# Hydrodynamics and jets in dialogue

## Abstract

Energy and momentum loss of jets in heavy ion collisions can affect the fluid dynamic evolution of the medium. We determine realistic event-by-event averages and correlation functions of the local energy-momentum transfer from hard particles to the soft sector using the jet-quenching Monte-Carlo code Jewel combined with a hydrodynamic model for the background. The expectation values for source terms due to jets in a typical (minimum-bias) event affect the fluid dynamic evolution mainly by their momentum transfer. This leads to a small increase in flow. The presence of hard jets in the event constitutes only a minor correction.

## Keywords

Source Term Fluid Velocity Hydrodynamic Evolution Bulk Viscous Pressure Momentum Deposition## 1 Introduction

For a phenomenological, fluid dynamic description of heavy ion collisions it is usually assumed that the bulk of the medium produced after a heavy ion collision is in local thermal equilibrium. While this may be a reasonable approximation for the low-\(p_\perp \) part, it is phenomenologically clear that high-\(p_\perp \) particles do not originate from a locally equilibrated or thermal distribution. It is an interesting question how this non-equilibrated part influences the hydrodynamical evolution of the bulk. The most important effect in this regard might be due to the energy loss of high-\(p_\perp \) particles propagating in the medium (jet quenching) which leads to a transfer of energy and momentum to the bulk part described by fluid dynamics. This could have important implications for the interpretation of soft observables (e.g. the anisotropic flow coefficients \(v_n\)) as well as jet measurements [1], which rely on background subtraction techniques assuming currently that the soft event and jets are uncorrelated. On the other side one expects that local fluid properties determine the strength of the energy loss. Generally speaking, energy loss is expected to be stronger for a denser medium. More specific, the jet-quenching parameter \(\hat{q}\) is expected to depend on the temperature and other parameters of the medium.

The energy loss of hard partons due to induced gluon radiation in a hydrodynamic background has been studied in various approaches [2, 3, 4, 5]. They are, however, not suited for quantifying the energy and momentum deposition into the bulk. Firstly, they operate in a high energy limit, where there is no collisional energy loss, and secondly, they do not keep track of radiated gluons. It thus has to be assumed that all radiated energy gets dissipated locally, which is clearly a bad approximation for energetic emissions. This is different in Monte Carlo codes aiming at a consistent description of the entire jet and its interactions in a background described by hydrodynamics [6, 7, 8]. They can trace all radiated partons, but also here the interactions between the jets and the bulk are accounted for in an effective way that cannot easily be translated into a local energy and momentum transfer between jets and background.

The influence of jets on the hydrodynamic evolution of the bulk was first discussed in the context of Mach cone formation [9, 10, 11, 12, 13, 14] and has been extended recently to other observables [15, 16]. Within AdS/CFT the interplay between the energy loss of a heavy quark and hydrodynamic excitations has been discussed in detail, see [17, 18] for an overview. In that context the source term can be extracted unambiguously from the energy and momentum loss of a (single) heavy quark. Arguably, holographic models provide currently the only consistent field theoretic framework for calculating source functions from first principles. The weakness of this approach is, however, that it is unclear to what extent it captures dominant features of jets propagating in a QCD medium.

The studies in QCD, on the other hand, feature an elaborate treatment of the hydrodynamic side of the problem, but have very simplified models for the energy and momentum deposited by the jets.

In general, a fully self-consistent description of the soft medium and high-\(p_\perp \) part of the spectrum with its mutual interactions in QCD can be a rather difficult task (first steps in this direction are taken by transport codes [19, 20, 21]). On the other side, the phenomenological success of the current fluid dynamic model, which neglects the influence of non-equilibrated hard particles completely, suggests that the influence of the latter is not too large. In order to investigate this question more quantitatively, we employ here a non-self-consistent description where the bulk medium is first described in terms of conventional fluid dynamics neglecting non-thermal components. This leads in particular to a temperature and fluid velocity profile as a function of the space-time coordinates. In a second step we use these results to estimate the local transfer of energy and momentum from the hard particles to the medium. This results effectively in an additional source term in the fluid dynamic evolution equations. The influence of this source for fluid dynamics can then be estimated in a third step. The effect of a fourth step, namely re-calculating the jets in the modified background, is expected to be numerically small and can thus be neglected.

In this study the jets are simulated with Jewel [22], which employs a microscopic description for the interactions of the jets in the background. As a first approximation one can thus interpret the energy and momentum flow in the individual scattering processes as the energy-momentum exchange between the jets and the background. This provides a realistic and well constrained model for the local energy-momentum transfer to the bulk. Moreover, Jewel provides a realistic ensemble of jets including their spatial and kinematic distributions and event-to-event fluctuations, e.g. in the jet fragmentation. This allows us to study statistical properties, in contrast to earlier approaches that have mainly focused on the medium response to single (simplified) jets.

This paper is organized as follows. In Sect. 2 we discuss the fluid dynamic evolution of the bulk including source terms for energy and momentum transferred from the hard sector. In Sect. 3 we introduce the description of jets and in the subsequent Sect. 4 we quantify the local energy and momentum transfer in terms of expectation values and correlation functions. Finally, we draw some conclusions in Sect. 5.

## 2 The hydrodynamic evolution

The fluid dynamic evolution equations (2.5) and (2.8) have to be supplemented by constitutive equations for the shear stress tensor and bulk viscous pressure. In a first order (Navier–Stokes type) formalism these are of the form (2.6), in a second order formalism these equations get supplemented by relaxation time terms. To solve the evolution equations one also needs an equation of state that relates pressure and energy density as well as the transport coefficients \(\eta \) and \(\zeta \) (and possible further coefficients such as relaxation times).

So far, we have not yet specified the source terms on the right hand side of Eqs. (2.5) and (2.8). If these correspond to high-momentum, non-equilibrated particles, they are in general different for each event. One might attempt at this point to implement an event-by-event description of fluid dynamics and a model for the high-momentum particles coupled to each other. On a technical level this becomes quickly rather involved. There is also a conceptual difficulty of drawing a line between the high-momentum part of the medium that is usually described in a microscopic way in terms of single particle excitations or partons and the low-momentum part that is described in a more macroscopic way in terms of fluid dynamics.

*statistical ensemble*of sources \(J^\nu \) or, equivalently, of a source component parallel to the fluid velocity,

Let us now specialize our considerations to a situation with Bjorken boost and azimuthal rotation symmetry. For the fluid dynamic fields (enthalpy density \(w=\epsilon +p\), fluid velocity \(u^\mu \), shear stress \(\pi ^{\mu \nu }\) and bulk viscous pressure \(\pi _\text {bulk}\)) this implies that they can depend only on Bjorken time \(\tau \) and radius \(r\) (but not on rapidity \(\eta \) and azimuthal angle \(\phi \)). For the fluid velocity only the components \(u^\tau \) and \(u^r\) can be non-zero and similar for the shear stress tensor \(\pi ^{\mu \nu }\). For the ensemble of sources \(J_S\) and \(J_V\) we assume that Bjorken boost and azimuthal rotation invariance are realized in a *statistical sense*. For the expectation values in (2.13) this has the same implications as for the hydrodynamical fields. For the correlation functions in (2.14) the situation is more complicated since they can depend also on the differences in rapidity and azimuthal angle between the two space-time points.

One observes that the source terms \(\bar{J}_S\) and \(\bar{J}_V\) have two effects: one is a slight increase in temperature at small radii and at early times, which is an expected effect from dissipation. The other is an increase of radial flow at intermediate times, the jets drag the fluid outwards. The slight decrease of the temperature in the center and the increase at large radii at later times are also a consequence of the larger flow. Both effects are relatively small. The change in the (averaged) temperature evolution seems to be negligible for practical purposes. The effect of the additional dissipated energy is hardly visible in Fig. 1, while the larger radial flow leads at larger radii to an increase in temperature of a few percent. For the radial component of the fluid velocity this effect is more direct and leads to an increase up to about 10 %.

In our setup the expectation values \(\bar{J}_S\) and \(\bar{J}_V\) are by construction symmetric under azimuthal rotations and can therefore not contribute to the harmonic flow coefficients \(v_m\). The effect of energy and momentum transfer from jets to the medium on these observables is encoded in correlation functions as in Eq. (2.14). We plan to investigate this more quantitatively in a separate publication.

## 3 Jet quenching in a hydrodynamic background

Let us now describe the formalism we use for the description of jets. Jets are simulated in Jewel [22] with the hydrodynamic calculation presented in Sect. 2 as background.

^{1}Keeping the leading terms only and introducing the infra-red regulator \(\mu _\text {D} \approx 3 T\) the partonic cross section reduces to

The parton shower thus generates all emissions—those associated to the QCD evolution of the jet (which would also take place in the absence of the background) and those initiated by re-scattering. In fact, it is generally impossible to assign an emission to a particular scattering process. The interplay of competing sources of radiation as well as the LPM interference are governed by the formation times of the emissions. When two emissions take place at the time the one with the shorter formation is formed as an independent particle and all scattering process within the formation time of an emission act coherently.

The local scattering rate is given by the product of the parton density and the scattering cross section (3.3) (taking care of the color factors for different parton species). When a scattering takes place a scattering center is generated from the local thermal distribution and the scattering process is simulated explicitly. The scattering centers are dynamical and recoil against the hard parton. This allows one to keep track of the energy and momentum exchange between the jet and the background.

When Jewel runs with the hydrodynamic background described in Sect. 2 it takes the temperature \(T(x)\) and transverse fluid rapidity \(\beta (x)\) (related to the radial component of the fluid velocity by \(u^r=\sinh \beta (x)\)) as input. The parton densities and momentum distributions are then computed assuming an ideal gas equation of state.

## 4 Characterizing the source term

For the most central collisions (\(b=0\)) the averaged source term \(\langle J^\mu \rangle \) is azimuthally symmetric. It is, however, not boost invariant, since the jet production cross section is rapidity dependent^{2} and the energy loss itself can in general also be rapidity dependent. For simplicity, we extract the source term only in the central unit of rapidity, where it varies only mildly, to preserve the symmetry of the background (the extension to a non-trivial rapidity dependence is straightforward). Consequently, \(\langle J^\mu \rangle \) depends only on \(\tau \) and \(r\) and not on \(\phi \) and \(\eta \). The projections of \(\langle J^\mu \rangle \) parallel and orthogonal to the fluid velocity are computed using for \(u^\mu \) the solution to the hydrodynamic equations without the source term. This is a good approximation as long as the source term is small, when it is not small the procedure may be iterated using the new solution. Concerning the source term from jet energy loss, we assume the nucleon–nucleon collisions in a nucleus–nucleus event to be independent. Then the expectation values \(\bar{J}_S\) and \(\bar{J}_V^\mu \) scale trivially with the number of di-jets. The results presented in this section are averaged over \(\phi \) and the central unit in \(\eta \).

In the soft QCD mode the average jet \(p_\perp \) is smaller than in the perturbative mode. The energy and momentum deposited in the medium per jet is thus lower in the former. In addition, the number of jets per event is also smaller due to the smaller cross section.

In the hard di-jet scenario a cut of \(p_{\perp ,\text {cut}} = 100\) GeV is placed on the matrix element. The final jet population looks very different due to quenching of the jets. When comparing to experimental data one would have to place the cut on the final jet \(p_\perp \), which is straightforward but not necessary for this exploratory study. In Fig. 7 the momentum deposition of such a hard di-jet is compared to a minimum-bias di-jet. As expected, the source term of hard jets is much larger in magnitude and extends to significantly later times. The energy transfer \(\bar{J}_S\) from hard di-jets follows the approximate scaling with \(T^2(\tau ,r)\cdot N_\text {coll}(r)\) for much longer since they do not reach thermal scales quickly. Deviations come from the dilution of the \(N_\text {coll}(r)\) distribution due to the propagation of the jets and an increase in the multiplicity of hard partons due to splitting. The same effects are also at work in \(\bar{J}_V\).

To obtain the source term for the entire event one has to add to the contribution of the hard di-jet \(N_\text {di-jet}-1\) times that of a minimum-bias jet. A \(\mathcal {O}(100)\) GeV di-jet deposits roughly a factor of 40 more energy and momentum than a minimum-bias jet. Since the number of minimum-bias di-jets per event is of the order 1500, the presence of a hard di-jet increases the energy transfer per event only by about 2–3 %.

The correlators generate potentially sizeable contributions to correlation observables on the fluid dynamic side such as the anisotropic flow coefficients \(v_n\). The calculations are somewhat more involved than for the averaged source terms and will be discussed in a separate publication.

## 5 Conclusions

We have studied the influence of the energy deposition by jets onto the evolution of the medium by combing a realistic microscopic jet quenching model with a fluid dynamic description of the bulk. The energy-momentum transfer from jets constitutes source terms in the hydrodynamic evolution equations, which we characterize in terms of event averages and correlation functions. The event-averaged source function that enters the time evolution of the energy density is largest at early times and for small radii. It leads to an increase of temperature due to the additional dissipated energy but the effect is numerically very small. The expectation value for the source function that enters the evolution of the radial fluid velocity peaks at intermediate times (a few fm/c) and for large radii (about 6 fm/c). This term gives the effect of the force opposing drag and leads to an increase of radial flow of up to about 10 %: The jets drag the fluid outwards. The momentum transfer causing the increase in radial flow was shown to have a non-trivial functional form, which is not easily captured by simple parametrizations of the source term. This highlights the advantage of constructing a realistic source term using a model based on microscopic dynamics.

Our formalism allows one to study also event-by-event fluctuations in the source terms. Here we quantify connected two-point correlation functions and find that they are largest at early times and that they are local (the correlation functions peak strongly for equal space-time arguments).

A conceptual difficulty of a formalism that combines a microscopic description of jets with a macroscopic description of the medium is that the separation between the two components is to some extent arbitrary. This becomes apparent in the difficulties related to defining the jet population. We chose to regularize the perturbative jet cross section such that jets are produced predominantly in the phase space region where they dominate over the thermal distribution. This leads to a rather low \(p_\perp \) cut-off of the order a few GeV. In this region the perturbative cross section has large uncertainties and multi-parton interactions may play a role. These difficulties are extenuated to some extent by the fact that—due to the fact that we employ a dynamical model of jet quenching—very soft partons on average do not lose energy and thus do not contribute to the fluid dynamic source terms. Nevertheless, we estimate the resulting uncertainties for the latter to be of the order of a factor 2 or 3.

We studied the source terms generated in a ‘typical’ event, i.e. without cuts on the jets, and the effect of a high-\(p_\perp \) (\(\mathcal {O}(100)\) GeV) di-jet. In events containing a hard di-jet the energy and momentum deposition is increased by only a few percent as compared to the minimum-bias scenario. The presence of a hard jet is thus negligible for global observables. This can be different for correlation observables such as harmonic flow coefficients, which receive potentially sizable contributions from jets. These will be studied in an upcoming publication.

## Footnotes

- 1.
In principle also the thermal scattering center can emit such radiation, but this is neglected in the current Jewel implementation due to very limited phase space.

- 2.
The jet production cross section depends on momentum rapidity, which is correlated to the space-time rapidity, since jets are produced at \(t=0\) and at (or close to) \(z=0\).

## Notes

### Acknowledgments

We would like to thank J. G. Milhano and U. A. Wiedemann for valuable comments on the manuscript.

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