# Local correlation among the chiral condensate, monopoles, and color magnetic
fields in Abelian projected QCD

Hideo Suganuma<EMAIL_ADDRESS>Department of Physics, Kyoto
University, Kitashirakawaoiwake, Sakyo, Kyoto 606-8502, Japan  Hiroki Ohata
<EMAIL_ADDRESS>Yukawa Institute for Theoretical Physics,
Kyoto University, Sakyo, Kyoto 606-8502, Japan

###### Abstract

Using the lattice gauge field theory, we study the relation among the local
chiral condensate, monopoles, and color magnetic fields in quantum
chromodynamics (QCD). First, we investigate idealized Abelian gauge systems of
1) a static monopole-antimonopole pair and 2) a magnetic flux without
monopoles, on a four-dimensional Euclidean lattice. In these systems, we
calculate the local chiral condensate on quasi-massless fermions coupled to
the Abelian gauge field, and find that the chiral condensate is localized in
the vicinity of the magnetic field. Second, using SU(3) lattice QCD Monte
Carlo calculations, we investigate Abelian projected QCD in the maximally
Abelian gauge, and find clear correlation of distribution similarity among the
local chiral condensate, monopoles, and color magnetic fields in the
Abelianized gauge configuration. As a statistical indicator, we measure the
correlation coefficient $r$, and find a strong positive correlation of
$r\simeq 0.8$ between the local chiral condensate and an Euclidean color-
magnetic quantity ${\cal F}$ in Abelian projected QCD. The correlation is also
investigated for the deconfined phase in thermal QCD. As an interesting
conjecture, like magnetic catalysis, the chiral condensate is locally enhanced
by the strong color-magnetic field around the monopoles in QCD.

††preprint: YITP-21-85

##  I Introduction

Quantum chromodynamics (QCD) is an SU($N_{c}$) gauge theory to describe the
strong interaction, and has presented many interesting subjects full of
variety and difficult problems in physics. Actually, in spite of the simple
form of the QCD action, this miracle theory creates hundreds of hadrons and
leads to various interesting nonperturbative phenomena, such as color
confinement and dynamical chiral-symmetry breaking [1].

This magic is due to the strong coupling of QCD in the low-energy region, and
this strong-coupling nature drastically changes the vacuum structure itself.
Therefore, perturbative technique is no more workable and analytical treatment
of QCD is fairly difficult in the strong-coupling region. As a reliable
standard technique, lattice QCD Monte Carlo simulations have been applied to
analyze nonperturbative QCD [2, 3].

Among the nonperturbative properties of QCD, spontaneous chiral-symmetry
breaking is particularly important in our real world. Indeed, chiral symmetry
breaking drastically influences the vacuum structure and gives a nontrivial
vacuum expectation value of the chiral condensate $\langle\bar{q}q\rangle$,
which plays the role of an order parameter. Also, it is considered that chiral
symmetry breaking leads to dynamical quark-mass generation [1, 4], and creates
most of the matter mass of our Universe, apart from the dark matter, because
only small masses of u, d current quarks and electrons are Higgs-origin in
atoms [5] and their contribution to the nucleon mass is estimated to be small
[6]. In addition, chiral symmetry breaking inevitably accompanies light pions
of the Nambu-Goldstone bosons, and their small mass gives range of the nuclear
force.

In nonperturbative QCD, color confinement is also one of the most important
phenomena in physics, and presents an extremely difficult mathematical
problem. Experiments for hadron spectra and lattice QCD studies for various
inter-quark potentials [7, 8, 9, 10] show that the quark confining force is
basically characterized by a universal physical quantity of the string tension
$\sigma\simeq$ 0.89 GeV/fm. This universal string tension is physically
explained by one-dimensional squeezing of the color electric flux, i.e., the
color flux-tube formation in hadrons, as is also indicated by lattice QCD for
both mesons [3] and baryons [11]. As for the relation between color
confinement and chiral symmetry breaking, it is not yet clarified directly
from QCD. While almost coincidence between deconfinement and chiral-
restoration temperatures [12] suggests their close correlation, a lattice QCD
analysis using the Dirac-mode expansion based on the Banks-Casher relation
[13] indicates some independence of these phenomena in QCD [14, 15].

For the quark confinement mechanism, Nambu [16], ’t Hooft [17], and Mandelstam
[18] proposed the dual superconductivity scenario, paying attention to analogy
with the Abrikosov vortex in the superconductivity, where Cooper-pair
condensation leads to the Meissner effect, and the magnetic flux is excluded
or squeezed like a one-dimensional tube as the Abrikosov vortex. If the QCD
vacuum can be regarded as the dual version of the superconductor, the
electric-type color flux is squeezed between (anti)quarks in hadrons, and
quark confinement can be physically explained by the dual Meissner effect.
Because of the electromagnetic duality, the dual Meissner effect inevitably
needs condensation of magnetic objects, i.e., color magnetic monopoles, which
correspond to the dual version of the electric Cooper-pair bosonic field.

In the dual-superconductor picture for the QCD vacuum, however, there are two
large gaps with QCD.

  1. 1.

While QCD is a non-Abelian gauge theory, the dual-superconductor picture is
based on an Abelian gauge theory subject to the Maxwell-type equations
including magnetic currents, where electromagnetic duality is manifest.

  2. 2.

While QCD includes only color electric variables, i.e., quarks and gluons, as
the elementary degrees of freedom, the dual-superconductor picture requires
condensation of color magnetic monopoles as a key concept.

Historically, to bridge between QCD and the dual-superconductivity, ’t Hooft
proposed Abelian gauge fixing [19], partial gauge fixing which only remains
Abelian gauge degrees of freedom in QCD. By Abelian gauge fixing, QCD reduces
into an Abelian gauge theory, where off-diagonal gluons behave as charged
matter fields similar to $W^{\pm}_{\mu}$ in the Weinberg-Salam model and give
the color electric current $j_{\mu}$ in terms of the residual Abelian gauge
symmetry. As a remarkable fact in the Abelian gauge, color-magnetic monopoles
appear as topological objects corresponding to the nontrivial homotopy group
$\Pi_{2}({\rm SU}(N_{c})/{\rm U}(1)^{N_{c}-1})={\bf Z}^{N_{c}-1}_{\infty}$ in
a similar manner to appearance of ’t Hooft-Polyakov monopoles [20] in the
SU(2) non-Abelian Higgs theory. Thus, in the Abelian gauge, QCD is reduced
into an Abelian gauge theory including both electric current $j_{\mu}$ and
magnetic current $k_{\mu}$, which is expected to give a theoretical basis of
the dual-superconductor picture for the confinement mechanism, although off-
diagonal gluons remain as charged matter fields.

From the viewpoint of Abelianzation of QCD, the maximally Abelian (MA) gauge
[21] is an interesting special Abelian gauge. In the MA gauge, off-diagonal
gluons have a large effective mass of about 1 GeV in both SU(2) and SU(3)
lattice QCD [22, 23, 24], so that off-diagonal gluons become infrared
inactive, and only the Abelian gluon is relevant at distances larger than
about 0.2 fm. Also, monopole condensation is suggested from appearance of long
entangled monopole worldlines [21, 25] and the magnetic screening in lattice
QCD [26, 27].

In this way, by taking the MA gauge, the QCD vacuum can be regarded as an
Abelian dual superconductor at a large scale, and color magnetic monopoles
seem to capture essence of nonperturbative QCD. Note however that, even
without gauge fixing, there is an evidence of monopole condensation in
nonabelian gauge theories [27], and therefore it might be possible to define
infrared-relevant monopoles in QCD and to construct the dual superconductor
system in more general manner. In fact, MA gauge fixing gives a concrete way
to extract infrared-relevant Abelian gauge manifold and monopoles from QCD.

In the context of the dual superconductir picture, close correlation between
monopoles and chiral symmetry breaking was pointed out in the dual Ginzburg-
Landau theory [28], in SU(2) lattice QCD in the MA gauge [29, 30], and in
SU(3) lattice QCD [31, 32]. Since most of the pioneering lattice studies were
done in SU(2) QCD or on a small lattice as $8^{3}\times 4$, we recently
investigated SU(3) QCD with a large volume, and find a clear correlation
between monopoles and the chiral condensate in SU(3) lattice QCD in the MA
gauge [33].

In this paper, as a continuation of Ref.[33], we proceed the lattice works for
the relation between chiral symmetry breaking and color magnetic objects
including monopoles. In particular, as a new point of this paper, we
quantitatively study correlation of the local chiral condensate with color
magnetic fields using the lattice gauge theory.

The organization of this paper is as follows. In Section 2, we review the MA
gauge and Abelianization of QCD in SU(3) lattice formalism. In Section 3, we
prepare magnetic objects in Abelian projected QCD. In Section 4, we consider
the local chiral condensate and chiral symmetry breaking in Abelian gauge
systems. In Section 5, we present idealized Abelian gauge systems of a static
monopole-antimonopole pair on a lattice, and investigate the relation of the
local chiral condensate with the magnetic objects. In Section 6, we perform
SU(3) lattice QCD Monte Carlo calculations and study the relation among
monopoles, magnetic fields, and the local chiral condensate in Abelian
projected QCD in the MA gauge. Section 7 is devoted for summary and
conclusion.

##  II Maximally Abelian gauge and Abelianization of QCD

To begin with, we briefly review the lattice formalism for maximally Abelian
(MA) gauge fixing and Abelianization in QCD.

Continuum QCD is described with the quark field $q(x)$, the gluon field
$A_{\mu}(x)\in{\rm su}(N_{c})$ and the QCD gauge coupling $g$. In SU($N_{c}$)
lattice QCD [3], the gluon field is described as the SU($N_{c}$) link variable
$U_{\mu}(s)\equiv\exp\left(iagA_{\mu}(s)\right)\in{\rm SU}(N_{c})$ on four-
dimensional Euclidean lattices with the spacing $a$ and the volume
$V=L_{x}L_{y}L_{z}L_{t}$.

Using the Cartan subalgebra $\vec{H}\equiv(T_{3},T_{8})$ in SU(3), MA gauge
fixing is defined so as to maximize

$R_{\rm MA}[U_{\mu}(s)]\equiv\sum_{s}\sum_{\mu=1}^{4}{\rm
tr}\left(U_{\mu}^{\dagger}(s)\vec{H}U_{\mu}(s)\vec{H}\right)=\sum_{s}\sum_{\mu=1}^{4}\left(1-\frac{1}{2}\sum_{i\neq
j}\left|U_{\mu}(s)_{ij}\right|^{2}\right)$ (1)

by the SU(3) gauge transformation, and therefore this gauge fixing strongly
suppresses all the off-diagonal fluctuation of the SU(3) gauge field. In the
MA gauge, the SU(3) gauge group is partially fixed remaining its maximal torus
subgroup ${\rm U(1)}_{3}\times{\rm U(1)}_{8}$ with the global Weyl (color
permutation) symmetry [34], and QCD is reduced to an Abelian gauge theory.

From the SU(3) link variable $U_{\mu}^{\rm MA}(s)\in{\rm SU(3)}$ in the MA
gauge, we extract the Abelian link variable

$u_{\mu}(s)=e^{i\vec{\theta}_{\mu}(s)\cdot\vec{H}}=\mathrm{diag}\left(e^{i\theta_{\mu}^{1}(s)},e^{i\theta_{\mu}^{2}(s)},e^{i\theta_{\mu}^{3}(s)}\right)\in{\rm
U}(1)_{3}\times{\rm U}(1)_{8}\subset{\rm SU}(3)$ (2)

by maximizing the overlap

$R_{\rm Abel}\equiv\frac{1}{3}{\rm Re\,tr}\left\\{U_{\mu}^{\rm
MA}(s)u_{\mu}^{\dagger}(s)\right\\}\in\left[-\frac{1}{2},1\right].$ (3)

Note that the distance between $u_{\mu}(s)$ and $U_{\mu}^{\rm MA}(s)$ becomes
the smallest in the SU(3) manifold, and there is a constraint
$\sum_{i=1}^{3}\theta_{\mu}^{i}(s)=0\,\left(\mathrm{mod}~{}2\pi\right)$
reflecting the uni-determinant of $u_{\mu}(s)$. Here, $\theta_{\mu}^{i}(s)$
($i$ = 1, 2, 3) is taken to be the principal value of
$-\pi\leq\theta_{\mu}^{i}(s)<\pi$.

The Abelian projection is defined by the simple replacement of the SU(3) link
variable $U_{\mu}(s)$ by the Abelian link variable $u_{\mu}(s)$ for each gauge
configuration, that is, $O[U_{\mu}(s)]\rightarrow O[u_{\mu}(s)]$ for QCD
operators. Abelian projected QCD is thus extracted from SU(3) QCD. The case of
$\langle O[U_{\mu}(s)]\rangle\simeq\langle O[u_{\mu}(s)]\rangle$ is called
“Abelian dominance” for the operator $O$ [35].

As a remarkable fact, Abelian dominance of quark confinement is shown in both
SU(2) [36] and SU(3) lattice QCD [37, 38, 39]. Also, Abelian dominance of
chiral symmetry breaking is observed in SU(2) [29, 30, 31] and SU(3) lattice
QCD [33].

##  III Magnetic Objects in Abelian projected QCD

In this section, we prepare magnetic objects in Abelian projected QCD in four-
dimensional Euclidean space-time.

###  III.1 Monopoles in Abelian projected QCD

In this subsection, we define monopoles in Abelian projected QCD in lattice
formalism [40]. Like the ordinary SU(3) plaquette, the Abelian plaquette
variable is defined as

$\displaystyle u_{\mu\nu}(s)$ $\displaystyle\equiv$ $\displaystyle
u_{\mu}(s)u_{\nu}(s+\hat{\mu})u_{\mu}^{\dagger}(s+\hat{\nu})u_{\nu}^{\dagger}(s)=e^{i\vec{\theta}_{\mu\nu}(s)\cdot\vec{H}}$
(4) $\displaystyle=$ $\displaystyle{\rm
diag}(e^{i\theta_{\mu\nu}^{1}(s)},e^{i\theta_{\mu\nu}^{2}(s)},e^{i\theta_{\mu\nu}^{3}(s)})\in{\rm
U(1)}^{2}\subset{\rm SU}(3),$ (5)

where $\hat{\mu}$ is the $\mu$-directed unit vector in the lattice unit. The
Abelian field strength $\theta_{\mu\nu}^{i}(s)$ ($i=1,2,3$) is the principal
value of the exponent in $u_{\mu\nu}(s)$, and is defined as

$\displaystyle\partial_{\mu}\theta_{\nu}^{i}(s)-\partial_{\nu}\theta_{\mu}^{i}(s)$
$\displaystyle=\theta_{\mu\nu}^{i}(s)-2\pi n_{\mu\nu}^{i}(s),$
$\displaystyle-\pi\leq\theta_{\mu\nu}^{i}(s)$ $\displaystyle<\pi,\quad
n_{\mu\nu}^{i}(s)\in\mathbb{Z},$ (6)

with the forward derivative $\partial_{\mu}$. Note that
$\theta_{\mu\nu}^{i}(s)$ is ${\rm U(1)}^{2}$ gauge invariant and corresponds
to the regular Abelian field strength in the continuum limit of $a\rightarrow
0$, while $n_{\mu\nu}^{i}(s)$ corresponds to the singular gauge-variant Dirac
string [40].

The electric current $j_{\mu}^{i}$ and the monopole current $k_{\mu}^{i}$ are
defined from the Abelian field strength $\theta_{\mu\nu}^{i}$ as

$\displaystyle j_{\nu}^{i}(s)$ $\displaystyle\equiv$
$\displaystyle\partial_{\mu}^{\prime}\theta_{\mu\nu}^{i}(s),$ (7)
$\displaystyle k_{\nu}^{i}(s)$ $\displaystyle\equiv$
$\displaystyle\partial_{\mu}\tilde{\theta}_{\mu\nu}^{i}(s)/2\pi=\partial_{\mu}\tilde{n}_{\mu\nu}^{i}(s)\in\mathbb{Z},$
(8)

where $\partial_{\mu}^{\prime}$ is the backward derivative and
$\tilde{\theta}_{\mu\nu}$ is the dual tensor of
$\tilde{\theta}_{\mu\nu}\equiv\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}{\theta}_{\alpha\beta}$.
Both electric and monopole currents are $\mathrm{U}(1)^{2}$ gauge invariant,
according to $\mathrm{U}(1)^{2}$ gauge invariance of
${\theta}_{\mu\nu}^{i}(s)$. In the lattice formalism, $k_{\mu}^{i}(s)$ is
located at the dual lattice $L_{\rm dual}^{4}$ of $s^{\alpha}+\frac{1}{2}$
with flowing in $\mu$ direction [41].

In this way, Abelian projected QCD includes both electric current
$j_{\mu}^{i}$ and monopole current $k_{\mu}^{i}$. Remarkably, lattice QCD
shows monopole dominance, i.e., dominant role of monopoles for quark
confinement in the MA gauge [42]. Also, lattice QCD shows monopole dominance
for chiral symmetry breaking, that is, monopoles in the MA gauge crucially
contribute to spontaneous chiral-symmetry breaking in both SU(2) [29, 31] and
SU(3) lattice QCD [33].

In the lattice formalism, the monopole current $k_{\mu}^{i}$ appears on the
dual lattice $L_{\rm dual}^{4}$ of $s^{\alpha}+\frac{1}{2}$, and therefore we
define the local monopole density

$\displaystyle\rho_{\mathrm{L}}(s)\equiv\frac{1}{3\cdot
2^{4}}\sum_{i=1}^{3}\sum_{s^{\prime}\in
P(s)}\sum_{\mu=1}^{4}\left|k_{\mu}^{i}(s^{\prime})\right|,$ (9)

where $P(s)$ denotes the dual lattices in the vicinity of $s$, i.e.,
$P(s)=\left\\{s^{\prime}\in L_{\rm
dual}^{4}\middle||s-s^{\prime}|=1\right\\}$. Note here that the distance
between the site $s$ and its closest dual site $s^{\prime}$ is
$|s-s^{\prime}|=\sqrt{\sum_{1}^{4}(\frac{1}{2})^{2}}=1$ in the four-
dimensional Euclidean space-time.

###  III.2 General argument for magnetic instability and magnetic objects in
the QCD vacuum

In QCD in the MA gauge, color magnetic monopoles generally appear, and play an
important role in nonperturbative properties, which migth looks curious, since
the original QCD action does not have monopoles.

However, some active roles of magnetic objects would be natural in QCD,
because QCD itself has color magnetic instability, and spontaneous generation
of color magnetic fields generally takes place, as Savvidy first pointed out
in 1977 [43, 44].

In fact, in the QCD vacuum in the Minkowski space-time, the gluon condensate
$\langle G_{\mu\nu}^{a}G_{\mu\nu}^{a}\rangle$ takes a large positive value,
which physically means that the QCD vacuum is filled with color magnetic
fields. Since the gluon condensate is expressed with color magnetic fields
$\vec{H}_{a}$ and color electric fields $\vec{E}_{a}$ as

$\displaystyle\langle
G_{\mu\nu}^{a}G_{\mu\nu}^{a}\rangle=2(\langle\vec{H}_{a}^{2}\rangle-\langle\vec{E}_{a}^{2}\rangle)>0,$
(10)

its large positivity means inevitable significant generation of color magnetic
fields. Thus, some superior role of magnetic objects is expected instead of
electric objects in the Minkowski QCD vacuum.

In the Euclidean space-time, because of the space-time $SO(4)$ symmetry, the
roles of magnetic and electric fields become similar. Actually, the gluon
condensate is written as
$G_{\mu\nu}^{a}G_{\mu\nu}^{a}=2(\vec{H}_{a}^{2}+\vec{E}_{a}^{2})$, where the
electromagnetic duality is manifest. Then, in Euclidean QCD, the electric
field often behaves as a magnetic field, and therefore we regard the Euclidean
electric field as a sort of the magnetic field in this paper.

###  III.3 Lorentz invariant quantities in Abelian projected QCD

In Abelian projected QCD, there are two Lorentz invariant quantities ${\cal
F}$ and ${\cal G}$ in the Euclidean space-time:

$\displaystyle{\cal F}$
$\displaystyle\equiv\frac{1}{3}\sum_{i=1}^{3}\frac{1}{4}F^{\mu\nu}_{i}F^{\mu\nu}_{i}=\frac{1}{3}\sum_{i=1}^{3}\frac{1}{2}(\vec{H}_{i}^{2}+\vec{E}_{i}^{2}),$
(11) $\displaystyle{\cal G}$
$\displaystyle\equiv\frac{1}{3}\sum_{i=1}^{3}\frac{1}{4}F^{\mu\nu}_{i}\tilde{F}^{\mu\nu}_{i}=\frac{1}{3}\sum_{i=1}^{3}\
\vec{H}_{i}\cdot\vec{E}_{i},$ (12)

with the color magnetic field
$(\vec{H}_{i})_{j}\equiv\frac{1}{2}\epsilon_{jkl}F_{i}^{kl}$ and the color
electric field $(\vec{E}_{i})_{j}\equiv F_{i}^{j4}$. These quantities are also
invariant under the residual U(1)2 gauge transformation and global Weyl
transformation [34], i.e., permutation of the color index, in the MA gauge.

Here, ${\cal F}$ is parity-even and expresses total magnitude of magnetic
fields in the Euclidean space-time, since the electric field behaves as a
magnetic field there. In this paper, we simply call ${\cal F}$ “magnetic
quantity” in Euclidean gauge theories. Note that ${\cal G}$ is parity-odd and
is just the Abelian projected quantity of the topological charge density on
instantons in QCD, which might relate to chiral symmetry breaking.

In the lattice formalism, the field strength tensor is a plaquette variable
spanning at $s$, $s+\hat{\mu}$, $s+\hat{\nu}$, and $s+\hat{\mu}+\hat{\nu}$, so
that we define the Abelian field strength $F_{\mu\nu}^{i}(s)$ as the local
average of clover-type four plaquettes,

$\displaystyle
a^{2}gF_{\mu\nu}^{i}(s)\equiv\frac{1}{4}\left(\theta^{i}_{\mu\nu}(s)+\theta^{i}_{\mu\nu}(s+\hat{\mu})+\theta^{i}_{\mu\nu}(s+\hat{\nu})+\theta^{i}_{\mu\nu}(s+\hat{\mu}+\hat{\nu})\right),$
(13)

and consider ${\cal F}$ and ${\cal G}$ as local quantities in each Abelian
gauge configuration.

##  IV Local chiral condensate and chiral symmetry breaking in gauge theories

In this section, we consider the chiral condensate and chiral symmetry
breaking in the gauge theory in terms of the quark propagator.

###  IV.1 Local chiral condensate in lattice QCD

In this subsection, we briefly review the local chiral condensate in lattice
formalism. The local chiral condensate can be calculated with the quark
propagator for each gauge configuration $U=\\{U_{\mu}(s)\\}$ generated with
the Monte Carlo method.

As the lattice fermion, we here adopt the Kogut-Susskind (KS) fermion [3]. For
the KS fermion, the Dirac operator $\gamma_{\mu}D_{\mu}$ is expressed by
$\eta_{\mu}D_{\mu}$ with the staggered phase
$\eta_{\mu}(s)\equiv(-1)^{s_{1}+\cdots+s_{\mu-1}}$ $(\mu\geq 2)$ with
$\eta_{1}(s)\equiv 1$. The KS Dirac operator is expressed as

$\displaystyle(\eta_{\mu}D_{\mu})_{ss^{\prime}}=\frac{1}{2}\sum_{\mu=1}^{4}\sum_{\pm}\pm\eta_{\mu}(s)U_{\pm\mu}(s)\delta_{s\pm\hat{\mu},s^{\prime}}$
(14)

with $U_{-\mu}(s)\equiv U^{\dagger}_{\mu}(s-\hat{\mu})$, and the KS Dirac
eigenvalue equation takes the form of

$\displaystyle\frac{1}{2}\sum_{\mu=1}^{4}\sum_{\pm}\pm\eta_{\mu}(s)U_{\pm\mu}(s)\chi_{n}(s\pm\hat{\mu})=i\lambda_{n}\chi_{n}(s).$
(15)

Here, the quark field $q^{\alpha}(s)$ is described by a spinless Grassmann
variable $\chi(s)$ [3], and the chiral condensate per flavor is evaluated as
$\langle\bar{q}q\rangle=\langle\bar{\chi}\chi\rangle$/4 in the continuum
limit.

The local chiral condensate can be calculated using the quark propagator of
the KS fermion with a small quark mass $m$. The chiral-limit value is
estimated by the chiral extrapolation of $m\rightarrow 0$. As a technical
caution, the chiral and continuum limits do not commute for the KS fermion at
the quenched level, although this problem would be absent in full QCD [45].

For the gauge configuration $U=\\{U_{\mu}(s)\\}$, the Euclidean KS fermion
propagator is given by

$\displaystyle
G_{U}^{ij}(x,y)\equiv\langle{\chi}^{i}(x)\bar{\chi}^{j}(y)\rangle_{U}=\langle
x,i|\left(\frac{1}{\eta_{\mu}D_{\mu}[U]+m}\right)|y,j\rangle$ (16)

with the color index $i$ and $j$. This propagator is numerically obtained by
solving the large-scale linear equation with a point source. The local chiral
condensate for the gauge configuration $\\{U_{\mu}(s)\\}$ is expressed with
the propagator as

$\displaystyle\langle\bar{\chi}(x)\chi(x)\rangle_{U}=-{\rm Tr}\,G_{U}(x,x).$
(17)

Here, we consider the net chiral condensate by subtracting the contribution
from the trivial vacuum $U=1$ as

$\displaystyle\langle\bar{\chi}\chi(x)\rangle_{U}\equiv\langle\bar{\chi}(x)\chi(x)\rangle_{U}-\langle\bar{\chi}\chi\rangle_{U=1},$
(18)

where the subtraction term is exactly zero in the chiral limit $m=0$. The
global chiral condensate is obtained by taking its average over the space-time
$x$ and the gauge ensembles $U_{1},U_{2},...,U_{N}$,

$\displaystyle\langle\bar{\chi}\chi\rangle\equiv\sum_{x,i}\langle\bar{\chi}\chi(x)\rangle_{U_{i}}/\sum_{x,i}1.$
(19)

###  IV.2 Chiral symmetry breaking in Abelian gauge system

In this subsection, we analytically investigate relation between chiral
symmetry breaking and the field strength in Euclidean Abelian gauge systems.
For the simple argument, we consider Euclidean U(1) gauge systems with quasi-
massless Dirac fermions coupled to the U(1) gauge field, although it is
straightforward to generalize this argument to Abelian projected QCD with
${\rm U}(1)^{2}$ gauge symmetry.

In the U(1) gauge system, the chiral condensate is proportional to the
functional trace of the fermion propagator,

$\displaystyle I\equiv{\rm Tr}\frac{1}{\not D+m}=-m{\rm
Tr}\frac{1}{D^{2}-m^{2}+\frac{g}{2}\sigma\cdot F},$ (20)

with the covariant derivative $D_{\mu}\equiv\partial_{\mu}+igA_{\mu}$, the
field strength $F_{\mu\nu}$, $\sigma\cdot F\equiv\sigma_{\mu\nu}F_{\mu\nu}$
and $\sigma_{\mu\nu}\equiv\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$. Note that
$D^{2}-m^{2}$ is a negative-definite operator, and all of its eigenvalues are
negative. Since the trace of any odd-number product of $\gamma$-matrices is
zero, we find

$\displaystyle I=-m{\rm Tr}\frac{D^{2}-m^{2}-\frac{g}{2}\sigma\cdot
F}{(D^{2}-m^{2})^{2}-2g^{2}({\cal F}-\gamma_{5}{\cal
G})-\frac{g}{2}[D^{2},\sigma\cdot F]},$ (21)

with

$\displaystyle{\cal
F}\equiv\frac{1}{4}F_{\mu\nu}F_{\mu\nu}=\frac{1}{2}(\vec{H}^{2}+\vec{E}^{2}),\quad{\cal
G}\equiv\frac{1}{4}F_{\mu\nu}\tilde{F}_{\mu\nu}=\vec{H}\cdot\vec{E}.$ (22)

Because of the overall factor $m$ in $I$, $I$ goes to zero in the chiral limit
$m\rightarrow 0$, unless the denominator becomes zero in this limit.

Since the operator $(D^{2}-m^{2})^{2}$ in the denominator is positive
definite, to realize the zero denominator in $I$ in Eq. (21), we need a
significant negative contribution from the other three terms including ${\cal
F}$, ${\cal G}$, or $[D^{2},\sigma\cdot F]$. For instance, in the absence of
the field strength, i.e., $F_{\mu\nu}\equiv 0$, one finds near $m\simeq 0$

$\displaystyle I_{F_{\mu\nu}\equiv 0}=m{\rm
Tr}\frac{1}{p^{2}+m^{2}}=m\gamma\int\frac{d^{4}p}{(2\pi)^{4}}\frac{1}{p^{2}+m^{2}}=m\frac{\gamma}{16\pi^{2}}\int^{\Lambda^{2}}dp^{2}\frac{p^{2}}{p^{2}+m^{2}}\simeq
m\frac{\gamma\Lambda^{2}}{16\pi^{2}}$ (23)

with the UV cutoff $\Lambda$ and the degeneracy $\gamma$. According to the
positive denominator in the integrand, $I_{F_{\mu\nu}\equiv 0}$ has no IR
singularity to cancel $m$ of the numerator, and therefore $I_{F_{\mu\nu}\equiv
0}$ goes to zero in the chiral limit of $m\rightarrow 0$.

To cancel $m$ in the numerator of $I$, we need a significantly large amount of
the field strength so as to present zero mode in the denominator of $I$ and to
keep $I$ non-zero in the chiral limit. Note here that ${\cal F}(\geq 0)$
always gives a negative (non-positive) contribution in the denominator of $I$,
while the contribution from ${\cal G}$ or $[D^{2},\sigma\cdot F]$ can be
positive and negative. In fact, the magnetic quantity ${\cal F}$ can give the
zero mode in the denominator of $I$, even without the contribution from ${\cal
G}$ and $[D^{2},\sigma\cdot F]$. In contrast, in Euclidean Abelian gauge
systems, ${\cal F}\equiv\frac{1}{4}F_{\mu\nu}^{2}=0$ means $F_{\mu\nu}=0$, and
then ${\cal G}=[D^{2},\sigma\cdot F]=0$.

To conclude, the magnetic quantity ${\cal F}$ is expected to be significantly
important to realize chiral symmetry breaking in Euclidean Abelian gauge
theories, although, in some cases, the contribution from ${\cal G}$ and
$[D^{2},\sigma\cdot F]$ can assist the realization of chiral symmetry
breaking.

In a special case of constant $F_{\mu\nu}$, one finds $[D^{2},\sigma\cdot
F]=0$ for the Abelian system, and obtains

$\displaystyle I=-m{\rm Tr}\frac{(D^{2}-m^{2})[(D^{2}-m^{2})^{2}-2g^{2}{\cal
F}]}{[(D^{2}-m^{2})^{2}-2g^{2}{\cal F}]^{2}-4g^{2}{\cal G}^{2}},$ (24)

because of ${\rm tr}\gamma_{5}={\rm tr}\sigma_{\mu\nu}={\rm
tr}\gamma_{5}\sigma_{\mu\nu}=0$. For more special case of a constant magnetic
field, there occurs the Landau-level quantization, and the spatial degrees of
freedom perpendicular to the magnetic field is frozen in the lowest Landau
level. This infrared effective low-dimensionalization of the charged spinor
dynamics induces chiral symmetry breaking in the chiral limit [46, 47, 48],
which is known as magnetic catalysis [49].

##  V Abelian gauge system with a static monopole-antimonopole pair on a
lattice

In the QCD vacuum, complicated monopole world-lines generally emerge in the MA
gauge [21, 25], and therefore it is difficult to clarify the primary
correlation with the chiral condensate among the magnetic objects such as
monopoles, ${\cal F}$ and ${\cal G}$.

In this section, to seek for the primary correlation with the chiral
condensate, we create idealized Abelian gauge system with a monopole-
antimonopole pair on a lattice, and investigate the relation among the local
chiral condensate, monoples, and magnetic fields. Also, we consider a magnetic
flux system without monopoles.

For simplicity, we here consider U(1) lattice gauge systems described by U(1)
link variables

$\displaystyle u_{\mu}(s)=e^{i\theta_{\mu}(s)}\in{\rm U}(1),$ (25)

and quasi-massless Dirac fermions coupled to U(1) gauge fields with the
coupling $g=1$.

###  V.1 Static monopole-antimonopole pair systems

To begin with, we deal with an idealized Abelian gauge system of a static
monopole-antimonopole pair on a periodic lattice of the four-dimensional
Euclidean space-time.

In the three-dimensional space ${\bf R}^{3}$, let us consider a static
monopole-antimonopole pair with the distance of $l$ in $z$-direction. To
realize such a lattice gauge system, we set the Abelian link-variable
$u_{\mu}(s)$ to be

$\displaystyle
u_{x}(s)=u_{y}(s+\hat{x})=u_{x}^{\dagger}(s+\hat{y})=u_{y}^{\dagger}(s)=i\quad\hbox{for}\quad
s_{x}=s_{y}=0,~{}~{}1\leq s_{z}\leq l,$ (26)

otherwise $u_{\mu}(s)=1$.

Figure 1 shows the building-block plaquette to realize a static monopole-
antimonopole pair on the lattice. Here, only the red link-variables take a
nontrivial value of $i$.

As for the phase variable $\theta_{\mu}(s)$, which corresponds to the Abelian
gluon, one finds

$\displaystyle\theta_{x}(s)=\theta_{y}(s+\hat{x})=-\theta_{x}(s+\hat{y})=-\theta_{y}(s)=\frac{\pi}{2}\quad\hbox{for}\quad
s_{x}=s_{y}=0,~{}~{}1\leq s_{z}\leq l,$ (27)

otherwise $\theta_{\mu}(s)=0$. For the all-red plaqutte with $s_{x}=s_{y}=0$
and $1\leq s_{z}\leq l$, one gets

$\displaystyle\partial_{x}\theta_{y}(s)-\partial_{y}\theta_{x}(s)=\theta_{x}(s)+\theta_{y}(s+\hat{x})-\theta_{x}(s+\hat{y})-\theta_{y}(s)=2\pi,$
(28)

which leads to the Dirac string of $n_{xy}(s)=-1$ and zero field strength
$\theta_{xy}(s)=0$, because of the definition of the field strength
$\theta_{\mu\nu}(s)$ and the Dirac string $n_{\mu\nu}(s)$,

$\displaystyle\partial_{\mu}\theta_{\nu}(s)-\partial_{\nu}\theta_{\mu}(s)=\theta_{\mu\nu}(s)-2\pi
n_{\mu\nu}(s),\quad-\pi\leq\theta_{\mu\nu}(s)<\pi,\quad
n_{\mu\nu}(s)\in\mathbb{Z}.$ (29)

Thus, the all-red plaquette induces the singular Dirac string at its center on
the dual lattice. In fact, for the idealized system in Fig. 1(b), a Dirac
string appears inside the all-red plaquette.

Figure 1: The building-block plaquette to realize a static monopole-
antimonopole pair on the lattice in (a) the $x$-$y$ plane and (b) spatial
${\bf R}^{3}$ for $s=(0,0,s_{z},s_{t})$ with $1\leq s_{z}\leq l$. Only the red
link-variables take a nontrivial value of $i$. The all-red plaquette induces
the singular Dirac string at its center on the dual lattice. A physical
magnetic field is also created in the neighboring plaquette $u_{xy}(s)$
including only one red link.

At the terminal of the Dirac string, a monopole or an anti-monopole appears on
the dual lattice, as shown in Fig. 2.

Figure 2: The link-variables to realize a static monopole-antimonopole pair on
the lattice in spatial ${\bf R}^{3}$. Only the red link-variables take a
nontrivial value of $i$. (a) The cube including only one all-red plaquette
induces a magnetic monopole at its inside on the dual lattice. (b) A monopole
(black diagmond) and an anti-monopole (white diamond) appear at the two
terminals of the red plaquette tower.

Actually, the three-dimensional spatial cube including only one all-red
plauette has a static (anti)monopole at its center (on the dual lattice),
because only one $n_{kl}(s)$ has nonzero value of $\pm 1$ among the six
independent plaquettes composing the cube,

$\displaystyle k_{4}(s)$ $\displaystyle=$
$\displaystyle\partial_{j}\tilde{n}_{j4}(s)=\frac{1}{2}\epsilon_{jkl}\partial_{j}n_{kl}(s)=\frac{1}{2}\epsilon_{jkl}\\{n_{kl}(s+\hat{j})-n_{kl}(s)\\}$
(30) $\displaystyle=$ $\displaystyle
n_{xy}(s+\hat{z})-n_{xy}(s)+n_{yz}(s+\hat{x})-n_{yz}(s)+n_{zx}(s+\hat{y})-n_{zx}(s)$
(31) $\displaystyle=$ $\displaystyle\pm 1.$ (32)

Thus, this idealized system includes a static monopole at
$(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ and a static anti-monopole at
$(\frac{1}{2},\frac{1}{2},l+\frac{1}{2})$ in spatial ${\bf R}^{3}$.

This monopole and anti-monopole system has also physical magnetic flux around
the line segment connecting the monopole pair. In fact, a physical magnetic
field is created in the neighboring plaquette $u_{xy}(s)$ of the all-red
plaquette in Fig. 1. In this idealized system, only the plaquette $u_{xy}(s)$
including one red link takes a nontrivial value as

$\displaystyle u_{xy}(s)=-i=e^{-i\pi/2}\quad\hbox{in case with one nontrivial
link},$ (33)

otherwise $u_{\mu\nu}(s)=1$. Note here that, by gauge transformation, the
location of the Dirac string is generally changed, but the physical field
strength is never changed.

Figure 3 shows the local chiral condensate
$\langle\bar{\chi}\chi(s)\rangle_{u}$ and the magnetic quantity ${\cal
F}\equiv\frac{1}{4}F_{\mu\nu}^{2}=\frac{1}{2}\vec{H}^{2}$ for $l=4$ in the
three dimensional space ${\bf R}^{3}$. In this demonstration, the quark mass
is taken to be $m=0.01$ in the lattice unit.

Figure 3:  An idealized Abelian gauge system of a static monopole and anti-
monopole pair with the distance of $l=4$. In the three dimensional space ${\bf
R}^{3}$, the value is visualized with the color graduation for (a) the local
chiral condensate and (b) the magnetic quantity ${\cal F}$.

For this idealized static system, there actually appears a magnetic field
$\vec{H}$, i.e., nonzero flux of ${\cal F}=\frac{1}{2}\vec{H}^{2}>0$, in space
between the monopole and the anti-monopole, and the local chiral condensate
takes a significant value in the vicinity of the magnetic field. In contrast,
one finds ${\cal G}=\vec{H}\cdot\vec{E}=0$ everywhere, since only spatial
plaquettes take a nontrivial value and $\vec{E}=\vec{0}$. Thus, in this
system, it is likely that the magnetic field stemming from monopoles has the
primary correlation with the local chiral condensate.

###  V.2 Static magnetic flux system

Next, let us investigate a static magnetic flux system without monopoles.
Owing to the spatial periodicity, the special case of $l=L_{z}$ in the static
monopole-antimonopole system has no (anti)monopoles, because of the magnetic-
charge cancellation. In this special case of $l=L_{z}$, there only exists a
physical static magnetic flux along $z$-direction.

Figure 4 shows the local chiral condensate
$\langle\bar{\chi}\chi(s)\rangle_{u}$ and the magnetic quantity ${\cal
F}\equiv\frac{1}{4}F_{\mu\nu}^{2}=\frac{1}{2}\vec{H}^{2}$ for $l=L_{z}$ in
spatial ${\bf R}^{3}$, taking the quark mass of $m=0.01$ in the lattice unit.

Figure 4:  An idealized Abelian gauge system of a static magnetic flux without
monopoles. In the three dimensional space ${\bf R}^{3}$, the value is
visualized with the color graduation for (a) the local chiral condensate and
(b) the magnetic quantity ${\cal F}$.

Again, the local chiral condensate takes a significant value in the vicinity
of the magnetic field $\vec{H}$, i.e., nonzero flux of ${\cal
F}=\frac{1}{2}\vec{H}^{2}>0$, even without (anti)monopoles. Note also that
this system has ${\cal G}=\vec{H}\cdot\vec{E}=0$ everywhere, because of
$\vec{E}=\vec{0}$. Therefore, in this idealized system, we conclude that the
magnetic field or ${\cal F}$ has the primary correlation with the local chiral
condensate.

##  VI Lattice QCD study for local chiral condensate, monopoles, and magnetic
fields

In our previous study with lattice QCD, we observed a strong correlation
between the local chiral condensate and monopoles in Abelian projected QCD
[33]. As a possible reason of this correlation, we conjectured that the strong
magnetic field around monopoles is responsible to chiral symmetry breaking in
QCD, similarly to the magnetic catalysis [46, 47, 48, 49].

In this section, using lattice QCD Monte Carlo simulations, we investigate the
relation among the local chiral condensate, monopoles, and magnetic fields in
Abelian projected QCD. In this paper, the SU(3) lattice QCD simulation is
performed using the standard plaquette action at the quenched level. In each
space-time direction, we impose the periodic boundary condition for link
variables, and the anti-periodic boundary condition for quarks in order to
describe also thermal QCD.

For the numerical Monte Carlo calculation, we basically adopt the lattice
parameter of $\beta\equiv 2N_{c}/g^{2}=6.0$ and the size $V=24^{4}$. The
lattice spacing $a\simeq$ 0.1 fm is obtained from the string tension
$\sigma=0.89$ GeV/fm [37]. Also, we adopt $\beta=6.0$ and $V=24^{3}\times 6$
for the high-temperature deconfined phase at $T\simeq$ 330 MeV above the
critical temperature.

Using the pseudo-heat-bath algorithm, we generate 100 and 200 gauge
configurations for $V=24^{4}$ and $24^{3}\times 6$, respectively. All the
gauge configurations are taken every 500 sweeps after thermalization of 5,000
sweeps. MA gauge fixing is performed with the stopping criterion that the
deviation $\Delta R_{\mathrm{MA}}/\left(4V\right)$ becomes smaller than
$10^{-5}$ in 100 iterations. For the calculation of the local chiral
condensate, we use the quark propagator of the KS fermion with the quark mass
of $m=0.01,0.015,0.02$ in the lattice unit, Here, the quark mass is taken to
be finite, since the chiral and continuum limits do not commute for the KS
fermion at the quenched level [45]. The jackknife method is used for
statistical error estimates.

For each lattice gauge configuration of Abelian projected QCD in the MA gauge,
we calculate the local monopole density $\rho_{\rm L}(s)$, the local chiral
condensate, and the Lorentz invariants ${\cal F}$ and ${\cal G}$, defined in
Section III

###  VI.1 Distribution similarity between local chiral condensate and magnetic
variables

To begin with, we pick up a gauge configuration generated in lattice QCD on
$V=24^{4}$ at $\beta=6.0$, and investigate correlation between the local
chiral condensate and magnetic variables.

Figure 5 shows the local chiral condensate
$\langle\bar{\chi}\chi(s)\rangle_{u}$ with the quark mass of $m=0.02$, the
local monopole density $\rho_{\rm L}(s)$, and the Lorentz invariants ${\cal
F}(s)$ and $|{\cal G}(s)|$, respectively, as well as the monopole location in
the space ${\bf R}^{3}$ at a time slice, for a typical gauge configuration of
Abelian projected QCD.

Figure 5: Lattice QCD results for (a) the local chiral condensate
$\langle\bar{\chi}\chi(s)\rangle_{u}$ with the quark mass of $m=0.02$, (b) the
local monopole density $\rho_{\rm L}(s)$, and the Lorentz invariants (c)
${\cal F}(s)$ and (d) $|{\cal G}(s)|$ in spatial ${\bf R}^{3}$ at a time
slice, for a typical gauge configuration of Abelian projected QCD. The value
is visualized with the color graduation. Monopoles at $t$ = 11.5 and 12.5 are
plotted with upper and lower triangles, respectively.

From Fig. 5(a), one finds that the local chiral condensate tends to take a
large value near the monopole location [33]. Since monopoles appear on the
dual lattice, we show the local monopole density $\rho_{\rm L}(s)$, as the
average on closest dual sites. Of course, $\rho_{\rm L}(s)$ takes a large
value near the monopole. The distribution of the the local monopole density
resembles that of the local chiral condensate, as was pointed out in Ref.
[33]. Figures 5(c) and (d) show the Lorentz invariants ${\cal F}$ and ${\cal
G}$, respectively. As a new result in this paper, we find that the
distributions of ${\cal F}$ and ${\cal G}$ also resemble that of the local
chiral condensate.

The close relation of monopoles with ${\cal F}$ and ${\cal G}$ might be
understood, since the field strength tensor relates to monopoles as
$\partial_{\mu}\tilde{F}_{\mu\nu}^{i}=k_{\nu}^{i}$. Roughly speaking, the
monopole can be a kind of source of ${\cal F}$ and ${\cal G}$. In contrast,
their similarity with the local chiral condensate is fairly nontrivial.

In any case, we find clear correlation of distribution similarity among the
local chiral condensate, the local monopole density, and the Lorentz
invariants ${\cal F}$ and ${\cal G}$ in Abelian projected QCD in the MA gauge.

###  VI.2 Correlation coefficients between local chiral condensate and
magnetic variables

In this subsection, we quantify the similarity between the local chiral
condensate $\langle\bar{\chi}\chi(s)\rangle_{u}$ and magnetic variables, i.e.,
$\rho_{\rm L}(s)$, ${\cal F}(s)$ and ${\cal G}(s)$, defined in Section III. To
this end, we use all the generated 100 gauge configurations in lattice QCD on
$V=24^{4}$ at $\beta=6.0$, and calculate the local chiral condensate at
$2^{4}$ distant space-time points for each gauge configuration, resulting 1600
data points at each quark mass.

Figure 6 shows the scatter plot between the local chiral condensate
$\langle\bar{\chi}\chi(s)\rangle_{u}$ and magnetic variables, i.e., the local
monopole density $\rho_{\rm L}(s)$, Lorentz invariants ${\cal F}(s)$ and
$|{\cal G}(s)|$, respectively, using 100 gauge configurations of Abelian
projected QCD in the MA gauge, with the quark mass of $m=0.01,0.015,0.02$ in
the lattice unit. In Fig. 6, positive correlation is qualitatively found
between the local chiral condensate and the magnetic variables, $\rho_{\rm
L}$, ${\cal F}$ and $|{\cal G}|$, respectively.

Figure 6: The scatter plot between the local chiral condensate
$\langle\bar{\chi}\chi(s)\rangle_{u}$ and (a) the local monopole density
$\rho_{\rm L}(s)$, (b) ${\cal F}(s)$ and (c) $|{\cal G}(s)|$, using 100
Abelianized gauge configurations in SU(3) lattice QCD with $\beta=6.0$ and
$V=24^{4}$ at each quark mass $m$.

Next, we consider a quantitative analysis using correlation coefficients
between the local chiral condensate and the magnetic variables, as a
statistical indicator of correlation. In general, for arbitrary two
statistical ensembles $\\{A_{i}\\}$ and $\\{B_{i}\\}$, their correlation
coefficient $r$ is defined as

$\displaystyle r\equiv\frac{\langle(A-\langle A\rangle)(B-\langle
B\rangle)\rangle}{\sigma_{A}\sigma_{B}},$ (34)

using the average notation $\langle~{}~{}\rangle$ and the standard deviation
$\sigma_{A}\equiv\sqrt{\langle(A-\langle A\rangle)^{2}\rangle}$ and
$\sigma_{B}\equiv\sqrt{\langle(B-\langle B\rangle)^{2}\rangle}$. Here, $r=1$
means perfect positive linear correlation, and $r\gtrsim 0.7$ indicates strong
positive linear correlation.

We measure correlation coefficients between the local chiral condensate
$\left|\langle{\bar{\chi}\chi}(s)\rangle_{u}\right|$ and three magnetic
variables, $\rho_{\mathrm{L}}(s)^{\alpha}$, $\mathcal{F}(s)^{\alpha}$ and
$\left|\mathcal{G}(s)\right|^{\alpha}$, at various exponent $\alpha$, using
100 gauge configurations of Abelian projected QCD in SU(3) lattice QCD at
$\beta=6.0$ on $V=24^{4}$, for the quark mass $m=0.01$ in the lattice unit.
Table 1 shows the result for the correlation coefficients.

Lattice | $\alpha$ | $\rho_{\mathrm{L}}^{\alpha}$ | $\mathcal{F}^{\alpha}$ | $\left|\mathcal{G}\right|^{\alpha}$
---|---|---|---|---
$V=24^{4},\,\beta=6.0$ | $0.25$ | $0.47$ | $0.63$ | $0.60$
| $0.5$ | $0.55$ | $0.71$ | $0.67$
| $1$ | $0.62$ | $0.79$ | $0.67$
| $1.5$ | $0.63$ | $0.81$ | $0.60$
| $2$ | $0.60$ | $0.80$ | $0.55$
Table 1:  Correlation coefficients between the local chiral condensate
$\left|\langle{\bar{\chi}\chi}(s)\rangle_{u}\right|$ and three magnetic
variables, $\rho_{\mathrm{L}}(s)^{\alpha}$, $\mathcal{F}(s)^{\alpha}$ and
$\left|\mathcal{G}(s)\right|^{\alpha}$ at various $\alpha$, using 100 gauge
configurations of Abelian projected QCD in SU(3) lattice QCD at $\beta=6.0$ on
$V=24^{4}$, for the quark mass $m=0.01$ in the lattice unit.

Quantitatively, the magnetic quantity ${\cal F}$ has the strongest correlation
with the chiral condensate rather than $\rho_{\rm L}$ and ${\cal G}$. As a
conclusion of this paper, we find a strong positive correlation of $r\simeq
0.8$ between the local chiral condensate
$\left|\langle{\bar{\chi}\chi}(s)\rangle_{u}\right|$ and the magnetic quantity
${\cal F}(s)$ in the confined vacuum of Abelian projected QCD.

###  VI.3 High-temperature deconfined phase

Finally, we investigate also a high-temperature deconfined phase in lattice
QCD on $V=24^{3}\times 6$ at $\beta=6.0$, where the temperature is $T\simeq$
330 MeV above the critical temperature. We generate 200 gauge configurations,
and calculate the local chiral condensate at $2^{3}$ distant space points at a
time slice for each gauge configuration, resulting 1600 data points at each
quark mass.

Figure 7 shows the scatter plot between the local chiral condensate
$\left|\langle{\bar{\chi}\chi}(s)\rangle_{u}\right|$ and magnetic variables,
i.e., the local monopole density $\rho_{\rm L}(s)$, and Lorentz invariants
${\cal F}(s)$ and $|{\cal G}(s)|$, respectively, using 200 gauge
configurations of Abelian projected QCD in the MA gauge, with the quark mass
of $m=0.01,0.015,0.02$ in the lattice unit.

Figure 7:  Result of the high-temperature deconfined phase for the scatter
plot between the local chiral condensate $\langle\bar{\chi}\chi(s)\rangle_{u}$
and (a) $\rho_{\rm L}(s)$, (b) ${\cal F}(s)$ and (c) $|{\cal G}(s)|$, using
200 Abelianized gauge configurations in SU(3) lattice QCD with $\beta=6.0$ and
$V=24^{3}\times 6$ at each quark mass $m$.

We show in Table 2 correlation coefficients between the local chiral
condensate $\left|\langle{\bar{\chi}\chi}(s)\rangle_{u}\right|$ and three
magnetic variables, $\rho_{\mathrm{L}}(s)^{\alpha}$, $\mathcal{F}(s)^{\alpha}$
and $\left|\mathcal{G}(s)\right|^{\alpha}$, at various exponent $\alpha$,
using 200 gauge configurations of Abelian projected QCD in SU(3) lattice QCD
at $\beta=6.0$ on $V=24^{3}\times 6$, for the quark mass $m=0.01$ in the
lattice unit.

Lattice | $\alpha$ | $\rho_{\mathrm{L}}^{\alpha}$ | $\mathcal{F}^{\alpha}$ | $\left|\mathcal{G}\right|^{\alpha}$
---|---|---|---|---
$V=24^{3}\times 6,\,\beta=6.0$ | $0.25$ | $0.37$ | $0.49$ | $0.49$
| $0.5$ | $0.43$ | $0.55$ | $0.55$
| $1$ | $0.49$ | $0.55$ | $0.52$
| $1.5$ | $0.50$ | $0.51$ | $0.45$
| $2$ | $0.49$ | $0.46$ | $0.38$
Table 2:  Correlation coefficients in the deconfined phase between the local
chiral condensate $\left|\langle{\bar{\chi}\chi}(s)\rangle_{u}\right|$ and
three magnetic variables, $\rho_{\mathrm{L}}(s)^{\alpha}$,
$\mathcal{F}(s)^{\alpha}$ and $\left|\mathcal{G}(s)\right|^{\alpha}$ at
various $\alpha$ in Abelian projected QCD of SU(3) lattice QCD at $\beta=6.0$
on $V=24^{3}\times 6$ for $m=0.01$.

From Fig. 7 and Table 2, all the correlations between the local chiral
condensate and the three magnetic variables, $\rho_{\rm L}$, ${\cal F}$ and
${\cal G}$, become weaker in the deconfined phase, where the chiral condensate
itself goes to zero in the chiral limit.

##  VII Summary and Conclusion

We have studied the relation among the local chiral condensate, monopoles, and
magnetic fields, using the lattice gauge theory, as a continuation of
Ref.[33].

First, we have created idealized Abelian gauge systems of 1) a static
monopole-antimonopole pair and 2) a magnetic flux without monopoles, on a
four-dimensional Euclidean lattice. In these systems, we have calculated the
local chiral condensate on quasi-massless fermions coupled to the Abelian
gauge field, and have found that the chiral condensate is localized in the
vicinity of the magnetic field.

Second, performing SU(3) lattice QCD Monte Carlo simulations, we have
investigated Abelian projected QCD in the maximally Abelian gauge, and have
found clear correlation of distribution similarity among the local chiral
condensate, color monopoles, and color magnetic fields in the Abelianized
gauge configuration.

As a statistical indicator, we have measured the correlation coefficient $r$,
and have found a strong positive correlation of $r\simeq 0.8$ between the
local chiral condensate and the Euclidean color-magnetic quantity ${\cal F}$.

We have also examined the local correlation in the deconfined phase of thermal
QCD, and have found that the correlation between the local chiral condensate
and magnetic variables becomes weaker.

Thus, in this paper, we have observed a strong correlation between the local
chiral condensate and magnetic fields in both idealized Abelian gauge systems
and Abelian projected QCD. From these results, we conjecture that the chiral
condensate is locally enhanced by the strong color-magnetic field around the
monopoles in Abelian projected QCD, like magnetic catalysis.

Note however that this correlation does not necessarily mean that chiral
symmetry breaking is caused by the non-uniform magnetic field. To realize
spontaneous chiral-symmetry breaking, as was discussed in subsection IV.2, we
need some zero mode in the denominator of $I$ in the chiral limit. In the
context of the dual superconductor picture, this might be realized by
condensation of monopoles, as was suggested in the dual Ginzburg-Landau theory
[28].

To conclude, once chiral symmetry is spontaneously broken, the local chiral
condensate is expected to have a strong correlation with the color magnetic
field.

###### Acknowledgements.

H.S. is supported in part by the Grants-in-Aid for Scientific Research
[19K03869] from Japan Society for the Promotion of Science. H.O. is supported
by a Grant-in-Aid for JSPS Fellows (Grant No.21J20089). Most of numerical
calculations have been performed on OCTOPUS, at the Cybermedia Center, Osaka
University. We have used PETSc to solve linear equations for the Dirac
operator [50, 51, 52].

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