“Theoretical and numerical study of QCD phase –What do high energy nuclear experiments tell us?”N. V. Gerasimeniuk
Pacific Quantum Center, Far Eastern Federal University, 690950 Vladivostok, Russia
V. A. Goy
Pacific Quantum Center, Far Eastern Federal University, 690950 Vladivostok, Russia
Institut Denis Poisson CNRS/UMR 7013, Université de Tours, 37200 France
A. Nakamura
Pacific Quantum Center, Far Eastern Federal University, 690950 Vladivostok, Russia

1Introduction

We study a phase transition in a dense matter for understanding the early Universe andthe inside of neutron stars. An understanding of hadron physics is very important forthe study of such phenomena. A hadron is a subatomic composite particle made of two
or more quarks held together by a strong force. For study hadrons, people use collisionof heavy ions which are composed of many nucleons, each of which includes quarks andgluons. After heavy ion collisions, many quarks and gluons are created. See Fig.1. Insuch collisions, high density QCD state is expected to be realized. But it is know that if
the incident energy is very high, like RHIC (USA) and LHC (Europe), very dense state
is not created. For a theoretical study, J-PARC (Japan) as well as NICA (Russia) is very
appropriate to study the very high density.
For it, we are using the first-principle, i.e., lattice QCD simulation which is a great
approach for studying non-perturbation physics. This was impossible for many years,
because (1) it requires very heavy numerical works and (2) at the finite density simulations,the sign problem makes numerical simulation impossible.
This simulation is possible only with high performance computational environments:
we use the GPGPU (General-purpose computing on graphics processing units) technology, which allows us to transfer mathematical calculations to a graphic card. RCNP
provides the resources from the Large-Scale Computer System operated by Cyber Media
Center of Osaka university, which includes GPGPU.

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Project Description

Our project is dedicated to the study of dense baryonic matter at zero and finite temperatures. To simulate baryonic matter, we use lattice calculations for several types of fermion
action and color number. We used two-color QCD with staggered quarks and three-color
1Figure 1: Conceptual rendering of the quark-gluon plasma created by RHIC.
https://www.bnl.gov/science/QCD-matter.php
QCD with Wilson fermions, Y. Iwasaki gauge action and clover improvements. Two-color
QCD, i.e. QCD with the SU(2) gauge group is the simplest non-Abelian gauge theory
without a sign problem for a finite quark density. For staggered quark Dirac operator wealso use stout smearing improvement. This project is aimed at studying the properties andcalculating a large number of physical observables in this theory at nonzero temperatureand high quark density, as well as developing and testing new computational methods
applicable in conventional 3-color lattice QCD at large values of the baryon chemical
potential. The results obtained will also allow us to create a platform for testing othernon-perturbative approaches at finite density, which include additional simplifications orassumptions.

3Highlights in 2020

As is well-known, QCD simulations in the dense system suffer from the sign problem.
There is a place where no sign problem appears, i.e., two-color QCD. We can perform
simulations at both imaginary and real quark chemical potentials.
First, we study two-color QCD with two flavors of staggered fermions at imaginary
and real quark chemical potential µq and T > Tc . We employ various methods of an2

Figure 2: View of NICA. https://nica.jinr.ru/physics.php

Figure 3: View of J-PARC. https://j-parc.jp/public/Acc/ja/index.html
alytic continuation of the quark number density from imaginary to real quark chemical
potentials µq on the basis of the numerical results for imaginary µq,I . Below the RobergeWeisse temperature TRW , we find that the cluster expansion model combined with thecanonical formalism provides rather good analytic continuation. Above TRW , we see that
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the analytic continuation to the real values of µq based on trigonometric functions works
equally well with the conventional method based on the Taylor expansion in powers of µq .
We also computed a lot of observables at zero temperature in two-color QCD. Largechemical potential corresponds to the high density state. In the figures 4 and 5, thesimulation results of the quark density and pressure are shown as a function of the quarkchemical potential. In the first figure (left,top) of Fig. 4, we can see this relation between
the chemical potential and the density. The results are stable when we change the latticesize.
The second figure (right,top) of Fig. 4 shows the thermodynamical quantity, the pressure P , increased rapidly when the chemical potential starts to increase, and it reached
to an asymptotic value. The 3rd and 4th figures (left, bottom and right bottom) arenormalized to the free quark gas values.
These results are calculated on OCTOPUS.

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0Figure 4: Quark density and pressure at finite µq a and zero temperature. Here nq,0 , P0
are the quark density and pressure of free fermion gas. Selected points by purple circles
are calculated on the OCTOPUS.

< qq >µ − < qq >0 , 10−3Then, we studied the QCD matter at finite density and temperature. The sign problem
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Figure 5: Diquark condensates at finite µq a and zero temperature. Selected points bypurple circles are calculated on the OCTOPUS.
4Figure 6: An illustration of the deconfinement in the space between the plates
is overcome by our new canonical approach. We compare RHIC energy scan data andlattice QCD simulations. We extracted from the RHIC data approximate value of chemical
potential and T (V ) dependence.
We have also started to study of the MIT bag model in hadron physics. It is a phenomenological model where hadrons considered as bags embedded into a non-perturbative
QCD vacuum. In this case space divided into 2 regions: interior of bag and exterior ofbag. Between two regions exist boundaries. When boundaries are introduced into thetheory, the Casimir force appears. In the vacuum, particles and anti-particles are paircreated and annihilated. The presence of physical objects affects this vacuum fluctuations
nearby. This is the Casimir effect. We studied this Casimir effect via machine-learningtechniques in the third article by numerical simulations. Vacuum fluctuations of quantum
fields between physical objects depend on the shapes, positions, and internal compositionof the latter. For objects of arbitrary shapes, even made from idealized materials, thecalculation of the associated zero-point (Casimir) energy is an analytically intractablechallenge. We propose a different numerical approach to this problem based on machinelearning techniques and illustrate the effectiveness of the method in a (2+1)-dimensional
scalar field theory. The Casimir energy is first calculated numerically using a Monte Carloalgorithm for a set of the Dirichlet boundaries of various shapes. Then, a neural network
is trained to compute this energy given the Dirichlet domain, treating the latter as blackand-white pixelated images. We show that after the learning phase, the neural network
is able to quickly predict the Casimir energy for new boundaries of general shapes withreasonable accuracy.

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Summary and perspectives

Based on these activities, we pursue the following projects.
1. N. Gerasimenyuk, A. Korneev ”Lee-Yang zeros for determining the QCD phasetransition point”.
2. V. Goy, A. Hosaka and A. Nakamura, ”Hadrons at finite baryon density”.
3. A. Molochkov, A. Hosaka and V. Goy, ”Hadrons at finite box”.
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In the first paper, we demonstrate that we can construct the Lee-Yang zero distribution
by the first principle calculation. The Lee-Yang zeros indicate the phase transition points
in T − µ plane. This is possible only when we can calculate the finite chemical potential
region reliably.
The 2nd paper shows that now we can calculate not only the phase boundary, butalso hadron behavior on the phase space.
In 3rd article, we are the first who started to use machine learning (ML) techniquesin the study of the Casimir effect. We have now developed two numerical approaches
for solving the problem of studying the Casimir effect. Using our developments, we willcontinue to study this effect in real QCD at the finite box.

References[1] A. Begun, V.G. Bornyakov, N.V. Gerasimeniuk, V.A. Goy, A. Nakamura, R.N. Rogalyov ”Quark Density in Lattice QC2 D at Imaginary and Real Chemical Potential”,(2021), e-Print: 2103.07442 [hep-lat][2] V. Bornyakov, D. Boyda, V. Goy, A. Molochkov, A. Nakamura ”Canonical partition
functions in lattice QCD at finite density and temperature”, PoS LATTICE2019 271(2020)[3] M.N. Chernodub, H. Erbin, I.V. Grishmanovskii, V.A. Goy, A.V. Molochkov”Casimir effect with machine learning”, Phys.Rev.Res. 2 033375 (2020), e-Print:
1911.07571 [hep-lat]6