Introduction
Integrated Hydro-Kinetic Model (iHKM) is the powerful tool for simulation of a relativistic heavy ion collision, which allows to describe or predict a wide variety of measured bulk observables. Naturally, a researcher is interested in finding the connection between different parameter values and the simulation results in order to have the possibility to better understand the nature of the investigated processes and to discover the properties and characteristics of hot dense matter, basing on the experimental data. In iHKM each stage is regulated by specific model parameters, and the aggregated effect of the matter’s evolving during all the stages defines the final state of the system and the behavior of various observables, measured in the experiments.
Hydrodynamics is considered now as the basic part of a spatiotemporal picture of the matter evolution in the processes of ultrarelativistic heavy ion collisions. To complete the description of A+A collision processes, hydrodynamics must be supplied with a generator of an initial nonequilibrated state, pre-thermal dynamics which forms the initial near locally equilibrated conditions for hydro-evolution, and prescription for particle production during the breakup of the continuous medium at the final stage of the matter expansion. Thus iHKM consists of such parts:
- generator of the initial state GLISSANDO
- pre-thermal dynamics of A+A collisions leading to thermalization
- subsequent relativistic viscous hydrodynamic expansion of quark-gluon and hadron medium
- its particlization
- hadronic cascade ultrarelativistic quantum molecular dynamics (UrQMD).
The Model Description
Initial state
The generation of the initial state is based on the GLISSANDO 2 package (http://www.ujk.edu.pl/homepages/mryb/GLISSANDO/index.html). This package works in the frame of the semiclassical model. Within this approach at the very initial stage of the collision, individual interactions between nucleons deposit transverse energy. Each deposition of the transverse energy at a certain space-time point or region is called a source and each source has its weight which is called relative deposited strength, or RDS. The simulation of a single event is made in three stages:
- Generation of the positions of nucleons in the two colliding nuclei according to the fluctuating nuclear density distribution. The form of the mean distribution depends on the mass of nuclei A. For sufficiently large nuclei this distribution has a Woods-Saxon form with an account of nuclear deformations, the latter are small in our case of Pb+Pb collisions.
- Generation of the transverse positions of the sources and their RDSs.
- Calculation of the physical quantities and writing the results in the output file.
Pre-thermal stage of the matter evolution
The generation of the initial energy density configuration with the help of GLISSANDO is supposed to be followed by a thermalization process. In iHKM we describe the matter evolution at the pre-thermal stage with the relaxation model. We assume a longitudinal boost invariance, which is a good approximation for the central rapidity region of the fireball at high collision energies. The relaxation model simulations are applied to the system for the early stage of the heavy ion collision, from the initial proper time to the termalization time. Thermalization time is supposed to be τ = 1.0 fm/c, the “conventional” time of the formation of strongly interaction matter, inverted mass of pion. At this stage the total energy-momentum tensor of the matter is taken in the form of combination of hydrodynamic (local equilibrium) and free (fully or almost free evolving) components.
For detailed description of relaxation model see e.g. S.V. Akkelin, Yu.M. Sinyukov, Phys. Rev. C 81, 064901 (2010) or V.Yu. Naboka, S.V. Akkelin, Iu.A. Karpenko, and Yu.M. Sinyukov, Phys. Rev. C 91, 014906 (2015), arXiv:1411.4490
Matter evolution in thermal and chemical locally near-equilibrated zone
After τ = 1 fm/c the the total energy-momentum tensor is equal to hydrodynamic one and further evolution is described by the relativistic viscous hydrodynamics. A numerical solution of the viscous hydrodynamic equations is constructed with the vHLLE code (https://arxiv.org/abs/1312.4160). Such an evolution describes the expansion of superdense quark-gluon and hadron matter close to local chemical and thermal equilibria with a baryon chemical potential equal to zero (which is a good approximation for LHC energies) until the temperature when such an approach breaks down. Then the system has lost the properties of local equilibrium, thermal and chemical as well, and another approximation should be used.
Particlization stage
iHKM describes particlization as a continuous process: particles gradually “escape” from the expanding fluid, forming the non-equilibrium Wigner function, which then can be used at some space-like hypersurface as the input for UrQMD hadronic cascade. At smooth initial conditions the results are very similar to a sudden particlization scenario and we are using the latter. We assume that the chemically and thermally locally equilibrated evolution takes place until temperature T = 165 MeV is reached, and switch to particle cascade at the hypersurface defined by this criterion. Such a switching surface is built during the hydrodynamic evolution with the help of the Cornelius routine (https://karman.physics.purdue.edu/OSCAR/index.php/CORNELIUS).
Hadronic cascade
The generated hadrons are then fed into the UrQMD cascade (https://urqmd.org/). Since the cascade accepts only a list of particles at an equal Cartesian time as an input, the created particles are propagated backward in time to when the first particle was created. The particles are not allowed to interact in the cascade until their trajectories cross the
particlization hypersurface. The Laine-Schroeder EoS, that is applied in our analysis corresponds to an equilibrium hadron-resonance gas consisting of about 360 hadrons in the low temperature limit. Many of those heavy hadrons are not included in the UrQMD hadron list. To prevent violation of the energy-momentum conservation even at space-like parts of
the isotherm T = 165 MeV, we decay the heavy resonances, which are not in the UrQMD list, just at the switching hypersurface. The particle propagation is stopped at Cartesian time 400 fm/c, where their coordinates and momenta are recorded. The generated sets of events are stored in ROOT trees and files. To describe or predict the different experimental observables, the special script-codes for each kind of them are applied.