# Simulating heavy ion collisions with MUSIC¶

By Jean-François Paquet, Duke University, Summer 2022

Collisions of large nuclei at relativistic energies * are used to study the many-body properties of Quantum Chromodynamics.

*

Relativistic energies mean that the energy of the nuclei is much larger than their mass.

The theoretical tools that are the usual workhorses of quantum field theory, such as perturbative methods and lattice calculations, can only describe very specific features of relativistic heavy ion collisions. As such, a first principles description of the complex dynamics of heavy ion collisions is not yet possible. On the other hand, very successful multi-stage models of heavy ion collisions have been built by combining lattice and perturbative calculations with effective models such as hydrodynamics.

This document is an introduction to the C++ code MUSIC, a relativistic viscous hydrodynamic simulation of heavy ion collisions. The aim of this document is to explain the physics that MUSIC is designed to describe, and discuss the corresponding code where this physics is implemented.

## A brief introduction to heavy ion physics¶

Quantum Chromodynamics (QCD) is the theory of the strong (nuclear) interaction. It is a quantum field theory whose elementary degrees of freedom are quarks and gluons, which carry a “colour” charge. These elementary degrees of freedom are not asymptotic states however: quarks and gluons are normally confined into colour-neutral states called hadrons (protons, neutrons, nuclei, …).

Heavy ion physics uses relativistic collisions of nuclei to deconfine, for a short amount of time, quarks and gluons from hadrons. This makes possible to study the many-body properties of deconfined matter.

Producing deconfined matter with relativistic heavy ion collisions

Nuclei are imparted with a huge amount of kinetic energy (”relativistic collisions”). When the nuclei collide, this energy is used to break up the nuclei and keep the produced coloured matter deconfined for a short period of time.

The figure above depicts different stages of the spacetime evolution of a heavy ion collision. The unit that is used to measure the size of the medium, and the amount of time that the medium remains deconfined, is the fermi .

1 fermi = 1 fm = $$10^{-15}$$ m.

The radius of a proton is approximately 1 fm, while large nuclei have radii of order 5-10 fm.

By dividing by the speed of light, “c”, we obtain a unit for time as well: $$1\textrm{ fm/c}\approx 3 \times 10^{-24}\textrm{s}$$.

In natural units, where c=1, time can be written in “fm/c” or just “fm”.

Five different stages are pictured, corresponding to:

• Before the collisions

The two nuclei are moving close to the speed of light along the beam axis (horizontal axis in the figure), which is the reason for their Lorentz contraction (i.e. the nuclei are flattened along that axis).

• Early stage dynamics

After the nuclei collide, a “soup” of deconfined QCD matter is produced.

• Hydrodynamic phase

After $$\sim 0.1-1$$ fm, the overall spacetime evolution of the deconfined QCD matter can begin to be described with relativistic hydrodynamics. This phase can be referred to as the strongly-coupled quark-gluon plasma (sQGP).

• Particlization

As the medium expands and becomes more dilute, the strongly-coupled quark-gluon plasma begins to reconfine into a gas of hadrons (a small amount of energy is also released as photons, leptons, …). This reconfinement begins at the edge of the plasma and usually ends in the central regions. After $$\sim 10$$ fm, most of the plasma is reconfined into hadrons.

• Late stage hadronic dynamics

The gas of hadrons that follows the hydrodynamic phase still undergoes a complex dynamic. Unstable hadrons decay, and hadrons undergo elastic and inelastic collisions that change the momentum distribution and species composition of the hadron gas. Once all unstable hadrons have decayed into stable ones and once the hadron gas is too dilute for even hadronic interactions to occur, hadrons continue their way unhindered toward the detectors.

MUSIC is primarily designed to describe the hydrodynamic phase of heavy ion collision. As explained in more details later, the early stage dynamics is not described in MUSIC, but is provided as an initial condition for the hydrodynamics phase. On the other hand, particlization and hadronic decays (one of the forms of late stage hadronic dynamics) are included in MUSIC, allowing for hadronic observables to be computed directly within MUSIC for comparison with data.

Since hydrodynamics forms the basis of MUSIC, it is preferable to describe it first. The early stage dynamics is then discussed, followed up particlization, late stage hadronic dynamics and a discussion of how to compute hadronic observables in MUSIC. Before entering into these discussions, a brief software-oriented overview of MUSIC is provided.