terms, while a system with does not support quasi-particles since their mean free path is much smaller than the typical size of their wave packet, the estimate obtained here is self-consistent but on the lower end of where a transport theory of quasi-particles is meaningful Danielewicz:1984ww (). Furthermore, it is remarkable that evaluating the perturbative QCD result Arnold:2000dr (); Arnold:2003zc (); Ghiglieri:2018dib () with realistic values for the QCD coupling constant results in a value for of similar magnitude. These considerations indicate that transport models of the type described here or modeled in transport codes implement indeed a picture of produced hot QCD matter that differs qualitatively from that of a perfect liquid.

We acknowledge helpful discussions with Ulrich Heinz.

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