# Introducing vortices in the continuum using direct and indirect methods

Zahra Asmaee<EMAIL_ADDRESS>Sedigheh Deldar<EMAIL_ADDRESS>Department of Physics, University of Tehran,
P. O. Box 14395/547, Tehran 1439955961, Iran  Motahareh Kiamari
<EMAIL_ADDRESS>School of Particles and Accelerators, IPM,
P. O. Box 19395-5531, Tehran, Iran

###### Abstract

Inspired by direct and indirect maximal center gauge methods which confirm the
existence of vortices in lattice calculations and by using the connection
formalism, we show that under some appropriate gauge transformations vortices
and chains appear in the QCD vacuum of the continuum limit. In the direct
method, by applying center gauge transformation and “center projection,”QCD is
reduced to a gauge theory including vortices, which corresponds to the non-
trivial first homotopy group $\Pi_{1}\left(\text{SO}(3)\right)=Z_{2}.$ On the
other hand, using the indirect method, in addition to the center gauge
transformation and “center projection,”an initial step called Abelian gauge
transformation and then Abelian projection are applied. Therefore, instead of
single vortices, chains that contain monopoles and vortices appear in the
theory.

PACS numbers

12.38.Aw, 12.38.Lg, 12.40.-y.

###### pacs:

Valid PACS appear here

##  I Introduction

Quantum chromodynamics is the non-Abelian gauge theory of the strong
interaction which describes the hadrons in terms of quarks and gluons. There
are many books that discuss QCD; for instance, see Refs. QCD1986 ; QCD2010 .
However, quarks have not been observed as isolated particles in the real
world. Only hadrons (mesons and baryons) are observed as some color singlet
combinations. This experimental fact reflects the confinement mechanism as one
of the most controversial unsolved issues in particle physics in the low-
energy regime or large distances confinement2004 ; conf2011 . During the past
decades, many ideas have been proposed to approach this problem. There are
many articles about this subject; for instance, see Refs. aconfinement1979 ;
a2confinement1999 ; a3confinement1999 ; aconfinement1980 ; aconfinement1981 ;
aconfinement2005 ; aconfinement2008 .

The area law of the Wilson loop average is a well-known gauge-invariant
criterion in studying quark confinement. It leads to a linear potential
between a pair of static quark-antiquark. To study the linear part of the
confinement potential, the quenched approximation is used where the dynamical
quarks are removed for the infrared regime conf2011 . In fact, one can obtain
some collective modes from gluons Ichie  which are associated with some
topological degrees of freedom of the QCD vacuum, and as a result it is
assumed that the QCD vacuum is filled with the topological objects obtained
from these collective modes. Magnetic monopoles and center vortices are among
the main candidates for describing confinement and each has its own fans.

For the non-perturbative description, people use lattice QCD simulations and
phenomenological models to look for the confinement and topological objects.
The results of the phenomenological models must be in agreement with the
results of the lattice QCD, though. In fact, lattice QCD can be served as a
laboratory for confirming the correctness or incorrectness of the
phenomenological models.

In the absence of matter fields, some various mechanisms of confinement have
been suggested to extract the topological degrees of freedom of pure Yang-
Mills theory. One of those mechanisms is the picture of the dual
superconductor and appearance of Abelian monopoles. It was proposed by Nambu
Nambu , Mandelstam Mandelstam , ’t Hooft 't Hooft  and Polyakov Polyakov  in
the 1970s. The idea is that the QCD vacuum can behave like a dual
superconductor and it is filled with magnetic monopoles. Just as the Meissner
effect leads to the condensation of the Cooper pairs as electrically charged
objects in an ordinary superconductor, the magnetic monopoles are condensed in
a dual superconductor and squeeze the chromoelectric flux between the quark-
antiquark pair inside a tube. Therefore, confinement of electric fields is
obtained as a result of the condensation of magnetic monopoles in this picture
Ichie ; Ripka ; kondo .

The second possible mechanism is given by the center vortex model DFG97 ; FGO
; DFG98 ; Greensite ; DFGO97 ; DFGO ; AGG . Historically, vortex-like
structures were introduced in superconductors in 1959. Even though they were
not observed at that time, they were recognized a few years later by Abrikosov
Abrikosov . It was proposed in various forms by ’t Hooft t Hooft78 ; t Hooft80
; t Hooft81 ; t Hooft82 , Nielsen and Olesen NO , Ambjorn and Olesen AO , Mack
and Petkova Mack ; MP , and Cornwall Cornwall  in the late 1970s with a field
theoretical approach. The idea is that the QCD vacuum is filled with closed
magnetic vortices, and it is assumed that the vortices are condensed in the
QCD vacuum. If a Wilson loop is linked to a vortex in an SU($N$) gauge group,
the Wilson loop obtains a phase difference equal to $e^{i2\pi n/N}$ ($n=0$ to
$N-1$) corresponding to the type of the vortex. As a result, some disorders
are created in the lattice which eventually lead to an area law fall-off and
confinement.

Vortices are defined by the center of the SU($N$) gauge group and there exist
$(N-1)$ distinct vortices, which are called non-Abelian ZN vortices. The
simplest vortices are defined by the Z2 gauge group and they have the topology
of tubes (in three Euclidean dimensions) or surfaces of finite thickness (in
four dimensions) carrying some well-defined magnetic fluxes aconfinement1979 ;
AO ; Mack ; MP ; Cornwall ; t Hooft78 ; t Hooft80 ; NO .

Lattice calculations show that ZN vortices produce full string tensions as the
Yang-Mills vacuum does. This is an encouraging motivation to study confinement
via center vortices. If the center vortices are removed from the lattice, the
string tension also disappears DFGO97 ; DFG97 ; DFG98 ; LTER ; ELRT ; GLR .

The vortex condensation picture relies upon center gauge fixing and center
projection. After performing center projection in lattice QCD, the full QCD
with SU($N$) gauge symmetry is reduced to a gauge theory with a Z($N$) gauge
symmetry. These vortices are called projection vortices (or p-vortices).
Unlike monopoles, the modified models of vortices like thick center vortices
can qualitatively explain the Casimir scaling dependence for all
representations FGO .

To study the confinement problem by center vortices, one first has to discuss
the existence of vortices in the continuum limit. The most common methods of
identifying vortices in the lattice simulation are direct maximal center gauge
(DMCG) DFG98  and indirect maximal center gauge (IMCG) DFG97 . Inspired by
these two methods of identifying vortices in lattice calculations and by the
help of the connection formalism Ichie , we discuss the appearance of vortices
in the continuum limit of QCD.

We review DMCG and IMCG methods in lattice QCD in Sec. II. In Sec. III,
motivated by the methods proposed in lattice calculations, we introduce the
vortices in the continuum by direct method for SU($N$) gauge group. As an
example, by applying an appropriate gauge transformation in the SU($2$) gauge
group and using the results of Sec. III, we show in Sec. IV that under the
center gauge transformation the vortex and anti-vortex can appear in the
theory. Then, we remove the term that represents the anti-vortex. The theory
has an SO($3$) symmetry containing the vortex, which corresponds to the non-
trivial first homotopy group of $\Pi_{1}\left(\text{SO}(3)\right)=Z_{2}.$
Removal of the contribution of the anti-vortex is called “center projection”
in our paper. In Sec. V, we introduce thin vortices in the continuum by the
indirect method for SU($N$) gauge group. Sec. VI is brought in two
subsections. In Sec. VI.1, applying an Abelian gauge transformation for
SU($2$) gauge theory, we show that the QCD vacuum is filled with monopoles and
anti-monopoles. It is shown that after Abelian projection the monopole appears
in the vacuum and the gauge group symmetry is reduced from SU($2$) to U($1$)
and we have a monopole vacuum. Then, in Sec. VI.2 we show that under a center
gauge transformation on the monopole vacuum the vortex and anti-vortex appear
in the gauge theory along with the monopole. After applying a “center
projection” we have a gauge theory that contains chains including monopoles
and vortices.

##  II DMCG and IMCG in lattice QCD

There are some methods to identify vortices in lattice calculations and by
using appropriate gauge transformations, which are in agreement with the
vortex condensation picture DFG97 ; FGO ; DFG98 ; DFGO97 ; DFGO .

In lattice QCD, the action is expressed in terms of link variables on which
the gluon fields are defined. The idea is that under an appropriate gauge
transformation the link variables $U_{\mu}(x)$ get as close as possible to the
center gauge group; $\text{center}\left(\text{SU}(N)\right)=Z_{N}$. Then,
after a projection, a smaller set of degrees of freedom remains. This job is
usually done via two methods in lattice QCD calculation. In the following, we
briefly review both methods.

###  II.1 Direct maximal center gauge method

This method was proposed by Del Debbio et al. DFG98 , who tried to maximize
the following quantity by determining the gauge transformation
$G(x)\in\text{SU}(N)$:

$R[U]=\underset{G}{\text{max}}\sum_{x,\mu}\lvert\text{Tr}\
U^{G}_{\mu}(x)\lvert^{2}.$ (1)

$U^{G}_{\mu}(x)=G(x)U_{\mu}(x)G^{\dagger}(x+\hat{\mu})$ shows the gauge
transformation of the link variables $U_{\mu}(x)$ and as a result of the above
maximization, $U^{G}_{\mu}$ becomes as close as possible to the center
elements. $\hat{\mu}$ is a unit vector along the $\mu$ direction. Then, by
performing the center projection, one replaces the transformed link variable
$U^{G}_{\mu}(x)$ by the closest associated center elements of the group ZN. As
an example, the center projection is defined for the SU($2$) gauge group as

$U^{G}_{\mu}(x)\rightarrow Z(2)=\text{sign}\left[\text{Tr}\
U^{G}_{\mu}(x)\right]\textbf{1}=\left\\{+1,-1\right\\}\times\textbf{1},$ (2)

where 1 represents a $2\times 2$ unit matrix. P-vortices identified by the
DMCG method, are related to the non-perturbative degrees of freedom; see Ref.
conf2011  for more details.

###  II.2 Indirect maximal center gauge method

The indirect maximal center gauge method was originally examined in lattice
QCD for the SU($2$) gauge group DFG97 . In general, for an SU($N$) gauge
group, the procedure is as the follows. For the first step, a non-Abelian
gauge configuration is fixed under Abelian gauge fixing, and after Abelian
projection, the SU($N$) gauge symmetry is reduced to
$\left[\text{U}(1)\right]^{N-1}$. In the second step, the remaining
$\left[\text{U}(1)\right]^{N-1}$ configuration is fixed under center gauge
fixing such that the transformed gauge fields become as close as possible to
the center elements. Finally, by performing a center projection, one gets the
center elements. For example, for the SU($2$) gauge group,

$Z(2)=\sum_{x,\mu}\text{sign}\left[cos\theta(x,\mu)\right]\textbf{1}=\left\\{+1,-1\right\\}\times\textbf{1},$
(3)

where $\theta(x,\mu)$ parametrizes the links. In general, in both methods,
identification of the vortices is done by using gauge fixing and then
projection.

##  III  direct method of introducing vortices in the Continuum

In this section, inspired by DMCG method in lattice QCD which confirms the
existence of vortices in the infrared regime, we show that vortices appear in
the theory by a singular gauge transformation. In this procedure, we use the
connection formalism, which is applied for the singular gauge transformation.
We would like to mention that we do not find a continuum formula for maximal
center gauge transformation, Eq.(1), which has already been done in Ref.
ER2017 . Instead, motivated by the fact that vortices exist in the infrared
part of the theory, as shown by the lattice calculations, we use the
connection formalism to make explicit the vortices, somehow similar to the
procedure done in Ref. Ichie  for monopoles.

###  III.1 Link variable and transformed gluon field

In lattice gauge theory, color confinement can be studied by quenched
approximation where only gluon fields exist in the theory. Gluons are defined
on link variables as follows:

$U_{\mu}(x)=e^{iagA_{\mu}(x)}\in\text{SU}(N).$ (4)

Under a gauge transformation $G(x)\in\text{SU}(N)$, the link variables are
transformed as

$U_{\mu}(x)\rightarrow U^{G}_{\mu}(x)=G(x)U_{\mu}(x)G^{\dagger}(x+\hat{\mu}).$
(5)

Using Eq.(4) in the above equation,

$U^{G}_{\mu}(x)=G(x)e^{iagA_{\mu}(x)}G^{\dagger}(x+\hat{\mu}).$ (6)

Since the lattice spacing $a$ is small enough, we can use the exponential
expansion of $e^{iagA_{\mu}(x)},$ and using the Taylor expansion for
$G^{\dagger}(x+\hat{\mu})$,

$\begin{split}U^{G}_{\mu}&=1+iag\left[G(x)A_{\mu}G^{\dagger}(x)-\dfrac{i}{g}G(x)\partial_{\mu}G^{\dagger}(x)\right]+\mathcal{O}(a^{2})\\\
&=e^{iagA^{G}_{\mu}}.\end{split}$ (7)

Thus, in the continuum limit where $a\rightarrow 0$, the gluon field is
transformed as

$A^{G}_{\mu}(x)=G(x)A_{\mu}(x)G^{\dagger}(x)-\dfrac{i}{g}G(x)\partial_{\mu}G^{\dagger}(x),$
(8)

where $A^{G}_{\mu}(x)\in\text{SU}(N)$. In terms of group generators,

$\vec{A}^{G}_{\mu}.\vec{T}=G(x)\left(A^{c}_{\mu}T^{c}\right)G^{\dagger}(x)-\dfrac{i}{g}G(x)\partial_{\mu}G^{\dagger}(x),$
(9)

$T^{c}$ are generators of the SU($N$) group, and $c$ is the color index.

Since we are interested in observing topological defects from the gluon
fields, we have to use an appropriate gauge transformation. Thin vortices
appear as topological defects after center gauge transformations and some
subsequent efforts.

Equation (9) can be used to study the vortices if $G(x)\equiv N(x)$ is defined
as a center gauge transformation,

$\vec{A}^{N}_{\mu}.\vec{T}=N(x)\left(A^{c}_{\mu}T^{c}\right)N^{\dagger}(x)-\dfrac{i}{g}N(x)\partial_{\mu}N^{\dagger}(x).$
(10)

To study the contribution of thin vortices in the continuum, we first recall
that in lattice QCD calculations when the Wilson loop links to the vortex it
receives a phase difference equal to $e^{i2\pi n/N}$ associated with the non-
trivial center element contribution Z$(k)$ kondo ,

$W(C)\rightarrow e^{i2\pi n/N}W(C),\qquad(n=1,2,...,N-1).$ (11)

(a)

(b)

Figure 1: Linking between the Wilson loop $C$ and a vortex with hypersurface
$\Sigma$. The boundary of the hypersurface is shown by $\partial\Sigma$,
indicating a thin vortex.

Under a center gauge transformation $N(x)$, a Wilson line should be
transformed as

$\begin{split}W(C^{\prime})\rightarrow
W^{N}(C^{\prime})&=N(x-\epsilon)W(C^{\prime})N^{\dagger}(x+\epsilon)\\\
&=N(x-\epsilon)N^{\dagger}(x+\epsilon)+\mathcal{O}(\epsilon)\\\ &\equiv
Z(k)+\mathcal{O}(\epsilon),\end{split}$ (12)

$W(C^{\prime})=1+\mathcal{O}(\epsilon)$. $C^{\prime}$ indicates an open circle
from $x-\epsilon$ to $x+\epsilon$, where $x$ indicates the location of the
intersection of $C^{\prime}$ and the hypersurface $\Sigma$. $\epsilon$ is an
infinitesimal quantity so that in the limit where $\epsilon\rightarrow 0$,
$C^{\prime}=C$.

We use Eq.(12) to obtain the appropriate gauge transformation $N(x)$, which
gives the non-trivial center elements indicating the existence of vortices in
the last line of Eq.(12).

From Fig. 1(a), an ideal vortex is defined on $(D-1)$-dimensional hypersurface
$\Sigma$, while the thin vortex is defined on $(D-2)$-dimensional boundary
$S=\partial\Sigma$ kondo ; ER2017 . Piercing the hypersurface by the Wilson
loop results in a discontinuity Z$(k)$. The center vortex in $D=2,3,$ and $4$
is defined as a string, surface, and volume, respectively.

The relation between an ideal vortex and a thin vortex is as follows ER2017 ;
kondo :

$\text{ideal\
vortex}=-\dfrac{i}{g}N(x)\partial_{\mu}N^{\dagger}(x)-\text{thin\ vortex}.$
(13)

In fact, intersecting the hypersurface $\Sigma$ of an ideal vortex with a
Wilson loop $C$ gives a phase to the Wilson loop proportional to a center
group element. The boundary $\partial\Sigma$ indicates the location of the
thin vortex, which is gauge equivalent to the ideal vortex. Thus, the ideal
vortex field is not unique and can be gauge transformed to a thin vortex field
which has the support only on the boundary $\partial\Sigma$ kondo .

Replacing $\left(-\dfrac{i}{g}N(x)\partial_{\mu}N^{\dagger}(x)\right)$ from
Eq.(13) in Eq.(10),

$\vec{A}^{N}_{\mu}.\vec{T}=N(x)\left(A^{c}_{\mu}T^{c}\right)N^{\dagger}(x)+\text{ideal\
vortex}+\text{thin\ vortex}.$ (14)

On the other hand, in analogy to the lattice calculation where the thin vortex
links to the Wilson loop [Fig.1(b)], one can define a gauge field in the coset
space by removing the ideal vortex ER2017  so that
$\vec{A}^{N}_{\mu}.\vec{T}\rightarrow\vec{A}^{\prime N}_{\mu}.\vec{T}$;

$\vec{A}^{\prime
N}_{\mu}.\vec{T}=N(x)\left(A^{c}_{\mu}T^{c}\right)N^{\dagger}(x)+\text{thin\
vortex}.$ (15)

We recall that $\vec{A}^{\prime N}_{\mu}.\vec{T}$ is still singular on
$\partial\Sigma$. The gauge field configuration $\vec{A}^{\prime
N}_{\mu}.\vec{T}$ induces the same behavior for arbitrary Wilson loop as the
configuration $\vec{A}^{N}_{\mu}.\vec{T}$ does. In other words, for $x\notin$
hypersurface, we only see the boundary of the vortex, called the thin vortex
field. In fact, by this choice of $x$, we have removed the hypersurface
$\Sigma$ from the space-time. So, the contribution of the ideal vortex defined
on the hypersurface vortex would be zero;

$\text{thin\ vortex}=-\dfrac{i}{g}N(x)\partial_{\mu}N^{\dagger}(x),\quad
x\notin\text{hypersurface}.$ (16)

###  III.2 Field strength tensor and connection formalism

In this subsection, we discuss the connection formalism, which has already
been used in some references, for instance, Refs. Ichie ; Suzuki . In fact, we
generalize the connection formalism, previously applied to the Abelian gauge
transformation, to the center gauge transformation.

The Yang-Mills Lagrangian has an SU($N$) symmetry and is given by

$\pounds_{\text{YM}}=-\dfrac{1}{2}Tr(F_{\mu\nu}F^{\mu\nu}),$ (17)

where the SU($N$) non-Abelian field strength tensor called
$F_{\mu\nu}=\vec{F}_{\mu\nu}.\vec{T}=F^{c}_{\mu\nu}T^{c}$ is defined by
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+ig\left[A_{\mu},A_{\nu}\right]$
and for a regular system can be written as

$F_{\mu\nu}=\dfrac{1}{ig}\left[\hat{D}_{\mu},\hat{D}_{\nu}\right],\quad\hat{D}_{\mu}=\hat{\partial}_{\mu}+igA_{\mu},$
(18)

where $\hat{D}_{\mu}$ is the covariant-derivative operator.

But topological defects appear as a result of singular gauge transformation.
To observe these defects explicitly, we rewrite the Yang-Mills gauge theory in
terms of the covariant-derivative operator $\hat{D}_{\mu}$ and the ordinary
derivative operator $\hat{\partial}_{\mu}$,

$\begin{split}\dfrac{1}{ig}\left[\hat{D}_{\mu},\hat{D}_{\nu}\right]&=\dfrac{1}{ig}\left[\hat{\partial}_{\mu}+igA_{\mu},\hat{\partial}_{\nu}+igA_{\nu}\right]\\\
&=\dfrac{1}{ig}\left[\hat{\partial}_{\mu},\hat{\partial}_{\nu}\right]+\left[\hat{\partial}_{\mu},A_{\nu}\right]+\left[A_{\mu},\hat{\partial}_{\nu}\right]\\\
&+ig\left[A_{\mu},A_{\nu}\right].\end{split}$ (19)

Using the $\left[\hat{\partial}_{\mu},f\right]=\partial_{\mu}f$,

$\dfrac{1}{ig}\left[\hat{D}_{\mu},\hat{D}_{\nu}\right]=\dfrac{1}{ig}\left[\hat{\partial}_{\mu},\hat{\partial}_{\nu}\right]+\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+ig\left[A_{\mu},A_{\nu}\right].$
(20)

For regular systems, the first term on the right-hand side of Eq.(20) is zero,
so we have Eq.(18). But this term is not zero for singular systems. Therefore,

$F_{\mu\nu}=\dfrac{1}{ig}\left[\hat{D}_{\mu},\hat{D}_{\nu}\right]-\dfrac{1}{ig}\left[\hat{\partial}_{\mu},\hat{\partial}_{\nu}\right],$
(21)

where $F_{\mu\nu}$ is the SU($N$) non-Abelian field strength tensor, and
Eq.(21) is applied when the singularity exists in the system. As a result of
singular gauge transformation, topological defects like monopoles and vortices
appear in the theory.

We study the behavior of the non-Abelian field strength tensor under singular
gauge transformations.

In general, if $G(x)\in\text{SU}(N)$ represents a regular gauge
transformation, the field strength tensor is transformed as
$F^{G}_{\mu\nu}=G(x)F_{\mu\nu}G^{\dagger}(x)$. Therefore, $F_{\mu\nu}$ is

$\begin{split}F^{G}_{\mu\nu}&=G(x)\left\\{\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+ig\left[A_{\mu},A_{\nu}\right]\right\\}G^{\dagger}(x)\\\
&=\left(\partial_{\mu}A^{G}_{\nu}-\partial_{\nu}A^{G}_{\mu}\right)+ig\left[A^{G}_{\mu},A^{G}_{\nu}\right].\end{split}$
(22)

On the other hand, for a singular system where $F_{\mu\nu}$ is defined by
Eq.(21), one gets

$\begin{split}F^{G}_{\mu\nu}&=\dfrac{1}{ig}G(x)\left[\hat{D}_{\mu},\hat{D}_{\nu}\right]G^{\dagger}(x)-\dfrac{1}{ig}G(x)\left[\hat{\partial}_{\mu},\hat{\partial}_{\nu}\right]G^{\dagger}(x)\\\
&=\dfrac{1}{ig}\left[\hat{D}^{G}_{\mu},\hat{D}^{G}_{\nu}\right]-\dfrac{1}{ig}G(x)\left[\hat{\partial}_{\mu},\hat{\partial}_{\nu}\right]G^{\dagger}(x),\end{split}$
(23)

where $\hat{D}^{G}_{\mu}=G(x)\hat{D}_{\mu}G^{\dagger}(x)$. The first term of
Eq.(23) can be written by the help of Eq.(20) for a gauge transformed field,

$\dfrac{1}{ig}\left[\hat{D}^{G}_{\mu},\hat{D}^{G}_{\nu}\right]=\dfrac{1}{ig}\left[\hat{\partial}_{\mu},\hat{\partial}_{\nu}\right]+\partial_{\mu}A^{G}_{\nu}-\partial_{\nu}A^{G}_{\mu}+ig\left[A^{G}_{\mu},A^{G}_{\nu}\right].$
(24)

Replacing Eq.(24) in Eq.(23),

$F^{G}_{\mu\nu}=\left(\partial_{\mu}A^{G}_{\nu}-\partial_{\nu}A^{G}_{\mu}\right)+ig\left[A^{G}_{\mu},A^{G}_{\nu}\right]+\dfrac{i}{g}G\left[{\partial}_{\mu},{\partial}_{\nu}\right]G^{\dagger}.$
(25)

This is a noticeable result. The last term of Eq.(25) shows the difference
between this equation and Eq.(22).

The advantage of using the connection formalism technique is that the gauge
theory will remain gauge invariant after the singular gauge transformation.
Equation (25) is valid for both the Abelian and center gauge transformations.
It has already been discussed for the Abelian gauge transformation Ichie ;
Suzuki  and we intend to use it for the center gauge transformation, as well.

If one uses Eq.(25) without applying any projection, a full QCD will be
obtained at the end. We discuss how we perform “center projection” in Sec. IV.

##  IV  direct method for introducing vortices in SU($2$) gauge group

The formation of center vortices in the QCD vacuum relies upon two steps:
center gauge transformation and “center projection”. Using the results of Sec.
III, we discuss these two steps for the SU($2$) gauge group.

#### Step 1: Center gauge transformation

In general, a $2\times 2$ gauge transformation $G(x)\in\text{SU}(2)$ is
written in terms of three Euler angles $\alpha$,$\beta$,$\gamma$,

$\begin{split}&G(x)=e^{i\gamma(x)T^{3}}e^{i\beta(x)T^{2}}e^{i\alpha(x)T^{3}}\\\
&=\left(\begin{matrix}e^{\dfrac{i}{2}\left[\gamma(x)+\alpha(x)\right]}cos\dfrac{\beta(x)}{2}&e^{\dfrac{i}{2}\left[\gamma(x)-\alpha(x)\right]}sin\dfrac{\beta(x)}{2}\\\
-e^{-\dfrac{i}{2}\left[\gamma(x)-\alpha(x)\right]}sin\dfrac{\beta(x)}{2}&e^{-\dfrac{i}{2}\left[\gamma(x)+\alpha(x)\right]}cos\dfrac{\beta(x)}{2}\end{matrix}\right)\\\
&\alpha(x)\in\left[0,2\pi\right),\ \beta(x)\in\left[0,\pi\right],\
\gamma(x)\in\left[0,2\pi\right),\ T^{c}=\dfrac{\sigma^{c}}{2},\end{split}$
(26)

where $T^{c}$’s are generators of the SU($2$) group and $\sigma^{c}$’s are
Pauli matrices. The center gauge transformation $G(x)\equiv
N(x)\in\text{SU}(2)$ is continuous everywhere except at the hypersurface of
the vortex. Therefore, the Euler angles are selected in a way that the
constraint of Eq.(12) is satisfied. There are different choices for the
angles. One can choose $\alpha=\gamma=\dfrac{\varphi}{2}$ and $\beta=0$,

$N=\left(\begin{matrix}e^{i\dfrac{\varphi}{2}}&0\\\
0&e^{-i\dfrac{\varphi}{2}}\end{matrix}\right),\quad\varphi\in\left[0,2\pi\right).$
(27)

It can be shown that

$N(\varphi=\epsilon)N^{\dagger}(\varphi=2\pi-\epsilon)=-\textbf{1}_{2\times
2}\in Z(2),\ \epsilon\rightarrow 0,$ (28)

where $\left(-\textbf{1}_{2\times 2}\right)$ represents the non-trivial
contribution of the Z($2$) gauge group. Thus, the contribution of an ideal
vortex is observed at $\varphi=0$. On the other hand, outside the
hypersurface, the contribution of the thin vortex is represented by a pure
gauge shown in Eq.(16),

$\text{thin\
vortex}\equiv\vec{V}_{\mu}.\vec{T}=-\dfrac{i}{g}N\partial_{\mu}N^{\dagger}=-\dfrac{1}{g}\partial_{\mu}\varphi
T^{3}.$ (29)

The spatial component of thin vortex is

$\vec{V}_{\varphi}.\vec{T}=-\dfrac{g^{-1}}{\rho}T^{3},\quad\vec{V}_{\rho}.\vec{T}=0.$
(30)

Equation (30) represents the gauge field associated with the thin vortex in
cylindrical coordinates. The thin vortex is observed at $\rho=0$ ER2017  in
the third direction of color space. Under a center gauge transformation, the
gluon field is defined by Eq.(15),

$\vec{A}^{\prime
N}_{\mu}.\vec{T}=N\left(\sum^{3}_{c=1}A^{c}_{\mu}T^{c}\right)N^{\dagger}+\text{thin\
vortex},$ (31)

where the first term on the right-hand side is regular and the second term
indicates a topological defect. Replacing Eqs.(27) and (29) in Eq.(31), one
obtains

$\begin{split}\vec{A}^{\prime
N}_{\mu}.\vec{T}&=\left[A^{1}_{\mu}cos\varphi+A^{2}_{\mu}sin\varphi\right]T^{1}\\\
&+\left[-A^{1}_{\mu}sin\varphi+A^{2}_{\mu}cos\varphi\right]T^{2}\\\
&+\left[A^{3}_{\mu}-\dfrac{1}{g}\partial_{\mu}\varphi\right]T^{3}\end{split}$
(32)

The magnetic vortex flux $\Phi^{\text{flux}}$ is

$\begin{split}\Phi^{\text{flux}}&=\int
dx^{\mu}\left(\vec{V}_{\mu}.\vec{T}\right)=-\dfrac{1}{2g}\int^{2\pi}_{0}d\varphi\left(\begin{matrix}1&0\\\
0&-1\end{matrix}\right)\\\ &=-\dfrac{2\pi}{g}T^{3}.\end{split}$ (33)

The total contribution of the magnetic flux is in the third direction in color
space.

Using Eq.(25) of Sec. III.2 for the transformed field strength,

$\vec{F}^{N}_{\mu\nu}.\vec{T}=\left(\partial_{\mu}\left(\vec{A}^{\prime
N}_{\nu}.\vec{T}\right)-\partial_{\nu}\left(\vec{A}^{\prime
N}_{\mu}.\vec{T}\right)\right)+ig\left[\vec{A}^{\prime
N}_{\mu}.\vec{T},\vec{A}^{\prime
N}_{\nu}.\vec{T}\right]+\dfrac{i}{g}N\left[{\partial}_{\mu},{\partial}_{\nu}\right]N^{\dagger}.$
(34)

With the help of Eq.(32), we rewrite the first term of Eq.(34) which is linear
in terms of $\vec{A}^{\prime N}_{\mu}.\vec{T}$,

$\begin{split}F&{}^{\text{linear}}_{\mu\nu}=\vec{F}^{\text{linear}}_{\mu\nu}.\vec{T}\equiv\partial_{\mu}\left(\vec{A}^{\prime
N}_{\nu}.\vec{T}\right)-\partial_{\nu}\left(\vec{A}^{\prime
N}_{\mu}.\vec{T}\right)\\\
&=\left[\left(\partial_{\mu}A^{1}_{\nu}-\partial_{\nu}A^{1}_{\mu}\right)cos\varphi+\left(\partial_{\mu}A^{2}_{\nu}-\partial_{\nu}A^{2}_{\mu}\right)sin\varphi\right]T^{1}+\left[-\left(\partial_{\mu}A^{1}_{\nu}-\partial_{\nu}A^{1}_{\mu}\right)sin\varphi+\left(\partial_{\mu}A^{2}_{\nu}-\partial_{\nu}A^{2}_{\mu}\right)cos\varphi\right]T^{2}\\\
&+\left[\partial_{\mu}A^{3}_{\nu}-\partial_{\nu}A^{3}_{\mu}\right]T^{3}+\left[-g\left(A^{1}_{\nu}\dfrac{1}{g}\partial_{\mu}\varphi-A^{1}_{\mu}\dfrac{1}{g}\partial_{\nu}\varphi\right)sin\varphi+g\left(A^{2}_{\nu}\dfrac{1}{g}\partial_{\mu}\varphi-A^{2}_{\mu}\dfrac{1}{g}\partial_{\nu}\varphi\right)cos\varphi\right]T^{1}\\\
&+\left[-g\left(A^{1}_{\nu}\dfrac{1}{g}\partial_{\mu}\varphi-A^{1}_{\mu}\dfrac{1}{g}\partial_{\nu}\varphi\right)cos\varphi-g\left(A^{2}_{\nu}\dfrac{1}{g}\partial_{\mu}\varphi-A^{2}_{\mu}\dfrac{1}{g}\partial_{\nu}\varphi\right)sin\varphi\right]T^{2}+\left(-\dfrac{1}{g}\left[\partial_{\mu},\partial_{\nu}\right]\varphi\right)T^{3}.\end{split}$
(35)

The first three sets of brackets of Eq.(35) are regular, and the fourth and
the fifth sets of brackets indicate some kind of interactions between thin
vortex and the off-diagonal gluon fields.
$\left(-\dfrac{1}{g}\left[\partial_{\mu},\partial_{\nu}\right]\varphi\right)T^{3}$
represents the field strength of a thin vortex field carrying a magnetic flux,
which is equal to $\Phi^{\text{flux}}=-\dfrac{2\pi}{g}T^{3}$. The second term
of Eq.(34) can be written with the help of Eq.(32),

$\begin{split}F&{}^{\text{bilinear}}_{\mu\nu}=\vec{F}^{\text{bilinear}}_{\mu\nu}.\vec{T}\equiv
ig\left[\vec{A}^{\prime N}_{\mu}.\vec{T},\vec{A}^{\prime
N}_{\nu}.\vec{T}\right]\\\
&=\left[g\left(A^{1}_{\mu}A^{3}_{\nu}-A^{1}_{\nu}A^{3}_{\mu}\right)sin\varphi-g\left(A^{2}_{\mu}A^{3}_{\nu}-A^{2}_{\nu}A^{3}_{\mu}\right)cos\varphi\right]T^{1}+\left[g\left(A^{1}_{\mu}A^{3}_{\nu}-A^{1}_{\nu}A^{3}_{\mu}\right)cos\varphi+g\left(A^{2}_{\mu}A^{3}_{\nu}-A^{2}_{\nu}A^{3}_{\mu}\right)sin\varphi\right]T^{2}\\\
&+g\left[A^{2}_{\mu}A^{1}_{\nu}-A^{1}_{\mu}A^{2}_{\nu}\right]T^{3}-\left[-g\left(A^{1}_{\nu}\dfrac{1}{g}\partial_{\mu}\varphi-A^{1}_{\mu}\dfrac{1}{g}\partial_{\nu}\varphi\right)sin\varphi+g\left(A^{2}_{\nu}\dfrac{1}{g}\partial_{\mu}\varphi-A^{2}_{\mu}\dfrac{1}{g}\partial_{\nu}\varphi\right)cos\varphi\right]T^{1}\\\
&-\left[-g\left(A^{1}_{\nu}\dfrac{1}{g}\partial_{\mu}\varphi-A^{1}_{\mu}\dfrac{1}{g}\partial_{\nu}\varphi\right)cos\varphi-g\left(A^{2}_{\nu}\dfrac{1}{g}\partial_{\mu}\varphi-A^{2}_{\mu}\dfrac{1}{g}\partial_{\nu}\varphi\right)sin\varphi\right]T^{2}.\end{split}$
(36)

The first three brackets of Eq.(36) represent interactions between gluon
fields and are regular. The fourth and the fifth brackets indicate
interactions between the thin vortex and off-diagonal gluon fields but with an
opposite sign compared with their counterparts in Eq.(35). Back to Eq.(34),
the last term can be rewritten with the help of Eq.(27),

$\vec{F}^{\text{singular}}_{\mu\nu}.\vec{T}\equiv\dfrac{i}{g}N\left[{\partial}_{\mu},{\partial}_{\nu}\right]N^{\dagger}=\dfrac{1}{g}\left[\partial_{\mu},\partial_{\nu}\right]\varphi
T^{3}.$ (37)

$F^{\text{singular}}_{\mu\nu}=\vec{F}^{\text{singular}}_{\mu\nu}.\vec{T}$ in
the above equation indicates the field strength of an anti-thin vortex
carrying a magnetic flux equal to $\Phi^{\text{flux}}=+\dfrac{2\pi}{g}T^{3}$.

#### Step 2: “Center projection”

Adding $F^{\text{linear}}_{\mu\nu}$, $F^{\text{bilinear}}_{\mu\nu}$, and
$F^{\text{singular}}_{\mu\nu}$ together, one finds out that the similar terms
with opposite signs cancel each other in $F^{\text{bilinear}}_{\mu\nu}$ and
$F^{\text{linear}}_{\mu\nu}$. On the other hand, the anti-thin vortex field
strength tensor contribution represented by $F^{\text{singular}}_{\mu\nu}$ is
canceled by the thin vortex field strength tensor contribution brought in the
last term of $F^{\text{linear}}_{\mu\nu}$, and finally, one is left with a
full QCD field strength tensor.

In fact, with the above parametrization, one can argue that the vacuum is
filled with thin vortices and anti-thin vortices, (see Fig.(2)).

To have only the contribution of the thin vortices, we remove the
$F^{\text{singular}}_{\mu\nu}$ term in Fig.(2). We call this procedure “center
projection” and the “center projected field strength tensor” is defined as
follows:

$\begin{split}\vec{F}&{}^{\text{CP}}_{\mu\nu}.\vec{T}\equiv\vec{F}^{\text{linear}}_{\mu\nu}.\vec{T}+\vec{F}^{\text{bilinear}}_{\mu\nu}.\vec{T}\\\
&=\left[\left(\partial_{\mu}A^{1}_{\nu}-\partial_{\nu}A^{1}_{\mu}\right)cos\varphi+\left(\partial_{\mu}A^{2}_{\nu}-\partial_{\nu}A^{2}_{\mu}\right)sin\varphi+g\left(A^{1}_{\mu}A^{3}_{\nu}-A^{1}_{\nu}A^{3}_{\mu}\right)sin\varphi-g\left(A^{2}_{\mu}A^{3}_{\nu}-A^{2}_{\nu}A^{3}_{\mu}\right)cos\varphi\right]T^{1}\\\
&+\left[-\left(\partial_{\mu}A^{1}_{\nu}-\partial_{\nu}A^{1}_{\mu}\right)sin\varphi+\left(\partial_{\mu}A^{2}_{\nu}-\partial_{\nu}A^{2}_{\mu}\right)cos\varphi+g\left(A^{1}_{\mu}A^{3}_{\nu}-A^{1}_{\nu}A^{3}_{\mu}\right)cos\varphi+g\left(A^{2}_{\mu}A^{3}_{\nu}-A^{2}_{\nu}A^{3}_{\mu}\right)sin\varphi\right]T^{2}\\\
&+\left[\partial_{\mu}A^{3}_{\nu}-\partial_{\nu}A^{3}_{\mu}+g\left(A^{2}_{\mu}A^{1}_{\nu}-A^{1}_{\mu}A^{2}_{\nu}\right)\right]T^{3}+\left(-\dfrac{1}{g}\left[\partial_{\mu},\partial_{\nu}\right]\varphi\right)T^{3}.\end{split}$
(38)

All the terms in Eq.(38) are regular except the last term, which represents
the field strength of a thin vortex field.

Therefore, the gauge symmetry is SO($3$), and thin vortices appear as the
topological defects corresponding to the non-trivial first homotopy group
$\Pi_{1}\left(\text{SO}(3)\right)=Z_{2}$.

Figure 2: Appearance of a vortex and an anti-vortex under center gauge
transformation.

##  V  indirect method of introducing vortices in the continuum

In this section, motivated by the IMCG method in lattice QCD which confirms
the existence of vortices, we study vortices in the continuum. As mentioned in
Sec. II.2, in the indirect method, in addition to the center gauge
transformation and center projection DFG97 , an initial step including Abelian
gauge transformation and then Abelian projection is done.

Two successive gauge transformations are performed such that the first one is
an Abelian gauge transformation $M(x)\in\text{SU}(N)$ and the second one is a
center gauge transformation $N(x)\in\text{SU}(N)$. The transformation of link
variables is

$U_{\mu}(x)\overset{M(x)}{\longrightarrow}U^{M}_{\mu}(x)\overset{N(x)}{\longrightarrow}U^{NM}_{\mu}(x)$
(39)

or

$\begin{split}U^{NM}_{\mu}(x)&=N(x)U^{M}_{\mu}(x)N^{\dagger}(x+\hat{\mu})\\\
&=N(x)M(x)U_{\mu}(x)M^{\dagger}(x+\hat{\mu})N^{\dagger}(x+\hat{\mu})\\\
&=N(x)M(x)e^{iagA_{\mu}}M^{\dagger}(x+\hat{\mu})N^{\dagger}(x+\hat{\mu}).\end{split}$
(40)

In the last equality, we used Eq.(4). Similar to what we have done in Sec.
III.1, we can use the exponential expansion and by using the Taylor expansion
for $M^{\dagger}(x+\hat{\mu})$ and $N^{\dagger}(x+\hat{\mu})$,

$\begin{split}U^{NM}_{\mu}&=1+iag\left(N(x)\left[M(x)A_{\mu}M^{\dagger}(x)-\dfrac{i}{g}M(x)\partial_{\mu}M^{\dagger}(x)\right]N^{\dagger}(x)-\dfrac{i}{g}N(x)\partial_{\mu}N^{\dagger}(x)\right)+\mathcal{O}(a^{2})\\\
&=e^{iagA^{NM}_{\mu}}.\end{split}$ (41)

For the continuum limit where $a\rightarrow 0$,

$A^{NM}_{\mu}(x)=N(x)\left[M(x)A_{\mu}M^{\dagger}(x)-\dfrac{i}{g}M(x)\partial_{\mu}M^{\dagger}(x)\right]N^{\dagger}(x)-\dfrac{i}{g}N(x)\partial_{\mu}N^{\dagger}(x).$
(42)

$N(x)$ indicates a center gauge transformation, and the contribution of
vortices must be obtained from this gauge transformation. Therefore, similar
to the Sec. III.1 and Eq.(15), the gauge field in the coset space is written
as

$\vec{A}^{\prime
NM}_{\mu}.\vec{T}=N(x)\left[M(x)\left(A^{c}_{\mu}T^{c}\right)M^{\dagger}(x)-\dfrac{i}{g}M(x)\partial_{\mu}M^{\dagger}(x)\right]N^{\dagger}(x)+\text{thin\
vortex}.$ (43)

This is somehow similar to an Abelian gauge fixing plus center gauge fixing of
IMCG method in lattice calculations. However, an intermediate step including
Abelian projection must be applied, and we discuss it for the SU($2$) gauge
group in Sec. VI.

##  VI  indirect method of introducing vortices in SU($2$) gauge group

In this section, we apply the procedure explained in the previous section to
the SU($2$) gauge group, and we show that, unlike Sec. IV where we have gotten
vortices as a result of center gauge transformation and a “center projection”,
we get a chain of vortices and monopole by the indirect method.

We discuss this section in two subsections. In Sec. VI.1, we study the Abelian
gauge transformation and Abelian projection, which leads to the emergence of
the monopole.

In Sec. VI.2, in addition to steps $1$ and $2$ in Sec. VI.1, a center gauge
transformation followed by a “center projection” which leads to the appearance
of chains containing vortices and monopoles, is discussed.

###  VI.1 Abelian gauge tTransformations and Abelian projection: Monopole

The appearance of monopoles relies upon Abelian gauge transformation followed
by an Abelian projection, which is discussed in this subsection for the
SU($2$) gauge group.

Lattice studies show that within a good approximation the string tension
between a pair of quark and anti-quark is described by Abelian variables of
the maximal Abelian gauge transformation. Therefore, in the continuum limit,
the idea of the Abelian gauge transformation is to repress the contribution of
the off-diagonal components of the gauge fields so that the contribution of
diagonal components is dominant in the low-energy regime.

We perform a local rotation in color space called an Abelian gauge
transformation. As a result, magnetic monopoles can be extracted from the
diagonal part of the non-Abelian gauge field.

#### Step 1: Abelian gauge transformation

Choosing $\alpha(x)=\varphi$, $\beta(x)=\theta$ and $\gamma(x)=\pm\varphi$ in
the gauge rotation matrix of Eq.(26), an appropriate gauge transformation
$M(x)\in\text{SU}(2)$ which leads to an Abelian gauge transformation, is
obtained.

In this paper, we choose $\gamma(x)=-\varphi$,

$M(\theta,\varphi)=\left(\begin{matrix}cos\dfrac{\theta}{2}&e^{-i\varphi}sin\dfrac{\theta}{2}\\\
-e^{i\varphi}sin\dfrac{\theta}{2}&cos\dfrac{\theta}{2}\end{matrix}\right).$
(44)

According to Sec. III.1, we define $G(x)\equiv M(x)$ as an Abelian gauge
transformation; then the transformation of gluon field is given by Eq.(9),

$\vec{A}^{M}_{\mu}.\vec{T}=M(x)\left(\sum^{3}_{c=1}A^{c}_{\mu}T^{c}\right)M^{\dagger}(x)-\dfrac{i}{g}M(x)\partial_{\mu}M^{\dagger}(x).$
(45)

The first term on the right-hand side of Eq.(45) is regular under Abelian
gauge transformation $M(x)$, but the second term is singular. Replacing
Eq.(44) in Eq.(45),

$\begin{split}\vec{A}^{M}_{\mu}.\vec{T}&=\left[A^{1}_{\mu}\left(1-2sin^{2}\dfrac{\theta}{2}cos^{2}\varphi\right)+A^{2}_{\mu}\left(-sin^{2}\dfrac{\theta}{2}sin2\varphi\right)+A^{3}_{\mu}\left(-sin\theta
cos\varphi\right)+\dfrac{1}{g}sin\varphi\partial_{\mu}\theta+\dfrac{1}{g}sin\theta
cos\varphi\partial_{\mu}\varphi\right]T^{1}\\\
&+\left[A^{1}_{\mu}\left(-sin^{2}\dfrac{\theta}{2}sin2\varphi\right)+A^{2}_{\mu}\left(1-2sin^{2}\dfrac{\theta}{2}sin^{2}\varphi\right)+A^{3}_{\mu}\left(-sin\theta
sin\varphi\right)-\dfrac{1}{g}cos\varphi\partial_{\mu}\theta+\dfrac{1}{g}sin\theta
sin\varphi\partial_{\mu}\varphi\right]T^{2}\\\
&+\left[A^{1}_{\mu}\left(sin\theta
cos\varphi\right)+A^{2}_{\mu}\left(sin\theta
sin\varphi\right)+A^{3}_{\mu}\left(cos\theta\right)+\dfrac{1}{g}\left(1-cos\theta\right)\partial_{\mu}\varphi\right]T^{3},\end{split}$
(46)

where the singularity appears in the inhomogeneous term of the above equation
defined by $A^{\text{singular}}_{\mu}(\theta,\varphi)=A^{c\
\text{singular}}_{\mu}(\theta,\varphi)T^{c}\equiv-\dfrac{i}{g}M(\theta,\varphi)\partial_{\mu}M^{\dagger}(\theta,\varphi)$
in spherical coordinates and is given by

$\begin{split}\textbf{A}^{\text{singular}}(\theta,\varphi)&=-\dfrac{i}{g}M(\theta,\varphi)\nabla
M^{\dagger}(\theta,\varphi)=\dfrac{1}{2g}\left(\begin{matrix}\left[1-cos\theta\right]\nabla\varphi&\left[i\nabla\theta+sin\theta\nabla\varphi\right]e^{-i\varphi}\\\
\left[-i\nabla\theta+sin\theta\nabla\varphi\right]e^{i\varphi}&-\left[1-cos\theta\right]\nabla\varphi\end{matrix}\right)\\\
&=\dfrac{g^{-1}}{r}\left(cos\varphi\textbf{e}_{\varphi}+sin\varphi\textbf{e}_{\theta}\right)T^{1}+\dfrac{g^{-1}}{r}\left(sin\varphi\textbf{e}_{\varphi}-cos\varphi\textbf{e}_{\theta}\right)T^{2}+\dfrac{g^{-1}}{r}\dfrac{1-cos\theta}{sin\theta}\textbf{e}_{\varphi}T^{3},\end{split}$
(47)

where $\textbf{A}^{\text{singular}}(\theta,\varphi)=\textbf{A}^{c\
\text{singular}}(\theta,\varphi)T^{c}$. It is observed from Eq.(47) that there
exists a magnetic monopole as a point defect at the origin, $r=0$ along with a
Dirac string at $\theta=\pi$.

The magnetic flux $\Phi^{\text{flux}}(\theta)$ of the inhomogeneous term is
given by

$\begin{split}\Phi^{\text{flux}}(\theta)&=\int_{c}dx^{\mu}A^{singular}_{\mu}(\theta,\varphi)=\dfrac{1}{2g}\int^{2\pi}_{0}d\varphi\left(\begin{matrix}1-cos\theta&\sin\theta
e^{-i\varphi}\\\ sin\theta
e^{i\varphi}&-\left(1-cos\theta\right)\end{matrix}\right)\\\
&=\dfrac{2\pi}{2g}\left(\begin{matrix}1-cos\theta&0\\\
0&-\left(1-cos\theta\right)\end{matrix}\right)=\dfrac{2\pi}{g}\left(1-cos\theta\right)T^{3}.\end{split}$
(48)

It is observed that the total contribution of the magnetic flux is located
along the third direction of the color space. At $\theta=\pi$, the magnetic
flux of a Dirac string that enters a monopole located at the origin $r=0$ is
equal to $\dfrac{4\pi}{g}T^{3}$.

We have discussed in Sec. III.2 that under a gauge transformation, the field
strength tensor is obtained from Eq.(25). We rewrite it as follows:

$\begin{split}\vec{F}^{M}_{\mu\nu}.\vec{T}&=\left(\partial_{\mu}\left(\vec{A}^{M}_{\nu}.\vec{T}\right)-\partial_{\nu}\left(\vec{A}^{M}_{\mu}.\vec{T}\right)\right)\\\
&+ig\left[\vec{A}^{M}_{\mu}.\vec{T},\vec{A}^{M}_{\nu}.\vec{T}\right]\\\
&+\dfrac{i}{g}M(\theta,\varphi)\left[{\partial}_{\mu},{\partial}_{\nu}\right]M^{\dagger}(\theta,\varphi).\end{split}$
(49)

Equation (49) can be calculated using Eqs.(44) and (46). Since the two color
directions $T^{1}$ and $T^{2}$ have no contribution in the magnetic flux, we
will suppress these non-diagonal components of the gauge fields in the
“Abelian projection” step. It can be easily confirmed that the first term of
the Abelian sector
$\left(F^{\text{linear}}_{\mu\nu}\right)^{3}\equiv\partial_{\mu}\left(A^{M}_{\nu}\right)^{3}-\partial_{\nu}\left(A^{M}_{\mu}\right)^{3}$
includes a magnetic monopole sitting at the origin along with a Dirac string
in $\theta=\pi$. The second term of the Abelian sector is called
$\left(F^{\text{bilinear}}_{\mu\nu}\right)^{3}\equiv-g\left\\{\left(A^{M}_{\mu}\right)^{1}\left(A^{M}_{\nu}\right)^{2}-\left(A^{M}_{\mu}\right)^{2}\left(A^{M}_{\nu}\right)^{1}\right\\}$
and contains an anti-monopole at the origin, and the third term of the Abelian
sector
$F^{\text{singular}}_{\mu\nu}\equiv\dfrac{i}{g}M(\theta,\varphi)\left[{\partial}_{\mu},{\partial}_{\nu}\right]M^{\dagger}(\theta,\varphi)$
includes an anti-Dirac string at $\theta=\pi$ with a magnetic flux equal to
$-\dfrac{4\pi}{g}T^{3}$. (See Appendix A and Fig.(3)),

$F^{M}_{\mu\nu}=F^{\text{linear}}_{\mu\nu}+F^{\text{singular}}_{\mu\nu}+F^{\text{bilinear}}_{\mu\nu}.$
(50)

Figure 3: Appearance of a monopole accompanying a Dirac-string, an anti-Dirac
string, and an anti-monopole as a result of an Abelian gauge transformation.

If we add the contents of the above three terms, the similar terms with
opposite signs are canceled, and a field strength tensor which gives a full
QCD is obtained Ichie .

#### Step 2: Abelian projection

The sum of the two terms
$F^{\text{linear}}_{\mu\nu}+F^{\text{singular}}_{\mu\nu}$ represents a gauge
configuration that only contains a monopole at $r=0$. However, it is exactly
canceled by the anti-monopole arisen from $F^{\text{bilinear}}_{\mu\nu}$.
Thus, one can claim that the vacuum is filled with monopoles and anti-
monopoles.

From Fig.(3), it is observed that, in order to have only the contribution of
the monopole, we discard the term $F^{\text{bilinear}}_{\mu\nu}$ so that Ichie

$\vec{A}^{M}_{\mu}.\vec{T}=\left(A^{M}_{\mu}\right)^{a}T^{a}\rightarrow\mathcal{A}_{\mu}\equiv\left(A^{M}_{\mu}\right)^{3}T^{3}.$
(51)

From Eq.(46), the gauge field is changed to

$\begin{split}\mathcal{A}_{\mu}=\left[A^{1}_{\mu}\left(sin\theta
cos\varphi\right)+A^{2}_{\mu}\left(sin\theta
sin\varphi\right)+A^{3}_{\mu}\left(cos\theta\right)+\dfrac{1}{g}\left(1-cos\theta\right)\partial_{\mu}\varphi\right]T^{3}.\end{split}$
(52)

As a result, $F^{\text{bilinear}}_{\mu\nu}$, which gives the anti-monopole
contribution, is equal to zero, and the remaining part
$F^{\text{linear}}_{\mu\nu}+F^{\text{singular}}_{\mu\nu}$ describes an Abelian
projected QCD, which contains a monopole at $r=0$ and is called the monopole
vacuum.

After Abelian projection, SU($2$) gauge symmetry is reduced to U($1$) gauge
symmetry, and monopoles appear as the topological defects corresponding to the
non-trivial second homotopy group
$\Pi_{2}\left(\text{SU}(2)/\text{U}(1)\right)=\mathbb{Z}$.

###  VI.2 Center gauge transformation and “Center Projection”: Chain

As mentioned before, the vortex recognition by the indirect method is done in
four steps. We have discussed the first two steps in Sec. VI.1 and we explain
the final two steps in the following.

#### Step 3: Center gauge transformation;

We have shown that under two successive gauge transformations the gluon field
is changed by Eq.(43). After Abelian projection, the bracket in Eq.(43) should
be replaced by Eq.(51),

$\vec{A}^{\prime
NM}_{\mu}.\vec{T}=N(x)\mathcal{A}_{\mu}N^{\dagger}(x)+\text{thin\ vortex}.$
(53)

For $x\notin$ hypersurface, the thin vortex is defined in Eq.(16).

In fact, Eq.(53) expresses that a center gauge transformation is applied on a
monopole vacuum. Using the center gauge transformation defined by Eq.(27) and
the Abelian projected field defined by Eq.(52), the above equation is changed
to

$\begin{split}\vec{A}^{\prime
NM}_{\mu}.\vec{T}&=\left[A^{1}_{\mu}\left(sin\theta
cos\varphi\right)+A^{2}_{\mu}\left(sin\theta
sin\varphi\right)+A^{3}_{\mu}\left(cos\theta\right)+\dfrac{1}{g}\left(1-cos\theta\right)\partial_{\mu}\varphi-\dfrac{1}{g}\partial_{\mu}\varphi\right]T^{3}\\\
&=\left[A^{1}_{\mu}\left(sin\theta
cos\varphi\right)+A^{2}_{\mu}\left(sin\theta
sin\varphi\right)+A^{3}_{\mu}\left(cos\theta\right)-\dfrac{1}{g}cos\theta\partial_{\mu}\varphi\right]T^{3}.\end{split}$
(54)

The first three terms on the right-hand side of Eq.(54) are regular, and the
last term is defined in the spherical coordinates as
$\left(-\dfrac{1}{g}cos\theta\partial_{\mu}\varphi
T^{3}=-\dfrac{g^{-1}}{r}\dfrac{cos\theta}{sin\theta}\textbf{e}_{\varphi}T^{3}\right)$.
It indicates a defect representing a monopole located at the origin $r=0$
along with the two vortices at $\theta=0,\pi$. This singular term remarkably
represents a chain containing monopole and vortices. In fact, the magnetic
potential of the chain defined by
$E_{\mu}\equiv-\dfrac{1}{g}cos\theta\partial_{\mu}\varphi T^{3}$ , can be
interpreted as the sum of two terms: a magnetic potential of a monopole along
with a Dirac string defined by
$B_{\mu}\equiv\dfrac{1}{g}\left(1-cos\theta\right)\partial_{\mu}\varphi T^{3}$
plus a magnetic potential of a vortex defined by
$V_{\mu}\equiv-\dfrac{1}{g}\partial_{\mu}\varphi T^{3}$.

The magnetic flux $\Phi^{\text{flux}}(\theta)$ passing through a closed
contour $C(r,\theta)$ is defined as

$\begin{split}\Phi^{\text{flux}}(\theta)&=\int_{c}dx^{\mu}A^{\prime\
\text{singular}}_{\mu}=\dfrac{2\pi}{2g}\left(\begin{matrix}1-cos\theta&0\\\
0&-\left(1-cos\theta\right)\end{matrix}\right)-\dfrac{2\pi}{2g}\left(\begin{matrix}1&0\\\
0&-1\end{matrix}\right)\\\
&=\dfrac{2\pi}{g}\left(1-cos\theta\right)T^{3}-\dfrac{2\pi}{g}T^{3}\\\
&=-\dfrac{2\pi}{g}cos\theta T^{3},\end{split}$ (55)

where the first term in the second line of Eq.(55) represents the magnetic
monopole flux plus a Dirac string. At $\theta=\pi$, the contribution of the
flux of the Dirac string is equal to $+\dfrac{4\pi}{g}T^{3}$. The second term
in the second line of Eq.(55) indicates a vortex extending on the $z$ axis
with a flux equal to $-\dfrac{2\pi}{g}T^{3}$. Finally, the chain flux is
obtained as the sum of the vortex flux and the magnetic monopole flux plus the
Dirac string and is equal to $-\dfrac{2\pi}{g}cos\theta T^{3}$.

Now, we have some discussions about the chain characteristic. From Eq.(55) for
$\theta=0$, we only have the contribution of a magnetic vortex flux equal to
$-\dfrac{2\pi}{g}T^{3}$, located in the positive direction of the $z$ axis
which enters the magnetic monopole placed at the origin, $r=0$. At
$\theta=\pi$, there exists a Dirac string flux equal to
$+\dfrac{4\pi}{g}T^{3}$ located in the negative direction of the $z$ axis and
entering the magnetic monopole. There is also a magnetic vortex whose flux is
equal to $-\dfrac{2\pi}{g}T^{3}$ at $\theta=\pi$. It is located in the
negative direction of the $z$ axis and exits from the magnetic monopole placed
at $r=0$. In fact, the sum of the two fluxes $\Phi^{\text{Dirac\
string}}+\Phi^{\text{line\ vortex}}$ represents the contribution of a vortex
equal to $+\dfrac{2\pi}{g}T^{3}$, which enters the magnetic monopole sitting
at the origin, $r=0$. As a result, the magnetic flux of the monopole is
obtained as the sum of the absolute values of the fluxes of the two vortices
entering into it.

Equation (25) is used to obtain the field strength tensor of the
transformation,

$\begin{split}\vec{F}^{NM}_{\mu\nu}.\vec{T}&=N(x)\left(\vec{F}^{M}_{\mu\nu}.\vec{T}\right)N^{\dagger}(x)=N(x)M(x)\left(\vec{F}_{\mu\nu}.\vec{T}\right)M^{\dagger}(x)N^{\dagger}(x)\\\
&=\dfrac{1}{ig}\left[\hat{D}^{NM}_{\mu},\hat{D}^{NM}_{\nu}\right]-\dfrac{1}{ig}N(x)M(x)\left[\hat{\partial}_{\mu},\hat{\partial}_{\nu}\right]M^{\dagger}(x)N^{\dagger}(x)\\\
&=\left(\partial_{\mu}\left(\vec{A}^{\prime
NM}_{\nu}.\vec{T}\right)-\partial_{\nu}\left(\vec{A}^{\prime
NM}_{\mu}.\vec{T}\right)\right)+ig\left[\vec{A}^{\prime
NM}_{\mu}.\vec{T},\vec{A}^{\prime
NM}_{\nu}.\vec{T}\right]+\dfrac{i}{g}N(x)M(x)\left[{\partial}_{\mu},{\partial}_{\nu}\right]M^{\dagger}(x)N^{\dagger}(x).\end{split}$
(56)

We use Eq.(56) to study some various topological defects. A full QCD is
obtained if one uses Eq.(56) without applying any projection. In this section,
we discuss the possible resulting defects after Abelian and “center
projections”.

Looking at the last line of Eq.(56), we define the first term by
$F^{\text{linear}}_{\mu\nu}=\vec{F}^{\text{linear}}_{\mu\nu}.\vec{T}\equiv\partial_{\mu}\left(\vec{A}^{\prime
NM}_{\nu}.\vec{T}\right)-\partial_{\nu}\left(\vec{A}^{\prime
NM}_{\mu}.\vec{T}\right)$ and rewrite it using Eq.(54),

$\begin{split}\vec{F}^{\text{linear}}_{\mu\nu}.\vec{T}&=\left\\{\left(\partial_{\mu}A^{1}_{\nu}-\partial_{\nu}A^{1}_{\mu}\right)sin\theta
cos\varphi+A^{1}_{\nu}\partial_{\mu}\left(sin\theta
cos\varphi\right)-A^{1}_{\mu}\partial_{\nu}\left(sin\theta
cos\varphi\right)\right\\}T^{3}\\\
&+\left\\{\left(\partial_{\mu}A^{2}_{\nu}-\partial_{\nu}A^{2}_{\mu}\right)sin\theta
sin\varphi+A^{2}_{\nu}\partial_{\mu}\left(sin\theta
sin\varphi\right)-A^{2}_{\mu}\partial_{\nu}\left(sin\theta
sin\varphi\right)\right\\}T^{3}\\\
&+\left\\{\left(\partial_{\mu}A^{3}_{\nu}-\partial_{\nu}A^{3}_{\mu}\right)cos\theta+A^{3}_{\nu}\partial_{\mu}\left(cos\theta\right)-A^{3}_{\mu}\partial_{\nu}\left(cos\theta\right)\right\\}T^{3}\\\
&+\left(\partial_{\mu}E_{\nu}-\partial_{\nu}E_{\mu}\right)T^{3}.\end{split}$
(57)

The last line of the above equation contains some defects as explained in the
following:

$\begin{split}\left(\partial_{\mu}E_{\nu}-\partial_{\nu}E_{\mu}\right)T^{3}&=\dfrac{1}{g}sin\theta\left(\partial_{\mu}\theta\partial_{\nu}\varphi-\partial_{\mu}\varphi\partial_{\nu}\theta\right)\
T^{3}\\\
&+\dfrac{1}{g}\left(1-cos\theta\right)\left[\partial_{\mu},\partial_{\nu}\right]\varphi\
T^{3}\\\ &-\dfrac{1}{g}\left[\partial_{\mu},\partial_{\nu}\right]\varphi\
T^{3}.\end{split}$ (58)

The first term of Eq.(58) represents the field strength of a magnetic monopole
located at $r=0$, the second term indicates the field strength of a Dirac
string at $\theta=\pi$, and the third term represents the field strength of a
thin vortex field that extended on the $z$ axis. (See Fig.(4).)

Figure 4: Appearance of a monopole attached to a Dirac-string and a vortex
after Abelian gauge transformation, Abelian projection, and center gauge
transformation.

It is clear that the second term of Eq.(56),
$\vec{F}^{\text{bilinear}}_{\mu\nu}.\vec{T}\equiv ig\left[\vec{A}^{\prime
NM}_{\mu}(x).\vec{T},\vec{A}^{\prime NM}_{\nu}(x).\vec{T}\right]$, is zero.
This happens because of the Abelian projection process which makes the
components of the gluon field zero for the two color directions $T^{1}$ and
$T^{2}$. Using the center gauge transformation defined in Eq.(27) and the
Abelian gauge transformation of Eq.(44), the third term of Eq.(56) is

$\begin{split}F^{\text{singular}}_{\mu\nu}&=\vec{F}^{\text{singular}}_{\mu\nu}.\vec{T}\\\
&=-\dfrac{1}{g}\left(1-cos\theta\right)\left[\partial_{\mu},\partial_{\nu}\right]\varphi\
T^{3}+\dfrac{1}{g}\left[\partial_{\mu},\partial_{\nu}\right]\varphi\ T^{3}\\\
&=\dfrac{1}{g}cos\theta\left[\partial_{\mu},\partial_{\nu}\right]\varphi\
T^{3},\end{split}$ (59)

where
$-\dfrac{1}{g}\left(1-cos\theta\right)\left[\partial_{\mu},\partial_{\nu}\right]\varphi\
T^{3}$ represents the field strength of an anti-Dirac string in $\theta=\pi$
with a flux equal to $-\dfrac{4\pi}{g}T^{3}$ and the term
$\dfrac{1}{g}\left[\partial_{\mu},\partial_{\nu}\right]\varphi\ T^{3}$
represents the field strength of an anti-vortex on the $z$ axis with a flux
equal to $+\dfrac{2\pi}{g}T^{3}$. (See Fig.(5))

Figure 5: Appearance of an anti-Dirac string and an anti-vortex after Abelian
gauge transformation, followed by an Abelian projection and then a center
gauge transformation.

In fact, the contribution of the vortex and the Dirac string appearing in
$F^{\text{linear}}_{\mu\nu}$ is exactly canceled by the contribution of the
anti-vortex and the anti-Dirac string in $F^{\text{singular}}_{\mu\nu}$. As a
result, a monopole vacuum is obtained unless we remove some of the
singularities from the theory.

#### Step 4: “Center projection”

As explained in Sec. IV, a “center projection” is done by removing
$F^{\text{singular}}_{\mu\nu}$ defined in Eq.(59). This means that “center
projection” is obtained by
$F^{\text{linear}}_{\mu\nu}+F^{\text{bilinear}}_{\mu\nu}$. On the other hand,
we have shown that $F^{\text{bilinear}}_{\mu\nu}$ is zero, and as a result,
the center projected field strength tensor is as follows:

$F^{\text{CP}}_{\mu\nu}=F^{\text{linear}}_{\mu\nu}.$ (60)

Therefore, only a monopole attached to a Dirac string and a vortex remain. We
can interpret these configuration as a chain as shown in Fig.(6).

Figure 6: Appearance of a chain after Abelian gauge transformation, Abelian
projection, center gauge transformation, and “center projection”.

The first plot on the left-hand side of Fig.(6) represents a monopole at $r=0$
plus a Dirac string located at $\theta=\pi$ carrying a magnetic flux equal to
$\dfrac{4\pi}{g}T^{3}$. The second plot on the left-hand side indicates a
vortex carrying a magnetic flux equal to $-\dfrac{2\pi}{g}T^{3}$ extending on
the $z$ axis. Combining these two plots, a chain shown on the right-hand side
of Fig.(6) is obtained. A chain contains a monopole at $r=0$ and two vortices
entering it. The flux of the vortex sitting at $\theta=0$ is equal to
$-\dfrac{2\pi}{g}T^{3}$, and the flux of the vortex sitting at $\theta=\pi$ is
equal to $+\dfrac{2\pi}{g}T^{3}$. The latter vortex is obtained as a result of
combining the flux of the Dirac string sitting in the negative $z$ direction
and the first vortex located in the $z$ direction.

Our arguments about the chains of monopoles and vortices are in agreement with
the work of Del Debbio et al. DFGO , which is done by lattice QCD (see
Fig.(7)), and also in agreement with the results of Reinhardt and Engelhardt
ER2000 , in which two vortex enter a monopole (see Fig.(8)).

(a)

(b)

(c)

Figure 7: Vortex field strength: (a) before gauge fixing, (b) after maximal
Abelian gauge fixing in the horizontal $\pm\sigma^{3}$ direction, and (c)
after Abelian projection DFGO . As shown in this figure, two vortex lines
enter a monopole or an anti-monopole, in agreement with what we have
introduced in this paper as a chain configuration in Fig. (6).

(a)

(b)

Figure 8:  Illustration of the connection between Dirac string shown in plot
(a) and the center vortex shown in plot (b) ER2000 . The interpretation of a
chain represented in Fig. (6) is the same as this figure where two vortex
lines enter a monopole.

We end this section by discussing the possible advantages of using chains. As
we mentioned at the beginning of the article, in both the dual superconductor
model and the center vortex model, monopoles and vortices can explain some
aspects of the color confinement like the linear potential between a quark and
anti-quark. However, none of these models nor the associated defects is able
to describe all the expected features of the confining potential between color
sources.

At intermediate distances, a well-defined linear confining potential is
expected, $V_{R}(r)\sim\sigma_{R}r$, in which $\sigma_{R}$ is the string
tension of representation $R$. The confining potential should agree with the
Casimir scaling at intermediate distances. It means that the string tension of
the potential between a quark and an anti-quark in representation $R$,
$\sigma_{R}$, is approximately proportional to the quadratic Casimir operator
$C_{R}$ of representation $R$, i.e.,

$\sigma_{R}=\dfrac{C_{R}}{C_{F}}\sigma_{F},$ (61)

where $F$ indicates the fundamental representation and $\sigma_{F}$ shows the
string tension of the fundamental representation. $C_{F}$ denotes the
eigenvalue of the Casimir operator of representation $F$. We recall that the
dependence of the potential slope to the Casimir scaling applies only for the
intermediate distances and it is valid and exact for the large $N$ limit GH ;
Witten . In addition, at large distances, the $k$-string tension depends on
the $N$-ality of the representation; it is equal to the fundamental
representation string tension for the non-zero N-ality representations and
zero for the zero $N$-ality representations kondo-n .

Proportionality with Casimir scaling for the intermediate distances and the
$N$-ality dependence of the potentials at large distances are confirmed by
lattice calculations for the fundamental and a variety of higher
representations deldar ; deldar1 ; bali . Therefore, any phenomenological
model which tries to describe the potential between static color sources is
expected to interpret these two features. Vortex based models have been able
to explain the $N$-ality dependence. However, to get the Casimir scaling for
all representations, the models have been modified by defining a thickness for
the vortex DFGO ; DFGO1 . On the other hand, lattice results confirm the
existence of chains of monopoles and vortices AGG ; monte  that may explain
the agreement of the potentials with Casimir scaling for higher
representations. In this article, we have followed this approach to study the
existence of chains of monopoles and vortices for the continuum.

We recall that an Abelian-projected theory gives the $N$-ality dependence
(after all, it can still contain vortices), but it does not give the Casimir
scaling dependence at intermediate distances DFGO97 ; final

In this paper, motivated by direct and indirect methods of identifying
vortices in lattice QCD, we have shown the existence of chains of monopoles
and vortices for the continuum.

##  VII Conclusions

Motivated by lattice QCD, which discusses the vortex contribution in color
confinement, we have tried to introduce vortices in the continuum.

In the absence of matter fields, we work in the quenched approximation where
dynamical quarks are removed from the theory. Therefore, the theory includes
only the gluon fields, in this limit.

In recent years, the identification of vortices in lattice QCD has seen
significant progress. Therefore, one expects to observe the same physics in
the continuum limit when one uses the lattice results for the limit where
$a\rightarrow 0$.

Inspired by direct and indirect maximal center gauge methods which have
studied vortices in lattice calculations and by using connection formalism
technique, we have tried to recognize the vortices in the continuum. We have
introduced the thin vortices from the gluon fields via both direct and
indirect methods for the SU($N$) gauge group. We also get some help from the
techniques proposed by Engelhardt and Reinhardt.

For an example, from the direct method, we have shown that under center gauge
transformation the QCD vacuum of the SU($2$) gauge group is filled with the
vortices and anti-vortices. Then, applying a “center projection” we reach a
theory that contains the thin vortex. The theory has an SO($3$) symmetry
containing the vortex, which corresponds to the non-trivial first homotopy
group of $\Pi_{1}\left(\text{SO}(3)\right)=Z_{2}$.

Then, using the indirect method, we have shown that under Abelian gauge
transformation for the SU($2$) case the gauge theory would contain monopoles
along with the Dirac strings and anti-Dirac strings as well as anti-monopoles.
Then, applying Abelian projection and removing the anti-monopole contribution,
we end up with a theory that includes only the monopoles. In other words,
SU($2$) gauge symmetry is reduced to a U($1$) gauge symmetry, and monopoles
appear as the topological defects corresponding to the non-trivial second
homotopy group $\Pi_{2}\left(\text{SU}(2)/\text{U}(1)\right)=\mathbb{Z}$.

Next, we have done a center gauge transformation on the Abelian vacuum. As a
result, we get the monopole along with the Dirac string, the vortex, and anti-
Dirac string, and the anti-vortex. Eventually, by applying “center
projection”, we end up with a theory that contains chains including monopole
and two vortices.

## Appendix

###  VII.1 Transformation of field strength tensor under an Abelian gauge
transformation

In subsection VI.1, we express that the field strength is changed by Eq.(49)
when Abelian gauge transformation is applied. One can also show that,

$\begin{split}\vec{F}^{M}_{\mu\nu}.\vec{T}&=\vec{F}^{\text{linear}}_{\mu\nu}.\vec{T}+\vec{F}^{\text{bilinear}}_{\mu\nu}.\vec{T}+\vec{F}^{\text{singular}}_{\mu\nu}.\vec{T}\\\
&\equiv\left(\partial_{\mu}\left(\vec{A}^{M}_{\nu}.\vec{T}\right)-\partial_{\nu}\left(\vec{A}^{M}_{\mu}.\vec{T}\right)\right)\\\
&+ig\left[\vec{A}^{M}_{\mu}.\vec{T},\vec{A}^{M}_{\nu}.\vec{T}\right]\\\
&+\dfrac{i}{g}M(\theta,\varphi)\left[{\partial}_{\mu},{\partial}_{\nu}\right]M^{\dagger}(\theta,\varphi).\end{split}$
(A.1)

Each term of Eq.(A.1) can be expressed in terms of the SU($2$) group
generators as the following:

$\begin{split}&\vec{F}^{\text{linear}}_{\mu\nu}.\vec{T}=\left(F^{\text{linear}}_{\mu\nu}\right)^{1}T^{1}+\left(F^{\text{linear}}_{\mu\nu}\right)^{2}T^{2}+\left(F^{\text{linear}}_{\mu\nu}\right)^{3}T^{3},\\\
&\vec{F}^{\text{bilinear}}_{\mu\nu}.\vec{T}=\left(F^{\text{bilinear}}_{\mu\nu}\right)^{1}T^{1}+\left(F^{\text{bilinear}}_{\mu\nu}\right)^{2}T^{2}+\left(F^{\text{bilinear}}_{\mu\nu}\right)^{3}T^{3},\\\
&\vec{F}^{\text{singular}}_{\mu\nu}.\vec{T}=\left(F^{\text{singular}}_{\mu\nu}\right)^{1}T^{1}+\left(F^{\text{singular}}_{\mu\nu}\right)^{2}T^{2}+\left(F^{\text{singular}}_{\mu\nu}\right)^{3}T^{3}.\end{split}$
(A.2)

From Eq.(46), $\vec{F}^{\text{linear}}_{\mu\nu}.\vec{T}$ is obtained for the
Abelian sector

$\begin{split}\left(F^{\text{linear}}_{\mu\nu}\right)^{3}&=\left(\partial_{\mu}A^{1}_{\nu}-\partial_{\nu}A^{1}_{\mu}\right)sin\theta
cos\varphi+A^{1}_{\nu}\partial_{\mu}\left(sin\theta
cos\varphi\right)-A^{1}_{\mu}\partial_{\nu}\left(sin\theta
cos\varphi\right)\\\
&+\left(\partial_{\mu}A^{2}_{\nu}-\partial_{\nu}A^{2}_{\mu}\right)sin\theta
sin\varphi+A^{2}_{\nu}\partial_{\mu}\left(sin\theta
sin\varphi\right)-A^{2}_{\mu}\partial_{\nu}\left(sin\theta
sin\varphi\right)\\\
&+\left(\partial_{\mu}A^{3}_{\nu}-\partial_{\nu}A^{3}_{\mu}\right)cos\theta+A^{3}_{\nu}\partial_{\mu}\left(cos\theta\right)-A^{3}_{\mu}\partial_{\nu}\left(cos\theta\right)\\\
&+\dfrac{1}{g}sin\theta\left(\partial_{\mu}\theta\partial_{\nu}\varphi-\partial_{\mu}\varphi\partial_{\nu}\theta\right)+\dfrac{1}{g}\left(1-cos\theta\right)\left[\partial_{\mu},\partial_{\nu}\right]\varphi,\end{split}$
(A.3)

where,
$+\dfrac{1}{g}sin\theta\left(\partial_{\mu}\theta\partial_{\nu}\varphi-\partial_{\mu}\varphi\partial_{\nu}\theta\right)$
represents the field strength of a magnetic monopole at $r=0$ and
$\dfrac{1}{g}\left(1-cos\theta\right)\left[\partial_{\mu},\partial_{\nu}\right]\varphi$
indicates the field strength of a Dirac string at $\theta=\pi$.

The Abelian sector $\vec{F}^{\text{bilinear}}_{\mu\nu}.\vec{T}$ is also
obtained by Eq.(46),

$\begin{split}\left(F^{\text{bilinear}}_{\mu\nu}\right)^{3}=&-g\left(A^{2}_{\mu}A^{3}_{\nu}-A^{3}_{\mu}A^{2}_{\nu}\right)sin\theta
cos\varphi-A^{1}_{\nu}\partial_{\mu}\left(sin\theta
cos\varphi\right)+A^{1}_{\mu}\partial_{\nu}\left(sin\theta
cos\varphi\right)\\\
&-g\left(A^{3}_{\mu}A^{1}_{\nu}-A^{1}_{\mu}A^{3}_{\nu}\right)sin\theta
sin\varphi-A^{2}_{\nu}\partial_{\mu}\left(sin\theta
sin\varphi\right)+A^{2}_{\mu}\partial_{\nu}\left(sin\theta
sin\varphi\right)\\\
&-g\left(A^{1}_{\mu}A^{2}_{\nu}-A^{2}_{\mu}A^{1}_{\nu}\right)cos\theta-A^{3}_{\nu}\partial_{\mu}\left(cos\theta\right)+A^{3}_{\mu}\partial_{\nu}\left(cos\theta\right)\\\
&-\dfrac{1}{g}sin\theta\left(\partial_{\mu}\theta\partial_{\nu}\varphi-\partial_{\mu}\varphi\partial_{\nu}\theta\right),\end{split}$
(A.4)

where
$-\dfrac{1}{g}sin\theta\left(\partial_{\mu}\theta\partial_{\nu}\varphi-\partial_{\mu}\varphi\partial_{\nu}\theta\right)$
represents the field strength of an anti-monopole at $r=0$.

Finally, $\vec{F}^{\text{singular}}_{\mu\nu}.\vec{T}$ is obtained for the
Abelian sector by Eq.(44),

$\left(F^{\text{singular}}_{\mu\nu}\right)^{3}=-\dfrac{1}{g}\left(1-cos\theta\right)\left[\partial_{\mu},\partial_{\nu}\right]\varphi.$
(A.5)

And the above singular term shows the field strength of an anti-Dirac string
at $\theta=\pi$.

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