# Origin of Symmetric Dimer Images of Si(001) Observed in Low-Temperature STM

Xiao-Yan Ren1,2, Hyun-Jung Kim3,4, Chun-Yao Niu1,5, Yu Jia1,5†, Jun-Hyung
Cho3,1,6∗ 1 International Laboratory for Quantum Functional Materials of
Henan, and School of Physics and Engineering, Zhengzhou University, Zhengzhou
450001, China
2 School of Mechanical and Electrical Engineering, Henan Institute of Science
and Technology, Xinxiang 453003, China
3 Department of Physics and Research Institute for Natural Sciences, Hanyang
University, 17 Haengdang-Dong, Seongdong-Ku, Seoul 133-791, Korea
4 Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul
130-722, Korea
5 Center for Advanced Analysis and Computational Science, Zhengzhou
University, Zhengzhou 45001, China
6 International Center for Quantum Design of Functional Materials (ICQD),
HFNL, University of Science and Technology of China, Hefei, Anhui 230026,
China

###### Abstract

It has been a long-standing puzzle why buckled dimers of the Si(001) surface
appeared symmetric below ${\sim}$20 K in scanning tunneling microscopy (STM)
experiments. Although such symmetric dimer images were concluded to be due to
an artifact induced by STM measurements, its underlying mechanism is still
veiled. Here, we demonstrate, based on a first-principles density-functional
theory calculation, that the symmetric dimer images are originated from the
flip-flop motion of buckled dimers, driven by quantum tunneling (QT). It is
revealed that at low temperature the tunneling-induced surface charging with
holes reduces the energy barrier for the flipping of buckled dimers, thereby
giving rise to a sizable QT-driven frequency of the flip-flop motion. However,
such a QT phenomenon becomes marginal in the tunneling-induced surface
charging with electrons. Our findings provide an explanation for low-
temperature STM data that exhibits apparent symmetric (buckled) dimer
structure in the filled-state (empty-state) images.

###### pacs:

73.20.At, 68.35.Ja, 68.47.Fg

Over the last 30 years the atomic and electronic structures of the Si(001)
surface have been extensively investigated because of the fundamental building
block for the fabrication of electronic devices as well as for the
prototypical model system of semiconductor surfaces Dabro ; Chadi ; Pollmann ;
yates ; Medeiros ; wol . From enormous experimental and theoretical studies,
it is well established that the basic reconstruction of Si(001) consists of
the formation of buckled dimers Tromp ; Ram ; Kipp ; Shir . However, at room
temperature scanning tunneling microscopy (STM) experiments showed symmetric
dimer images because of a thermally activated flip-flop motion of buckled
dimers. Such apparent symmetric dimer images disappear below ${\sim}$120 K
Wolkow , forming the $c(4{\times}2)$ or $p(2{\times}2)$ reconstruction
structure [see Fig. 1(a)]. Surprisingly, further cooling below ${\sim}$20 K
causes the buckled dimers to appear symmetric again kondo ; Yokoyama1 . Such
symmetric-dimer STM images at low temperature have been explained in terms of
various origins such as a dynamical flip-flop motion of buckled dimers
Yokoyama1 ; Mitsui , local surface charging effects Ono , a possible
asymmetric $p(2{\times}1)$ reconstruction Perdigao , and a contribution of
bulk states Hata ; Manzano . However, the microscopic mechanism underlying the
low-temperature symmetric dimer images has remained an open question.

Figure 1: (Color on line) (a) Top and side views of the optimized
$c$(4${\times}$2) structure of Si(001). The $c$(4${\times}$2) unit cell is
indicated by the dashed line. The ${\bf x}$ (${\bf y}$) axis is perpendicular
(parallel) to dimer rows, while the ${\bf z}$ axis is along the [001]
direction. For distinction, the Si-dimer atoms are drawn with dark circles. In
(b), the symmetric double-well potential for the flipping of buckled dimers is
schematically drawn. Here, $E_{b}$ denotes the energy barrier, obtained by the
energy difference between the $p$(2${\times}$1) and $c$(4${\times}$2)
structures.

There have so far been a number of low-temperature STM experiments Yokoyama1 ;
Mitsui ; Ono ; Manzano  to characterize the apparent symmetric dimer images.
Yokoyama and Takayanagi Yokoyama1  observed that the symmetric dimer images
measured at 5 K have flicker noise, which was explained by slow dynamical
flip-flop motion of the buckled dimers during the STM scan. Mitsui and
Takayanagi Mitsui  found that at 65 K higher tunneling currents increase not
only the area of symmetric dimer images but also the flip-flop rate of buckled
dimers. Below 10 K, Ono $et$ $al$. Ono  observed both buckled and symmetric
dimer images depending on the polarity of the bias voltage: i.e., the buckled
dimer images, locally forming $c$(4${\times}$2) or $p$(2${\times}$2)
periodicity, were observed with positive bias voltages (empty-state images),
while most of the dimers appear symmetric with negative bias voltages (filled-
state images). Recently, Manzano $et$ $al$. Manzano  reported that at 7 K the
negative bias voltages smaller than $-$1.5 V remained a $c$(4${\times}$2)
reconstruction, but those larger than $-$1.5 V produced symmetric dimer
images. On the basis of existing low-temperature STM data Yokoyama1 ; Mitsui ;
Ono ; Manzano , the following questions on the appearance of symmetric dimer
images can be raised: i.e., Why does the activation barrier ($E_{b}$) for the
flipping of buckled dimers become much reduced at low temperature? What is the
reason why the filled-state and empty-state STM images exhibit symmetric and
buckled dimer structures, respectively? How does the tunneling-induced surface
charging at low temperature Ono ; Bri ; Pol  or the electric field via bias
voltage affect STM imaging to show apparent symmetric dimer structure?

In this Letter, we perform first-principles density-functional theory (DFT)
calculations to investigate the energy difference (equivalently $E_{b}$)
between the symmetric-dimer structure and the $c$(4${\times}$2) structure
under electron or hole doping as well as in the presence of external electric
field applied along the [001] direction. We find that, as the amount of hole
doping increases, $E_{b}$ decreases more dominantly than the case of electron
doping. Compared to such surface charging effects, the application of electric
field is found to give a relatively small change in $E_{b}$. As $E_{b}$
decreases with hole doping, the thermally activated flipping rate of buckled
dimers is still negligible below 10 K, but the quantum tunneling (QT) driven
flip-flop motion can be enabled to produce the symmetric-dimer STM images.
Such a QT phenomenon of buckled dimers is, however, marginal with electron
doping. Thus, a long-standing puzzle about the appearance of symmetric dimer
images in low-temperature STM experiments can be solved in terms of the QT-
driven flip-flop motion of buckled dimers, which can be facilitated by the
tunneling-induced surface charging with holes.

We begin to optimize both the symmetric dimer structure, forming a
$p$(2${\times}$1) periodicity (hereafter, designated as the $p$(2${\times}$1)
structure), and the $c$(4${\times}$2) structure by using the DFT calculation
within the generalized-gradient approximation (GGA) method . The optimized
$c$(4${\times}$2) structure is displayed in Fig. 1(a). We find that the
$c$(4${\times}$2) structure consisting of alternatively buckled dimers along
and perpendicular to the dimer rows has a dimer bond length of $d_{D}$ = 2.357
Å and a dimer buckling angle of ${\theta}$ = 18.0∘. This $c$(4${\times}$2)
structure is found to be more stable than the symmetric-dimer structure by 255
meV per dimer, yielding $E_{b}$ = 255 meV [see Fig. 1(b)]. As shown in Fig.
2(a) and 2(b), the calculated band structure of $p$(2${\times}$1) has a
metallic band crossing the Fermi level $E_{F}$, whereas that of
$c$(4${\times}$2) exhibits a semiconducting feature with a band gap of 0.27
eV. The present results for the geometry, energetics, and band structure of
the $c$(4${\times}$2) structure are in good agreement with those of previous
DFT calculations Ram ; Sei .

Figure 2: (Color on line) Calculated surface band structures of (a) the
$p$(2${\times}$1) and (b) $c$(4${\times}$2) structures. The bands projected
onto the $p_{x}$, $p_{y}$, and $p_{z}$ orbitals of Si-dimer atoms are
displayed with circles whose radii are proportional to the weights of such
orbitals. The energy zero represents $E_{F}$. The inset in (a) shows the
surface Brilloiun zones of the $p$(2${\times}$1) and $c$(4${\times}$2) unit
cells. The total DOS and the local DOS of Si dimers are displayed with solid
and dotted lines, respectively. The charge characters of the ${\pi}$ and
${\pi}^{*}$ surface states at the ${\Gamma}$ point are drawn with an
isosurface of 0.05 $e$/Å, while that of the bulk state of $c$(4${\times}$2) at
the ${\Gamma}$ point (just below $E_{F}$) is drawn with an isosurface of 0.02
$e$/Å.

It has been known that below ${\sim}$40 K electrons or holes, injected through
tunneling current in STM, result in surface charging due to a slow carrier
relaxation between the surface layer and the semiconducting bulk Si Ono ; Bri
; Pol . In order to examine the influence of surface charging on the
energetics of the $p$(2${\times}$1) and $c$(4${\times}$2) structures, we
perform total-energy calculations for the two structures with electron or hole
doping. For the simulation of surface charging, we use the virtual crystal
approximation VCM  to compensate excess electrons $n_{e}$ or holes (whose
amount is represented as a negative value of $n_{e}$). Figure 3 shows the
calculated values of $E_{b}$ as a function of $n_{e}$ ranging from $-$0.6$e$
to 0.6$e$ per $p$(2${\times}$1) unit cell. We find that both the electron and
hole dopings reduce the energy difference between the $p$(2${\times}$1) and
$c$(4${\times}$2) structures. The resulting decrease of $E_{b}$ with electron
or hole doping can be attributed to the metallic and semiconducting features
of the $p$(2${\times}$1) and $c$(4${\times}$2) structures, respectively. As
shown in Fig. 2(a) and 2(b), for instance of $|n_{e}|$ = 0.3$e$, excess
electrons in $p$(2${\times}$1) are filled in the relatively lower unoccupied
electronic states compared to those in $c$(4${\times}$2), whereas holes in
$p$(2${\times}$1) are created in the relatively higher occupied electronic
states compared to those in $c$(4${\times}$2). Therefore, as electron (hole)
doping increases, the total energy of the $p$(2${\times}$1)
[$c$(4${\times}$2)] structure is expected to decrease (increase) more largely
compared to the $c$(4${\times}$2) [$p$(2${\times}$1)] structure, leading to a
decrease of $E_{b}$.

As shown in Fig. 3, $E_{b}$ decreases more significantly with increasing hole
doping, compared to the case of electron doping. This difference between
electron and hole dopings may be due to the different characters of the
unoccupied and occupied electronic states in the $c$(4${\times}$2) structure:
i.e., the lowest unoccupied states are mostly the surface states of
${\pi}^{*}$ orbitals, while the occupied states below $E_{F}$ consist of the
surface states of ${\pi}$ orbitals as well as the bulk states [see the total
density of states (DOS) and the local DOS of Si dimers in Fig. 2(b)]. We note
that, for hole doping with $n_{e}$ = $-$0.3$e$, the majority of the holes in
the $c$(4${\times}$2) structure is created in the bulk states around the
${\Gamma}$ point (see Fig. 1S of the Supplemental Material Supp ), possibly
giving rise to a relatively larger strain energy compared to the
$p$(2${\times}$1) structure where holes are created mostly in the surface
states. This fact may cause a more significant decrease of $E_{b}$ with hole
doping, compared to electron doping where both the $c$(4${\times}$2) and
$p$(2${\times}$1) structures occupy excess electrons mostly in their surface
states.

Figure 3: (Color on line) Calculated energy barrier $E_{b}$ [see Fig. 1(b)]
for the flip-flop motion of buckled dimers as a function of electron and hole
dopings as well as external electric field. The unit of $e$ in $n_{e}$ is
given per $p$(2${\times}$1) unit cell.

Next, we examine the influence of external electric field ${\bf E}$ on the
energetics of the $p$(2${\times}$1) and $c$(4${\times}$2) structures. Here,
${\bf E}$ is simulated by superimposing an additional sawtooth potential along
the [001] direction (taken as the +$z$ direction) with discontinuity at the
mid-plane of the vacuum region of the supercell. Note that an STM bias voltage
of 1.5 V and a tip-sample distance of ${\sim}$5 Å would give rise to an
electrical field of ${\sim}$0.3 V/Å. Figure 3 also shows the calculated values
of $E_{b}$ as a function of ${\bf E}$ ranging between $-$0.5 and +0.5 V/Å. We
find that $E_{b}$ increases (decreases) as ${\bf E}$ increases along the +$z$
($-z$) direction. These different behaviors of $E_{b}$ depending on the
direction of ${\bf E}$ can be explained in terms of the different
contributions of electrostatic energy due to external electric field between
the $p$(2${\times}$1) and $c$(4${\times}$2) structures. Since the surface
dipole moment pz (pointing $-z$ direction) of the metallic $p$(2${\times}$1)
structure is larger in magnitude by 0.038 $e$Å than that of the semiconducting
$c$(4${\times}$2) structure, an electric field applied along the +$z$ ($-z$)
direction gives a positively (negatively) larger electrostatic energy $U$ =
$-{\bf p}{\cdot}{\bf E}$ of surface dipole in $p$(2${\times}$1) compared to in
$c$(4${\times}$2), leading to an increase (decrease) of $E_{b}$. We find that
the variation of $E_{b}$ with respect to the external electric field of 0
${\leq}$ $|{\bf E}|$ ${\leq}$ 0.5 V/Å is less than ${\sim}$20 meV, much
smaller than that (${\sim}$160 meV) obtained from hole doping (see Fig. 3).
Thus, we can say that the influence of hole doping on $E_{b}$ is much more
pronounced than that arising from external electric field.

To account for the symmetric dimer images observed from low-temperature STM
experiments Yokoyama1 ; Mitsui ; Ono ; Manzano , we investigate the flip-flop
motion of buckled dimers driven by either thermal activation Hata2001  or
quantum tunneling. For this, we assume a symmetric double-well potential [see
Fig. 1(b)] that describes the potential energy surface of flipping dimers as a
function of ${\theta}$. Using a harmonic approximation, we obtain a vibration
frequency for this potential well as $f_{0}$ = ${\frac{\omega}{2{\pi}}}$ =
${\frac{1}{2{\pi}}}\sqrt{\frac{k}{I}}$ ${\approx}$ 0.3${\times}10^{13}$ sec-1
in the absence of electron or hole doping, where the torsion constant $k$ and
the inertia moment $I$ of flipping dimer can be estimated from $E_{b}$ =
$k{{\theta}_{0}}^{2}$ (${\theta}_{0}$: dimer buckling angle at the lowest-
energy configuration) and $I$ = $\frac{1}{2}m_{\rm si}{d_{D}}^{2}$ ($m_{\rm
si}$: mass of Si atom). Based on an Arrhenius-type activation process, a
thermally excited flipping rate can be expressed as $f_{T}$ =
$f_{0}exp(\frac{-E_{b}}{kT})$. With the calculated values of $E_{b}$ and
$f_{0}$ as a function of $|n_{e}|$ ${\leq}$ 0.6$e$, we obtain $f_{T}$ smaller
than 0.8${\times}10^{-36}$ sec-1 at 10 K. This thermal flipping rate is too
small to explain the observed symmetric-dimer STM images with flicker noise
Yokoyama1 ; Mitsui . As an alternative explanation for the flip-flop motion of
buckled dimers, we consider quantum tunneling (QT) within the double-well
potential, whose flipping rate can be approximated Fab ; Mer  as

$\displaystyle f_{\rm
QT}=\sqrt{\frac{8E_{b}{\omega}}{{\hbar}{\pi}}}exp({\frac{-2E_{b}}{{\hbar}{\omega}}}).$
(1)

Contrasting with $f_{T}$, $f_{\rm QT}$ is independent of temperature, while it
is determined by the ratio of $E_{b}$ and the zero-point energy
${\frac{1}{2}}{\hbar}{\omega}$. The estimated values of $f_{\rm QT}$ are
plotted as a function of $n_{e}$. We find that $f_{\rm QT}$ sharply increases
with increasing hole doping, while it is nearly flat with respect to electron
doping. Here, note that electron doping decreases ${\omega}$ due to an
increase of ${\theta}_{0}$, thereby hardly changing the ratio of $E_{b}$ and
${\frac{1}{2}}{\hbar}{\omega}$. For hole doping with $|n_{e}|$ $>$ 0.3$e$,
$f_{\rm QT}$ becomes greater than ${\sim}$1.9${\times}10^{-1}$ sec-1.
Considering that it takes about 10-2 sec to obtain an STM image of a dimer,
such a hole-doping induced flip-flop motion can produce the observed symmetric
dimer images in low-temperature STM experiments Yokoyama1 ; Mitsui ; Ono ;
Manzano . It is noticeable that the application of ${\bf E}$ along the $-z$
(+$z$) direction decreases (increases) $E_{b}$. Consequently, one expects that
negative sample bias (equivalently, negative electric field) inducing hole
doping at low temperature enhances the magnitude of $f_{\rm QT}$. On the other
hand, positive sample bias (positive electric field) inducing electron doping
suppresses $f_{\rm QT}$. These drastically different aspects of negative and
positive bias voltages in low-temperature STM experiments account for the
observations of symmetric and buckled dimer images in filled-state and empty-
state images, respectively Ono ; Manzano .

Figure 4: (Color on line) Calculated QT-driven flipping rate of buckled dimers
as a function of electron and hole dopings.

Although we present a simple picture of the QT-driven flip-flop motion of
buckled dimers with a double-well potential, we believe that it captures the
microscopic mechanism underlying low-temperature symmetric-dimer STM images,
as explained above. It is noted that the present DFT-GGA calculation may tend
to somewhat overestimate the energy gain due to buckling. Indeed, the quantum
Monte Carlo (QMC) calculation QMC  which accurately describes electronic
correlations extrapolates the value of $E_{b}$ up to ${\sim}$150 meV per
dimer. Obviously, this reduction of $E_{b}$ should enhance the QT-driven flip-
flop motion of buckled dimers. More rigorous QMC simulations with sufficiently
large clusters or slab geometries will be a subject of future work.

In summary, we have performed a DFT-GGA calculation for the Si(001) surface to
investigate the energy difference between the symmetric-dimer structure and
the $c$(4${\times}$2) structure under electron or hole doping as well as
applied external electric field along the [001] direction. This energy
difference corresponding to the energy barrier for the flipping of buckled
dimers was found to decrease more significantly with respect to hole doping
compared to electron doping. Consequently, we found that hole doping gives
rise to a sizable QT-driven frequency of the flip-flop motion of buckled
dimers while electron doping shows the marginal QT effects. These different QT
aspects of hole and electron dopings are most likely to yield the imaging
difference between the filled- and empty-state STM images at low temperature.
Thus, we concluded that quantum tunneling enhanced by the tunneling-induced
hole doping causes the observation of symmetric dimer images in low-
temperature STM experiments.

ACKNOWLEDGEMENTS We thank Prof. Changgan Zeng, Prof. Shun-Fang Li, and Prof.
Zhenyu Zhang for helpful discussions. This work was supported by the National
Basic Research Program of China (Grant No. 2012CB921300), National Natural
Science Foundation of China (Grants No. 11274280 and No. 61434002), and
National Research Foundation of Korea (Grant No. 2015R1A2A2A01003248). The
calculations were performed by KISTI supercomputing center through the
strategic support program (KSC-2015-C3-017) for the supercomputing application
research.

∗Corresponding author<EMAIL_ADDRESS>
†Corresponding author<EMAIL_ADDRESS>
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††institutetext: Research Center for Nuclear Physics, Osaka University,
Osaka 567-0047, Japan

# Color confinement due to spontaneous breaking of magnetic $U(1)_{m}^{8}$

Tsuneo Suzuki<EMAIL_ADDRESS>
###### Abstract

It is possible to understand non-Abelian color confinement in the framework of
the Abelian dual Meissner effect, if the non-Abelian Bianchi identity is
violated. The violation is equal to 8 Abelian monopole currents of the Dirac
type satisfying Abelian conservation rules kinematically. There exist magnetic
$U(1)_{m}^{8}$ symmetries in non-Abelian $SU(3)$ QCD. When the magnetic
$U(1)_{m}^{8}$ symmetries are broken spontaneously, only states which are
invariant under all $U(1)_{e}$ subgroups of $SU(3)$ can exist as a physical
state. Such states are just $SU(3)$ singlets. The QCD vacuum in the
confinement phase is characterized by one long percolating monopole loop
running over the whole lattice volume in both quenched and full QCD. The long
loop in full QCD is on average a few times longer in comparison with that in
quenched QCD case at a similar lattice distance on the same volume.
Surprisingly, the monopole behaviors in full QCD seem to be independent of the
bare quark mass suggesting irrelevance of Abelian monopoles to the chiral
symmetry breaking mechanism in QCD. Existence of such Abelian magnetic
monopoles in the continuum limit is studied in detail in $SU(3)$ by means of a
block spin transformation of monopoles and the inverse Monte-Carlo method. The
monopole density $\rho$ and the infrared effective monopole action $S(k)$ of
$n$ blocked monopoles are determined for $a(\beta)=(0.04\sim 2)$fm and
$n=1\sim 12$ blockings on $48^{4}$ lattice in quenched QCD and for
$a(\beta)=0.0846(7)$fm and $n=1\sim 24$ on $96^{4}$ in full QCD at
$m_{\pi}=146$MeV. Originally $\rho$ and $S(k)$ are a two-point function of
$a(\beta)$ and the number of times of the blocking transformation $n$.
However, both are found to be a function of $b=na(\beta)$ alone in the
quenched QCD which suggests the existence of the continuum limit. In the full
QCD, the renormalization flow is observed similarly but the scaling is not yet
proved since the vacuum ensembles are only at $\beta=1.82$. The distributions
of the long loops show that monopole condensation occurs due to the entropy
dominance over the energy for all $b$ considered.

###### Keywords:

Color confinement, lattice QCD, Abelian monopole, dual Meissner effect, block-
spin transformation, inverse Monte-Carlo

##  1 Introduction

Color confinement in the presence of light quarks and massless gluons in QCD
is not yet clarified. The problem is known to be related to the Yang-Mills
(YM) existence and mass gap problemCMI:2000mp . It is very interesting to find
a fundamental mechanism behind these problems. The dual Meissner picture
proposed by ’tHoofttHooft:1975pu  and MandelstamMandelstam:1974pi  is
promising when color magnetic monopoles which condense in the vacuum are found
naturally in QCD. However, it is theoretically difficult to find such magnetic
monopoles in QCD composed of color electric fields alone without any
artificial assumption as done say in Ref.tHooft:1981ht .

In 2014, motivated by a workBonati:2010tz  which found violation of non-
Abelian Bianchi identity (VNABI) exists behind the ’tHooft Abelian
monopolestHooft:1981ht , the present authorSuzuki:2014wya ; Suzuki:2017lco ;
Suzuki:20220422  found an interesting fact that violation of non-Abelian
Bianchi identity (VNABI) if it occurs is equivalent to 8 Abelian magnetic
monopole currents satisfying Abelian conservation
$\partial_{\mu}k_{\mu}^{a}(x)=0\ \ (a=1\sim 8)$ kinematicallyArafune:1974uy .
VNABI can exist when original gauge fields contain a singularity of the Dirac
typeDirac:1931 . If condensation of such 8 magnetic monopoles occurs, any
state which has an electric color charge with respect to any electric
$U(1)_{e}$ subgroup in $SU(3)$ is confined due to the Abelian dual Meissner
effect. States neutral under all $U(1)_{e}$ subgroups are $SU(3)$ singlets
alone. Hence non-Abelian $SU(3)$ color confinement is proved through the
Abelian dual Meissner mechanism due to the Abelian monopole condensation with
respect to $U(1)_{m}^{8}$.

We show again how VNABI leads us to existence of 8 Abelian monopole currents
satisfying conservation rulesSuzuki:2014wya . Define a covariant derivative
operator $D_{\mu}=\partial_{\mu}-igA_{\mu}$. The Jacobi identities are
expressed as $\epsilon_{\mu\nu\rho\sigma}[D_{\nu},[D_{\rho},D_{\sigma}]]=0$.
By direct calculations, we get
$[D_{\rho},D_{\sigma}]=-igG_{\rho\sigma}+[\partial_{\rho},\partial_{\sigma}]$,
where the second commutator term of the partial derivative operators cannot be
discarded in general, since gauge fields may contain a line singularity.
Actually, it is the origin of the violation of the non-Abelian Bianchi
identities (VNABI) as shown in the following. The non-Abelian Bianchi
identities and the Abelian-like Bianchi identities are, respectively:
$D_{\nu}G^{*}_{\mu\nu}=0$ and $\partial_{\nu}f^{*}_{\mu\nu}=0$, where
$f_{\mu\nu}$ is defined as
$f_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}=(\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu})\lambda^{a}/2$.
The relation $[D_{\nu},G_{\rho\sigma}]=D_{\nu}G_{\rho\sigma}$ and the Jacobi
identities lead us to

$\displaystyle
D_{\nu}G^{*}_{\mu\nu}=-\frac{i}{2g}\epsilon_{\mu\nu\rho\sigma}[D_{\nu},[\partial_{\rho},\partial_{\sigma}]]=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}[\partial_{\rho},\partial_{\sigma}]A_{\nu}=\partial_{\nu}f^{*}_{\mu\nu}.$
(1)

Namely Eq.(1) shows that VNABI, if exists, is equivalent to a set of Abelian-
like Bianchi identities. Denote VNABI as $J_{\mu}=D_{\nu}G^{*}_{\mu\nu}$ and
Abelian-like monopole currents $k_{\mu}\equiv k_{\mu}^{a}\lambda^{a}/2$
without any gauge-fixing as the violation of the Abelian-like Bianchi
identities:
$k_{\mu}=\partial_{\nu}f^{*}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\partial_{\nu}f_{\rho\sigma}.$
Eq.(1) shows that $J_{\mu}=k_{\mu}$. The Abelian-like monopole currents
satisfy an Abelian conservation rule kinematically,
$\partial_{\mu}k_{\mu}(x)=0$Arafune:1974uy . There exist exact Abelian (but
kinematical) $U(1)_{m}^{8}$ symmetries in non-Abelian QCD. However the
existence of the above Abelian monopoles is not related to any topological
property like a case in Ref.tHooft:1981ht  and hence the Dirac quantization
conditionDirac:1931  is not shown a priori. It must be proved rigorously if
such Abelian monopoles play an important role in nature.

##  2 Non-Abelian color confinement due to spontaneous breaking of
$U(1)_{m}^{8}$

Now we show explicitly why the Abelian dual Meissner effects due to
spontaneous breaking of the Abelian magnetic $U(1)_{m}^{8}$ from
$\partial_{\mu}k_{\mu}^{a}=0\ \ (a=1\sim 8)$ lead us to non-Abelian color
confinement. To say about the Abelian dual Meissner effect, it is necessary to
define a local $U(1)_{e}$ symmetry having a photon-like field from $SU(3)$. An
arbitrary $SU(3)$ gauge transformation is expressed as

$\displaystyle V(x)=e^{i\sum_{i=1,8}\alpha^{i}(x)X^{i}},$ (2)

where $X^{i}$ is a generator of $su(3)$ Lie algebra. There are eight
$U(1)_{e}^{i}$ subgroups expressed in terms of $\alpha^{i}$ alone among
$SU(3)$, although they are not commutative unless they belong to the same
maximal torus group $U(1)^{2}$. Consider for example, one $U(1)_{m}^{3}$
corresponding to the monopole current conservation
$\partial_{\mu}k_{\mu}^{3}(x)=0$, where $k_{\mu}^{3}(x)$ comes from the
singularity of $A_{\mu}^{3}(x)$. Now consider a $U(1)_{e}^{3}$ subgroup where
$A_{\mu}^{3}(x)$ plays a role of a photon. If the monopoles $k_{\mu}^{3}(x)$
condense in the vacuum, $U(1)_{m}^{3}$ is broken spontaneously. The Abelian
dual Meissner effect due to the monopole condensation leads us to that only
the neutral states $|H>$ under $U(1)_{e}^{3}$ can exist in nature, that is ,

$\displaystyle e^{i\alpha^{3}(x)X^{3}}|H>=|H>.$ (3)

The same situations occur with respect to all color components similarly.
Hence physical states which are neutral with respect to any $U(1)_{e}^{i}$
satisfy $e^{i\alpha^{i}(x)X^{i}}|H>=|H>$ for $i=1\sim 8$. Then it is possible
to show that

$\displaystyle\tilde{V}(x)|H>=|H>,$ (4)

where $\tilde{V}(x)\equiv\Pi_{i=1}^{8}e^{i\alpha^{i}(x)X^{i}}$ and the
ordering of the product is fixed arbitrary. Note that $\tilde{V}(x)$ has eight
parameters satisfying $\tilde{V}(x)\tilde{V}^{{\dagger}}(x)=1$ and
$det(\tilde{V}(x))=1$. Hence $\tilde{V}(x)\in SU(3)$. Eq.(4) indicates that
the state $|H>$ is $SU(3)$ invariant. Hence the Abelian dual Meissner effect
due to spontaneous breaking of magnetic $U(1)_{m}^{8}$ can explain non-Abelian
color confinement of QCD.

$\beta$ | $m_{\pi}$ | color | n | maxlp | lpnum | lp4 | n | maxlp | lpnum | lp4
---|---|---|---|---|---|---|---|---|---|---
2.7 | quench | 1 | 1 | 969444(420) | 22620(22) | 20440(21) | 8 | 3652(10) | 6(0) | 5(0)
|  | 3 | 1 | 1016422(212) | 15446(13) | 14623(13) | 8 | 3809(11) | 6(0) | 5(0)
1.9 | 702(1) | 1 | 1 | 3914480(209) | 8112(9) | 8111(9) | 8 | 9210(22) | 6(0) | 5(0)
|  | 3 | 1 | 3914649(189) | 8084(10) | 8083(10) | 8 | 9220(22) | 6(0) | 5(0)
| 384(3) | 1 | 1 | 3914572(183) | 8100(11) | 8098(11) | 8 | 9237(22) | 5(0) | 4(0)
|  | 3 | 1 | 3914908(173) | 8083(10) | 8082(10) | 8 | 9262(18) | 6(0) | 5(0)
| 156(6) | 1 | 1 | 3914682(188) | 8079(9) | 8078(9) | 8 | 9233(22) | 6(0) | 5(0)
|  | 3 | 1 | 3915081(156) | 8086(10) | 8085(10) | 8 | 9217(18) | 6(0) | 5(0)
Table 1:  Loop length distribution for 80 original and blocked monopole configurations. (The coupling constant $\beta$, the pion mass, color, blocking number of times $n$, longest loop length (maxlp), total number of loops (lpnum) and number of loops with length=4 (lp4)).  $m_{\pi}$ | $n$ | density | $n$ | density | $n$ | density | $n$ | density
---|---|---|---|---|---|---|---|---
quench | 1 | 0.76337(47) | 2 | 1.91557(123) | 4 | 4.11464(175) | 8 | 8.2855(73)
702(1) | 1 | 1.91166(2) | 2 | 3.83134(23) | 4 | 7.66345(129) | 8 | 15.3113(127)
384(3) | 1 | 1.91173(2) | 2 | 3.83120(19) | 4 | 7.65998(141) | 8 | 15.3275(122)
156(6) | 1 | 1.91171(2) | 2 | 3.83130(25) | 4 | 7.66164(177) | 8 | 15.3300(120)
Table 2:  Monopole density of $\beta=2.7$ quenched and $\beta=1.9$ full QCD on
$32^{3}\times 64$. $m_{\pi}$ is the pion mass and $n$ is the blocking number.

##  3 Monopole behaviors in the lattice QCD framework

It is very interesting to study whether the above scenario is realized in
nature or not in the lattice QCD framework rigorously.

First we have to define an Abelian monopole on lattice. In compact QED,
DeGrand and Toussaint(DGT)DeGrand:1980eq  defined Abelian monopoles on lattice
on the basis of the Dirac quantization condition corresponding to Abelian
monopoles of the Dirac type. A plaquette variable
$\theta_{\mu\nu}(s)\equiv\partial_{\mu}\theta_{\nu}(s)-\partial_{\nu}\theta_{\mu}(s)$
where $\theta_{\mu}(s)$ is an Abelian link field is decomposed into
$\theta_{\mu\nu}(s)=\bar{\theta}_{\mu\nu}(s)+2\pi n_{\mu\nu}(s)$, where
$\bar{\theta}_{\mu\nu}(s)\in[-\pi,\pi]$. Then the integer
$n_{\mu\nu}(s)\in[-2,2]$ could be regarded as a number of the Dirac string
penetrating the plaquette, although the number can take $[-\infty,\infty]$ in
the continuum theory. Then DGT DeGrand:1980eq  defined a monopole current on
lattice as

$\displaystyle
k_{\mu}(s)=-\frac{1}{4\pi}\epsilon_{\mu\nu\rho\sigma}\partial_{\nu}\bar{\theta}_{\rho\sigma}(s+\hat{\mu}).$
(5)

Since the Dirac quantization conditionDirac:1931  is essential for existence
of Abelian monopoles in QCD, we adopt here in $SU(3)$ the above lattice
Abelian monopole definitionDeGrand:1980eq  for each global color and study the
continuum limit.

Now let us study monopole distributions both in full QCD and quenched $SU(3)$
QCD of the Iwasaki gauge actionIwasaki:1985 ; IKKY:1997 ; Takeda:2004 . The
full $2+1$ flavor QCD configurations in this section are cited from the works
done by PACS-CS collaborationAoki:2009  using the nonperturbatively
$O(a)$-improved Wilson quark action and Iwasaki gauge action on $32^{3}\times
64$ lattice. There exist six data with different pion masses
($m_{\pi}=702(1),569(2),411(2),384(3),295(3),156(6)$MeV) at $\beta=1.9$
corresponding to the lattice spacing $a=0.0907(13)$ fm. For comparison, we
adopt quenched $SU(3)$ configurations of the Iwasaki gauge action at
$\beta=2.7$ which has a similar lattice spacing $a=0.0904(74)$. The
configuration numbers are restricted to 80 in all cases. To reduce lattice
artifact monopoles, we introduce an additional gauge fixing called Maximally
Abelian gauge with subsequenct $U(1)^{2}$ Landau gauge (MAU12)Kronfeld:1987ri
; Kronfeld:1987vd .

Since the Abelian monopole currents satisfy the conservation condition at each
site, they make a closed loop on the dual lattice. Loop length distributions
are measured numerically. The results are summarized in Table 1.

  1. 1.

There exist one long monopole loop both in most quenched and full QCD cases.
For example, see the full QCD case in Table 1. The total loop number minus the
number of the shortest loops is almost always one.

  2. 2.

The partial gauge fixing introduced here to make the vacuum smooth violates
global color invariance. The MA gauge enhances the diagonal color components.
But as seen from the diagonal color=3 and the off-diagonal color=1 examples,
large difference is not seen after the subsequent $U(1)^{2}$ fixing is done.

  3. 3.

The length of the long loop in full QCD is a few times larger than those in
quenched case.

  4. 4.

The most interesting fact is that above monopole behaviors are almost
unaffected by quark and pion masses as long as the dynamical quark effect is
taken into account. This is totally unexpected to the present author. Since
monopoles are expected to be important in color confinement, this fact
strongly suggests that the mechanisms of color confinement and chiral symmetry
breaking in QCD are quite different, since the chiral symmetry depends
strongly on the bare quark mass.

  5. 5.

Monopole density in full QCD is a few times larger than that in quenched QCD
as seen from Table2. Also in this case there exist essentially no difference
among full QCD configurations with different quark masses.

Figure 1:  Density of color=1 (left) and color=3 (right) monopoles

##  4 Study of the continuum limit of the Abelian monopoles

For the purpose of studying the continuum limit of lattice monopoles, the
block spin transformation method is very powerful. The idea of the block spin
with respect to Abelian monopoles on lattice was first introduced by Ivanenko
et al.Ivanenko:1991wt  and applied to the study obtaining an infrared
effective monopole action in Ref.Shiba:1994db . The $n$ blocked monopole has a
total magnetic charge inside the $n^{3}$ cube and is defined on a blocked
reduced lattice with the spacing $b=na$. The respective magnetic currents for
each color are defined as

$\displaystyle
k_{\mu}^{(n)}(s_{n})=\sum_{i,j,l=0}^{n-1}k_{\mu}(ns_{n}+(n-1)\hat{\mu}+i\hat{\nu}+j\hat{\rho}+l\hat{\sigma}),$
(6)

where $s_{n}$ is a site number on the reduced lattice and the color indices
are not shown explicitly. These equations show for example that the relation
between $k_{\mu}^{(4)}(s_{4})$ and $k_{\mu}^{(2)}(s_{2})$ is similar to that
between $k_{\mu}^{(2)}(s_{2})$ and $k_{\mu}(s)$ and hence the above equation
(6) corresponds to the usual block spin transformation. After the block spin
transformation, the number of short lattice artifact monopole loops decreases
while loops having larger magnetic charges appear. For details, see
Ref.Suzuki:2017lco .

###  4.1 Monopole density

First we calculate the density of a monopole with color $a$ defined as

$\displaystyle\rho^{a}=\frac{\sum_{\mu,s_{n}}|k_{\mu}^{a}(s_{n})|}{4\sqrt{8}V_{n}b^{3}},$
(7)

where $V_{n}=V/n^{4}$ is the 4 dimensional volume of the reduced lattice,
$b=na(\beta)$ is the spacing of the reduced lattice after $n$ times of the
block spin transformation. To reduce lattice artifact monopoles, we have to
adopt some partial gauge-fixing smoothing the vacuum. In previous works in
$SU(2)$Suzuki:2017lco  we adopted the maximal center
gauge(MCG)DelDebbio:1996mh ; DelDebbio:1998uu  and maximal Abelian
gauge(MAG)Kronfeld:1987ri ; Kronfeld:1987vd . Very clear scaling and gauge
independence are seen. However in $SU(3)$, we adopt only the MAG gauge and the
subsequent $U(1)^{2}$ Landau gauge, where the global $SU(3)$ is broken. The
color-averaged density has been observed in the previous workSuzuki:2023 ,
where preliminary analyses of $S(k)$ in $SU(3)$ are made for the first time.
To see the color dependence, we here observe both densities of color 1 and
color 3 monopoles separately as a typical example of off-diagonal and diagonal
components. They are shown in Figure 1. In the quenched $SU(3)$ case, we adopt
$48^{4}$ Iwasaki gauge action at $\beta=2.3\sim 3.5$, where
$a(\beta)=0.04fm\sim 2.0fm$. To study volume dependence, we also study
configurations of $24^{4}$ Iwasaki action at $\beta=1.9\sim 2.1$. For
comparison, the full QCD data are also plotted, where the configurations using
the 6-APE stout smeared Wilson clover action and Iwasaki gauge action on
$96^{4}$ at $\beta=1.82$ and $m_{\pi}=146$MeV are cited from the work done by
PACS collaborationIshikawa:2016 . The densities are shown for various $n=1\sim
24$. Clear scaling and volume independent results are seen in the quenched
case. The monopole density in the full QCD is about 3 or 4 times larger than
those in the quenched case, although the $b$ dependence looks similar except
for very small $b$ region.

Figure 2: Self-coupling in quenched QCD for the color=1 (left) and for the
color=3 (right).

Figure 3: Nearest neighbor-coupling $F(2)$ (left) and $F(3)$ (right) for the
color=3.

Figure 4: Comparison between $SU(2)$ and $SU(3)$. Self $F(1)$ (left) and
neighbor-coupling $F(2)$ (right) for color=3. $SU(2)$ data are from
Ref.Suzuki:2017zdh .

###  4.2 The infrared effective monopole action

It is important to know the dual magnetic theory from QCD. But contrary to the
compact QEDVillain:1975 ; Janke:1986 ; Banks:1977 ; Peskin:1978 ; Frolich:1986
; Smit:1991 ; Shiba:1994pu ; Jersak:1985 , the exact dual transformation
cannot be done in QCD. However, it is possible to derive numerically the
infrared effective monopole action $S(k)$ defined as

$\displaystyle e^{-{\cal S}[k]}=\int
DU{\psi}e^{-S(U)}\prod_{a,\mu,s}\delta(k_{\mu}^{a}(s)+\frac{1}{4\pi}\epsilon_{\mu\nu\rho\sigma}\partial_{\nu}\bar{\theta}_{\rho\sigma}^{a}(s+\hat{\mu})),$

where $S(U)$ is the QCD action. The effective action for blocked monopoles
$S(k^{(n)})$ is defined similarly. Making use of the inverse Monte-Carlo
method first proposed by SwendsenSwendsen:1984  and extended to closed
monopole currents by Shiba and Suzuki Shiba:1994db , we get the
renormalization flow of infrared effective monopole actions as ${\cal
S}[k]\to{\cal S}[k^{(1)}]\to{\cal S}[k^{(2)}]\ldots$. The extensive studies in
quenched $SU(2)$ caseKato:1998ur ; Fujimoto:2000 ; Chernodub:2000 ;
Chernodub:2003 ; Chernodub:200308  show that interactions between two monopole
currents being not far apart are dominant except for small $b<0.5fm$, so that
we adopt here first 10 two-point interactions
$S[k]=\sum_{i}^{10}F(i)S_{i}[k]$. For example,
$S_{1}(k)=\sum_{s,\mu}(k_{\mu}(s))^{2}$,
$S_{2}(k)=\sum_{s,\mu}k_{\mu}(s)k_{\mu}(s+\mu)$ and
$S_{3}(k)=\sum_{s,\mu\neq\nu}k_{\mu}(s)k_{\mu}(s+\nu)$. Others are shown in
Table VI in the previous workSuzuki:2023 . Since we deal with only two-point
interactions between monopoles of the same color, we omit the color index $a$.
For details of the inverse Monte-Carlo method, see Ref.Suzuki:2017zdh .

The results are as follows:

  1. 1.

Self ($F(1)$) and two nearest neighbor ($F(2)$, $F(3)$) coupling constants
both in quenched and full cases are shown in Figure 2 and in Figure 3. These
coupling constants $F(i)$ are originally a two-point function of $a(\beta)$
and the number of blocking times $n$, but actually they are found to be a
function of $b=na(\beta)$ alone. The beautiful scaling behaviors in the
quenched case suggest that we are seeing the continuum limit, since
$n\to\infty$ corresponds to $a(\beta)\to 0$ for fixed $b$. The coupling
constants in full QCD are always lower than those in the quenched data.
Moreover the full QCD data show stronger decreasing behaviors as $b$ becomes
larger, although the scaling is not yet checked in the full QCD case taken at
$\beta=1.82$ alone.

  2. 2.

Both quenched $SU(2)$Suzuki:2017zdh  and $SU(3)$ results are found to be very
similar as shown in Figure 4, although different gauge-fixings are adopted.
But note that both deal with only a $U(1)$ effective monopole action.

  3. 3.

The results on $24^{4}$ and $48^{4}$ lattices are found to be almost the same.
Volume dependence is not seen as seen from Figure 5 shown later.

Figure 5: Action/Length ($S(k)/L$) of the color=1 (left) and the color=3
(right) monopoles

Figure 6: Loop length histogram of percolating monopoles on $24^{4}$ of color
1 at $\beta=1.9$ (left) and of color 3 at $\beta=2.1$

Figure 7: Entropy-Energy $\gamma=\textrm{ln}\mu-f_{1}$ of the color=1 (left)
and the color 3 (right) components

###  4.3 Energy entropy balance and monopole condensation

Now that infrared effective monopole actions are derived numerically as above
in quenched and full QCD, let us go to the discussion about the monopole
condensation on the basis of the energy-entropy balance as a clue to color
confinement.

Since the monopole currents are conserved, they form closed loops on the dual
links. The distribution of the monopole loops are known to be composed of a
very long percolating loop running over the whole volume and some short
ultraviolet loops. See Chernodub:200308  in $SU(2)$ and in $SU(3)$ see Table1.
Figure 5 shows that the total monopole action is proportinal to the
percolating monopole length $L$. The entropy of monopole currents of length
$L$ can be expressed essentially as $\textrm{ln}(\mu^{L})$, where $\mu$ is
some positive number. When the monopoles obey the random walk, $\mu=7$ in four
dimensions as seen in compact QEDSmit:1991 . Here since we are considering
also the blocked monopoles, $\mu$ is not fixed theoretically. The monopole
partition function is hence well approximated by the sum of the monopole loop
distribution function $D(L)$ over the loop lengths of the monopoles:

$\displaystyle D(L)\propto\mu^{L}e^{-S(L)}=e^{\gamma(b)L},$ (8)

where $\gamma(b)=\textrm{ln}\mu-f_{1}(b)$ and $S(L)=f_{1}(b)L$. When the
lattice volume is finite, the monopole length $L$ cannot become infinite and
so there must exist a certain cut depending on the lattice volume. We choose
it following Ref.Chernodub:200308  as follows:

$\displaystyle D(L)=e^{-\alpha(b,V)L^{\eta}+\gamma(b)L},$ (9)

where $\alpha,\eta,\gamma$ are some parameters. In (9), there can exist a
power law term coming from the ultraviolet short loops, but it is negligible
as checked in Ref.Chernodub:200308 . The function (9) can reproduce well the
actual distribution of percolating loops as shown in Figure 6 where 9600
monopole configurations on $24^{4}$ of color 1 at $\beta=1.9$ and of color 3
at $\beta=2.1$ are used as typical examples. The parameter $\eta$ is found to
be almost 2.0 showing the Gaussian distribution. The same situations are seen
in all other cases irrespectively of color, $\beta$ and number of blocking
times $n$ on $24^{4}$. Hence we fix $\eta=2.0$ hereafter. The peak of the
distribution is then given by

$\displaystyle L_{max}=\gamma(b)/(2\alpha(b,V)).$ (10)

The monopole density $\rho$ showing the scaling as seen from Figure 1 and the
definition of the monopole density (7) give

$\displaystyle\rho\propto L_{max}/V,$ (11)

so that we see $\alpha(b,V)=A(b)/V$. These parameters are fixed from the
analyses of loop distributions as seen from an example of Figure 6. However,
on larger lattices, it is difficult to adopt so many configurations. Hence on
$48^{4}$, we first generate 2000 configurations as usual. Assuming the
Gaussian distribution, we can fix the parameters as

$\displaystyle\alpha=\frac{1}{2}\frac{1}{<L^{2}>-<L>^{2}},\ \ \ \
\gamma=\frac{<L>}{<L^{2}>-<L>^{2}},$ (12)

where $<L^{2}>$ and $<L>$ are determined from the histograms. To evaluate the
errors of the parameters, we adopt the bootstrap methodChernodub:200308  by
generating randomly 2000 sets of 2000 configurations. In Figure 7, we plot
$\gamma(b)=\textrm{ln}\mu-f_{1}(b)$. $\gamma(b)$ looks to be a function of
$b=na(\beta)$ alone for large $b$ and almost volume independent. Note that
these data of $\gamma(b)$ are all positive indicating that the entropy of the
monopole loops dominates over the energy. Hence monopole condensation is
realized for all $b=na(\beta)$ regions in quenched QCD. In the case of full
QCD, the present number of configurations are too few to analyze the monopole
loop distribution clearly. However the distribution of long loops is
consistent with the above Gaussian form having a positive $\gamma$.

##  5 Concluding remarks

If VNABI exists in QCD, there appear magnetic $U(1)_{m}^{8}$ kinematical
symmetries corresponding to 8 Abelian conserved monopole currents. It is
explicitly shown that if the magnetic $U(1)_{m}^{8}$ are broken spontaneously,
non-Abelian color confinement is explained by the Abelian dual Meissner
effect. Next it is shown in the framework of lattice QCD that clear scaling
behaviors especially in quenched QCD with respect to the monopole density and
the infrared effective action are observed. Finally analyzing the monopole
loop distribution, we prove that the monopole condensation occurs for all $b$
regions discussed here due to the energy-entropy balance. These results show
that the new Abelian monopoles corresponding to the violation of non-Abelian
Bianchi identity are very hopeful candidates for color confinement. It is also
interesting to check if the energy-entropy arguments work also in explaining
the confinement-deconfinement transition in finite-temperature QCD. Analyses
of the effective action of the spacelike monopoles projected onto the three
dimensional time-slice show strongly that the entropy-energy balance is the
mechanism of the confinement-deconfinement transition in $SU(2)$CIS:2005 .
Similar studies are expected in quenched and dynamical $SU(3)$.

Although we have discussed only infrared effective monopole actions, various
infrared effective theories of QCD can be derived from the above Abelian
monopole actionsChernodub:1999 . The monopole action composed of quadratic
interactions alone corresponds to the dual Abelian Higgs (DAH) model in the
London limit, where the DAH modelSuzuki:1988 ; Maedan:1989  is given by

$\displaystyle
L_{DAH}=\frac{1}{4g_{m}^{2}}(\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu})^{2}+\frac{1}{2}|(\partial_{\mu}+iB_{\mu})\Phi|^{2}+\lambda(|\Phi|^{2}-\eta^{2})^{2}.$
(13)

Here $B_{\mu}$ is a dual Abelian gauge field and $\Phi$ is a complex scalar
monopole field. In the London limit $\lambda\to\infty$, the radial part of the
monopole field is frozen, that is, $|\Phi|=\eta$. In this limit, the mass of
the scalar field becomes infinite. But from the analyses of the penetration
and the coherence lengths, we expect that the masses of the dual gauge and the
scalar fields are roughly equal both in $SU(2)$Suzuki:2009  and
$SU(3)$IHS:2022  cases. Hence higher-order interactions such as four- and six-
point monopole interactions are necessary. Actually from the extensive studies
in quenched $SU(2)$ QCD, we know such higher-order interactions become
relevant for $b<0.5fm$ regionsChernodub:2000 . Previous results on
$24^{3}\times 4$IHS:2022  in quenched QCD suggest that the vacuum type in
$SU(3)$ is the weak type I (dual) superconductor near the border. Also the
mass of the dual gauge boson determined from the penetration length is about
1GeV. If the quenched results are not changed much in full QCD at the physical
point, both axial and scalar bosons are predicted to have a mass roughly
around 1 GeV. To find such new scalar and axial-vector bosons experimentally
is crucial to the confinement picture proposed here. Note that both the scalar
and the axial vector bosons are color-singlet and eightfold degenerate
exactly. In addition to the dual Abelian Higgs model, the hadronic string
model can be derived exactly from the Abelian monopole actionChernodub:1999
as naturally expected.

These results obtained in this work are consistent with the perfect Abelian
and monopole dominances with respect to the string tension showing the area-
law behaviors of non-Abelian Wilson loopsSuzuki:2009 ; IHS:2022 ; Suzuki:2023
.

###### Acknowledgements.

The numerical simulations of this work were done using High Performance
Computing resources at Research Center for Nuclear Physics (RCNP) of Osaka
University. The author would like to thank RCNP for their support of computer
facilities. This work is finacially supported by JSPS KAKENHI Grant Number
JP19K03848. This work is supported by the JLDG constructed over the SINET5 of
NII. The author thanks the PACS-CS and the PACS Collaborations for making
their 2 +1 flavour configurations available via the International Lattice Data
Grid (ILDG).

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