# Thermo-magnetic Effects in an External Magnetic Field in the Logarithmic-
Quark Sigma Model

M. Abu-Shady Department of Applied Mathematics, Faculty of Science, Menoufia
University, Egypt

###### Abstract

The phenomenon of magnetic catalysis of chiral symmetry breaking in the
quantum chromodynamic theory in the framework of logarithmic quark sigma model
is studied. Thermodynamic properties are calculated in the mean-field
approximation such as the pressure, the entropy density, the energy density,
and measure interaction. The pressure, the entropy density, and the energy
density increase with increasing temperature and an external magnetic field.
The critical temperature increases with increasing an external magnetic field.
In addition, the chiral phase transition is crossover in the presence of an
external magnetic field with absent of baryonic chemical potential when
explicit symmetry breaking is included. A comparison is presented with the
original sigma model and other works. A conclusion indicates that the
logarithmic quark model enhances the magnetic catalysis phenomenon.

Keywords: Chiral Lagrangian density, Magnetic catalysis, Mean-field
approximation

Mean-field approximation, Quark sigma model, Magnetic Catalysis

###### pacs:

PACS number

††preprint:

LABEL:FirstPage1 LABEL:LastPage#1

##  I Introduction

The study of the influence of an external magnetic fields on the fundamental
properties of quantum chromodynamic (QCD) theory such as the confinement of
quark and gluons at low energy and asymptotic freedom at high energy is still
a matter of great theoretical and experimental $\left[1\right]$. The
transition from composite objects to colored quarks and gluons has several
characteristics of both theoretical and phenomenological relevance. One such
characteristic is the equation of state which is the fundamental relation
encoding the thermodynamic properties of the system $\left[1-4\right]$. So
for, most estimates have been carried out at vanishing chemical potential with
the aid of effective theories such as the linear sigma model (LSM)
$\left[5\right]$ and the Numbu-Jona-Lasinio (NJL) model $\left[6,7\right]$ in
the mean field approximation.

The linear sigma model exhibits many of global symmetries of QCD theory. The
model was originally introduced by Gell-Mann and Levy $\left[8\right]$ with
the purpose of describing pion-nucleon interactions. During the last years an
impressive amount of work has been done with this model. The idea is to
consider it as an effective low energy approach for QCD theory that the model
has some aspects of QCD theory such as the chiral symmetry. Thus, the model is
successfully to describe the most of hadron properties at low energy such as
in Refs. $\left[5,9,10\right]$. Some observables that are calculated in this
model are conflict with experimental data. So researchers interest to modify
this model to provide a good description of hadron properties such as in Refs.
$\left[11-15\right].$

In Ref. $\left[16\right],$ the authors have modified the linear sigma model by
including the logarithmic mesonic potential and study its effect on the phase
transition at finite temperature. In addition, the comparison with other
models is done. On the same hand, the logarithmic sigma model successfully
describes nucleon properties at finite temperature and chemical potential
$\left[17\right]$.

To continue the investigation that started in Ref. $\left[18\right]$. In this
paper, we investigate the effect of external magnetic field on the
thermodynamic properties in the framework of logarithmic quark sigma model at
finite temperature and chemical potential. So for no attempts have done to
investigate the thermodynamic properties in the framework of the logarithmic
sigma model.

This paper is organized as follows: The original sigma model is briefly
presented in Sec. 2. Next, the effective logarithmic mesonic potential in the
presence of external magnetic field is presented in Sec. 3. The results are
discussed and are compared with other works in Secs. 4 and 5, respectively.
Finally, the summary and conclusion are presented in Sec. 6.

##  II The chiral sigma model with original effective potential

The interactions of quarks via the exchange of $\sigma-$ and $\mathbf{\pi}$ \-
meson fields are given by the Lagrangian density $[5]$ as follows:

${\small
L}\left(r\right){\small=i}\overline{\Psi}{\small\gamma}^{\mu}{\small\partial}_{\mu}{\small\Psi+}\frac{1}{2}\left(\partial_{\mu}\sigma\partial^{\mu}\sigma+\partial_{\mu}\mathbf{\pi}.\partial^{\mu}\mathbf{\pi}\right){\small+g}\overline{\Psi}\left(\sigma+i\gamma_{5}\mathbf{\tau}.\mathbf{\pi}\right){\small\Psi-U}_{1}{\small(\sigma,\pi),}$
(1)

with

${\small
U}_{1}{\small(\sigma,\pi)=}\frac{\lambda^{2}}{4}\left(\sigma^{2}+\mathbf{\pi}^{2}-\nu^{2}\right)^{2}+{\small
m}_{\pi}^{2}{\small f}_{\pi}{\small\sigma.}$ (2)

where $U\left(\sigma,\mathbf{\pi}\right)$ is the meson-meson interaction
potential and $\Psi,\sigma$ and $\mathbf{\pi}$ are the quark, sigma, and pion
fields, respectively. In the mean-field approximation, the meson fields are
treated as time-independent classical fields. This means that we replace the
power and products of the meson fields by corresponding powers and the
products of their expectation values. The meson-meson interactions in Eq. (2)
lead to hidden chiral $SU(2)\times SU(2)$ symmetry with $\sigma\left(r\right)$
taking on a vacuum expectation value

$\ \ \ \ {\small\ \
}\left\langle\sigma\right\rangle{\small=-f}_{\pi}{\small,}$ (3)

where $f_{\pi}=93$ MeV is the pion decay constant. The final term in Eq. (2)
included to break the chiral symmetry explicitly. It leads to the partial
conservation of axial-vector isospin current (PCAC). The parameters
$\lambda^{2}$and $\nu^{2}$ can be expressed in terms of$\ f_{\pi}$, sigma and
pion masses as,

${\small\lambda}^{2}{\small=}\frac{m_{\sigma}^{2}-m_{\pi}^{2}}{2f_{\pi}^{2}}{\small,}$
(4)
${\small\nu}^{2}{\small=f}_{\pi}^{2}{\small-}\frac{m_{\pi}^{2}}{\lambda^{2}}{\small.}$
(5)

##  III The effective logarithmic potential in the presence of magnetic field

In this section, the logarithmic mesonic potential $U_{2}(\sigma,\pi)$ is
applied. In Eq. (6), the logarithmic potential is included with the external
magnetic field at finite temperature and baryonic chemical potential
$\left[2\right]$ as follows,

$U_{eff}(\sigma,\pi)=U_{2}(\sigma,\pi)+U_{Vaccum}+U_{Matter}+U_{Medium},$ (6)

where

$\ \ \ \ \ \
U_{2}\left(\sigma,\mathbf{\pi}\right)=-\lambda_{1}^{2}\left(\sigma^{2}+\mathbf{\pi}^{2}\right)+\lambda_{2}^{2}\left(\sigma^{2}+\mathbf{\pi}^{2}\right)^{2}\log\left(\frac{\sigma^{2}+\mathbf{\pi}^{2}}{f_{\pi}^{2}}\right)+m_{\pi}^{2}f_{\pi}\sigma,$
(7)

In Eq. 7, the logarithmic potential satisfies the chiral symmetry when
$m_{\pi}\longrightarrow 0$ as well as in the original potential in Eq. 2.
Spontaneous chiral symmetry breaking gives a nonzero vacuum expectation for
$\sigma$ and the explicit chiral symmetry breaking term in Eq. 7 gives the
pion its mass.

$\left\langle\sigma\right\rangle=-f_{\pi}.$ (8)

Where

$\ \ \ \ \ \lambda_{1}^{2}=\frac{m_{\sigma}^{2}-7m_{\pi}^{2}}{12},$ (9) $\ \ \
\ \ \lambda_{2}^{2}=\frac{m_{\sigma}^{2}-m_{\pi}^{2}}{12f_{\pi}^{2}}.$ (10)

For details, see Refs. $\left[17,18\right]$. To include the external magnetic
field in the present model, we follow Ref. $\left[2\right]$ by including the
pure fermionic vacuum contribution in the free potential energy. Since this
model is renormalizable the usual procedure is to regularize divergent
integrals using dimensional regularization and to subtract the ultra violet
divergences. This procedure gives the following result

${\small
U}_{Vac\operatorname{u}m}=\frac{N_{c}N_{f}~{}g^{4}}{(2\pi)^{2}}(\sigma^{2}+\pi^{2})^{2}(\frac{3}{2}-\ln(\frac{g^{2}(\sigma^{2}+\pi^{2})}{\Lambda^{2}})),$
(11)

where $N_{c}=3$ and $N_{f}=2$ are color and flavor degrees of freedom,
respectively and $\Lambda$ is mass scale,

$U_{Matter}=\frac{N_{c}}{2\pi^{2}}\sum_{f~{}=u}^{d}(\left|q_{f}\right|B)^{2}[\zeta^{(1,0)}(-1,x_{f})-\frac{1}{2}(x_{f}^{2}-x_{f})\ln
x_{f}+\frac{x_{f}^{2}}{4}]$ (12)

In Eq. 12, we have used
$x_{f}=\frac{g^{2}(\sigma^{2}+\pi^{2})}{(2\left|q_{f}\right|B)}$ and
$\zeta^{(1,0)}(-1,x_{f})=\frac{d~{}\zeta(z,x_{f})}{dz}\mid_{z~{}=-1}$ that
represents the Riemann-Hurwitz function, and also $\left|q_{f}\right|$ is the
absolute value of quark electric charge in the external magnetic field with
intense $B$.

$U_{Medium}=\frac{N_{c}}{\left(2\pi\right)^{2}}\sum_{f~{}=u}^{d}\sum_{k=0}^{\infty}\alpha_{k}(\left|q_{f}\right|B){\displaystyle\int\limits_{\infty}^{-\infty}}dp_{z}[T\ln\left[1+e^{-\frac{{}^{[E_{P,k}(B)+\mu]}}{T}}\right]+T\ln\left[1+e^{-\frac{{}^{[E_{P,k}(B)-\mu]}}{T}}\right]].$
(13)

where $E_{P,k}(B)=\sqrt{P_{z}^{2}+2k\left|q_{f}\right|B+M^{2}}$, $M$ is the
effective self-consistent quark mass and $\mu$ is baryonic chemical potential.

##  IV  Discussion of results

In this section, we study the effective potential of the logarithmic sigma
model. For this purpose, we numerically calculate the effective potential in
Eq. (6). The parameters of the present model are the coupling constant $g$ and
the sigma mass $m_{\sigma}$. The choice of free parameters of coupling
constant $g$ and sigma mass $m_{\sigma}$ based on Ref. $\left[2\right].$ The
parameters are usually chosen so that the chiral symmetry is spontaneously
broken in the vacuum and the expectation values of the meson fields. In this
work, we consider set of parameters, $m_{\pi}=138$ MeV, $m_{\sigma}=600$ MeV,
and $g=4.5$. We start by calculating the quantities such as the dimensionless
of pressure $\frac{P_{n}}{T^{4}}$ , the dimensionless energy density
$\frac{E_{n}}{T^{3}}$, and the dimensionless entropy density
$\frac{S}{T^{3}}$. These quantities can be readily obtained by the effective
potential that defines in Eq. $\left(6\right)$, gives the negative pressure
$U_{eff}(\sigma,\pi)=-P_{n}$, then the net quark number density is obtained
from $\rho=\frac{dP_{n}}{d\mu}$, and the entropy density from
$s=\frac{dP_{n}}{dT}$ while the energy density is $E_{n}=-P_{n}+TS+\mu\rho$
see Ref. $\left[2\right]$. These quantities are displayed in Figs. (1, 2, 3,
4, 5, and 6). First of all, one observes that the quantities are calculated at
zero and finite strong magnetic field in the two cases at vanishing of
chemical potential and non-vanishing of chemical potential.

Fig. 1. The dimensionless pressure is plotted as a function of temperature at
$\mu=0~{}$ and $\mu=0.1$ GeV at vanishing magnetic field

Fig. 2. The dimensionless pressure is plotted as a function of temperature for
at $\mu=0$ and $\mu=0.1$GeV at strong magnetic field eB = 0.4 GeV${}^{2}.$

In Fig. 1, the dimensionless pressure is plotted as a function of temperature
at vanishing of magnetic field $(B)$. At vanishing of chemical potential
($\mu=0)$, the pressure increases with increasing temperature. The behavior is
good qualitative agreement with recent lattice calculation in Ref.
$\left[1\right]$, in which the generalized integral method is used and the
pressure increases with increasing temperature. Also, the behavior of pressure
is in qualitative agreement the original sigma model and NJL model
$\left[2\right]$. On the same hand, the behavior of pressure is qualitative
agreement work of Tawfik et al. $\left[3\right],$ in which the hadron gas and
the Polyakov linear sigma models are applied in their calculations. At
$\mu\neq 0$, we note that the curve shifts to higher values, in particular, at
higher-values of temperature. At finite baryonic chemical potential is not
considered in many works such as in Ref. $\left[3\right]$. In Fig. 2, the
effect of strong magnetic field is studied on the behavior of pressure, then
we choose magnetic field $eB=0.2$ GeV2 as in Refs. $\left[1,2,3\right]$. We
note that qualitative agreement between Fig. 1 represents the case of
vanishing magnetic field and Fig. 2 represents the presence of an external
magnetic field. The effect of magnetic field appears on the pressure value
that the pressure increases strongly at higher-values of temperature. in
comparison with vanishing magnetic field as in Fig. 1. This behavior is
qualitative agreement with Refs. $\left[1,3\right]$.

Fig. 3. The dimensionless energy density is plotted as a function of
temperature for at $\mu=0$ and $\mu=0.2$GeV at vanishing magnetic field

Fig. 4. The dimensionless energy density is plotted as a function of
temperature at $\mu=0~{}$ and $\mu=0.1$ GeV at non-vanishing magnetic field eB
=0.4 GeV2

In Fig. 3, the dimensionless energy density is plotted as a function of
temperature. We note that the energy density increases with increasing
temperature at vanishing and nonvanishing chemical potential. At vanishing of
baryonic chemical potential ($\mu=0)$, the energy density increases with
increasing temperature up to 0.225 GeV and then slowly decreases at larger
values of temperature above critical temperature due to taking ratio
$\frac{E_{n}}{T^{4}}$. By increasing magnetic field as in Fig 4, the behavior
of the energy density is a qualitative agreement with the behavior at
vanishing magnetic field as in Fig. 3 and supporting with Refs.
$\left[1,3\right].$

Fig. 5. The dimensionless entropy density is plotted as a function of
temperature for at $\mu=0$ and $\mu=0.1$GeV at vanishing magnetic field

Fig. 6. The dimensionless entropy density is plotted as a function of
temperature for at $\mu=0$ and $\mu=0.1$ GeV at magnetic field eB = 0.4 GeV2

In Fig. 5, the dimensionless entropy density is plotted as a function of
temperature. The entropy density increases with increasing temperature at
vanishing and nonvanishing chemical potential. The effect of temperature
appears on entropy density at higher values of temperatures. By increasing
magnetic field as in Fig. 6, the behavior of the entropy density is a
qualitative agreement with the behavior at vanishing magnetic field as in Fig.
5\. Also, we note that entropy density decreases at higher temperature above
critical temperature $T_{c}=0.2$ GeV. This due to taking ratio
$\frac{S}{T^{3}}$

Fig. 7. The dimensionless interaction measure is plotted as a function of
temperature for different values of magnetic field eB = 0.2, 0.3, and 0.4
GeV2are arranged from down to up.

Fig. 8. Critical of temperature is plotted as a function of magnetic field at
vanishing of baryonic chemical potential.

In Fig. $\left(7\right)$, the measure interaction is plotted as a function of
temperature. The measure interaction increases with increasing temperature up
to the top curve and then gradually decreases with increasing temperature. By
increasing magnetic field, we note that the top shifts in direction of
increasing temperature. This behavior is an agreement with original sigma
model and NJL model $\left[2\right]$. This behavior is interpreted as
crossover phase transition. In Fig. (8), the critical temperature increases
with increasing magnetic field. Thus, the logarithmic quark enhances magnetic
catalysis as well as in the original sigma model and NJL model
$\left[2\right]$. By including Polyakov loop to take confinement into account.
The results show that $T_{c}$ related to deconfinment also increases with
increasing magnetic field $\left[26\right]$. On other hand, the first lattice
attempt to solve the problem taken two quark flavors and high values of pion
mass $(m_{\pi}=200-400$ MeV) confirming that $T_{c}$ increasing with
increasing magnetic field $\left[27\right]$.

##  V Summary and conclusion

In this work, we have employed the effective logarithmic potential to study
thermodynamic properties in the presence of an external magnetic field. So,
the novelty in this work, thermodynamic properties are investigated in the
framework of logarithmic quark sigma model. The present results are agreement
with first lattice calculations as in Ref. $\left[27\right]$ and original
sigma model and NJL model as in Refs. $\left[2\right]$. In addition, the
present calculations are carried out beyond the zero chemical potential which
are not taken many recent works. Our conclusion indicates that the logarithmic
quark sigma model enhances magnetic catalysis at finite temperature and
baryonic chemical potential.

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