### Microscopic Transport Model Primer

Microscopic transport models aim at describing the time-evolution of a heavy-ion collision, either in part or from beginning to end, using microscopic degrees of freedom, i.e. nucleons, hadrons or quarks and gluons. They are based on the solution of a *transport equation*, which can be derived within the framework of kinetic theory. The two most common transport equations upon which such transport models are based on are the *Boltzmann* and the *Vlasov* equations. Both can be derived from the Liouville equation using the BBKGY hierarchy.

The earliest incarnations of microscopic transport models for relativistic heavy-ion collisions date back to 1980 and are the so-called Intra-Nuclear Cascade (INC) models, which are essentially based on a Boltzmann equation and describe the evolution of the reaction as a sequence of binary scatterings among the nucleons and produced hadrons. The first INC codes were developed by Y. Yariv and Z. Fraenkel in 1979 as well as by J. Cugnon in 1980. However, the development of the INC model dates as far back as the work by M. Goldberger in 1948 and earlier work by Chen & Fraenkel et al. in 1968.

In parallel to the development of the INC models, which are based on the assumption that interactions among the hadrons occurs solely via two-body scattering, the development of mean-field models progressed, which assume that all interactions can be described via potentials. Classically this can be described via a Vlasov equation. On the quantum level such an assumption leads to time-dependent Hartree-Fock calculations (TDHF). Note that particle production poses a large conceptual problem for TDHF calculations and severely limits their applicability in the relativistic domain. The first viable models combining interactions based on two-particle scattering with the nuclear mean-field (or in terms of transport equations, combining the Boltzmann and Vlasov equations) were the Vlasov-Uehling-Uhlenbeck (VUU) and Boltzmann-Uehling-Uhlenbeck (BUU) models:

- Vlasov-Uehling-Uhlenbeck Theory Of Medium-Energy Heavy Ion Reactions: Role Of Mean Field Dynamics And Two-Body Collisions
- Numerical simulation of medium energy heavy ion reactions

Other notable VUU/BUU implementations, which are used to this day, include:

- Energetic photons from intermediate energy proton- and heavy-ion-induced reactions
- Relativistic Vlasov Equation For Heavy Ion Collisions
- The Relativistic BUU approach: Analysis of retardation effects and meson field radiation
- Formation of superdense hadronic matter in high-energy heavy ion collisions

Among the drawbacks of the VUU/BUU model was that the underlying Vlasov-Boltzmann equation is a one-body equation, i.e. that correlations among particles were not properly treated in that equation. In order to overcome this problem and be able to describe the production of nuclear fragments in intermediate energy heavy-ion reactions, the Quantum-Molecular-Dynamics (QMD) model was developed by Aichelin and Peilert:

- Importance Of Momentum Dependent Interactions For The Extraction Of The Nuclear Equation Of State From High-Energy Heavy Ion Collisions
- ‘Quantum’ molecular dynamics: A Dynamical microscopic n body approach to investigate fragment formation and the nuclear equation of state in heavy ion collisions

Note that the QMD approach can be seen as an extension to the classical Molecular Dynamics (MD) approach, well known in chemistry, which was first applied to the physics of heavy-ion collisions by Bodmer & Panos in 1977 and Yariv & Wilets in 1978. A very good review on (microscopic) transport theory and their application to probing the equation of state of hot and dense matter created in relativistic heavy-ion collisions can be found here.

The first QMD version to explicitly incorporate isospin degrees of freedom was the IQMD model, developed by C. Hartnack on the basis of H. Stöcker’s original VUU code. IQMD then served as the basis for the development of the Relativistic Quantum Molecular Dynamics (RQMD) model, which sought to extend the (I)QMD model’s reach to ultra-relativistic energies, by expanding its collision term and introducing Hamiltonian Constrained Dynamics for the proper relativistic propagation of particles under the influence of a two-body potential. The RQMD model served for many years as the workhorse for the analysis of experiments performed at the AGS and SPS accelerator facilities.

The Ultra-relativistic Quantum Molecular Dynamics Model (UrQMD) was developed on the basis of the RQMD principles, but with an improved and vastly extended collision term, (see also here for a description of included cross sections), and to this day is being employed from SIS energies up to RHIC energies. Other 3rd generation hadronic string/transport models include HSD (developed by W. Cassing, W. Ehehalt and E. Bratkovskaya), URASiMA (purely hadronic, developed by S. Date et al.) and JAM (developed by Y. Nara). UrQMD and JAM also serve as hadronic “afterburners” to describe the late hadronic (break-up) stage of hydrodynamic calculations (in so-called hybrid hydro+micro models) and UrQMD is used as a module in cosmic air-shower simulations (for the CORSIKA and SENECA packages).

All of the above mentioned transport models treat the heavy-ion reaction in terms of hadronic degrees of freedom. In the case of RQMD, HSD, JAM and UrQMD these are augmented by constituent quarks and di-quarks formed as intermediate states of a string fragmentation. However, these only exist within the confines of the pre-formed hadrons they will ultimately end up in and not as deconfined states. For that reason the medium described by all of these approaches does not constitute a Quark-Gluon-Plasma (QGP), but a string/hadron gas. Applying these models to ultra-relativistic collisions therefore constitutes a null-hypothesis, namely the assumption that no QGP has been formed. The idea is that if the model predictions vary significantly from the data, this could be interpreted as a sign of new physics phenomena (e.g. QGP formation) evidence in the data.

In the early 1990’s the Parton Cascade Model (PCM), based on the Boltzmann equation, was developed by K. Geiger and B. Müller in order to describe the evolution of an ultra-relativistic nucleus-nucleus collision based quark and gluon degrees of freedom:

- Dynamics of parton cascades in highly relativistic nuclear collisions
- Space-time description of ultrarelativistic nuclear collisions in the QCD parton picture

After the tragic and untimely death of Klaus Kinder-Geiger in the crass of Swissair flight #111, further development of Geiger’s PCM implementation was taken over by S.A. Bass, B. Müller and D.K. Srivastava. Other notable PCM implementations or transport models including a PCM stage are:

- ZPC 1.0.1: A Parton cascade for ultrarelativistic heavy ion collision
- Elliptic flow and freezeout from the parton cascade MPC
- A multiphase transport model for nuclear collisions at RHIC (AMPT)

One of the shortcomings of the original PCM approach is its restriction to cross sections calculated in the framework of perturbative QCD, which are at most on the order of a few millibarn. Cross sections of this magnitude are not large enough to thermalize the medium on the time-scale of a relativistic heavy-ion reaction and to account for the amount of collectivity and the near ideal fluid like behavior observed in collisions at RHIC, if one restricts the PCM to binary scatterings and subsequent radiation. Also, due to the momentum cut-off needed to regulate the IR-divergence of the pQCD cross section, soft parton-parton interactions which contribute considerably to the total hadron-hadron cross section are not taken into account. It has been demonstrated by the practitioners of the MPC and AMPT approaches that one way of generating the collectivity observed at RHIC is to increase the parton cross sections by factors of 4-10. However, it remains unclear what the physics underlying this change in cross section would be.

One way to avoid the ad-hoc introduction of a momentum cut-off in order to regulate the IR-divergence of the pQCD cross section is to take the screening properties of the colored medium into account – this had lead to the formulation of the Selfscreened Parton Cascade Model (SPCM).

Another issue microscopic transport models at ultra-relativistic energies have to deal with is the occurrence of acausalities due to the superluminous propagation of information: this takes place over length-scales associated with the finite size of the cross section, since standard collision term implementations execute an interaction at a fixed time either in the hard sphere limit when the colliding particles *touch* or at the point of closest approach among the colliding degrees of freedom. In very dense systems with large cross sections this can lead to significant violations of covariance. One possible solution to this problem utilized in the MPC approach is the subdivision of physical particles into test particles which only carry a fraction of the cross section: in the limit of infinite test particles per physical particle and thus vanishing cross sections this approach becomes fully covariant. Another method of dealing with this problem is the introduction of cells and collision-rates based on the test-particle densities in those cells instead of the tracking of individual trajectories and two-particle interactions. This method has been pioneered by the Giessen group in the context of their RBUU model and has now been taken up by Greiner et. al (see below).

Recently, Xu and Greiner introduced a PCM implementation which includes multi-parton rescattering, i.e. a consistent treatment of partonic 2 -> 3 and 3 -> 2 interactions. Their key finding is that the inclusion of these processes can account for rapid thermalization and the observed flow collectivity at RHIC:

- Thermalization of gluons in ultrarelativistic heavy ion collisions by including three-body interactions in a parton cascade
- Transport rates and momentum isotropization of gluon matter in ultrarelativistic heavy-ion collisions

Note that the key physics component for achieving the rapid thermalization in these calculations is the particular implementation of the Landau Pomeranchuk Migdal (LPM) effect, which yields a nearly isotropic emission of the radiated gluon in the 2 -> 3 process. This LPM implementation differs significantly from the ones chosen in the other PCM models.

In order to properly account for soft non-perturbative interactions among partons, one would need to incorporate a Vlasov term into the PCM, i.e. a partonic VUU model. Such an approach has been formulated in:

- Quark – Gluon Transport Theory. Part 1. the Classical Theory
- Quark – Gluon Transport Theory. Part 2. Color Response And Color Correlations In A Quark – Gluon Plasma
- Lattice gauge description of colliding nuclei

For many years, the implementation of such a scheme presented a methodological and computational challenge, but recently significant progress has been made by the Frankfurt group: