# Detection of monopoles and vortices in SU($2$) Yang-Mills theory

N. Karimimanesh<EMAIL_ADDRESS>S. Deldar<EMAIL_ADDRESS>Department of
Physics, University of Tehran,
P.O. Box 14395-547, Tehran 1439955961, Iran

###### Abstract

Motivated by the correlated monopoles and vortices observed in lattice QCD,
using Cho decomposition method and two successive gauge transformations which
lead to the observation of configurations that include both monopoles and
center vortices simultaneously, we obtain a Lagrangian density that explicitly
indicates these two magnetic defects and the interaction between them.

PACS numbers

14.80.Hv, 12.38.Aw, 12.38.Lg, 12.39.Pn

###### pacs:

Valid PACS appear here

##  I Introduction

One of the unsolved and challenging problems in particle physics is the
confinement of quarks in Quantum Chromodynamics. There exist various
suggestions to approach this difficult problem. There are many articles which
discuss about the magnetic objects that play significant roles in confinement
of quarks. Some of them are listed in references gbook  to e . Monopoles and
vortices are among the major candidates which have been studied to describe
confinement of color sources.

In the absence of matter fields, two main methods have been presented to
extract magnetic monopole degrees of freedom in the YM (Yang-Mills) theories.
The first one is the Abelian projection proposed by ’t Hooft hooft  and the
second one is a field decomposition method which is introduced by Duan, Ge,
Cho, Faddeev and Niemi duan ; cho1 ; cho2 ; cho3 ; fadev ; shaban .

On the other hand, some lattice calculations show that line-like (surface-
like) objects in three (four) dimensions vor  are responsible for the
phenomenon of confinement. A method called Center Projection has been proposed
faber  to examine the role of center vortices in the Yang-Mills lattice gauge
configurations, which has been very successful. However, scenarios which are
solely based on either monopoles or center vortices are not able to describe
all the expected details of the confining potential between color sources. The
confining potential should be linear as well as proportional to the Casimir
scaling at intermediate distances. In addition, N-ality dependence should be
observed for all representations at large distances. The advantage of the
center vortex picture versus the monopole picture is that it explains N-ality
and Casimir scaling dependence. On the other hand, the monopole picture along
with Abelian dominance exhibits some numerical success. (for a review, see
ref. greens ). A clever idea to overcome these problems and to reconcile these
two candidates of confinement is to present a theory which establishes a
correlation between center vortices and monopoles. The corresponding
configurations have been supported by lattice simulations greensite ; pepe ;
zakharov  where center vortices have been observed to end at monopole world-
lines.

Some phenomenological attempts have been done in using both vortices and
monopoles to describe the color confinement. For example, in reference oxman ,
by two successive gauge transformations and by means of a careful treatment of
Cho decomposition, some configurations which simultaneously include monopoles
and vortices have been discussed. A nontrivial transformation leads to the
appearance of monopoles, and an SU($2$) gauge transformation that is not
single valued along a closed loop was shown to be responsible for the
existence of center vortices. In another paper val , a Yang-Mills-Higgs model
containing three external adjoint Higgs fields was presented. This model leads
to Z(2) vortices and the junctions which are formed by a pair of vortices, are
attached to a monopole-like configurations. The ultimate goal of the model was
to use it to describe hybrid hadrons. Some other works shabani  and 4Densem
in reconciling vortices and monopoles have been reported.

In this article, motivated by the method presented in reference oxman  and by
comparison with Abelian Projection scenario which has been discussed in
reference ichie , we obtain a Lagrangian density for QCD vacuum in the
confining regime and discuss explicitly the contributions of various defects
such as vortex, anti vortex, monopoles, chains and eventually the interaction
between monopoles and vortices. In Section II, by limiting ourselves to
SU($2$) Yang-Mills fields, we carefully review the Cho decomposition method.
In section III, magnetic monopoles are discussed by a nontrivial gauge
transformation which results to a local color frame including monopoles. Then,
the gauge transformed field strength tensor is calculated and its various
terms are interpreted. In section IV, using a gauge transformation that is not
single valued along any closed loop, center vortices are extracted. In section
V, it is shown that magnetic monopoles and vortices can simultaneously appear
as two defects in the color frame $\hat{n}_{a},a=1,2,3$. Then, we obtain a
Lagrangian density for correlated monopoles and vortices. Finally, in Section
VI, we present our conclusions. In this article, we study SU($2$) gauge group
for simplicity. With some more efforts, it can be extended to higher gauge
groups like SU($3$).

##  II Cho decomposition method

In this section we review Cho decomposition method for the SU($2$) gauge
group. For higher gauge groups like SU($3$), that have SU($2$) subgroups,
field decomposition can be written in a Weyl symmetric way frank18 . Our main
discussion in this paper is about SU($2$) group and we will not go into
SU($3$).

The Yang-Mills action of SU($2$) is defined as the following

$S_{YM}=\frac{1}{2}\int d^{4}x\ tr(F_{\mu\nu}F_{\mu\nu})\ \ ,\
F_{\mu\nu}=F_{\mu\nu}^{a}T_{a},$ (1)
$F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+g\epsilon^{abc}A_{\mu}^{b}\times
A_{\nu}^{c},$ (2)

where $g$ is the coupling constant, $T^{a}=\frac{\tau^{a}}{2},a=1,2,3$ are
generators of SU($2$), $\tau^{a}$ are the Pauli matrices, and $\epsilon^{abc}$
is the Levi-Civita symbol. The field strength tensor can be written in terms
of the gauge fields $\vec{A}_{\nu}$

$\vec{F}_{\mu\nu}=\partial_{\mu}\vec{A}_{\nu}-\partial_{\nu}\vec{A}_{\mu}+g\vec{A}_{\mu}\times\vec{A}_{\nu},$
(3) $\vec{A}_{\mu}=A_{\mu}^{a}\hat{e}_{a}\ \ ,\
\vec{F}_{\mu\nu}=F^{a}_{\mu\nu}\hat{e}_{a},$ (4)

where $\hat{e}_{a}$ is the basis in the color space.

A general local frame is considered in the color space, $\hat{n}_{a},a=1,2,3$
which can be defined by means of an orthogonal local transformation $R$

$\hat{n}_{a}=R\hat{e}_{a}\ \ ,\ R\in SO(3).$ (5)

This frame is used to represent the gauge field $\vec{A}_{\mu}$ in terms of
Cho decomposition elements

$\vec{A}_{\mu}=A_{\mu}^{(n)}\hat{n}-\frac{1}{g}\hat{n}\times\partial_{\mu}\hat{n}+\vec{X}_{\mu}^{(n)}\
\ ,\ \hat{n}.\vec{X}_{\mu}^{(n)}=0,$ (6) $\hat{n}_{a}.\hat{n}_{b}=\delta_{ab}\
\ ,\ a,b=1,2,3\ \ ,\ \hat{n}\equiv\hat{n}_{3},$ (7)

where $A_{\mu}^{(n)}$ is called the electric potential or the Abelian
component of $\vec{A}_{\mu}$ directed along $\hat{n}\equiv\hat{n}_{3}$ and
$\vec{X}_{\mu}^{(n)}$ is a gauge covariant potential orthogonal to $\hat{n}$
called the valence potential.

The restricted potential is defined as, (see cho2000  and its references)

$\hat{A}_{\mu}=A_{\mu}^{(n)}\hat{n}-\frac{1}{g}\hat{n}\times\partial_{\mu}\hat{n}.$
(8)

Using $\hat{A}_{\mu}$ of Eqn. (8), one can easily calculate the field strength
tensor

$\vec{F}_{\mu\nu}=(F_{\mu\nu}^{(n)}+H_{\mu\nu}^{(n)})\hat{n},$ (9)

where

$F_{\mu\nu}^{(n)}=\partial_{\mu}A_{\nu}^{(n)}-\partial_{\nu}A_{\mu}^{(n)},$
(10)
$H_{\mu\nu}^{(n)}=-\frac{1}{g}\hat{n}.(\partial_{\mu}\hat{n}\times\partial_{\nu}\hat{n}).$
(11)

Eqn. (9) shows that $\vec{F}_{\mu\nu}$ is parallel to $\hat{n}$, and it is
made of two parts: $F_{\mu\nu}^{(n)}$ which is called the electric strength
tensor and $H_{\mu\nu}^{(n)}$ which is called the magnetic strength tensor.

Choosing a hedgehog configuration for $\hat{n}$ and choosing
$A_{\mu}^{(n)}=0$, one can obtain a Wu-Yang monopole wu

$\hat{n}=\hat{r}=\begin{pmatrix}\sin\theta\cos\varphi\\\
\sin\theta\sin\varphi\\\ \cos\theta\end{pmatrix},$ (12)

Using Eqn. (12) in Eqn. (11), $H_{\mu\nu}^{(n)}$ can be written in the
following format

$H_{\mu\nu}^{(n)}=\partial_{\mu}C_{\nu}^{(n)}-\partial_{\nu}C_{\mu}^{(n)},$
(13)

where

$C_{\mu}^{(n)}=\frac{1}{g}\cos\theta\partial_{\mu}\varphi,$ (14)

$C_{\mu}^{(n)}$ is called magnetic potential.

Therefore, using Cho decomposition method and the restricted potential of Eqn.
(8), the Wu-Yang monopole is extracted.

In order to show that center vortices can be obtained by Cho decomposition
method, we do not limit ourselves to the restricted section of the field, and
the role of each of the three fields in Eqn. (6) is examined. Using
$\vec{A}_{\mu}$ of Eqn. (6) obtained from Cho decomposition method to compute
the field strength tensor, we get

$\vec{F}_{\mu\nu}=(F_{\mu\nu}^{(n)}+H_{\mu\nu}^{(n)}+K_{\mu\nu})\hat{n}+\vec{G}_{\mu\nu}+\vec{L}_{\mu\nu},$
(15)
$F_{\mu\nu}^{(n)}=\partial_{\mu}A_{\nu}^{(n)}-\partial_{\nu}A_{\mu}^{(n)}\ \
,\
H_{\mu\nu}^{(n)}=-\frac{1}{g}\hat{n}.(\partial_{\mu}\hat{n}\times\partial_{\nu}\hat{n}),$
(16) $K_{\mu\nu}=g\hat{n}.(\vec{X}_{\mu}\times\vec{X}_{\nu})\ \ ,\
\vec{X}_{\mu}=X_{\mu}^{1}\hat{n}_{1}+X_{\mu}^{2}\hat{n}_{2},$ (17)
$\begin{split}\vec{G}_{\mu\nu}&=\hat{D}_{\mu}\vec{X}_{\nu}^{(n)}-\hat{D}_{\nu}\vec{X}_{\mu}^{(n)},\\\
\hat{D}_{\mu}\vec{X}_{\nu}^{(n)}&=\partial_{\mu}\vec{X}_{\nu}^{(n)}+g\hat{A}_{\mu}\times\vec{X}_{\nu}^{(n)},\end{split}$
(18)
$\vec{L}_{\mu\nu}=-\frac{1}{g}\hat{n}\times[\partial_{\mu},\partial_{\nu}]\hat{n}=L_{\mu\nu}^{1}\hat{n}_{1}+L_{\mu\nu}^{2}\hat{n}_{2}.$
(19)

It is clear that $\vec{L}_{\mu\nu}$ which is orthogonal to $\hat{n}$, is
concentrated on two-dimensional surfaces. It could be nontrivial only for
local frames containing defects in the color direction $\hat{n}$. The tensor
$\vec{G}_{\mu\nu}$ have been computed in cho1 ; cho2 ; cho3 , cho2000 ; lee ;
bae ; pak :
$\vec{G}_{\mu\nu}=G_{\mu\nu}^{1}\hat{n}_{1}+G_{\mu\nu}^{2}\hat{n}_{2}$, being
orthogonal to $\hat{n}$.

In the presence of singularities, this result may remain unchanged. For
simplicity, a new “Abelianized” form for the field strength tensor is defined:
$\partial_{\mu}C_{\nu}^{(n)}-\partial_{\nu}C_{\mu}^{(n)}$ which should be
revised when the gauge fields containing defects.

Using $\hat{A}_{\mu}$ of Eqn. (8) and $\vec{X}_{\mu}$ of Eqn. (17) in Eqn.
(18), and defining

$C_{\mu}^{(n)}=-\frac{1}{g}\hat{n}_{1}.\partial_{\mu}\hat{n}_{2},$ (20)

the covariant derivative of $\vec{X}_{\mu}^{(n)}$ is obtained as the following

$\begin{split}\hat{D}_{\mu}\vec{X}_{\nu}^{(n)}&=[\partial_{\mu}X_{\nu}^{1}-g(A_{\mu}^{(n)}+C_{\mu}^{(n)})X_{\nu}^{2}]\hat{n}_{1}\\\
&+[\partial_{\mu}X_{\nu}^{2}-g(A_{\mu}^{(n)}+C_{\mu}^{(n)})X_{\nu}^{1}]\hat{n}_{2}.\end{split}$
(21)

Now, we can redefine $H_{\mu\nu}^{(n)}$ in terms of $C_{\mu}^{(n)}$

$H_{\mu\nu}^{(n)}=\partial_{\mu}C_{\nu}^{(n)}-\partial_{\nu}C_{\mu}^{(n)}+D_{\mu\nu}^{(n)},$
(22)

where $D_{\mu\nu}^{(n)}$ is obtained to be

$D_{\mu\nu}^{(n)}=\frac{1}{g}\hat{n}_{1}.[\partial_{\mu},\partial_{\nu}]\hat{n}_{2}.$
(23)

In the following we show how $D_{\mu\nu}^{(n)}$ is obtained. From the
definition of Eqn. (20),

$\begin{split}\partial_{\mu}C_{\nu}^{(n)}&-\partial_{\nu}C_{\mu}^{(n)}=-\frac{1}{g}[\partial_{\mu}(\hat{n}_{1}.\partial_{\nu}\hat{n}_{2})-\partial_{\nu}(\hat{n}_{1}.\partial_{\mu}\hat{n}_{2})]\\\
&=-\frac{1}{g}(\partial_{\mu}\hat{n}_{1}.\partial_{\nu}\hat{n}_{2}-\partial_{\nu}\hat{n}_{1}.\partial_{\mu}\hat{n}_{2})-\frac{1}{g}\hat{n}_{1}.[\partial_{\mu},\partial_{\nu}]\hat{n}_{2},\end{split}$
(24)
$\begin{split}\partial_{\mu}\hat{n}_{1}.\hat{n}_{1}=0&\Rightarrow\partial_{\mu}\hat{n}_{1}=\alpha_{\mu}^{1}\hat{n}_{2}+\beta_{\mu}^{1}\hat{n},\\\
\partial_{\mu}\hat{n}_{2}.\hat{n}_{2}=0&\Rightarrow\partial_{\mu}\hat{n}_{2}=\alpha_{\mu}^{2}\hat{n}_{1}+\beta_{\mu}^{2}\hat{n},\\\
\end{split}$ (25)
$\begin{split}\hat{n}=\hat{n}_{1}\times\hat{n}_{2}\Rightarrow\partial_{\mu}\hat{n}&=\partial_{\mu}\hat{n}_{1}\times\hat{n}_{2}+\hat{n}_{1}\times\partial_{\nu}\hat{n}_{2}\\\
&=-\beta_{\mu}^{1}\hat{n}_{1}-\beta_{\mu}^{2}\hat{n}_{2}.\end{split}$ (26)

By replacing Eqn. (26) in Eqn. (24)

$\partial_{\mu}C_{\nu}^{(n)}-\partial_{\nu}C_{\mu}^{(n)}=-\frac{1}{g}(\beta_{\mu}^{1}\beta_{\nu}^{2}-\beta_{\nu}^{1}\beta_{\mu}^{2})-\frac{1}{g}\hat{n}_{1}.[\partial_{\mu},\partial_{\nu}]\hat{n}_{2}.$
(27)

On the other hand, by replacing Eqn. (26) in Eqn. (16)

$\begin{split}\partial_{\mu}\hat{n}\times\partial_{\nu}\hat{n}&=(-\beta_{\mu}^{1}\hat{n}_{1}-\beta_{\mu}^{2}\hat{n}_{2})\times(-\beta_{\nu}^{1}\hat{n}_{1}-\beta_{\nu}^{2}\hat{n}_{2})\\\
&=(\beta_{\mu}^{1}\beta_{\nu}^{2}-\beta_{\nu}^{1}\beta_{\mu}^{2})\hat{n}\end{split}$
(28)
$\hat{n}.(\partial_{\mu}\hat{n}\times\partial_{\nu}\hat{n})=(\beta_{\mu}^{1}\beta_{\nu}^{2}-\beta_{\nu}^{1}\beta_{\mu}^{2}).$
(29)

By comparing Eqn. (27) and Eqn. (29), $D_{\mu\nu}^{(n)}$ in Eqn. (23) is
obtained.

For a framework that has no singularity in two-dimensional surface orthogonal
to $\hat{n}$, $D_{\mu\nu}^{(n)}$ is zero. In fact, it can be stated that if
$C_{\mu}^{(n)}$ is the magnetic monopole potential in the third color
direction, then the associated Dirac worldsheet is localized on its orthogonal
surface. Therefore, when $D_{\mu\nu}^{(n)}$ is non-zero, the contribution of
the Dirac worldsheets should be considered .

If the frames were regular, we would have $\vec{L}_{\mu\nu}=\vec{0}$,
$D_{\mu\nu}^{(n)}=0$, and substituting them in Eqn. (15), we would obtain the
Abelianized form given in cho2000 ; lee ; bae ; pak

$\begin{split}\vec{F}_{\mu\nu}&=(\partial_{\mu}(A_{\nu}^{(n)}+C_{\nu}^{(n)})-\partial_{\nu}(A_{\mu}^{(n)}+C_{\mu}^{(n)})+K_{\mu\nu})\hat{n}\\\
&+\hat{D}_{\mu}\vec{X}_{\nu}^{(n)}-\hat{D}_{\nu}\vec{X}_{\mu}^{(n)}.\end{split}$
(30)

Therefore, taking into account a general configuration containing monopoles,
the associated Dirac worldsheets, and center vortices and introducing a local
frame containing these defects, the field strength tensor is calculated by Cho
decomposition method. In the next sections, using the results of this section,
possible defects and the associated consequences are studied.

##  III Monopoles in SU($2$) gauge group

In the previous section, it is shown that magnetic monopoles can be extracted
using Cho decomposition method and by attributing the defects to the local
color frame. In order to make contact with Cho decomposition, using a
nontrivial gauge transformation oxman , a local color frame is defined and the
monopoles are studied as defects of this local color frame. At the end, we
interpret the components of the ultimate field strength tensors.

In general, the Yang-Mills theory can be affected in two different ways from
gauge transformations. Under regular gauge transformations $S\in SU(N)$, the
Yang-Mills action is invariant and

$\begin{split}\vec{A}_{\mu}^{S}.\vec{T}&=S\vec{A}_{\mu}.\vec{T}S^{-1}+\frac{i}{g}S\partial_{\mu}S^{-1},\\\
\vec{F}_{\mu\nu}^{S}.\vec{T}&=S\vec{F}_{\mu\nu}.\vec{T}S^{-1}.\end{split}$
(31)

It should be noted that by a regular gauge transformation, the transformation
and its derivative lack any kind of singularity.
Under nontrivial gauge transformations, the Yang-Mills gauge field and the
field strength change as the following

$\vec{A}_{\mu}^{U}.\vec{T}=U\vec{A}_{\mu}.\vec{T}U^{-1}+\frac{i}{g}U\partial_{\mu}U^{-1},$
(32)
$\vec{F}_{\mu\nu}^{U}.\vec{T}=U\vec{F}_{\mu\nu}.\vec{T}U^{-1}+\frac{i}{g}U[\partial_{\mu},\partial_{\nu}]U^{-1}.$
(33)

These transformations indicate some singularities for the nontrivial gauge
transformations. The singularities may appear in the transformation or its
derivative. Because of the second term in Eqn. (33), the fields
$\vec{A}_{\mu}^{U}$ and $\vec{A}_{\mu}$ are not physically equivalent. As a
result, Eqn. (32) does not show a simple gauge transformation and it may
contain monopole-like defects.

In order to relate the above discussions to Cho decomposition, we need to
define a color frame $\hat{m}_{a},a=1,2,3,$ which is created by the nontrivial
single valued gauge transformation $U$

$UT^{a}U^{-1}=\hat{m}_{a}.\vec{T}\ \ or,\ \hat{m}_{a}=R(U)\hat{e}_{a}.$ (34)

This transformation can be expressed in terms of the Euler angles

$U=e^{-i\alpha T_{3}}e^{-i\beta T_{2}}e^{-i\gamma T_{3}}\ \ ,\ R(U)=e^{\alpha
M_{3}}e^{\beta M_{2}}e^{\gamma M_{3}},$ (35)

where $M_{a}$ are the generators of $SO(3)$ gauge group.

For a general nontrivial gauge transformation, the second term of Eqn. (32) is
obtained as the following oxman ,

$\frac{i}{g}U\partial_{\mu}U^{-1}=-(C_{\mu}^{(m)}\hat{m}+\frac{1}{g}\hat{m}\times\partial_{\mu}\hat{m}).\vec{T}.$
(36)

where $C_{\mu}^{(m)}$ is defined in Eqn. (20).

By applying Eqn. (36) and

$\begin{split}U\vec{A}_{\mu}.\vec{T}U^{-1}&=U(A_{\mu}^{1}T^{1}+A_{\mu}^{2}T^{2}+A_{\mu}^{3}T^{3})U^{-1}\\\
&=A_{\mu}^{1}\hat{m}_{1}.\vec{T}+A_{\mu}^{2}\hat{m}_{2}.\vec{T}+A_{\mu}^{3}\hat{m}.\vec{T},\end{split}$
(37)

to Eqn. (32), we obtain

$\vec{A}_{\mu}^{U}.\vec{T}=[(A_{\mu}^{3}-C_{\mu}^{(m)})\hat{m}-\frac{1}{g}\hat{m}\times\partial_{\mu}\hat{m}+A_{\mu}^{1}\hat{m}_{1}+A_{\mu}^{2}\hat{m}_{2}].\vec{T}.$
(38)

Defining

$A_{\mu}^{(m)}=A_{\mu}^{3}-C_{\mu}^{(m)}\ \ ,\
\vec{X}_{\mu}^{(m)}=A_{\mu}^{1}\hat{m}_{1}+A_{\mu}^{2}\hat{m}_{2},$ (39)

we get

$\vec{A}_{\mu}^{U}=A_{\mu}^{(m)}\hat{m}-\frac{1}{g}\hat{m}\times\partial_{\mu}\hat{m}+\vec{X}_{\mu}^{(m)}.$
(40)

Both representations (32) and (40) are equivalent in describing monopoles
oxman . Monopole-like singularities of the connection $\vec{A}_{\mu}$ are
described in terms of a defect located in the color direction $\hat{m}$
cho2000 ; lee ; bae ; pak . If we choose $A_{\mu}^{1}=A_{\mu}^{2}=0$, the
gauge field $\vec{A}_{\mu}^{U}$ in Eqn. (40) changes to the restricted
potential mentioned in section II. We recall from section II that by
considering a hedgehog configuration for $\hat{m}$ and defining
$A_{\mu}^{(m)}=0$, a Wu-Yang monopole wu  is obtained.

###  III.1 Interpreting the transformed gauge field and field strength tensor

Now, we use the gauge field $\vec{A}_{\mu}^{U}$ obtained in Eqn. (38) and
calculate the field strength tensor and interpret its different terms. Then,
we discuss about the field strength tensor in terms of magnetic defects.

We consider a hedgehog form for $\hat{m}$ as $\hat{m}=\hat{r}$ and calculate
the components of $\vec{A}_{\mu}^{U}$. The hedgehog configuration can be
obtained by choosing $\alpha=\varphi,\beta=\theta,\gamma=\varphi$ for $U$ in
Eqn. (35), where $\theta,\varphi$ are the polar angles associated to
$\hat{r}$.

$U=\left(\begin{array}[]{cc}\cos\frac{\theta}{2}e^{-i\varphi}&-\sin\frac{\theta}{2}\\\
\sin\frac{\theta}{2}&\cos\frac{\theta}{2}e^{i\varphi}\end{array}\right)$ (41)

Using $U$ of Eqn. (41) in Eqn. (34)

$\begin{split}UT^{1}U^{-1}=\hat{m}_{1}.\vec{T}\Longrightarrow\hat{m}_{1}&=\begin{pmatrix}-\sin^{2}\frac{\theta}{2}+\cos^{2}\frac{\theta}{2}\cos
2\varphi\\\ \cos^{2}\frac{\theta}{2}\sin 2\varphi\\\
-\sin\theta\cos\varphi\end{pmatrix}\\\ &=\cos\varphi\
\hat{\theta}+\sin\varphi\ \hat{\varphi}\end{split}$ (42)
$\begin{split}UT^{2}U^{-1}=\hat{m}_{2}.\vec{T}\Longrightarrow\hat{m}_{2}&=\begin{pmatrix}-\cos^{2}\frac{\theta}{2}\sin
2\varphi\\\ \sin^{2}\frac{\theta}{2}+\cos^{2}\frac{\theta}{2}\cos 2\varphi\\\
\sin\theta\sin\varphi\end{pmatrix}\\\ &=-\sin\varphi\
\hat{\theta}+\cos\varphi\ \hat{\varphi}\end{split}$ (43)
$UT^{3}U^{-1}=\hat{m}.\vec{T}\Longrightarrow\hat{m}=\begin{pmatrix}\sin\theta\cos\varphi\\\
\sin\theta\sin\varphi\\\ \cos\theta\end{pmatrix}=\hat{r}$ (44)

$\hat{r}$, $\hat{\theta}$ and $\hat{\varphi}$ are the unit vectors of
spherical coordinates.

By replacing the above equations in Eqn. (38), the components of
$\vec{A}_{\mu}^{U}$ is obtained

$\begin{split}(A_{\mu}^{U})^{1}&=(-\sin^{2}\frac{\theta}{2}+\cos^{2}\frac{\theta}{2}\cos
2\varphi)A_{\mu}^{1}-\cos^{2}\frac{\theta}{2}\sin 2\varphi
A_{\mu}^{2}+\sin\theta\cos\varphi
A_{\mu}^{3}+\frac{1}{g}\sin\varphi\partial_{\mu}\theta-\frac{1}{g}\sin\theta\cos\varphi\partial_{\mu}\varphi\\\
(A_{\mu}^{U})^{2}&=\cos^{2}\frac{\theta}{2}\sin 2\varphi
A_{\mu}^{1}+(\sin^{2}\frac{\theta}{2}+\cos^{2}\frac{\theta}{2}\cos
2\varphi)A_{\mu}^{2}+\sin\theta\sin\varphi
A_{\mu}^{3}-\frac{1}{g}\cos\varphi\partial_{\mu}\theta-\frac{1}{g}\sin\theta\sin\varphi\partial_{\mu}\varphi\\\
(A_{\mu}^{U})^{3}&=-\sin\theta\cos\varphi A_{\mu}^{1}+\sin\theta\sin\varphi
A_{\mu}^{2}+\cos\theta
A_{\mu}^{3}-\frac{1}{g}(1+\cos\theta)\partial_{\mu}\varphi\end{split}$ (45)

The first three terms in all three components are regular fields. We discuss
about the remaining terms and their physical interpretation. The last two
terms in $(A_{\mu}^{U})^{1}$ are proportional to $\frac{1}{r}$ and describes
the contribution of monopoles. For the static monopoles, $\partial_{\mu}$ can
be written as $\vec{\nabla}$.

$\begin{split}\frac{1}{g}\sin\varphi\vec{\nabla}\theta&=\frac{1}{g}\frac{1}{r}\sin\varphi\
\hat{\theta}\\\
-\frac{1}{g}\sin\theta\cos\varphi\vec{\nabla}\varphi&=-\frac{1}{g}\frac{1}{r}\cos\varphi\
\hat{\varphi}\end{split}$ (46)

Monopoles are also found in the last two terms of $(A_{\mu}^{U})^{2}$

$\begin{split}-\frac{1}{g}\cos\varphi\vec{\nabla}\theta&=-\frac{1}{g}\frac{1}{r}\cos\varphi\
\hat{\theta}\\\
-\frac{1}{g}\sin\theta\sin\varphi\vec{\nabla}\varphi&=-\frac{1}{g}\frac{1}{r}\sin\varphi\
\hat{\varphi}\end{split}$ (47)

The last term of $(A_{\mu}^{U})^{3}$ is proportional to $\frac{1}{r}$ and it
diverges at $\theta=0$. Therefore it describes a monopole attached to a Dirac
string.

$-\frac{1}{g}(1+\cos\theta)\vec{\nabla}\varphi=-\frac{1}{g}\frac{1}{r}\frac{1+\cos\theta}{\sin\theta}\
\hat{\varphi}$ (48)

We can obtain the magnetic flux corresponding to the above monopole terms
which penetrates the area inside the closed contour
$c(r,\theta)\equiv\\{(r,\theta,\varphi)|0\leq\varphi<2\pi\\}$

$\begin{split}\Phi^{\text{flux}}&=\frac{1}{g}\int_{0}^{2\pi}r\sin\theta
d\varphi\
\hat{\varphi}\cdot(\sin\varphi\vec{\nabla}\theta-\sin\theta\cos\varphi\vec{\nabla}\varphi)T^{1}\\\
&=\frac{1}{g}\int_{0}^{2\pi}-\sin\theta\cos\varphi\ d\varphi\
T^{1}=0\end{split}$ (49)
$\begin{split}\Phi^{\text{flux}}&=-\frac{1}{g}\int_{0}^{2\pi}r\sin\theta
d\varphi\
\hat{\varphi}\cdot(\cos\varphi\vec{\nabla}\theta+\sin\theta\sin\varphi\vec{\nabla}\varphi)T^{2}\\\
&=-\frac{1}{g}\int_{0}^{2\pi}\sin\theta\sin\varphi\ d\varphi\
T^{2}=0,\end{split}$ (50)

It means that the magnetic flux of monopoles have no contribution in
directions $T^{1}$ and $T^{2}$.

$\begin{split}&\Phi^{\text{flux}}=-\frac{1}{g}\int_{0}^{2\pi}r\sin\theta
d\varphi\ \hat{\varphi}\cdot(1+\cos\theta)\vec{\nabla}\varphi T^{3}\\\
&=-\frac{1}{g}(1+\cos\theta)\int_{0}^{2\pi}d\varphi\
T^{3}=-\frac{4\pi}{g}\frac{1+\cos\theta}{2}T^{3},\end{split}$ (51)

which illustrates the magnetic flux of the monopole attached to a Dirac
string. The total magnetic flux is obtained from the contribution of monopole
in direction $T^{3}$.

Next, we calculate the components of field strength tensor. If we use Eqn.
(33) to obtain the field strength, the Yang-Mills action would not be
invariant under the gauge transformation $U$. In order to get an invariant
Yang-Mills action, we use the following equation for the field strength
tensor.

$\begin{split}\vec{\mathcal{F}}_{\mu\nu}^{U}.\vec{T}&=U\vec{F}_{\mu\nu}.\vec{T}U^{-1}=\vec{F}_{\mu\nu}^{U}.\vec{T}-\frac{i}{g}U[\partial_{\mu},\partial_{\nu}]U^{-1}\\\
&=(\partial_{\mu}\vec{A}_{\nu}^{U}-\partial_{\nu}\vec{A}_{\mu}^{U}+g\vec{A}_{\mu}^{U}\times\vec{A}_{\nu}^{U}).\vec{T}-\frac{i}{g}U[\partial_{\mu},\partial_{\nu}]U^{-1}\end{split}$
(52)

In fact the second term in the first line of the above equation cancels the
existence of the singularity of the first term and as a result a gauge
invariant action is obtained. $\vec{\mathcal{F}}_{\mu\nu}^{U}$ can be written
in three parts

$\begin{split}&\vec{F}_{\mu\nu}^{\text{linear}}=\partial_{\mu}\vec{A}_{\nu}^{U}-\partial_{\nu}\vec{A}_{\mu}^{U},\\\
&\vec{F}_{\mu\nu}^{\text{bilinear}}=g(\vec{A}_{\mu}^{U}\times\vec{A}_{\nu}^{U}),\\\
&\vec{F}_{\mu\nu}^{\text{sing}}=-\frac{i}{g}U[\partial_{\mu},\partial_{\nu}]U^{-1}\end{split}$
(53)

Which agrees with what is obtained in Reference ichie  using Abelian
Projection. Finally, with an Abelian projection, the contribution of the Dirac
and anti-Dirac strings cancel each other and the monopole appears in the
theory.

Figure 1: Emergence of monopoles in Abelian projected QCDichie

Now, using the gauge field $\vec{A}_{\mu}^{U}$ defined in Eqn. (40), we
calculate the linear and bilinear terms of field strength tensor.

$\begin{split}\vec{F}_{\mu\nu}^{\text{linear}}&=(\partial_{\mu}A_{\nu}^{3}-\partial_{\nu}A_{\mu}^{3})\hat{m}+(\partial_{\mu}A_{\nu}^{1}-\partial_{\nu}A_{\mu}^{1})\hat{m}_{1}+(\partial_{\mu}A_{\nu}^{2}-\partial_{\nu}A_{\mu}^{2})\hat{m}_{2}\\\
&+(A_{\nu}^{(m)}\partial_{\mu}\hat{m}-A_{\mu}^{(m)}\partial_{\nu}\hat{m})+(A_{\nu}^{1}\partial_{\mu}\hat{m}_{1}-A_{\mu}^{1}\partial_{\nu}\hat{m}_{1})+(A_{\nu}^{2}\partial_{\mu}\hat{m}_{2}-A_{\mu}^{2}\partial_{\nu}\hat{m}_{2})\\\
&+\frac{1}{g}(\hat{m}_{1}.[\partial_{\mu},\partial_{\nu}]\hat{m}_{2})\hat{m}-\frac{1}{g}\hat{m}\times[\partial_{\mu},\partial_{\nu}]\hat{m}\end{split}$
(54)
$\begin{split}\vec{F}_{\mu\nu}^{\text{bilinear}}&=g(A_{\mu}^{2}A_{\nu}^{3}-A_{\nu}^{2}A_{\mu}^{3})\hat{m}_{1}+g(A_{\mu}^{3}A_{\nu}^{1}-A_{\nu}^{3}A_{\mu}^{1})\hat{m}_{2}+g(A_{\mu}^{1}A_{\nu}^{2}-A_{\nu}^{1}A_{\mu}^{2})\hat{m}\\\
&-(A_{\nu}^{(m)}\partial_{\mu}\hat{m}-A_{\mu}^{(m)}\partial_{\nu}\hat{m})-(A_{\nu}^{1}\partial_{\mu}\hat{m}_{1}-A_{\mu}^{1}\partial_{\nu}\hat{m}_{1})-(A_{\nu}^{2}\partial_{\mu}\hat{m}_{2}-A_{\mu}^{2}\partial_{\nu}\hat{m}_{2})\\\
\end{split}$ (55)

Using the gauge function $U$ of Eqn. (41), the singular term of the field
strength is obtained

$\begin{split}\vec{F}_{\mu\nu}^{\text{sing}}&=(-\frac{1}{g}\sin\varphi[\partial_{\mu},\partial_{\nu}]\theta+\frac{1}{g}\sin\theta\cos\varphi[\partial_{\mu},\partial_{\nu}]\varphi)T^{1}+(\frac{1}{g}\cos\varphi[\partial_{\mu},\partial_{\nu}]\theta+\frac{1}{g}\sin\theta\sin\varphi[\partial_{\mu},\partial_{\nu}]\varphi)T^{2}\\\
&+\frac{1}{g}(1+\cos\theta)[\partial_{\mu},\partial_{\nu}]\varphi T^{3}\\\
&=-\frac{1}{g}(\hat{m}_{1}.[\partial_{\mu},\partial_{\nu}]\hat{m}_{2})\hat{m}+\frac{1}{g}\hat{m}\times[\partial_{\mu},\partial_{\nu}]\hat{m}\end{split}$
(56)

We recall that $\vec{F}_{\mu\nu}^{\text{linear}}$ represents a monopole
attached to a Dirac string, $\vec{F}_{\mu\nu}^{\text{bilinear}}$ describes an
anti-monopole and $\vec{F}_{\mu\nu}^{\text{sing}}$ indicates an anti-Dirac
string. If we add the linear, bilinear, and singular terms, the singularities
are canceled and at the end, only the regular fields are remained and
therefore the full QCD is restored.

$\begin{split}\vec{F}_{\mu\nu}&=(\partial_{\mu}A_{\nu}^{3}-\partial_{\nu}A_{\mu}^{3})\hat{m}+g(A_{\mu}^{1}A_{\nu}^{2}-A_{\nu}^{1}A_{\mu}^{2})\hat{m}\\\
&+(\partial_{\mu}A_{\nu}^{1}-\partial_{\nu}A_{\mu}^{1})\hat{m}_{1}+g(A_{\mu}^{2}A_{\nu}^{3}-A_{\nu}^{2}A_{\mu}^{3})\hat{m}_{1}\\\
&+(\partial_{\mu}A_{\nu}^{2}-\partial_{\nu}A_{\mu}^{2})\hat{m}_{2}+g(A_{\mu}^{3}A_{\nu}^{1}-A_{\nu}^{3}A_{\mu}^{1})\hat{m}_{2}\end{split}$
(57)

It is also possible to obtain the linear and bilinear field strengths using
the gauge field components in Eqn. (45) in a different way. One can easily
show that the result is the same as what is obtained in Eqns. (54) and
(LABEL:beline).

Therefore, using a nontrivial gauge transformation $U$, a local color frame
$\hat{n}_{a}$ is defined and it is shown that it includes magnetic monopole.
The transformed gauge field $\vec{A}_{\mu}^{U}$ in Eqn. (40) takes the form of
Cho decomposition. After calculating the transformed field strength tensor, we
find that it consists of three parts: linear, bilinear and singular. The
linear part describes a monopole attached to a Dirac string, the bilinear part
describes anti-monopole and the singular part describes anti-Dirac string.
When we sum these three parts together, we get full QCD. This result is in
agreement with what has been done in Reference ichie .

###  III.2 Agreement with the results obtained by Cho

So far, monopoles have been extracted for SU($2$) gauge group. Monopole have
already been extracted in a different way by Cho, and their condensation have
been investigated for SU($2$) frank  and SU($3$) frank18  gauge groups. By
imposing the maximal Abelian isometry on $\vec{A}_{\mu}$ which makes $\hat{m}$
a covariant constant, the restricted potential is obtained by Cho et al.

$D_{\mu}\hat{m}=(\partial_{\mu}+gA_{\mu}\times)\hat{m}=0\Rightarrow\hat{A}_{\mu}=A_{\mu}^{(m)}\hat{m}-\frac{1}{g}\hat{m}\times\partial_{\mu}\hat{m}$
(58)

When $A_{\mu}^{(m)}=0$ and $\hat{m}=\hat{r}$, the restricted potential
describes precisely the Wu-Yang monopole.

In this work, using a nontrivial gauge transformation U, we define a local
color frame $\hat{m}_{a}$ that contains monopoles. Under U transformation, the
gauge field is transformed to Eqn. (40) which is in the form of extended Cho
decomposition written in the frame $\hat{m}_{a}$, which contains monopoles. By
setting $A_{\mu}^{1}$ and $A_{\mu}^{2}$ to zero, we reach the restricted Cho
decomposition with $\hat{m}=\hat{r}$ that describes magnetic monopoles.

We are looking for configurations that include monopole and vortex
simultaneously (chains). In the following sections, we define another color
frame $\hat{n}_{a}$ that contains monopole and vortex by applying another
gauge transformation V on top of the previously considered monopole
description. In this paper, we study SU($2$) gauge group. This way can be
extended to higher gauge groups like SU($3$) with some more efforts.

We recall that $\hat{m}=\hat{r}$ corresponds to a Wu-Yang monopole. This
configuration can be obtained with $\alpha=\varphi,\beta=\theta$ in Eqn. (35),
where $\theta,\varphi$ are the polar angles associated to $\hat{r}$. Since
$R\hat{e}_{3}$ is independent of $\gamma$, any choice for $\gamma$ can be
chosen. Using Eqns. (35) and (20) and selecting $\gamma=\varphi$, we have
oxman

$\begin{split}C_{\mu}^{(m)}&=\frac{1}{g}(\cos\beta\partial_{\mu}\alpha+\partial_{\mu}\gamma)\bigg{|}_{\beta=\theta,\alpha=\gamma=\varphi}\\\
&=\frac{1}{g}(\cos\theta+1)\partial_{\mu}\varphi,\end{split}$ (59)

The components of this magnetic potential are

$C_{r}=C_{\theta}=0\ ,\
C_{\varphi}=\frac{1}{g}\frac{1+\cos\theta}{r\sin\theta}.$ (60)

The above magnetic potential has a singularity at $\theta=0$. Therefore, this
magnetic potential describes a Wu-Yang monopole located at the origin and
attached to a Dirac string along the positive z-axis.

The choice of $\gamma$ is associated with the position of the Dirac string
oxman . If one select $\gamma=-\varphi$, magnetic potential changes to

$C_{\mu}^{(m)}=\frac{1}{g}(\cos\theta-1)\partial_{\mu}\varphi.$ (61)

The above magnetic potential has a singularity at $\theta=\pi$. Therefore,
this magnetic potential describes a Wu-Yang monopole located at the origin and
attached to a Dirac string along the negative z-axis.

As explained earlier, assuming that $A_{\mu}^{(m)}=0$, $A_{\mu}^{a}=0$ for
$a=1,2$ and also considering a hedgehog form $\hat{m}=\hat{r}$, a structure
which indicates Wu-Yang monopoles in $4D$ wu  is found. Therefore by replacing
$A_{\mu}^{1}=A_{\mu}^{2}=0$ and $A_{\mu}^{3}=C_{\mu}^{(m)}$ in Eqn. (57), the
field strength is

$\vec{F}_{\mu\nu}=(\partial_{\mu}C_{\nu}^{(m)}-\partial_{\nu}C_{\mu}^{(m)})\hat{m}=F_{\mu\nu}^{(m)}\hat{m}$
(62)

Thus, the Lagrangian density for Wu-Yang monopoles is

$\begin{split}\mathcal{L}&=-\frac{1}{4}\vec{F}_{\mu\nu}.\vec{F}_{\mu\nu}\\\
&=-\frac{1}{4}(\partial_{\mu}C_{\nu}^{(m)}-\partial_{\nu}C_{\mu}^{(m)})^{2},\end{split}$
(63)

where $C_{\mu}^{(m)}$ is the magnetic potential for Wu-Yang monopoles.

The fact that the magnetic potential has a Dirac string singularity is an
undesirable feature for Lagrangian (63). In order to provide a field
theoretical description of the theory, we have to remove the Dirac string
singularity. To do this, we consider the dual field strength tensor
$F_{\mu\nu}^{*(m)}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma(m)}$
that can be described by a regular potential with no Dirac string singularity
cho1 . So the dual magnetic potential $C_{\mu}^{*(m)}$ is defined by

$F^{*(m)}_{\mu\nu}=\partial_{\mu}C_{\mu}^{*(m)}-\partial_{\nu}C_{\mu}^{*(m)}.$
(64)

$C_{\mu}^{*(m)}$ can describe the monopole and does not contain the Dirac
string singularity anymore. Now we can replace $C_{\mu}^{(m)}$ in favor of
$C_{\mu}^{*(m)}$ in the Lagrangian (63).

For a field theoretical description of the monopole, a new field for the
monopole should be introduced. Since a monopole is a point-like object and
does not have any obvious spin structure, it may be described by a complex
scalar field $\phi$. Naturally we would expect that $\phi$ should couple to
$C_{\mu}^{*(m)}$ minimally. By adding a potential and a kinetic term for
$\phi$ cho1 , the Lagrangian for monopoles is completed.

$\begin{split}\mathcal{L}=&-\frac{1}{4}(\partial_{\mu}C_{\nu}^{*(m)}-\partial_{\nu}C_{\mu}^{*(m)})^{2}\\\
&+|(\partial_{\mu}+igC_{\mu}^{*(m)})\phi|^{2}-\frac{m^{2}}{2}(\phi\phi^{*})-\lambda(\phi\phi^{*})^{2}.\end{split}$
(65)

By adding the scalar potential, the spontaneous symmetry breaking would be
possible and one can examine the condensation of monopoles and confinement.

##  IV Center vortices as defects of the local color frame

Center vortices are color magnetic line-like (surface-like) defects in three
(four) dimensions. Their quantization is done in terms of center elements of
the gauge group. When a center vortex is linked by a Wilson loop, the Wilson
loop variable gets an element of the center $Z(N)$.

To observe vortices in the theory, it is not necessary to use a nontrivial
gauge transformation as it is done for monopoles in Eqn. (32). Rather, by
proposing the following configuration, closed thin center vortices are
introduced

$\vec{A}_{\mu}^{\textsl{thin}}.\vec{T}=VA_{\mu}^{a}T^{a}V^{-1}+\frac{i}{g}V\partial_{\mu}V^{-1}-\text{ideal
vortex}$ (66)

where $V\in SU(N)$. The ideal vortex is localized on the three-volume where
the transformation $V$ is discontinuous engel ; rein .

For example for the SU($2$) gauge group, the gauge transformation can be
parametrized by $V=e^{i\varphi T_{3}}$ and we introduce a local basis
$\hat{n}^{\prime}_{a}$ in the color space

$VT^{a}V^{-1}=\hat{n}^{\prime}_{a}.\vec{T}\ \ ,\
\hat{n}^{\prime}_{a}=R_{3}\hat{e}_{a},$ (67) $V=e^{i\varphi
T_{3}}=\left(\begin{array}[]{cc}e^{i\frac{\varphi}{2}}&0\\\
0&e^{-i\frac{\varphi}{2}}\end{array}\right).$ (68)

Using matrix multiplication, the components of the color frame
$\hat{n}^{\prime}$ can be obtained as follows

$VT^{1}V^{-1}=\hat{n}^{\prime}_{1}.\vec{T}\Rightarrow\hat{n}^{\prime}_{1}=\begin{pmatrix}\cos\varphi\\\
-\sin\varphi\\\ 0\end{pmatrix}$ (69)
$VT^{2}V^{-1}=\hat{n}^{\prime}_{2}.\vec{T}\Rightarrow\hat{n}^{\prime}_{2}=\begin{pmatrix}\sin\varphi\\\
\cos\varphi\\\ 0\end{pmatrix}$ (70)
$VT^{3}V^{-1}=\hat{n}^{\prime}_{3}.\vec{T}\Rightarrow\hat{n}^{\prime}_{3}=\begin{pmatrix}0\\\
0\\\ 1\end{pmatrix}$ (71)

Using the definition of (20) and the above definitions for
$\hat{n}^{\prime}_{1}$ and $\hat{n}^{\prime}_{2}$, we calculate the magnetic
potential of vortices

$C_{\mu}^{(v)}=-\frac{1}{g}\hat{n}^{\prime}_{1}.\partial_{\mu}\hat{n}^{\prime}_{2}=-\frac{1}{g}\partial_{\mu}\varphi$
(72) $C_{\mu}^{(v)}=-\frac{1}{g}\partial_{\mu}\varphi\rightarrow
C_{\rho}=C_{z}=0\ \ ,\ C_{\varphi}=-\frac{1}{g}\frac{1}{\rho}$ (73)

where $\rho$ is the distance from the $z$-axis in the cylindrical coordinate
system.

The mapping $V$ is not single valued and

$\frac{i}{g}e^{i\varphi T_{3}}\partial_{\mu}e^{-i\varphi
T_{3}}=\frac{1}{g}\partial_{\mu}\varphi\delta^{a3}T^{a}+\text{ideal vortex}.$
(74)

Replacing Eqn. (74) and
$VA_{\mu}^{a}T^{a}V^{-1}=A_{\mu}^{a}\hat{n}^{\prime}_{a}.\vec{T}$ in Eqn. (66)
leads to the following representation for thin center vortices thinv

$\vec{A}_{\mu}^{\textsl{V}}.\vec{T}=\bigg{(}(\frac{1}{g}\partial_{\mu}\varphi+A_{\mu}^{3})\hat{n}^{\prime}_{3}+A_{\mu}^{1}\hat{n}^{\prime}_{1}+A_{\mu}^{2}\hat{n}^{\prime}_{2}\bigg{)}.\vec{T}.$
(75)

Using the above equation and setting $\hat{n}^{\prime}=\hat{n}^{\prime}_{3}$,
we calculate the field strength tensor. According to what was mentioned in the
previous sections about the effect of gauge transformations on Lagrangian
invariance, if we define the field strength tensor as follows, the Lagrangian
does not remain invariant under gauge transformations

$\begin{split}\vec{F}_{\mu\nu}^{V}&=\partial_{\mu}\vec{A}_{\nu}^{V}-\partial_{\nu}\vec{A}_{\mu}^{V}+g\vec{A}_{\mu}^{V}\times\vec{A}_{\nu}^{V}\\\
&=(\partial_{\mu}A_{\nu}^{3}-\partial_{\nu}A_{\mu}^{3})\hat{n}^{\prime}+g(A_{\mu}^{1}A_{\nu}^{2}-A_{\mu}^{2}A_{\nu}^{1})\hat{n}^{\prime}\\\
&+(\partial_{\mu}A_{\nu}^{1}-\partial_{\nu}A_{\mu}^{1})\hat{n}^{\prime}_{1}+g(A_{\nu}^{3}A_{\mu}^{2}-A_{\mu}^{3}A_{\nu}^{2})\hat{n}^{\prime}_{1}\\\
&+(\partial_{\mu}A_{\nu}^{2}-\partial_{\nu}A_{\mu}^{2})\hat{n}^{\prime}_{2}+g(A_{\mu}^{3}A_{\nu}^{1}-A_{\nu}^{3}A_{\mu}^{1})\hat{n}^{\prime}_{2}\\\
&+\frac{1}{g}[\partial_{\mu}.\partial_{\nu}]\varphi\hat{n}^{\prime},\end{split}$
(76)

where the last term describes the vortex. But if we use the following equation
to calculate the field strength tensor, Lagrangian will be invariant under
gauge transformations.

$\begin{split}\vec{F}_{\mu\nu}^{V}.&\vec{T}=V\vec{F}_{\mu\nu}.\vec{T}V^{-1}\\\
&=(\partial_{\mu}\vec{A}_{\nu}^{V}-\partial_{\nu}\vec{A}_{\mu}^{V}+g\vec{A}_{\mu}^{V}\times\vec{A}_{\nu}^{V}).\vec{T}-\frac{i}{g}V[\partial_{\mu}.\partial_{\nu}]V^{-1}\end{split}$
(77)

where,

$-\frac{i}{g}V[\partial_{\mu}.\partial_{\nu}]V^{-1}=-\frac{1}{g}[\partial_{\mu}.\partial_{\nu}]\varphi
T^{3}=-\frac{1}{g}[\partial_{\mu}.\partial_{\nu}]\varphi\hat{n}^{\prime}$ (78)

Therefore

$\begin{split}\vec{F}_{\mu\nu}^{V}&=(\partial_{\mu}A_{\nu}^{3}-\partial_{\nu}A_{\mu}^{3})\hat{n}^{\prime}+g(A_{\mu}^{1}A_{\nu}^{2}-A_{\mu}^{2}A_{\nu}^{1})\hat{n}^{\prime}\\\
&+(\partial_{\mu}A_{\nu}^{1}-\partial_{\nu}A_{\mu}^{1})\hat{n}^{\prime}_{1}+g(A_{\nu}^{3}A_{\mu}^{2}-A_{\mu}^{3}A_{\nu}^{2})\hat{n}^{\prime}_{1}\\\
&+(\partial_{\mu}A_{\nu}^{2}-\partial_{\nu}A_{\mu}^{2})\hat{n}^{\prime}_{2}+g(A_{\mu}^{3}A_{\nu}^{1}-A_{\nu}^{3}A_{\mu}^{1})\hat{n}^{\prime}_{2},\end{split}$
(79)

which includes regular gauge fields and describes the full QCD.

Now we use Eqn. (76) and calculate the Lagrangian density for describing
center vortices

$\begin{split}\mathcal{L}&=-\frac{1}{4}\vec{F}_{\mu\nu}^{V}.\vec{F}_{\mu\nu}^{V}\\\
&=-\frac{1}{4}\big{(}\partial_{\mu}(A_{\nu}^{3}+\frac{1}{g}\partial_{\nu}\varphi)-\partial_{\nu}(A_{\mu}^{3}+\frac{1}{g}\partial_{\mu}\varphi)\big{)}^{2}\\\
&-\frac{1}{4}(\partial_{\mu}A_{\nu}^{1}-\partial_{\nu}A_{\mu}^{1})^{2}-\frac{1}{4}(\partial_{\mu}A_{\nu}^{2}-\partial_{\nu}A_{\mu}^{2})^{2}\\\
&-\frac{1}{4}g^{2}\bigg{(}(A_{\mu}^{1}A_{\nu}^{2}-A_{\mu}^{2}A_{\nu}^{1})^{2}+(A_{\nu}^{3}A_{\mu}^{2}-A_{\mu}^{3}A_{\nu}^{2})^{2}\\\
&+(A_{\mu}^{3}A_{\nu}^{1}-A_{\nu}^{3}A_{\mu}^{1})^{2}\bigg{)}\end{split}$ (80)

The first term illustrates the kinetic energy of center vortices, and the
other terms illustrate the kinetic energy and interaction between regular
fields. By adding proper scalar field for center vortices as Higgs field, one
can investigate the spontaneous symmetry breaking and condensation of
vortices. Since the vortices are line-like objects, their condensation needs
further investigation.

##  V Correlated monopoles and vortices

In this section, it is shown that by using Cho decomposition method and
parameterizing the gauge fields appropriately, structures involving vortices
attached to monopoles, called chains, appear in the theory. We use an extended
class of frames $\hat{n}_{a}$, obtained by introducing a $V$-sector on top of
the previously considered monopole description oxman . Then, by extending
$U\rightarrow VU$, the frame $\hat{n}_{a},a=1,2,3$ is defined, where $U$ is
the same transformation used in Section III for describing the monopoles.

$(VU)T^{a}(VU)^{-1}=VUT^{a}U^{-1}V^{-1}=\hat{n}_{a}.\vec{T},$ (81)
$\hat{n}_{a}=R(VU)\hat{e}_{a}=R(V)R(U)\hat{e}_{a}=R(V)\hat{m}_{a}.$ (82)

$V\in SU(2)$ is not single valued along any closed loop. For instance, we can
choose $V$ including a rotation that leaves $\hat{m}$ invariant

$V=e^{-i\xi\hat{m}.\vec{T}},$ (83)

where if $\xi$ changes by $2\pi$, a rotation around the vortex is understood.
We can also write

$VU=e^{-i\xi UT_{3}U^{-1}}U=UV_{3}\ \ ,\ V_{3}=e^{-i\xi T_{3}}.$ (84)

With this parametrization, the frame $\hat{n}_{a}$ changes as the following

$\hat{n}_{a}=R_{m}(\xi)\hat{m}_{a}\ \ ,\ R_{m}(\xi)=e^{\xi\hat{m}.\vec{M}}\ \
,\ \hat{n}=\hat{m}.$ (85)

Applying the definition of $\hat{n}_{a}$ to
$C_{\mu}^{(n)}=-\frac{1}{g}\hat{n}_{1}.\partial_{\mu}\hat{n}_{2}$ in Eqn.
(20), the magnetic potential changes as follows

$C_{\mu}^{(n)}=-\frac{1}{g}R_{m}\hat{m}_{1}.[R_{m}(\partial_{\mu}\hat{m}_{2})+(\partial_{\mu}R_{m})\hat{m}_{2}]=C_{\mu}^{(m)}+C_{\mu}^{(v)},$
(86)

where

$\begin{split}C_{\mu}^{(m)}&=-\frac{1}{g}\hat{m}_{1}.\partial_{\mu}\hat{m}_{2},\\\
C_{\mu}^{(v)}&=-\frac{1}{g}\hat{m}_{1}.(R_{m}^{-1}\partial_{\mu}R_{m})\hat{m}_{2}=\frac{1}{g}\partial_{\mu}\xi.\end{split}$
(87)

The magnetic potential $C_{\mu}^{(n)}$ in Eqn. (86) is written as a sum of two
parts: magnetic potentials of monopoles and vortices. $C_{\mu}^{(v)}$
represents the potential of a center vortex in the outside of the core without
any additional ideal vortex, while $C_{\mu}^{(m)}$ represents the magnetic
potential of a monopole discussed in the Section III. Therefore, by
introducing a local frame constructed by $VU$ transformation, the relationship
between monopoles and vortices is determined. That is, the vortex surfaces are
connected by the monopole worldlines.

Using Eqn. (41) and selecting $\xi=-\varphi$, $VU$ is obtained as follows

$VU=\left(\begin{array}[]{cc}\cos\frac{\theta}{2}e^{-i\frac{\varphi}{2}}&-\sin\frac{\theta}{2}e^{-i\frac{\varphi}{2}}\\\
\sin\frac{\theta}{2}e^{i\frac{\varphi}{2}}&\cos\frac{\theta}{2}e^{i\frac{\varphi}{2}}\end{array}\right)$
(88)

Therefore the components of the local frame $\hat{n}_{a}$ is obtained as
follows

$(VU)T^{1}(VU)^{-1}=\hat{n}_{1}.\vec{T}\Rightarrow\hat{n}_{1}=\begin{pmatrix}\cos\theta\cos\varphi\\\
\cos\theta\sin\varphi\\\ -\sin\theta\end{pmatrix}=\hat{\theta}$ (89)
$(VU)T^{2}(VU)^{-1}=\hat{n}_{2}.\vec{T}\Rightarrow\hat{n}_{2}=\begin{pmatrix}-\sin\varphi\\\
\cos\varphi\\\ 0\end{pmatrix}=\hat{\varphi}$ (90)
$(VU)T^{3}(VU)^{-1}=\hat{n}.\vec{T}\Rightarrow\hat{n}=\begin{pmatrix}\sin\theta\cos\varphi\\\
\sin\theta\sin\varphi\\\ \cos\theta\end{pmatrix}=\hat{m}=\hat{r}$ (91)

Using Eqn. (LABEL:cmum)

$C_{\mu}^{(m)}=\frac{1}{g}(1+\cos\theta)\partial_{\mu}\varphi,$ (92)

and selecting $\xi=-\varphi$

$C_{\mu}^{(v)}=\frac{1}{g}\partial_{\mu}\xi\bigg{|}_{\xi=-\varphi}=-\frac{1}{g}\partial_{\mu}\varphi,$
(93)

we have

$C_{\mu}^{(n)}=C_{\mu}^{(m)}+C_{\mu}^{(v)}=\frac{1}{g}\cos\theta\partial_{\mu}\varphi.$
(94)

Eqn. (94) can also be obtained directly using
$C_{\mu}^{(n)}=-\frac{1}{g}\hat{n}_{1}.\partial_{\mu}\hat{n}_{2}$ and using
the $\hat{n}_{1}$ and $\hat{n}_{2}$ components in Eqns. (89) and (90).

Eqn. (94) becomes non-zero for $\theta=0$ and $\pi$. It means that there are
some defects on the positive and negative z-axis. Since this potential is of
vortex type potential, Eqn. (94) represents two vortex-line, one in the
positive z-axis and the other one in the negative z-axis, and they are
connected by a monopole located at the origin oxman

Now we continue the work done in reference oxman  to obtain an explicit
Lagrangian for the defects and their interactions. By calculating the field
strength tensor for a configuration involving monopoles and vortices, we
obtain a Lagrangian density which clearly illustrates the interaction between
monopoles and vortices.

Under $VU$ gauge transformation, the gauge field is transformed as follows

$\begin{split}&\vec{A}_{\mu}^{VU}.\vec{T}=(VU)\vec{A}_{\mu}.\vec{T}(VU)^{-1}+\frac{i}{g}(VU)\partial_{\mu}(VU)^{-1}\\\
&=\bigg{(}(A_{\mu}^{3}-C_{\mu}^{(m)}-C_{\mu}^{(v)})\hat{n}-\frac{1}{g}\hat{n}\times\partial_{\mu}\hat{n}+A_{\mu}^{1}\hat{n}_{1}+A_{\mu}^{2}\hat{n}_{2}\bigg{)}.\vec{T}\\\
&=(A_{\mu}^{VU})^{1}T^{1}+(A_{\mu}^{VU})^{2}T^{2}+(A_{\mu}^{VU})^{3}T^{3},\end{split}$
(95)

where

$\begin{split}(A_{\mu}^{VU})^{1}&=\cos\theta\cos\varphi
A_{\mu}^{1}-\sin\varphi A_{\mu}^{2}+\sin\theta\cos\varphi
A_{\mu}^{3}+\frac{1}{g}\sin\varphi\partial_{\mu}\theta\\\
(A_{\mu}^{VU})^{2}&=\cos\theta\sin\varphi A_{\mu}^{1}+\cos\varphi
A_{\mu}^{2}+\sin\theta\sin\varphi
A_{\mu}^{3}-\frac{1}{g}\cos\varphi\partial_{\mu}\theta\\\
(A_{\mu}^{VU})^{3}&=-\sin\theta A_{\mu}^{1}+\cos\theta
A_{\mu}^{3}-\frac{1}{g}\partial_{\mu}\varphi\end{split}$ (96)

The first three terms of $(A_{\mu}^{VU})^{1}$ and $(A_{\mu}^{VU})^{2}$ include
regular fields and we would like to interpret their last terms

$\begin{split}\frac{1}{g}\sin\varphi\vec{\nabla}\theta&=\frac{1}{g}\frac{1}{r}\sin\varphi\
\hat{\theta}\\\
-\frac{1}{g}\cos\varphi\vec{\nabla}\theta&=-\frac{1}{g}\frac{1}{r}\cos\varphi\
\hat{\theta}.\end{split}$ (97)

These terms are proportional to $\frac{1}{r}$ and describe contribution of a
monopole at the origin. The first two terms of $(A_{\mu}^{VU})^{3}$ include
regular fields and the last term describes contribution of monopole and center
vortex.

Now we calculate all terms of field strength tensor and interpret them.
Similar to what is mentioned in the previous sections, in order to have an
invariant Lagrangian under gauge transformations, we use the following
equation to obtain the field strength tensor

$\begin{split}\vec{\mathcal{F}}_{\mu\nu}^{VU}.\vec{T}&=(VU)\vec{F}_{\mu\nu}(VU)^{-1}\\\
&=(\partial_{\mu}\vec{A}_{\nu}^{VU}-\partial_{\nu}\vec{A}_{\mu}^{VU}+g\vec{A}_{\mu}^{VU}\times\vec{A}_{\nu}^{VU}).\vec{T}-\frac{i}{g}(VU)[\partial_{\mu}.\partial_{\nu}](VU)^{-1},\end{split}$
(98)

Which can be written as a sum of three sentences: linear, bilinear and
singular.

$\begin{split}\vec{F}_{\mu\nu}^{\text{linear}}&=\partial_{\mu}\vec{A}_{\nu}^{VU}-\partial_{\nu}\vec{A}_{\mu}^{VU}\\\
&-(\partial_{\mu}A_{\nu}^{3}-\partial_{\nu}A_{\mu}^{3})\hat{n}+(\partial_{\mu}A_{\nu}^{1}-\partial_{\nu}A_{\mu}^{1})\hat{n}_{1}+(\partial_{\mu}A_{\nu}^{2}-\partial_{\nu}A_{\mu}^{2})\hat{n}_{2}\\\
&+(A_{\nu}^{(n)}\partial_{\mu}\hat{n}-A_{\mu}^{(n)}\partial_{\nu}\hat{n})+(A_{\nu}^{1}\partial_{\mu}\hat{n}_{1}-A_{\mu}^{1}\partial_{\nu}\hat{n}_{1})+(A_{\nu}^{2}\partial_{\mu}\hat{n}_{2}-A_{\mu}^{2}\partial_{\nu}\hat{n}_{2})\\\
&+\frac{1}{g}(\hat{n}_{1}.[\partial_{\mu},\partial_{\nu}]\hat{n}_{2})\hat{n}-\frac{1}{g}\hat{n}\times[\partial_{\mu},\partial_{\nu}]\hat{n}\end{split}$
(99)
$\begin{split}\vec{F}_{\mu\nu}^{\text{bilinear}}&=g\vec{A}_{\mu}^{VU}\times\vec{A}_{\nu}^{VU}\\\
&=g(A_{\mu}^{2}A_{\nu}^{3}-A_{\nu}^{2}A_{\mu}^{3})\hat{n}_{1}+g(A_{\mu}^{3}A_{\nu}^{1}-A_{\nu}^{3}A_{\mu}^{1})\hat{n}_{2}+g(A_{\mu}^{1}A_{\nu}^{2}-A_{\nu}^{1}A_{\mu}^{2})\hat{n}\\\
&-(A_{\nu}^{(n)}\partial_{\mu}\hat{n}-A_{\mu}^{(n)}\partial_{\nu}\hat{n})-(A_{\nu}^{1}\partial_{\mu}\hat{n}_{1}-A_{\mu}^{1}\partial_{\nu}\hat{n}_{1})-(A_{\nu}^{2}\partial_{\mu}\hat{n}_{2}-A_{\mu}^{2}\partial_{\nu}\hat{n}_{2})\end{split}$
(100)
$\begin{split}\vec{F}_{\mu\nu}^{\text{sing}}&=-\frac{i}{g}(VU)[\partial_{\mu}.\partial_{\nu}](VU)^{-1}\\\
&=-\frac{1}{g}\sin\varphi[\partial_{\mu}.\partial_{\nu}]\theta
T^{1}+\frac{1}{g}\cos\varphi[\partial_{\mu}.\partial_{\nu}]\theta
T^{2}+\frac{1}{g}[\partial_{\mu}.\partial_{\nu}]\varphi T^{3}\\\
&=-\frac{1}{g}(\hat{n}_{1}.[\partial_{\mu},\partial_{\nu}]\hat{n}_{2})\hat{n}+\frac{1}{g}\hat{n}\times[\partial_{\mu},\partial_{\nu}]\hat{n},\end{split}$
(101)

where $A_{\mu}^{(n)}=A_{\mu}^{3}-C_{\mu}^{(m)}-C_{\mu}^{(v)}$.
$\vec{F}_{\mu\nu}^{\text{linear}}$ describes the monopole and vortex,
$\vec{F}_{\mu\nu}^{\text{bilinear}}$ describes an anti-monopole and
$\vec{F}_{\mu\nu}^{\text{sing}}$ describes an anti-vortex. However, if we add
the linear, bilinear, and singular field strengths together, eventually the
regular fields remain which describe full QCD.

$\begin{split}\vec{F}_{\mu\nu}&=(\partial_{\mu}A_{\nu}^{3}-\partial_{\nu}A_{\mu}^{3})\hat{n}+g(A_{\mu}^{1}A_{\nu}^{2}-A_{\nu}^{1}A_{\mu}^{2})\hat{n}\\\
&+(\partial_{\mu}A_{\nu}^{1}-\partial_{\nu}A_{\mu}^{1})\hat{n}_{1}+g(A_{\mu}^{2}A_{\nu}^{3}-A_{\nu}^{2}A_{\mu}^{3})\hat{n}_{1}\\\
&+(\partial_{\mu}A_{\nu}^{2}-\partial_{\nu}A_{\mu}^{2})\hat{n}_{2}+g(A_{\mu}^{3}A_{\nu}^{1}-A_{\nu}^{3}A_{\mu}^{1})\hat{n}_{2}\end{split}$
(102)

If we select $A_{\mu}^{1}=A_{\mu}^{2}=0$ and
$A_{\mu}^{3}=C_{\mu}^{(m)}+C_{\mu}^{(v)}$

$\vec{F}_{\mu\nu}=\big{(}\partial_{\mu}(C_{\nu}^{(m)}+C_{\nu}^{(v)})-\partial_{\nu}(C_{\mu}^{(m)}+C_{\mu}^{(v)})\big{)}\hat{n}$
(103)

According to Eqn. (94),

$C_{\mu}^{(m)}+C_{\mu}^{(v)}=\frac{1}{g}\cos\theta\partial_{\mu}\varphi\Rightarrow
C_{r}=C_{\theta}=0,C_{\varphi}=\frac{1}{g}\frac{\cos\theta}{r\sin\theta}$
(104)

$C_{\varphi}$ diverges for $r=0$, $\theta=0$ and $\theta=\pi$. Therefore,
there are some defects at the origin, positive and negative z-axis that
include a monopole at the origin and two vortex-line on positive and negative
z-axis which form a chain. We can write the Lagrangian density for this
configuration

$\begin{split}\mathcal{L}&=-\frac{1}{4}\vec{F}_{\mu\nu}.\vec{F}_{\mu\nu}\\\
&=-\frac{1}{4}\big{(}\partial_{\mu}(C_{\nu}^{(m)}+C_{\nu}^{(v)})-\partial_{\nu}(C_{\mu}^{(m)}+C_{\mu}^{(v)})\big{)}^{2}\\\
&=-\frac{1}{4}(\partial_{\mu}C_{\nu}^{(m)}-\partial_{\nu}C_{\mu}^{(m)})^{2}-\frac{1}{4}(\partial_{\mu}C_{\nu}^{(v)}-\partial_{\nu}C_{\mu}^{(v)})^{2}\\\
&-\frac{1}{2}(\partial_{\mu}C_{\nu}^{(m)}-\partial_{\nu}C_{\mu}^{(m)})(\partial_{\mu}C_{\nu}^{(v)}-\partial_{\nu}C_{\mu}^{(v)})\end{split}$
(105)

The above Lagrangian describes correlated monopole and vortices. The first
term describes kinetic energy of monopole, the second term describes kinetic
energy of vortices and the third term describes the interaction between
monopole and vortices. By adding proper scalar fields for monopole and
vortices as Higgs fields, the spontaneous symmetry breaking and condensations
can be investigated. The components of $F_{\mu\nu}^{\text{linear}}$ and
$F_{\mu\nu}^{\text{bilinear}}$ are calculated using $A_{\mu}^{VU}$ components.
We present these components in Appendix A.

##  VI Conclusions

Even though monopoles and vortices as two of the successful candidates for QCD
magnetic defects, have been able to describe some of the confinement
properties, however neither of them present a complete description of
confinement properties and both have shown some shortcomings in explaining
some other properties of confinement potentials. Correlated monopoles and
center vortices observed in lattice simulations may be the key to the solution
of these shortcomings. A theory which reconciles these two defects may
correctly predict the confining potential behaviors including N-ality
dependence at large distances and Casimir scaling at intermediate distances.

In this paper, we use the Cho decomposition method cho1 ; cho2 ; cho3  and we
apply the successive gauge transformations proposed by Oxman oxman  to observe
the monopole vortex junctions. The correlated monopoles and vortices can be
constructed in terms of a local frame made by the two gauge transformations,
such that the vortex worldsheets is connected by the Monopole worldlines. A
nontrivial gauge transformation that leads to the appearance of monopoles, and
an SU($2$) gauge transformation that is single valued around each closed loop
leads to appearance of center vortices. Next, by comparison with Abelian
Projection scenario which has been discussed in reference ichie , a Lagrangian
density for QCD vacuum in the confining regime is obtained. The Lagrangian
includes kinetic energy of monopoles and vortices as well as a contribution
which describes the interaction between monopole and vortices.

##  Appendix A calculating the components of $F_{\mu\nu}^{\text{linear}}$ and
$F_{\mu\nu}^{\text{bilinear}}$ for correlated monopoles and vortices

In this Appendix using the components of $A_{\mu}^{VU}$ in Eqn. (96), linear
and bilinear field strengths are calculated. Finally we show that summing the
linear, bilinear and singular field strengths leads to full QCD.

$\begin{split}(F_{\mu\nu}^{\text{linear}})^{1}&=\partial_{\mu}(A_{\nu}^{VU})^{1}-\partial_{\nu}(A_{\mu}^{VU})^{1}\\\
&=\cos\theta\cos\varphi(\partial_{\mu}A_{\nu}^{1}-\partial_{\nu}A_{\mu}^{1})-\sin\varphi(\partial_{\mu}A_{\nu}^{2}-\partial_{\nu}A_{\mu}^{2})+\sin\theta\cos\varphi(\partial_{\mu}A_{\nu}^{3}-\partial_{\nu}A_{\mu}^{3})\\\
&-\sin\theta\cos\varphi(A_{\nu}^{1}\partial_{\mu}\theta-
A_{\mu}^{1}\partial_{\nu}\theta)-\cos\theta\sin\varphi(A_{\nu}^{1}\partial_{\mu}\varphi-
A_{\mu}^{1}\partial_{\nu}\varphi)-\cos\varphi(A_{\nu}^{2}\partial_{\mu}\varphi-
A_{\mu}^{2}\partial_{\nu}\varphi)\\\
&+\cos\theta\cos\varphi(A_{\nu}^{3}\partial_{\mu}\theta-
A_{\mu}^{3}\partial_{\nu}\theta)-\sin\theta\sin\varphi(A_{\nu}^{3}\partial_{\mu}\varphi-
A_{\mu}^{3}\partial_{\nu}\varphi)\\\
&+\frac{1}{g}\cos\varphi(\partial_{\mu}\varphi\partial_{\nu}\theta-\partial_{\nu}\varphi\partial_{\mu}\theta)+\frac{1}{g}\sin\varphi[\partial_{\mu},\partial_{\nu}]\theta\end{split}$
(106)
$\begin{split}(F_{\mu\nu}^{\text{linear}})^{2}&=\partial_{\mu}(A_{\nu}^{VU})^{2}-\partial_{\nu}(A_{\mu}^{VU})^{2}\\\
&=\cos\theta\sin\varphi(\partial_{\mu}A_{\nu}^{1}-\partial_{\nu}A_{\mu}^{1})+\cos\varphi(\partial_{\mu}A_{\nu}^{2}-\partial_{\nu}A_{\mu}^{2})+\sin\theta\sin\varphi(\partial_{\mu}A_{\nu}^{3}-\partial_{\nu}A_{\mu}^{3})\\\
&-\sin\theta\sin\varphi(A_{\nu}^{1}\partial_{\mu}\theta-
A_{\mu}^{1}\partial_{\nu}\theta)+\cos\theta\cos\varphi(A_{\nu}^{1}\partial_{\mu}\varphi-
A_{\mu}^{1}\partial_{\nu}\varphi)-\sin\varphi(A_{\nu}^{2}\partial_{\mu}\varphi-
A_{\mu}^{2}\partial_{\nu}\varphi)\\\
&+\cos\theta\sin\varphi(A_{\nu}^{3}\partial_{\mu}\theta-
A_{\mu}^{3}\partial_{\nu}\theta)+\sin\theta\cos\varphi(A_{\nu}^{3}\partial_{\mu}\varphi-
A_{\mu}^{3}\partial_{\nu}\varphi)\\\
&+\frac{1}{g}\sin\varphi(\partial_{\mu}\varphi\partial_{\nu}\theta-\partial_{\nu}\varphi\partial_{\mu}\theta)-\frac{1}{g}\cos\varphi[\partial_{\mu},\partial_{\nu}]\theta\end{split}$
(107)
$\begin{split}(F_{\mu\nu}^{\text{linear}})^{3}&=\partial_{\mu}(A_{\nu}^{VU})^{3}-\partial_{\nu}(A_{\mu}^{VU})^{3}\\\
&=-\sin\theta(\partial_{\mu}A_{\nu}^{1}-\partial_{\nu}A_{\mu}^{1})+\cos\theta(\partial_{\mu}A_{\nu}^{3}-\partial_{\nu}A_{\mu}^{3})-\cos\theta(A_{\nu}^{1}\partial_{\mu}\theta-
A_{\mu}^{1}\partial_{\nu}\theta)-\sin\theta(A_{\nu}^{3}\partial_{\mu}\theta-
A_{\mu}^{3}\partial_{\nu}\theta)\\\
&-\frac{1}{g}[\partial_{\mu},\partial_{\nu}]\varphi\end{split}$ (108)
$\begin{split}(F_{\mu\nu}^{\text{bilinear}})^{1}&=g\big{(}(A_{\mu}^{VU})^{2}(A_{\nu}^{VU})^{3}-(A_{\nu}^{VU})^{2}(A_{\mu}^{VU})^{3}\big{)}\\\
&=g\bigg{(}\cos\varphi\cos\theta(A_{\mu}^{2}A_{\nu}^{3}-A_{\nu}^{2}A_{\mu}^{3})-\sin\varphi(A_{\mu}^{3}A_{\nu}^{1}-A_{\nu}^{3}A_{\mu}^{1})+\sin\theta\cos\varphi(A_{\mu}^{1}A_{\nu}^{2}-A_{\nu}^{1}A_{\mu}^{2})\bigg{)}\\\
&+\sin\theta\cos\varphi(A_{\nu}^{1}\partial_{\mu}\theta-
A_{\mu}^{1}\partial_{\nu}\theta)+\cos\theta\sin\varphi(A_{\nu}^{1}\partial_{\mu}\varphi-
A_{\mu}^{1}\partial_{\nu}\varphi)+\cos\varphi(A_{\nu}^{2}\partial_{\mu}\varphi-
A_{\mu}^{2}\partial_{\nu}\varphi)\\\
&-\cos\theta\cos\varphi(A_{\nu}^{3}\partial_{\mu}\theta-
A_{\mu}^{3}\partial_{\nu}\theta)+\sin\theta\sin\varphi(A_{\nu}^{3}\partial_{\mu}\varphi-
A_{\mu}^{3}\partial_{\nu}\varphi)-\frac{1}{g}\cos\varphi(\partial_{\mu}\varphi\partial_{\nu}\theta-\partial_{\nu}\varphi\partial_{\mu}\theta)\end{split}$
(109)
$\begin{split}(F_{\mu\nu}^{\text{bilinear}})^{2}&=g\big{(}(A_{\mu}^{VU})^{3}(A_{\nu}^{VU})^{1}-(A_{\nu}^{VU})^{3}(A_{\mu}^{VU})^{1}\big{)}\\\
&=g\bigg{(}\sin\varphi\cos\theta(A_{\mu}^{2}A_{\nu}^{3}-A_{\nu}^{2}A_{\mu}^{3})+\cos\varphi(A_{\mu}^{3}A_{\nu}^{1}-A_{\nu}^{3}A_{\mu}^{1})+\sin\theta\sin\varphi(A_{\mu}^{1}A_{\nu}^{2}-A_{\nu}^{1}A_{\mu}^{2})\bigg{)}\\\
&+\sin\theta\sin\varphi(A_{\nu}^{1}\partial_{\mu}\theta-
A_{\mu}^{1}\partial_{\nu}\theta)-\cos\theta\cos\varphi(A_{\nu}^{1}\partial_{\mu}\varphi-
A_{\mu}^{1}\partial_{\nu}\varphi)+\sin\varphi(A_{\nu}^{2}\partial_{\mu}\varphi-
A_{\mu}^{2}\partial_{\nu}\varphi)\\\
&-\cos\theta\sin\varphi(A_{\nu}^{3}\partial_{\mu}\theta-
A_{\mu}^{3}\partial_{\nu}\theta)-\sin\theta\cos\varphi(A_{\nu}^{3}\partial_{\mu}\varphi-
A_{\mu}^{3}\partial_{\nu}\varphi)-\frac{1}{g}\sin\varphi(\partial_{\mu}\varphi\partial_{\nu}\theta-\partial_{\nu}\varphi\partial_{\mu}\theta)\end{split}$
(110)
$\begin{split}(F_{\mu\nu}^{\text{bilinear}})^{3}&=g\big{(}(A_{\mu}^{VU})^{1}(A_{\nu}^{VU})^{2}-(A_{\nu}^{VU})^{1}(A_{\mu}^{VU})^{2}\big{)}\\\
&=g\bigg{(}-\sin\theta(A_{\mu}^{2}A_{\nu}^{3}-A_{\nu}^{2}A_{\mu}^{3})+\cos\theta(A_{\mu}^{1}A_{\nu}^{2}-A_{\nu}^{1}A_{\mu}^{2})\bigg{)}+\cos\theta(A_{\nu}^{1}\partial_{\mu}\theta-
A_{\mu}^{1}\partial_{\nu}\theta)+\sin\theta(A_{\nu}^{3}\partial_{\mu}\theta-
A_{\mu}^{3}\partial_{\nu}\theta)\end{split}$ (111)

Now if we sum $F_{\mu\nu}^{\text{linear}}$, $F_{\mu\nu}^{\text{bilinear}}$ in
the above equations and $F_{\mu\nu}^{\text{sing}}$ in Eqn. (101), we have

$\begin{split}F_{\mu\nu}&=F_{\mu\nu}^{\text{linear}}+F_{\mu\nu}^{\text{bilinear}}+F_{\mu\nu}^{\text{sing}}\\\
&=\big{(}\cos\theta\cos\varphi(\partial_{\mu}A_{\nu}^{1}-\partial_{\nu}A_{\mu}^{1})-\sin\varphi(\partial_{\mu}A_{\nu}^{2}-\partial_{\nu}A_{\mu}^{2})+\sin\theta\cos\varphi(\partial_{\mu}A_{\nu}^{3}-\partial_{\nu}A_{\mu}^{3})\big{)}T^{1}\\\
&+\big{(}\cos\theta\sin\varphi(\partial_{\mu}A_{\nu}^{1}-\partial_{\nu}A_{\mu}^{1})+\cos\varphi(\partial_{\mu}A_{\nu}^{2}-\partial_{\nu}A_{\mu}^{2})+\sin\theta\sin\varphi(\partial_{\mu}A_{\nu}^{3}-\partial_{\nu}A_{\mu}^{3})\big{)}T^{2}\\\
&+\big{(}-\sin\theta(\partial_{\mu}A_{\nu}^{1}-\partial_{\nu}A_{\mu}^{1})+\cos\theta(\partial_{\mu}A_{\nu}^{3}-\partial_{\nu}A_{\mu}^{3})\big{)}T^{3}\\\
&+g\big{(}\cos\theta\cos\varphi(A_{\mu}^{2}A_{\nu}^{3}-A_{\mu}^{3}A_{\nu}^{2})-\sin\varphi(A_{\mu}^{3}A_{\nu}^{1}-A_{\mu}^{1}A_{\nu}^{3})+\sin\theta\cos\varphi(A_{\mu}^{1}A_{\nu}^{2}-A_{\nu}^{1}A_{\mu}^{2})\big{)}T^{1}\\\
&+g\big{(}\cos\theta\sin\varphi(A_{\mu}^{2}A_{\nu}^{3}-A_{\mu}^{3}A_{\nu}^{2})+\cos\varphi(A_{\mu}^{3}A_{\nu}^{1}-A_{\mu}^{1}A_{\nu}^{3})+\sin\theta\sin\varphi(A_{\mu}^{1}A_{\nu}^{2}-A_{\nu}^{1}A_{\mu}^{2})\big{)}T^{2}\\\
&+g\big{(}-\sin\theta(A_{\mu}^{2}A_{\nu}^{3}-A_{\mu}^{3}A_{\nu}^{2})+\cos\theta(A_{\mu}^{1}A_{\nu}^{2}-A_{\nu}^{1}A_{\mu}^{2})\big{)}T^{3}\\\
\end{split}$ (112)

After summing linear, bilinear and singular field strengths, only regular
gauge fields remain and this shows full QCD. One can use the definitions of
$\hat{n}_{1}$, $\hat{n}_{2}$ and $\hat{n}$ in Eqns. (89), (90) and (91) and
rewrite the above equation as follows

$\begin{split}\vec{F}_{\mu\nu}&=(\partial_{\mu}A_{\nu}^{3}-\partial_{\nu}A_{\mu}^{3})\hat{n}+g(A_{\mu}^{1}A_{\nu}^{2}-A_{\nu}^{1}A_{\mu}^{2})\hat{n}\\\
&+(\partial_{\mu}A_{\nu}^{1}-\partial_{\nu}A_{\mu}^{1})\hat{n}_{1}+g(A_{\mu}^{2}A_{\nu}^{3}-A_{\nu}^{2}A_{\mu}^{3})\hat{n}_{1}\\\
&+(\partial_{\mu}A_{\nu}^{2}-\partial_{\nu}A_{\mu}^{2})\hat{n}_{2}+g(A_{\mu}^{3}A_{\nu}^{1}-A_{\nu}^{3}A_{\mu}^{1})\hat{n}_{2}\end{split}$
(113)

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# Contributions of the Cartan generators in potentials between static sources

Seyed Mohsen Hosseini Nejad<EMAIL_ADDRESS>Faculty of Physics,
Semnan University, P.O. Box 35131-19111, Semnan, Iran

###### Abstract

We investigate the contributions of the Cartan generators in the static
potentials for various representations in the framework of the domain model of
center vortices for SU($3$) gauge theory. Using the center domains with the
cores corresponding to only one Cartan generator $H_{8}$, already given as a
particular proposal, leads to some concavities in the potentials for higher
representations. Furthermore, the string tension of the fundamental
representation is the same at Casimir scaling and N-ality regimes. We add the
contribution of the other Cartan generator $H_{3}$ to the potentials and
therefore these shortcomings can be eliminated. However, we discuss that at
intermediate range of distances the potentials induced by only $H_{8}$ agree
with the Casimir scaling better than those corresponding to both Cartan
generators.

PACS. 11.15.Ha, 12.38.Aw, 12.38.Lg, 12.39.Pn

##  I INTRODUCTION

Non-perturbative regime of QCD is dominated by the phenomena of quark
confinement and spontaneous chiral symmetry breaking. Lattice simulations
DelDebbio:1996lih ; Langfeld:1997jx ; DelDebbio:1997ke ; Langfeld:1998cz ;
Engelhardt:1999fd ; Kovacs:1998xm  and infrared models Faber:1997rp ;
Greensite:2006sm ; Engelhardt:1999wr ; Engelhardt:2003wm ; Deldar:2010hw ;
Deldar:2011fh ; Deldar:2009aw ; Nejad:2014hka ; Nejad:2017hka  have shown that
center vortices tHooft:1977nqb ; Vinciarelli:1978kp ; Yoneya:1978dt ;
Cornwall:1979hz ; Mack:1978rq ; Nielsen:1979xu , quantized magnetic flux tubes
in terms of the non-trivial center elements, cause the area law of the Wilson
loop and therefore a linearly rising potential in the infrared regime. In
addition, lattice simulations have indicated that center vortices are also
responsible for the topological charge and spontaneous chiral symmetry
breaking Faber:2017pfl ; Faber:2020 ; deForcrand:1999our ; Engelhardt:2002qs ;
Hollwieser:2008tq ; Bowman:2010zr ; Nejad:2016fcl ; Hollwieser:2013xja ;
Nejad:2018pfl ; Altarawneh:2020pfl . Neglecting dynamical quarks in the
vacuum, there is two non-perturbative ranges of interquark distances with
distinct behaviors. At intermediate distances, the string tension follows the
Casimir scaling rule Deldar:1999vi ; Bali:2000un ; Piccioni:2005un  while the
asymptotic string tension only depends on $N$-ality Kratochvila:2003zj .

The domain structure model can account for the color confinement in terms of
the interaction of the Wilson loops with the center domains Faber:1997rp ;
Greensite:2006sm . In the model, it is assumed that the QCD vacuum is filled
with the domains of the center-vortex type corresponding to the non-trivial
center element and of the vacuum type corresponding to the identity element.
In some literatures given as a particular proposal for SU($3$) Deldar:2001dd
and SU($4$) Deldar:2005ss ; Deldar:2007dr , the core of domains is
corresponding to only one Cartan generator which lead to some shortcomings.

In this article, without this constraint and therefore using all Cartan
generators within the core of domains, we study the static potentials. In the
framework of the domain structure model, we investigate the Yang–Mills vacuum
of the SU($3$) gauge theory including two Cartan generators $H_{8}$ and
$H_{3}$. A convex potential without any concavity should be observed when
interpolating between the short distances and the asymptotic regimes
Bachas:1985xs  while using only one Cartan generator $H_{8}$ for some higher
representations unexpected concavity occurs Deldar:2001dd ;
HosseiniNejad:2019dd . In addition, applying only one Cartan generator
$H_{8}$, the string tension of the fundamental representation sources for
SU($N$)($N=2,3,4$) is the same at intermediate and asymptotic distances
Greensite:2006sm ; Deldar:2001dd ; HosseiniNejad:2019dd ; Deldar:2005ss ;
Deldar:2007dr . However there is no explanation for why the string tension of
the fundamental representation should be the same in these regimes and it is
mentioned as a weakness of the model in Ref. Greensite:2006sm . This equality
can not be explained by fine-tuning one of the parameters in the model. We
analyze the influence of increasing the number of Cartan generators within the
core of domains on these shortcomings. Furthermore we investigate the
contributions of the Cartan generators on the Casimir scaling of potentials at
intermediate distances.

In Sec. II, we briefly explain the domain structure model. We analyze the
contributions of Cartan generators in the potentials in Sec. III. The string
tension of the fundamental representation sources is discussed in Subsec.
III.1. Then, in Subsec. III.2, the Casimir scaling property is investigated.
We discuss about the concavity of the potentials in Subsec. III.3. We
summarize the main points of our study in Sec. IV. Also, in appendix, the
functions of the group factors for several representations which give some
information about the details of potentials are presented.

##  II Domain structures and quark potentials

Any theory of confinement for SU($N$) gauge group should explain some features
of the confining force which can be found in lattice simulations. The quark
potential at intermediate distances, from the onset of the confinement to the
onset of color screening, is expected to rise linearly. In this regime, the
string tension of the potential for the representation $r$ is proportional to
$C_{r}$, the quadratic Casimir of the quark representation $r$, i.e.
$\sigma_{r}\approx\frac{C_{r}}{C_{F}}\sigma_{F}$ where $F$ denotes the
fundamental representation Deldar:1999vi ; Bali:2000un ; Piccioni:2005un . At
large distances, a pair of gluon-anti gluon is created in the vacuum and
combined with initial sources, and Casimir scaling breaks down and the
potential depends on the $N$-ality $k_{r}$ of the representation i.e.
$\sigma_{r}=\sigma(k_{r})$ where the string tension $\sigma(k_{r})$ is related
to the lowest dimensional representation of SU($N$) with $N$-ality $k_{r}$. In
addition, Lattice simulations Bachas:1985xs  have found that the static quark
potential must be everywhere convex i.e.

$\frac{dV}{dr}>0\quad\text{and}\quad\frac{d^{2}V}{dr^{2}}\leq 0.$ (1)

Therefore, a convex potential without any concavity should be observed when
interpolating between the short distances and the asymptotic regimes.

Any theory of the quark confinement should be able to explain these features
of the confining force. The domain structure model has been fairly successful
in describing the mechanism of confinement in QCD Faber:1997rp ;
Greensite:2006sm . The model assumes that the vacuum is filled with domains.
In $SU(N)$, there are $N$ types of domains including $N-1$ types of center
vortices corresponding to the non-trivial center elements of $z_{n}=\exp(i2\pi
n/N)$ where the value $n=1,...,N-1$ and vacuum type corresponding to the
trivial center element $z_{0}=1$ ($n=0$). The effect of a domain on a planar
Wilson loop is to multiply the loop by a group factor

${G}_{r}(\alpha^{n}_{C}(x))={1}/{d_{r}}\text{Tr}~{}\exp[i\vec{\alpha}^{n}_{C}\vec{{H}}],$
(2)

where $d_{r}$ is the dimension of the representation $r$, $\\{H_{i}\\}$ are
generators of the Cartan subalgebra, and the angle $\alpha^{n}_{C}(x)$ depends
on both the Wilson loop $C$ and the position of the domain $x$. If the domain
is all contained within the Wilson loop

$\exp[i\vec{\alpha}^{n}_{C}\vec{{H}}]=(z_{n})^{k_{r}}\mathbb{I},$ (3)

where $k_{r}$ is the $N$-ality of representation $r$. Using this constraint,
one can obtain the maximum value of the angle ${\alpha}^{n}_{max}$. If the
domain is outside the loop

$\exp[i\vec{\alpha}^{n}_{C}\vec{{H}}]=\mathbb{I},$ (4)

and therefore the domain has no effect on the loop.

The quark potential induced by the domains is as follows Faber:1997rp ;
Greensite:2006sm :

$V_{r}(R)=-\sum_{x}\ln(1-\sum^{N-1}_{n=0}f_{n}[1-{\mathrm{Re}}{G}_{r}(\vec{\alpha}^{n}_{C}(x))]),$
(5)

where $f_{n}$ is the probability that any given plaquette is pierced by an nth
domain. An ansatz for the angle $\vec{\alpha}^{n}_{C}$ was introduced by
Greensite $\it{et~{}al.}$ Greensite:2006sm . The color magnetic flux within
each domain with square cross section $A_{d}=L_{d}\times L_{d}$ fluctuates
almost independently in small subregions of area $l^{2}\ll A_{d}$ which $l$ is
a short length of correlation. The only restriction is that the total fluxes
of the subregions must correspond to a center element of the gauge group. This
square ansatz is as follows:

$({\alpha}^{n}_{i}(x))^{2}=\frac{A_{d}}{2\mu}\left[\frac{A}{A_{d}}-\frac{A^{2}}{A_{d}^{2}}\right]+\left(\alpha^{n}_{i\\{max\\}}\frac{A}{A_{d}}\right)^{2},$
(6)

where $A$ is the cross section of the domain overlapping with the minimal area
of the Wilson loop and $\mu$ is a free parameter.

In the next section, we focus on the contribution of the Cartan generators to
the static potentials between color sources for SU($3$) gauge theory.

##  III Contributions of the Cartan generators in the static potentials

The group factor given in Eq. (2) plays an important role in the static
potential and it is included $N-1$ Cartan generators $\\{H_{i}\\}$ in SU($N$)
gauge group. For SU($3$) case, two Cartan generators $H_{3}$ and $H_{8}$ in
the fundamental representation are as the following

${H}_{3}=\frac{1}{2}\mathrm{diag}\big{(}1,-1,0\big{)},$ (7)
${H}_{8}=\frac{1}{2\sqrt{3}}\mathrm{diag}\big{(}1,1,-2\big{)}.$ (8)

All diagonal elements of the Cartan generator $H_{8}$ are non-zero in the
fundamental representation and the other Cartan generator $H_{3}$ comes from
the $SU(2)$ gauge group. Because of this fact, as a particular proposal in
Refs. Deldar:2001dd ; HosseiniNejad:2019dd , only one Cartan generator $H_{8}$
is used in the group factor and the Eq. (2) is reduced as

${G}_{r}(\alpha^{n}_{C}(x))=\frac{1}{d_{r}}{\mathrm{Tr}}\exp[{\mathrm{i}}\alpha^{n}_{8}H_{8}].$
(9)

Therefore, the core of a domain is related to only one Cartan generator
$H_{8}$. In Refs.Deldar:2005ss ; Deldar:2007dr , for SU($4$) with three Cartan
generators is also used only one Cartan generator with all diagonal elements
non-zero in the group factor and the other two Cartan generators come from
$SU(2)$ and $SU(3)$.

Now, we analyze the behavior of static potentials without the constraint that
only one Cartan generator is used in the group factor. Therefore, in $SU(3)$,
we include both $H_{3}$ and $H_{8}$ in the group factor as

${G}_{r}(\alpha^{n}_{C}(x))=\frac{1}{d_{r}}{\mathrm{Tr}}\exp[{\mathrm{i}}(\alpha^{n}_{3}H_{3}+\alpha^{n}_{8}H_{8})].$
(10)

In the next subsections, we investigate the contributions of the Cartan
generators on the properties of the confining force for SU($3$) gauge theory.

###  III.1 The string tension of SU($3$) fundamental representation sources

Applying the model in Refs. Greensite:2006sm ; Deldar:2001dd ;
HosseiniNejad:2019dd ; Deldar:2005ss ; Deldar:2007dr , using only one Cartan
generator in the group factor of the potential, the string tension of the
fundamental representation sources for SU($N$)($N=2,3,4$) is obtained the same
at Casimir scaling and N-ality regimes. This equality can not be explained by
fine-tuning one of the parameters in the model Greensite:2006sm  and there is
no explanation for why the string tension of the fundamental representation
should be the same in these regimes. Now, we analyze the influence of
increasing the number of Cartan generators within the core of domains on the
potential of the fundamental representation. For SU($3$) gauge theory, we add
the second Cartan generator in the potential. In SU($3$), there are two non-
trivial center elements in addition to the trivial center element

$\mathbb{Z}(3)=\\{z_{0}=1,z_{1}=e^{\frac{2\pi i}{3}},z_{2}=e^{\frac{4\pi
i}{3}}\\}.$ (11)

Since $z_{1}=(z_{2})^{*}$, the vortex magnetic flux corresponding to $z_{1}$
is equivalent to an oppositely oriented vortex magnetic flux corresponding to
$z_{2}$. Using the Eq. (3) as well as the Cartan generators of $SU(3)$ given
in Eq. (7) and Eq. (8), the maximum values of the angles
$\alpha_{8\\{max\\}}^{(1)}$ and $\alpha_{3\\{max\\}}^{(1)}$ for the center
vortices are equal to ${4\pi}/{\sqrt{3}}$ and zero respectively. Furthermore,
these maximum values of the angles for the vacuum domains are zero. Therefore,
the square ansatz given in Eq. (6) for the angles of the center vortices are:

$\Bigl{(}\alpha^{1}_{8}(x)\Bigr{)}^{2}=\frac{A_{d}}{2\mu}\left[\frac{A}{A_{d}}-\frac{A^{2}}{A_{d}^{2}}\right]+\left(\frac{4\pi}{\sqrt{3}}\frac{A}{A_{d}}\right)^{2},~{}~{}~{}\\\
\Bigl{(}\alpha^{1}_{3}(x)\Bigr{)}^{2}=\frac{A_{d}}{2\mu}\left[\frac{A}{A_{d}}-\frac{A^{2}}{A^{2}_{d}}\right],$
(12)

and those for the vacuum domains are:

$\Bigl{(}\alpha^{0}_{8}(x)\Bigr{)}^{2}=\frac{A_{d}}{2\mu}\left[\frac{A}{A_{d}}-\frac{A^{2}}{A_{d}^{2}}\right],~{}~{}~{}\\\
\Bigl{(}\alpha^{0}_{3}(x)\Bigr{)}^{2}=\frac{A_{d}}{2\mu}\left[\frac{A}{A_{d}}-\frac{A^{2}}{A^{2}_{d}}\right].$
(13)

Using the square ansatz, the static potential induced by all domains given in
Eq. (5) for SU($3$) is as the following Greensite:2006sm :

$V_{r}(R)=-\sum^{{{L_{d}}/2+R}}_{{x=-{L_{d}}/2}}\ln\Bigl{\\{}(1-f_{1}-f_{0})+f_{1}{\mathrm{Re}}G_{r}[\vec{\alpha}_{C}^{1}(x)+f_{0}{\mathrm{Re}}G_{r}[\vec{\alpha}_{C}^{0}(x)]\Bigr{\\}},$
(14)

where $f_{1}$ and $f_{0}$ are the probabilities that any given plaquette is
pierced by a center vortex and a vacuum domain respectively. Figure 1 shows
the static potentials $V(R)$ for the fundamental representation using all
domains. The free parameters $L_{d}$, $f_{1}$, $f_{0}$, and $L^{2}_{d}/(2\mu)$
are chosen to be $100$, $0.01$, $0.03$, and $4$, respectively. We take the
correlation length $l=1$ and therefore the static potentials are linear from
the beginning ($R=l$). As show, when we use only the Cartan generator $H_{8}$
in the group factor (Eq. (9), the string tension of the fundamental
representation is almost the same at intermediate and asymptotic distances
while by adding the Cartan generator $H_{3}$ within the group factor (Eq.
(10), the string tension of the fundamental representation at intermediate
distances is different with the one at asymptotic distances. At intermediate
distances, as shown in Eq. (12) and Eq. (13), one can find the contribution of
$H_{3}$ and the core of domains is corresponding to $H_{3}$ and $H_{8}$.

Figure 1:  The static potentials induced by all domains in the fundamental
representation. Using only the Cartan generator $H_{8}$, the string tension is
almost the same at intermediate and asymptotic distances. Although $H_{3}$
contribution for large distances is zero, its contribution is non-zero at
intermediate distances. Therefore, adding $H_{3}$ contribution to the
potential, the string tension at intermediate distances is obtained different
value with the one at asymptotic distances. The free parameters are
$L_{d}=100$, $f_{1}=0.01,f_{0}=0.03$, and $L^{2}_{d}/(2\mu)=4$.

At large distances, the angle corresponding to $H_{3}$ is zero
($\alpha_{3}=0$) and therefore the string tension of N-ality regimes through
adding $H_{3}$ contribution to the potential do not change. As a result, only
Cartan generator $H_{8}$ contributes at large regimes while there are the
contributions of both Cartan generators at intermediate regimes. Therefore,
adding $H_{3}$ contribution to the potential, the string tension of the
fundamental representation at intermediate distances can be achieved different
value with the one at asymptotic distances, as shown in Fig. 1.

###  III.2 Casimir scaling of SU($3$) static potentials

Now, we study the contributions of Cartan generators on the Casimir scaling of
potentials for some representations at intermediate distances. As mentioned
above, although the potentials at large distances are governed by only
$H_{8}$, one can add the contribution of $H_{3}$ at intermediate regimes. The
group factors of the potentials within Eq. (14) for the lowest representations
of SU($3$) can be found in the appendix. Figure 2 plots the potential ratios
$V_{\\{6\\}}(R)/V_{\\{3\\}}(R)$ and $V_{\\{8\\}}(R)/V_{\\{3\\}}(R)$ for
different contributions of Cartan generators for center vortices at
intermediate distances. The free parameters are $L_{d}=100$, $f_{1}=0.01$ and
$L^{2}_{d}/(2\mu)=4$. Figure 3 is the same as Fig. 2 but for the vacuum
domains where any given plaquette is pierced by a vacuum domain with the
probability $f_{0}=0.01$. In the range $R\in[0,20]$, these potential ratios
start out at the Casimir ratios ${C_{\\{6\\}}}/{C_{\\{3\\}}}=2.5$ and
${C_{\\{8\\}}}/{C_{\\{3\\}}}=2.25$.

a) b)

Figure 2:  Potential ratios induced by center vortices for different
contributions of the Cartan generators at intermediate distances a) for
${V_{\\{6\\}}(R)}/{V_{\\{3\\}}(R)}$ b) for
${V_{\\{8\\}}(R)}/{V_{\\{3\\}}(R)}$. In the diagrams, ”Vor.” means center
vortices. Using only the Cartan generator $H_{8}$, the potential ratios which
starts from the Casimir ratios violate slowly from the Casimir ratios. Adding
the contribution of $H_{3}$ to the potentials, the slope of the ratios
increases. The potentials induced by the center vortices corresponding to
$H_{8}$ agree with the Casimir scaling better than those corresponding to both
Cartan generators. The free parameters are $L_{d}=100$, $f_{1}=0.01$ and
$L^{2}_{d}/(2\mu)=4$.

a) b)

Figure 3:  The same as Fig. 2 but for the vacuum domains. In the diagrams,
”Vac.” means vacuum domains. The results are the same as those of center
vortices. The free parameters are $L_{d}=100$, $f_{0}=0.01$ and
$L^{2}_{d}/(2\mu)=4$.

As shown in Fig. 2, the potential ratios $V_{\\{6\\}}(R)/V_{\\{3\\}}(R)$ and
$V_{\\{8\\}}(R)/V_{\\{3\\}}(R)$ induced by the center vortices corresponding
to only $H_{8}$ drop slowly from $2.5$ and $2.25$ to about $2.2$ and $2.02$,
respectively. Adding the contribution of $H_{3}$ within the potential ratios
$V_{\\{6\\}}(R)/V_{\\{3\\}}(R)$ and $V_{\\{8\\}}(R)/V_{\\{3\\}}(R)$ obtained
by the center vortices, the slope of the curve increases (the potential ratios
change from $2.5$ and $2.25$ to about $2.12$ and $1.97$). In addition, as
shown in Fig. 3, for the same interval, the potential ratios
$V_{\\{6\\}}(R)/V_{\\{3\\}}(R)$ and $V_{\\{8\\}}(R)/V_{\\{3\\}}(R)$ induced by
the vacuum domains corresponding to only $H_{8}$ drop very slowly from $2.5$
and $2.25$ to about $2.42$ and $2.18$, respectively. Adding the contribution
of $H_{3}$ within the potential ratios $V_{\\{6\\}}(R)/V_{\\{3\\}}(R)$ and
$V_{\\{8\\}}(R)/V_{\\{3\\}}(R)$ obtained by the vacuum domains, the slope of
the curve again increases (the potential ratios change from $2.5$ and $2.25$
to about $2.35$ and $2.14$). It should be noted that the same behavior occurs
for the potential ratios for the higher representations.

As a result, the potentials induced by the domains with the core corresponding
to $H_{8}$ agree with the Casimir scaling better than those obtained from the
domains with the core corresponding to both $H_{8}$ and $H_{3}$.

Figure 4 plots the static potentials $V(R)$ induced by all domains
corresponding to only Cartan generator $H_{8}$ for the lowest representations
in the range $R\in[0,100]$. As shown, Casimir scaling and $N$-ality regimes
for the representation $\\{15s\\}$ do not connect smoothly and some kind of
unexpected concavity occurs which explicitly disagrees with lattice results.
In Ref. Deldar:2010hw , using only center vortices corresponding to $H_{8}$,
the concavity is observed independent of the ansatz for the angle in the
representation $\\{15s\\}$.

Figure 4:  The static potentials induced by all domains corresponding to
$H_{8}$ for several representations. Using only $H_{8}$ contribution in the
potentials, the concavity is observed for the potential of the higher
representation $\\{15s\\}$. The free parameters are $L_{d}=100$,
$f_{1}=f_{0}=0.01$, and $L^{2}_{d}/(2\mu)=4$.

In the next subsection, for removing the concavity of the representation
$\\{15s\\}$, we study the effect of adding the contribution of $H_{3}$ on the
potential.

###  III.3 Concavity in the potential of the representation $\\{15s\\}$

In the static potentials for all representations, the Casimir scaling and
$N$-ality regimes should be connected smoothly to each other without any
concavity. In Ref. HosseiniNejad:2019dd  using only one Cartan generator
$H_{8}$, we add the contribution of the vacuum domain in the potential and
therefore the concavity can be reduced. However this artifact stay and there
is some concavities for higher representations. Now, we analyze the influence
of adding the contribution of $H_{3}$ on the potential with the concavity
especially for the representation $\\{15s\\}$. In Fig. 5, the static potential
for the representation $\\{15s\\}$ is plotted for various contributions of
domains and Cartan generators.

Figure 5:  The potential of the representation $\\{15s\\}$ for various
contributions of domains and Cartan generators. Using the center vortex
contribution corresponding to $H_{8}$ on the vacuum, the concavity occurs for
the potential. Adding the vacuum domain contribution corresponding to $H_{8}$,
the concavity is reduced. However, this artifact stays on the potential. Using
center vortices and vacuum domains corresponding to both $H_{8}$ and $H_{3}$,
the concavity can be eliminated. The free parameters are $L_{d}=100$,
$f_{1}=f_{0}=0.01$, and $L^{2}_{d}/(2\mu)=4$.

As shown, the concavity of the potential induced by the domains with the core
corresponding to $H_{8}$, can be removed through adding $H_{3}$ contribution
in the potential. For the details of the potential in the representation
$\\{15s\\}$, we analyze its group factors. In the appendix, the function of
group factor of the representation $\\{15s\\}$ using both Cartan generators is
given. Figure 6 plots the group factors of the representation $\\{15s\\}$
induced by center vortices for different Cartan generator contributions at
large distances ($R=100$). The Wilson loop legs are located at $x=0$ and
$x=100$. The group factor using only $H_{8}$ or both Cartan generators like
the one of the fundamental representation, interpolates from $-0.5$, when the
vortex core is located entirely within the Wilson loop, to $1$, when the core
is entirely outside the loop. Also, in the same interval, when the core of the
vacuum domain is located entirely within the loop, the group factor reaches to
the trivial value $1$.

Figure 6: The group factors ${\mathrm{Re}}G_{\\{15s\\}}(\alpha)$ of center
vortices versus $x$ for the large size Wilson loop with $R=100$. The group
factor, using only $H_{8}$ or both Cartan generators like the one of the
fundamental representation, interpolates from $-0.5$, when the core of vortex
is contained entirely within the Wilson loop, to $1$, when the core is
entirely outside the loop. It shows that the string tension of N-ality regimes
through adding $H_{3}$ contribution to the potential do not change. The free
parameters are $L_{d}=100$ and $L^{2}_{d}/(2\mu)=4$.

a) b)

Figure 7:  a) The group factors ${\mathrm{Re}}G_{\\{15s\\}}(\alpha)$ of the
center vortices versus $x$ at $R=60$ close to the concavity regime
corresponding to Fig. 5. There are fluctuations with equal large sizes of
maxima and minima around any time-like leg within the center vortex group
factor corresponding to $H_{8}$ leading to the concavity behavior in the
potentials. Adding $H_{3}$ contribution to the center vortex group factor, it
changes smoothly around any time-like leg removing the concavity in the
potential. b) The same as a) but for the vacuum domains where the group factor
using only $H_{8}$ or both Cartan generators changes smoothly. The free
parameters are $L_{d}=100$ and $L^{2}_{d}/(2\mu)=4$.

Figure 7 a) depicts the group factors induced by center vortices for different
Cartan generator contributions in the medium size Wilson loop close to
concavity regime (about $R=60$) for the representation $\\{15s\\}$. Figure 7
b) is the same as Fig. 7 a) but for the vacuum domains. The Wilson loop legs
are located at $x=0$ and $x=60$. For the group factor obtained by center
vortices corresponding to only $H_{8}$, the fluctuations of the center vortex
group factor with equal large sizes of maxima and minima around any time-like
leg lead to the concavity behavior in the potential while these fluctuations
are eliminated somewhat by adding the $H_{3}$ contribution and the group
factor changes almost smoothly around any time-like leg. In addition, as shown
in Fig. 7 b) the group factor obtained by vacuum domains using both Cartan
generators $H_{8}$ and $H_{3}$ changes almost smoothly around any time-like
leg. Therefore the concavity of the potential could be removed by including
the $H_{3}$ contribution to the potential.

Figure 8 a) plots the static potentials induced by all domains using both
Cartan generators in the various representations for the range $R\in[0,200]$.
Furthermore, the potential ratios are shown in Fig. 8 b). Therefore, the
satisfactory potentials for the different representations can be achieved.
Using both Cartan generators in the potential is crucial for obtaining
different string tensions in the intermediate and large distances in
particular for the fundamental representation and also the concavity of the
potentials for higher representations can be eliminated. The potential ratios
start out at the Casimir ratios ${C_{\\{6\\}}}/{C_{\\{3\\}}}=2.5$,
${C_{\\{8\\}}}/{C_{\\{3\\}}}=2.25$, ${C_{\\{10\\}}}/{C_{\\{3\\}}}=4.5$, and
${C_{\\{15s\\}}}/{C_{\\{3\\}}}=7$ and drop slowly from the exact Casimir
scaling for all representations. However the potentials induced by the domains
corresponding to $H_{8}$ agree with the Casimir scaling better than those
obtained from the domains corresponding to both $H_{8}$ and $H_{3}$.

a) b)

Figure 8:  a) The static potentials induced by center vortices and vacuum
domains using both Cartan generators for the various representations of
$SU(3)$. The satisfactory potentials without any concavity can be achieved for
the different representations. b) Potential ratios
$V_{\\{r\\}}(R)/V_{\\{3\\}}(R)$ at the intermediate distances agreeing with
the Casimir scaling. The free parameters are $L_{d}=100$, $f_{1}=0.01$,
$f_{0}=0.05$, and $L^{2}_{d}/(2\mu)=4$.

##  IV Conclusion

In the framework of the domain model of center vortices, we analyze the static
potentials for various representations in SU($3$) Yang-Mills theory. The
contributions of the Cartan generators in potentials between static sources
are investigated. Using only one Cartan generator $H_{8}$ in the potentials,
already given as a particular proposal, there are some concavities in the
potential for higher representations. In addition, the string tension of the
fundamental representation is the same at Casimir scaling and N-ality regimes
while there is no explanation for why the string tension of the fundamental
representation should be the same in these regimes. For analyzing these
shortcomings, we add the contribution of the other Cartan generator $H_{3}$ in
the potentials and therefore the core of a center domain is corresponding to
both Cartan generators. We show that adding $H_{3}$ contribution to the group
factor of the potential the fluctuations of the group factor are removed
somewhat and the group factor changes almost smoothly around any time-like leg
of the Wilson loop. Therefore, the concavities in the potentials can be
eliminated by adding $H_{3}$ contribution on the potentials. Increasing the
number of Cartan generators within the core of domain, the string tension of
the fundamental representation at intermediate distances can be achieved
different value with the one at asymptotic distances. Although the static
potential is governed by only $H_{8}$ at large regimes, there are the
contributions of both Cartan generators at intermediate regimes. Furthermore,
we investigate the contributions of the Cartan generators on the Casimir
scaling of potentials at intermediate distances. The potentials induced by the
domains with the core corresponding to $H_{8}$ agree with the Casimir scaling
better than those obtained from the domains with the core corresponding to
both Cartan generators.

Therefore including all Cartan generators is crucial in obtaining the convex
potentials in agreement with Casimir scaling at intermediate regimes and also
getting different string tensions at Casimir scaling and N-ality regimes.

##  Appendix A  Group factors for several representations of SU($3$)

The Cartan generators $H_{8}^{\\{r\\}}$ and $H_{3}^{\\{r\\}}$ for the
representation $r$ within the group factors of the static potential given in
Eq. (14) can be obtained using the tensor method. The real part of the group
factors using both Cartan generators $H_{8}^{\\{r\\}}$ and $H_{3}^{\\{r\\}}$
in several representations for a center domain can be calculated as:

${\mathrm{Re}}{G}_{\\{3\\}}(\alpha^{n})=\frac{1}{3}[cos(\frac{\alpha_{3}^{n}}{2}+\frac{\alpha_{8}^{n}}{2\sqrt{3}})+cos(\frac{\alpha_{3}^{n}}{2}-\frac{\alpha_{8}^{n}}{2\sqrt{3}})+cos(\frac{\alpha_{8}^{n}}{\sqrt{3}})],$
(15)
${\mathrm{Re}}{G}_{\\{6\\}}(\alpha^{n})=\frac{1}{6}[cos(\alpha_{3}^{n}+\frac{\alpha_{8}^{n}}{\sqrt{3}})+cos(\frac{\alpha_{8}^{n}}{\sqrt{3}})+cos(\frac{\alpha_{3}^{n}}{2}-\frac{\alpha_{8}^{n}}{2\sqrt{3}})+cos(\alpha_{3}^{n}-\frac{\alpha_{8}^{n}}{\sqrt{3}})+cos(\frac{\alpha_{3}^{n}}{2}+\frac{\alpha_{8}^{n}}{2\sqrt{3}})+cos(\frac{2\alpha_{8}^{n}}{\sqrt{3}})],$
(16)
${\mathrm{Re}}{G}_{\\{8\\}}(\alpha^{n})=\frac{1}{8}[2+2cos(\alpha_{3}^{n})+2cos(\frac{\alpha_{3}^{n}}{2}+\frac{3\alpha_{8}^{n}}{2\sqrt{3}})+2cos(\frac{\alpha_{3}^{n}}{2}-\frac{3\alpha_{8}^{n}}{2\sqrt{3}})],$
(17)
${\mathrm{Re}}{G}_{\\{10\\}}(\alpha^{n})=\frac{1}{10}[1+cos(\frac{3\alpha_{3}^{n}}{2}+\frac{3\alpha_{8}^{n}}{2\sqrt{3}})+2cos(\frac{\alpha_{3}^{n}}{2}+\frac{3\alpha_{8}^{n}}{2\sqrt{3}})+2cos(\alpha_{3}^{n})+2cos(\frac{\alpha_{3}^{n}}{2}-\frac{3\alpha_{8}^{n}}{2\sqrt{3}})+cos(\frac{3\alpha_{3}^{n}}{2}-\frac{3\alpha_{8}^{n}}{2\sqrt{3}})+cos(\frac{3\alpha_{8}^{n}}{\sqrt{3}})],$
(18) $\displaystyle{\mathrm{Re}}{G}_{\\{15s\\}}(\alpha^{n})$
$\displaystyle=\frac{1}{15}[cos(2\alpha_{3}^{n}+\frac{2\alpha_{8}^{n}}{\sqrt{3}})+cos(\alpha_{3}^{n}+\frac{2\alpha_{8}^{n}}{\sqrt{3}})+cos(\frac{3\alpha_{3}^{n}}{2}+\frac{\alpha_{8}^{n}}{2\sqrt{3}})+cos(-\alpha_{3}^{n}+\frac{2\alpha_{8}^{n}}{\sqrt{3}})+cos(2\alpha_{3}^{n}-\frac{2\alpha_{8}^{n}}{\sqrt{3}})+$
(19)
$\displaystyle+cos(\frac{3\alpha_{3}^{n}}{2}-\frac{\alpha_{8}^{n}}{2\sqrt{3}})+cos(\frac{\alpha_{3}^{n}}{2}-\frac{5\alpha_{8}^{n}}{2\sqrt{3}})+cos(\frac{\alpha_{3}^{n}}{2}+\frac{5\alpha_{8}^{n}}{2\sqrt{3}})+cos(\frac{4\alpha_{8}^{n}}{\sqrt{3}})+cos(\frac{2\alpha_{8}^{n}}{\sqrt{3}})+$
$\displaystyle+cos(\alpha_{3}^{n}-\frac{\alpha_{8}^{n}}{\sqrt{3}})+cos(\alpha_{3}^{n}+\frac{\alpha_{8}^{n}}{\sqrt{3}})+cos(\frac{\alpha_{3}^{n}}{2}+\frac{\alpha_{8}^{n}}{2\sqrt{3}})+cos(\frac{\alpha_{3}^{n}}{2}-\frac{\alpha_{8}^{n}}{2\sqrt{3}})+cos(\frac{\alpha_{8}^{n}}{\sqrt{3}})].$

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