## Periodic Reporting for period 1 - IMFDforHIC (Improved Fluid Dynamics for Relativistic Heavy Ion Collisions)

**Reporting period:**2016-01-01

**to**2017-12-31

## Summary of the context and overall objectives of the project

One of the fundamental questions in high-energy physics concerns the properties of strongly interacting matter at high temperature or density. At sufficiently high temperature one expects a transition from hadronic matter to a deconfined state, the so-called quark-gluon plasma (QGP), where the degrees of freedom are quarks and gluons, instead of their bound states, hadrons.

Such matter can be experimentally studied in relativistic heavy-ion collisions. There are currently two major heavy-ion colliders, the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) and the Large Hadron Collider (LHC) at CERN, where large-scale experiments investigate such matter. Currently, there are strong indications that a small droplet of nearly thermalized QGP is indeed formed in these collisions. Extracting the properties of the matter from experimental data is, however, challenging, and requires a good understanding of the dynamical evolution of the system. With the present computational techniques it is not possible to solve the evolution directly from the theory of strong interactions, QCD, but phenomenological models are needed to describe the evolution, and determine how the properties of the matter are reflected in the experimental observables.

In order to reliably extract the properties of the formed matter, it is essential that the models describe simultaneously as many experimental observables as possible. Furthermore, it is important that the validity of the theoretical models and uncertainties associated with the used approximations and input parameters are properly addressed. The main goals of the proposed research are: (i) reduce and quantify the uncertainties in the modeling of the space-time evolution of the system formed in the collisions, and (ii) find constraints for the unknown properties of strongly interacting matter from the currently available experimental data.

Such matter can be experimentally studied in relativistic heavy-ion collisions. There are currently two major heavy-ion colliders, the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) and the Large Hadron Collider (LHC) at CERN, where large-scale experiments investigate such matter. Currently, there are strong indications that a small droplet of nearly thermalized QGP is indeed formed in these collisions. Extracting the properties of the matter from experimental data is, however, challenging, and requires a good understanding of the dynamical evolution of the system. With the present computational techniques it is not possible to solve the evolution directly from the theory of strong interactions, QCD, but phenomenological models are needed to describe the evolution, and determine how the properties of the matter are reflected in the experimental observables.

In order to reliably extract the properties of the formed matter, it is essential that the models describe simultaneously as many experimental observables as possible. Furthermore, it is important that the validity of the theoretical models and uncertainties associated with the used approximations and input parameters are properly addressed. The main goals of the proposed research are: (i) reduce and quantify the uncertainties in the modeling of the space-time evolution of the system formed in the collisions, and (ii) find constraints for the unknown properties of strongly interacting matter from the currently available experimental data.

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

A conventional way to derive fluid-dynamical equations of motion from the Boltzmann equation is to expand the momentum distribution function around an isotropic local equilibrium distribution function. However, especially in the early stages of the stages of relativistic heavy-ion collisions, the deviations from equilibrium can be significant, and such an expansion is expected to break down. Nevertheless, one can still derive a fluid-dynamical theory, called anisotropic dissipative fluid dynamics, in terms of an expansion around an anisotropic single-particle distribution function, which incorporates at least parts of the momentum anisotropy. We have constructed the basic formalism for such an expansion in a way that can be systematically improved. We derived the equations of motion of anisotropic dissipative fluid dynamics starting from the Boltzmann equation.

We have then applied the above formalism for the one-dimensional Bjorken expansion, which is in particular relevant for the early stages of a heavy-ion collision. In order to close the equations of motion of anisotropic fluid dynamics, one needs to choose an additional moment of the Boltzmann equation. In this work we considered several choices and compared the corresponding fluid-dynamical solutions to the exact solution of the Boltzmann equation. We were then able to identify the best choice for the moment, and showed that the formalism can be extended to situations where the system is strongly out-of-equilibrium.

We analyzed the directed flow in high-energy heavy-ion collisions in the energy range from 7.7 to 27 GeV as a signature for the softening of the equation state (EOS), i.e., as a signature of the QCD phase transition. These lower-energy collision probe in particular the properties of strongly interacting matter at higher baryon density. We gave a detailed analysis of how directed flow is generated during the collision. In particular, we found that the softening of effective EOS in the overlapping region of the two colliding nuclei, i.e., the reaction stages where the system reaches a high-baryon density state, is needed to explain the observed collapse of the proton directed flow within a hadronic transport approach.

This study was further extended to study the sensitivity of the directed flow on the EOS. The EOS was modified by introducing a new collision term in order to control the pressure of a system by appropriately selecting an azimuthal angle in two-body collisions according to a given EOS. The beam-energy dependence of the directed flow of protons was examined with two different EOS, one with a first-order phase transition and one with a crossover. It was found that our approach yields quite similar results as hydrodynamical predictions on the beam-energy dependence of the directed flow: Transport theory predicts a minimum in the excitation function of the slope of proton directed flow and does indeed yield negative directed flow, if the EOS with a first-order phase transition is employed. Our result strongly suggests that the highest sensitivity for the critical point can be seen at beam energies from 4.7 to 11.5 GeV.

In addition, we have compared numerical solutions of conventional fluid dynamics and with solutions of the Boltzmann equation in several situations resembling the ones encountered in heavy-ion collisions. A most significant technical challenge has been to solve the Boltzmann equation numerically with sufficient numerical accuracy. We believe that we have now reached such accuracy, and that we can make meaningful comparisons with fluid-dynamical solutions. We are currently finishing a first publication of this part of the project, where we analyze the conditions for the applicability of fluid dynamics in describing the space-time evolution of the small amount of matter formed in nuclear collisions.

We have also made several predictions for different flow correlations measured by the ALICE Collaboration at the LHC and STAR Collabor

We have then applied the above formalism for the one-dimensional Bjorken expansion, which is in particular relevant for the early stages of a heavy-ion collision. In order to close the equations of motion of anisotropic fluid dynamics, one needs to choose an additional moment of the Boltzmann equation. In this work we considered several choices and compared the corresponding fluid-dynamical solutions to the exact solution of the Boltzmann equation. We were then able to identify the best choice for the moment, and showed that the formalism can be extended to situations where the system is strongly out-of-equilibrium.

We analyzed the directed flow in high-energy heavy-ion collisions in the energy range from 7.7 to 27 GeV as a signature for the softening of the equation state (EOS), i.e., as a signature of the QCD phase transition. These lower-energy collision probe in particular the properties of strongly interacting matter at higher baryon density. We gave a detailed analysis of how directed flow is generated during the collision. In particular, we found that the softening of effective EOS in the overlapping region of the two colliding nuclei, i.e., the reaction stages where the system reaches a high-baryon density state, is needed to explain the observed collapse of the proton directed flow within a hadronic transport approach.

This study was further extended to study the sensitivity of the directed flow on the EOS. The EOS was modified by introducing a new collision term in order to control the pressure of a system by appropriately selecting an azimuthal angle in two-body collisions according to a given EOS. The beam-energy dependence of the directed flow of protons was examined with two different EOS, one with a first-order phase transition and one with a crossover. It was found that our approach yields quite similar results as hydrodynamical predictions on the beam-energy dependence of the directed flow: Transport theory predicts a minimum in the excitation function of the slope of proton directed flow and does indeed yield negative directed flow, if the EOS with a first-order phase transition is employed. Our result strongly suggests that the highest sensitivity for the critical point can be seen at beam energies from 4.7 to 11.5 GeV.

In addition, we have compared numerical solutions of conventional fluid dynamics and with solutions of the Boltzmann equation in several situations resembling the ones encountered in heavy-ion collisions. A most significant technical challenge has been to solve the Boltzmann equation numerically with sufficient numerical accuracy. We believe that we have now reached such accuracy, and that we can make meaningful comparisons with fluid-dynamical solutions. We are currently finishing a first publication of this part of the project, where we analyze the conditions for the applicability of fluid dynamics in describing the space-time evolution of the small amount of matter formed in nuclear collisions.

We have also made several predictions for different flow correlations measured by the ALICE Collaboration at the LHC and STAR Collabor

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The formalism of anisotropic fluid dynamics that we have developed forms a basis for relativistic fluid dynamics in situations where either the equilibrium state is not isotropic, e.g. due to the presence of magnetic field, or that the system deviates strongly from the isotropy due the external conditions. It can be used as a starting point for developing more advanced theories, since in this formalism they can be systematically improved.

We have also extended the theory of relativistic fluid dynamics to situations where a strong magnetic field affects the dynamics, and analyzed the conditions for the applicability of fluid dynamics.

Moreover, we have shown that we can describe a wide range of different collision systems and in particular we can constrain the temperature dependence of the QCD shear viscosity by analyzing several collision energies simultaneously.

We have also extended the theory of relativistic fluid dynamics to situations where a strong magnetic field affects the dynamics, and analyzed the conditions for the applicability of fluid dynamics.

Moreover, we have shown that we can describe a wide range of different collision systems and in particular we can constrain the temperature dependence of the QCD shear viscosity by analyzing several collision energies simultaneously.