e2e-mail<EMAIL_ADDRESS>
11institutetext:  Department of Physics, Norwegian University of Science and
Technology, Høgskoleringen 5, N-7491 Trondheim, Norway

# QCD phase diagram in a constant magnetic background

Inverse magnetic catalysis: where models meet the lattice

Jens O. Andersene2,addr2

(Received: date / Revised version: date)

###### Abstract

Magnetic catalysis is the enhancement of a condensate due to the presence of
an external magnetic field. Magnetic catalysis at $T=0$ is a robust phenomenon
in low-energy theories and models of QCD as well as in lattice simulations. We
review the underlying physics of magnetic catalysis from both perspectives.
The quark-meson model is used as a specific example of a model that exhibits
magnetic catalysis. Regularization and renormalization are discussed and we
pay particular attention to a consistent and correct determination of the
parameters of the Lagrangian using the on-shell renormalization scheme. A
straightforward application of the quark-meson model and the NJL model leads
to the prediction that the chiral transition temperature $T_{\chi}$ is
increasing as a function of the magnetic field $B$. This is in disagreement
with lattice results, which show that $T_{\chi}$ is a decreasing function of
$B$, independent of the pion mass. The behavior can be understood in terms of
the so-called valence and sea contributions to the quark condensate and the
competition between them. We critically examine these ideas as well recent
attempts to improve low-energy models using lattice input.

††journal: Eur. Phys. J.

##  1 Introduction

The phase diagram of QCD has received a lot of attention since the first ideas
appeared in the 1970s. At that time, it was thought that QCD has two phases, a
hadronic phase at low temperatures and a deconfined phase of quarks and gluons
at high temperatures. In 1984, Bailin and Love colorfirst  suggested that at
high density, quark matter should be a color superconductor. The ideas are
analogous to those of ordinary superconductivity and BCS theory bcs , namely
the instability of the Fermi surface to form Cooper pairs under an attractive
interaction. In QCD, an attractive interaction is provided by one-gluon
exchange in the triplet channel. Since then, there have been large efforts to
map out the phase diagram of QCD and study the properties of its different
phases raja ; alford ; fukurev . The phase diagram has shown to be
surprisingly rich at high baryon density and low temperatures. It includes a
quarkyonic phase rob1  as well as a number of superconducting phases, some of
them being inhomogeneous. Most of these results have been obtained using low-
energy models of QCD, notably the quark-meson (QM) model and the Nambu-Jona-
Lasinio (NJL) model, with or without coupling to the Polyakov loop. The reason
is that lattice simulations are notoriously difficult to perform at finite
baryon chemical potential $\mu_{B}$ due to the sign problem, so that one
cannot use techniques involving importance sampling.

The temperature $T$ and baryon chemical potential $\mu_{B}$ are not the only
relevant parameters of QCD. For example, one can introduce a separate chemical
potential $\mu_{f}$ for each quark flavor $f$. For two flavors, this leads to
another independent chemical potential besides $\mu_{B}$, namely the isospin
chemical potential $\mu_{I}$. For the three flavors, $\mu_{S}={1\over
2}(\mu_{u}+\mu_{d}-2\mu_{s})$ is added. The addition of these chemical
potentials gives rise to pion and kaon condensation. At $T=0$, pion
condensation occurs for $\mu_{I}>m_{\pi}$, while kaon condensation takes place
for $|\pm{1\over 2}\mu_{I}+\mu_{S}|>m_{K}$ (upper sign for charged kaons and
lower sign for neutral kaons). The former is particularly interesting since
finite $\mu_{I}$ and vanishing $\mu_{B}$ has no sign problem and is therefore
amenable to lattice simulations.

The final example of an external parameter, which is the topic of this review,
is a (constant) magnetic background. There are several areas of high-energy
physics, where such a background is relevant. One is non-central heavy-ion
collisions, where large, time-dependent fields are generated. These fields are
short-lived and have a maximum value of approximately $eB=6m_{\pi}^{2}$ kar1 .
The basis mechanism is simply that (in the center-of mass frame) the two
nuclei represent electric currents that according to Maxwell’s equations
generate a magnetic field. Another example where strong magnetic fields
appear, are magnetars duncan . This is a special class of neutron stars with
relatively low rotation frequencies. It is believed that the magnetic fields
on the surface are $10^{14}-10^{15}$ Gauss, while in the interior they can be
as strong as $10^{16}-10^{19}$ Gauss.

We consider QCD with an $SU(3)$ gauge group, a global $SU(N_{f})$ vector
symmetry and quark masses $m_{f}$. The QCD Lagrangian is

$\displaystyle{\cal L}_{\rm QCD}$ $\displaystyle=$ $\displaystyle-{1\over
4}F_{\mu\nu}^{a}F_{a}^{\mu\nu}+i\bar{\psi}_{f}\gamma^{\mu}D_{\mu}\psi_{f}-m_{f}\bar{\psi}_{f}\psi_{f}$
(1) $\displaystyle+{\cal L}_{\rm gf}+{\cal L}_{\rm ghost}\;,$

where the gluon field strength tensor is
$F^{a}_{\mu\nu}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A^{a}_{\mu}-gf^{abc}A_{\mu}^{b}A_{\nu}^{c}$,
$f^{abc}$ are the structure constants the covariant derivative in the presence
of an abelian background field $A_{\mu}^{\rm EM}$ is

$\displaystyle D_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}+iq_{f}A_{\mu}^{\rm EM}+igA_{\mu}\;.$ (2)

Moreover, $m_{f}$ is the mass of a quark of flavor $f$ and there is a sum of
flavors in Eq. (1). The nonabelian gauge field is $A_{\mu}=t_{a}A_{\mu}^{a}$,
$t_{a}={1\over 2}\lambda_{a}$, and $\lambda_{a}$ are the Gell-Mann matrices.
Finally ${\cal L}_{\rm gf}$ and ${\cal L}_{\rm ghost}$ are the gauge-fixing
and ghost part of the Lagrangian, respectively.

The partition function in QCD can be written as

$\displaystyle{\cal Z}$ $\displaystyle=$ $\displaystyle\int{\cal
D}A_{\mu}{\cal D}\bar{\psi}_{f}{\cal D}\psi_{f}\,e^{-S_{\rm QCD}}$ (3)
$\displaystyle=$ $\displaystyle\int{\cal
D}A_{\mu}e^{-S_{g}}\det(/\\!\\!\\!\\!\\!\\!D(B)+m_{f})\;,$

where $S_{\rm QCD}$ is the Euclidean action for QCD. In the second line, we
have integrated over the fermions which can be done exactly since ${\cal
L}_{\rm QCD}$ is bilinear in the quark fields. Moreover, $S_{g}$ is the
Euclidean action for the gluons and

$\displaystyle/\\!\\!\\!\\!\\!\\!D(B)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&iX\\\ iX^{\dagger}&0\end{pmatrix}\;,$ (4)
$\displaystyle iX$ $\displaystyle=$ $\displaystyle
D_{0}+i{\bm{\sigma}}\cdot{\bf D}\;.$ (5)

This yields

$\displaystyle\det\left(/\\!\\!\\!\\!\\!\\!D(B)+m_{f}\right)$ $\displaystyle=$
$\displaystyle\det\left[X^{\dagger}X+m_{f}^{2}\right]\;.$ (6)

The last equation shows that the fermion determinant is manifestly positive.
As in the case of finite isospin chemical potential, QCD in a magnetic field
is also free of the sign problem, and one can therefore carry out lattice
simulations. Interestingly, the combination of finite isospin and magnetic
field is free of the sign problem only if the charges of the $u$ and $d$-quark
are the same. This is of course not real QCD, but it offers the possibility to
compare lattice predictions with those of low-energy effective theories and
models.

In this review, we will discuss (inverse) magnetic catalysis and the phase
diagram of QCD in a strong magnetic background, paying attention to recent
developments. There are other reviews review ; dima ; ourrev ; fariasrev
focusing on different aspects of the field. The paper is organized as follows.
In the next section, we discuss the physics of magnetic catalysis at $T=0$. In
Sec. 3, we introduce the Polyakov loop and discuss magnetic catalysis in model
calculations at nonzero temperature. In Sec. 4, we review inverse magnetic
catalysis on the lattice focusing on the competing sea and valence effects. In
Sec. 5, the improvement of models to incorporate inverse magnetic catalysis is
discussed and in Sec. 6, we summarize. In A, we discuss renormalization of the
quark-meson model in the on-shell scheme, while in B, we show how the
parameters of the model are fixed.

##  2 Magnetic catalysis at zero temperature

Magnetic catalysis can be defined as

  1. 1.

The magnitude of a condensate is enhanced by the presence of an external
magnetic field $B$ if the condensate already is present at vanishing field.

  2. 2.

An external magnetic field induces symmetry breaking and a nonzero value of a
condensate when the symmetry is intact for $B=0$.

The condensate is the expectation value of a field, which can be either
fundamental or composite. The expectation of a scalar field $\phi$ in low-
energy models is an example of the former, while $\bar{\psi}\psi$ is the
chiral condensate in e.g. the NJL model or QCD is an example of the latter.
One refers to the second case as dynamical symmetry breaking by a magnetic
field. We will discuss both cases below. The first papers on magnetic
catalysis at $T=0$ appeared three decades ago in the study of the NJL model in
three dimensions lemmer . Shortly thereafter in the linear sigma model sugu
and the NJL model in two dimensions klim1 ; klim2 ; klim3 . Since then it has
been demonstrated in QED qedgus , chiral perturbation theory smilga ; werbos ;
werbos2 , in the Walecka model in nuclear physics andreas , and also on the
lattice, see e.g. cherno ; chern2 ; braguta ; delia ; quarkcon .

In this section, we will use the two-flavor quark-meson model as an explicit
example of a low-energy effective model of QCD that displays magnetic
catalysis. The Lagrangian is

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle-{1\over
2}B^{2}+{1\over
2}\left[(\partial_{\mu}\sigma)(\partial^{\mu}\sigma)+(\partial_{\mu}\pi_{0})(\partial^{\mu}\pi_{0})\right]$
(7) $\displaystyle+D_{\mu}^{*}\pi^{-}D^{\mu}\pi^{+}-{1\over
2}m^{2}(\sigma^{2}+\pi_{0}^{2}+2\pi^{+}\pi^{-})$ $\displaystyle-{\lambda\over
24}(\sigma^{2}+{\pi}_{0}^{2}+2\pi^{+}\pi^{-})^{2}+h\sigma$
$\displaystyle+\bar{\psi}\left[i\gamma^{\mu}D_{\mu}-g(\sigma+i\gamma^{5}{\bf\tau}\cdot{\bm{\pi}})\right]\psi\;,$

where $D_{\mu}=\partial_{\mu}+iqA_{\mu}$ is the covariant derivative $\sigma$,
${\bm{\pi}}=(\pi_{0},\pi_{1},\pi_{2})$ are the meson fields,
$\pi^{\pm}={1\over\sqrt{2}}(\pi_{1}\mp i\pi_{2})$ , $\tau_{a}$ are the Pauli
matrices, $\psi$ is a color $N_{c}$-plet, a four-component Dirac spinor as
well as a flavor doublet

$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}u\\\ d\end{array}\right)\;.$ (10)

In the absence of an abelian gauge field in Eq. (7), the symmetry is
$SU(2)_{L}\times SU(2)_{R}$ for $h=0$, otherwise it is $SU(2)_{V}$. In its
presence, the Lagrangian Eq. (7) has a $U(1)_{L}\times U(1)_{R}$ symmetry for
$h=0$, otherwise it is $U(1)_{V}$. The reason is that one cannot transform a
$u$-quark into a $d$-quark due to their different electric charges. Defining
$\Delta^{\pm}={1\over\sqrt{2}}(\sigma\pm i\gamma^{5}\pi_{0})$, the two sets of
transformations are 1) $u\rightarrow e^{-i\gamma^{5}\alpha}u$, $d\rightarrow
e^{i\gamma^{5}\alpha}d$, $\Delta^{\pm}\rightarrow\Delta^{\pm}e^{\pm
2i\gamma^{5}\alpha}$, and $\pi^{\pm}\rightarrow\pi^{\pm}$, and 2)
$u\rightarrow e^{i\alpha}u$, $d\rightarrow e^{-i\alpha}d$,
$\Delta^{\pm}\rightarrow\Delta^{\pm}$, and
$\pi^{\pm}\rightarrow\pi^{\pm}e^{\pm 2i\alpha}$.

After symmetry breaking, the sigma field has a nonzero expectation value
$\phi_{0}$. The classical potential is

$\displaystyle V_{0}$ $\displaystyle=$ $\displaystyle{1\over 2}B^{2}+{1\over
2}m^{2}\phi_{0}^{2}+{\lambda\over 24}\phi_{0}^{4}-h\phi_{0}\;.$ (11)

The tree-level relations between the parameters of the Lagrangian $m^{2}$,
$\lambda$, $g$, and $h$ and the physical masses $m_{\sigma}$ and $m_{\pi}$,
the pion decay constant $f_{\pi}$, and the quark mass $m_{q}$ are

$\displaystyle m^{2}$ $\displaystyle=-{1\over
2}\left(m_{\sigma}^{2}-3m_{\pi}^{2}\right)\;,$ $\displaystyle\lambda$
$\displaystyle=3{(m_{\sigma}^{2}-m_{\pi}^{2})\over f_{\pi}^{2}}\;,$ (12)
$\displaystyle g^{2}$ $\displaystyle={m_{q}^{2}\over f_{\pi}^{2}}\;,$
$\displaystyle h$ $\displaystyle=m_{\pi}^{2}f_{\pi}\;.$ (13)

Using the relations (12)–(13), we obtain

$\displaystyle V_{0}$ $\displaystyle=$ $\displaystyle{1\over
2}B^{2}+\frac{3}{4}m_{\pi}^{2}f_{\pi}^{2}\frac{\Delta^{2}}{m_{q}^{2}}-\frac{1}{4}m_{\sigma}^{2}f_{\pi}^{2}\frac{\Delta^{2}}{m_{q}^{2}}+\frac{1}{8}m_{\sigma}^{2}f_{\pi}^{2}\frac{\Delta^{4}}{m_{q}^{4}}$
(14)
$\displaystyle-\frac{1}{8}m_{\pi}^{2}f_{\pi}^{2}\frac{\Delta^{4}}{m_{q}^{4}}-m_{\pi}^{2}f_{\pi}^{2}\frac{\Delta}{m_{q}}\;,$

where we have introduced $\Delta=g\phi_{0}$. The minimum of the classical
potential is given by $\Delta=gf_{\pi}$.

The classical potential has by construction its minimum at $\Delta=m_{q}$ or
$\phi_{0}=f_{\pi}$. In the large-$N_{c}$ limit, the mesons are included at
tree level, while we include the Gaussian fluctuations of the fermions.
Including the one-loop corrections from the fermions using a minimal
subtraction scheme, leaves a renormalized one-loop effective potential that
depends on the renormalization scale $\Lambda$. The minimum of the effective
potential therefore depends on $\Lambda$. In order to ensure that the one-loop
effective potential has its minimum at $\phi_{0}=f_{\pi}$ for zero magnetic
field $B$, several methods have been used in the literature. One method is
simply to subtract the one-loop contribution to the effective potential for
$B=0$. Then the renormalization scale dependence drops out and the correction
to Eq. (14) is a finite $B$-dependent term that vanishes for $B=0$. However
this is inconsistent since one includes fermion fluctuations in the effective
potential at finite magnetic field, but not for $B=0$. Moreover, it is also
incorrect since Eqs. (12)–(13) are tree-level relations that receive radiative
corrections. One can also choose a specific value for $\Lambda$ such that the
one-loop correction to the position of the minimum of the effective potential
vanishes. In this case, one has included quantum fluctuations also for $B=0$,
but again, the tree-level relations between the parameters of the Lagrangian
and physical quantities receive loop corrections. In order to be consistent,
the parameters of the Lagrangian must be determined to the same order in the
loop expansion as one calculates the effective potential. The solution to the
problem is to combine the minimal subtraction scheme with the on-shell scheme
sirlin ; sirlin2 ; bohm ; hollik . In this way one includes loop corrections
to Eqs. (12)–(13), while at the same time ensures that the effective potential
has its minimum at $\Delta=gf_{\pi}$. Details of the renormalization of the
one-loop effective potential in the large-$N_{c}$ limit can be found in A and
the parameter fixing in B. It reads

The first four lines of the one-loop effective potential are independent of
the magnetic field and this part was first calculated in Ref. allofus . The
last line is the $B$-dependent correction to $V_{\rm 0+1}$. Note also that
final result is independent of the renormalization scale $\Lambda$.

In Fig. 1, we show the effective potential divided by $f_{\pi}^{4}$ at $T=0$.
The black line is the tree-level potential Eq. (14), while the green and blue
lines are the one-loop effective potential Eq. (LABEL:fullb) for $|eB|=0$ and
$|eB|=10m_{\pi}^{2}$, respectively. We have used $m_{\sigma}=600$ MeV,
$m_{\pi}=140$ MeV, $f_{\pi}=93$ MeV, and $m_{q}=300$ MeV. The classical
potential as well the one-loop effective potential with $|eB|=0$ both have a
minimum at $\Delta=gf_{\pi}$ by construction. Notice, however, that the latter
is significantly deeper. The blue line with $|eB|=10m_{\pi}^{2}$, shows that
the minimum of the effective moves to a larger value, i. e. the system
exhibits magnetic catalysis.

Figure 1: Effective potential as a function of $\Delta$ normalized by
$f_{\pi}^{4}$ at $T=0$. The black line is the tree-level result, the green and
blue lines are the one-loop result for zero magnetic field and for
$eB=10m_{\pi}^{2}$. See main text for details.

While the above clearly demonstrates magnetic catalysis numerically, we would
like to gain insight in the mechanism behind the effect. Instead of analyzing
Eq. (LABEL:fullb), we will discuss the gap equation in the NJL model.

In order to simplify the discussion, we will consider the NJL model with a
single quark flavor and color, $N_{f}=N_{c}=1$ with electric charge $q_{f}$.
In the chiral limit, the Lagrangian is reg1

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\bar{\psi}i\gamma^{\mu}\partial_{\mu}\psi+{G\over
2}\left[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma^{5}\psi)^{2}\right]\;.$ (16)

This Lagrangian has a $U(1)_{V}\times U(1)_{A}$ symmetry. We introduce the gap
$M=-G\langle\bar{\psi}\psi\rangle$ and linearize the interaction terms,
writing
$(\bar{\psi}\psi)^{2}\approx\langle\bar{\psi}\psi\rangle^{2}+2\langle\bar{\psi}\psi\rangle\bar{\psi}\psi$
and $(\bar{\psi}i\gamma^{5}\psi)^{2}\approx 0$. $M$ is now an effective quark
mass arising after breaking the axial symmetry spontaneously, i.e. when
$\langle\bar{\psi}\psi\rangle\neq 0$. In the mean-field approximation, we
perform the Gaussian integral over the fermion field giving rise to the
following one-loop effective potential,

$\displaystyle V_{0+1}$ $\displaystyle=$ $\displaystyle{M^{2}\over
2G}-2\int{d^{4}p\over(2\pi)^{4}}\log\left[p^{2}+M^{2}\right]\;.$ (17)

The gap equation for $M$ is found by extremizing $V_{0+1}$ which yields

$\displaystyle{M\over 4G}$ $\displaystyle=$ $\displaystyle
M\int{d^{4}p\over(2\pi)^{4}}{1\over p^{2}+M^{2}}\;.$ (18)

Conventionally, since the NJL model is non-renormalizable, one has used a
three-dimensional or a four-dimensional momentum cutoff $\Lambda$ to regulate
divergences. If $\Lambda$ is a four-dimensional cutoff, the gap equation (18)
reads for $M\ll\Lambda$

$\displaystyle M\left[{4\pi^{2}\over G}-\Lambda^{2}+M^{2}\log{\Lambda^{2}\over
M^{2}}\right]$ $\displaystyle=$ $\displaystyle 0\;.$ (19)

$M=0$ is always a solution, however for $G>G_{c}={4\pi^{2}\over\Lambda^{2}}$
there is also a nontrivial solution. Thus for $G$ larger than the critical
value ${4\pi^{2}\over\Lambda^{2}}$, quantum fluctuations induce symmetry
breaking in the model.

At finite magnetic field, the partial derivative in Eq. (16) is replaced by
the covariant derivative and we add a term ${1\over 2}B^{2}$ to the effective
potential. The gap equation becomes

$\displaystyle{M\over 2G}$ $\displaystyle=$ $\displaystyle M{|q_{f}B|\over
2\pi}\sum_{s=\pm
1}\sum_{k=0}^{\infty}\int{d^{2}p_{\parallel}\over(2\pi)^{2}}{1\over
p_{0}^{2}+p_{z}^{2}+M_{B}^{2}}\;,$ (20)

where $M_{B}^{2}=M^{2}+|q_{f}B|(2k+1-s)$,
$p_{\parallel}^{2}=p_{0}^{2}+p_{z}^{2}$ and $q_{f}$ is the charge. The
divergences in Eq. (20) can be isolated by adding and subtracting the right-
hand side of Eq. (18). The right-hand side of Eq. (20) minus the subtracted
term is finite and is conveniently evaluated using dimensional regularization
in the same way as done appendix A menes ; farias . Finally, we impose a four-
dimensional cutoff on the added term as in Eq. (18). Factoring out the trivial
solution $M=0$, this yields the regularized gap equation

$\displaystyle{4\pi^{2}\over G}-\Lambda^{2}+M^{2}\log{\Lambda^{2}\over
M^{2}}-{2|q_{f}B|}\left[\zeta^{(1,0)}(0,x_{f})+x_{f}\right.$
$\displaystyle\left.-{1\over 2}(2x_{f}-1)\log{x_{f}}\right]$ $\displaystyle=$
$\displaystyle 0\;,$ (21)

where $x_{f}={M^{2}\over 2|q_{f}B|}$. This equation has only a nonzero $M$ as
solution. For $G<G_{c}$, the solution is reg1 ; gus2

$\displaystyle M^{2}$ $\displaystyle=$
$\displaystyle{|q_{f}B|\over\pi}\exp\left[{-{1\over|q_{f}B|}}\left({4\pi^{2}\over
G}-\Lambda^{2}\right)\right]\;.$ (22)

In the limit $|q_{f}B|\rightarrow 0$, this solution connects to the trivial
solution $M=0$. In the lowest Landau level approximation, the gap equation has
solution $M^{2}=\Lambda^{2}e^{-4\pi^{2}/G|q_{f}B|}$, which is reminiscent of
Eq. (22) if we identify the cutoff $\Lambda$ with $\sqrt{|q_{f}B|}$. We can
then think of magnetic catalysis as a 1+1 dimensional phenomenon, i.e. a
dimensional reduction from 3+1 dimensions has taken place. The functional form
of the gap equation is the same as for the gap equation in BCS theory of
superconductivity as well as the gap equation found in the large-$N$ limit of
$O(N)$-symmetric nonlinear sigma model in 1+1 dimensions. The 1+1 dimensional
nature of magnetic catalysis raises the question of whether this phenomenon is
in conflict with the Coleman theorem, which forbids spontaneous symmetry
breaking in less than two spatial dimensions at zero temperature coleman . As
pointed out in Ref. gus2 , the field $\bar{\psi}\psi$ is neutral with respect
to the magnetic field. The neutral pion is the associated Goldstone boson that
appears after breaking the $U(1)$ symmetry. The charged pions are now massive
even in the chiral limit.

There are other ways of regularizing the gap equation (20) or the fermion
contribution to the one-loop effective potential (LABEL:deter), for example
Schwinger’s proper time method swing . Let us illustrate this by computing the
corresponding bosonic functional determinant, which shows up in chiral
perturbation theory. It is based on the representation in Euclidean space

$\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle\log\det(-D_{\mu}D^{\mu}+m^{2})$ (23) $\displaystyle=$
$\displaystyle-\int_{0}^{\infty}{ds\over s}{\rm
Tr}\,e^{-s(-D_{\mu}D^{\mu}+m^{2})}$ $\displaystyle=$ $\displaystyle-{|qB|\over
2\pi}\sum_{k=0}^{\infty}\int_{p_{\parallel}}\int_{0}^{\infty}{ds\over
s}e^{-s(p_{\parallel}^{2}+|qB|(2k+1)+m^{2})}\;,$

where the sum over Landau levels $k$ as well the momentum integral over
$p_{\parallel}$ is convergent. The result is

$\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle-{(e^{\gamma_{E}}\Lambda^{2})^{\epsilon}\over(4\pi)^{2}}\int_{0}^{\infty}{ds\over
s^{3-\epsilon}}e^{-m^{2}s}{|qB|s\over\sinh(|qB|s)}\;.$ (24)

For $\epsilon=0$, the integral is divergent for small $s$, i.e. for large
momentum. By adding and subtracting the divergent terms, we can isolate the
divergences. One finds

$\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle-{(e^{\gamma_{E}}\Lambda^{2})^{\epsilon}\over(4\pi)^{2}}\int_{0}^{\infty}{ds\over
s^{3-\epsilon}}e^{-m^{2}s}+{(qB)^{2}\over
6(4\pi)^{2}}\int_{0}^{\infty}{ds\over s^{1-\epsilon}}e^{-m^{2}s}$
$\displaystyle-{(e^{\gamma_{E}}\Lambda^{2})^{\epsilon}\over(4\pi)^{2}}\int_{0}^{\infty}{ds\over
s^{3-\epsilon}}e^{-m^{2}s}\left[{|qB|s\over\sinh(|qB|s)}-1+{(qBs)^{2}\over
6}\right]\;.$

The integrals in the first line are divergent for $\epsilon=0$, while the last
integral is convergent. The divergences show up as poles in $\epsilon$. The
first term in Eq. (LABEL:swing) is a vacuum energy counterterm while the
second term corresponds to charge and wavefunction renormalization swing . The
last integral in Eq. (LABEL:swing) can be calculating exactly and involves the
Hurwitz zeta function. Using the proper time method with the momentum
integrals evaluated in $d=2-2\epsilon$ dimensions yields the same results as
those obtained by combining dimensional regularization and zeta-function
regularization, as done in A. Alternatively, one can evaluate Eq.
(LABEL:swing) with $\epsilon=0$ using a cutoff $1/\Lambda^{2}$ as the lower
limit of the $s$-integration in the divergent integrals.

The regularization methods discussed so far separates in clean way the
$B$-independent divergences from the $B$-dependent terms whether they are
finite or divergent. 111They are finite in the gap equation Eq. (20), but
divergent in the effective potential, cf. Eq. (LABEL:swing). There are other
regularization methods that do not separate this contributions, for example a
sharp cutoff imposed directly on the integral in Eq. (24) or a form factor
that is a function of e.g. $p_{z}^{2}+2k|qB|(2k+1-s)$. One has to be careful
choosing such regulators since nonphysical oscillations may result farreg0 ;
farreg .

The on-shell scheme used to obtain the final result Eq. (LABEL:fullb) has two
important virtues as first pointed out in Ref. hrg . By considering the
small-$B$ (large-$x_{f}$) behavior of the Hurwitz zeta-function, one finds
that the only contributions at order $B^{2}$ comes from the renormalized
magnetic field term ${1\over 2}B^{2}$,

$\displaystyle{\zeta^{(1,0)}(-1,x_{f})}$ $\displaystyle=$
$\displaystyle-{1\over 4}x_{f}^{2}+{1\over 2}x_{f}^{2}\log x_{f}-{1\over
2}x_{f}\log x_{f}$ (26) $\displaystyle+{1\over 12}\log x_{f}+{1\over
12}+{1\over 720}{1\over x_{f}^{2}}+...\;.$

It also ensures that the magnetic-field contribution to the effective
potential and the magnetization vanishes in the limit
$m_{q}\rightarrow\infty$. Both properties are expected from a physical point
of view. It leads to a paramagnetic vacuum, in agreement with the hadron
resonance gas model calculations hrg  and lattice QCD lattice simulations para
. Other renormalization schemes, such as the (modified) minimal subtraction
scheme are connected to the above by a finite renormalization. However, the
effective potential and the magnetization then grow logarithmically in the
limit $m_{q}\rightarrow\infty$. Similar remarks apply to the (P)NJL model.
Regularizing the model by imposing a UV cutoff $\Lambda$ on the divergent
integrals in the fermionic version of Eq. (LABEL:swing), the authors of sidney
show that it predicts diamagnetic behavior for low values of $B$ and
paramagnetic behavior for large magnetic fields. By defining a subtraction
procedure that resembles the renormalization of the magnetic field in the on-
shell scheme, their effective potential leads to paramagnetic behavior as seen
on the lattice.

After having discussed magnetic catalysis in models, we now turn to lattice
gauge theory. The first lattice simulations were carried out for an $SU(2)$
gauge group for magnetic field strengths up to $\sqrt{eB}\sim 3$ GeV in the
quenched approximation, i.e. setting the quark determinant to unity cherno .
The simulations confirmed that the quark condensates is enhanced by the
magnetic field and that the enhancement is qualitative linear with $eB$. The
quark condensate itself was calculated using the Banks-Casher relation banks ,
which relates the density of eigenvalues close to zero of the Dirac operator
and the condensate. Their calculations showed a monotonic increase of the
spectral density for typical gauge field configurations. This enhancement
induced by the magnetic field can be considered the basic mechanism behind
magnetic catalysis. Below, we will discuss this mechanism further, here it
suffices to add that the enhancement of the spectral density as a function of
$B$ for typical gauge configuration is also seen in full QCD bruck . Even in
the free case, there is a proliferation of small eigenvalues due to the
degeneracy of states, which in a constant magnetic field is proportional to
$|qB|$ smilga .

##  3 Magnetic catalysis at nonzero temperature

In the previous section, we reviewed magnetic catalysis at $T=0$ in some
detail. A survey of the literature shows that it is a robust feature of
lattice simulations as well model calculations: Magnetic catalysis does not
depend on particular values of the masses or couplings. Since the condensate
increases as a function of the magnetic field, it should raise the transition
temperature for the chiral transition. This expectation was made explicit a
long time ago in Ref. smilga . A number of model calculations have confirmed
this expectation, e.g. lemmer ; fraga0 ; fraga ; gatto ; gatto2 ; skokov ;
marco1 ; william , although some of them suggest that the chiral and
deconfinement transition split for larger values of $eB$.

The effective potential of the quark-meson model in the large-$N_{c}$
approximation at finite temperature is

$\displaystyle V_{0+1}^{T}$ $\displaystyle=$ $\displaystyle
V_{0+1}-\sum_{f}{2N_{c}|q_{f}B|\over 2\pi}T\sum_{s=\pm
1}\sum_{k=0}^{\infty}\int_{p_{z}}\log\left[1+e^{-\beta E_{f}}\right]\;,$

where $E_{f}=\sqrt{p_{z}^{2}+\Delta^{2}+|q_{f}B|(2k+1-s)}$ and $V_{0+1}$ is
given by Eq. (LABEL:fullb). In Fig. 2, we show the results of a typical
calculation where the quark-meson model was used. The curves show the
transition temperature for the chiral transition as a function of
$|qB|/m_{\pi}^{2}$ in the chiral limit (green points) and at the physical
point (red points). At $B=0$, the gap between the two critical temperatures is
approximately 10 MeV, which decreases as $|qB|$ grows. In both cases, it is
clear that the transition temperature increases with the magnetic field. Here
the transition temperature was defined as the inflection point of the curve
$\phi_{0}(T)$ at the physical point and $\phi_{0}(T)=0$ in the chiral limit.
An alternative definition of the critical temperature is the peak of the
chiral susceptibility 222Using the peak of ${\partial\Phi\over\partial T}$,
where $\Phi$ is the Polyakov loop, yields a transition temperature for
deconfinement, which is very close to the chiral transition temperature.

$\displaystyle\chi$ $\displaystyle=$
$\displaystyle{\partial\langle\bar{\psi}\bar{\psi}\rangle\over\partial T}\;.$
(28)

Figure 2: $T_{pc}(B)$ as a function of $|qB|$ in units of $m_{\pi}^{2}$ in the
quark-meson model. The green points are in the chiral limit and the red points
are at the physical point. See main text for details. Figure taken from Ref.
ourrev .

In QCD, two transitions take place as one increases the temperature, namely
the chiral transition and the deconfinement transition. Lattice calculations
suggest that chiral symmetry is “restored” at a temperature of approximately
$T^{\chi}_{c}$= 155 MeV aoki ; aoki2 ; borsa ; baza0 ; baza  though strictly
speaking the transition is only a crossover. The crossover temperature is
defined by the peak of the chiral susceptibility. This temperature is slightly
less than the crossover temperature for the deconfinement transition,
$T_{c}^{\rm dec}=$170 MeV. However this temperature difference is observable
dependent. In most cases, $T_{c}^{\rm dec}$ has been determined by the
behavior of the Polyakov loop. Recently, it has been defined by the behavior
of the quark entropy and in this case the two crossover temperatures agree
within errors baza .

We will next discuss the Polyakov loop and how it can be incorporated in model
calculations. The Wilson line is defined as

$\displaystyle L({\bf x})$ $\displaystyle=$ $\displaystyle{\cal
P}\exp\left[i\int_{0}^{\beta}d\tau A_{4}({\bf x},\tau)\right]\;,$ (29)

where ${\cal P}$ denotes path ordering, $A_{4}=iA_{0}$ and
$A_{0}=t_{a}A_{0}^{a}$. The Polyakov loop operator $l$ is the trace of the
Wilson line (29). Together with its Hermitian conjugate, it is defined as

$\displaystyle l$ $\displaystyle={1\over N_{c}}{\rm Tr}L\;,$ $\displaystyle
l^{\dagger}={1\over N_{c}}{\rm Tr}L^{\dagger}\;,$ (30)

where $N_{c}$ is the number of colors. The expectation values of $l$ and
$l^{\dagger}$ are denoted by $\Phi$ and $\bar{\Phi}$. Under the center group
$Z_{N_{c}}$ of the gauge group $SU(N_{c})$, the Polyakov loop transforms as
$\Phi\rightarrow e^{2\pi in\over N_{c}}\Phi$ with $n=0,1,2...N_{c}-1$. In
pure-glue QCD it is an order parameter for confinement, while for QCD with
dynamical fermions it is only an approximate order parameter yaffe . Note also
that $\Phi=\bar{\Phi}$ at zero density, i.e. for $\mu_{f}=0$.

For $N_{c}=3$ and in the Polyakov gauge, one can write a nonabelian background
gauge field as

$\displaystyle A_{4}$ $\displaystyle=$ $\displaystyle
t_{3}A_{4}^{3}+t_{8}A_{4}^{8}\;.$ (31)

Introducing the fields $\phi_{1}={1\over 2}\beta A_{4}^{3}$ and
$\phi_{2}={1\over 2\sqrt{3}}\beta A_{4}^{8}$, the thermal Wilson line reads
for constant gauge fields

$\displaystyle L$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}e^{i(\phi_{1}+\phi_{2})}&0&0\\\
0&e^{i(-\phi_{1}+\phi_{2})}&\\\ 0&0&e^{-2i\phi_{2}}\\\ \end{array}\right)\;.$
(35)

Since the Polyakov loop is an approximate order parameter for deconfinement,
the strategy put forward in Ref. polyakov1  is to write down a
phenomenological effective potential for $\Phi$, $\bar{\Phi}$ and the chiral
condensate that describes the thermodynamics of the system. This potential
consists of a gluonic part $U(\Phi,\bar{\Phi})$ as well as a matter part. The
term $U(\Phi,\bar{\Phi})$ is constructed such that it reproduces the pure-glue
pressure calculated on the lattice glue . A number of different forms of
$U(\Phi,\bar{\Phi})$ have been proposed ratti ; sjafer ; ratti2 ; ratti3 . In
Ref. ratti , they used a Polynomial expansion incorporating the $Z_{3}$ center
symmetry,

$\displaystyle{{U}\over T^{4}}$ $\displaystyle=$ $\displaystyle-{1\over
2}b_{2}(T)\Phi\bar{\Phi}-{1\over
6}b_{3}(T)\left[\Phi^{3}+\bar{\Phi}^{3}\right]+{1\over
4}b_{4}(\Phi\bar{\Phi})^{2}\;.$

Here the coefficients are

$\displaystyle b_{2}(T)$ $\displaystyle=$ $\displaystyle
6.75-1.95\left({T_{0}\over T}\right)+2.625\left({T_{0}\over
T}\right)^{2}-7.44\left({T_{0}\over T}\right)^{3}\;,$ $\displaystyle b_{3}$
$\displaystyle=$ $\displaystyle 0.75\;,$ (38) $\displaystyle b_{4}$
$\displaystyle=$ $\displaystyle 7.5\;,$ (39)

and $T_{0}=270$ MeV, the transition temperature for pure-glue QCD glue . A
drawback of the proposed pure-glue potentials is that they are independent of
the number of flavors $n_{f}$. The transition temperature for $B=0$ depends on
the number of flavors and one should incorporate the back-reaction from the
fermions to the gluonic sector sjafer . This is done by using an
$n_{f}$-dependent $T_{0}$. Once the coupling between the gluonic sector and
the matter sector has been implemented, the two transitions take place at
approximately the same temperature: The chiral transition moves to larger
temperatures, while the deconfinement transition moves to lower temperatures.
Finally, the Polyakov-loop potential is coupled to the matter sector via
replacing the partial derivatives in the fermionic part of the Lagrangian by
covariant ones including the constant background gauge field. This is
implemented by making the substitution

$\displaystyle\log\left[1+e^{-\beta E_{f}}\right]$ $\displaystyle\rightarrow$
$\displaystyle{1\over 6}\log\left[1+3\Phi e^{-\beta
E_{f}}+3\bar{\Phi}e^{-2\beta E_{f}}\right.$ (40)
$\displaystyle\left.+e^{-3\beta E_{f}}\right]$ $\displaystyle+{1\over
6}\log\left[1+3\bar{\Phi}e^{-\beta E_{f}}+3{\Phi}e^{-2\beta E_{f}}+e^{-3\beta
E_{f}}\right]$

in Eq. (LABEL:tempcont). In the same way, the Fermi-Dirac distribution
function is generalized,

$\displaystyle n_{F}(\beta E_{f})$ $\displaystyle=$
$\displaystyle{1+2\bar{\Phi}e^{\beta E_{f}}+\Phi e^{2\beta E_{f}}\over
1+3\bar{\Phi}e^{\beta E_{f}}+3\Phi e^{2\beta E_{f}}+e^{3\beta E_{f}}}\;.$ (41)

For small values of the Polyakov loop, $\Phi\approx 0$, $\bar{\Phi}\approx 0$
Eq. (41) reduces to a Fermi-Dirac distribution with excitation energy
$3E_{f}$, i.e. that of three quarks. For large temperatures, when $\Phi\approx
1$, $\bar{\Phi}\approx 1$, the excitation energy is $E_{f}$, which is the
distribution function of deconfined quarks.

To the best of our knowledge, there are no systematic studies of the
transition temperature as a function of the magnetic field $B$ in various
approximations. However, some interesting results using the quark-meson model
exist.

Figure 3: $T_{\rm pc}(B)/T_{\rm pc}$ as a function of $eB/m_{\pi}^{2}$ in the
quark-meson model. Solid line is the mean-field result and the dashed line is
the result from the functional renormalization group. See main text for
details. Figure taken from Ref. skokov .

Fig. 3 shows the normalized transition temperature from Ref. skokov  in two
approximations using the functional renormalization group (FRG) wetterich . In
this approach, one solves a flow equation for the effective potential
numerically by lowering a sliding scale $k$ from an initial UV cutoff
$k=\Lambda$ (where the effective potential is equal to the classical
potential) down to $k=0$. The bare parameters at $k=\Lambda$ are tuned such
that one obtains the physical values of the masses and the pion decay constant
in the vacuum (i. e. for $k=0$). In this way, all quantum and thermal
fluctuations are included. The black solid line is the mean-field result, i.e.
the bosons are excluded from the flow equation, whereas the brown line is the
result using the functional renormalization group. Clearly, the addition of
bosonic fluctuations increases the transition temperature significantly.

Figure 4: Normalized chiral transition temperature $T_{\phi}(B)$ as a function
of $|qB|$ in units of $m_{\pi}^{2}$ in the quark-meson model. The green points
are without the Polyakov loop and the blue points are with the Polyakov loop.
See main text for details. Figure taken from Ref. william .

In Fig. 4, we show the transition temperature at the physical point using the
functional renormalization group william . The green points are the results
without the Polyakov loop, whereas the blue points are the results including
it. Clearly, the Polyakov loop lowers the transition temperature for fixed
$B$, but it is still increasing as we increase the magnetic field. The above
FRG results are obtained in the so-called local-potential approximation. In
Ref. kamikado , the authors added the effects of wavefunction renormalization
and the curve for the critical temperature lies between mean-field and the
local-potential approximation. Thus the coupling of the Polyakov loop to the
chiral sector is not sufficient to reproduce (qualitatively) the results seen
on the lattice.

##  4 (Inverse) Magnetic catalysis on the lattice

After having discussed magnetic catalysis in low-energy models and theories of
QCD, we next consider QCD lattice simulations. In the past decade, there have
been a number of lattice calculations of QCD in a magnetic field cherno ;
chern2 ; braguta ; bali0 ; quarkcon ; bruck ; delia ; delia1 ; ani ; delia2 ;
heavypi ; ding0 ; ding1 , which have improved our understanding of QCD in a
magnetic background.

In order to discuss (inverse) magnetic catalysis as seen on the lattice, it is
advantageous to take a look at the path-integral representation of a number of
expectation values. The QCD Lagrangian is bilinear in the quark fields
$\psi_{f}$ and so one can integrate over them, giving for the partition
function as a path integral over gauge configurations $A_{\mu}$

$\displaystyle{\cal Z}(B)$ $\displaystyle=$ $\displaystyle\int
d{A}_{\mu}e^{-S_{g}}\det(D\\!\\!\\!\\!\\!/(B)+m)\;,$ (42)

where $S_{g}$ is the Euclidean gluon action and $\det(D\\!\\!\\!\\!\\!/(B)+m)$
is the fermion functional determinant (suppressing flavors). The operator
$D\\!\\!\\!\\!\\!/(B)$ contains the nonabelian gauge field, which we have
suppressed, as well as the abelian background $B$ that we have indicated. The
quark condensate is given by

$\displaystyle\langle\bar{\psi}\psi\rangle$ $\displaystyle=$
$\displaystyle{\partial\over\partial m}\log{\cal Z}(B)$ $\displaystyle=$
$\displaystyle{1\over{\cal Z}(B)}\int
d{A}_{\mu}e^{-S_{g}}\det(D\\!\\!\\!\\!\\!/(B)+m){\rm
Tr}(D\\!\\!\\!\\!\\!/(B)+m)^{-1}\;.$

We can think of $P={1\over{\cal Z}(B)}e^{-S_{g}}\det(D\\!\\!\\!\\!\\!/(B)+m)$
as a measure that depends on the gauge-field configuration $A_{\mu}$, the
magnetic field, and the quark masses. Note that the $B$-dependence is in the
functional determinant as well as the the trace of the propagator. In order to
study the contributions to the quark condensate coming separately from the
change of the operator and the change of the measure, it is convenient to
introduce the valence and sea contributions defined as

$\displaystyle\langle\bar{\psi}\psi\rangle^{\rm val}$ $\displaystyle=$
$\displaystyle{1\over{\cal Z}(0)}\int
d{A}_{\mu}e^{-S_{g}}\det(D\\!\\!\\!\\!\\!/(0)+m){\rm
Tr}(D\\!\\!\\!\\!\\!/(B)+m)^{-1}\;,$
$\displaystyle\langle\bar{\psi}\psi\rangle^{\rm sea}$ $\displaystyle=$
$\displaystyle{1\over{\cal Z}(B)}\int
d{A}_{\mu}e^{-S_{g}}\det(D\\!\\!\\!\\!\\!/(B)+m){\rm
Tr}(D\\!\\!\\!\\!\\!/(0)+m)^{-1}\;.$

This can be thought of as an expansion of the quark condensate around $B=0$. A
priori, the sum of the two contributions needs not add up to the total quark
condensate unless we are at small fields. However, it turns out that writing
the condensate as a sum of the valence and sea contribution is remarkably
good. This is clearly demonstrated in Fig. 5 from Ref. delia , which shows the
relative increment $r$ of the valence and sea contributions, their sum as well
as the complete results for the quark condensate as a function of a
dimensionless quantity $b$. The relative increment is defined as

$\displaystyle r$ $\displaystyle=$
$\displaystyle{\langle\bar{\psi}\psi\rangle_{B}\over\langle\bar{\psi}\psi\rangle}-1\;,$
(46)

where $\langle\bar{\psi}\psi\rangle$ is the average of the $u$ and $d$ quark
condensates. Within error, the additivity is confirmed for values of $b$ up to
8, which corresponds to magnetic fields up to $eB=(500{\rm MeV})^{2}$ delia .
It is also of interest to notice that both contributions work in the same
direction, namely to increase the quark condensate as $B$ grows. This is
unlike what happens at temperatures around the critical temperature $T_{c}$,
as we shall see below. As pointed out in Ref. bruck ,
$\langle\bar{\psi}\psi\rangle^{\rm sea}$ can be thought of as the quark
condensate of an electrically neutral fermion flavor coupled to an
electrically charged fermion flavor, since the magnetic field only appears in
the functional determinant and not in the propagator. On the other hand,
$\langle\bar{\psi}\psi\rangle^{\rm val}$ is reminiscent of the expression of
the quark condensate in model calculations, except in models one does not
integrate over gauge-field configurations.

Figure 5: Relative increment of the average of the $u$ and $d$ quark
condensates as a function of $b$. Valence (red points) and dynamical (sea)
(blue points) contributions, the sum of them (open circles), and the full
quark condensate as a function of the dimensionless quantity $b$. See main
text for details. Figure taken from Ref. delia .

Let us now turn to finite temperature. Inverse magnetic catalysis seems to
have two somewhat different meanings in the literature. The first meaning
corresponds directly to the concept magnetic catalysis discussed above: it
simply means that a condensate, for example $\langle\bar{\psi}\psi\rangle$,
decreases with the magnetic field at a fixed temperature. The second meaning
is that the transition temperature itself is a decreasing function of the
magnetic field.

The first finite-temperature lattice simulations were carried out in cherno ;
chern2  for $SU(2)$ gauge theory in the quenched approximation, focusing on
the $B$-dependence of the chiral condensate for temperatures below the
transition. In two-flavor QCD, simulations at finite temperature were carried
out for pion masses in the range $200-480$ MeV in Ref. delia  and it was
concluded that the chiral and deconfinement transitions take place at the same
temperature and that they increase slightly with the external magnetic field.
The increase of the transition temperature with $B$ is, at least
qualitatively, in agreement with model calculations. Bali et al quarkcon ;
bali0  carried out lattice simulations at the physical point with 2+1 flavors,
i.e. for quark masses that correspond to $m_{\pi}=140$ MeV, and the result was
somewhat surprising: The transition temperature turned out to be decreasing as
$B$ increases. The different behavior of $T_{c}$ is not a consequence of the
different pion masses, rather it results from lattice artefacts and that the
results of delia  were not continuum extrapolated. Today there is consensus
that the chiral transition temperature is a decreasing function of the
magnetic field. This behavior is illustrated in Fig. 6, which shows the
results of a recent lattice simulation delia2 , namely the transition
temperature in MeV as a function of the magnetic field $eB$ in GeV for three
different pion masses. The pion mass is 343 MeV (red points), 440 MeV (blue
points), and 664 MeV (green points), which is much larger than the physical
pion mass of 140 MeV. The transition temperature increases as a function of
the pion mass for fixed value of $B$, which is also known from $B=0$
calculations.

Figure 6: Transition temperature in GeV for the chiral transition as a
function of $eB$ for different values of the pion mass. See main text for
details. Figure taken from Ref. delia2 .

In Ref. bruck , the authors carried out a thorough analysis of the quark
condensate around the critical temperature to understand the behavior of the
transition temperature, focusing on disentangling the valence and sea effects.
The valence contribution Eq. (LABEL:valence) can also be written as

$\displaystyle\langle\bar{\psi}\psi\rangle^{\rm val}$ $\displaystyle=$
$\displaystyle\langle{\rm Tr}(D\\!\\!\\!\\!\\!/(B)+m)^{-1}\rangle_{0}\;,$ (47)

where the subscript indicates that the quark determinant is without a magnetic
field. The spectral density of the quark operator for different values of the
magnetic field is shown in Fig. 7. From the figure, it is evident that there
is an increase in the spectral density around zero with increasing magnetic
field. The corresponding ensemble was generated at finite temperature, $T=142$
MeV and for vanishing magnetic background bruck . The Banks-Casher relation
banks  then implies an increase of the valence contribution.

Figure 7: Spectral density of the Dirac operator for three different values of
the magnetic field. See main text for details. Figure taken from Ref. bruck .

Defining the quantity

$\displaystyle-\Delta
S_{f}(B)=\log\det(D\\!\\!\\!\\!\\!/(B)+m)-\log\det(D\\!\\!\\!\\!\\!/(0)+m)\;,$
(48)

the full condensate can be written as

$\displaystyle\langle\bar{\psi}\psi\rangle$ $\displaystyle=$
$\displaystyle{\langle e^{-\Delta S_{f}(B)}{\rm
Tr}(D\\!\\!\\!\\!\\!/(B)+m)^{-1}\rangle_{0}\over\langle e^{-\Delta
S_{f}(B)}\rangle_{0}}\;.$ (49)

Note that Eq. (49) reduces to the valence contribution Eq. (47) if one
replaces $\Delta S_{f}(B)$ by unity. Fig. 8 from Ref. bruck  shows a scatter
plot of the condensate as a function of the change in the action $\Delta
S_{f}(B)$ due to the magnetic field. In this plot, the magnetic field strength
is $eB\approx 5$ GeV2 and $T$ close to the transition temperature. Each point
represents a gauge configuration and they were generated at vanishing magnetic
field. The plot suggests that larger values of the condensates correspond to
larger values of the weight $e^{-\Delta S_{f}(B)}$ and therefore suppresses
the weight of the associated gauge configuration. As a result, this
counteracts the valence effect, and leads to a decrease in the critical
temperature. For pion masses that are not too large, it also leads to a
decrease of the condensate itself (see discussion below).

Figure 8:  Scatter plot of the down-quark condensate as a function of $\Delta
S_{f}(B)$. See main text for details. Figure taken from Ref. bruck .

In Refs. delia2 ; heavypi , the effects of varying the pion mass on the quark
condensate as a function of the temperature have been studied in detail (again
2+1 flavors). Fig. 9 from delia2  shows the difference between the quark
condensates as a function of the temperature at $B=0$ and $eB=0.425$ GeV2
(blue data points) and $eB=0.85$ GeV2 (red points) for three values of the
pion mass. The authors find that the sea contribution is a decreasing function
of $B$ around $T_{c}$ for the different values of the pion masses, while the
valence contribution is on the other hand an increasing function of the
magnetic field for all temperatures and pion masses. In the upper panel it is
clear that the sea contribution wins the competition around the transition
temperature implying inverse magnetic catalysis in the strict sense of the
word. This effect can barely been seen in the middle panel and is completely
absent in the lower panel. In other words, a decreasing function of the
transition temperature does not imply that the chiral condensate decreases as
a function of temperature and it is therefore not clear that the latter is the
driving mechanism of the former delia2 . This nontrivial behavior was also
demonstrated in heavypi , where the authors fixed the magnetic field to
$eB=0.6$ GeV2 and varied the pion mass. In QCD, there is inverse magnetic
catalysis for pion masses up to 500 MeV, and magnetic catalysis for larger
values 333 In Ref. ding0 , the authors find no sign of inverse catalysis in
their $N_{f}=3$ simulations with a pion mass of $280$ MeV.

Figure 9: Difference between the quark condensates at zero magnetic field and
non-vanishing $B$ for three different values of the pion mass. See main text
for details. Figure taken from Ref. delia2 .

We finally comment on the nature of the chiral transition and the temperature
as a function of $B$. The simulations have been done with magnetic fields up
to $eB=1$ GeV2. They all show an analytic crossover and that the transition
temperature is a decreasing function of $B$. However, it has been conjectured
that the transition would start increasing again for sufficiently large
temperatures, a phenomenon dubbed delayed magnetic catalysis. In Ref. ani  the
author went as high as $eB=3.25$ GeV in the simulations. The transition
remains a crossover (albeit sharper), there is no sign of delayed magnetic
catalysis, and the chiral and deconfinement transitions coincide. The sharper
crossover suggests that there may be a critical point for even larger values
of the magnetic field. For asymptotically large fields, QCD can be mapped onto
an anistropic pure-glue theory asymp . This theory was simulated on the
lattice and strong evidence for a first-order transition was found ani . This
implies the existence of a critical point and its position was estimated to be
at $eB\simeq 10$ GeV2.

##  5 Improvements of models

The failure of models to correctly describe the behavior of QCD around the
critical temperature, even after the introduction of the Polyakov loop, has
lead to significant efforts to improve them, see e.g. ferr2 ; fraga22 ;
farias0 ; ayala ; ferrer ; jos ; ferrer1 ; loewe1 ; maoinverse ; skok ;
farias2 . The temperature $T_{0}$ that enters the Polyakov-loop potential
depends on the number of flavors, it is therefore a reasonable assumption that
it also depends on the magnetic field. In Ref. ferr2 , the authors fitted the
strange-quark susceptibilities from their calculations in the entangled PNJL
model to the lattice results of Ref. bali0 . Their ansatz for the field
dependence was a simple polynomial in $(eB)^{2}$ up to quadratic order, giving
two fitting parameters,

$\displaystyle T_{0}(eB)$ $\displaystyle=$ $\displaystyle
T_{0}(0)+\zeta(eB)^{2}+\xi(eB)^{4}\;.$ (50)

An interesting feature here is that the model predicts a first-order
transition for magnetic fields larger than approximately $eB=0.25$ GeV2. As
mentioned above, such a critical point is expected in QCD, albeit at much
larger magnetic fields ani .

The crossover nature of the chiral transition was a guide for the authors of
Ref. fraga22 , trying to incorporate the decreasing behavior of the transition
temperature in the (Polyakov-loop extended) two-flavor quark-meson model. In
their mean-field analysis, they allowed the Yukawa coupling to vary with the
magnetic field, $g=g(B)$ using the boundary value $g(0)=3.3$. This value is
indicated by the vertical dotted line in Fig. 10 and corresponds to a fixed
quark mass in the vacuum. The critical temperature as a function of the Yukawa
coupling for three values of the magnetic field is also shown in Fig. 10. The
grey shaded region indicates the values of $g$ for which the transition is
first order.

Figure 10: $T_{c}$ as a function of the Yukawa coupling $g$ for various values
of the magnetic field. See main text for details. Figure taken from Ref.
fraga22 .

However, to obtain a transition temperature which is decreasing with the
magnetic field, any curve $g(B)$ must start at $g(0)=3.3$ and successively
cross the dashed (red) and solid (black) curves. One therefore soon enters the
shaded region which indicates a first-order transition in the QM model. Thus a
function $g(B)$ can simply not describe the correct $B$-dependence of the
transition temperature, while at the same time having a crossover transition.

Similar approaches have been used in the NJL model allowing the coupling $G$
to depend on both $T$ and $B$. For example, in Ref. farias0 , the authors fix
a set of parameters to get a reasonable fit to the lattice data for the sum of
the light quark condensates. The form of the $B$-dependent coupling was
motivated by the running of the coupling in QCD for strong magnetic fields
showrun .

One particular appealing idea was recently put forward in Ref. marko  (see
also Ref. costi1 ). Only $T=0$ physics from the lattice is used as input to
improve the model. In other words, there is no fitting to lattice data for
$T_{c}$ or a coupling that depends both upon $B$ and $T$. The authors first
performed a determination of the baryon spectrum at the physical point as a
function of magnetic field using lattice simulations. The authors focused on
strong magnetic fields, which are relevant for the phase diagram. Making the
simple assumption that the baryon masses can be written as the sum of the
masses of their constituents, they derived $B$-dependent constituent quark
masses. This was used as input in the PNJL model at zero temperature: Using
the gap equation with the $B$-dependent quark masses, a $B$-dependent four-
fermion coupling was obtained. The $B$-dependent constituent quark masses as
well as $G(B)$ are decreasing functions of the magnetic field. For $B=0$,
$M^{2}=0.097$ MeV2 and $G(0)=12.8$ GeV-2 and for the largest magnetic field
used ($|eB|=0.6$ GeV2), $M^{2}=0.079$ MeV2 and $G(B)=6.7$ GeV-2.

Figure 11: Normalized average quark condensate in the PNJL model as a function
of $eB$. See main text for details. Figure taken from Ref. marko .

Fig. 11 shows the normalized average quark condensate at $T=0$ as a function
of the magnetic field. The dashed black line is obtained from a next-to-
leading order (one-loop) calculation in chiral perturbation theory smilga ;
werbos , the light-blue points are obtained from the standard PNJL model,
while the green points are obtained from the lattice-improved PNJL model.
Finally, the red band shows the results from lattice simulations including
errors. The plot has several interesting features. Firstly, all low-energy
approaches are in good agreement with lattice results for low values of the
magnetic field. For larger values, both $\chi$PT and PNJL underestimate the
quark condensate. This is in contrast with the lattice-improved PNJL model,
which is in quantitative very good agreement with the simulations. It would be
of interest to see the predictions if one would include the effects of the
$s$-quark in the model calculations.

We next consider the finite-temperature calculations of Ref. marko . Fig. 12
shows the quark condensate as a function of $T$ for different strengths of the
magnetic field. The dashed lines are the predictions from the standard PNJL
model without error bands, while the solid bands are the lattice-improved PNJL
model predictions. We first note that quark condensate at $T=0$ for fixed
magnetic field is higher for the lattice-improved PNJL model in agreement with
Fig. 11. This behavior persists for low temperature, for larger temperatures,
however, the condensate drops faster compared to that calculated using the
standard PNJL model. Thus for fixed magnetic field, the transition temperature
defined as the inflection point of the quark condensate moves to a lower value
of $T$ compared to the PNJL model. This in itself is not enough to conclude
that we have inverse magnetic catalysis, but the effect is so strong that the
inflection point moves to the left as a function of $B$ so the transition
temperature decreases.

Figure 12: Quark condensate in the PNJL model as a function of $T$ for
different values of the magnetic field. See main text for details. Figure
taken from Ref. marko .

Fig. 13 shows the normalized transition temperature for the chiral transition,
as defined by the inflection point of the quark condensate, as a function of
$eB$ in units of GeV2. The pink band is from the lattice results of Refs.
bali0 . The width of the band indicates the errors of the simulations. The
light-blue points are the results of a calculation from the standard PNJL
model showing that the transition temperature increases as the magnetic field
grows. This is in sharp contrast to the lattice-improved PNJL model, where the
results are shown by the green points including uncertainties coming from the
lattice determination of the baryon masses. Given the large uncertainties, the
results for the transition temperature are in good agreement with the
simulations. Similarly, the analytic crossover found also agrees with lattice
results.

Figure 13: Normalized transition temperature from the lattice and in the PNJL
model as a function of $eB$. See main text for details. Figure taken from Ref.
marko .

##  6 Summary and final remarks

The idea of magnetic catalysis at zero temperature has been around for three
decades after its discovery in the NJL model. It is a robust phenomenon. For
large values of the magnetic field, it can be understood in terms of
dimensional reduction from $3+1$ to $1+1$ dimensions (or from 2+1 to 0+1
dimensions). Lattice simulations in the last decade have improved our
understanding of the effect significantly by showing that both the valence and
sea effect contribute in a nontrivial way. Since the discovery of the the
decrease of the chiral transition temperature with increasing magnetic field,
a lot of work has been devoted to incorporate this feature in models. This
includes $T$ \- and $B$-dependent couplings and Polyakov-loop potentials.
Fitting parameters can be considered an indirect way of incorporating the sea
effect and requires input from lattice simulations at finite temperature. In
our opinion, a cleaner approach is provided by the work marko , which uses
lattice input at $T=0$ only. In much the same way as one uses experimentally
measured meson masses in the vacuum, they use $B$-dependent baryon masses
measured on the lattice as input in their PNJL-model calculations, although
assuming that the baryon mass is the sum of its constituents perhaps is
somewhat simplistic.

## Acknowledgements

The author would like to thank Prabal Adhikari, Patrick Kneschke, William
Naylor, and Anders Tranberg for discussions and collaboration on related
topics. The author would like to thank Massimo D’elia, Gergely Endródi,
Eduardo Fraga, and Vladimir Skokov for permission to use their figures.

##  Appendix A Renormalization of the one-loop effective potential in the
quark-meson model

In this appendix, we will discuss renormalization of the one-loop effective
potential in the quark-meson model using the on-shell scheme. The starting
point is the one-loop contribution to the effective potential of a fermion of
mass $m_{f}$ in a constant magnetic field, which is given by

$\displaystyle V_{1}$ $\displaystyle=$ $\displaystyle-{|q_{f}B|\over
2\pi}\sum_{s=\pm
1}\sum_{k=0}^{\infty}\int_{p_{\parallel}}\log\left[p_{\parallel}^{2}+m_{f}^{2}+|q_{f}B|(2k+1-s)\right]\;,$

where $p_{\parallel}^{2}=p_{0}^{2}+p_{z}^{2}$ and the integral is defined in
$d$ dimensions using dimensional regularization,

$\displaystyle\int_{p}$ $\displaystyle=$
$\displaystyle\left({e^{\gamma_{E}}\Lambda^{2}\over
4\pi}\right)^{\epsilon}\int{d^{d}p\over(2\pi)^{d}}\;.$ (52)

and where $\Lambda$ is the renormalization scale associated with the
$\overline{\rm MS}$-scheme. The sum is over spin $s$ and Landau levels $k$.
The integral over $p_{\parallel}$ can be evaluated using dimensional
regularization in $d=2-2\epsilon$ dimensions. The result is

$\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle{2|q_{f}B|\over(4\pi)^{2}}\Gamma(-1+\epsilon)\left({e^{\gamma_{E}}\Lambda^{2}}\right)^{\epsilon}\sum_{s=\pm
1}\sum_{k=0}^{\infty}M_{B}^{2-2\epsilon}\;,$ (53)

where $M_{B}^{2}=m_{f}^{2}+|q_{f}B|(2k+1-s)$. The sum over spin $s$ and Landau
levels $n$ can be expressed in terms of the Hurwitz $\zeta$-function as

$\displaystyle\sum_{s=\pm 1}\sum_{k=0}^{\infty}M_{B}^{2-2\epsilon}$
$\displaystyle=$ $\displaystyle
2(2|q_{f}B|)^{1-\epsilon}\sum_{k=0}^{\infty}\left[k+{m_{f}^{2}\over
2|q_{f}B|}\right]^{1-\epsilon}-m_{f}^{2-2\epsilon}$ (54) $\displaystyle=$
$\displaystyle
2(2|q_{f}B|)^{1-\epsilon}\zeta(-1+\epsilon,x_{f})-m_{f}^{2-2\epsilon}\;,$

where $x_{f}={m_{f}^{2}\over 2|q_{f}B|}$ is a dimensionless variable. The
effective potential can then be written as

$\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle{8(q_{f}B)^{2}\over(4\pi)^{2}}\left({e^{\gamma_{E}}\Lambda^{2}\over
2|q_{f}B|}\right)^{\epsilon}\Gamma(-1+\epsilon)\bigg{[}\zeta(-1+\epsilon,x_{f})$
(55) $\displaystyle-{1\over 2}x_{f}^{1-\epsilon}\bigg{]}\;.$

We next expand the result (55) in powers of $\epsilon$ to order
$\epsilon^{0}$. This yields

$\displaystyle{V}_{1}$ $\displaystyle=$
$\displaystyle{1\over(4\pi)^{2}}\left({\Lambda^{2}\over
2|q_{f}B|}\right)^{\epsilon}\bigg{[}\left({2(q_{f}B)^{2}\over
3}+m_{f}^{4}\right)\left({1\over\epsilon}+1\right)$ (56)
$\displaystyle-8(q_{f}B)^{2}\zeta^{(1,0)}(-1,x_{f})-2|q_{f}B|m_{f}^{2}\log
x_{f}\bigg{]}\;.$

For renormalization purposes, it is convenient to isolate in Eq. (56) the
terms in the functional determinant that equal the $B=0$ result, cf. Eq. (26).
Adding the tree-level potential $V_{0}$, setting $m_{f}=\Delta$, and summing
over quark flavors and colors yields

$\displaystyle V_{0+1}$ $\displaystyle=$ $\displaystyle{1\over 2}B^{2}+{1\over
2}{m^{2}\over g^{2}}\Delta^{2}+{\lambda\over 24g^{4}}\Delta^{4}-{h\over
g}\Delta$ (57)
$\displaystyle+{2N_{c}\Delta^{4}\over(4\pi)^{2}}\left[{1\over\epsilon}+{3\over
2}+\log{\Lambda^{2}\over\Delta^{2}}\right]$
$\displaystyle+{N_{c}\over(4\pi)^{2}}\sum_{f}{2(q_{f}B)^{2}\over
3}\left[{1\over\epsilon}+\log{\Lambda^{2}\over 2|q_{f}B|}\right]$
$\displaystyle-{8N_{c}\over(4\pi)^{2}}\sum_{f}(q_{f}B)^{2}\left[{\zeta^{(1,0)}(-1,x_{f})}+{1\over
4}x_{f}^{2}\right.$ $\displaystyle\left.-{1\over 2}x_{f}^{2}\log x_{f}+{1\over
2}x_{f}\log x_{f}-{1\over 12}\right]\;.$

Eq. (57) has simple poles in epsilon. The pole proportional to $(q_{f}B)^{2}$
is eliminated by wavefunction renormalization of $B$ and the charge $q_{f}$
such that $q_{f}B$ is invariant. The pole propertional to $\Delta^{4}$ is
eliminated by renormalization of the parameters in the Lagrangian. The bare
parameters are replaced by the parameters in $\overline{\rm MS}$-scheme, e.g.
$m^{2}\rightarrow m^{2}_{\overline{\text{\tiny MS}}}+\delta
m^{2}_{\overline{\text{\tiny MS}}}$ and the running parameters given by Eqs.
(21)–(24) are substituted into the renormalized expression for the effective
potential. The couplings at the reference scale $\Lambda_{0}$ are determined
using Eqs. (LABEL:osm1)–(LABEL:osm4) and expressed in terms of physical
quantities. The result is Eq. (LABEL:fullb).

##  Appendix B Parameter fixing

The mass of a particle is given by the pole of the propagator. In the on-shell
scheme, the sum of the self-energy evaluated on-shell and the counterterms
vanishes,

$\displaystyle\Sigma_{\sigma,\pi}(p^{2}=m^{2}_{\sigma,\pi})+{\rm
counterterms}$ $\displaystyle=$ $\displaystyle 0\;.$ (1)

In addition, the residue of the propagator evaluated on shell is unity. This
implies

$\displaystyle\Sigma^{\prime}(p^{2}=m_{\sigma,\pi}^{2})+{\rm counterterms}$
$\displaystyle=$ $\displaystyle 0\;.$ (2)

The self-energies are

$\displaystyle\Sigma_{\sigma}(p^{2})$ $\displaystyle=$
$\displaystyle-8ig^{2}N_{c}\left[A(m_{q}^{2})-\mbox{$1\over
2$}\left(p^{2}-4m_{q}^{2}\right)B(p^{2})\right]\;,$ (3)
$\displaystyle\Sigma_{\pi}(p^{2})$ $\displaystyle=$
$\displaystyle-8ig^{2}N_{c}\left[A(m_{q}^{2})-\mbox{$1\over
2$}p^{2}B(p^{2})\right]\;,$ (4)

where the integrals $A(m^{2})$ and $B(p^{2})$ are defined in Eqs. (27)–
(LABEL:bdef). We have omitted the tadpole diagram which is one-particle
reducible. It is cancelled by a counterterm, when we impose the condition that
$\phi_{0}=f_{\pi}$. The counterterms are given by the expressions

$\displaystyle\delta m_{\sigma,\pi}^{2}$ $\displaystyle=$
$\displaystyle-\Sigma_{\sigma}(p^{2})\Big{|}_{p^{2}=m_{\sigma,\pi}^{2}}\;,\hskip
2.84526pt\delta
Z_{\sigma,\pi}=\Sigma_{\sigma,\pi}^{\prime}(p^{2})\Big{|}_{p^{2}=m_{\sigma,\pi}^{2}}\;.\;.$
(5)

This yields

$\displaystyle\delta m_{\sigma}^{2}$ $\displaystyle=$ $\displaystyle
8ig^{2}N_{c}\left[A(m_{q}^{2})-\mbox{$1\over
2$}(m_{\sigma}^{2}-4m_{q}^{2})B(m_{\sigma}^{2})\right]\;,$ (6)
$\displaystyle\delta m_{\pi}^{2}$ $\displaystyle=$ $\displaystyle
8ig^{2}N_{c}\left[A(m_{q}^{2})-\mbox{$1\over
2$}m_{\pi}^{2}B(m_{\pi}^{2})\right]\;,$ (7) $\displaystyle\delta Z_{\sigma}$
$\displaystyle=$ $\displaystyle
4ig^{2}N_{c}\left[B(m_{\sigma}^{2})+(m_{\sigma}^{2}-4m_{q}^{2})B^{\prime}(m_{\sigma}^{2})\right]\;,$
(8) $\displaystyle\delta Z_{\pi}$ $\displaystyle=$ $\displaystyle
4ig^{2}N_{c}\left[B(m_{\pi}^{2})+m_{\pi}^{2}B^{\prime}(m_{\pi}^{2})\right]\;,$
(9) $\displaystyle\delta t$ $\displaystyle=$
$\displaystyle-8ig^{2}N_{c}f_{\pi}A(m_{q}^{2})\;,$ (10)

where we have added the counterterm $\delta t$ for the one-point function. We
next need to relate the above counterterms to the counterterms of the
parameters of the Lagrangian. These relations follow immediately from Eqs.
(12)–(13),

$\displaystyle\delta m^{2}_{\text{\tiny OS}}$ $\displaystyle=$
$\displaystyle-{1\over 2}(\delta m_{\sigma}^{2}-3\delta m_{\pi}^{2})\;,$ (11)
$\displaystyle\delta\lambda_{\text{\tiny OS}}$ $\displaystyle=$ $\displaystyle
3{(\delta m_{\sigma}^{2}-\delta m_{\pi}^{2})\over f_{\pi}^{2}}-\lambda{\delta
f_{\pi}^{2}\over f_{\pi}^{2}}\;,$ (12) $\displaystyle\delta g^{2}_{\text{\tiny
OS}}$ $\displaystyle=$ $\displaystyle{\delta m_{q}^{2}\over
f_{\pi}^{2}}-g^{2}{\delta f_{\pi}^{2}\over f_{\pi}^{2}}\;.$ (13)

In the large-$N_{c}$ limit, $\delta m_{q}^{2}=0$ implying that $\delta
g^{2}_{\text{\tiny OS}}=-g^{2}{\delta f_{\pi}^{2}\over f_{\pi}^{2}}$. There is
also no loop correction to the quark-pion vertex and $\delta Z_{\psi}=1$. This
implies that the associated counterterms must cancel as well, leading to
$\delta g^{2}_{\text{\tiny OS}}=-g^{2}\delta Z_{\pi}^{\text{\tiny OS}}$. We
can therefore write

$\displaystyle\delta\lambda_{\text{\tiny OS}}$ $\displaystyle=$ $\displaystyle
3{(\delta m_{\sigma}^{2}-\delta m_{\pi}^{2})\over f_{\pi}^{2}}-\lambda\delta
Z_{\pi}^{\text{\tiny OS}}\;.$ (14)

The counterterm $\delta h_{\text{\tiny OS}}$ is found from the one-point
function. At tree level, we have $t=h-m_{\pi}^{2}f_{\pi}=0$, which yields
$\delta t=\delta h_{\text{\tiny OS}}-\delta
m_{\pi}^{2}f_{\pi}-m_{\pi}^{2}\delta f_{\pi}=$ or $\delta h_{\text{\tiny
OS}}=\delta t+\delta m_{\pi}^{2}f_{\pi}+{1\over 2}m_{\pi}^{2}f_{\pi}\delta
Z_{\pi}^{\text{\tiny OS}}$. Finally, we need the counterterm for the
electromagnetic field,

$\displaystyle\delta Z_{A}^{\text{\tiny OS}}$ $\displaystyle=$ $\displaystyle
i{N_{c}\over(4\pi)^{2}}\sum_{f}{4q_{f}^{2}\over 3}B(0)\;,$ (15)

with the integral $B(p^{2})$ defined in Eq. (LABEL:bdef). Renormalization is
then carried out by making the substitution $B^{2}\rightarrow B^{2}(1+\delta
Z_{A}^{\text{\tiny OS}})$. We note that the on-shell scheme is not well-
defined when the fermions are massless. In that case, the (modified) minimal
subtraction scheme may be used. Since the bare parameters are independent of
the renormalization scheme, we can immediately write down the relations
between the renormalized parameters in the on-shell and $\overline{\rm MS}$
schemes. For example $g_{\overline{\text{\tiny MS}}}^{2}+\delta
g_{\overline{\text{\tiny MS}}}^{2}=g^{2}_{\text{\tiny OS}}+\delta
g^{2}_{\text{\tiny OS}}$. From Eq. (9), we find

$\displaystyle\delta g^{2}_{\rm os}$ $\displaystyle=$
$\displaystyle{4g^{4}N_{c}\over(4\pi)^{2}}\left[{1\over\epsilon}+\log{\Lambda^{2}\over
m_{q}^{2}}+F(m_{\pi}^{2})+m_{\pi}^{2}F^{\prime}(m_{\pi}^{2})\right]\;,$

where $F(p^{2})$ and $F^{\prime}(p^{2})$ are defined in Eqs. (29)–(30). The
counterterm in the $\overline{\rm MS}$-scheme is simply the pole part, $\delta
g^{2}_{\overline{\text{\tiny
MS}}}={4N_{c}g^{4}\over(4\pi)^{2}}{1\over\epsilon}$. From this, one finds the
running coupling $g^{2}_{\overline{\text{\tiny MS}}}$ using
$g_{\overline{\text{\tiny MS}}}^{2}=g^{2}_{\text{\tiny OS}}+\delta
g^{2}_{\text{\tiny OS}}-\delta g_{\overline{\text{\tiny MS}}}^{2}$ and given
by Eq. (LABEL:osm3). The running parameters are

$\displaystyle m_{\overline{\text{\tiny MS}}}^{2}(\Lambda)$ $\displaystyle=$
$\displaystyle{m_{0}^{2}\over
1-{4g_{0}^{2}N_{c}\over(4\pi)^{2}}\log{\Lambda^{2}\over m_{q}^{2}}}\;,$ (21)
$\displaystyle g_{\overline{\text{\tiny MS}}}^{2}(\Lambda)$ $\displaystyle=$
$\displaystyle{g_{0}^{2}\over
1-{4g_{0}^{2}N_{c}\over(4\pi)^{2}}\log{\Lambda^{2}\over m_{q}^{2}}}\;,$ (22)
$\displaystyle\lambda_{\overline{\text{\tiny MS}}}(\Lambda)$ $\displaystyle=$
$\displaystyle{\lambda_{0}-{48g_{0}^{4}N_{c}\over(4\pi)^{2}}\log{\Lambda^{2}\over
m_{q}^{2}}\over\left(1-{4g_{0}^{2}N_{c}\over(4\pi)^{2}}\log{\Lambda^{2}\over
m_{q}^{2}}\right)^{2}}\;,$ (23) $\displaystyle h_{\overline{\text{\tiny
MS}}}(\Lambda)$ $\displaystyle=$ $\displaystyle{h_{0}\over
1-{2g_{0}^{2}N_{c}\over(4\pi)^{2}}\log{\Lambda^{2}\over m_{q}^{2}}}\;,$ (24)

where $m_{0}^{2}$, $g_{0}^{2}$, $\lambda_{0}$, and $h_{0}$ are the values of
the running mass and couplings at the scale $\Lambda_{0}$ determined by

$\displaystyle\left[\log{\Lambda_{0}^{2}\over
m_{q}^{2}}+F(m_{\pi}^{2})+m_{\pi}^{2}F^{\prime}(m_{\pi}^{2})\right]$
$\displaystyle=$ $\displaystyle 0\;.$ (25)

This equation in conjunction with Eqs. (LABEL:osm1)–(LABEL:osm4) can be used
to determine the values of the couplings at the scale $\Lambda_{0}$ expressed
in terms of physical quantities. For example, it follows that
$g^{2}_{0}=g^{2}_{\overline{\text{\tiny MS}}}(\Lambda_{0})={m_{q}^{2}\over
f_{\pi}^{2}}$.

We need a few divergent integrals space in four dimensions. Going to Euclidean
space via Wick rotation, we can use dimensional regularization in
$d=4-2\epsilon$ dimensions. The integrals needed are

$\displaystyle\int_{k}\log\left[k^{2}+m^{2}\right]$ $\displaystyle=$
$\displaystyle-{m^{4}\over 2(4\pi)^{2}}\left({\Lambda^{2}\over
m^{2}}\right)^{\epsilon}\left[{1\over\epsilon}+{3\over 2}+{\cal
O}(\epsilon)\right]\;,$ (26) $\displaystyle A(m^{2})$ $\displaystyle=$
$\displaystyle\int_{k}{1\over k^{2}-m^{2}}$ (27) $\displaystyle=$
$\displaystyle{im^{2}\over(4\pi)^{2}}\left({\Lambda^{2}\over
m^{2}}\right)^{\epsilon}\left[{1\over\epsilon}+1+{\cal O}(\epsilon)\right]\;,$
$\displaystyle B(p^{2})$ $\displaystyle=$
$\displaystyle\int_{k}{1\over(k^{2}-m_{q}^{2})[(k+p)^{2}-m_{q}^{2})}$
$\displaystyle=$ $\displaystyle{i\over(4\pi)^{2}}\left({\Lambda^{2}\over
m_{q}^{2}}\right)^{\epsilon}\left[{1\over\epsilon}+F(p^{2})+{\cal
O}(\epsilon)\right]\;,$

where $\Lambda$ is the renormalization scale associated with the
$\overline{\rm MS}$ scheme and

$\displaystyle F(p^{2})$ $\displaystyle=$ $\displaystyle
2-2r\arctan\left({1\over r}\right)\;,$ (29) $\displaystyle B^{\prime}(p^{2})$
$\displaystyle=$ $\displaystyle F^{\prime}(p^{2})={4m_{q}^{2}\over
p^{4}r}\arctan\left({1\over r}\right)-{1\over p^{2}}\;,$ (30)

where $r=\sqrt{{4m_{q}^{2}\over p^{2}}-1}$.

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