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E. L. Bratkovskaya et al.
properties of the medium generated on short time scales. Since relativistic heavy-ion
collisions start with impinging nuclei in their ground states a proper non-equilibrium
description of the entire dynamics through possibly different phases up to the final
asymptotic hadronic states—eventually showing some degree of equilibration—is
mandatory.
About 40 years ago cascade calculations have been employed for the description
of nucleus–nucleus collisions in the 1-2 AGeV range [1] which provided already
some good idea about the reaction dynamics including essentially nucleons, -
resonances, pions, and kaons. These calculations have been based on the Boltzmann equation which, however, is entirely classical and lacks quantum statistics
appropriate for fermions and bosons. In particular the Pauli-blocking for nucleons
was found to be essential at lower bombarding energies and cascade calculations
were extended in line with the Uehling-Ulenbeck equation for fermions [2] incorporating also some mean-field potential calculated in Hartree approximation with
various two-body Skyrme forces. These types of transport models are denoted as
Boltzmann-Uehling-Uhlenbeck (BUU) or Vlasov-Uehling-Uhlenbeck (VUU) models [3, 4] and are still in use nowadays by some groups. Independently, Quantum
Molecular Dynamical (QMD) models [5] have been proposed in which the test
particles of the BUU/VUU approaches are replaced by Gaussians allowing for the
simulation of single events while keeping the fluctuations. Explicit isospin degrees
of freedom have been incorporated in IQMD [6], too. Since these types of models are based on a Hamiltonian with fixed two-body forces one could evaluate the
nuclear equation of state (EoS) at zero temperature or in thermal equilibrium and
one of the primary issues was to extract the nuclear EoS from heavy-ion data by
means of BUU/VUU or QMD calculations. Later on higher baryonic resonances as
well as mesons like η, K
±
, K
0
, ¯
K
0
, ρ, ω, φ have been incorporated which led to
coupled-channel BUU (CBUU) approaches.
Apart from adding more hadronic degrees of freedom in BUU/VUU fully relativistic formulations have been carried out on the basis of some Lagrangian density
including a selected set of hadronic degrees of freedom [7–9]. All baryons in such
relativistic BUU (RBUU) models were propagated with scalar and vector self energies that were matched to reproduce collective flow data from heavy-ion collisions as
well as particle spectra. This was a necessary step to go ahead in bombarding energy
to ultra-relativistic p+A and A+A collisions, which were studied experimentally at
the CERN SPS in the nineties. However, when increasing the number of degrees of
freedom and adding high-mass short-lived resonances a lot of ambiguities entered
the RBUU models since the couplings between the different hadronic species were
unknown experimentally to a large extent. A way out was to incorporate the particle
production by string formation and decay in line with the LUND model [10] which
included only a formation time of hadrons (τ F ≈ 0.8 fm/c) and a fragmentation function primarily fitted to hadron spectra from e
+ e
− annihilation, where only a single
string is formed. Familiar versions are the Hadron-String-Dynamics (HSD) [11, 12]
or Ultra-relativistic Quantum Molecular Dynamics (UrQMD) [13] approaches that
have been applied to p+A and A+A reactions in a wide range of energies up to the
top SPS energy of 158 AGeV. In fact, a direct comparison between these two models
E. L. Bratkovskaya et al.
properties of the medium generated on short time scales. Since relativistic heavy-ion
collisions start with impinging nuclei in their ground states a proper non-equilibrium
description of the entire dynamics through possibly different phases up to the final
asymptotic hadronic states—eventually showing some degree of equilibration—is
mandatory.
About 40 years ago cascade calculations have been employed for the description
of nucleus–nucleus collisions in the 1-2 AGeV range [1] which provided already
some good idea about the reaction dynamics including essentially nucleons, -
resonances, pions, and kaons. These calculations have been based on the Boltzmann equation which, however, is entirely classical and lacks quantum statistics
appropriate for fermions and bosons. In particular the Pauli-blocking for nucleons
was found to be essential at lower bombarding energies and cascade calculations
were extended in line with the Uehling-Ulenbeck equation for fermions [2] incorporating also some mean-field potential calculated in Hartree approximation with
various two-body Skyrme forces. These types of transport models are denoted as
Boltzmann-Uehling-Uhlenbeck (BUU) or Vlasov-Uehling-Uhlenbeck (VUU) models [3, 4] and are still in use nowadays by some groups. Independently, Quantum
Molecular Dynamical (QMD) models [5] have been proposed in which the test
particles of the BUU/VUU approaches are replaced by Gaussians allowing for the
simulation of single events while keeping the fluctuations. Explicit isospin degrees
of freedom have been incorporated in IQMD [6], too. Since these types of models are based on a Hamiltonian with fixed two-body forces one could evaluate the
nuclear equation of state (EoS) at zero temperature or in thermal equilibrium and
one of the primary issues was to extract the nuclear EoS from heavy-ion data by
means of BUU/VUU or QMD calculations. Later on higher baryonic resonances as
well as mesons like η, K
±
, K
0
, ¯
K
0
, ρ, ω, φ have been incorporated which led to
coupled-channel BUU (CBUU) approaches.
Apart from adding more hadronic degrees of freedom in BUU/VUU fully relativistic formulations have been carried out on the basis of some Lagrangian density
including a selected set of hadronic degrees of freedom [7–9]. All baryons in such
relativistic BUU (RBUU) models were propagated with scalar and vector self energies that were matched to reproduce collective flow data from heavy-ion collisions as
well as particle spectra. This was a necessary step to go ahead in bombarding energy
to ultra-relativistic p+A and A+A collisions, which were studied experimentally at
the CERN SPS in the nineties. However, when increasing the number of degrees of
freedom and adding high-mass short-lived resonances a lot of ambiguities entered
the RBUU models since the couplings between the different hadronic species were
unknown experimentally to a large extent. A way out was to incorporate the particle
production by string formation and decay in line with the LUND model [10] which
included only a formation time of hadrons (τ F ≈ 0.8 fm/c) and a fragmentation function primarily fitted to hadron spectra from e
+ e
− annihilation, where only a single
string is formed. Familiar versions are the Hadron-String-Dynamics (HSD) [11, 12]
or Ultra-relativistic Quantum Molecular Dynamics (UrQMD) [13] approaches that
have been applied to p+A and A+A reactions in a wide range of energies up to the
top SPS energy of 158 AGeV. In fact, a direct comparison between these two models