The QCD Phase Diagram and Equation of State Gauss Centre for Supercomputing e.V.


The QCD Phase Diagram and Equation of State

Principal Investigator:
Zoltán Fodor

University of Wuppertal

Local Project ID:
chwu08, GCS-POSR, pn34mu

HPC Platform used:
JUQUEEN and JUWELS (JSC), Hazel Hen (HLRS), SuperMUC and SuperMUC-NG (LRZ)

Date published:


Only microseconds after the Big Bang, the whole Universe consisted of a tiny droplet of the hottest and most dense liquid known to humankind: the Quark Gluon Plasma (QGP). Under normal conditions, such as those in our everyday life, the quarks and gluons - the most fundamental building blocks of matter - are confined into protons and neutrons. When matter is extremely hot, or extremely compressed, these nucleons melt and quarks and gluons form the Quark Gluon Plasma, in which they are free to move. When the Universe expanded and cooled down, eventually the quarks and gluons coalesced into nucleons, that in turn formed the atoms, then molecules and slowly formed matter as we see it today.

Thanks to extraordinary experimental efforts, the QGP is now routinely produced in particle accelerator facilities such as the Relativistic Heavy Ion Collider - in the US, and the Large Hadron Collider at CERN - in Switzerland.

Heavy nuclei are accelerated to almost the speed of light and collided; the enormous amount of energy produced is sufficient to create the QGP in the laboratory, which then undergoes the same transition as the early Universe itself.

Intense theoretical as well as experimental investigations are in place in order to study the properties of this new form of matter, and in particular its transition to ordinary matter. A delicate question regards the study of the nature of this transition. In the early Universe there was a virtually perfect symmetry between matter and anti-matter. This is not the case in today's universe, especially so in ultra-dense astronomical objects such as neutron stars, which are also the subject of intense investigations. A key question is today how does this transition change when conditions more similar to these high-density objects are considered - i.e. when the density of quarks is much larger than that of the early Universe.

The theory of strong interactions, which governs the behavior of quarks and gluons - and thus of protons and neutrons - is called Quantum Chromodynamics (QCD). The nature of strongly interacting matter under different conditions is commonly depicted in the so-called QCD phase diagram, which is sketched in Fig. 1. The different phases of matter are shown in the plane of temperature and chemical potential. The chemical potential is a thermodynamic coordinate like the temperature, which can be best understood as the energy required by the system to change its chemical composition. It is tightly connected with the density of quarks: when the former is zero, the latter is zero as well. This two-dimensional depiction is completely analogous to what is done in the phase diagram of water: we know that for certain values of the thermodynamic coordinates water is liquid, gaseous (vapor) or solid (ice).

Most of the features one sees in Fig. 1 are in reality conjectured, based on model-dependent calculations and not on first-principle determinations. The major first-principle tool of investigation is represented by the sophisticated numerical simulations - like the ones we perform - of lattice QCD. Most notably, they have led to the outstanding discovery that the transition at zero quark density is extremely smooth, very much unlike e.g., the boiling of water [1]. On the other hand, this transition is expected to become much abrupt at larger chemical potentials, which would imply the presence in the phase diagram of a so-called critical point. The search for such point is possibly the main endeavor in today's high-energy nuclear physics research.

Direct simulations at non-zero chemical potentials are hindered by the infamous, so-called “sign problem'', which in essence is a problem of calculating averages from quantities that fluctuate violently. However, tools have been developed to reconstruct the physics at non-zero chemical potential from available calculations, performed at zero (or imaginary) chemical potential. (Fig 2.) The impact of these results is mainly twofold. Firstly, knowledge of the phase diagram of QCD is inherently a fundamental achievement, which deepens our understanding of matter and the Universe, with applications that range from the smallest (nuclear physics) to the largest scales (astrophysics and cosmology). Secondly, their are crucial input for many other simulations, such as the evolution of the matter created in nuclear collisions, the early Universe or dense objects as e.g. neutron stars.

On-going Research / Outlook

The main focus of our further research is the study of the so-called equation of state (EoS) of QCD. This is simply the relation between quantities such as the pressure and density, and how they change in different regions of the phase diagram. Because of the “sign problem'', results at non-zero density are difficult to obtain, but nonetheless extremely valuable, especially in a new era of astrophysical observations inaugurated by the very first detection of gravitational waves. 

Results and Methods

In 2006 we calculated the transition temperature at zero net baryon density, as it was realized in the Early Universe. In that work we have seen that although there is no real transition, one can find a transition range around 150 MeV temperature [2].

In our latest work [3] we determined the so-called “transition line'' of QCD. In short, we calculated with very high precision the temperature at which the QCD transition takes place, for quite a broad range of small but non-zero chemical potential. In order to do so, we have exploited simulations at zero and imaginary chemical potential, which do not suffer from the “sign problem”. Studying observables which are sensitive to the phase transition, we have determined the temperature at which the transition occurs for each chemical potential we simulated. We then analytically continued the dependence of this transition temperature on the chemical potential, extending it to the relevant regime of real chemical potential. The extrapolated transition line is shown in Fig. 3 with a green band which represents the uncertainty of the result. Other comparable results, although obtained with other, non first-principle methods, are shown along with it. It is worth mentioning that this determination is the most precise both at zero and non-zero chemical potential available in the literature.

References and Links

[1] Y. Aoki et al, Nature 443 (2006) 675-678
[2] Y. Aoki et al, Phys. Lett. B 643 (2006) 46-54
[3] S.Borsányi et al., Phys. Rev. Lett. 125 (2020) no.5, 052001, doi:10.1103/PhysRevLett.125.052001

Research Team

Szabolcs Borsányi1, Zoltán Fodor1,2,3 (PI), P. Parotto1 , C. Ratti4

1University of Wuppertal
2Jülich Supercomputing Centre, Forschungszentrum Jülich
3Eötvös University, Budapest
4University of Houston, Texas, USA

Scientific Contact

Prof. Dr. Szabolcs Borsányi
Theoretische Physik, Fachbereich C - Bergische Universität Wuppertal
D-42097 Wuppertal/Germany
e-mail: borsanyi [@]

Local project IDs: chwu08 (JSC), pn34mu (LRZ)

February 2021

Tags: JSC LRZ QCD Large-Scale Project