# Calculation of SU($2$) string tension in the continuum limit by an effective
theory of center vortices

Zahra Asmaeea<EMAIL_ADDRESS>Motahareh Kiamarib<EMAIL_ADDRESS>Sedigheh Deldara<EMAIL_ADDRESS>Corresponding author aDepartment of
Physics, University of Tehran,
P. O. Box 14395/547, Tehran 1439955961, Iran
bSchool of Particles and Accelerators, IPM,
P. O. Box 19395-5531, Tehran, Iran

###### Abstract

The area law fall-off for the Wilson loop average has been confirmed by
lattice calculations. Using an effective theory for an ensemble of center
vortices, we observe the area law fall-off in the continuum limit for the
SU($2$) gauge group in three-dimensional Euclidean space-time. The string
tension is obtained in terms of the intrinsic properties of the vortices and
the parameter describing their interactions. A good qualitative agreement
between our results and the lattice ones is observed. In addition, we show
that the repulsive force between the vortices increases with temperature. This
behavior is expected due to the reduction of vortex structures at higher
temperatures, required for the deconfinement regime.

PACS numbers

74.25.Ha, 74.25.Uv, 12.38.Aw, 12.38.Lg.

###### pacs:

Valid PACS appear here

##  I Introduction

Quantum Chromodynamics $\left(\text{QCD}\right)$ describes the strong
interaction explaining the underlying structure of hadrons in terms of quarks
and gluons QCD1 ; QCD2 ; QCD3 . The dynamics of gluons as the mediate
particles of strong interactions are governed by the Yang-Mills theory Yang-
Mills .

In the ultraviolet regime, the gauge-coupling constant of QCD becomes small so
that the asymptotic freedom behavior is dominant, and the perturbative methods
can provide a direct description of QCD vacuum in terms of quarks and gluons.
Therefore, non-divergent scattering amplitudes for the high-energy regime or
the ultraviolet region can be computed WH .

On the other hand, in the infrared regime, the strong gauge-coupling constant
nature of QCD leads to non-perturbative problems which are not easily resolved
Nambo-Joan1 ; Nambo-Joan2 ; K. Higashijima ; U-Bar ; E. V. Shuryak ; diakonov
. Quarks and gluons as the fundamental degrees of freedom of QCD, have not
been observed as isolated particles in the low-energy regime. So far, only
hadrons including mesons and baryons, have been observed as color singlet
objects in nature. This experimental fact reflects the hypothesis of color
confinement mechanism as one of the most controversial unsolved issues in
particle physics confinement2004 ; Greensite . During the past few decades,
many ideas have been proposed to approach this problem based on non-
perturbative methods. cho-restricted ; cho-extented ; Faddeev-Niemi ; t
Hooft78 ; Cornwall ; t Hooft80 ; t Hooft81 ; t Hooft82 ; Shabanov ; Faddeev-
Niemi2

Lattice QCD and phenomenological models can be introduced as the most popular
non-perturbative methods to explain the quark confinement cho-restricted ;
cho-extented ; Ripka ; Ichie ; lattice1 ; lattice2 ; lattice3 ; lattice3.1 ;
Greensite2003 ; lattice4 ; lattice5 ; lattice6 ; lattice7 ; lattice8 ; deldar
; Deldar1 . The attempt to clarify the mechanism of the confinement phenomenon
has led to the fact that the QCD vacuum has some non-trivial structure and
this structure is responsible for the confinement of quarks inside the hadrons
Ichie ; bruckmann . Topological objects like magnetic monopoles, vortices,
instantons, dyons, and calorons are among the candidates that can be used to
describe the confinement via some phenomenological models Ichie ; Deldar2 ;
Deldar3 ; Deldar4 .

Any theory that investigates the confinement mechanism must predict some
features like N-ality dependence and Casimir scaling of string tension, and a
linear potential between a pair of static quark-antiquark, which are the
important properties of the confining force Greensite . However, there is
still no comprehensive mechanism that explains all the properties of the QCD
in the infrared regime. Therefore, the study of quark confinement is still of
main interest to particle physicists.

From lattice QCD simulations, the potential between a pair of quark-antiquark
is kondo : $V^{q\bar{q}}(r)=V^{\text{ele}}(r)+V^{\text{mag}}(r)$. Where
$V^{\text{ele}}(r)=-\dfrac{A}{r}$ indicates the Coulombic potential which is
inversely proportional to the distance between the quark-antiquark pair $r$
columb , and $V^{\text{mag}}(r)=\sigma r$ shows the linear potential which
increases with the distance and is proportional to the string tension Ripka ;
Ichie .

The area law fall-off for the Wilson loop average is a well-known gauge-
invariant criterion to study quark confinement. It leads to a linear potential
between a pair of static quark-antiquark Wilson . To study the linear part of
the confinement potential, the quenched approximation is used where the
dynamical quarks are removed for the infrared regime Greensite , and the
confinement mechanism is described by the dynamics of gluons.

One can introduce some collective modes from gluons that are associated with
the topological degrees of freedom of the QCD vacuum, and as a result, it is
assumed that the QCD vacuum is filled with the topological defects obtained
from these collective modes. Ichie ; kondo

In the absence of matter fields, the center vortex model is a promising
scenario for quark confinement. Historically, vortex-like structures were
introduced in superconductors in 1959 Gorkov . 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 which can be condensed in the confined phase. If a Wilson
loop is linked to a vortex in an SU($N$) gauge group, the Wilson loop obtains
a phase equal to $e^{i2\pi k/N}$ ($k=1$ 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 the confinement at the end.

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. Lattice
calculations show that the ZN type vortices produce full string tension 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 picture relies upon center gauge fixing and center projection DFG97
; DFG98 . The vortices which appear after performing center projection on the
lattice, are called projection vortices (or p-vortices).

For SU($2$) gauge group, the string tension obtained from the projected thin
center-vortex ensemble reproduces almost $97.7\%$ of the fundamental string
tension calculated from the non gauge fixed situation. For SU($3$), this value
is $\sim 62\%$ Langfeld . Another important feature of the center vortex
picture is that it naturally explains N-ality dependence of the asymptotic
string tension. On the other hand, the modified models of vortices like thick
center vortices can explain qualitatively the Casimir scaling dependence for
higher representations t Hooft80 ; faber ; Debbio ; Deldar2001 ; Deldar2005 .
Therefore, the center vortex models can be a promising scenario for quark
confinement.

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 . Using phenomenological models, vortices have been identified by
different methods. There are many articles about this subject; for instance,
see Refs. t Hooft78 ; samuel . We have also conducted some studies on
identifying vortices in the continuum limit of QCD for the SU($2$) and SU($3$)
gauge group asmaee ; karimimanesh .

After reviewing the maximal center gauge fixing to identify vortices in Sec.
(II), we review the partition function in Sec. (III) by using a technique that
has long been used by polymer physicists. The idea is that a condensate of
oriented closed strings can be mapped onto a complex scalar field in a field
theoretical approach. Some particle physicists have also used this trick; for
instance, see Refs. samuel ; stone ; stone78 ; samuel78 ; oxman . Assuming the
intrinsic properties for the center vortex loops such as the stiffness and the
tension, as well as defining an interaction between the vortices, one can
consider an effective partition function for an ensemble of vortices using the
polymer technique. In Sec. (IV), we calculate the Wilson loop average in the
continuum limit using an ensemble of closed center vortices in three-
dimensional Euclidean space-time. We show that the Wilson loop average
displays an area law fall-off and the string tension is derived for the
SU($2$) gauge group in terms of the intrinsic properties of the center
vortices and the parameter which describes the interaction between the
vortices. A good qualitative agreement between the plot of the stiffness
versus the tension of this calculation and the lattice results is obtained.
The repulsive potential between vortices is studied and it is discussed that
the strength of the potential shows an appropriate behavior concerning the
temperature.

##  II Introducing vortices in the continuum limit

It has been more than three decades since the properties of the center
vortices and their role in quark confinement have been confirmed by lattice
QCD; for a good review see reference Greensite2003 .

Direct maximal center gauge $\left(\text{DMCG}\right)$ and indirect maximal
center gauge $\left(\text{IMCG}\right)$ are among the most common methods of
identifying vortices in lattice simulations. DMCG and IMCG are introduced to
perform maximal center gauge fixing and center projection to identify
vortices. Other methods for identifying vortices were also proposed in lattice
QCD. Examples are Laplacian center gauge fixing by de Forcrand et al. LCGF ;
LCGF1  and direct Laplacian center gauge fixing by Langfeld et al.
Langfeld2001  and Faber et al. Faber . All of these methods yield
qualitatively similar results.

Center vortices are defined by the centers of the SU($N$) gauge group and
carry magnetic fluxes corresponding to the centers, Z($N$). The magnetic
fluxes are associated with the non-trivial center elements
$z_{k}=\text{exp}\left(i\dfrac{2\pi}{N}k\right)\textbf{1}_{N\times
N},\left(k=1,2,...,N-1\right)$ of Z($N$) discrete gauge group. For SU($2$)
gauge group, the non-trivial center element corresponds to -1 and the trivial
center element to +1, where 1 represents a $2\times 2$ unit matrix. The center
vortices are defined as a string, surface and volume in $D=2$, $3$ and $4$,
respectively.

It is assumed that the center vortices carrying magnetic fluxes, squeeze the
electric fields between a quark-antiquark pair inside a flux tube. If a Wilson
loop is linked to a vortex in an SU($N$) gauge group, it obtains a phase equal
to the center element, 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.

In this section, we review the emergence of vortices in the continuum limit
which we have reported in reference asmaee . Inspired by DMCG method in
lattice QCD which confirms the existence of vortices using the maximal center
gauge fixing and center projection, we have shown that vortices are identified
by performing a singular gauge transformation. Linking the Wilson loop and the
vortices, the Wilson loop gets a phase equal to the non-trivial center
elements associated with $z$($k$),

$W(C)\longrightarrow z(k)W(C).$ (1)

The non-trivial center elements $z(k)$ of SU($N$) group are given as,

$z(k)=e^{i\frac{2\pi}{N}k}\textbf{1},\quad\left(k=1,2,...,N-1\right),$ (2)

where 1 represents an $N\times N$ unit matrix. On the other hand, the center
elements $z(k)$ can be rewritten as the following oxman

$z(k)=e^{i2\pi\nu(k)},\quad\nu(k)=\nu^{a}(k)H_{a},$ (3)

where $\nu(k)$ are co-weights and $H_{a}$’s indicate the generators of the
Cartan subalgebra.

In three-dimensional space-time, a non-trivial contribution to the Wilson loop
operator is obtained by the linking number $L\left(C,l\right)$ between the
Wilson loop $C$ and the vortex loop $l$. Where $l$ indicates the boundary of
the hypersurface $\Sigma$ $\left(\text{ i.e.}\ l=\partial\Sigma\right)$ shown
in Fig. 1(b). The linking number can also be described in other alternative
ways: by the intersection number $I\left(C,\Sigma\right)$ between the Wilson
loop $C$ and the hypersurface $\Sigma$ shown in Fig. 1(a), or by the
intersection number $I\left(S,l\right)$ between the Wilson surface $S$ and the
vortex loop $l$ (see Fig. 1(c)). These three definitions are equivalent:
$L\left(C,l\right)=I\left(C,\Sigma\right)=I\left(S,l\right)$ kondo . We recall
that the linking number and the intersection number are integers.

(a)

(b)

(c)

Figure 1: (a) Intersection between the Wilson loop $C$ and hypersurface
$\Sigma$. (b) Linking between the Wilson loop $C$ and a vortex loop $l$ which
is the boundary of hypersurface, $l=\partial\Sigma$. (c) Intersection between
the Wilson surface $S$ and the vortex loop $l$.

Under a center gauge transformation $N(x)\in$ SU($2$), the contribution of the
thin vortex defined on the closed loop $l=\partial\Sigma$ for $x\neq\Sigma$ is
asmaee ; ER2017 ,

$\text{thin\ vortex}=\dfrac{i}{g}N(x)\partial_{\mu}N^{\dagger}(x),\quad
x\notin\Sigma.$ (4)

For SU($2$) group, the appropriate center gauge transformation is asmaee ,

$N=e^{i\varphi T^{3}}=\left(\begin{matrix}e^{i\frac{\varphi}{2}}&0\\\
0&e^{-i\frac{\varphi}{2}}\end{matrix}\right),\quad\varphi\in\left[0,2\pi\right),$
(5)

where $\varphi$ is the azimuth angle in cylindrical coordinates and $T^{3}$ is
equal to the half of the Pauli matrix $\sigma^{3}$. Replacing Eq. (5) in Eq.
(4), the contribution of the thin vortex is,

$\text{thin\ vortex}\equiv
A_{\mu}^{l}(x)=\vec{A}_{\mu}^{l}.\vec{T}=\dfrac{i}{g}N\partial_{\mu}N^{\dagger}=\dfrac{1}{g}\partial_{\mu}\varphi
T^{3}.$ (6)

The spatial component of the thin vortex is,

$\vec{A}_{\varphi}^{l}.\vec{T}=\dfrac{1}{g\rho}T^{3},\quad\vec{A}_{\rho}^{l}.\vec{T}=0.$
(7)

Equation (7) represents the gauge field associated with the thin vortex in
cylindrical coordinates. The thin vortex is observed at $\rho=0$ in the third
direction of color space. The magnetic vortex flux $\Phi^{\text{flux}}$ is,

$\begin{split}\Phi^{\text{flux}}&=\int
dx_{\mu}\left(\vec{A}_{\mu}^{l}.\vec{T}\right)=\dfrac{1}{2g}\int^{2\pi}_{0}d\varphi\left(\begin{matrix}1&0\\\
0&-1\end{matrix}\right)\\\ &=\dfrac{2\pi}{g}T^{3}.\end{split}$ (8)

We would like to calculate the Wilson loop average from the vortex ensemble.
The center gauge fixing imposed on the SU($2$) Yang-Mills theory is assumed to
be similar to the center gauge fixing on a lattice. Therefore, the
contribution of the vortex field $A_{\mu}^{l}$ to the Wilson loop is as the
following,

$W_{\text{vortex}}(C)\equiv e^{i\oint_{C}dx_{\mu}A_{\mu}^{l}(x)}.$ (9)

On the other hand, when the thin vortex loop $l$ intersects the Wilson surface
$S$, the Wilson loop obtains a factor as the following,

$W_{\text{vortex}}(C)=z^{I(S,l)},$ (10)

where $z$ is the non-trivial center element of the SU($2$) gauge group.
Replacing Eq. (3) for SU($2$) group in Eq. (10),

$W_{\text{vortex}}(C)=e^{i2\pi\nu I(S,l)}.$ (11)

Using Eqs. (9) and (11), the relationship between intersection number and the
line integral of a vortex field $A_{\mu}^{l}$ is as follows,

$\oint_{C}dx_{\mu}A_{\mu}^{l}(x)=2\pi\nu I(S,l).$ (12)

An explicit integral formula for the intersection $I\left(S,l\right)$ between
the Wilson surface $S$ and the vortex loop $l$ is defined kondo ,

$\begin{split}&I\left(S,l\right)=\dfrac{1}{4\pi}\oint_{l=\partial\Sigma}dx_{\mu}\int_{S}d^{2}{\sigma}_{\mu}\delta^{\left(3\right)}\left(x-\bar{x}\left(\sigma\right)\right),\\\
&d^{2}{\sigma}_{\mu}=\epsilon_{\mu\nu\rho}\dfrac{\partial\bar{x}_{\nu}}{\partial\sigma_{1}}\dfrac{\partial\bar{x}_{\rho}}{\partial\sigma_{2}}d\sigma_{1}d\sigma_{2},\end{split}$
(13)

where $d^{2}{\sigma}_{\mu}$ is the surface element and $S$ is parameterized by
$\bar{x}\left(\sigma\right)=\bar{x}\left(\sigma_{1},\sigma_{2}\right)$.

Replacing Eq. (13) in Eq. (12),

$\oint_{C}dx_{\mu}A_{\mu}^{l}(x)=\dfrac{\nu}{2}\oint_{l=\partial\Sigma}dx_{\mu}\int_{S}d^{2}{\sigma}_{\mu}\delta^{\left(3\right)}\left(x-\bar{x}\left(\sigma\right)\right).$
(14)

The vector field $J_{\mu}^{C}$ is defined on the Wilson surface $S$ as the
following,

$J_{\mu}^{C}(x)\equiv\dfrac{\nu}{2}\int_{S}d^{2}\sigma_{\mu}\delta^{\left(3\right)}\left(x-\bar{x}\left(\sigma\right)\right).$
(15)

Replacing Eq. (15) in Eq. (14),

$\oint_{C}dx_{\mu}A_{\mu}^{l}(x)=\oint_{l}dx_{\mu}J_{\mu}^{C}(x).$ (16)

The linking number $L(C,l)$ is symmetric with respect to the interchange of
the loops $C$ and $l$: $L(C,l)=L(l,C)$. Applying this interchange to the
previous equation, one obtains a noticeable result. The vortex field
$A_{\mu}^{l}$ and the vector field $J_{\mu}^{l}$ are somehow equivalent. The
vortex field $A_{\mu}^{l}$ which is defined on the closed loop $l$ can be
gauge transformed to the vector field $J_{\mu}^{l}$. Indeed, when $n$ vortices
intersect the Wilson surface, the flux of all $n$ vortices passes through the
surface $S$ $\left(\text{see\ Fig.}\ \ref{fig2}\right)$, therefore, the vector
field $J_{\mu}^{C}$ in Eq. (15) indicates the gauge potential of the vortex
field which its worldline $l$ is replaced by $C$ oxman .

Figure 2: $n$ vortices intersecting the Wilson surface $S$ bounded by the
closed loop $C$.

It is useful to rewrite Eq. (16) as the following,

$\begin{split}\oint_{C}dx_{\mu}A_{\mu}^{l}(x)=\oint_{l}dx_{\mu}J_{\mu}^{C}(x)&=\int_{0}^{L}ds\dfrac{dx_{\mu}}{ds}J_{\mu}^{C}(x(s))\\\
&=\int_{0}^{L}dsu_{\mu}(s)J_{\mu}^{C}\left(x\left(s\right)\right),\end{split}$
(17)

where $L$ is the total length of the vortex worldline and $s$ is the arc
length parameter of the vortex loop which runs from $0$ to $L$, and
$u_{\mu}(s)$ is a tangent vector to the $s$.

Replacing (17) in Eq. (9), the Wilson loop average is,

$W_{\text{vortex}}(C)=e^{i\int_{0}^{L}dsu_{\mu}(s)J_{\mu}^{C}\left(x(s)\right)}.$
(18)

From the above equation and Fig. 2, if $n$ vortices intersect the Wilson
surface $S$,

$W_{\text{vortex}}(C)=e^{i\sum_{i=1}^{n}\int_{0}^{L_{i}}ds_{i}u_{\mu}(s_{i})J_{\mu}^{C}(x(s_{i}))}.$
(19)

In fact, $J_{\mu}^{C}(x)$ describes a general vector field containing the flux
of the $n$ vortices passing through surface $S$.

##  III Partition function for an ensemble of center vortices

To calculate the linear potential between a pair of quark-antiquark and the
string tension in three-dimensional Euclidean space-time, we use the partition
function of an ensemble of center vortices. To obtain an effective partition
function, one can use a trick that for a long time has been used by polymer
physicists, and particle physicists have used that, as well. The idea is that
a condensate of oriented closed strings can be mapped onto a complex scalar
field. The action is written in terms of the interacting vortices and
intrinsic properties of the vortex loops such as the tension and stiffness.

The action of an ensemble of $n$ vortices can be written as the following
oxman ; stiff1 ; stiff2 ; stiff3 ,

$S^{0}_{n}=\sum_{i=1}^{n}\int_{0}^{L_{i}}ds_{i}\left[\mu+\dfrac{1}{2\kappa}\left(\dfrac{du_{\mu}(s_{i})}{ds_{i}}\right)^{2}\right],$
(20)

where $\mu$ indicates the action per length and is called the tension of the
center vortex. The parameter $\dfrac{1}{\kappa}$ shows the stiffness of the
vortices and it is small for flexible vortices. The structure of a vortex is
described by a space curve $x(s)$ in which $s\in\left[0,L\right]$ is a
parameter denoting the arc length along the vortex backbone. The vector
$u\left(s\right)=\frac{dx\left(s\right)}{ds}$ is a tangent vector and
$\frac{du\left(s\right)}{ds}=\frac{d^{2}x\left(s\right)}{ds^{2}}$ is
interpreted as the local curvature of the vortex. Fredrickson

To make the model more physical, one should assume a potential that represents
the interaction between the vortices. Therefore, a delta-function type
potential $V(x_{i}-x_{j})$ is assumed oxman ,

$V(x_{i}-x_{j})=\dfrac{1}{\zeta}\delta^{\left(3\right)}(x_{i}-x_{j}),\quad\zeta>0.$
(21)

We call the parameter $\dfrac{1}{\zeta}$ as the strength of the potential. The
action corresponding to Eq. (21), shown the vortices interaction is,

$\begin{split}S^{\text{int}}_{n}&=\dfrac{1}{2}\sum_{i,j=1}^{n}\int_{0}^{L_{i}}\int_{0}^{L_{j}}ds_{i}ds_{j}V(x_{i}(s_{i}),x_{j}(s_{j}))\\\
&=\dfrac{1}{2\zeta}\sum_{i,j=1}^{n}\int_{0}^{L_{i}}\int_{0}^{L_{j}}ds_{i}ds_{j}\delta^{\left(3\right)}(x_{i}(s_{i})-x_{j}(s_{j})).\end{split}$
(22)

Using Eqs. (20) and (22), the action of the vortex ensemble is obtained as the
sum of the non-interacting part $S^{0}_{n}$ and the interacting part
$S^{\text{int}}_{n}$,

$S^{\text{total}}_{n}=S^{0}_{n}+S^{\text{int}}_{n}.$ (23)

Next, an interaction between the Wilson loop, defined by the vector field
$J_{\mu}^{C}(x)$ in Eq. (19), and the vortex ensemble is assumed. Using Eqs.
(19) and (23), the partition function of the ensemble of $n$ vortices and the
Wilson loop is written as,

$Z\left[J_{\mu}^{C}\right]=\sum_{n}\int\left[Dl\right]_{n}e^{i\sum_{i=1}^{n}\int_{0}^{L_{i}}ds_{i}u_{\mu}(s_{i})J^{C}_{\mu}(x(s_{i}))}e^{-S^{\text{total}}_{n}},$
(24)

where,

$\left[Dl\right]_{n}\equiv\dfrac{1}{n!}\prod_{i=1}^{n}\int_{0}^{\infty}\dfrac{dL_{i}}{L_{i}}\prod_{j=1}^{n}\int
dv_{j}\prod_{k=1}^{n}\int\left[Dv\left(s_{k}\right)\right]_{v_{k},v_{k}}^{L_{k}},$
(25)

The measure $\left[Dl\right]_{n}$ integrates over all possible states of the
$n$ vortices. The measure $dL_{i}$ integrates on vortex worldlines which have
a fixed length of $L$. The measures $dv_{j}=d^{3}x_{j}d^{2}u_{j}$ and
$\left[Dv\left(s_{k}\right)\right]_{v_{k},v_{k}}^{L_{k}}$ correspond to the
vortex loop oxman .

For no intersection between the Wilson surface and the vortices, the vector
field $J^{C}_{\mu}$ is zero and the partition function of the vortex ensemble
is $Z\left[J^{C}_{\mu}=0\right]\equiv
Z\left[0\right]=\sum_{n}\int\left[Dl\right]_{n}e^{-S^{\text{total}}_{n}}$.
Therefore, one can define the Wilson loop average in terms of the two
partition functions,

$\left\langle W(C)\right\rangle=\dfrac{Z[J^{C}_{\mu}]}{Z[0]}.$ (26)

In Sec. IV, using Eq. (26) we obtain the area law fall-off for the Wilson loop
average.

To get a more efficient statement for the partition function,
$e^{-S^{\text{int}}_{n}}$ in Eq. (22) can be rewritten in terms of a scalar
vortex density $\rho(x)$, defined as the following,

$\rho(x)\equiv\sum_{i=1}^{n}\int_{0}^{L_{i}}ds_{i}\delta^{(3)}(x-x_{i}(s_{i})).$
(27)

Using Eq. (27),

$e^{-\frac{1}{2\zeta}\int d^{3}x\rho^{2}}=e^{-S^{\text{int}}_{n}}.$ (28)

${W[\phi]}$ is defined in terms of the scalar field $\phi$,

$W[\phi]\equiv\int d^{3}x\phi^{2}(x).$ (29)

Using Eq. (29) and the Gaussian formula, Eq. (28) is obtained from the left
hand side of the following expression,

$\dfrac{1}{\sqrt{\pi}}\int\left[D\phi\right]e^{-W[\phi]}e^{i\sqrt{\frac{2}{\zeta}}\int
d^{3}x\rho(x)\phi(x)}=e^{-S^{\text{int}}_{n}}.$ (30)

Replacing Eqs. (20), (23) and (30) in Eq. (24),

$Z\left[J^{C}_{\mu}\right]=\dfrac{1}{\sqrt{\pi}}\int\left[D\phi\right]e^{-W[\phi]}\sum_{n}\int\left[Dl\right]_{n}e^{-\sum_{i=1}^{n}\int_{0}^{L_{i}}ds_{i}\left[w(x_{i},u_{\mu}(s_{i}))+\frac{1}{2\kappa}\dot{u}_{\mu}^{2}(s_{i})\right]},$
(31)

where

$w(x_{i},u_{\mu}(s_{i}))\equiv\mu-i\left(u_{\mu}(s_{i})J^{C}_{\mu}(x(s_{i}))+\sqrt{\dfrac{2}{\zeta}}\phi(x_{i})\right).$
(32)

In order to calculate the partition function of Eq. (31), particle physicists
use the trick introduced by polymer physicists which we are going to briefly
explain.

(a)

(b)

(c)

(d)

Figure 3: (a) A continuous curve between $x^{i}$ and $x^{f}$, (b) the polymer
approximation of (a). (c) a closed curve, (d) the polymer approximation of
(c).

The continuous curve of Fig. 3(a), is divided into $N$ parts shown in Fig.
3(b). It resembles a polymer with the vertices acting as atoms (or monomers)
and the segments acting as bonds. The polymer of Fig. 3(b) extends from
$x^{i}$ to $x^{f}$ via $x^{1},x^{2},...,x^{\left(N-1\right)}$ and each bond
has a fixed length equal to $\epsilon$. Therefore, the total length of the
polymer is equal to $N\epsilon.$ Similarly, a vortex of worldline of length
$L$ can be divided into $N$ parts with fixed length $\epsilon$ and the
extension of the vortex is assumed to be from $x^{0}$ to $x^{N}$. To get the
continuum, $\epsilon\rightarrow 0$ and $N\rightarrow\infty$, and finally the
vortex loop is obtained if $x^{i}=x$, $x^{f}=x$ (see Fig. 3(c)). The polymer
approximation of a loop shown in Fig. 3(c) is brought in Fig. 3(d)).

Replacing the measure $[Dl]_{n}$ of Eq. (25) in Eq. (31), the partition
function for the vortex worldline with length $L$ which extends from $x^{0}$
identified by $u_{\mu}^{0}$ to $x^{N}$ identified by $u_{\mu}^{N}$ is,

$Z\left[J^{C}_{\mu}\right]=\dfrac{1}{\sqrt{\pi}}\int\left[D\phi\right]\
e^{-W[\phi]}\ \text{exp}\left\\{\int_{0}^{\infty}\dfrac{dL}{L}\int
dvq\left(v,v,L\right)\right\\}$ (33)

$q\left(v^{N},v^{0},L\right)$ is defined as the following,

$q\left(v^{N},v^{0},L\right)=\int\left[Dv\left(s_{k}\right)\right]_{v^{N},v^{0}}^{L}e^{-\int_{0}^{L}ds\left[w(x,u_{\mu}(s))+\frac{1}{2\kappa}\left(\dot{u}_{\mu}(s)\right)^{2}\right]}.$
(34)

It is called the end-to-end distance probability distribution for the vortex
worldline with length $L$, which goes from $x^{0}$ to $x^{N}$ (see Fig. 3(a)).
Therefore, $\int dvq\left(v,v,L\right)$ in the last equality of Eq. (33)
represent the partition function of a vortex loop with fixed length $L$.
Setting $v^{N}=v^{0}=v$ in Eq. (34), we obtain $q$ for a loop needed in Eq.
(33).

$q\left(x^{N},u^{N};x^{0},u^{0};L\right)=\int\prod_{i=1}^{N-1}d^{3}x^{i}d^{2}u^{i}\prod_{j=1}^{N}e^{-\epsilon
w\left(x^{j}(s^{j}),u^{j}\right)}\prod_{k=1}^{N}\psi\left(u^{k}-u^{k-1}\right)\delta^{(3)}\left(x^{k}-x^{k-1}-u^{k}\epsilon\right),$
(35)

where $\psi\left(u^{k}-u^{k-1}\right)\equiv
e^{-\frac{1}{2\kappa}\epsilon\left(\frac{u^{k}-u^{k-1}}{\epsilon}\right)^{2}}$.

On the other hand, the relation between $q\left(v^{N},v^{0},L\right)$ and
$q\left(x^{N},u^{N};x^{0},u^{0};L\right)$ is defined as,

$q\left(v^{N},v^{0},L\right)=\lim_{N\rightarrow\infty}q\left(x^{N},u^{N};x^{0},u^{0};L\right).$
(36)

Separating one of the segments, for example, the last one, from the others and
keeping the rest as a string with the length $L-\epsilon$, we can rewrite (35)
as the following,

$q\left(x^{N},u^{N};x^{0},u^{0};L\right)=e^{-\epsilon
w\left(x^{N}(s^{N}),u^{N}\right)}\int
d^{2}u^{N-1}\psi\left(u^{N}-u^{N-1}\right)q\left(x^{N}-u^{N}\epsilon,u^{N-1};x^{0},u^{0};L-\epsilon\right).$
(37)

where $q\left(x^{N-1},u^{N-1};x^{0},u^{0},L-\epsilon\right)$ is the end-to-end
distance probability distribution for the vortex worldline with length
$L-\epsilon$ extending from $x^{0}$ to $x^{N-1}$.

Taylor expanding both sides of Eq. (37), and keeping the terms to the first
order in $\epsilon$, one finds that,

$\partial_{L}q\left(x^{N},u^{N};x^{0},u^{0};L\right)=\left[\dfrac{\kappa}{2}\nabla^{2}_{u}-\mu+i\sqrt{\frac{2}{\zeta}}\phi(x_{N})-u^{N}.D_{\mu}\right]q\left(x^{N},u^{N};x^{0},u^{0};L\right),\quad
D_{\mu}=\nabla_{x^{N}}-iJ^{C}_{\mu}(x^{N}),$ (38)

where $D_{\mu}$ is the covariant derivative and $\nabla_{u}^{2}$ is the
Laplacian operator on the unit sphere.

Equation (38) can be solved with the following initial conditions,

$q\left(x^{N},u^{N};x^{0},u^{0};L=0\right)=\delta^{\left(3\right)}\left(x^{N}-x^{0}\right)\delta^{\left(2\right)}\left(u^{N}-u^{0}\right).$
(39)

For the small stiffness limit, corresponding to large $\kappa$, we can use the
methods of Refs. oxman ; Fredrickson ; oxman1  and finally, the solution can
be approximated by,

$q\left(x^{N},u^{N};x^{0},u^{0};L\right)\approx\dfrac{1}{4\pi}\left\langle
x^{N}\left|e^{-L\hat{O}}\right|x^{0}\right\rangle,$ (40)

where

$\hat{O}=-\dfrac{1}{3\kappa}D_{\mu}D_{\mu}+\mu-i\sqrt{\dfrac{2}{\zeta}}\phi.$
(41)

Replacing Eq. (40) in the exponential of Eq. (33) for $x^{N}=x^{0}=x$,

$\begin{split}\int_{0}^{\infty}\dfrac{dL}{L}\int
dvq\left(v,v;L\right)&=\int_{0}^{\infty}\dfrac{dL}{L}\int d^{3}x\left\langle
x\left|e^{-L\hat{O}}\right|x\right\rangle\\\ &=\int_{0}^{\infty}\dfrac{dL}{L}\
\text{Tr}\ e^{-L\hat{O}}=\text{Tr}\ln\hat{O}.\end{split}$ (42)

And we use the following equation,

$\exp\left\\{-\text{Tr}\ln\hat{O}\right\\}=\int D\bar{V}DV\ e^{-\int
d^{3}x\bar{V}\hat{O}V}.$ (43)

Replacing Eqs. (29),(41), (42) and (43) into Eq. (33), the partition function
$Z[J^{C}_{\mu}]$ is,

$\begin{split}Z\left[J^{C}_{\mu}\right]&=\dfrac{1}{\sqrt{\pi}}\int\left[D\phi\right]\
e^{-W[\phi]}\int D\bar{V}DV\ e^{-\int
d^{3}x\bar{V}\left(-\frac{1}{3\kappa}D_{\mu}D_{\mu}+\mu-i\sqrt{\frac{2}{\zeta}}\phi\right)V}\\\
&=\dfrac{1}{\sqrt{\pi}}\int D\bar{V}DV\ e^{-\int
d^{3}x\left(\frac{1}{3\kappa}\bar{D}_{\mu}\bar{V}D_{\mu}V+\mu\bar{V}V\right)}\int\left[D\phi\right]\
e^{\int d^{3}x\left(-\phi^{2}+i\sqrt{\frac{2}{\zeta}}\bar{V}V\phi\right)}\\\
&=\int D\bar{V}DV\ e^{-\int
d^{3}x\left(\frac{1}{3\kappa}\bar{D}_{\mu}\bar{V}D_{\mu}V+\mu\bar{V}V+\frac{1}{2\zeta}\left(\bar{V}V\right)^{2}\right)},\end{split}$
(44)

in the last equality, the Gaussian formula is used to calculate the second
integral and $V$ is a complex field.

In this section, we have explained how one can obtain an effective partition
function for a vortex ensemble using the trick of polymer physicists. In the
next section, we use the effective partition function Eq. (44) to compute the
Wilson loop average and we obtain the area law fall-off which leads to the
linear potential between a pair of static quark-antiquark and at the end the
string tension is extracted. The dependence of the string tension on the
vortex parameters and the temperature are discussed in detail.

##  IV the Wilson loop average in the presence of the center vortices - the
area law fall-off

From the partition function of Eq. (44), the Lagrangian of the vortex ensemble
is understood to be,

$\mathcal{L}=\dfrac{1}{3\kappa}\bar{D}_{\mu}\bar{V}D_{\mu}V+\underbrace{\mu\bar{V}V+\dfrac{1}{2\zeta}\left(\bar{V}V\right)^{2}}_{\equiv\mathbf{V}\left(\bar{V},V\right)},$
(45)

The Lagrangian $\mathcal{L}$ is invariant under a global U($1$) gauge
transformation. The vacuum expectation value $v$ is obtained by minimizing the
potential $\mathbf{V}\left(\bar{V},V\right)$,

$v^{2}=-\mu\zeta>0,\quad\mu<0.$ (46)

The complex scalar field $V(x)$ can be defined as,

$V(x)=ve^{i\gamma(x)}.$ (47)

The scalar field $\gamma(x)$ is a modulo $2\pi$ function so that a set of
degenerate vacua are related to each other. Rewriting the potential
$\mathbf{V}\left(\bar{V},V\right)$ in terms of the vacuum expectation value
$v$, the partition function is,

$Z\left[J^{C}_{\mu}\right]=\int D\bar{V}DV\ e^{-\int
d^{3}x\left(\frac{1}{3\kappa}\bar{D}_{\mu}\bar{V}D_{\mu}V+\frac{1}{2\zeta}\left(\bar{V}V-v^{2}\right)^{2}\right)}.$
(48)

To compute the Wilson loop average in the presence of the vortices, we use
Eqs. (26) and (48).

We recall that when there is no intersection between the Wilson surface and
the vortices, the vector field $J^{C}_{\mu}$ is zero and the covariant
derivative $D_{\mu}=\partial_{\mu}-igJ^{C}_{\mu}$ is replaced by the partial
derivative $\partial_{\mu}$. Therefore, using Eq. (48), the partition function
$Z[0]$ is,

$Z[0]=\int D\bar{V}DVe^{-\int
d^{3}x\left(\frac{1}{3\kappa}\partial_{\mu}\bar{V}\partial_{\mu}V+\frac{1}{2\zeta}\left(\bar{V}V-v^{2}\right)^{2}\right)}.$
(49)

Then, the Wilson loop average is calculated as the following,

$\left\langle W(C)\right\rangle=\dfrac{Z[J^{C}_{\mu}]}{Z[0]}=\dfrac{\int
D\bar{V}DV\ e^{-\int
d^{3}x\left(\frac{1}{3\kappa}\bar{D}_{\mu}\bar{V}D_{\mu}V+\frac{1}{2\zeta}\left(\bar{V}V-v^{2}\right)^{2}\right)}}{\int
D\bar{V}DVe^{-\int
d^{3}x\left(\frac{1}{3\kappa}\partial_{\mu}\bar{V}\partial_{\mu}V+\frac{1}{2\zeta}\left(\bar{V}V-v^{2}\right)^{2}\right)}}=\dfrac{\int
D\bar{V}DV\ e^{-\int
d^{3}x\frac{1}{3\kappa}\bar{D}_{\mu}\bar{V}D_{\mu}V}}{\int D\bar{V}DVe^{-\int
d^{3}x\frac{1}{3\kappa}\partial_{\mu}\bar{V}\partial_{\mu}V}}.$ (50)

The last equality is obtained by expansion of the complex scalar field V
around the vacuum expectation value.

The partition function $Z[0]$ in Eq. (50) is defined in the absence of any
linking between the vortices and the Wilson loop,

$Z[0]=\int D\bar{V}DVe^{-\int
d^{3}x\frac{1}{3\kappa}\partial_{\mu}\bar{V}\partial_{\mu}V}.$ (51)

Using Eq. (47),

$\partial_{\mu}\bar{V}(x)\partial_{\mu}V(x)=v^{2}\left(\partial_{\mu}\gamma(x)\right)^{2}.$
(52)

Replacing Eq. (52) in Eq. (51), the partition function $Z[0]$ is,

$Z[0]=N^{\prime}v^{2}\int D\gamma e^{-\frac{v^{2}}{3\kappa}\int
d^{3}x\left(\partial_{\mu}\gamma(x)\right)^{2}},$ (53)

where $N^{\prime}$ is an overall normalization factor and is irrelevant.
Equation (53) expresses the partition function $Z[0]$ as a functional integral
on $\gamma$. In three-dimensional Euclidean space-time kapusta ,

$\begin{split}S_{0}&\equiv-\frac{v^{2}}{3\kappa}\int
d^{3}x\left(\partial_{\mu}\gamma(x)\right)^{2}\\\
&=-\frac{v^{2}}{3\kappa}\int^{\beta}_{0}dt\int
d^{2}x\left[\left(\frac{\partial\gamma(x)}{\partial
t}\right)^{2}+\left(\nabla\gamma(x)\right)^{2}\right]\\\
&=-\frac{v^{2}}{3\kappa}\int^{\beta}_{0}dt\int
d^{2}x\gamma(x)\left(-\frac{\partial^{2}}{\partial
t^{2}}-\nabla^{2}\right)\gamma(x).\end{split}$ (54)

Next, we use a Fourier expansion for the scalar field $\gamma(x)$ kapusta ,

$\gamma(x)=\gamma\left(t,x,y\right)=\gamma\left(t,\vec{x}\right)=\sum_{n=-\infty}^{\infty}\sum_{\vec{p}}e^{i\left(w_{n}t+\vec{p}.\vec{x}\right)}\gamma_{n}(\vec{p}),$
(55)

where $w_{n}=2\pi nT$ and $T$ is the temperature. Using Eq. (55),

$\begin{split}&\frac{\partial^{2}\gamma(x)}{\partial
t^{2}}=-\sum_{n=-\infty}^{\infty}\sum_{\vec{p}}w_{n}^{2}e^{i\left(w_{n}t+\vec{p}.\vec{x}\right)}\gamma_{n}(\vec{p}).\\\
&\nabla^{2}\gamma(x)=\frac{\partial^{2}\gamma(x)}{\partial
x^{2}}+\frac{\partial^{2}\gamma(x)}{\partial y^{2}}\\\ &\quad\quad\quad\
=-\sum_{n=-\infty}^{\infty}\sum_{\vec{p}}\vec{p}^{2}e^{i\left(w_{n}t+\vec{p}.\vec{x}\right)}\gamma_{n}(\vec{p}),\quad\vec{p}^{2}=\vec{p}^{2}_{x}+\vec{p}^{2}_{y}.\end{split}$
(56)

Replacing Eq. (56) in Eq. (54),

$S_{0}=-\dfrac{v^{2}\mathtt{V}}{3\kappa}\sum_{n=-\infty}^{\infty}\sum_{\vec{p}}\left(w_{n}^{2}+\vec{p}^{2}\right)\left|\gamma_{n}(\vec{p})\right|^{2},$
(57)

where $\mathtt{V}\equiv\beta L^{2}$ is the volume and $L$ indicates the
spatial length and $\beta$ represents the imaginary time. Using Eq. (57) in
Eq. (53) for the action, the partition function $Z[0]$ is,

$\begin{split}Z[0]&=N^{\prime}v^{2}\prod_{n=-\infty}^{\infty}\prod_{\vec{p}}\
\int d\gamma_{n}(\vec{p})\
e^{-\frac{v^{2}\mathtt{V}}{3\kappa}\left(w_{n}^{2}+\vec{p}^{2}\right)\left|\gamma_{n}(\vec{p})\right|^{2}}\\\
&=N^{\prime}v^{2}\prod_{n=-\infty}^{\infty}\prod_{\vec{p}}\left(\dfrac{3\kappa\pi}{v^{2}\mathtt{V}}\right)^{\frac{1}{2}}\left[w_{n}^{2}+\vec{p}^{2}\right]^{-\frac{1}{2}}.\end{split}$
(58)

In the second equality, the Gaussian integral formula is used.

Next, we calculate the partition function $Z[J^{C}_{\mu}]$ in Eq. (50). We
recall that it shows the case where the Wilson loop links the vortices,

$Z[J^{C}_{\mu}]=\int D\bar{V}DV\ e^{-\frac{1}{3\kappa}\int
d^{3}x\bar{D}_{\mu}\bar{V}D_{\mu}V}.$ (59)

Replacing the definition of the covariant derivative
$D_{\mu}=\partial_{\mu}-iJ^{C}_{\mu}$ in the above equation,

$Z[J^{C}_{\mu}]=\int D\bar{V}DV\ e^{-\frac{1}{3\kappa}\int
d^{3}x\partial_{\mu}\bar{V}(x)\partial_{\mu}V(x)}\ e^{-\frac{1}{3\kappa}\int
d^{3}xJ^{C}_{\mu}\bar{V}(x)J^{C}_{\mu}V(x)}\ e^{\frac{i}{3\kappa}\int
d^{3}x\left(\partial_{\mu}\bar{V}(x)\right)J^{C}_{\mu}V(x)}\
e^{-\frac{i}{3\kappa}\int d^{3}xJ^{C}_{\mu}(x)\bar{V}(x)\partial_{\mu}V(x)}.$
(60)

To obtain the partition function $Z[J^{C}_{\mu}]$, we first calculate the four
exponential functions appeared in Eq. (60).

The first one is,

$\begin{split}I_{1}&\equiv e^{-\frac{1}{3\kappa}\int
d^{3}x\partial_{\mu}\bar{V}(x)\partial_{\mu}V(x)}=e^{S_{0}}\\\
&=\prod_{n=-\infty}^{\infty}\prod_{\vec{p}}e^{-\frac{v^{2}\mathtt{V}}{3\kappa}\left(w_{n}^{2}+\vec{p}^{2}\right)\left|\gamma_{n}(\vec{p})\right|^{2}},\end{split}$
(61)

where we have used Eq. (57) in the second line.

The second exponential function in the integral of Eq. (60) is,

$I_{2}\equiv e^{-\frac{1}{3\kappa}\int
d^{3}xJ^{C}_{\mu}\bar{V}(x)J^{C}_{\mu}V(x)}.$ (62)

The Wilson surface whose boundary is the Wilson loop is not unique and
different Wilson surfaces are related to each other by gauge transformations.
On the other hand, the string tension which is calculated by the Wilson loop
average, is a gauge invariant quantity and is independent of the choice of the
Wilson surface shape. Each point on the Wilson surface is characterized by
$\left(t,x,y\right)$. For simplicity, we consider the Wilson surface as a
rectangular shape on the $t-x$ plane and each point on this Wilson surface is
parameterized by $\left(\bar{t},\bar{x}\right)$. Fig. 4 shows a schematic
picture of a three dimensional Wilson loop which is projected on $t-x$ plane.
The origin is located on one of the corners of the rectangle. Projection on
the $t-x$ plane is assumed to be equivalent to a gauge transformation.

Figure 4: Intersection between the Wilson surface and the vortex loop.

Replacing Eqs. (15) and (47) in Eq. (62),

$\begin{split}I_{2}&=e^{-\frac{v^{2}\nu^{2}}{12\kappa}\int_{S}d^{2}\sigma_{\mu}\int_{S}d^{2}\sigma^{\prime}_{\mu}\int
d^{3}x\delta^{\left(3\right)}\left(x-\bar{x}(\sigma)\right)\delta^{\left(3\right)}\left(x-\bar{x}^{\prime}(\sigma^{\prime})\right)}\\\
&=e^{-\frac{v^{2}\nu^{2}}{12\kappa}\int_{S}d^{2}\sigma_{\mu}\int_{S}d^{2}\sigma^{\prime}_{\mu}\delta^{\left(3\right)}\left(\bar{x}^{\prime}(\sigma^{\prime})-\bar{x}(\sigma)\right)}\\\
&=e^{-\frac{v^{2}\nu^{2}}{12\kappa}\int_{S}d^{2}\sigma_{\mu}\int_{S}d\bar{t}^{\prime}d\bar{x}^{\prime}\delta\left(\bar{t}^{\prime}-\bar{t}\right)\delta\left(\bar{x}^{\prime}-\bar{x}\right)\delta\left(\bar{y}^{\prime}-\bar{y}\right)}\\\
&=e^{-\frac{v^{2}\nu^{2}}{12\kappa}\int_{S}d\bar{t}d\bar{x}\delta\left(\bar{y}^{\prime}-\bar{y}\right)}.\end{split}$
(63)

We have chosen $\bar{y}^{\prime}=0$ since the projected Wilson loop is indeed
located at the $t-x$ plane. Therefore the above equation is,

$I_{2}=e^{-\frac{v^{2}\nu^{2}}{12\kappa}\delta\left(\bar{y}\right)\int_{0}^{\tau}d\bar{t}\int_{0}^{l_{x}}d\bar{x}}=e^{-\frac{v^{2}\nu^{2}}{12\kappa}\delta\left(\bar{y}\right)\times
S},$ (64)

where $S\equiv\tau l_{x}$ is the area of the Wilson loop in the $t-x$ plane.
$\delta\left(\bar{y}\right)$ shows the intersection location of the Wilson
surface and the vortex loop (see Fig. 1.(c)). In other words, we have a
singularity at $\bar{y}=0$.

The third exponential function of the integral of Eq. (60) is,

$I_{3}\equiv e^{\frac{i}{3\kappa}\int
d^{3}x\left(\partial_{\mu}\bar{V}(x)\right)J^{C}_{\mu}V(x)}.$ (65)

Replacing Eqs. (15) and (47) in Eq. (65),

$\begin{split}I_{3}&=e^{\frac{i}{3\kappa}\int
d^{3}x\partial_{\mu}\left(ve^{-i\gamma(x)}\right)\frac{\nu}{2}\int_{S}d^{2}\sigma_{\mu}\delta^{\left(3\right)}\left(x-\bar{x}(\sigma)\right)ve^{i\gamma(x)}}\\\
&=e^{\frac{v^{2}\nu}{6\kappa}\int_{S}d^{2}\sigma_{\mu}\int
d^{3}x\partial_{\mu}\gamma(x)\delta^{\left(3\right)}\left(x-\bar{x}(\sigma)\right)}\\\
&=e^{\frac{v^{2}\nu}{6\kappa}\int_{S}d^{2}\sigma_{\mu}\partial_{\mu}\gamma(\bar{x}(\sigma))}.\end{split}$
(66)

On the other hand,

$d^{2}\sigma_{\mu}\partial_{\mu}\gamma(x(\sigma))=d^{2}\sigma_{0}\partial_{0}\gamma(x)+d^{2}\sigma_{1}\partial_{1}\gamma(x)+d^{2}\sigma_{2}\partial_{2}\gamma(x).$
(67)

From Fig. (4), we choose the following parameterization,

$\begin{split}&\bar{x}_{0}=\bar{t},\quad\bar{x}_{1}=\bar{x},\quad\bar{x}_{2}=\bar{y}\\\
&\sigma_{1}=\bar{t},\quad\sigma_{2}=\bar{x}.\end{split}$ (68)

Using Eqs. (13) and (68), the surface elements $d^{2}\sigma_{0}$ and
$d^{2}\sigma_{1}$ are zero, and the surface element $d^{2}\sigma_{2}$ is equal
to $d\bar{t}d\bar{x}$. Thus,

$d^{2}\sigma_{\mu}\partial_{\mu}\gamma(x(\sigma))=d^{2}\sigma_{2}\partial_{2}\gamma(x(\sigma))=d\bar{t}d\bar{x}\dfrac{\partial\gamma(x)}{\partial
y}.$ (69)

From Eq. (55),

$\begin{split}&\dfrac{\partial\gamma(x)}{\partial
y}=\sum_{n=-\infty}^{\infty}\sum_{\vec{p}}ip_{y}\
e^{i\left(w_{n}t+\vec{p}.\vec{x}\right)}\gamma_{n}(\vec{p}),\\\
&\dfrac{\partial\gamma(\bar{x})}{\partial
y}\mid_{S}=\sum_{n=-\infty}^{\infty}\sum_{\vec{p}}ip_{y}\
e^{i\left(w_{n}\bar{t}+p_{x}\bar{x}\right)}\gamma_{n}(\vec{p}).\end{split}$
(70)

Finally,

$\int_{S}d^{2}\sigma_{\mu}\partial_{\mu}\gamma(\bar{x}(\sigma))=\sum_{n=-\infty}^{\infty}\sum_{\vec{p}}ip_{y}\gamma_{n}(\vec{p})\int_{S}d\bar{t}d\bar{x}e^{i\left(w_{n}\bar{t}+p_{x}\bar{x}\right)}.$
(71)

Replacing Eq. (71) in Eq. (66),

$\begin{split}I_{3}&=e^{\frac{v^{2}\nu}{6\kappa}\sum_{n=-\infty}^{\infty}\sum_{\vec{p}}ip_{y}\gamma_{n}(\vec{p})\int_{0}^{\tau}d\bar{t}e^{iw_{n}\bar{t}}\int_{0}^{l_{x}}d\bar{x}e^{ip_{x}\bar{x}}}\\\
&=e^{\frac{v^{2}\nu}{6\kappa}\sum_{n=-\infty}^{\infty}\sum_{\vec{p}}ip_{y}\gamma_{n}(\vec{p})\tau
l_{x}\delta_{w_{n},0}\delta_{p_{x},0}},\end{split}$ (72)

where $w_{n}=2\pi nT$. Therefore, for finite temperature
$\delta_{w_{n},0}=\delta_{n,0}$.

$I_{3}=e^{i\frac{v^{2}\nu}{6\kappa}\tau
l_{x}\sum_{\vec{p}}p_{y}\gamma_{0}\left(\vec{\mathtt{P}}\right)}=\prod_{\vec{p}}e^{i\frac{v^{2}\nu}{6\kappa}\tau
l_{x}p_{y}\gamma_{0}\left(\vec{\mathtt{P}}\right)},$ (73)

where, we define
$\gamma_{0}\left(\vec{\mathtt{P}}\right)\equiv\gamma_{0}\left(0,\vec{p}_{y}\right)$.

The fourth exponential of the integral of Eq. (60) is,

$I_{4}\equiv e^{-\frac{i}{3\kappa}\int
d^{3}xJ^{C}_{\mu}(x)\bar{V}(x)\partial_{\mu}V(x)}.$ (74)

Replacing Eqs. (15) and (47) in Eq. (74),

$\begin{split}I_{4}&=e^{-\frac{i}{3\kappa}\int
d^{3}x\frac{\nu}{2}\int_{S}d^{2}\sigma_{\mu}\delta^{\left(3\right)}\left(x-\bar{x}(\sigma)\right)ve^{-i\gamma(x)}\partial_{\mu}\left(ve^{i\gamma(x)}\right)}\\\
&=e^{\frac{v^{2}\nu}{6\kappa}\int_{S}d^{2}\sigma_{\mu}\partial_{\mu}\gamma(\bar{x}(\sigma))}.\end{split}$
(75)

Comparing Eqs. (66) and (75), we conclude that $I_{3}=I_{4}$.

Now we are ready to calculate the partition function for the linking Wilson
loop and vortices using the above calculations. Replacing Eqs. (61), (64) and
(73) in Eq. (60),

$\begin{split}Z\left[J^{C}_{\mu}\right]&=N^{\prime}v^{2}e^{-\frac{v^{2}\nu^{2}}{12\kappa}\delta\left(\bar{y}\right)\times
S}\prod_{n=-\infty}^{\infty}\prod_{\vec{p}}\int d\gamma_{n}(\vec{p})\
e^{-\frac{v^{2}\mathtt{V}}{3\kappa}\left(w_{n}^{2}+\vec{p}^{2}\right)\left|\gamma_{n}(\vec{p})\right|^{2}}e^{i\frac{v^{2}\nu}{3\kappa}\tau
l_{x}p_{y}\gamma_{0}\left(\vec{\mathtt{P}}\right)}\\\
&=N^{\prime}v^{2}e^{-\frac{v^{2}\nu^{2}}{12\kappa}\delta\left(\bar{y}\right)\times
S}\times\left\\{\prod_{\overset{n=-\infty}{n\neq
0}}^{\infty}\prod_{\overset{\vec{p}}{p_{x}\neq 0}}\int d\gamma_{n}(\vec{p})\
e^{-\frac{v^{2}\mathtt{V}}{3\kappa}\left(w_{n}^{2}+\vec{p}^{2}\right)\left|\gamma_{n}(\vec{p})\right|^{2}}\right\\}\\\
&\qquad\qquad\quad\quad\quad\quad\quad\times\left\\{\prod_{\overset{\vec{p}}{p_{x}=0}}\int
d\gamma_{0}\left(\vec{\mathtt{p}}\right)e^{\left[-\frac{v^{2}\mathtt{V}}{3\kappa}p^{2}_{y}\left|\gamma_{0}(\vec{\mathtt{P}})\right|^{2}+i\frac{v^{2}\nu}{3\kappa}\tau
l_{x}p_{y}\gamma_{0}\left(\vec{\mathtt{P}}\right)\right]}\right\\}.\end{split}$
(76)

Using the Gaussian formula

$Z\left[J^{C}_{\mu}\right]=N^{\prime}v^{2}e^{-\frac{v^{2}\nu^{2}}{12\kappa}\delta\left(\bar{y}\right)\times
S}\times\left\\{\prod_{\overset{n=-\infty}{n\neq
0}}^{\infty}\prod_{\overset{\vec{p}}{p_{x}\neq
0}}\left(\dfrac{3\kappa\pi}{v^{2}\mathtt{V}}\right)^{\frac{1}{2}}\left[w_{n}^{2}+\vec{p}^{2}\right]^{-\frac{1}{2}}\right\\}\times\left\\{\prod_{\overset{\vec{p}}{p_{x}=0}}\left(\dfrac{3\kappa\pi}{v^{2}\mathtt{V}}\right)^{\frac{1}{2}}\dfrac{1}{p_{y}}\right\\}\times
e^{-\frac{v^{2}\nu^{2}}{12\kappa}\ \frac{\tau^{2}l^{2}_{x}}{\mathtt{V}}}.$
(77)

Following the same method of calculation $Z[J^{C}_{\mu}]$, we obtain $Z[0]$
from Eq. (58),

$\begin{split}Z[0]&=N^{\prime}v^{2}\prod_{n=-\infty}^{\infty}\prod_{\vec{p}}\
\int d\gamma_{n}(\vec{p})\
e^{-\frac{v^{2}\mathtt{V}}{3\kappa}\left(w_{n}^{2}+\vec{p}^{2}\right)\left|\gamma_{n}(\vec{p})\right|^{2}}\\\
&=N^{\prime}v^{2}\left\\{\prod_{\overset{n=-\infty}{n\neq
0}}^{\infty}\prod_{\overset{\vec{p}}{p_{x}\neq 0}}\int
d\gamma_{n}(\vec{p})e^{-\frac{v^{2}\mathtt{V}}{3\kappa}\left(w_{n}^{2}+\vec{p}^{2}\right)\left|\gamma_{n}(\vec{p})\right|^{2}}\right\\}\times\left\\{\prod_{\overset{\vec{p}}{p_{x}=0}}d\gamma_{0}(\vec{\mathtt{P}})e^{-\frac{v^{2}\mathtt{V}p^{2}_{y}}{3\kappa}\left|\gamma_{0}(\vec{\mathtt{P}})\right|^{2}}\right\\}\\\
&=N^{\prime}v^{2}\left\\{\prod_{\overset{n=-\infty}{n\neq
0}}^{\infty}\prod_{\overset{\vec{p}}{p_{x}\neq
0}}\left(\dfrac{3\kappa\pi}{v^{2}\mathtt{V}}\right)^{\dfrac{1}{2}}\left[w_{n}^{2}+\vec{p}^{2}\right]^{-\dfrac{1}{2}}\right\\}\times\left\\{\prod_{\overset{\vec{p}}{p_{x}=0}}\left(\dfrac{3\kappa\pi}{v^{2}\mathtt{V}}\right)^{\dfrac{1}{2}}\dfrac{1}{p_{y}}\right\\}.\end{split}$
(78)

Next, we compute the Wilson loop average for the vortex ensemble by replacing
Eqs. (77) and (78) in Eq. (50),

$\begin{split}\left\langle
W(C)\right\rangle&=\dfrac{N^{\prime}v^{2}e^{-\frac{v^{2}\nu^{2}}{12\kappa}\delta\left(\bar{y}\right)\times
S}\times\left\\{\prod_{\overset{n=-\infty}{n\neq
0}}^{\infty}\prod_{\overset{\vec{p}}{p_{x}\neq
0}}\left(\dfrac{3\kappa\pi}{v^{2}\mathtt{V}}\right)^{\frac{1}{2}}\left[w_{n}^{2}+\vec{p}^{2}\right]^{-\frac{1}{2}}\right\\}\times\left\\{\prod_{\overset{\vec{p}}{p_{x}=0}}\left(\dfrac{3\kappa\pi}{v^{2}\mathtt{V}}\right)^{\frac{1}{2}}\dfrac{1}{p_{y}}\right\\}\times
e^{-\frac{v^{2}\nu^{2}}{12\kappa}\
\frac{\tau^{2}l^{2}_{x}}{\mathtt{V}}}}{N^{\prime}v^{2}\left\\{\prod_{\overset{n=-\infty}{n\neq
0}}^{\infty}\prod_{\overset{\vec{p}}{p_{x}\neq
0}}\left(\dfrac{3\kappa\pi}{v^{2}\mathtt{V}}\right)^{\frac{1}{2}}\left[w_{n}^{2}+\vec{p}^{2}\right]^{-\frac{1}{2}}\right\\}\times\left\\{\prod_{\overset{\vec{p}}{p_{x}=0}}\left(\dfrac{3\kappa\pi}{v^{2}\mathtt{V}}\right)^{\frac{1}{2}}\dfrac{1}{p_{y}}\right\\}}\\\
\\\ &=e^{-\frac{v^{2}\nu^{2}}{12\kappa}\delta\left(\bar{y}\right)\times
S}\times e^{-\frac{v^{2}\nu^{2}}{12\kappa}\
\frac{\tau^{2}l^{2}_{x}}{\mathtt{V}}}.\end{split}$ (79)

In the infinite-volume limit, the second exponential in Eq. (79) is,

$\lim_{\mathtt{V}\rightarrow\infty}e^{-\frac{v^{2}\nu^{2}}{12\kappa}\
\frac{\tau^{2}l^{2}_{x}}{\mathtt{V}}}=1.$ (80)

Then, simplifying Eq. (79), the Wilson loop average $\left\langle
W(C)\right\rangle$ is obtained,

$\left\langle
W(C)\right\rangle=e^{-\frac{v^{2}\nu^{2}}{12\kappa}\delta\left(\bar{y}\right)\times
S}.$ (81)

This is what we have planned to obtain, the area law fall-off for the Wilson
loop which corresponds to a linear potential between a pair of static quark-
antiquark.

Next, we obtain the string tension and discuss its characteristics in terms of
the stiffness, tension and the parameter defined as the potential strength
between the vortices. As mentioned in section II, when a Wilson loop pierces
the hypersurface $\Sigma$, it observes a singularity (see Fig. 1.(a)), which
multiplies the Wilson loop by a phase (see Eq. (1)). On the other hand, we
recall that the linking number $L\left(C,l\right)$ is symmetric concerning the
interchange of the loops $C$ and $l$. Therefore, it can be interpreted in such
a way that the Wilson surface $S$ pierces by the vortex loop $l$ and the
singularity on $S$ is observed (see Fig. 1.(c)). This singularity is shown as
$\delta(\bar{y})$ in Eq. (81). For simplicity, we consider the Wilson surface
as a rectangular shape on the $t-x$ plane, therefore this singularity is
located on the $\bar{y}=0$ plane.

The area $S$ and the string tension $\sigma$ have dimensions $L^{2}$ and
$L^{-2}$, respectively; where $L$ indicates the length. On the other hand, the
delta function $\delta(\bar{y})$ has dimension $L^{-1}$. Using temperature $T$
which has dimension $L^{-1}$, the Wilson loop is rewritten as,

$\left\langle
W(C)\right\rangle=e^{-\frac{v^{2}\nu^{2}T}{12\kappa}\frac{\delta\left(\bar{y}\right)}{T}\times
S}.$ (82)

Where the parameter $\dfrac{\delta(\bar{y})}{T}$ is dimensionless. As a
result, the string tension is obtained in terms of the parameters $v$, $\nu$
and $\dfrac{1}{\kappa}$ which are the vacuum expectation value, the co-weights
and stiffness, respectively,

$\sigma=\dfrac{v^{2}\nu^{2}}{12\kappa}T.$ (83)

Using Eq. (46) and the fact that for the fundamental representation
$\nu=\dfrac{1}{2}$, the string tension of Eq. (83) is obtained as the
following,

$\sigma=\dfrac{\left|\mu\right|\zeta}{48\kappa}T.$ (84)

Where the sign of $\mu$ is applied by using the absolute value in the above
equation. Since all the parameters $\mu$, $\zeta$, $\kappa$ and $T$ have
dimension $L^{-1}$, the string tension in Eq. (84) has dimension $L^{-2}$.

From Eq. (84), the relation between the tension $\left|\mu\right|$ and the
stiffness $\dfrac{1}{\kappa}$ is,

$\dfrac{1}{\kappa}=\dfrac{48\sigma}{\zeta T}\dfrac{1}{\left|\mu\right|}.$ (85)

This equation shows that the stiffness $\dfrac{1}{\kappa}$ decreases by
increasing the tension $\left|\mu\right|$. Reduction of the stiffness results
in a more flexible vortex loop. This behavior is in good qualitative agreement
with the results reported in stiff1 .

To plot the stiffness $\dfrac{1}{\kappa}$ versus the tension
$\left|\mu\right|$, one must know the coefficient $\dfrac{48\sigma}{\zeta T}$
in Eq. (85). We have used figure $2$ of reference stiff1  and the coefficient
is obtained to be approximately equal to 0.012. Fig. 5 shows the plot of
$\dfrac{1}{\kappa}$ versus $\left|\mu\right|$ from our calculation and using
the coefficient $\dfrac{48\sigma}{\zeta T}$ obtained from reference stiff1 .
The stiffness and the tension are scaled with the temperature and are
dimensionless. A magnification is done in the upper right to compare the data
with the range of the data reported in figure 2 of reference stiff1 .

Assuming $\dfrac{48\sigma}{\zeta T}=0.012$ from the above discussion, the
dimensionless string tension is rewritten from Eq. (85),

$\dfrac{\sigma(T)}{T^{2}}=\dfrac{0.012}{48}\dfrac{\zeta(T)}{T}.$ (86)

Figure 5: stiffness $\dfrac{1}{\kappa}$ versus the tension $\left|\mu\right|$.
A magnification is done in the upper right of the plot to compare the data
with the range of the data reported in ref. stiff1 .

On the other hand, the temperature dependence of SU($2$) string tension has
been computed by lattice QCD zeta ,

$\dfrac{\sigma(T)}{T^{2}}=\dfrac{\sigma_{0}}{T^{2}_{c}}\left(\dfrac{T_{c}}{T}\right)^{2}A\left(1-\dfrac{T}{T_{c}}\right)^{0.63}\left(1+B\left(1-\dfrac{T}{T_{c}}\right)^{\frac{1}{2}}\right),$
(87)

where the constants $A=1.39$ and $B=\dfrac{1}{A}-1$ zeta . The ratio
$\dfrac{T_{c}}{\sqrt{\sigma_{0}}}\approx 0.69$ with
$\sigma_{0}=\left(440\text{Mev}\right)^{2}$ stiff1 , gives
$T_{c}=303.6\text{Mev}$. From Eqs. (86) and (87), the dimensionless quantity
$\dfrac{\zeta(T)}{T}$ is concluded,

$\dfrac{\zeta(T)}{T}=\dfrac{48}{0.012}\dfrac{\sigma_{0}}{T^{2}}A\left(1-\dfrac{T}{T_{c}}\right)^{0.63}\left(1+B\left(1-\dfrac{T}{T_{c}}\right)^{\frac{1}{2}}\right),$
(88)

where the string tension and $\dfrac{\zeta(T)}{T}$ are scaled with the
temperature. Fig. 6 plots $\dfrac{1}{\zeta(T)}$ versus $\dfrac{T}{T_{c}}$.

Figure 6: Dependence of the parameter which indicates the potential strength
between the vortices, to the temperature.

The potential between the vortices represented in Eq. (21) is repulsive and it
is a function of the parameter $\dfrac{1}{\zeta}$. On the other hand, Fig. 6
shows that $\dfrac{1}{\zeta}$ increases by increasing the temperature. Thus,
the repulsive potential between the vortices increases by increasing the
temperature. This result is in good agreement with the result of lattice QCD
zeta lattice  and the fact that by increasing the temperature one gets close
to the deconfinement regime and expects the disappearance of vortex
configurations. Increasing the repulsive forces between the vortices at higher
temperatures may prevent the vortices from having stable topological
structures required for quark confinement.

##  V Summary and Conclusions

Inspired by lattice results confirming the area law fall-off for the Wilson
loop average, we compute the Wilson loop in the continuum limit for SU($2$)
gauge group in three-dimensional Euclidean space-time. The area law fall-off
is observed and the linear potential between a pair of quark-antiquark and the
string tension is obtained.

A technique used by polymer physicists and later by particle physicists is
applied to an ensemble of vortices to get the appropriate partition function
containing a free complex scalar field representing the vortex field. The
action of an ensemble of $n$ non-interacting vortices is written in terms of
two intrinsic parameters of the vortices called the tension and the stiffness.
Then, a potential between the vortices is assumed and the partition function
of the ensemble of vortices linking the Wilson loop, is obtained.

We have used the above partition function to compute the Wilson loop. The area
law fall-off for the Wilson loop is observed and the string tension is
extracted in terms of the vortices parameters. The stiffens versus the tension
plot of this calculation is in good qualitative agreement with the lattice
results reported by stiff1 .

A comparison between our results and the lattice ones stiff1  is done to
obtain a coefficient of our calculations which contains the string tension and
the potential strength. As a result, a dimensionless string tension is defined
in terms of the potential strength. This string tension is compared with the
string tension reported by zeta , and the potential strength in terms of the
temperature is obtained. It is shown that the repulsive force between the
vortices increases with the temperature. This is what one expects from the
vortex models if one wants to see the reduction of vortex effects at higher
temperatures. It is also in agreement with lattice results where the effect of
topological defects must be decreased at smaller distances.

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