11institutetext: Egyptian Center for Theoretical Physics (ECTP), Juhayna
Square of 26th–July–Corridor, 12588 Giza, Egypt 22institutetext: Modern
University for Technology and Information (MTI), Faculty of Engineering, 11571
Cairo, Egypt

# Chiral magnetic properties of QCD phase–diagram

Abdel Nasser Tawfik _Present address:_ Goethe University, Institute for
Theoretical Physics (ITP), Max-von-Laue-Str. 1, D-60438 Frankfurt am Main,
Germany11 Abdel Magied Diab 22

(Received: date / Revised version: date)

###### Abstract

The QCD phase–diagram is studied, at finite magnetic field. Our calculations
are based on the QCD effective model, the SU($3$) Polyakov linear–sigma model
(PLSM), in which the chiral symmetry is integrated in the hadron phase and in
the parton phase, the up–, down– and strange–quark degrees of freedom are
incorporated besides the inclusion of Polyakov loop potentials in the pure
gauge limit, which are motivated by various underlying QCD symmetries. The
Landau quantization and the magnetic catalysis are implemented. The response
of the QCD matter to an external magnetic field such as magnetization,
magnetic susceptibility and permeability has been estimated. We conclude that
the parton phase has higher values of magnetization, magnetic susceptibility,
and permeability relative to the hadron phase. Depending on the contributions
to the Landau levels, we conclude that the chiral magnetic field enhances the
chiral quark condensates and hence the chiral QCD phase–diagram, i.e. the
hadron–parton phase–transition likely takes place, at lower critical
temperatures and chemical potentials.

###### pacs:

11.10.WxChiral transition and 25.75.NqQuark deconfinement, quark–gluon plasma
production, and phase transitions and 98.62.EnElectric and magnetic fields

##  1 Introduction

Exploring the quantum chromodynamic (QCD) phase–diagram and studying the phase
structures and the deconfinement phase–transitions of strongly interacting
matter are among the fundamental issues in nuclear physics. Studying QCD
matter in laboratory is one of the greatest challenges of the experimental
nuclear physics as the parton matter is not directly accessible. There are
different experimental methods implemented in accelerating and colliding ions
(hadrons). Due to the center–of–mass energy of the collision achieved,
different domains of the QCD phase–diagram could be drawn. The first
prediction of end of the hadron domain, at high temperatures, was fomulated
long time before the invention of QCD, where the partons are assumed as the
effective degrees–of—freedom (dof), at temperatures larger than the Hagedorn
temperature $T_{H}$ Hagedorn:1965st ; Hagedorn:1984hz . The hadron matter
forms fireballs of new particles, which can again produce new fireballs. In
1975, Cabibbo proposed a QCD phase diagram in $T-n_{B}$ plane Cabibbo:1975ig ,
where $T_{H}$ in the statistical Bootstrap model (SBM) Hagedorn:1965st  was
interpreted as the critical temperature $T_{c}$ and is conjectured to be
associated to second–order phase–transition into the deconfinement state. At
large baryon density, $n_{B}$, a weak interaction between quarks and gluons -
due to the asymptotic dof - has been recognized Collins:1974ky . A Historical
summary on the QCD phase diagram and its investigation in the heavy–ion
collisions (HIC) experiments are available, for instance, in refs. Baym:2001in
; fukushima2008chiral .

In statistical physics, the phase transitions are defined as singularities or
non–analyticity in the free energy as a function of thermodynamic quantities.
In lattice QCD simulations, the corresponding partition functions are taken as
functional integrals over compact groups described and evaluated in dependence
on the temperature $T$, the chemical potential $\mu$, the volume $V$, the
magnetic field $eB$, $\cdots$, etc.

In HIC and due to oppositely ultra–relativistic motion of colliding
heavy–ions, a huge magnetic field can be generated. Their motion generates an
electric current which in turn induces magnetic field to the system. In a
non–central HIC, the two counter–propagating nuclei collide, at finite impact
parameter $b$. In Fig. 1 the magnetic field in the center of the over–lapping
surface in $Au+Au$ collisions, for instance, at $b=10~{}$fm and
$\sqrt{s_{NN}}=200~{}$GeV, is visualized. This is perpendicular to the
reaction plane owing to the symmetry of the collision. Let us assume that all
colliding nuclei are located, at the center of the nucleus, by applying
Biot–Savart law,

$\displaystyle-eB_{y}\sim
2Z_{Au}\gamma\frac{e^{2}}{4\pi}\,v_{z}\left(\frac{2}{b}\right)^{2}\approx
10^{19}\,\mbox{Gauss},$ (1)

where the negative sign appears due to the assumption that the magnetic field
is pointing in $-y$ direction, $v_{z}=(1-(2m_{N}/\sqrt{s})^{2})^{1/2}\,\approx
0.99$ is the velocity of accelerated nuclei, $m_{N}$ is the nucleon mass,
$\gamma=1/\sqrt{1-v_{z}^{2}}\approx\,100$ is the Lorentz gamma, and
$Z_{Au}=79$ is the change number of the gold nucleus. It is assumed that such
a huge magnetic field would have great impacts on the dynamics of the parton
(quarks and gluons) matter produced in HIC.

Fig. 1: Left panel shows the geometry of non–central HIC. $b$ is the impact
parameter and $R_{A}$ is the radius of the nucleus. The magnetic field $B$ is
expected to be perpendicular to the reaction plane due to the left–right
symmetry of the collision geometry. The figure is taken from
huang2016electromagnetic . Right panel illustrates a schematic QCD
phase–diagram.

Over the last few decades, great efforts have been done to map out the QCD
phase–diagram. Right panel of Fig. 1 illustrates a schematic version. At low
temperatures and baryon chemical potentials, the quarks and gluons are still
confinement forming colorless hadrons. At $T\sim 150~{}$MeV, there is a
crossover to the partonic colored phase; quark–gluon plasma. With increasing
baryon–chemical potential, at low temperatures, the quarks shall be grouped in
pairs known as correlated Cooper–pairs, which likely condense. Various
experiments aim at studying the phase structures and the QCD phase–diagram
such as RHIC at BNL $\sqrt{s_{NN}}=200~{}$GeV and LHC at CERN with energies up
$\sqrt{s_{NN}}=13~{}$Tev. At $\sqrt{s_{NN}}=4-11~{}$GeV, the large density
regime shall be explored by CBM at FAIR and MPD at JINR.

In the heavy–ion experiments, as a result of the non–central heavy–ion
collisions, a huge magnetic field could be created. For instance, at RHIC and
LHC energies, the magnetic field ranges between $m_{\pi}^{2}$ and
$10-15m_{\pi}^{2}$, respectively Vachaspati:1991nm ; bzdak2012event , where
$m_{\pi}^{2}\sim 10^{8}~{}$Gauss. It is worth mentioning that this value of
course is just a snapshot, since the magnetic field is strongly
time–dependent. Detailed discussion on such a dynamical system does not lay
within the scope of the present study. On the other hand, the proposed
approach, the PLSM, take into account the evolution of such dynamical system.
The magnetic field lifetime in HIC including the electric and chiral magnetic
effects McLerran:2013hla ; Tuchin:2013apa  and the electromagnetic impacts on
the heavy–ion phenomenology are reviewed in ref. Tuchin:2014hza . The features
of the electromagnetic fields in HIC shall be addressed, quantitatively.

So far, there are various numerical approaches supporting the concept of
magnetic catalysis in hot quark–matter and well agreeing with the recent PLSM
result in strong magnetic–field such as the numerical lattice QCD simulations
Bali:2011qj ; Levkova:2013qda ; Bonati:2013vba ; Bruckmann:2013oba ;
Bali:2013txa , the hadron resonance gas (HRG) model Endrodi:2013cs ;
Bhattacharyya:2015pra , and the Polyakov–Nambu–Jona–Lasinio model Ratti:2005jh
; Ferreira:2015gxa ; Ferreira:2014kpa ; Farias:2014eca ; Chao:2013qpa ;
Mei:2020jzn ; Mao:2016fha ; Andersen:2021lnk . Great details on understanding
the phase structure of the QCD matter in strong magnetic–field are reviewed,
for instance, in refs. Shovkovy:2012zn ; Andersen:2014xxa ; Kharzeev:2012ph ;
Gatto:2012sp .

In 1960s, the linear sigma model (LSM), a low–energy model, was introduced by
Gell–Mann and Levey gell1960axial , long before the invertion of the Quantum
Chromdynamics (QCD), the theory of the strong interaction. Many studies have
been performed on LSM $\mathcal{O}(4)$ at (non)–zero temperature
Lenaghan:1999si ; Petropoulos:1998gt  and for $N_{f}=2,3,$ and $4$ quark
flavors levy1967currents ; hu1974chiral ; geddes1980spin . Moreover, the LSM
is coupled with the Polyakov loop fields, known as the PLSM, to include the
interaction and dynamics of the colored gluons. We have a solid term in
developing the PLSM to obtain reliable results. For instance, in estimating
the features of the moments in thermal QCD medium Tawfik:2014uka , obtaining
the thermal spectrum for masses of meson states and QCD equation–s–tate (EoS)
in thermal and dense QCD medium at (non)–zero magnetic backgrounds
Tawfik:2014gga ; Tawfik:2019rdd ; Tawfik:2016lih , Furthermore, the magnetic
properties of the QCD matter such as the magnetization, magnetic
susceptibility and inverse magnetic catalysis could be estimated
Tawfik:2016gye ; AbdelAalDiab:2016rje ; Tawfik:2017cdx . We have also improved
the PLSM to study the thermal structure of the transport properties, the bulk
and the shear viscosity, the thermal and the electric conductivity of the QCD
matter Tawfik:2016edq ; Tawfik:2016ihn . The extension to $N_{f}=4$)–PLSM
Diab:2016iig ; AbdelAalDiab:2018hrx  is an essential improvement in order to
match with the recent lattice QCD simulations.

The present paper is divided into two main sections. Section 2 summarizes the
main features of the effective QCD approach, the Polyakov linear–Sigma model.
Section 3 outlines the results obtained.

##  2 The Approach

###  2.1 Polyakov Linear–Sigma Model

Assuming that the hadronic degrees–of–freedom (dof) are the colors of their
quark constituents, various effective models relying on the chiral symmetry of
QCD have been proposed Scavenius:2000qd . The linear sigma model (LSM)
implements the chiral symmetry in the hadronic sector and incorporates
additional partonic (quarks) dof. We briefly summary the basic concepts of the
SU($N_{f}$) Polyakov LSM (PLSM) in section 2.2. Various approaches for the
Polyakov loop potentials shall be discussed in section 2.3. The inclusion of
the magnetic effects in the mean field approximation by means of the Landau
quantization shall be elaborated in section 2.4.

###  2.2 SU(Nf) Lagrangian

For LSM with SU($3$)${}_{L}\times$ SU($3$)R and $N_{f}=2,\,3,\,4$ quark
flavors and $N_{c}$ color dof, the symmetric LSM Lagrangian ${\cal
L}_{chiral}={\cal L}_{f}+{\cal L}_{m}$, where the fermionic part is given as

$\displaystyle\mathcal{L}_{f}=\bar{\psi}\left[i\partial/-g\;T_{a}\,\left(\sigma_{a}+i\,\gamma_{5}\,\pi_{a}+\gamma_{\mu}V_{a}^{\mu}+\gamma_{\mu}\gamma_{5}A_{a}^{\mu}\right)\,\right]\psi,\hskip
14.22636pt$ (2)

with $\mu$ is an additional Lorentz index Koch:1997ei , $g$ is the Yukawa
coupling of the quarks to the mesonic contributions ${\cal L}_{m}={\cal
L}_{SP}+{\cal L}_{VA}+{\cal L}_{Int}+\mathcal{L}_{U(1)_{A}}$ represented to
${\cal L}_{SP}$ scalars ($J^{PC}=0^{++}$) and pseudo–scalars
($J^{PC}=0^{-+}$), ${\cal L}_{VA}$ to vectors ($J^{PC}=1^{-}$) and
axial–vectors ($J^{PC}=1^{++}$) mesons and ${\cal L}_{Int}$ being the
interaction between them. The Lagrangian of the anomaly term is given by
$\mathcal{L}_{U(1)_{A}}$ gell1960axial ; gasiorowicz1969effective ; Ko:1994en
; Parganlija:2008jf ; Pisarski:1994yp ; Kovacs:2013xca ; Tawfik:2014gga .

$\displaystyle\mathcal{L}_{SP}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}(\partial_{\mu}\Phi^{{\dagger}}\partial^{\mu}\Phi-m^{2}\Phi^{{\dagger}}\Phi)-\lambda_{1}[\mathrm{Tr}(\Phi^{{\dagger}}\Phi)]^{2}$
(3) $\displaystyle-$
$\displaystyle\lambda_{2}\mathrm{Tr}(\Phi^{{\dagger}}\Phi)^{2}+\mathrm{Tr}[H(\Phi+\Phi^{{\dagger}})],$
$\displaystyle\mathcal{L}_{AV}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\mathop{\mathrm{Tr}}(L_{\mu\nu}^{2}+R_{\mu\nu}^{2})+\mathop{\mathrm{Tr}}\left[\left(\frac{m_{1}^{2}}{2}+\Delta\right)(L_{\mu}^{2}+R_{\mu}^{2})\right]$
(4) $\displaystyle+$ $\displaystyle
i\frac{g_{2}}{2}(\mathop{\mathrm{Tr}}\\{L_{\mu\nu}[L^{\mu},L^{\nu}]\\}+\mathop{\mathrm{Tr}}\\{R_{\mu\nu}[R^{\mu},R^{\nu}]\\}){}$
$\displaystyle+$ $\displaystyle
g_{3}[\mathop{\mathrm{Tr}}(L_{\mu}L_{\nu}L^{\mu}L^{\nu})+\mathop{\mathrm{Tr}}(R_{\mu}R_{\nu}R^{\mu}R^{\nu})]$
$\displaystyle+$ $\displaystyle
g_{4}[\mathop{\mathrm{Tr}}\left(L_{\mu}L^{\mu}L_{\nu}L^{\nu}\right)+\mathop{\mathrm{Tr}}\left(R_{\mu}R^{\mu}R_{\nu}R^{\nu}\right)]{}$
$\displaystyle+$ $\displaystyle
g_{5}\mathop{\mathrm{Tr}}\left(L_{\mu}L^{\mu}\right)\,\mathop{\mathrm{Tr}}\left(R_{\nu}R^{\nu}\right)$
$\displaystyle+$ $\displaystyle
g_{6}[\mathop{\mathrm{Tr}}(L_{\mu}L^{\mu})\,\mathop{\mathrm{Tr}}(L_{\nu}L^{\nu})+\mathop{\mathrm{Tr}}(R_{\mu}R^{\mu})\,\mathop{\mathrm{Tr}}(R_{\nu}R^{\nu})],$
$\displaystyle\mathcal{L}_{Int}$ $\displaystyle=$
$\displaystyle\frac{h_{1}}{2}\mathop{\mathrm{Tr}}(\Phi^{\dagger}\Phi)\mathop{\mathrm{Tr}}(L_{\mu}^{2}+R_{\mu}^{2})+h_{2}\mathop{\mathrm{Tr}}[|L_{\mu}\Phi|^{2}+|\Phi
R_{\mu}|^{2}]$ (5) $\displaystyle+$ $\displaystyle
2h_{3}\mathop{\mathrm{Tr}}(L_{\mu}\Phi R^{\mu}\Phi^{\dagger}),$
$\displaystyle\mathcal{L}_{U(1)_{A}}$ $\displaystyle=$ $\displaystyle
c[\mathrm{Det}(\Phi)+\mathrm{Det}(\Phi^{{\dagger}})]+c_{0}[\mathrm{Det}(\Phi)-\mathrm{Det}(\Phi^{{\dagger}})]^{2}$
(6) $\displaystyle+$ $\displaystyle
c_{1}[\mathrm{Det}(\Phi)+\mathrm{Det}(\Phi^{{\dagger}})]\,\mathrm{Tr}[\Phi\Phi^{{\dagger}}].\hskip
28.45274pt$

Equation (3) represents kinetic and potential terms for the scalar meson
nonets. The third term - in this expression - gives the explicit symmetry
breaking, which is defined in Eq. (11). This part of the Lagrangian creates
scalar and pseudo–scalar mesonic states defined in $\Phi$ nonets, Eq. (10).
Equation (4) represents the vector meson nonets involving explicit symmetry
breaking as given in the second term in Eq. (11). It is obvious that the
$3\times 3$ matrix of the vector meson nonets involves vector and axial–vector
fields, Eq. (10). Thus, this part creates vector and axial–vector mesonic
states and expresses the interactions between (pseudo)–scalar and
(axial)–vector as outlined in Eq. (5). Because of the explicit and spontaneous
symmetry breaking, an anomaly term $\mathcal{L}_{U(1)_{A}}$ in
SU(3)${}_{r}\times$ SU(3)ℓ is included in the effective Lagrangian. The
parameters $c,c_{0},c_{1}$ have to be determined, experimentally
Parganlija:2012fy . The first two terms in this Lagrangian approximate the
original axial anomaly term Fariborz:2008bd ; Rosenzweig:1979ay , while the
third term, which is proportional to the first term, is a mixed one. The first
anomaly term, in which other terms are used to compare with other effects of
different anomaly terms on the hadronic structure Kovacs:2013xca , is the one
taken into account in the present calculations.

In order to repreoduce the related experimental results, the higher–order
terms with local chiral symmetry have been included in Parganlija:2012fy . It
is worthy highlighting that $\mathcal{L}_{U(1)_{A}}$ symmetry is anomalous
weinberg1975u  and known as the QCD vacuum anomaly weinberg1975u ;
Schaefer:2008hk , i.e. broken by quantum effects. Without this term, a ninth
pseudo–scalar Goldstone boson corresponding to the spontaneous breaking of the
chiral U(3)${}_{\ell}\times$ U(3)r symmetry likely unfold weinberg1975u ;
Schaefer:2008hk . Therefore, the anomaly term is essential and the local
chiral symmetry would not cause further numerical problems, at a mass scale of
$1-2~{}$GeV Parganlija:2012fy . The constraint terms are conjetured to have
great influences Parganlija:2012fy . Thus, it is assumed that the
$\mathcal{L}_{U(1)_{A}}$ problem is effectively controlled by the inclusion of
the $c$–term. Also, it should be noticed that $m^{2}$ (squared tree–level
masses of mesons) and $m_{1}^{2}$ have contributions from the spontaneous
symmetry breaking Parganlija:2012fy .

The inclusion of scalar and vector meson nonets in the Lagrangian of PLSM is
possible with a redefinition for the contra–covariant derivative of the
quark–meson contributions, Eq. (7), where dof of scalar $\Phi$ and vector
$L^{\mu}$ and $R^{\mu}$ meson nonets are coupled to $A^{\mu}$, the
electromagnetic field. Eqs. (8) and (9), the left– and right–handed field
strength tensors, respectively, represent self interaction between vector and
axial–vector mesons $A^{\mu}$. Emerging from the globally invariant PLSM
Lagrangian, the local chiral invariance requires that
$g_{1}=g_{2}=g_{3}=g_{4}=g_{5}=g_{6}=g$ Parganlija:2012fy

$\displaystyle D^{\mu}\Phi$ $\displaystyle\equiv$
$\displaystyle\partial^{\mu}\Phi-ig_{1}(L^{\mu}\Phi-\Phi
R^{\mu})-ieA^{\mu}[T_{3},\Phi],$ (7) $\displaystyle L^{\mu\nu}$
$\displaystyle\equiv$
$\displaystyle\partial^{\mu}L^{\nu}-ieA^{\mu}[T_{3},L^{\nu}]-\left\\{\partial^{\nu}L^{\mu}-ieA^{\nu}[T_{3},L^{\mu}]\right\\},$
(8) $\displaystyle R^{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\partial^{\mu}R^{\nu}-ieA^{\mu}[T_{3},R^{\nu}]-\left\\{\partial^{\nu}R^{\mu}-ieA^{\nu}[T_{3},R^{\mu}]\right\\}.\hskip
14.22636pt$ (9)

It is obvious that $T_{a}=\hat{\lambda}_{a}/2$ with $a=0\dots 8$ are nine
U($3$) generators, where $\hat{\lambda}_{a}$ are Gell–Mann matrices with
fields $\Phi$ of $3\times 3$ complex matrix comprising of scalars $\sigma_{a}$
($J^{PC}=0^{++}$), pseudo–scalars $\pi_{a}$ ($J^{PC}=0^{-+}$), $V_{a}^{\mu}$,
vectors ($J^{PC}=1^{--}$), and $A_{a}^{\mu}$ axial–vectors ($J^{PC}=1^{++}$)
meson states, which are given as

$\displaystyle\Phi=\sum_{a=0}^{8}T_{a}(\sigma_{a}+i\pi_{a}),\quad$
$\displaystyle\quad L^{\mu}=\sum_{a=0}^{8}\,T_{a}\,(V_{a}^{\mu}+A_{a}^{\mu}),$
$\displaystyle R^{\mu}=\sum_{a=0}^{8}\,T_{a}\,(V_{a}^{\mu}-A_{a}^{\mu}).\quad$
(10)

The chiral symmetry is explicitly broken by

$\displaystyle H=\sum_{a=0}^{8}T_{a}h_{a},\qquad$
$\displaystyle\Delta=\sum_{a=0}^{8}T_{a}\delta_{a}.$ (11)

A non–vanishing vacuum expectation value for $\Phi$,
$\langle\Phi\rangle=T_{a}\sigma_{a}$ breaks the chiral symmetry,
spontaneously. Because of the parity is not broken in the vacuum, there are no
non–vanishing vacuum expectation values for fields $\pi_{a}$. In
$U(3)_{V}\times U(3)_{A}$ symmetry, these patterns of the explicit symmetry
breaking have been obtained as given in ref. Lenaghan:2000ey .

Due to finite quark masses in the (pseudo)–scalar and (axial)–vector sectors,
the breaking of U$(3)_{A}$ if $H_{0},\Delta_{0}\neq 0$ and the symmetry
breaking of $U(3)_{V}\rightarrow$ SU(2)${}_{V}\times U(1)_{V}$ if
$H_{8},\Delta_{8}\neq 0$ Lenaghan:2000ey . The symmetry breaking terms are
originated from U($3$)${}_{L}\times$ U($3$)${}_{R}=$U($3$)${}_{V}\times$
U($3$)A. They are proportional to the matrices $H$ and $\Delta$, Eq. (11). The
spontaneous chiral symmetry breaking is conjectured to take place in vacuum
state. Therefore, a finite vacuum expectation value for $\Phi$ and
$\bar{\Phi}$ are assumed to carry the quantum numbers of the vacuum, itself
gasiorowicz1969effective . Thus, the explicit symmetry breaking components
(diagonal) $h_{0}$, $h_{3}$ and $h_{8}$ and $\delta_{0}$, $\delta_{3}$ and
$\delta_{8}$ vanish gasiorowicz1969effective , leading to extracting three
finite condensates $\bar{\sigma_{0}}$, $\bar{\sigma_{3}}$ and
$\bar{\sigma_{8}}$. On the other hand, $\bar{\sigma_{3}}$ breaks the isospin
symmetry SU($2$) gasiorowicz1969effective . To avoid this situation, we
restrict ourselves to SU($3$). This can be $N_{f}=2+1$ Schaefer:2008hk
flavors. Correspondingly, two degenerate light (up– and down–quarks) and one
heavier strange–quark are assumed, i.e. $m_{u}=m_{d}\neq m_{s}$, where the
violation of the isospin symmetry is neglected. This facilitates the choice of
$h_{a}$ ($h_{0}\neq 0$, $h_{3}=0$ and $h_{8}\neq 0$) and for $\delta_{a}$
($\delta_{0}\neq 0$, $\delta_{3}=0$ and $\delta_{8}\neq 0$).

###  2.3 Polyakov loops

The LSM Lagrangian can be coupled to the Polyakov loops Schaefer:2008hk ;
Mao:2009aq ,

$\displaystyle\mathcal{L}=\mathcal{L}_{chiral}-\mathbf{\mathcal{U}}(\phi,\phi^{*},T),$
(12)

where the second term, $\mathbf{\mathcal{U}}(\phi,\phi^{*},T)$, represents the
effective Polyakov loop potential Polyakov:1978vu . There are various
proposals motivated by different underlying QCD symmetries in the pure gauge
limit. These are different parameterizations reproducing first–order
transition, at $T\sim 187~{}$MeV, $N_{c}=3$, and $N_{f}=2+1$ Ratti:2005jh ;
Roessner:2006xn ; Fukushima:2008wg .

  * •

A simple choice based on Ginzburg–Landau ansatz Ratti:2005jh ; Schaefer:2007pw
. The underlying $Z(3)$ center symmetry which is spontaneously broken should
be conserved. Hence, an expansion in terms of the order–parameter can be
expressed as

$\displaystyle\frac{\mathcal{U}_{\mbox{poly}}}{T^{4}}$ $\displaystyle=$
$\displaystyle-\frac{b_{2}}{4}\left(\,|\phi|^{2}+|\phi^{*}|^{2}\right)-\frac{b_{3}}{6}\,(\phi^{3}+\phi^{*3})$
(13) $\displaystyle+$
$\displaystyle\frac{b_{4}}{16}\left(|\phi|^{2}+|\phi^{*}|^{2}\right)^{2}$

with the temperature–dependent coefficients
$b_{2}(T)=a_{0}+a_{1}(T_{0}/T)+a_{2}(T_{0}//T)^{2}+a_{3}(T_{0}/T)^{3}$. The
parameters are estimated from the pure gauge lattice simulations, such that
the equation of state and the Polyakov loop expectation values are well
reproduced. The $\Phi$ terms are required to break the U($1$) symmetry of the
remaining terms to Z($3$). $a_{0}=6.75$, $a_{1}=-1.95$, $a_{2}=2.625$,
$a_{3}=-7.44$, $b_{3}=0.75$ and $b_{4}=7.5$ Ratti:2005jh  and deconfinement
temperature $T_{0}=270~{}$MeV, can be used to estimate the pure gauge QCD
thermodynamics and the Polyakov loop potential as functions of temperature.

  * •

An improved ansatz for the logarithmic form constrains $\phi$ and $\phi^{*}$
Roessner:2006xn .

$\displaystyle\frac{\mathcal{U}_{\mbox{log}}}{T^{4}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,a(T)\phi^{*}\phi+b(T)\ln\Big{[}1-6\phi^{*}\phi+4\left(\phi^{3}+\phi^{*3}\right)$
(14) $\displaystyle-$ $\displaystyle 3\left(\phi^{*}\phi\right)^{2}\Big{]},$

with the temperature–dependent coefficients
$a(T)=a_{0}+a_{1}(T_{0}/T)+a_{2}(T_{0}/T)^{2}$ and $b(T)=b_{3}(T_{0}/T)^{3}$.
This potential is qualitatively consistent with the leading–order results from
strong coupling expansion langelage2011centre . Equation (14) diverges as
$\phi^{*}\rightarrow 1$. This sets on limits to the Polyakov loop variables,
i.e. they remain small, especially at high temperatures.
$a_{0},\,a_{1},\,a_{2}$, and $b_{3}$ can be determined from lattice QCD
simulations, Tab. 1. Higher–order terms have been included Lo:2013hla ,

$\displaystyle\frac{\mathbf{\mathcal{U}}_{\mathrm{PolyLog}}}{T^{4}}$
$\displaystyle=$
$\displaystyle\frac{-a(T)}{2}\;\phi^{*}\phi+b(T)\;\ln\Big{[}1-6\,\phi^{*}\phi$
(15) $\displaystyle+$ $\displaystyle
4\,(\phi^{*3}+\phi^{3})-3\,(\phi^{*}\phi)^{2}\Big{]}$ $\displaystyle+$
$\displaystyle\frac{c(T)}{2}\,(\phi^{*3}+\phi^{3})+d(T)\,(\phi^{*}\phi)^{2}.$

It should be noticed that if $c(T)$ and $d(T)$ vanish, Eq. (15) is reduced to
Eq. (14). The various coefficients in Eq. (15) are determined Lo:2013hla
$x(T)=(x_{0}+x_{1}\left(T0/T\right)+x_{2}\left(T0/T\right)^{2})(1+x_{3}\left(T0/T\right)+x_{4}\left(T0/T\right)^{2})$
and $b(T)=b_{0}(T0/T)^{b_{1}}(1-e^{b_{2}(T0/T)^{b_{3}}})$, where $x=(a,c,d)$.
The different parameters are also summarized in Tab. 1.

  * •

A potential inspired by a strong–coupling analysis Fukushima:2008wg

$\displaystyle\frac{\mathcal{U}_{\mbox{Fuku}}}{T^{4}}$ $\displaystyle=$
$\displaystyle-\frac{b}{T^{3}}\Big{[}\,54\exp{(-a/T)}\phi^{*}\phi$ (16)
$\displaystyle+$
$\displaystyle\ln\Big{(}1-6\phi^{*}\phi-3(\phi^{*}\phi)^{2}+4(\phi^{3}+\phi^{*3})\Big{)}\Big{]}.\hskip
14.22636pt$

The nearest–neighbor interaction is given in the first term, while the
logarithm term is the Haar measure, Eq. (14). There are only two parameters,
$a$ determines the deconfinement transition, i.e. the transition temperature
in pure gauge theory and $b$ controls the mixing of the chiral and the
deconfinement transition. At deconfinement temperature $T_{0}\simeq 270$MeV,
$a=664$MeV and $b=196.2$MeV3.

Tab. 1: Fit parameters of logarithmic Roessner:2006xn  and polynomial–logarithmic Polyakov loop potentials Lo:2013hla  deduced from recent lattice QCD simulations.  Ref. Roessner:2006xn  |  $a_{0}$ |  $a_{1}$ |  $a_{2}$ |  $b_{3}$ |
---|---|---|---|---|---
|  $3.51$ |  $-2.47$ |  $15.2$ |  $-1.75$ |
Ref. Lo:2013hla  |  $a_{0}$ |  $a_{1}$ |  $a_{2}$ |  $a_{3}$ |  $a_{4}$
|  $-44.14$ |  $151.4$ |  $-90.0677$ |  $2.77173$ |  $3.56403$
|  $b_{0}$ |  $b_{1}$ |  $b_{2}$ |  $b_{3}$ |
|  $-0.32665$ |  $5.85559$ |  $-82.9823$ |  $3.0$ |
|  $c_{0}$ |  $c_{1}$ |  $c_{2}$ |  $c_{3}$ |  $c_{4}$
|  $-50.7961$ |  $114.038$ |  $-89.4596$ |  $3.08718$ |  $6.72812$
|  $d_{0}$ |  $d_{1}$ |  $d_{2}$ |  $d_{3}$ |  $d_{4}$
|  $27.0885$ |  $-56.0859$ |  $71.2225$ |  $2.9715$ |  $6.61433$

In the present work, we have utilized the higher–order parameterization of the
Polyakov loop fields based on the alternatively–improved extension of $\phi$
and $\phi^{*}$, for instance, the polynomial–logarithmic Polyakov loop
potentials Eq. (15).

###  2.4 Landau quantization

The quantity $2n+1-\sigma$ can be replaced by a summation over the Landau
Levels $0\,\leq\nu\,\leq\nu_{max_{f}}$. The earlier is the Lowest Landau Level
(LLL), while the latter stands for Maximum Landau Level (MLL), i.e.
$\nu_{max}$. For the sake of completeness, we recall that $2-\delta_{0\nu}$
represents degenerate Landau Levels and $\nu_{max_{f}}$ contributes to the
maximum quantization number, i.e. $\nu_{max_{f}}\rightarrow\infty$).

$\nu_{max_{f}}=\left\lfloor\frac{\tau_{f}^{2}-\Lambda^{2}_{QCD}}{2|q_{f}|B}\right\rfloor,$
(17)

where the brackets represent floor of enclosed quantities and the parameter
$\tau$ is related to $\mu$, at varying $T$, i. e. for results given in varying
$T$, we should take into consideration $\tau\equiv\mu_{f}$. Concretely, when
analyzing the results in thermal medium, $\mu_{f}$ is given by $\tau_{f}$,
while when analyzing the results in dense medium, $T$ is given as function of
$\tau$.

Equation (17) refers to the contribution of single and double degenerates for
the upper Landau levels. For the present study, we merely need to highlight
that various works should have been analyzed Menezes:2009uc ; Wen:2016atg ;
Avancini:2011zz ; Boomsma:2009yk ; Tawfik:2015tga . Their results in MLL,
$\nu_{max_{f}}$, at different temperatures and densities, for instance, can be
proposed in terms of the medium parameters such as $\mu$, $T$ and $eB$. For
the present calculations, we have to determine the maximum LL in order to
infinity the contributions of MLL. We assume MLL as $\nu_{max_{f}}$, where
$f\rightarrow\infty$. For details, interested readers are kindly advised to
consult ref. Tawfik:2017cdx .

At finite magnetic field, the dispersion relation contributes the Landau
Level, so that

$\displaystyle
E_{B,f}(B)=\left(p_{z}^{2}+m_{f}^{2}+|q_{f}|(2n+1-\Sigma)B\right)^{1/2}.$ (18)

Thus, the dispersion relation itself is modified by a quantization number,
$n$, known as the Landau quantum number. $\sigma$ is related to the spin
quantum number, $\Sigma=\pm S_{z}/2$ and $m_{f}(q_{f})$ is the quark flavor
mass (charge). The present study counts for different contributions from
Landau levels. The chiral condensates and deconfinement order–parameters shall
be analyzed in a wide range of temperatures, baryon chemical potentials,
magnetic fields, so that the chiral QCD phase–diagram could be mapped out in
various directions.

A more challenging question is how the Landau quantization is fixed by the
magnetic field, let us consider, for example, a Fermi sphere of quarks, where
the system is considered as discrete energy levels with respect to the
momentum space. At magnetic field background the all spin directions are
aligned though the transverse magnetic field, $\vec{B}=B\hat{e_{z}}$
”polarization”. With increasing the magnetic field, the order of Fermi energy
increases and the energy levels are discretizied. This is called the ”Landau
levels”. Moreover, the phenomena of magnetic catalysis is mainly defined as an
enhancement of the dynamical symmetry breaking by an external magnetic field.
For instance, upon increasing the magnetic field tends to enhance or
”catalysis” the quark–antiquark condensate. This chiral condensate is strongly
associated with the breaking of chiral symmetries and also creates masses.
Apparently, this is magnetic catalysis Tawfik:2017cdx .

The phenomena of (inverse)–magnetic catalysis are reviewed in great detail in
ref. Shovkovy:2012zn ; Kharzeev:2012ph , where the magnetic field enhances the
spontaneous symmetry breaking and the chiral critical temperature decreases as
increases the magnetic field strength, i.e. inverse magnetic catalysis.

###  2.5 Mean–field approximation

For a spatially uniform system in a thermal equilibrium, at finite temperature
$T$ and finite quark chemical potential $\mu_{f}$, where $f$ stands for $u,d$
and $s$ quarks, the partition function can be constructed. The grand canonical
partition function governs the change in numbers of particles and
antiparticles. Then, a path integral over quark, antiquark and meson fields
leads to Schaefer:2008hk

$\displaystyle\mathcal{Z}$ $\displaystyle=$
$\displaystyle\mbox{Tr}\,\exp{\left[-(\hat{\mathcal{H}}-\sum_{f=u,d,s}\mu_{f}\hat{\mathcal{N}}_{f})/T\right]}$
(19) $\displaystyle=$
$\displaystyle\int\prod_{a}\mathcal{D}\sigma_{a}\mathcal{D}\pi_{a}\int\mathcal{D}\psi\mathcal{D}\bar{\psi}\;\exp\Big{[}\int_{x}\Big{(}\hat{\mathcal{H}}-\mu_{f}\hat{\mathcal{N}}$
$\displaystyle+$
$\displaystyle\sum_{f=u,d,s}\mu_{f}\bar{\psi}_{f}\gamma^{0}\psi_{f}\Big{)}\Big{]},$

in which, we have abbreviated the space–time integration as

$\displaystyle\int_{x}\equiv\int^{\beta}_{0}d\tau\int_{V}d^{3}x.$ (20)

The integration runs over imaginary time $\tau=i\,t$ from $0$ to $\beta=1/T$.
For a symmetric quark matter, uniform independent chemical potentials are
imposed $\mu_{f}\equiv\mu_{u}=\mu_{d}=\mu_{s}$ Schaefer:2007pw ;
Schaefer:2008hk ; Scavenius:2000qd .

$\displaystyle\mu_{u}$ $\displaystyle=$
$\displaystyle\frac{\mu_{B}}{3}+\frac{\mu_{I}}{2}+\frac{1}{3}\mu_{Y},$
$\displaystyle\mu_{d}$ $\displaystyle=$
$\displaystyle\frac{\mu_{B}}{3}-\frac{\mu_{I}}{2}+\frac{1}{3}\mu_{Y},$ (21)
$\displaystyle\mu_{s}$ $\displaystyle=$
$\displaystyle\frac{\mu_{B}}{3}-\frac{2}{3}\mu_{Y},$

where $\mu_{B}$, $\mu_{I}$ and $\mu_{Y}$ are the baryon, isospin and
hyper–charge chemical potentials, respectively.

Converting the condensates $\sigma_{0}$ and $\sigma_{8}$ into a pure
non–strange $\sigma_{l}$ and a pure strange $\sigma_{s}$ quark flavor leads to
Kovacs:2006ym

$\displaystyle\left({\begin{array}[]{c}\sigma_{l}\\\
\sigma_{s}\end{array}}\right)=\frac{1}{\sqrt{3}}\left({\begin{array}[]{cc}\sqrt{2}&1\\\
1&-\sqrt{2}\end{array}}\right)\left({\begin{array}[]{c}\sigma_{0}\\\
\sigma_{8}\end{array}}\right).$ (28)

Then, the purely mesonic potential is given as

$\displaystyle U(\sigma_{l},\sigma_{s})$ $\displaystyle=$ $\displaystyle-
h_{l}\sigma_{l}-h_{s}\sigma_{s}+\frac{m^{2}}{2}(\sigma^{2}_{l}+\sigma^{2}_{s})-\frac{c}{2\sqrt{2}}\sigma^{2}_{l}\sigma_{s}$
(29)
$\displaystyle+\frac{\lambda_{1}}{2}\sigma^{2}_{l}\sigma^{2}_{s}+\frac{(2\lambda_{1}+\lambda_{2})}{8}\sigma^{4}_{l}+\frac{(\lambda_{1}+\lambda_{2})}{4}\sigma^{4}_{s}.\hskip
17.07164pt$

The quarks and anti–quark contributions can be divided into two regimes:

  * •

At vanishing magnetic field ($eB=0$) but finite temperature ($T$) and baryon
chemical potential ($\mu_{f}$),

$\displaystyle\Omega_{\bar{q}q}(T,\mu_{f})=-2T\sum_{f=l,s}\int_{0}^{\infty}\frac{d^{3}\vec{p}}{(2\pi)^{3}}\times$
$\displaystyle\left\\{\ln\left[1+3\left(\phi+\phi^{*}e^{-\frac{E_{f}-\mu_{f}}{T}}\right)e^{-\frac{E_{f}-\mu_{f}}{T}}+e^{-3\frac{E_{f}-\mu_{f}}{T}}\right]\right.$
$\displaystyle\left.+\ln\left[1+3\left(\phi^{*}+\phi
e^{-\frac{E_{f}+\mu_{f}}{T}}\right)e^{-\frac{E_{f}+\mu_{f}}{T}}+e^{-3\frac{E_{f}+\mu_{f}}{T}}\right]\right\\},$
(30)

where the $E_{f}=[p^{2}+m_{f}^{2}]^{1/2}$ is the flavor–dependent
single–particle energies of quark mass $(f=l,s,c)$ as,

$\displaystyle m_{l=u,d}=g\frac{\sigma_{l}}{2}\quad
m_{s}=g\frac{\sigma_{s}}{\sqrt{2}}\quad m_{s}=g\frac{\sigma_{c}}{\sqrt{2}},$
(31)

where the $\sigma$–field coupled through Yukawa coupling $g$ and the subscript
$l$ being the degenerate light up and down quarks. The SU($3$) Polyakov
linear–sigma model (PLSM) in mean–field approximation is utilized in analyzing
the thermodynamic properties of quark matter in thermal and dense QCD medium
at finite isospin asymmetry in Tawfik:2019tkp .

  * •

At finite magnetic background ($eB\neq 0$) where the magnetic field
$\vec{B}=B\hat{e_{z}}$, all the spin directions should be aligned in parallel
with the magnetic field, and the momentum directions are only determined
according to chirality. The concepts of Landau quantization and magnetic
catalysis, where the magnetic field is assumed to be oriented along the
$z$–direction Fukushima:2008xe ; Fukushima:2009ft . One can notice that the
spin (Polarization) plays an essential role in the dimensional reduction of
Dirac particles such as quarks. This latter process is known as dimension
reduction or magnetic catalysis effect Shovkovy:2012zn ; Kharzeev:2012ph .
Moreover, the thermodynamic potential should be implemented in a finite
temperature, $T$, chemical potential, $\mu_{B}$, and $B$ as,

$\displaystyle\Omega_{\bar{q}q}(T,\mu_{f},B)=-\sum_{f=l,s}\frac{|q_{f}|B\,T}{(2\pi)^{2}}\sum_{\nu=0}^{\nu_{max_{f}}}(2-\delta_{0\nu})\int_{0}^{\infty}dp_{z}\times$
$\displaystyle\left\\{\ln\left[1+3\left(\phi+\phi^{*}e^{-\frac{E_{B,f}-\mu_{f}}{T}}\right)e^{-\frac{E_{B,f}-\mu_{f}}{T}}+e^{-3\frac{E_{B,f}-\mu_{f}}{T}}\right]\right.$
$\displaystyle\left.+\ln\left[1+3\left(\phi^{*}+\phi
e^{-\frac{E_{B,f}+\mu_{f}}{T}}\right)e^{-\frac{E_{B,f}+\mu_{f}}{T}}+e^{-3\frac{E_{B,f}+\mu_{f}}{T}}\right]\right\\}.$
(32)

Accordingly, one can solve the Dirac equation to obtain the modified
dispersion relation, Eq. 18. At finite volume, $V$, and finite magnetic field,
$eB$, the free energy is defined as $\mathcal{F}=-T\cdot\log[\mathcal{Z}]/V$
or

$\displaystyle\mathcal{F}$ $\displaystyle=$ $\displaystyle
U(\sigma_{l},\sigma_{s})+\mathbf{\mathcal{U}}(\phi,\phi^{*},T)+\Omega_{\bar{q}q}(T,\mu_{f},B)$
(33) $\displaystyle+$
$\displaystyle\delta_{0,eB}\,\Omega_{\bar{q}q}(T,\mu_{f}),$

in which,the LSM mesonic potential $U(\sigma_{l},\sigma_{s})$ counts for the
contributions of the valence quarks. At very low temperatures, this part of
the potential could be excluded, especially in the regime of temperatures
typical for the QCD phase transition Tawfik:2014uka

$\displaystyle\mathcal{F}$ $\displaystyle=$
$\displaystyle\mathbf{\mathcal{U}}(\phi,\phi^{*},T)+\Omega_{\bar{q}q}(T,\mu_{f},B)+\delta_{0,eB}\Omega_{\bar{q}q}(T,\mu_{f}).\hskip
17.07164pt$ (34)

At vanishing magnetic field $B=0$, the second term vanishes, Eq. (32), while
$\delta=1$ in the third term, i.e. $\Omega_{\bar{q}q}(T,\mu_{f})$, Eq. (30),
should be evaluated with the standard dispersion–relation, $E_{f}$. Therefore,
Eq. (34), can be given as

$\displaystyle\mathcal{F}$ $\displaystyle=$
$\displaystyle\mathbf{\mathcal{U}}(\phi,\phi^{*},T)+\Omega_{\bar{q}q}(T,\mu_{f}).$
(35)

When the magnetic field is switched on, $\delta=0$ and therefore the fourth
term vanishes, as well. In this case, Eq. (34) can be reduced to

$\displaystyle\mathcal{F}$ $\displaystyle=$
$\displaystyle\mathbf{\mathcal{U}}(\phi,\phi^{*},T)+\Omega_{\bar{q}q}(T,\mu_{f},B).$
(36)

The potential of quark and antiquark contribution, at finite magnetic filed,
$\Omega_{\bar{q}q}(T,\mu_{f},B)$, can be divided into two regimes as shown in
Eq. (32):

  1. 1.

The first integral in Eq. (32) refers to the contribution of the magnetic
field in zero Landau level, $\nu=0$. In this case, the modified
dispersion–relation, Eq (18), tends to the standard one, $E_{f}$.

  2. 2.

The second integral in Eq. (32) gives the contribution of the magnetic field
for the upper Landau levels, $\nu=1\rightarrow\infty$, where the dispersion
relation will be modified, $E_{B,f}$, as given in Eq. (18).

Furthermore, the excluding of $U(\sigma_{l},\sigma_{s})$ potential from the
free energy, especially, at high temperature, tends to maintain dominant
contributions of the sea quarks than the valence quarks. In order to evaluate
the expectation values of the PLSM order–parameters

$\displaystyle\left.\frac{\partial\mathcal{F}}{\partial\sigma_{l}}=\frac{\partial\mathcal{F}}{\partial\sigma_{s}}=\frac{\partial\mathcal{F}}{\partial\phi}=\frac{\partial\mathcal{F}}{\partial\phi^{*}}\right|_{min}=0.$
(37)

In nonzero chemical–potential ($\mu\neq 0$) and finite Ployakov loop
variables, the PLSM free energy, at finite $V$, Eq. (34), is complex. A
minimization of a such function would be seen as void of meaning.

An analysis of the order parameters is given by minimizing the real part of
the PLSM free energy ($\mbox{Re}\;\mathcal{F}$). In principle, the (thermal)
expectation values of the Ployakov loop $\bar{\phi}$ and its conjugate
$\bar{\phi}^{*}$ must be real quantities as discussed in ref. Dumitru:2005ng .
The solutions of these equations can be determined by minimizing the real part
pf the PLSM free energy ($\mbox{Re}\;\mathcal{F}$) at a saddle point. The
remaining parameters are the chiral order–parameters $\bar{\sigma_{l}}$,
$\bar{\sigma_{s}}$ and the Polyakov–loop expectation values
$\bar{\phi},\;\bar{\phi}^{*}$ as functions of $T$, $\mu$ and $eB$.

##  3 Results

###  3.1 Chiral and deconfinement order–parameters

The estimation of the chiral condensates ($\sigma_{l}$ and $\sigma_{s}$) and
the deconfinement order–parameters ($\phi$ and $\phi^{*}$) in dense and
thermal medium should be first computed by minimizing the free energy, Eq.
(37). The parameters of PLSM, as discussed in sections 2.2 and 2.3, are
estimated, at mass of (vacuum) sigma–meson $\sigma=800~{}$MeV, are measured
(vacuum) light and strange chiral condensates are $\sigma_{l_{o}}=92.5~{}$ MeV
and $\sigma_{s_{o}}=94.2~{}$MeV, respectively.

Fig. 2: Left–hand panels (a) and (c) show normalized chiral condensate with
respect to the vacuum value as functions of temperature. Right–hand panels (b)
and (d) give the expectation values of the Polyakov loop fields ($\phi$ and
$\phi^{*}$) as functions of temperatures, as well. The upper panels presents
the PLSM results, at magnetic fields $eB=m^{2}_{\pi}~{}$, while the bottom
panels at $eB=10\,m^{2}_{\pi}~{}$ and $\mu=0$ (solid curves), $100$ (dashed
curves), $200~{}$MeV (dotted curves).

Figure 2 presents both PLSM chiral–condensates and order–parameters, at finite
baryon chemical potentials and magnetic fields $eB=m^{2}_{\pi}$ (upper panel)
and $eB=10\,m^{2}_{\pi}~{}$ (bottom panel) as functions of $T$. In the
left–hand panels, (a) and (c) for normalized chiral–condensates,
$\sigma_{l}/\sigma_{l_{o}}$ and $\sigma_{s}/\sigma_{s_{o}}$, we notice that
the chiral condensates are shifted to lower values with increasing baryon
chemical potential. This means that the chiral critical temperature
($T_{\chi}$) decreases with increasing $eB$ and with increasing $\mu$, as
well. Therefore, we can draw a conclusion that the effect of strong magnetic
field is almost the same as that of the baryon chemical potentials, namely on
decreasing the phase transition (crossover).

In right–hand panels (b) and (d) for the Polyakov loop parameters, $\phi$ and
$\phi^{*}$ as functions of $T$, at diffident baryon chemical potentials, we
find that both order parameters become differentiable, at finite $\mu$;
$\mu=100~{}$MeV (dashed curves) and $\mu=200~{}$MeV (dotted curves), while at
$\mu=0~{}$MeV (solid curves), $\phi=\phi^{*}$. The deconfinement critical
temperature, $T_{\phi}$, is shifted to lower values as the magnetic field and
baryon chemical potential increase. We thus draw the conclusion that the
$\phi$ and $\phi^{*}$ have opposite dependence on temperatures. The latter
increases as the magnetic field and the baryon chemical potential $\mu$
increase, while the earlier decreases.

Fig. 3: Left–hand panel: the chiral condensates normalized to the vacuum value
are given as functions of temperature. Right–hand panel: the expectation
values of Polyakov loop parameters, $\phi$ and $\phi^{*}$, are calculated in
dependence on $T$, at $eB=0$ (solid curves), $0.1$ (dashed curves), $0.2$
(dotted curves) and $0.4~{}$GeV2 (dot–dashed curves) and vanishing baryon
chemical potential.

Figure 3 depicts the temperature dependence of normalized chiral condensates
(a) and deconfinement order parameters (b), at $eB=0$ (solid curves), $0.1$
(dashed curves), $0.2$ (dotted curves) and $0.4~{}$GeV2 (dot–dashed curves)
and vanishing baryon chemical potential. The left–hand panel (a) depicts the
same as the left-hand panel of Fig. 2 but for different values of the magnetic
field, at fixed values of baryon chemical potential. We notice that the chiral
critical temperature deceases and the crossover phase–transition becomes
sharper with increasing magnetic field. This can be interpreted due to the
maximum occupation of the Landau levels ($\nu_{max}\rightarrow\infty$). We
conclude that the phase transition seems to be of first order, whenever the
chiral condensate passes through a meta–stable phase, in which light quarks
become massless and move freely.

In right–hand panel (b), the temperature dependence of the deconfinement
phase–transition is depicted, at $\mu=0$. $\phi=\phi^{*}$, at $eB=0$ (solid
curves), but having different values, ar $0.1$ (dashed curves), $0.2$ (dotted
curves) and $0.4~{}$GeV2 (dot–dashed curves). It is obvious that the
deconfinement critical temperature, $T_{\phi}$, slightly decreases as the
magnetic field increases.

It would be noticed that if one compares the curve for $eB=0.1~{}$GeV2 with
the curve of $eB=10m_{\pi}^{2}\simeq 0.2~{}$GeV2, which would be nearly
similar values, one would find a slight difference. This is an artifact. It
comes from the effect of the occupation Landau levels. The Landau levels and
the quantization number are determined for the medium parameters such as $T$,
$\mu$, $eB$.

Fig. 4: The normalized chiral condensates are given as functions of
temperatures, at $eB=0.2~{}$GeV2. Different MLL are taken into considration.

As discussed, PLSM is well–suited to study the chiral limit. The inclusion of
magnetic field can be achieved by modifying the dispersion relation of quarks
and antiquarks, Eq. (30). In doing this, the dimension of the momentum–space
should be reduced from three to one and scaled via quark charge and magnetic
field. This process is known as magnetic catalysis effect of dimension
reduction Shovkovy:2012zn . Furthermore, the introduction of the magnetic
field requires suitable implementation of the Landau quantization, section
2.4.

Figure 4 shows the effects of the occupation number of the Landau levels on
the temperature dependence of the chiral condensate, $\sigma_{l}$ and
$\sigma_{s}$, at a finite magnetic field $eB=0.2~{}$GeV and a vanishing baryon
chemical potential. We observe that the change in the Landau levels is
relatively significant only in crossover phase–transition region and seems to
disappear otherwise. At $MLL=14$ (solid curves), $51$ (dotted curves) and
$\infty$ (dashed curves), the normalized chiral condensates for light and
strange quarks are analyzed as functions of temperatures, at finite magnetic
field and vanishing chemical potential. We conclude that the increase in the
Landau levels very slightly sharpens the phase transition or the crossover and
decreases the critical temperature $T_{\chi}$.

Fig. 5: Left–hand panels (a) and (c) show the chiral condensates, $\sigma_{l}$
and $\sigma_{s}$, normalized to the vacuum value as functions of the baryon
chemical potentials, at $eB=1$ (solid curves) $10$ (dashed curves), $15$
(dotted curves), $20$ (dot–dashed curves), and $25\,m^{2}_{\pi}~{}$ (double
dotted curves). Right–hand panels (b) and (d) give the expectation values of
the Polyakov loop parameters, $\phi$ and $\phi^{*}$, as functions of baryon
chemical potentials, at finite temperatures. Top panel shows the results, at
$T=50\,$MeV, while the bottom panel at $T=100\,$MeV.

Furthermore, the temperature dependence of the chiral condensates,
$\sigma_{l}$ and $\sigma_{s}$, and the deconfinement order–parameters, $\phi$
and $\phi^{*}$, are estimated, at different values of magnetic fields ($eB$)
and baryon chemical potential ($\mu$). Fig. 5 shows the deconfinement
order–parameters as functions of baryon chemical potentials, at $T=50~{}$MeV
(upper–panel) and $T=100~{}$MeV (bottom–panel), from which we notice that the
magnetic effect is very obvious. In left–hand panels (a) and (c), the chiral
condensates are given in dependence on magnetic fields; $eB=1$ (solid curves)
$10$ (dashed curves), $15$ (dotted curves), $20$ (dot–dashed curves), and
$25\,m^{2}_{\pi}~{}$ (double dotted curves). Firstly, we conclude that the
temperature causes a rapid decrease in the chiral condensates around the
chiral phase–transition similar to what was observed in a previous study from
PLSM without magnetic field Tawfik:2014gga . Secondly, a small sudden drop in
the chiral condensates referring to first–order phase–transition takes place.
There is a gap difference between light and strange chiral condensates, at
very high density. This could be understood because of the inclusion of the
anomaly term in Eq. (29), where the fit parameters are accordingly modified
Schaefer:2008hk ; Tawfik:2014gga . This was conjectured as an evident on
numerical estimation of the chiral condensates. The magnetic field seems to
have a sudden drop in the chiral phase–structure. This causes an increase in
the critical temperatures. We can draw a conclusion that increasing magnetic
field tends to sharpen the phase transition and to accelerate the formation of
metastable phase.

The right–hand panels (b) and (d) give the deconfinement phase–transition in
dependence on magnetic fields; $eB=1$ (solid curves), $10$ (dashed curves),
$15$ (dotted curves), $20$ (dot–dashed curves), and $25\,m^{2}_{\pi}$ (double
dotted curves). The increase in temperature increases the deconfinement
phase–transition to larger baryon chemical potential. The thermal and magnetic
effects of the hadronic medium on the evolution of Polyakov loop parameters
seem to be very smooth. The slope of $\phi$ and $\phi^{*}$ to the baryon
chemical potential can approximately be estimated. It is obvious that the
magnetic field decreases these slopes (increases the critical temperature),
while the temperature increases them (decreases the critical temperature).

###  3.2 Thermodynamics

We start this analysis with a brief introduction to the basic quantities of
thermodynamics, at non–vanishing magnetic field strength. For a statistical
system in equilibrium with volume $V$, temperature $T$ and chemical potential
$\mu$, the grand–canonical density operator, $\hat{\rho}$, the grand–canonical
partition function, $\mathcal{Z}(T,V,\mu)$, and the grand potential,
$\Omega(T,V,\mu)$ can be introduced in the natural units
$\kappa_{B}=\hbar=c=1$. The pressure as a function of finite magnetic field
$eB$ reads

$\displaystyle P(T,\mu,eB)=-\Omega(T,\mu,eB),$ (38)

which enables us to characterize the phenomenology to the strong interacting
QCD matter, at finite magnetic fields, as the case in HIC, including
magnetization, magnetic susceptibility and permeability.

Then, the free energy density can be written as Aad:2012ew

$\displaystyle\mathbf{f}$ $\displaystyle=$ $\displaystyle\epsilon-
Ts=\epsilon^{\mbox{tot}}-\epsilon^{\mbox{field}}-Ts$ (39) $\displaystyle=$
$\displaystyle\epsilon^{\mbox{tot}}-Ts-eB\;\mathcal{M},$ (40)

where $\mathcal{F}=-T\log\cdot\mathcal{Z}$, the total energy density
$\epsilon^{\mbox{tot}}=\epsilon+\epsilon^{\mbox{field}}$, which in turn is
divided into two terms; one for the energy density of the medium $\epsilon$
and another one of the magnetic field $\epsilon^{\mbox{field}}=eB\mathcal{M}$.

We notice that the partition function in vanishing magnetic field is given by
an integral over six–dimensional phase–space. The dispersion relations follow
the Lorentz invariance principle. But, in finite magnetic field, the integral
dimensional is reduced and simultaneously accompanied by a considerable
modification in the dispersion relation.

The velocity of a test particle with momentum
$\partial\epsilon^{\mbox{tot}}/\partial P$ in finite magnetic field $B$ can be
expressed as

$\displaystyle v_{p}$ $\displaystyle=$ $\displaystyle
c\left[\frac{c\,p}{c\,p+2|q_{f}|(\kappa+\frac{1}{2}-\frac{\sigma}{2})B}\right],$
(41)

where $\sigma=\pm S/2$. Then, the causality is guaranteed for $v_{p}$
not–exceeding the speed of light $c$, i.e. as long as the $B$–term is finite
positive, which should be quantitatively estimated as a function of
temperature and magnetic field. From Eq. (40), the entropy density and the
magnetization, for instance, could be derived as

$\displaystyle s=-\frac{1}{V}\frac{\partial\mathcal{F}}{\partial
T},\quad\quad\quad\mathcal{M}=-\frac{1}{V}\frac{\partial\mathcal{F}}{\partial(eB)}.$
(42)

Other thermodynamic quantities, such as, pressure can be derived, as well.
Because the magnetic field is conjectured to mark a preferred direction,
$P_{i}$ might be different along the geometry effect of the magnetic field. As
$V=L_{x}L_{y}L_{z}$ and the magnetic field is distributed along $z$ direction,
we can distinguish between two different systems.

  * •

$B$–scheme, in which the magnetic field remains constant in direction, results
in an isotropic pressure

$\displaystyle P_{x}=P_{y}=P_{z}=-\mathbf{f}$ (43)

  * •

$\varphi$–scheme, which sets up magnetic flux $\varphi=eB\cdot L_{x}L_{y}$,
remains constant and results in an anisotropic pressure

$\displaystyle P_{x}=P_{y}=P_{z}-eB\mathcal{M}$ (44)

Accordingly, the thermodynamic quantities should be modified. For instance, in
$\Phi$–scheme, the trace anomaly (the interaction measure) reads

$\displaystyle I$ $\displaystyle=$
$\displaystyle\epsilon-3p_{z}+2eB\cdot\mathcal{M}.$ (45)

The speed of sound squared, at constant entropy, is given as Tawfik:2012ty ,

$\displaystyle c_{s}^{2}$ $\displaystyle=$ $\displaystyle\left(\frac{\partial
p}{\partial\epsilon}\right)_{s}=\frac{\partial p}{\partial
T}/\frac{\partial\epsilon}{\partial T}=\frac{s}{c_{v}},$ (46)

where the specific heat, $c_{v}$, gives the thermal rate change of the energy
density, at constant volume. In finite magnetic field, Stefan–Boltzmann (SB)
limits can be deduced from lowest–order perturbation theory Bali:2014kia

$\displaystyle T\,\log\,\mathcal{Z}(V,T,\mu,eB)$ $\displaystyle=$
$\displaystyle\frac{19\,V\pi^{2}}{36}T^{4}$ (47) $\displaystyle+$
$\displaystyle
b_{1}^{\mbox{free}}(eB)^{2}\,V\,\log\left(\frac{T}{\Lambda_{H}}\right)+\cdots,\hskip
17.07164pt$

Fig. 6: Left–hand panel (a): the normalized pressure $p/T^{4}$ versus
$T/T_{c}$ is calculated in PLSM, at finite magnetic fields $eB=0.0$ (solid
curve), $eB=0.1$ (dotted curve) and $eB=0.3~{}$GeV2 (double–dotted curve) and
compared with recent lattice QCD (open square), (close square) and (open
triangle), respectively Bali:2014kia . Right–hand panel (b): The same as in
the left–hand panel but for the normalized trace–anomaly
$(\epsilon^{\mbox{tot}}-3P)/T^{4}$.

Figure 6 depicts the normalized pressure $p/T^{4}$ (left–hand panel) and the
normalized trace–anomaly $(\epsilon^{\mbox{tot}}-3P)/T^{4}$ (right–hand panel)
as functions of temperatures, at vanishing baryon chemical potential but
finite values of magnetic fields $eB=0.0$ (solid curve), $eB=0.1$ (dotted
curve) and $eB=0.3~{}$GeV2 (double–dotted curve). The results are compared
with recent lattice QCD Bali:2014kia  (open square), (close square) and (open
triangle), respectively. It is obvious that the pressure increases with
increasing magnetic field, especially, at low temperatures. At high
temperatures, $p/T^{4}$ is limited to the SB limits, which apparently
negligibly are affected by the magnetic field.

The right–hand panel of Fig. 6 presents the modified normalized trace–anomaly,
Eq. (45), as a function of temperature and magnetic field strengths, $eB=0.0$
(solid curve), $eB=0.1$ (dotted curve) and $eB=0.3~{}$GeV2 (double–dotted
curve). These are also compared with recent lattice QCD Bali:2014kia  (open
square), (close square) and (open triangle), respectively.

The normalized entropy density, $s/T^{3}$, which is derived from $p_{z}$ with
respect to $T$, vanishes at $T=0$. This may be understood from the fact that
the vacuum contribution is conjectured to be a pure quantum effect. This
emerges from the interaction of virtual quarks with the external field. Thus,
it likely doesn’t contribute to entropy Bali:2014kia . We also notice that, at
$T>0$, the magnetic field changes the thermal distributions and is expected to
modify entropy,

$\displaystyle s=\frac{\epsilon+p_{z}}{T}.$ (48)

We find that near the $T_{c}$–regime, $s/T^{3}$ excellently agrees with the
lattice QCD calculations. This might not be also the case, at higher
temperatures.

Fig. 7: Left–hand panel (a): the normalized entropy $s/T^{3}$ versus $T/T_{c}$
calculated in PLSM, at finite magnetic fields, $eB=0.0$ (solid curve),
$eB=0.1$ (dotted curve) and $eB=0.3~{}$GeV2 (double–dotted curve). The results
are compared with recent lattice QCD (open square), (close square) and (open
triangle), respectively Bali:2014kia . Right–hand panel (b) shows the same as
in the left–hand panel but here $P\epsilon^{\mbox{tot}}$.

Figure 7 presents $s/T^{3}$ and the equation of state $\epsilon(p)$ included
in $\epsilon^{\mbox{tot}}$ in dependence on $T$, at $\mu=0~{}$MeV and the same
values of the magnetic field as depicted in Fig. 6, namely $eB=0.0$ (solid
curve), $eB=0.1$ (dotted curve) and $eB=0.3~{}$GeV2 (double–dotted curve). The
corresponding curves and lattice points are the same as in Fig. 6. We find a
reasonable agreement with the lattice QCD simulations The phase transition
seems to smoothly take place. The temperature dependence continues even above
$T_{c}$. $s/T^{3}$ keeps its increase with increasing $T/T_{c}$ so that it
becomes slightly lower than the lattice results.

The right–hand panel of Fig. 7 (b) presents the PLSM calculations for
$p/\epsilon$, at $eB=0.0$ (solid curve), $eB=0.1$ (dotted curve) and
$eB=0.3~{}$GeV2 (double–dotted curve). We also compare with recent lattice QCD
Bali:2014kia  (open square), (close square) and (open triangle), respectively.
A fair agreement is also obtained, especially at $eB=0.1$, at low temperature.
It is obvious that such an agreement could be improved with increasing
magnetic fields.

Now, we are able to estimate some fundamental properties of the strongly
interacting QCD matter in finite magnetic field such as the magnetization, the
magnetic susceptibility, and the permeability. The response of QCD matter to
an external magnetic field can be estimated from the free energy density. The
magnetic susceptibility with proper renormalization has been introduced in
litrature Kamikado:2014bua . The magnetic permeability measures the ability of
the QCD matter to generate magnetic field or the ability to store magnetic
potential energy, which is proportionally constant for the magnetic flux. The
magnetic flux - in turn - is produced from influences of the magnetic field.
The magnetic permeability can be calculated along the magnetic field on the
transverse direction to momentum space $p_{z}$.

In thermal, dense and magnetic medium, the partition function
$\ln\,\mathcal{Z}$ is to be properly modified, from which the magnetization
can be derived, Eq. (42). The sign of magnetization defines an essential
magnetic property, namely whether QCD matter is para– or dia–magnetic, i.e.
$M>0$ (bara) or $M<0$ (dia).

  * •

For dia–magnetized QCD matter, the color charges align oppositely to the
direction of the magnetic field and produce an induced current, which spreads
as small loops attempting to cancel the effects of the applied magnetic field,
and

  * •

For para–magnetized QCD matter, most color charges align towards the direction
of the magnetic field.

Fig. 8: The magnetization $\mathcal{M}$ is calculated as a function of $T$, at
$eB=0.0-0.4$ GeV2 and compared with recent lattice QCD simulations (symbols)
Bali:2014kia .

As discussed, the magnetization greatly affects the thermodynamic properties
of the QCD matter, as the magnetization measures the response of the system of
interest to finite magnetic field. The latter is likely extremly generated in
HIC, at least in very short time intervals. Aslo, because of the relativistic,
off–center motion of spectators, i.e. peripheral collisions, the rapid motion
of electric charges generates magnetic field perpendicular to plane of both
motion direction and the electric field. Also, because of local imbalance in
the momenta carried by the colliding nucleons in peripheral and central
collisions. This local imbalance leads to an angular momentum and thus a
magnetic field Tawfik:2016cot . As discussed, such a magnetic field is
typically very huge, ${\cal O}(m_{\pi}^{2})$. It largely exceeds the
detector’s magnet field.

The magnetization, $\mathcal{M}$, can be derived from Eq. (34). The values
obtained can be given in GeV2 in the natural units. The sign of $\mathcal{M}$
refers to para– or dia–magnetic QCD matter. If $\mathcal{M}>0$ or
$\mathcal{M}<0$, the QCD matter is either para– or dia–magnetized,
respectively.

In Fig. 8, $\mathcal{M}$ is given as a function of $T$, at $eB=0.1$ (dotted),
$0.2$ (dashed), $0.3$ (double–dotted) and $0.4$ GeV2 (dash–dotted curve) at
vanishing $\mu$. The results are compared with recent lattice QCD Bali:2014kia
at $eB=0.1$ (closed square), $0.2$ (circle), $0.3$ (triangle) and $0.4$ GeV2
(astride). It is obvious that $M>0$ and increases as the magnetic field
increases. This result indicates that paramagnetic properties of the QCD
matter. At temperatures below the critical value, the PLSM results resemble
the lattice data in an excellent way. At temperatures characterizing QGP
(higher than the critical temperature), the PLSM curve becomes larger than the
lattice data, especially at very high temperatures. In this range of
temperatures, the colorless hadrons are conjectured to deconfine into colored
quarks and gluons. The discrepancies suggest that the corresponding dof are
not sufficient enough to achieve a good agreement, especially at very high
temperature. Furthermore, we notice that the PLSM calculations give an evident
on paramagnetic features of the hot QCD matter.

Fig. 9: Left–hand panel (a): the magnetic susceptibility $\chi_{B}$ versus
$T/T_{c}$ calculated in PLSM, at $eB=0~{}$GeV2. Right–hand panel (b): The same
as in the left–hand panel but for the magnetic permeability.

Also, the magnetic susceptibility and permeability reflect the magnetic
response of the hot QCD matter. In other words, the response of QCD matter to
the magnetic field can be determined by the slope of $\mathcal{M}$ with
respect to the magnetic field. The second derivative of the free energy
density with respect to the finite magnetic field is the magnetic
susceptibility,

$\displaystyle\chi_{B}=-\frac{1}{V}\frac{\partial^{2}\mathcal{F}}{\partial(eB)^{2}}|_{eB=0}.$
(49)

In response to the magnetic field, the magnetic susceptibility is a
dimensionless proportionality parameter indicating the degree of magnetization
of the QCD matter.

Furthermore, the relative magnetic permeability, $\mu_{r}$, normalized to the
vacuum magnetic permeability $\mu_{0}$ can be translated as the magnetic
effect in thermal QCD medium. This can be determined by different methods.
With a direct relation to the magnetic susceptibility, we have

$\mu_{r}=1+\chi_{B}.$ (50)

As shall be introduced, this relation agrees well with the lattice QCD
simulations, in which the magnetic permeability is expressed in terms of the
magnetic susceptibility,

$\mu_{B}\equiv\frac{B^{ind}}{B^{ext}}=\frac{1}{1-4\pi\alpha_{m}\cdot\chi_{B}},$
(51)

where $\alpha_{m}=e^{2}/4\pi$ is the fine structure constant. This expression
distinguishes between dimensionless proportionality constants, namely external
$B^{ext}$ and induced magnetic field $B^{ind}$. We highlight that the
higher–order permeability seems to be limited by the magnetic susceptibility,
which is given by the reciprocal of the square of elementary charge $e$, i.e.
$\chi_{B}\rightarrow 1/e^{2}$ when $\mu\rightarrow\infty$.

Left–hand panel of Fig. 9 (a) shows the magnetic susceptibility as functions
of temperature, at $eB=0.0~{}$GeV2 and vanishing $\mu$. The results deduced
from PLSM are compared with various lattice simulations (symbols), in which
different calculation methods and algorithms are applied. We also compare with
the resonance gas model (HRG) model. The recent lattice QCD simulations
Bali:2014kia  (open circle) were estimated by using half–half method in
$24^{3}\times 32$ lattice (closed triangle) and integral method in
$28^{3}\times 10$ lattice (open triangle). The diamonds represent lattice
simulations for $N_{f}=2+1$ and when using HISQ/tree action with light quark
masses $m_{l}/m_{s}=0.05$ and temporal dimension $N_{\tau}=8$, the lattice
results Levkova:2013qda . The closed circles stand for results obtained from
isotropy lattice Bali:2013owa .

Right–panel of Fig. 9 (b) depicts the relative permeability with respect to
that of the vacuum compared to recent lattice QCD calculations (open
triangles) Bali:2014kia  in a wide range of temperatures, at $eB=0.0~{}$GeV2
and at vanishing $\mu$. It is apparent that the agreement between PLSM and
lattice QCD calculations is excellent.

Features of PLSM and lattice QCD results can be summarized as follows.

  * •

The magnetic susceptibility obtained from the HRG model Bali:2014kia  (dashed
curve) confirms the nature of the QCD matter as dia–magnetized, especially at
low temperatures. Here, the free energy density is the sum over all
contributions from the colorless hadrons and their resonances contributes to
the hadron interactions in order to assure negative magnetic susceptibility
Bali:2014kia .

  * •

In PLSM, the free energy density is divided into three terms. The first one is
the pure mesonic potential which is obtained from the Lagrangian for the pure
gauge. The second one gives the contributions of the quark and antiquarks,
which - as the name says - have mesonic fluctuations from the quarks and the
antiquarks. The third term represents the interactions of the color charges
and the gluons. This means that there two types of contributions to the
hadronic fluctuations, while only one type contributes to the gluon
interactions.

  * •

At very low temperatures, the slope of $\chi(T)$ is apparently negative
(inside–box in left–hand panel of Fig. 9) Steinert:2013fza . This gives a
signature that the QCD matter is dia–magnetized and also well reproduces the
different lattice simulations. When switching to the high temperature regime,
i.e. restoring the broken chiral symmetry, we observe a transition between
dia– and para–magnetic properties, where QCD para–magnetism is likely, at high
temperature.

  * •

We notice that the PLSM results confirm that the strongly interacting QCD
matter has para–magnetic properties. The magnetic susceptibility steeply
increases when increasing temperatures towards the deconfinement
phase–transition. These conclusions have been found in a wide range of
temperatures $100\leq\,T\,\leq 250$ MeV Aoki:2009sc ; Borsanyi:2010cj .

###  3.3 QCD phase–diagram at finite magnetic field

There are two different mechanisms manifesting the influences of the magnetic
field on the QCD phase–diagram. The first one is that the magnetic field
improves the QCD phase–transition due to its contributions to the Landau
quantizations or the Landau levels. The second one is that the magnetic field
contributes to the suppression in the chiral condensate due to the restoration
of the broken chiral symmetry. This suppression is known as inverse magnetic
catalysis. It is a decrease in the chiral critical temperature with the
increase in the magnetic field.

The PLSM has two main types of the order parameters; the chiral condensates,
which are connected with two light quarks, $\sigma_{l}$, and one strange
quark, $\sigma_{s}$, and the Polyakov loop variables, $\phi$ and $\phi^{*}$.
The intersects of $\phi$ and $\phi^{*}$ with the chiral condensates are used
in determining the quasi–critical temperatures (dotted curve) in Fig. 10. The
solid curve is estimated from higher–order moments of quark multiplicity (not
shown in the present paper), where the critical temperatures are estimated at
the peak of normalized quark susceptibility $\chi_{q}/T^{2}$. From both
methods, we find that the critical temperatures decrease with increasing
magnetic fields referring to an inverse magnetic catalysis.

Fig. 10: Three chiral QCD phase–diagrams. Left–hand panel: critical
temperature $T$ to magnetic field $eB$. Middle–panel: critical baryon chemical
potential $\mu$ to $eB$. Right–hand panel: $T$ to $\mu$. The vertical bands
refer to magnetic fields likely generated, at RHIC and LHC energies.

In Fig. 10, the influences of the magnetic field on the chiral QCD
phase–diagram are depicted. Left–hand panel shows the variation of the
critical temperature with increasing magnetic field, at almost vanishing
$\mu$. The PLSM calculations are compared with recent lattice simulations
given (circles with errorbars) Bali:2014kia . The vertical bands refer to the
magnetic fields generated, at RHIC (orders of per cent of GeV2 or $\sim
m_{\pi}^{2}$) and LHC energies (orders of per ten GeV2 or $\sim
10-15\,m_{\pi}^{2}$). The PLSM calculations take into consideration two
methods in order to estimate the critical temperatures. One is based in
normalized quark susceptibility $\chi_{q}/T^{2}$ and the other one is based on
the freeze–out condition $s/T^{3}$, where $s$ is the entropy density.

There is an excellent agreement, especially at $0\leq\,eB\,$ [GeV2]$\,\leq
0.2$. The solid curve matches well with the lattice simulations, at a wider
range of magnetic fields $0.13\leq\,eB\,$ [GeV2]$\,\leq 0.55$. The first
method of determining the critical temperature apparently overestimates the
lattice results, at low temperature, while the second method slightly
underestimates these, especially at high temperatures. We conclude that the
chiral magnetic field improves the agreement as it enhances the chiral
condensates. This depends also on the type of contributions to the Landau
levels introduced to the QCD effective approach.

The middle–panel draws the dependence of the critical baryon chemical
potential on the magnetic fields, at $T=50$ (solid curve) and $100~{}$MeV
(dashed curve). At constant magnetic field as that at RHIC or LHC energy,
large critical baryon chemical potential can be reached, even at low
temperatures, i.e. $\mu$ decreases with increasing $eB$. The intersection
between the deconfinement phase–transition and the light quark chiral
condensate is utilized to determine $\mu$, at which the broken chiral symmetry
is restored. There is no lattice QCD simulations to compare with.

The right–hand panel presents $T$-$\mu$ phase–diagram, at $eB=m_{\pi}^{2}$
(solid curve), $10\,m_{\pi}^{2}$ (dashed curve) and $20\,m_{\pi}^{2}$ (dotted
curve). In determining the critical temperature, various methods have been
utilized, for example, the intersection between the Polyakov loop variables
$\phi$ and $\phi^{*}$, which are related to the deconfinement
phase–transition, and the chiral condensates of light and strange quarks,
$\sigma_{l}$ and $\sigma_{s}$, respectively. The latter is related to the
restoration of the broken chiral symmetry. For example, the critical
temperature corresponding to the chiral restoration of light quark,
$T_{c}^{\chi_{l}}$, can be determined by the intersection between $\phi$ and
$\sigma_{l}$, while the critical temperature corresponding to the chiral
restoration of strange quark, $T_{c}^{\chi_{s}}$, can be defined from the
intersection between $\phi^{*}$ and $\sigma_{s}$. We observe that increasing
magnetic field enhances the chiral QCD phase–diagram, i.e. the chiral
phase–transition takes place at lower temperature. The estimation of
freeze–out parameters, $T$ and $\mu$, in dependence on the heavy–ion
centralities or the impact parameters allows to analyze the influence of the
magnetic field, experimentally bzdak2012event . There are various experimental
results on chemical and thermal freeze–out as reviewed in ref. Tawfik:2014eba
.

Fig. 11: As in Fig. 10 but here the multi–dimensional chemical freeze–out
boundary combines the chemical freeze–out parameters $T$ with the magnetic
field $eB$ and the baryon chemical potentila $\mu$.

In Fig. 11, the chemical freeze–out condition $s(T,eB,\mu)/T^{3}=7$ is
implemented in order to estimate the feeze–out temperature $T$. The entropy
density is calculated, at different temperatures, baryon chemical potentials,
and magnetic fields. At a given value of the entropy density normalized to
$T^{3}$, the related baryon chemical potential, $\mu$, and the corresponding
magnetic field, $eB$, are determined. These three parameters are then depicted
in Fig. 11. It is a multi–dimensional chemical freeze–out boundary showing the
dependence the freeze–out diagram ($T-\mu$), which can directly be related to
the one analyzed from ethe experimental measurements of various particle
ratios Tawfik:2014eba , on the magnetic field. It is obvious that, at small
$\mu$, the effect of the magnetic field is almost negligible. At higher
temperatures, the drop in $T_{c}$ around the chiral phase–transition moves to
lower values with increasing $eB$. Again, this refers to an inverse catalysis.
At very high temperatures, there a slight increase in $T$ with increasing
$eB$. Increasing $eB$ makes the chiral phase–transition smother. It is
important to notice that the shape of $T$–$\mu$ phase–diagram looks different
from the one at vanishing $eB$ Tawfik:2014eba .

##  4 Conclusions

In relativistic HIC, the off–center motion of spectators in the peripheral
collisions and the rapid motion of electric charges generate huge magnetic
fields perpendicular to the plane of the motion direction and the electric
field. Also, the local imbalance in momenta carried by the colliding nucleons
in both peripheral and central collisions generates a huge magnetic field, as
well. The response of QCD matter modeled in PLSM to these external magnetic
fields such as magnetization, magnetic susceptibility and permeability is
determined. The chiral magnetic properties of the QCD phase–diagram are
analyzed. The inclusion of finite magnetic field in PLSM can be accomplished
by firstly modifying the dispersion relation of the quarks and antiquarks, so
that the dimension of the momentum–space is reduced from three to one and then
scaled via quark charge and magnetic field (magnetic catalysis) and secondly
Landau quantization should be integrated in. The latter modifies the
dispersion relation by a quantization number; the Landau quantum number.
$\sigma$, which is related to the spin quantum number and to the masses of
quark flavors. Both chiral condensates and deconfinement order–parameters have
been analyzed in a wide range of temperatures, baryon chemical potentials,
magnetic fields, so that the chiral QCD phase–diagram could be mapped out in
various directions.

The magnetization remarkably affects the thermodynamic properties of the QCD
matter. The PLSM results give an evident on paramagnetic features of the hot
QCD matter. The magnetic susceptibility, at low temperatures, is negative
indicating that the QCD matter is dia–magnetized. At higher temperatures
regime, i.e. restoring the broken chiral symmetry, there is a transition from
dia– to para–magnetic properties. The permeability is a characterizing
property of a magnetic material measuring the ability to create magnetic field
and to store magnetic potential energy. The latter is proportionally constant
for magnetic flux, which is produced from the influences of the magnetic
field. At low temperatures (hadron phase) the QCD permeability is small but
rapidly increases around the deconfinement phase–transition. At high
temperatures (parton phase) the QCD permeability is large. We conclude that in
a wide range of temperatures, the magnetic permeability normalized to the
vacuum value agrees well with the lattice QCD simulations, so that this
quantity can be utilized as a magnetic order–parameter.

To estimate the variation of the chiral critical temperature with the magnetic
field, the PLSM results are confronted to the recent lattice QCD data. We
found that the chiral critical temperature decreases as the magnetic field
increases. We conclude that the magnetic catalysis of the thermal QCD medium
is inverse and an inverse interrelation between the chiral chemical potential
and the magnetic field is obtained. The chiral phase–diagram is shifted to
lower temperatures due to the increase in the magnetic field. This result is
confirmed by two different methods, one from the thermal and dense
phase–transition of the PLSM parameters and the another one by applying a
condition from freeze–out parameters $s/T^{3}$.

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# Jacob’s ladders and the nonlocal interaction of the function $Z(t)$ with the
function $\tilde{Z}^{2}(t)$ on the distance $\sim(1-c)\pi(t)$ for a collection
of disconnected sets

Jan Moser  Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
<EMAIL_ADDRESS>
###### Abstract.

It is shown in this paper that there is a fine correlation of the third order
between the values of the functions $Z[\varphi_{1}(t)]$ and $\tilde{Z}^{2}(t)$
which corresponds to two collections of disconnected sets. The corresponding
new asymptotic formula cannot be obtained within known theories of
Balasubramanian, Heath-Brown and Ivic.

###### Key words and phrases:

Riemann zeta-function

##  1\. Result

### 1.1.

Let (see [2], (3), (4))

(1.1) $\begin{split}G_{1}(x)&=G_{1}(x;T,H)=\bigcup_{T\leq t_{2\nu}\leq
T+H}\left\\{t:\ t_{2\nu}(-x)<t<t_{2\nu}(x),\ 0<x\leq\frac{\pi}{2}\right\\}\\\
G_{2}(y)&=G_{2}(y;T,H)=\\\ &=\bigcup_{T\leq t_{2\nu+1}\leq T+H}\left\\{t:\
t_{2\nu+1}(-y)<t<t_{2\nu+1}(y),\ 0<y\leq\frac{\pi}{2}\right\\}\end{split}$
(1.2) $H=T^{1/6+2\epsilon},$

and the collection of the sequences $\\{t_{\nu}(\tau)\\},\ \tau\in[-\pi,\pi],\
\nu=1,2,\dots$ be defined by the equation (see [2], (1))

(1.3) $\vartheta[t_{\nu}(\tau)]=\pi\nu+\tau;\ t_{\nu}(0)=t_{\nu}.$

### 1.2.

In this paper we obtain some new properties of the signal

$Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right)$

generated by the Riemann zeta-function. Namely, let

(1.4) $G_{1}(x)=\varphi_{1}[\mathring{G}_{1}(x)],\
G_{2}(y)=\varphi_{1}[\mathring{G}_{2}(y)]$

where $y=\varphi_{1}(T),\ T\geq T_{0}[\varphi_{1}]$ is the Jacob’s ladder. The
following theorem holds true.

######  Theorem.

(1.5)
$\begin{split}\int_{\mathring{G}_{1}(x)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t&=\frac{2}{\pi}H\sin x+\mathcal{O}(T^{1/6+\epsilon}),\\\
\int_{\mathring{G}_{2}(y)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t&=-\frac{2}{\pi}H\sin y+\mathcal{O}(T^{1/6+\epsilon}),\end{split}$

where

(1.6) $t-\varphi_{1}(t)\sim(1-c)\pi(t),\qquad t\to\infty,$

and $c$ is the Euler’s constant and $\pi(t)$ is the prime-counting function.

Let us remind another representation of the signal $Z(t)$ given by the
Riemann-Siegel formula (in its local form)

$Z(t)=2\sum_{n<P_{0}}\frac{1}{\sqrt{n}}\cos\\{\vartheta(t)-t\ln
n\\}+\mathcal{O}(T^{-1/4}),\ t\in[T,T+H],\ P_{0}=\sqrt{\frac{T}{2\pi}},$

i.e. in the form of the resultant oscillation of the system of nonlinear
oscillators

$\frac{2}{\sqrt{n}}\cos\\{\vartheta(t)-t\ln n\\};\
\vartheta(t)=\frac{t}{2}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}+\mathcal{O}\left(\frac{1}{t}\right)$

(at the same time see, for example, the expression $Z[t_{\nu}(\tau)]$ by
(1.3)).

### 1.3.

Let (comp. (1.4))

(1.7) $T=\varphi_{1}(\mathring{T}),\ T+H=\varphi_{1}(\widering{T+H}).$

Since (see (1.6), (1.7))

$\mathring{T}-\varphi_{1}(\mathring{T})\sim(1-c)\frac{\mathring{T}}{\ln\mathring{T}}\
\Rightarrow\ \mathring{T}-T\sim(1-c)\frac{\mathring{T}}{\ln\mathring{T}}\
\Rightarrow\ \mathring{T}\sim T$

we have seen (see (1.2))

$\mathring{T}-(T+H)\sim(1-c)\frac{\mathring{T}}{\ln\mathring{T}}-H\sim(1-c)\frac{\mathring{T}}{\ln\mathring{T}},$

i.e. $\mathring{T}>T+H$. Then we have

(1.8) $\begin{split}&[T,T+H]\cap[\mathring{T},\widering{T+H}]=\emptyset;\
T+H<\mathring{T},\\\
&\rho\\{[T,T+H];[\mathring{T},\widering{T+H}]\\}\sim(1-c)\pi(T),\end{split}$

where $\rho$ stands for the distance of the corresponding segments (comp. [5],
(1.3), (1.6)).

######  Remark 1.

Some nonlocal interaction of the functions $Z[\varphi_{1}(t)]$ and
$\tilde{Z}^{2}(t)$ is expressed by the formula (1.5) where (see (1.7))

$t\in\mathring{G}_{1}(x)\cup\mathring{G}_{2}(y)\cap[\mathring{T},\widering{T+H}]\
\Rightarrow\ \varphi_{1}(t)\in G_{1}(x)\cup G_{2}(y)\cap[T,T+H].$

Such an interaction is connected with two collections of disconnected sets
unboundedly receding each from other (see (1.8), $\rho\to\infty$ as
$T\to\infty$) - like mutually receding galaxies (the Hubble law). Compare this
remark with the Remark 3 in [5].

######  Remark 2.

The asymptotic formulae (1.5) cannot be obtained by methods of
Balasubramanian, Heath-Brown and Ivic (comp. [1]).

This paper is a continuation of the series [3]-[15].

##  2\. The first corollaries

First of all, we obtain from (1.5)

######  Corollary 1.

(2.1)
$\int_{\mathring{G}_{1}(x)\cup\mathring{G}_{2}(y)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\left\\{\begin{array}[]{lcr}\frac{2}{\pi}(\sin x-\sin
y)H+\mathcal{O}(T^{1/6+\epsilon})&,&x\not=y\\\
\mathcal{O}(T^{1/6+\epsilon})&,&x=y,\end{array}\right.$
$\displaystyle\int_{\mathring{G}_{1}(x)}Z[\varphi_{2}(t)]\tilde{Z}^{2}(t){\rm
d}t-\int_{\mathring{G}_{2}(y)}Z[\varphi_{2}(t)]\tilde{Z}^{2}(t){\rm d}t=$
$\displaystyle\frac{2}{\pi}(\sin x+\sin y)H+\mathcal{O}(T^{1/6+\epsilon}).$

Next, in the case $x=y=\frac{\pi}{2}$ we obtain

######  Corollary 2.

$\begin{split}&\int_{\mathring{G}_{1}\left(\frac{\pi}{2}\right)\cup\mathring{G}_{2}\left(\frac{\pi}{2}\right)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\mathcal{O}(T^{1/6+\epsilon}),\\\
&\int_{\mathring{G}_{1}\left(\frac{\pi}{2}\right)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t-\int_{\mathring{G}_{2}\left(\frac{\pi}{2}\right)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\\\ &\frac{4}{\pi}H+\mathcal{O}(T^{1/6+\epsilon}),\end{split}$

where
$[\mathring{T},\widering{T+H}]\subset\mathring{G}_{1}\left(\frac{\pi}{2}\right)\cup\mathring{G}_{2}\left(\frac{\pi}{2}\right)$.

##  3\. Law of the asymptotic equality of the areas of the positive and
negative parts of the graph of the function
$Z[\varphi_{1}(t)]\tilde{Z}^{2}(t)$

Let

(3.1) $\begin{split}\mathring{G}_{1}^{+}(x)&=\left\\{t:\
t\in\mathring{G}_{1}(x),\ Z[\varphi_{1}(t)]\tilde{Z}^{2}(t)>0\right\\},\\\
\mathring{G}_{1}^{-}(x)&=\left\\{t:\ t\in\mathring{G}_{1}(x),\
Z[\varphi_{1}(t)]\tilde{Z}^{2}(t)<0\right\\},\\\
\mathring{G}_{2}^{+}(x)&=\left\\{t:\ t\in\mathring{G}_{2}(x),\
Z[\varphi_{1}(t)]\tilde{Z}^{2}(t)>0\right\\},\\\
\mathring{G}_{2}^{-}(x)&=\left\\{t:\ t\in\mathring{G}_{2}(x),\
Z[\varphi_{1}(t)]\tilde{Z}^{2}(t)<0\right\\}.\end{split}$

Then we obtain from (2.1) by (1.1), (3.1) the following

######  Corollary 3.

(3.2)
$\begin{split}&\int_{\mathring{G}_{1}^{+}(x)\cup\mathring{G}_{2}^{+}(x)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t\sim\\\
&-\int_{\mathring{G}_{1}^{-}(x)\cup\mathring{G}_{2}^{-}(x)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t,\quad T\to\infty.\end{split}$

Indeed, from (1.5) by (3.1) we have

$0<(1-\epsilon)\frac{2}{\pi}H\sin
x<\int_{\mathring{G}_{1}(x)}\leq\int_{\mathring{G}_{1}^{+}(x)}\leq\int_{\mathring{G}_{1}^{+}(x)\cup\mathring{G}_{2}^{+}(x)},$

and similarly

$0<(1-\epsilon)\frac{2}{\pi}H\sin
x<-\int_{\mathring{G}_{1}^{-}(x)\cup\mathring{G}_{2}^{-}(x)}.$

Hence, from (2.1), $x=y$, we get

$\int_{\mathring{G}_{1}^{+}(x)}+\int_{\mathring{G}_{1}^{-}(x)}+\int_{\mathring{G}_{2}^{+}(x)}+\int_{\mathring{G}_{2}^{-}(x)}=\mathcal{O}(T^{1/6+\epsilon})=o(H)$

(see (1.2), i.e. (3.2)).

######  Remark 3.

The formula (3.2) represents the law of the asymptotic equality of the areas
(measures) of the figures which correspond to the positive part and negative
part, respectively, of the graph of the function

(3.3) $Z[\varphi_{1}(t)]\tilde{Z}^{2}(t),\
t\in\mathring{G}_{1}(x)\cup\mathring{G}_{2}(x)$

with respect to the disconnected sets
$\mathring{G}_{1}^{+}(x)\cup\mathring{G}_{2}^{+}(x),\
\mathring{G}_{1}^{-}(x)\cup\mathring{G}_{2}^{-}(x)$. This is one of the laws
governing _chaotic_ behavior of the positive and negative values of the
function (3.3).

##  4\. Proof of the Theorem

### 4.1.

Let is remind that

$\tilde{Z}^{2}(t)=\frac{{\rm d}\varphi_{1}(t)}{{\rm d}t},\
\varphi_{1}(t)=\frac{1}{2}\varphi(t)$

where

$\tilde{Z}^{2}(t)=\frac{Z^{2}(t)}{2\Phi^{\prime}_{\varphi}[\varphi(t)]}=\frac{Z^{2}(t)}{\left\\{1+\mathcal{O}\left(\frac{\ln\ln
t}{\ln t}\right)\right\\}\ln t}$

(see [3], (3.9); [5], (1.3); [9], (1.1), (3.1), (3.2)). Thus, the following
lemma holds true (see [8], (2.5); [9], (3.3)).

######  Lemma.

For every integrable function (in the Lebesgue sense) $f(x),\
x\in[\varphi_{1}(T),\varphi_{1}(T+U)]$ the following is true

(4.1) $\int_{T}^{T+U}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{\varphi_{1}(T)}^{\varphi_{1}(T+U)}f(x){\rm d}x,\
U\in\left.\left(0,\frac{T}{\ln T}\right.\right],$

where $t-\varphi_{1}(t)\sim(1-c)\pi(t)$.

######  Remark 4.

The formula (4.1) remains true also in the case when the integral on the
right-hand side of eq. (4.1) is only relatively convergent improper integral
of the second kind (in the Riemann sense).

In the case (comp. (1.7)) $T=\varphi_{1}(\mathring{T}),\
T+U=\varphi_{1}(\widering{T+U})$ we obtain from (4.1)

(4.2)
$\int_{\mathring{T}}^{\widering{T+U}}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{T}^{T+U}f(x){\rm d}x.$

### 4.2.

First of all, we have from (4.2), for example,

(4.3)
$\int_{\mathring{t}_{2\nu}(-x)}^{\mathring{t}_{2\nu}(x)}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{t_{2\nu}(-x)}^{t_{2\nu}(x)}f(t){\rm d}t,$

(see (1.4). Next, in the case

$f(t)=Z[\varphi_{1}(t)]\tilde{Z}^{2}(t),\
t\in\mathring{G}_{1}(x)\cup\mathring{G}_{2}(y)$

we have the following $\tilde{Z}^{2}$-transformation

(4.4)
$\begin{split}&\int_{\mathring{G}_{1}(x)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{G_{1}(x)}Z(t){\rm d}t,\\\
&\int_{\mathring{G}_{2}(y)}Z[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{G_{2}(y)}Z(t){\rm d}t,\end{split}$

(see (1.1), (4.3)). Let us remind that we have proved the following mean-value
formulae (see [2])

(4.5) $\begin{split}&\int_{G_{1}(x)}Z(t){\rm d}t=\frac{2}{\pi}H\sin
x+\mathcal{O}(T^{1/6+\epsilon}),\\\ &\int_{G_{2}(y)}Z(t){\rm
d}t=-\frac{2}{\pi}H\sin y+\mathcal{O}(T^{1/6+\epsilon}).\end{split}$

Now, our formulae (1.5) follow from (4.4) by (4.5).

I would like to thank Michal Demetrian for helping me with the electronic
version of this work.

## References

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