# Monopoles of the Dirac type and color confinement in QCD
\- Study of the continuum limit -

Tsuneo Suzuki<EMAIL_ADDRESS>Research Center for Nuclear Physics,
Osaka University, Osaka 567-0047, Japan

###### Abstract

Non-Abelian gauge fields having a line-singularity of the Dirac type lead us
to violation of the non-Abelian Bianchi identity. The violation as an operator
is equivalent to violation of Abelian-like Bianchi identities corresponding to
eight Abelian-like conserved magnetic monopole currents of the Dirac type in
$SU(3)$ QCD. It is very interesting to study if these new Abelian-like
monopoles are responsible for color confinement in the continuum $SU(3)$ QCD,
since any reliable candidate of color magnetic monopoles is not known yet. If
these new Abelian-like monopoles exist in the continuum limit, the Abelian
dual Meissner effect occurs, so that the linear part of the static potential
between a quark-antiquark pair is reproduced fully by those of Abelian and
monopole static potentials. These phenomena are called here as perfect Abelian
and monopole dominances. It is shown that the perfect Abelian dominance is
reproduced fairly well, whereas the perfect monopole dominance seems to be
realized for large $\beta$ when use is made of the smooth lattice
configurations in the maximally Abelian (MA) gauge. Making use of a block spin
transformation with respect to monopoles, the scaling behaviors of the
monopole density and the effective monopole action are studied. Both monopole
density and the effective monopole action which are usually a two-point
function of $\beta$ and the number of times $n$ of the block spin
transformation are a function of $b=na(\beta)$ alone for $n=1,2,3,4,6,8,12$.
If the scaling behavior is seen for up to larger $n$, it shows the existence
of the continuum limit, since $a(\beta)\to 0$ when $n\to\infty$ for fixed
$b=na(\beta)$. Along with the previous results without any gauge fixing, these
new results obtained in MA gauge suggest that the new Abelian-like monopoles
play the role of color confinement in $SU(3)$ QCD.

###### pacs:

12.38.AW,14.80.Hv

##  I Introduction

Color confinement in quantum chromodynamics (QCD) is still an important
unsolved problem. As a picture of color confinement, ’t Hooft [1] and
Mandelstam [2] conjectured that the QCD vacuum is a kind of a magnetic
superconducting state caused by condensation of magnetic monopoles and an
effect dual to the Meissner effect works to confine color charges. However to
find color magnetic monopoles which condense is not straightforward in QCD. If
the dual Meissner effect picture is correct, it is absolutely necessary to
derive such color-magnetic monopoles from gluon dynamics of QCD.

An interesting idea to introduce such an Abelian monopole in QCD is to project
QCD to the Abelian maximal torus group by a partial (but singular) gauge
fixing [3]. In $SU(3)$ QCD, the maximal torus group is Abelian $U(1)^{2}$.
Then Abelian magnetic monopoles appear as a topological object at the space-
time points corresponding to the singularity of the gauge-fixing matrix.
Condensation of the monopoles causes the dual Meissner effect with respect to
$U(1)^{2}$. Numerically, an Abelian projection in various gauges such as the
maximally Abelian (MA) gauge [4, 5] seems to support the conjecture [7, 8, 9,
10]. Although numerically interesting, the idea of Abelian projection [3] is
theoretically unsatisfactory. Especially there are infinite ways of such a
partial gauge-fixing and whether the ’t Hooft scheme depends on gauge choice
or not is not known.

Motivated by an interesting work by Bonati et al.[11] which found volation of
non-Abelian Bianchi identity (VNABI) exists behind the ’tHooft Abelian
monopoles, the present author found in 2014 [12] an interesting and more
fundamental fact that, when original gluon fields have a singularity where
partial derivatives are not commutative, the non-Abelian Bianchi identity is
broken and VNABI is just equal to the violation of Abelian-like Bianchi
identities. The latter just corresponds to the existence of Abelian-like
monopoles. For more details, see also Ref.[13].

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, one gets
$[D_{\rho},D_{\sigma}]=-igG_{\rho\sigma}+[\partial_{\rho},\partial_{\sigma}]$,
where the second commutator term of the partial derivative operators can not
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 $\partial_{\nu}f^{*}_{\mu\nu}=0$. The relation
$[D_{\nu},G_{\rho\sigma}]=D_{\nu}G_{\rho\sigma}$ and the Jacobi identities
lead us to

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

where $f_{\mu\nu}$ is defined as
$f_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}=(\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu})\lambda^{a}/2$.
Namely Eq.(1) shows that the violation of the non-Abelian Bianchi identities,
if exists, is equivalent to that of the Abelian-like Bianchi identities.
Denote the violation of the non-Abelian Bianchi identities (VNABI) as
$J_{\mu}=D_{\nu}G^{*}_{\mu\nu}$ and Abelian-like monopole currents $k_{\mu}$
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,
$\partial_{\mu}k_{\mu}^{a}(x)=0$[14]. There can exist exact Abelian (but
kinematical) symmetries in non-Abelian QCD. This is an extension of the Dirac
idea [15] of monopoles in Abelian QED to non-Abelian QCD.

In the framework of simpler $SU(2)$ QCD, interesting numerical results were
obtained. Abelian and monopole dominances as well as the Abelian dual Meissner
effect are seen clearly without any additional gauge-fixing already in 2009
[16, 17], although at that time, no theoretical explanation was clarified with
respect to Abelian-like monopoles without any gauge-fixing. They are now found
to be just Abelian-like monopoles proposed in the above paper [12]. Also, the
existence of the continuum limit of this new kind of Abelian-like monopoles
was discussed with the help of the block spin renormalization group concerning
the Abelian-like monopoles. The beautiful scaling behaviors showing the
existence of the continuum limit are observed with respect to the monopole
density [18] and the infrared effective monopole action [19]. The scaling
behaviors seem also to be independent of gauges smoothing the lattice vacuum.

Table 1: The ratio of the Abelian and the non-Abelian string tensions $\sigma_{a}/\sigma_{F}$ determined by applying the multilevel method in the Wilson action. The data are cited from Ref.[20]. Lattice size | $\beta$ | $\sigma_{a}/\sigma_{F}$
---|---|---
$12^{4}$ | 5.6 | 0.87(13)
$16^{4}$ | 5.6 | 1.05(9)
$12^{4}$ | 5.7 | 0.91(8)
$12^{4}$ | 5.8 | 1.01(11)
Table 2: String tensions from Polyakov-loop correlations in the Wilson action at $\beta=5.6$ on $24^{3}\times 4$. The data are from Ref.[20]. Types of the potential | $\sigma a^{2}$
---|---
non-Abelian | 0.178(1)
Abelian | 0.16(3)
monopole | 0.17(2)
photon | -0.0007(1)

It is very interesting to study the new Abelian-like monopoles in $SU(3)$ QCD.
To check if the Dirac-type monopoles are a key quantity of color confinement
in the continuum $SU(3)$ QCD, it is necessary to study monopoles numerically
in the framework of lattice $SU(3)$ QCD and to study then if the continuum
limit exists. It is not so straightforward, however, to extend the previous
$SU(2)$ studies to $SU(3)$. How to define Abelian-like link fields and
monopoles without gauge-fixing is not so simple as in $SU(2)$, since a $SU(3)$
group link field is not expanded in terms of Lie-algebra elements defining
Abelian link fields as simply done as in $SU(2)$. There are theoretically many
possible definitions which have the same naive continuum limit in $SU(3)$. In
the previous work [20], we found a natural definition as shown later
explicitly. Using the definition, we showed as cited in Table 1 that the
perfect Abelian dominance exists with the help of the multilevel method[21,
22] but without introducing additional smoothing techniques like partial gauge
fixings. Table 2 shows that the perfect monopole dominance holds good again
without any additional gauge fixing. In the latter, we had to evaluate huge
number of correlations between non-local gauge-variant quantites in order to
extract probable gauge-invariant results[23].

The dual Meissner effect around a pair of static quark and antiquark was
studied. Abelian electric fields are squeezed due to solenoidal monopole
currents and the penetration length for an Abelian electric field of a single
color is the same as that of non-Abelian electric field. The coherence length
was also measured directly through the correlation of the monopole density and
the Polyakov loop pair. The Ginzburg-Landau parameter indicates that the
$SU(3)$ vacuum in the confinement phase is that of the weak type I (dual)
superconductor. But these previous results in Ref[20] are all only on very
small lattices and at restricted $\beta$. However they suggest that the new
idea of monopoles are also important in real $SU(3)$ QCD. It is necessary to
study on larger lattices at more different $\beta$ in order to show the
existence of the new monopoles actually in the continuum $SU(3)$.

Here the aim of this note is to study the scaling behaviors of the Abelian
monopoles with the help of additional technique reducing lattice artifact
monopoles as much as possible. First the most popular partial gauge fixing,
the maximally Abelian gauge is adopted for the Iwasaki improved gluon action
[24, 25, 26] on $48^{4}$ lattices for various coupling constants between
$\beta=2.3$ and $\beta=3.5$. It is studied if the Abelian dominance and the
monopole dominance expected from the Abelian dual Meissner picture[27] are
realized. Next introducing the block spin transformation, we measure the
renormalization flows of the monopole density and the effective monopole
action and study directly if the Abelian-like monopoles have the continuum
limit.

Table 3:  Simulation parameters. (The coupling constant $\beta$, the lattice size, the gauge configuration number, the lattice spacing $a(\beta)$ from Ref.[26], $ns$ the number of the Abelian smearing steps, ($t_{1}^{a,m}$, $t_{2}^{a,m}$) the fitting ranges in lattice unit.  $\beta$ | volume | $N_{\rm conf}$ |  $a(\beta)$ [fm] |  $ns$, $(t_{1}^{a},t_{2}^{a})$ |  $ns$, $(t_{1}^{m},t_{2}^{m})$
---|---|---|---|---|---
2.9 | $48^{4}$ | 160 | 0.1420(116) | 6, (1, 10) | 2, (5, 10)
3.0 | $48^{4}$ | 160 | 0.1312(99) | 4, (3, 11) | 0, (10, 17)
3.1 | $48^{4}$ | 160 | 0.1143(46) | 8, (2, 15) | 2, (5, 17)
3.2 | $48^{4}$ | 160 | 0.1080(60) | 6, (10, 18) | 0, (14, 24)
3.3 | $48^{4}$ | 160 | 0.0918(65) | 8, (1, 20) | 60, (15, 20)
3.4 | $48^{4}$ | 160 | 0.0855(75) | 10, (4, 18) | 0, (15, 24)
3.5 | $48^{4}$ | 160 | 0.0809(130) | 12, (3, 18) | 2, (4, 20)
Figure 1: Examples of the effective mass plots at $\beta=3.5$ on $48^{4}$ for
the Abelian (up) and the monopole (down) parts.

Figure 2: Examples of the static potentials at $\beta=3.5$ on $48^{4}$ for the
Abelian (up) and the monopole (down) parts.

Figure 3: Ratio of Abelian string tensions versus non-Abelian one on $48^{4}$.
Figure 4: Ratio of monopole string tensions versus non-Abelian one on
$48^{4}$.

##  II Lattice settup of $SU(3)$ QCD

To study the continuum limit clearly on large lattice volume, it is important
to reduce the lattice artifact monopoles as much as possible and for that
purpose, we adopt the maximally Abelian gauge (MA) [4, 5, 6] in which

$\displaystyle R_{MA}(U(s,\mu))=\sum_{s}\sum_{\mu=1}^{4}\mathop{\rm
Tr}(U^{{\dagger}}(s,\mu)\vec{H}U(s,\mu)\vec{H})$ (2)

is maximized under $SU(3)$ gauge transformations, where $\vec{H}$ is the
diagonal Cartan subalgebra. After the MA gauge-fixing, we perform gauge-fixing
with respect to the residual $U(1)^{2}$ symmetry in Landau gauge. Here we
denote such serial gauge-fixings as MAU12.

Then Abelian link fields ${\theta^{a}_{\mu}(s)}$ and Abelian Dirac-type
monopoles on $SU(3)$ lattice are defined from non-Abelian link fields
${U_{\mu}(s)}$ as in the previous work [20]. Maximizing the following quantity

$\displaystyle RA=\mathrm{Re}\mathop{\rm
Tr}\left\\{\exp(i\theta_{\mu}^{a}(s)\lambda^{a})U_{\mu}^{{\dagger}}(s)\right\\},$
(3)

where $\lambda^{a}$ is the Gell-Mann matrix leads us to, say, in the
$\lambda^{1}$ case,

$\displaystyle\theta^{1}_{\mu}(s)=\tan^{-1}\left\\{\frac{\mathrm{Im}(U_{12}(s,\mu)+U_{21}(s,\mu))}{\mathrm{Re}(U_{11}(s,\mu)+U_{22}(s,\mu))}\right\\}.$
(4)

To improve the overlapping, we perform the following smearings:
(1) The hypercubic smearing is done with respect to the temporal direction of
non-Abelian link fields similarly as done in [29]. But the results are found
to be not so sensitive on the hypercubic blocking.
(2) With respect to spaticial link variables, we perform Abelian smearing with
the fixed smearing parameter $\alpha=2.3$ similarly as done with respect to
non-Abelian link fields in Ref.[30]. We check the dependence of the iteration
numbers of smearing $ns$ for $0\leq ns\leq 60$ on the behaviors of the
effective mass and the overlap parameter. The results are not so different
except for the small $t<3a$ or large $t>20a$. We show in Table 3 the
simulation parameters.

We next define Abelian-like lattice monopoles. The unique reliable method ever
known to define a lattice Abelian monopole is the one proposed in compact QED
by DeGrand and Toussaint [31] who utilize the fact that the Dirac monopole has
a Dirac string with a magnetic flux satisfying the Dirac quantization
condition [15]. Hence we adopt the method here, since the Abelian-like
monopoles here are of the Dirac type in QCD.

It is known that MA gauge fixing in $SU(3)$ has some ambiguities especially in
defining Abelian monopoles correponding to the diagonal color components[28].
Here we adopt the simplest method in which two diagonal Gell-Mann matrices
$\lambda_{3}$ and $\lambda_{8}$ are used.

First we define Abelian plaquette variables from the above Abelian link
variables:

$\displaystyle\theta_{\mu\nu}^{a}(s)$ $\displaystyle\equiv$
$\displaystyle\partial_{\mu}\theta_{\nu}^{a}(s)-\partial_{\nu}\theta_{\mu}^{a}(s),$
(5)

where $\partial_{\nu}(\partial^{\prime}_{\nu})$ is a forward (backward)
difference. Then the plaquette variable can be decomposed as follows:

$\displaystyle\theta_{\mu\nu}^{a}(s)$ $\displaystyle=$
$\displaystyle\bar{\theta}_{\mu\nu}^{a}(s)+2\pi n_{\mu\nu}^{a}(s)\ \
(|\bar{\theta}_{\mu\nu}^{a}|<\pi),$ (6)

where $n_{\mu\nu}^{a}(s)$ is an integer corresponding to the number of the
Dirac string. Then VNABI as Abelian-like monopoles is defined by

$\displaystyle k_{\mu}^{a}(s)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\epsilon_{\mu\alpha\beta\gamma}\partial_{\alpha}\bar{\theta}_{\beta\gamma}^{a}(s+\hat{\mu})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon_{\mu\alpha\beta\gamma}\partial_{\alpha}n_{\beta\gamma}^{a}(s+\hat{\mu}),$
$\displaystyle J_{\mu}(s)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}k_{\mu}^{a}(s)\lambda^{a}.$ (7)

This definition (7) of VNABI satisfies the Abelian conservation condition and
takes an integer value which corresponds to the magnetic charge obeying the
Dirac quantization condition[15].

Table 4:  Simulation results of the Abelian and monopole string tensions $\sigma_{a,m}$ versus non-Abelian one $\sigma_{F}$. $FR(r/a)$ is the fitting range. $V(r)=\sigma*r+c$ is used in the monopole fit at $\beta=2.9$ and $3.1$.  | Abelian string tension | monopole string tension | non-Abelian string tension[26]
---|---|---|---
$\beta$ | $\sigma_{a}$ | $FR(r/a)$ | $\chi^{2}/N_{d.o.f}$ | $\sigma_{m}$ | $FR(r/a)$ | $\chi^{2}/N_{d.o.f}$ | $\sigma_{F}$
2.9 | 0.02044(5) | (3,16) | 0.12 | 0.01531(8) | (4,24) | 1.16 | 0.02017(47)
3.0 | 0.01670(34) | (4,24) | 0.79 | 0.01380(7) | (4,24) | 0.96 | 0.01722(34)
3.1 | 0.01312(4) | (4,24) | 1.67 | 0.00986(5) | (4,24) | 1.24 | 0.01306(12)
3.2 | 0.01126(22) | (8,18) | 1.38 | 0.01132(13) | (7,21) | 0.96 | 0.01167(14)
3.3 | 0.00928(3) | (4,24) | 0.89 | 0.00818(6) | (6,24) | 0.874 | 0.00842(11)
3.4 | 0.007662(4) | (3,24) | 0.06 | 0.00679(5) | (7,24) | 0.93 | 0.00731(11)
3.5 | 0.00664(4) | (5,12) | 1.01 | 0.00653(18) | (4,11) | 0.72 | 0.00655(17)

##  III Abelian and monopole static potentials

We evaluate the static potentials from Abelian Wilson loops and their monopole
contributions. Here, we take into account only a simple Abelian Wilson loop,
say, of size $I\times J$. Then such an Abelian Wilson loop operator is
expressed as

$\displaystyle W^{a}_{A}=\mathrm{exp}\\{i\sum
J^{ext,a}_{\mu}(s)\theta_{\mu}^{a}(s)\\},$ (8)

where $J^{ext,a}_{\mu}(s)$ is an external electric current having a color $a$
taking $\pm 1$ along the Wilson loop. Since $J^{ext,a}_{\mu}(s)$ is conserved,
it is rewritten for such a simple Wilson loop in terms of an antisymmetric
variable $M^{a}_{\mu\nu}$ as
$J^{ext,a}_{\nu}=\partial^{\prime}M^{a}_{\mu\nu}(s)$ with a forward (backward)
difference $\partial_{\nu}(\partial^{\prime}_{\nu})$. Note that
$M^{a}_{\mu\nu}(s)$ take $\pm 1$ on a surface with the Wilson loop boundary.
Although we can choose any surface type, we adopt a minimal flat surface here.
We get

$\displaystyle W^{a}_{A}=\mathrm{exp}\\{-\frac{i}{2}\sum
M^{a}_{\mu\nu}(s)\theta^{a}_{\mu\nu}(s)\\}.$ (9)

We investigate the monopole contribution to the static potential in order to
examine the role of monopoles for confinement. The monopole part of the
Abelian Wilson loop operator is extracted as follows [9, 10]. Using the
lattice Coulomb propagator $D(s-s^{\prime})$, which satisfies
$\partial_{\nu}\partial^{\prime}_{\nu}D(s-s^{\prime})=-\delta_{ss^{\prime}}$,
we get

$\displaystyle W^{a}_{A}$ $\displaystyle=$ $\displaystyle
W^{a}_{mon}W^{a}_{ph},$ (10) $\displaystyle W^{a}_{mon}$ $\displaystyle=$
$\displaystyle\mathrm{exp}\\{2\pi i\sum k^{a}_{\beta}(s)$ (11)
$\displaystyle\times$ $\displaystyle
D(s-s^{{}^{\prime}})\frac{1}{2}\epsilon_{\alpha\beta\rho\sigma}\partial_{\alpha}M^{a}_{\rho\sigma}(s^{{}^{\prime}})\\},$
$\displaystyle W^{a}_{ph}$ $\displaystyle=$
$\displaystyle\mathrm{exp}\\{-i\sum\partial^{{}^{\prime}}_{\mu}\bar{\theta}^{a}_{\mu\nu}(s)D(s-s^{{}^{\prime}})J^{a}_{\nu}(s^{{}^{\prime}})\\}.$
(12)

We then compute the static potential from the Abelian Wilson loops and the
monopole Wilson loops in the MAU12 gauge on the $48^{4}$ lattices at
$\beta=2.9,3.0,3.1,3.2,3.3,3.4$ and $3.5$. They are shown as follows

$\displaystyle V^{a}_{A}(r)$ $\displaystyle=$
$\displaystyle-\textrm{lim}_{t\to\infty}\textrm{ln}<W^{a}_{A}>$ (13)
$\displaystyle V^{a}_{m}(r)$ $\displaystyle=$
$\displaystyle-\textrm{lim}_{t\to\infty}\textrm{ln}<W^{a}_{m}>.$ (14)

We extract $V(r)$ from the least-squares fit with the single- exponential form

$\displaystyle W(r,t)=C(r)e^{-V(r)t}$ (15)

and choose the fit range of $t_{1}\leq t\leq t_{2}$ such that the stability of
the so-called effective mass

$\displaystyle V^{eff}(r,t)=\textrm{ln}\frac{W(r,t)}{W(r,t+1)}$ (16)

is observed in the range $t_{1}\leq t\leq t_{2}$[32]. We also measure the
overlap coefficient $C(r)$ in (15) to check if the ground-state part is
extracted or not. Then we fit the potential to the usual functional form

$\displaystyle V_{fit}(r)=\sigma r-c/r+\mu,$ (17)

where $\sigma$ denotes the string tension, $c$ the Coulombic coefficient, and
$\mu$ the constant. Since the MA gauge breaks global color invariance, the
Abelian and the monopole potentials depend on the color chosen. Here we show
explicitly the color 3 diagonal components alone. Examples of the effective
mass of the Abelian and the monopole Wilson loops are plotted in Fig.2. The
fitting ranges as well as other lattice parameters in each case are described
in Table 3.

Examples of the Abelian and the monopole static potentials in the MAU12 at
$\beta=3.5$ are shown in Fig.2.

The results of the string tensions on $48^{4}$ in the MAU12 gauge are
summarized for various $\beta$ in Table 4 and in Figs. 3 and 4. Here
$\sigma_{a}$ and $\sigma_{m}$ are Abelian, and monopole string tensions which
are compared with non-Abelian string tensions $\sigma_{F}$ determined in
Ref[26]. Perfect Abelian dominance is seen quite well for all
$2.9\leq\beta\leq 3.5$ considered. The perfect Abelian dominance in MAU12 was
shown also on $32^{4}$ lattices in the Wilson action[32]. Table 4 and Fig.3
show that the asymptotic scaling seems quite well satisfied. On the other
hand, perfect monopole dominance seems satisfied for $\beta\geq 3.2$ as seen
from Table 4 and Fig.4. These results along with the previous results[20]
without any additional gauge fixing but on smaller lattices are consistent
with the expectation that the Abelian dual Meissner effect due to the new
Abelian-like monopoles is the color confinement mechanism in the continuum
limit.

Some comments are in order.

  1. 1.

The errorbars of the Abelian and monopole potentials of color 3 in MAU12 are
very small and so they are not clearly seen in Fig.2.

  2. 2.

The errors in Table 4 are only statistical. There are systematic errors.
Changes of the fitting range giving rise to one of the systematic errors are
checked to be less than $10$ percent.

  3. 3.

In the above, we show only the results with respect to the color 3 diagonal
components. We also measure another color 8 diagonal components for all
$\beta$. To extract the color 8 Abelian link fields from non-Abelian one needs
to solve a quartic equation, so that a bit more complicated. But still we get
almost good Abelian and monopole dominances.

  4. 4.

We also measure off-diagonal color components. But then the overlap
coefficients $C(r)$ become smaller rapidly for large $r$ regions and then no
Abelian and monopole dominances are observed.

  5. 5.

The $SU(3)$ results in MAU12 obtained here seem better than those studied in
the serial maximally Abelian and $U(1)$ Landau gauge (MAU1) and the maximally
center gauge (MCG) in the $SU(2)$ case[33]. Note, however, in $SU(2)$, perfect
Abelian and monopole dominances are clearly shown in the works without any
additional gauge fixing[16, 17].

Table 5:  Simulation parameters (Coupling constants $\beta$ of the Iwasaki action, the lattice size, the gauge configuration number used. The lattice spacing $a(\beta)$ are from Ref.[26]. $\beta$ | volume | $N_{\rm conf}$ |  $a(\beta)$ [fm]
---|---|---|---
2.3 | $48^{4}$ | 80 | 0.1143(46)
2.4 | $48^{4}$ | 80 | 0.1143(46)
2.5 | $48^{4}$ | 80 | 0.1143(46)
2.6 | $48^{4}$ | 80 | 0.1080(60)
2.7 | $48^{4}$ | 80 | 0.0918(65)
2.8 | $48^{4}$ | 80 | 0.0855(75)
Figure 5: Monopole density behaviors for fixed $n=1$ monopoles versus $\beta$
(up) and for fixed $\beta$ versus $n$ (down).

Figure 6: Monopole density versus $b=na(\beta)$

##  IV Block spin transformation studies of the monopoles

###  IV.1 The block spin transformation method

Since Abelian monopoles considered here correspond to violation of non-Abelian
Bianchi identity (VNABI) in the continuum[12, 18], it is impossible to study
the continuum limit of such quantities on lattice in the framework of the
asymptotic scaling of usual continuum QCD where VNABI is assumed not to occur.
Namely existence of line-like singulariries leading to VNABI is not assumed in
the usual framework of QCD. To study the continuum limit of Abelian monopoles,
therefore, one needs to adopt a completely different method.

The renormalization-group method based on the block spin transformation is
known to be a powerful tool for studying the continuum limit and critical
phenomena especially in various spin-systems[34, 35, 36]. When the original
lattice has a volume $V$ with the lattice spacing $a$, the blocked lattice is
defined as that having a lattice spacing $na$ on the lattice volume $V/n^{3}$
and the blocked spin is defined by integrating out the original spins on the
original lattice inside the blocked lattice. An infrared effective action is
obtained describing the physics of the blocked spins leading us to the
renormalization-group flow.

The idea of the block spin with respect to Abelian monopoles on lattice was
first introduced by Ivanenko et al.[37] and applied to the study obtaining an
infrared effective monopole action in Ref.[38]. 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

$\displaystyle k_{\mu}^{(n)}(s_{n})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\partial_{\nu}n_{\rho\sigma}^{(n)}(s_{n}+\hat{\mu})$
(18) $\displaystyle=$ $\displaystyle\sum_{i,j,l=0}^{n-1}k_{\mu}(ns_{n}$
$\displaystyle+(n-1)\hat{\mu}+i\hat{\nu}+j\hat{\rho}+l\hat{\sigma}),$
$\displaystyle n_{\rho\sigma}^{(n)}(s_{n})$ $\displaystyle=$
$\displaystyle\sum_{i,j=0}^{n-1}n_{\rho\sigma}(ns_{n}+i\hat{\rho}+j\hat{\sigma}),$

where $s_{n}$ is a site number on the reduced lattice and the color indices
are not shown explicitly. For example,

$\displaystyle k_{\mu}^{(2)}(s_{2})$ $\displaystyle=$
$\displaystyle\sum_{i,j,l=0}^{1}k_{\mu}(2s_{2}+\hat{\mu}+i\hat{\nu}+j\hat{\rho}+l\hat{\sigma}),$
$\displaystyle k_{\mu}^{(4)}(s_{4})$ $\displaystyle=$
$\displaystyle\sum_{i,j,l=0}^{3}k_{\mu}(4s_{4}+3\hat{\mu}+i\hat{\nu}+j\hat{\rho}+l\hat{\sigma})$
$\displaystyle=$
$\displaystyle\sum_{i,j,l=0}^{1}k_{\mu}^{(2)}(2s_{4}+\hat{\mu}+i\hat{\nu}+j\hat{\rho}+l\hat{\sigma}).$

These equations show 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 one can see the above equation (18) 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.[18].

For the purpose of studying the scaling behaviors for wide range of $\beta$,
we adopt the vacuum ensembles of the Iwasaki action from $\beta=2.3$ till
$\beta=2.8$ as shown in Table 5 in addition to those in Table 3. For the
additional range of $\beta$, we adopt only 80 configurations in Table 3, since
the errors are very small in the case of monopole density and the effective
action studies.

###  IV.2 Monopole density

The first observable is the gauge-invariant monopole density. If the Abelian
monopoles exist in the continuum limit, the monopole density must exist non-
vanishing in the continuum. In $SU(2)$, this seems to be realized actually
[18].

In $SU(3)$ we have eight Abelian-like conserved monopole currents instead of
three in $SU(2)$. Since monopoles are three-dimensional objects, the monopole
density is defined as follows:

$\displaystyle\rho=\frac{\sum_{\mu,s_{n}}\sqrt{\sum_{a}(k_{\mu}^{a}(s_{n}))^{2}}}{4\sqrt{8}V_{n}b^{3}},$
(19)

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 $n$-step block spin
transformation. The superscript $a$ denotes a color component. It is to be
noted that we do not restrict ourselves only to the Abelian monopoles of color
diagonal components as usually adopted in MAU12 gauge. Here we adopt Abelian
monopoles of all color components and take the sum over all color components.
Then $\sum_{a}(k_{\mu}^{a})^{2}$ is gauge-invariant in the continuum limit,
since $J_{\mu}=k_{\mu}$ is an adjoint operator. Note that we are studying the
new Abelian-like monopoles of the Dirac type which must be independent of
additional partial gauge fixing.

In general, the density $\rho$ is a function of two variables $\beta$ and $n$,
i.e., $\rho=\rho(n,a(\beta))$. When we change $\beta$ larger for fixed number
of blocking step, the monopole density decreases as shown in the upper part of
Fig 5 in the case of original unblocked monopole currents. No asymptotic
scaling is seen for fixed number of blocking. On the otherhand, we change the
number of blocking steps from $n=1$ to $n=12$, the monopole density increases
monotonously for fixed $\beta$.
But it is interesting to show that, if we plot the monopole density versus
blocked lattice distance $b=na(\beta)$, we get a universal curve
$\rho(n,a(\beta))\to\rho(b=na(\beta))$ depending on $b$ alone as shown in Fig.
6. There is a beautiful scaling function similarly as observed in $SU(2)$[18],
although the latter $SU(2)$ results have smaller errorbars and more appealing.
If the same behavior $\rho(b=na(\beta))$ is kept for $n\to\infty$, it
correponds to the non-zero monopole density at $a(\beta)\to 0$, i.e., the
continuum limit. Although we have studied the block spin transformation up to
$n=12$, the results obtained support strongly existence of the continuum limit
of the Abelian-like monopoles considered here , since the asymptotic universal
scaling function depending only on $b$ is realized.

In $SU(2)$, we have studied three other smooth gauge fixings as well as MAU1
and no gauge-dependence is seen as expected from the new type of Abelian-like
monopoles[12]. On the otherhand in $SU(3)$, we have not yet obtained another
reliable gauge-fixed smooth vacuum ensemble except for those in MAU12. Hence
to prove existence of the new type of Abelian-like monopoles in $SU(3)$, the
scaling behavior in MAU12 alone is not enough. Gauge independence is still to
be studied.

Table 6: The quadratic interactions used for the modified Swendsen method. Color index $a$ of the monopole current $k_{\mu}^{a}$ is omitted. coupling $\left\\{F(i)\right\\}$ | distance |  type
---|---|---
$F(1)$ | (0,0,0,0) | $k_{\mu}(s)k_{\mu}(s)$
$F(2)$ | (1,0,0,0) | $k_{\mu}(s)k_{\mu}(s+\hat{\mu})$
$F(3)$ | (0,1,0,0) | $k_{\mu}(s)k_{\mu}(s+\hat{\nu})$
$F(4)$ | (1,1,0,0) | $k_{\mu}(s)k_{\mu}(s+\hat{\mu}+\hat{\nu})$
$F(5)$ | (0,1,1,0) | $k_{\mu}(s)k_{\mu}(s+\hat{\nu}+\hat{\rho})$
$F(6)$ | (1,1,1,0) | $k_{\mu}(s)k_{\mu}(s+\hat{\mu}+\hat{\nu}+\hat{\rho})$
$F(7)$ | (0,1,1,1) | $k_{\mu}(s)k_{\mu}(s+\hat{\nu}+\hat{\rho}+\hat{\sigma})$
$F(8)$ | (2,0,0,0) | $k_{\mu}(s)k_{\mu}(s+2\hat{\mu})$
$F(9)$ | (1,1,1,1) | $k_{\mu}(s)k_{\mu}(s+\hat{\mu}+\hat{\nu}+\hat{\rho}+\hat{\sigma})$
$F(10)$ | (0,2,0,0) | $k_{\mu}(s)k_{\mu}(s+2\hat{\nu})$
Figure 7: The self-coupling constant $F(1)$ versus $b=na(\beta)$. Figure 8:
The nearest-neighbor coupling constant $F(2)$ versus $b=na(\beta)$. Figure 9:
The another nearest-neighbor coupling constant $F(3)$ versus $b=na(\beta)$.
Figure 10: The next to nearest-neighbor coupling constant $F(4)$ versus
$b=na(\beta)$.

###  IV.3 Infrared effective monopole action

The next observable is the infrared effective monopole action. The effective
action $S(k)$ for original monopoles $\\{k_{\mu}^{a}(s)\\}$ is defined as
follows:

$\displaystyle e^{-{\cal S}[k]}$ $\displaystyle=$ $\displaystyle\int
DU(s,\mu)e^{-S(U)}$
$\displaystyle\times\prod_{a}\delta(k_{\mu}^{a}(s)-\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\partial_{\nu}n_{\rho\sigma}^{a}(s+\hat{\mu})),$

where $S(U)$ is the Iwasaki gauge action. The effective action for blocked
monopoles $\\{k_{\mu}^{(n)}(s_{n})\\}$ is evaluated as

$\displaystyle e^{-{\cal
S}[k^{(n)}]}=\Pi_{s,\mu}\sum_{k_{\mu}(s)=-\infty}^{\infty}\delta(\partial^{\prime}_{\mu}k_{\mu})e^{-{\cal
S}[k]}$
$\displaystyle\times\delta(k_{\mu}^{(n)}(s_{n})-\sum_{i,j,l=0}^{n-1}k_{\mu}(ns_{n}+(n-1)\hat{\mu}+i\hat{\nu}+j\hat{\rho}+l\hat{\sigma})).$

Then we get the renormalization flow of infrared effective monopole actions as
${\cal S}[k]\to{\cal S}[k^{(1)}]\to{\cal S}[k^{(2)}]\ldots$.

Practically, we have to restrict the number of interaction terms of monopoles.
It is natural to assume that monopoles which are far apart do not interact
strongly and to consider only short-ranged local interactions of monopoles.

We determine the monopole action (20), that is, the set of couplings $F(i)$
from the monopole current ensemble $\left\\{k_{\mu}^{a}(s)\right\\}$ with the
aid of an inverse Monte-Carlo method first developed by Swendsen [39] and
extended to closed monopole currents by Shiba and Suzuki [38]. The details of
the inverse Monte-Carlo method are reviewed in AppendixA of Ref. [19].

Also in $SU(3)$, we are dealing with Abelian-like monopoles of each color
separately, the method is the same as done in $SU(2)$ [19]. For simplicity,
here we consider only the most important two-point interactions between
monopole currents composed of first 10 couplings as infrared effective
monopole action, since they are known as most important from the careful
studies of $SU(2)$ case:

$\displaystyle{\cal S}[k]=\sum_{i}^{10}F(i){\cal S}_{i}[k],$ (20)

where $F(i)$ are first 10 coupling constants shown explicitly in Table 6.

Since we now consider vacuum configurations in the smooth MAU12 gauge, only
the diagonal components are important. Hence, we consider only the monopole
currents having a color 3.

As studied in the previous section discussing the monopole density, we perform
the block spin transformation of monopole currents for $n=1,2,3,4,6,8,12$ on
$48^{4}$ at $\beta=2.3\sim 3.5$ and try to fix the infrared monopole actions
for all blocked monopoles.

Contrary to the beautiful $SU(2)$ results [19], the coupling constants for
small steps of blocking $n=1\sim 3$ can not be determined well for
$\beta=2.3\sim 3.5$. We may need more delicate tuning of inital conditions for
$F(i)$. Here we discuss only the results of the results of $F(i)$ for $n\geq
4$. All coupling constants are in general a function of $a(\beta)$ and $n$.
But similarly as in the monopole density, the scaling behaviors are seen only
when we plot $F(i)$ versus $b=na(\beta)$. The most dominant self-coupling
constant $F(1)$ is shown in Fig.7. The result show that the coupling constant
$F(1)$ is a function of $b=na(\beta)$ alone, namely the scaling behavior is
seen. Behaviors of other important coupling constants are shown in Figs.8
$\sim$ 10. All data show similar scaling behaviors.

##  V Summary and discussions

In this note, the scaling behaviors of the new Abelian-like monopoles in pure
$SU(3)$ QCD are studied adopting the Iwasaki improved gauge action for wide
range of $\beta$ and the number of blocking transformations from
$n=1,2,3,4,6,8,12$. To reduce lattice-artifact monopoles, we adopt here the
maximally Abelian gauge and $U1^{2}$ Landau gauge.

  1. 1.

The perfect Abelian dominance and the perfect monopole dominance are seen
fairly well with respect to Abelian and monopole string tensions. The
asymptotic scaling behaviors are observed roughly in these cases. The $SU(3)$
results here look better than those in $SU(2)$[33].

  2. 2.

The block spin transformation studies with respect to Abelian monopoles are
done. The behaviors of the monopole densities $\rho(n,a(\beta))$ of the
blocked monopole currents show the beautiful scaling behavior:
$\rho(n,a(\beta))\to\rho(b=na(\beta))$, i.e. $\rho$ is a function of
$b=na(\beta)$ alone. The scaling behaviors are seen here for
$n=1,2,3,4,6,8,12$. If on larger lattices, similar scaling behaviors are seen
for $n\to\infty$, it means $a(\beta)\to 0$, the continuum limit. It is
stressed that, although we adopt MAU12 gauge, the scaling behavior of the
monopole density is seen with respect to $SU(3)$ invariant combination summing
over all color components.

  3. 3.

Adopting the inverse Monte Carlo method, we determine the coupling constant
flow of the effective monopole action under the blocking transformation.
Although we restrict ourselves to important two-point monopole current
interactions, we get the scaling behaviors also. Namely, all coupling
constants which usually a two-point function of $n$ and $a(\beta)$ are
actually found to be a function of $b=a(\beta)$ alone.

  4. 4.

It is interesting to know what is the continuum theory of Abelian monopoles.
The present author along with colleagues has studied the continuum theory of
Abelian monopoles. An Abelian dual Higgs model[40] seems to be the theory of
Abelian monopoles in the continuum limit. See the references[41, 42].

  5. 5.

These results are all on $48^{4}$ lattice for various coupling constants of
the Iwasaki gauge action, adopting MAU12 gauge for reducing the lattice-
artifact monopoles. It is absolutely necessary to show gauge independence to
prove the new type of Abelian monopoles coming from the violation of non-
Abelian Bianchi identity at least as done in $SU(2)$[16, 17] without adopting
any additional gauge fixing. But such studies in $SU(3)$ seem at present
almost impracticable except for the previous study on a small lattice[20].
Hence it is desirable to study in smooth gauges other than MAU12 as done in
$SU(2)$ case[18, 19]. We have tried the Maximal Center (MCG) gauge[43, 44],
since in $SU(2)$ it shows after the simulated annealing[45] a similar scaling
behavior as in MAU1. But in $SU(3)$ at present the simple MCG gauge fixing is
too difficult to find the real maximum point. There seem to exist so many
local maxima in the MCG gauge funtional. Such a work will be done in future.

## Acknowledgements

This work used High Performance Computing resources provided by Cybermedia
Center of Osaka University through the JHPCN System Research Project (Project
ID: jh220002). The numerical simulations of this work were done also using
High Performance Computing resources at Research Center for Nuclear Physics of
Osaka University, at Cybermedia Center of Osaka University. The author would
like to thank these centers for their support of computer facilities. This
work is finacially supported by JSPS KAKENHI Grant Number JP19K03848.

## References

  * [1] G. ’t Hooft,  in Proceedings of the EPS International, edited by A. Zichichi, p. 1225, 1976.
  * [2] S. Mandelstam,  Phys. Rept. 23, 245 (1976).
  * [3] G. ’t Hooft,  Nucl. Phys. B190, 455 (1981).
  * [4] A. S. Kronfeld, M. L. Laursen, G. Schierholz, and U. J. Wiese,  Phys. Lett. B198, 516 (1987).
  * [5] A. S. Kronfeld, G. Schierholz, and U. J. Wiese,  Nucl. Phys. B293, 461 (1987).
  * [6] The reader may wonder why some smearing or cooling methods smoothing the vacuum are not used instead of introducing a partial gauge-fixing. But these methods are not useful in reducing lattice artifact monopoles but keeping physical monopoles unchanged. Only smooth non-local gauge-fixngs like MAG seem to be able to do the work.
  * [7] T. Suzuki,  Nucl. Phys. Proc. Suppl. 30, 176 (1993).
  * [8] M. N. Chernodub and M. I. Polikarpov,  in ”Confinement, Duality and Nonperturbative Aspects of QCD”, edited by P. van Baal, p. 387, Cambridge, 1997, Plenum Press.
  * [9] H. Shiba and T. Suzuki,  Phys. Lett. B333, 461 (1994).
  * [10] J. D. Stack, S. D. Neiman, and R. J. Wensley, Phys. Rev. D 50, 3399 (1994).
  * [11] C. Bonati, A. Di Giacomo, L. Lepori and F. Pucci,  Phys. Rev. D81, 085022 (2010).
  * [12] T. Suzuki, A new scheme for color confinement due to violation of the non-Abelian Bianchi identities, hep-lat: arXiv:1402.1294 (2014)
  * [13] T.Suzuki,  Monopoles of the Dirac type and color confinement in QCD - $SU(3)$ invariant picture of color confinement -, arXiv:2204.11514.
  * [14] J. Arafune, P.G.O. Freund and C.J. Goebel,  J.Math.Phys. 16, 433 (1975). J. Arafune, P.G.O. Freund and C.J. Goebel,  J.Math.Phys. 16, 433 (1975).
  * [15] P. Dirac, Proc. Roy. Soc. (London) A 133, 60 (1931).
  * [16] T. Suzuki, K. Ishiguro, Y. Koma and T. Sekido,  Phys. Rev. D77, 034502 (2008).
  * [17] T. Suzuki, M. Hasegawa, K. Ishiguro, Y. Koma and T. Sekido,  Phys. Rev. D80, 054504 (2009).
  * [18] T. Suzuki, K. Ishiguro and V. Bornyakov, Phys. Rev. D97, 034501 (2018); Phys. Rev. D97, 099905(E) (2018).
  * [19] T. Suzuki, Phys. Rev D97, 034509 (2018).
  * [20] K.Ishiguro, A.Hiraguchi and T. Suzuki,Phys Rev. D106 .014515 (2022), arXiv:2207.04436.
  * [21] M. Lüscher and P. Weisz, JHEP 0109, 010 (2001), hep-lat/0108014.
  * [22] M. Lüscher and P. Weisz, JHEP 0207, 049 (2002), hep-lat/0207003.
  * [23] S. Elitzur, tat  Phys. Rev. D12, 3978 (1975).
  * [24] Y. Iwasaki, Nucl. Phys. B258 (1985) 141; Univ. of Tsukuba report UTHEP-118 (1983), unpublished.
  * [25] Y. Iwasaki, K. Kanaya, T. Kaneko and T. Yoshie,  Phys. Rev. D56, 151 (1997). arXiv:hep-lat/9610023.
  * [26] S. Takeda et al. (CP-PACS Collaboration), Phys. Rev. D 70, 074510 (2004).
  * [27] Y. Koma, M. Koma, E.-M. Ilgenfritz, T. Suzuki and M.I. Polikarpov, Phys.Rev. D68 (2003) 094018, arXiv: hep-lat/0302006.
  * [28] J. D. Stack, W. W. Tucker, R. J. Wensley,  The Maximal Abelian Gauge, Monopoles, and Vortices in $SU(3)$ Lattice Gauge Theory, arXiv:hep-lat/0110196, 2001.
  * [29] A. Hasenfratz and F. Knecht, Phys. Rev. D 64, 034504 (2001). [arXiv:hep-lat/0103029].
  * [30] G. S. Bali and K. Schilling, Phys. Rev. D 47, 661 (1993).
  * [31] T. A. DeGrand and D. Toussaint,  Phys. Rev. D22, 2478 (1980).
  * [32] N. Sakumichi and H. Suganuma, Phys. Rev. 90, 111501 (2014).
  * [33] A.Hiraguchi,K.Ishiguro and T.Suzuki, Phys. Rev. D102 114504 (2020),arXiv:2011.14377
  * [34] L.P. Kadanoff, Physics 2 2, 1966.
  * [35] F. Jegerlehner, An Introduction to the Theory of Critical Phenomena and the Renoramlization Group, Lecture in ”Troisème cycle de la physique en Suisse Romande” (the Ecole Polytechnique Fèdèrale de Lausanne, May 1976)
  * [36] G. Mack, Multigrid Methods in Quantum Field Theory, Cargese Lecturs 1987, in Nonpertubative Quantum Field Theory (Plenum, NY, 1989)
  * [37] T.L. Ivanenko, A. V. Pochinsky and M.I. Polikarpov,  Phys. Lett. B302, 458 (1993).
  * [38] H. Shiba and T. Suzuki,  Phys. Lett. B351, 519 (1995).
  * [39] R.H. Swendsen,Phys. Rev. Lett. 52,1165 (1984).
  * [40] T. Suzuki, Prog. Theor. Phys. 80, 929 (1988). S. Maedan and T. Suzuki, Prog. Theor. Phys. 81, 229 (1989).
  * [41] Y. Koma, M. Koma, E.-M. Ilgenfritz and T. Suzuki, Phys. Rev. D68, 114504 (2003).
  * [42] M.N. Chernodub, K. Ishiguro and T. Suzuki, Phys. Rev. D69, 094508 (2004).
  * [43] L. Del Debbio, M. Faber, J. Greensite and S. Olejnik,  Phys. Rev. D55, 2298 (1997)
  * [44] L. Del Debbio, M. Faber, J. Giedt, J. Greensite and S. Olejnik,  Phys. Rev. D58, 094501 (1998)
  * [45] V. G. Bornyakov, D. A. Komarov and M.I. Polikarpov,  Phys. Lett. B497, 151 (2001).