Preface to "Selected Papers from "Theory of Hadronic Matter under Extreme Conditions""

We are happy to present this reprint book edited from the special issue of the journal *Particles* with selected contributions to the second International Workshop on "Theory of Hadronic Matter under Extreme Conditions" that took place at JINR Dubna in September 16-19, 2019, see https://indico.jinr.ru/event/834/overview , with the group photo of the participants shown in Figure 1.

Figure 1: Group photo of the participants at the workshop "Hadronic Matter under Extreme Conditions", Dubna, September 16-19, 2019. From left to right, front row: Andrei Radzhabov, Vladimir Goy, Alina Czajka, Natalia Kolomoets, Nikita Astrakhantsev, Nikita Lebedev, Alexandra Friesen, Vladimir Voronin, Konstantin Maslov, Pandiat Saumia, Kalman Szabo, Francesca Cuteri, Paula Hillmann, Tom Reichert, Atsushi Nakamura, 2nd row: Ming-Tai Yang, Artem Roenko, Andrey Kotov, Stanislaw Mrowczynski, Jan Cleymans, Vyacheslav Toneev, Jan Pawlowski, Sergei Nedelko, Jörg Aichelin, Diana Alvear Terrero, Kenji Fukushima, 3rd row: Vitaly Bornyakov, Boris Kerbikov, Alexei Larionov, V. Nguen, Yuri Ivanov, Yuri Sinyukov, Trambak Bhattacharyya, Elena Bratkovskaya, Aleksandr Andrianov, Manfried Faber, Alexander Nikolaev, Lucia Oliva, Vladislav Tainov, back row: Roman Zhokhov, Pawel Lukyanov, Michael Bordag, Gennady Zinovyev, Aleksei Nikolskii, George Prokhorov, Evgeni Kolomeitsev, Dmitry Voskresensky, Bernd-Jochen Schaefer, Christof Gattringer, Roman Rogaliov, Lorenz von Smekal, Mikhail Nalimov, Kyrill Bugaev, Marina Komarova, Rudolf Golubich, Masayasu Hasegawa, Vadim Voronyuk, David Blaschke

In its nature, theoretical investigations in the field of relativistic heavy-ion collisions have a multidisciplinary character involving physics at various energy scales. They ask not only for the resolution of a number of fundamental problems but also phenomenological studies directly connected with experiments. The progress in this field relies on a coherent implementation of a wide range of methods of quantum chromodynamics, relativistic

nuclear physics, kinetic theory, hydrodynamics and physics of critical phenomena in finite short-lived systems. The construction of the Nuclotron-based Ion Collider fAcility (NICA), see Figure 2, based on superconducting rings for performing experiments with heavy-ion beams (see Figure 3) in the collider as well as the fixed target mode sets up an auspicious environment for enhancement of theoretical physics activities at the Joint Institute for Nuclear Research (JINR) related to relativistic heavy ion physics.

Figure 2: Aerial view of the NICA accelerator complex as of December 2020. The oval ring in front is the collider for protons and nuclei with two interaction points for the MPD and SPD experiments to be hosted in the rectangular buildings. In the back to the left is the synchrophasotron building which hosts the nuclotron superconducting accelerator serving as the injector to the collider ring, together with the ion source, linear accelerator and booster systems. The adjacent rectangular building is the fixed target hall where the baryonic matter at nuclotron (BM @ N) experiment is located.

Figure 3: Booster ring system based on superconducting magnet technology developed at JINR Dubna for the Nuclotron accelerator. The booster is commissioned inside the iron yoke of the former synchrophasotron at the Veksler-Baldin Laboratory for High-Energy Physics of the JINR Dubna.

As the Guest Editors, we would like to thank all participants of the meeting for their active role in making this event as inspiring as it was for the future development of the field of hadronic matter under extreme conditions and for the stimulating role it played for fostering the theoretical physics community supporting both the theoretical and experimental research in this field. These thanks concern in particular the authors of the contributions in this reprint book. We would also like to acknowledge the support in funding the meeting which came from the Directorate of JINR Dubna and various funding organisations that gave in-kind support that allowed to bring 84 participants from 13 countries together at the JINR Dubna, see the group photo in Figure 1.

It has been our big pleasure to collaborate with the MDPI journal "Particles" and its Editorial office which provided invaluable professional support throughout the realisation of this project.

> **David Blaschke, Victor Braguta, Evgeni Kolomeitsev, Sergei N. Nedelko, Alexandra Friesen, Vladimir E. Voronin** *Editors*

*Article*

#### **New Canonical and Grand Canonical Density of States Techniques for Finite Density Lattice QCD**

**Christof Gattringer \*,†, Michael Mandl † and Pascal Törek †**

Institute of Physics, University of Graz, 8010 Graz, Austria; mi.mandl@gmx.at (M.M.); pascal.toerek@uni-graz.at (P.T.)

**\*** Correspondence: christof.gattringer@uni-graz.at

† Member of NAWI Graz.

Received: 10 December 2019; Accepted: 5 February 2020; Published: 10 February 2020

**Abstract:** We discuss two new density of states approaches for finite density lattice QCD (Quantum Chromo Dynamics). The paper extends a recent presentation of the new techniques based on Wilson fermions, while here, we now discuss and test the case of finite density QCD with staggered fermions. The first of our two approaches is based on the canonical formulation where observables at a fixed net quark number *N* are obtained as Fourier moments of the vacuum expectation values at imaginary chemical potential *θ*. We treat the latter as densities that can be computed with the recently developed functional fit approach. The second method is based on a direct grand canonical evaluation after rewriting the QCD partition sum in terms of a suitable pseudo-fermion representation. In this form, the imaginary part of the pseudo-fermion action can be identified and the corresponding density may again be computed with the functional fit approach. We develop the details of the two approaches and discuss some exploratory first tests for the case of free fermions where reference results for assessing the new techniques may be obtained from Fourier transformation.

**Keywords:** lattice QCD; finite density; density of states techniques
