Search for Traces of the Critical Endpoint in QCD

**Principal Investigator:**

Dénes Sexty

**Affiliation:**

Bergische Universität Wuppertal, IAS/JSC Forschungszenturm Jülich

**Local Project ID:**

chwu32

**HPC Platform used:**

JUQUEEN and JUWELS of JSC

**Date published:**

The matter that makes up the nuclei of the atoms we are built of is made up of hadronic particles such as protons and neutrons. At large temperatures, this nuclear matter is expected to “melt” just like the solid phase of water, ice, melts at 0 degrees Celsius. This transition can be observed in particle accelerators in heavy ion collision experiments. The various phases this matter can take up is depicted on a phase diagram, which shows the phases as a function of the temperature and the chemical potential μ (which controls baryon density of the system). At small baryon density this transition is expected to be a crossover, meaning there is no latent heat taken up by the system as it goes through the transition. At larger densities various model calculations suggest that the transition turns into a first order phase transition, with nonzero latent heat. As the density is decreased, the first order line ends in a so-called critical point, which is characterized by large fluctuations.

This proposed critical point in the QCD phase diagram has been the subject of a vigorous search effort experimentally as well as theoretically. A first-principle theoretical tool, called lattice-QCD where the space-time is discretised on a finite lattice, has proved to be very successful in the past decades. Various algorithms based on the so called ‘importance sampling’ are used to carry out the calculation. These use the fact that the system can be described with an ensemble of configurations which describe an instance of gluon and quark fields. Each configuration has a weight which is interpreted as a probability depending on the temperature, quark masses and so on. This probabilistic description however breaks down as soon as there is a nonzero baryonic density in the system, because the weight of each configuration can now become negative. This is the infamous ‘sign problem’, a long standing challenge in the field of lattice QCD as well as other fields such as condensed matter physics, non-equilibrium physics, etc.

During this project we have tried various methods to circumvent the sign problem [1-5], which go by the names: Taylor expansion, imaginary chemical potential, density of states, reweighting, complex Langevin equation. If the system is sufficiently close to zero chemical potential (where there is no sign problem), there are various ways to circumvent the sign problem. One such method is the ‘Taylor expansion’ method, where the rate of change (with respect to the chemical potential) of various quantities are measured at zero chemical potential, and this allows the extrapolation of the results to nonzero densities. An other method uses the trick that we can insert a complex number into the parameter describing the chemical potential, and the sign problem disappears at imaginary values. Extrapolating back to real chemical potential values is possible again at small densities. This trick also helps in the calculation of the fluctuations of the baryonic charge in the system, which potentially is influenced by critical fluctuations in case a critical point is nearby.

Normally even calulations at zero chemical potential require a supercomputer as we need to use large lattices and small lattices spacings to get results which are sufficiently close to the real, continous world. The sign problem in this case makes the problem even harder, where extra effort is needed in the various sign-problem circumventing proposals.

We did not find the critical point during this study but we have managed to develop the methods to circumvent the sign problem, and might lead to the discovery (or the discovery of the absence) of the critical point of QCD.

We show here the result which calculated the equation of state of the high temperature nuclear matter at previously unaccessible large baryonic densities. This calculation used the Langevin equation which describes a random walk in configuration space, and can be used as a simulation method also at zero densities. This method however does not require the interpretation of weights as probabilities, and therefore is generalizable to non-positive weights as well. This generalization is based on the structure of the complex numbers, hence it has the name ‘Complex Langevin equation (CLE)’ . It was invented more than 30 years ago, however its status was unclear as sometimes this method gives unreliable results. In recent years however important results have clarified what conditions must be satisfied and what technical improvements are needed to make the method and its results trustworthy.

We have calculated the pressure and energy density change of the nuclear matter as the chemical potential is increased. The results were compared to the Taylor expansion method. We observed good agreement at small chemical potential. At larger chemical potential the errors of the Taylor expansion start to blow up, thus this method loses predictive power, while the results of the Complex Langevin approach remain precise.

**References**

[1] G. Endrodi et. al.: Phys.Rev.D 98 (2018) 7, 074508

[2] Dénes Sexty: Phys.Rev.D 100 (2019) 7, 074503

[3] M. Scherzer et. al.: Phys.Rev.D 102 (2020) 1, 014515

[4] Z. Fodor et. al.: Nucl.Phys.A 982 (2019), 843-846

[5] S. Borsanyi et.al.: Phys.Rev.Lett. 125 (2020) 5, 052001

**Research Team**

Principal Investigator: Dénes Sexty (Bergische Universität Wuppertal, IAS/JSC Forschungszenturm Jülich)

Project Contributiors: Dr. Dániel Nógrádi, Attila Pásztor, Csaba Török, Zoltán Varga (Eötvös University, Budapest)

**Scientific Contact**

Prof. Dr. Dénes Sexty

now at: Universität Graz

Institut für Physik

Universitätsplatz 5, A-Graz (Austria)

e-mail: denes.sexty [@] uni-graz.at

*Local project ID: chwu32*

*April 2021*