Monopole and Polyakov loop

TL;DR

This paper introduces a new order parameter for $Z_N$ symmetry in SU(N) gauge theories, analyzing how monopole condensation affects symmetry breaking using a dual superconductor model.

Contribution

It proposes a novel $Z_N$ order parameter based on the Polyakov loop and calculates monopole contributions, linking monopole condensation to symmetry breaking.

Findings

Monopole condensation leads to a non-zero order parameter.

The order parameter vanishes when $Z_N$ symmetry is preserved.

Calculations are performed within a dual superconductor model.

Abstract

We propose a new order parameter of a $Z_N$ symmetry in SU(N) gauge theories in $4$ dimensional Minkowski space-time, assuming spatial periodic boundary conditions. It is given by $Tr(P\exp(i\int_c A_{\mu}dx^{\mu}))$ where the spatial path $c$ is taken, for example, along $x_1$ axis. The parameter vanishes when the $Z_N$ symmetry is preserved. We calculate the contribution of QCD monopoles to the order parameter and show that when the monopoles condense $\langle\Phi\rangle\neq 0$, it vanishes, while it does not vanish when they do not condense. These calculations are performed using a monopole field $\Phi$ canonically quantized in a model of dual superconductor.

Equations

$$A_{\varphi }=g_m(1-\cos (\theta )),A_0=A_r=A_{\theta }=0$$

$$P_{c}=\frac{1}{2}Tr\big{(}P\exp(i\int_{c}dx^{\mu}gA_{\mu})\big{)}=\frac{1}{2}\Bigg{(}\exp(\frac{i}{2}\int_{c}dx^{1}gA_{1}^{a=3})+\exp(-\frac{i}{2}\int_{c}dx^{1}gA_{1}^{a=3})\Bigg{)}$$

$$A_1^3=-\frac{g_m\sin \varphi (1-\cos \theta )}{r\sin \theta }.$$

$${\int }_{-\infty }^{+\infty }d{x}^1{A}_1^3(x^1,x^2=y,x^3=z)=-g_m(\pi -2\arctan (z/y))=-g_m(\pi -2{\varphi }^{\prime }(y,z)),$$

$$\int d{x}^1{A}_1^3(x^1,y,z)=2g_m\int {\varphi }^{\prime }(\vec{x}-{\vec{x}}^{\prime })\rho ({\vec{x}}^{\prime })d^3{x}^{\prime }\text{mbox}with\rho (\vec{x})=\sum _{i=1,2,\cdots ,N}{\eta }_i{\delta }^3(\vec{x}-{\vec{x}}_i),$$

$$P_{c}(y,z,t)=\frac{1}{2}\Big{(}\exp\big{(}i\int\phi^{\prime}(\vec{x}-\vec{x}^{\prime})\rho(\vec{x}^{\prime})d^{3}x^{\prime}\big{)}+c.c.\Big{)}.$$

$$\langle P_{c}(\vec{x})\rangle_{\rm thermal}=\frac{1}{Z}\int\prod_{i}^{N}d^{3}x_{i}\exp\Big{(}i\sum_{i}\eta_{i}\phi^{\prime}(\vec{x}-\vec{x}_{i})-\sum_{i\neq j}\beta V(\vec{x}_{i}-\vec{x}_{j})\Big{)}$$

$$\rho ={\Phi }^{†}(i{\partial }_t+g_m{B}_t)\Phi +\Phi (-i{\partial }_t+g_m{B}_t){\Phi }^{†}=2v^2{g}_m{B}_t+4v{g}_m\delta \varphi B_t+2g_m(\delta \varphi {)}^2{B}_t$$

$$\displaystyle\langle\exp\Big{(}i\int\phi^{\prime}(\vec{x}-\vec{x}^{\prime})\rho(\vec{x}^{\prime})d^{3}x^{\prime}\Big{)}\rangle$$

$$⟨\rho ({\vec{x}}^{\prime })\rho ({\vec{y}}^{\prime })⟩=4v^4{g}_m^2⟨B_t({\vec{x}}^{\prime })B_t({\vec{y}}^{\prime })⟩+16v^2{g}_m^2⟨\delta \varphi ({\vec{x}}^{\prime })\delta \varphi ({\vec{y}}^{\prime })⟩⟨B_t({\vec{x}}^{\prime })B_t({\vec{y}}^{\prime })⟩,$$

$$⟨\delta \varphi ({\vec{x}}^{\prime })\delta \varphi ({\vec{y}}^{\prime })⟩=\int \frac{d^{3}k\exp (i\vec{k}\cdot ({\vec{x}}^{\prime }-{\vec{y}}^{\prime }))}{(2\pi {)}^{3}2\sqrt{{\vec{k}}^2+M^2}}\text{mbox}and⟨B_t({\vec{x}}^{\prime })B_t({\vec{y}}^{\prime })⟩=\int \frac{d^{3}k{\vec{k}}^2\exp (i\vec{k}\cdot ({\vec{x}}^{\prime }-{\vec{y}}^{\prime }))}{(2\pi {)}^{3}2m^2\sqrt{{\vec{k}}^2+m^2}}$$

$$⟨P_c⟩$$

$$\text{mbox}with|A_k{|}^2$$

$${\varphi }^{\prime }\to {\varphi }_{\mathrm{reg}}^{\prime }\equiv {\varphi }^{\prime }\exp (-a{r}^{\prime })$$

$${\int }_0^{\infty }{r}^{\prime }d{r}^{\prime }{\int }_0^{2\pi }d{\varphi }^{\prime }{\varphi }^{\prime }\exp (-ik{r}^{\prime }\cos {\varphi }^{\prime }-ar)\equiv \frac{1}{a^2}Q(\frac{k}{a})\text{mbox}withQ(x)={\int }_0^{2\pi }d{\varphi }^{\prime }\frac{{\varphi }^{\prime }(1-ix\cos {\varphi }^{\prime }{)}^2}{(1+x^2{\cos }^2{\varphi }^{\prime }{)}^2}.$$

$$a_k|0⟩=b_k|0⟩=0\text{mbox}where\Phi =\int \frac{d^{3}k}{\sqrt{(2\pi {)}^{3}2{\omega }_k}}(a_k\exp (-ikx)+b_k^{†}\exp (ikx))$$

$$\rho =$$

$$\int d^3{x}^{\prime }{d}^3{x}^{\prime \prime }{\varphi }_{\mathrm{reg}}^{\prime }(\vec{x}-{\vec{x}}^{\prime }){\varphi }_{\mathrm{reg}}^{\prime }(\vec{x}-{\vec{x}}^{\prime \prime })⟨\rho ({\vec{x}}^{\prime })\rho ({\vec{x}}^{\prime \prime })⟩$$

$$F(q,k)\equiv \frac{2(2M_0^2+q^2+k^2)K(1-\frac{M_0^2+k^2}{M_0^2+q^2})}{\sqrt{M_0^2+q^2}}-4\sqrt{M_0^2+k^2}E(1-\frac{M_0^2+q^2}{M_0^2+k^2})$$

Full text

Magnetic Monopole, Polyakov Loop, Confinement

Understanding the quark confinement is a long standing problem in SU(N) gauge theories. The confinement is characterized by the expectation value of the Polyakov loop $P(A)$ poly ; poly2 which represents the self energy of a quark classically put in the vacuum of the gauge theories. That is, $P(A)$ is defined in $4$ dimensional Euclidean space using the fundamental representation such that $P(A)=Tr(P\exp(i\int_{0}^{1/T}dx_{0}gA^{0}))$ with the temperature $T$ and the [math]th component $A_{0}$ of the gauge fields. Since the expectation value behaves such that $\langle P(A)\rangle\sim\exp(-E/T)$ with the self energy $E$ of the quark, the confinement is realized when $\langle P(A)\rangle=0$.

The fact can be understood in a different view point. The SU(N) gauge theories are invariant under the gauge transformation $A_{\mu}\to UA_{\mu}U^{-1}+\frac{1}{g}U\partial_{\mu}U^{-1}$ with the use of the gauge function such as $U(\vec{x},x_{0}=0)=Z_{N}U(\vec{x},x_{0}=1/T)$ where $Z_{N}=\exp(in\pi/N)$ ( $n=$ integer ). The transformation with $Z_{N}=1$ is the standard gauge transformation. But, the transformation with $Z_{N}\neq 1$ is non-trivial one associated with $Z_{N}$ symmetry. Under the transformation, $P(A)$ transforms such as $P(A)\to Z_{N}P(A)$. Thus, when the thermal state of the gauge theories is invariant under the transformation, $\langle P(A)\rangle$ automatically vanishes. The symmetry is called as $Z_{N}$ symmetry. The realization of the confinement can be seen by examining the value $\langle P(A)\rangle$. We note that the Polyakov loop is sensitive to the color electric excitations because it is defined using the temporal component $A_{0}$ of the gauge fields.

On the other hand, it is well known that magnetic monopolesmono arise in the gauge theories and their condensation causes the quark confinementnambu ; man ; thooft . That is, when the magnetic monopoles condense in the vacuum of the gauge theories, dual superconducting vacuumdual is realized so that color electric fields are squeezed into vortices. Thus, a quark ( color current ) put in the vacuum carries the infinitely long vortex of the color flux so that its energy is infinite. The physical view of the confinement is obvious. The picture is based on the Minkowski space-time. That is, the condensation is well formulated using a coherent state in Hamiltonian formalism. But, it is not obvious that the monopole condensation causes the vanishing of the Polyakov loop although there are some indications in numerical calculationssuzuki . Here we would like to address a question where or not there exists an order parameter described only by the gauge fields in the Minkowski space-time. The parameter plays a role similar to the one the Polyakov loop does. That is, it characterizes the confinement phase.

In this paper we define an new order parameter such as $P_{c}(A)=Tr(P\exp(i\int_{0}^{L}dx_{1}gA^{1}))$ expoly in $4$ dimensional Minkowski space-time where we assume the periodic boundary conditions in spatial directions with the length $L$ of the system. We show that $\langle P_{c}\rangle=0$ when the monopoles condense, while $\langle P_{c}\rangle\neq 0$ when they do not condense. Hence, it may characterize the confinement phase. The order parameter transforms such that $P_{c}(A)\to Z_{N}P_{c}(A)$ under the gauge transformation with the gauge function $U(x_{1},x_{2},x_{3},x_{0})$; $U(x_{1}=0,x_{2},x_{3},x_{0})=Z_{N}U(x_{1}=L,x_{2},x_{3},x_{0})$. When the expectation value of this parameter $\langle P_{c}(A)\rangle$ vanishes, we find that the $Z_{N}$ symmetry holds. We call the parameter as an extended Polyakov loop. ( Sometimes it is called as spatial Pokyakov loop in Euclidean spaces. But we discuss it only in the Minkowski space-time in this paper. ) The path of the line integral in $P_{c}=Tr(P\exp(i\int_{0}^{L}dx_{1}gA^{1}))$ is close since the periodic boundary condition is used.

The order parameter is sensitive to magnetic excitations because it is defined using the spatial component $A_{1}$ of the gauge fields. Indeed, we can represent it in terms of the canonically quantized monopole field $\Phi$ and show that $\langle P_{c}(A)\rangle$ vanishes when $\langle\Phi\rangle\neq 0$, while it does not vanish when $\langle\Phi\rangle=0$. Thus, the extended Polyakov loop is an order parameter of the $Z_{N}$ symmetry as well as one characterizing the confinement phase in $4$ dimensional Minkowski space-time. The physical meaning of the extended Polyakov loop is that it may represents a self energy of color electric current of the fundamental representation flowing in $x_{1}$ direction; $\langle P_{c}(A)\rangle\propto\exp(-EL)$. The current instantaneously flows from an edge in the $x_{1}$ direction to the other edge. ( Actually, we can check it by the calculation of $\int_{0}^{L}dx_{1}dy_{1}\langle gA^{1}(x_{1},x_{2},x_{3})gA^{1}(y_{1},x_{2},x_{3})\rangle$ using the translational invariance in the direction $x_{1}$. )

Here we should make a comment that the $Z_{N}$ symmetry defined in our paper is different from $Z_{N}$ symmetry used in the discussion of the Polyakov loop. The $Z_{N}$ symmetry in our paper is defined in the spatial direction of $4$ dimensional Minkowski space-time, while $Z_{N}$ symmetry in the Polyakov loop is defined in the temporal direction of $4$ dimensional Euclidean space. We may define the extended Polyakov loop in the Euclidean space and calculate it using lattice gauge theories. But we need to notice that in order to obtain the physically meaning behavior of the order parameter in lattice gauge theories, the extension $L$ in the spacial direction should be chosen to be larger than $\beta=1/T$. Otherwise, there is no distinction between the standard ( temporal ) Polyakov loop and the extended ( spatial ) Polyakov loop defined in $4$ dimensional Euclidean space. Namely, the standard ( temporal ) Polyakov loop is a relevant order parameter characterizing the confinement phase as long as $L>\beta$. Similarly the extended ( spatial ) Polyakov loop would play a role of a relevant order parameter as long as $L>\beta$.

For simplicity, we use SU(2) gauge theory. The generalization to SU(N) gauge theories is straightforward. We first explain Dirac monopoles. The Dirac monopolesmono are defined in U(1) gauge theory and their configuration is given such that

[TABLE]

where $\vec{A}\cdot d\vec{x}=A_{r}dr+A_{\theta}d\theta+A_{\phi}d\phi$ with polar coordinates $r,\theta$ and $\phi=\arctan(y/x)$. $g_{m}$ denotes a magnetic charge with which the magnetic field is given by $\vec{B}=g_{m}\vec{r}/r^{3}$. The magnetic charge satisfies the Dirac quantization condition $g_{m}g=n/2$ with integer $n$ where $g$ denotes the U(1) gauge coupling. Hereafter, we assume the monopoles with the magnetic charge $g_{m}=\pm 1/2g$.

The Dirac monopoles play a role in SU(2) gauge theory under the assumption of the Abelian dominanceabelian ; abelian2 . According to the assumption, the physical properties with low energies such as confinement, chiral symmetry breaking, etc, are described by massless gauge fields in the maximal Abelian gauge group of SU(2), i.e. U(1) gauge group and the magnetic monopoles. The monopoles are Dirac monopoles in the U(1) gauge theory. Thus, the monopoles are described by diagonal components $A_{\mu}^{3}\sigma^{3}$ ( Pauli matrices $\sigma^{a}$ ) of the SU(2) gauge fields. We calculate the contribution of the monopoles to extended Polyakov loops $P_{c}$,

[TABLE]

where we have taken the path $c$ of the integral along $x^{1}$ axis and have taken a gauge in which the component of the gauge field $A_{1}$ is along $\sigma^{3}$ direction in the isospin space; $A_{1}\equiv A_{1}^{a=3}\sigma^{3}/2$. For simplicity, we take the path $c$ as $(-\infty,+\infty)$.

The gauge field $A_{1}^{3}$ ( the $x_{1}$ component of the gauge fields in eq(1) ) of the monopole configuration is given by

[TABLE]

Thus, the integral $\int_{-\infty}^{+\infty}dx^{1}A_{1}^{3}$ is given by

[TABLE]

where we use the standard notation of spatial coordinates; $x^{1}=x,x^{2}=y,x^{3}=z$. The function $\phi^{\prime}(y,z)$ is defined by $\phi^{\prime}\equiv\arctan(z/y)$, i.e. azimuthal angle. Namely, the angle is defined as an angle measured from $y$ axis to the reference point in $y-z$ plane. The integral eq(4) represents a contribution from a monopole located at $\vec{x}=0$. When there are N monopoles located at $\vec{x}=\vec{x}_{i}$ with $i=1,2,\cdots,N$, the integral $\int dx^{1}A_{1}^{3}(x^{1},y,z)$ is given by

[TABLE]

where $\eta_{i}=\pm 1$ denotes positive or negative magnetic charge. ( Because the Dirac monopoles are solutions in the Abelian gauge theory, multi monopole solutions are trivially obtained by superposing the solutions. ) We have assumed that our system is magnetically neutral; $\int d^{3}x\rho(\vec{x})=0$. The angle $\phi^{\prime}(\vec{x})$ depends only on the coordinates $y$ and $z$. In general, the coordinates $\vec{x}_{i}$ of the monopoles may depend on the time $t=x_{0}$. Therefore, it follows that

[TABLE]

The formula represents how the monopoles with their number density $\rho(\vec{x})$ contribute to $P_{c}$. The dependence on the time coordinate $t$ arises through the coordinate $x_{i}(t)$ of each monopole. It is natural to consider that these monopoles interact with each other by the magnetic Coulomb interaction $\sum_{i\neq j}V(\vec{x_{i}}-\vec{x_{j}})\equiv\sum_{i\neq j}\pm g_{m}^{2}/|\vec{x_{i}}-\vec{x_{j}}|$.

Before examining the quantum effects of the monopoles on $P_{c}$, we discuss thermal average of $P_{c}$ using a toy model of classical monopole gas. The purpose is to see how monopoles contribute to $P_{c}$ when they freely move with weak magnetic interactions or when they make magnetic dipole with strong magnetic interactions.

The thermal average of $P_{c}$ in a gas of non relativistic monopoles interacting with each other by the magnetic Coulomb interaction. is given by

[TABLE]

where Z=\int\prod_{i}^{N}d^{3}x_{i}\exp\Big{(}-\sum_{i\neq j}\beta V(\vec{x}_{i}-\vec{x}_{j})\Big{)} with the temperature $\beta^{-1}$. We assume that the system is neutral; the number of the monopoles is equal to the number of the anti monopoles.

We present a physical picture of the monopole gas. When the gas is in a plasma phase at high temperature $\beta\ll 1$ or weak magnetic interaction $\sim\beta g_{m}^{2}/|\vec{x}-\vec{x^{\prime}}|$ with small $g_{m}$, each monopole in the phase can almost freely move since the interaction is very weak. We find that the angle $\phi^{\prime}(\vec{x}-\vec{x}^{\prime})$ can take any values ( i.e. there are no favored values ) when the monopole interaction $V$ is weak enough for each monopole to freely move. Thus, the average $\int d\phi^{\prime}\exp(i\phi^{\prime})$ vanishes. It implies that $\langle P_{c}\rangle_{\rm thermal}=0$. On the other hand, at low temperature or strong magnetic interaction, each monopole can not move freely. A positively charged monopole and a negatively charged monopole would form a bound state making a magnetic dipole. That is, the monopole gas is in a dipole phase. In the phase the correlation between a monopole and an anti monopole is much stronger than the one among monopoles with identical magnetic charges. Then, the angle $\phi^{\prime}(\vec{x}-\vec{x}_{1}^{\prime})-\phi^{\prime}(\vec{x}-\vec{x}_{2}^{\prime})$ between a monopole at $\vec{x}_{1}^{\prime}$ and anti-monopole at $\vec{x}_{2}^{\prime}$ measured from a point $\vec{x}$ is very small when they form a small dipole. That is, $\phi^{\prime}(\vec{x}-\vec{x}_{1}^{\prime})-\phi^{\prime}(\vec{x}-\vec{x}_{2}^{\prime})\ll 1$ for $|\vec{x}_{1}^{\prime}-\vec{x}_{2}^{\prime}|/|\vec{x}|\ll 1$. In other words, the angle takes small values in general when the dipole is small. It implies that $\langle P_{c}\rangle_{\rm thermal}\simeq 1$. Therefore, we conclude from the classical argument that when the monopole gas is in the plasma phase, $Z_{2}$ symmetry is preserved, i.e. $\langle P_{c}\rangle_{\rm thermal}=0$, while the symmetry does not hold when the gas is in the dipole phase, i.e. $\langle P_{c}\rangle_{\rm thermal}\neq 0$. The plasma phase of the monopole gas corresponds to the confinement phase, while the dipole phase does to the deconfinement phase. In other words, when the magnetic interaction $g_{m}^{2}\simeq g^{-2}$ is weak ( large $g$ ), the monopoles can freely move. On the other hand, when the magnetic interaction $g_{m}^{2}$ is strong ( small $g$ ). the monopoles would form the magnetic dipoles. It seems that the picture is reasonable.

In the above discussion the number density of the monopoles are fixed. When the density is larger, the magnetic interactions among the monopoles are stronger. This is because the average distance among the monopoles is smaller. Thus, the transition temperature between the plasma and dipole phases becomes lower.

Quantum mechanically, we expect that the monopoles condensemonogas at low temperature as a Bose gas when the magnetic interaction is weak. The phase corresponds to the plasma phase. The condensed monopoles freely move with zero momentum. Even at non zero but small temperature a fraction of the monopoles excited also freely move. Then, it follows that $\langle P_{c}\rangle_{\rm thermal}=0$. On the other hand, The number of the excited monopoles increases as the temperature increases. Then, at a critical temperature the condensation of the monopoles disappears. They would form a dipole because the number density of the monopoles are large so that magnetic interactions become large. The phase corresponds to the dipole phase. But we should mention that it is not clear whether an appropriate order parameter characterizing the dipole phase is present. Probably, the average distance between a monopole and an anti monopole is smaller than the distance between a monopole ( anti monopole ) and a monopole ( anti monopole ). As a consequence, the angle $\phi^{\prime}(\vec{x}-\vec{x}_{1}^{\prime})-\phi^{\prime}(\vec{x}-\vec{x}_{2}^{\prime})$ could not take arbitrary values. Then, it follows that $\langle P_{c}\rangle_{\rm thermal}\neq 0$.

Now we discuss how $P_{c}$ behaves in quantum field theory. Especially, we show that $P_{c}=0$ when the monopoles condense ( $\langle\Phi\rangle\neq 0$ ), while $P_{c}\neq 0$ when they do not condense ( $\langle\Phi\rangle=0$ ). Our results will show that the above naive argument on the phases of the monopole is correct.

Because we are concerned with the monopoles causing quark confinement, the monopoles couple with dual gauge fields $B_{\mu}$ in a model of dual superconductormaedan ; dual2 . They acquire mass $m$ owing to the monopole condensation. Taking a quantum monopole field $\Phi$ we express the monopole density $\rho$ in terms of $\Phi$ and $B_{\mu}$; $\rho=\Phi^{\dagger}(i\partial_{t}+g_{m}B_{t})\Phi+\Phi(-i\partial_{t}+g_{m}B_{t})\Phi^{\dagger}$. The field $\Phi$ is a complex scalar field. When the monopole condenses $\langle\Phi\rangle\neq 0$, the density is given by

[TABLE]

with $\Phi=|\Phi|\exp(i\theta)=(v+\delta\phi)\exp(i\theta)$ and $v=\langle\Phi\rangle\neq 0$, where $\theta$ is absorbed into $B_{t}$ by shifting $B_{t}\to B_{t}+g_{m}^{-1}\partial_{t}\theta$. We should stress that when the monopoles condense $\langle\Phi\rangle\neq 0$, the density of the monopoles involves the linear term of the field operator $\delta\phi$, while it does not involve the linear term when $\langle\Phi\rangle=0$. The term creates a monopole, while the quadratic terms $\delta\phi\delta\phi$ create a magnetic dipole ( a pair of a monopole and an anti monopole ). As we discussed above, each monopole which does not form a dipole, plays important roles in making $\langle P_{c}\rangle$ vanish.

First, we will show that $\langle P_{c}\rangle\neq 0$ when $\langle\Phi\rangle\neq 0$. We take only a contribution of the linear term of $\delta\phi$ and $B_{t}$. Furthermore, we approximate the formula \langle\exp\Big{(}i\int\phi^{\prime}(\vec{x}-\vec{x}^{\prime})\rho(\vec{x}^{\prime})d^{3}x^{\prime}\Big{)}\rangle such that

[TABLE]

where $\langle\rho(\vec{x}^{\prime})\rho(\vec{y}^{\prime})\rangle$ is given by

[TABLE]

with

[TABLE]

where $M$ denotes mass of the monopole excitation $\delta\phi$. The vacuum expectation values are taken using field operators with identical time coordinate.

Thus, we obtain

[TABLE]

with $k=\sqrt{k_{y}^{2}+k_{z}^{2}}$, where $\vec{k}=(k_{x},k_{y},0)$. In the above calculation, we regularize the angle $\phi^{\prime}$ such that

[TABLE]

with $r^{\prime}=\sqrt{y^{2}+z^{2}}$ and $\phi^{\prime}=\arctan(z/y)$, and set

[TABLE]

where the parameter $a$ should be taken to vanish in the final stage of calculations. We can show that $Q(x)\to x^{-(2+\epsilon)}$ with $\epsilon>0$ for $x\to\infty$. Thus, $Q(\frac{k}{a})/a^{2}\to 0$ for $a\to 0$.

The term $|A_{k}|^{2}$ ( $|B_{k}|^{2}$ ) arises from the term $\langle B_{t}(\vec{x}^{\prime})B_{t}(\vec{y}^{\prime})\rangle$ ( $\langle\delta\phi(\vec{x}^{\prime})\delta\phi(\vec{y}^{\prime})\rangle\langle B_{t}(\vec{x}^{\prime})B_{t}(\vec{y}^{\prime})\rangle$) in $\langle\rho(\vec{x}^{\prime})\rho(\vec{y}^{\prime})\rangle$. The integral over $q$ in $B_{k}$ is divergent so that we need to properly regularize the integral. The divergence arises owing to the singular behaviors in $\langle\delta\phi(\vec{x}^{\prime})\delta\phi(\vec{y}^{\prime})\rangle\langle B_{t}(\vec{x}^{\prime})B_{t}(\vec{y}^{\prime})\rangle$ as $\vec{x}^{\prime}-\vec{y}^{\prime}\to 0$. Here we simply use a cut off of the divergence. The divergence is irrelevant to the problem we are concerned with.

The integral $\int d^{2}k|A_{k}|^{2}\propto\int d^{2}k\,k^{2}Q^{2}(k/a)/(a^{4}\sqrt{k^{2}+m^{2}})$ represents the contribution of a gauge field excitation $B_{t}$ in the vacuum with the monopole condensation. The integral is finite as $a\to 0$. Thus, $\exp(-\frac{v^{2}g_{m}^{2}L}{(2\pi)^{2}}\int dk_{y}dk_{z}v^{2}|A_{k}|^{2})$ does not vanish as $a\to 0$. On the other hand, the integral $\int d^{2}k|B_{k}|^{2}$ represents the contribution of a monopole excitation $\delta\phi$. It is easy to see that the integral becomes infinity such that $\int d^{2}k|B_{k}|^{2}\to a^{-2}$ as $a\to 0$. Thus, we find that $\langle P_{c}\rangle=0$ when the monopoles condense and that $Z_{2}$ symmetry is not broken. The result that $\langle P_{c}\rangle=0$ as $a\to 0$ can be understood based on the physical meaning of the parameter $a$ as we explain below.

We would like to explain the physical meaning of the regularization of the angle $\phi^{\prime}\to\phi^{\prime}_{\rm reg}\equiv\phi^{\prime}\exp(-ar^{\prime})$. The angle is defined as an angle $\phi^{\prime}=\arctan(z/y)$ measured from the $y$ axis in the $y-z$ plane. Thus, when a monopole is outside of the region $r^{\prime}=\sqrt{z^{2}+y^{2}}<a^{-1}$, its contribution to the integral $\int d^{2}k|B_{k}|^{2}$ vanishes since $\phi^{\prime}_{\rm reg}=0$. Remember that $P_{c}\propto\exp(i\int d^{3}x^{\prime}\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime})\rho(\vec{x}^{\prime}))$. On the other hand, when a monopole is inside of the region, the monopole can make a nontrivial contribution. It corresponds to $\int d\phi^{\prime}\exp(i\phi^{\prime})=0$ in the case of the classical monopole gas. We note that the volume of the region $r^{\prime}=\sqrt{z^{2}+y^{2}}<a^{-1}$ is proportional to $a^{-2}$. Therefore, the integral $\int d^{2}k|B_{k}|^{2}$ receive the contributions equally from all monopoles inside the region, whose volume increases as $a^{-2}$ for $a\to 0$. Because the monopoles are uniformly distributed, the integral becomes infinite such as $\int d^{2}k|B_{k}|^{2}\to a^{-2}$, in other words, it becomes infinite with the volume $\propto a^{-2}$ for $a\to 0$. This is the result expected in the plasma phase of the monopole gas.

On the other hand, when the monopoles are in the dipole phase, the monopoles make dipole pairs of positive charged monopoles and negative charged ones. Then, the angle $\phi_{\rm reg}(\vec{x}-\vec{x}^{\prime})-\phi_{\rm reg}(\vec{x}-\vec{x}^{\prime\prime})$ of the pair becomes much smaller as the pair goes more distant; $|\vec{x}|\gg|\vec{x}^{\prime}-\vec{x}^{\prime\prime}|$. Hence, when the monopoles are in the dipole phase, there are no contribution from the monopole pairs located at large distance even if we remove the regularization. It would imply that the integral $\int\phi^{\prime}(\vec{x}-\vec{x}^{\prime})\phi^{\prime}(\vec{x}-\vec{y}^{\prime})\langle\rho(\vec{x}^{\prime})\rho(\vec{y}^{\prime})\rangle d^{3}x^{\prime}d^{3}y^{\prime}$ is finite as $a\to 0$. Then, it would follows that $\langle P_{c}\rangle\neq 0$ as $a\to 0$ in the phase.

In order to confirm the physical picture, we proceed to show that $\langle P_{c}\rangle\neq 0$ when there are no monopole condensation $\langle\Phi\rangle=0$. We note that there are no linear term of the monopole field $\Phi$. We take only quadratic terms of the monopole $\Phi$ in the density operator $\rho$ in eq(8) and neglect the term $|\Phi|^{2}B_{t}$ for simplicity. Then, the density operator $\rho=i\Phi^{\dagger}\partial_{t}\Phi+h.c.$ create magnetic dipoles ( a pair of a positive charged and a negative charged monopole ). We take the vacuum expectation value of $P_{c}=\exp(i\int d^{3}x\phi^{\prime}\rho)$ using the vacuum $|0\rangle$ such that

[TABLE]

with creation operators $a_{k}^{\dagger}$ ( $b_{k}^{\dagger}$ ) of magnetic monopoles ( anti-monopoles ); $[a_{k},a_{q}^{\dagger}]=[b_{k},b_{q}^{\dagger}]=\delta^{3}(\vec{k}-\vec{q})$ and the others vanish. Here $\omega_{k}=\sqrt{\vec{k}^{2}+M^{2}_{0}}$ with monopole mass $M_{0}$ in the phase without monopole condensation. We approximate the formula $\langle P_{c}\rangle$ in eq(9) and simply calculate the term $\int\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime})\phi_{\rm reg}^{\prime}(\vec{x}-\vec{y}^{\prime})\langle\rho(\vec{x}^{\prime})\rho(\vec{y}^{\prime})\rangle$. Namely, expanding $\langle P_{c}\rangle$ in $\rho$ such that $\langle P_{c}\rangle=1+\frac{i^{2}}{2}\int d^{3}x^{\prime}d^{3}x^{\prime\prime}\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime})\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime\prime})\langle\rho(\vec{x}^{\prime})\rho(\vec{x}^{\prime\prime})\rangle,\cdots$, we calculate the formula $\langle P_{c}\rangle\simeq\exp(-\frac{1}{2}\int\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime})\phi_{\rm reg}^{\prime}(\vec{x}-\vec{y}^{\prime})\langle\rho(\vec{x}^{\prime})\rho(\vec{y}^{\prime})\rangle d^{3}x^{\prime}d^{3}y^{\prime})$. The term in the exponent represents the contribution of a pair of a monopole and an anti-monopole created and then annihilated in the vacuum. We will show that the term $\int d^{3}x^{\prime}d^{3}x^{\prime\prime}\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime})\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime\prime})\langle\rho(\vec{x}^{\prime})\rho(\vec{x}^{\prime\prime})\rangle$ is finite as $a\to 0$. Hence, we find $\langle P_{c}\rangle\neq 0$ in the phase without the monopole condensation.

To calculate the integral $\int d^{3}x^{\prime}d^{3}x^{\prime\prime}\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime})\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime\prime})\langle\rho(\vec{x}^{\prime})\rho(\vec{x}^{\prime\prime})\rangle$, we need to take the normal ordering of the operator $\rho$ in order for $\langle\rho\rangle$ to vanish,

[TABLE]

Then, we calculate $\int d^{3}x^{\prime}d^{3}x^{\prime\prime}\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime})\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime\prime})\langle\rho(\vec{x}^{\prime})\rho(\vec{x}^{\prime\prime})\rangle$ such that

[TABLE]

where

[TABLE]

with complete elliptic integral of the first kind $K(x)$ and the second kind $E(x)$. Here we denote $q=\sqrt{q_{2}^{2}+q_{3}^{2}}$ and $k=\sqrt{k_{2}^{2}+k_{3}^{2}}$. In the equation we have used the regularized angle $\phi^{\prime}_{\rm reg}(\vec{x})$ in eq(13) and have replaced the delta function $\delta(0)$ such as $\frac{1}{2\pi}\int dx=\frac{L}{2\pi}$.

We can show that the integral with respect to the variable $|\vec{q}-\vec{k}|$ in eq(17) is infinite at $|\vec{q}-\vec{k}|=\infty$ but the integral with respect to $|\vec{q}+\vec{k}|$ is finite. So we need a cut off parameter $\Lambda$ in the integral with respect to $|\vec{q}-\vec{k}|$. The infinity arises from the singular behavior of $\langle\rho(\vec{x})\rho(\vec{y})\rangle$ as $\vec{x}-\vec{y}\to 0$ and has also arisen in the above calculation with $\langle\Phi\rangle\neq 0$. The point in our discussion is that the integral with respect to $|\vec{q}+\vec{k}|$ is finite and the limit $a\to 0$ is also finite. That is, the integral $\int d^{3}x^{\prime}d^{3}x^{\prime\prime}\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime})\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime\prime})\langle\rho(\vec{x}^{\prime})\rho(\vec{x}^{\prime\prime})\rangle$ is finite as $a\to 0$ and is proportional to $L$. Therefore, we find that $\langle P_{c}\rangle\neq 0$ when the monopoles do not condense, i.e. $\langle\Phi\rangle=0$.

The finiteness of the integral $\int d^{3}x^{\prime}d^{3}x^{\prime\prime}\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime})\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime\prime})\langle\rho(\vec{x}^{\prime})\rho(\vec{x}^{\prime\prime})\rangle$ as $a\to 0$ corresponds to the case of the dipole phase. Actually, we have taken into account the contributions from the monopole pairs created in the vacuum with no monopole condensation. ( Because $\rho=i\Phi^{\dagger}\partial_{t}\Phi+h.c.$, the pairs are created and then annihilated in the vacuum. Thus, the pairs behave like dipoles. ) This should be contrasted with the previous result in the vacuum with the monopole condensation. To find the result we have taken into account the single monopole excitations. ( Because $\rho$ involves a linear term in $\delta\phi$, a single monopole is created and then annihilated in the vacuum. The single monopole excitations behave like the monopole plasma. ) The infinity of the integral $\int d^{3}x^{\prime}d^{3}x^{\prime\prime}\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime})\phi_{\rm reg}^{\prime}(\vec{x}-\vec{x}^{\prime\prime})\langle\rho(\vec{x}^{\prime})\rho(\vec{x}^{\prime\prime})\rangle$ as $a\to 0$ corresponds to the plasma phase. Consequently, we find that $\langle P_{c}\rangle=0$ in the vacuum with the monopole condensation $\langle\Phi\rangle\neq 0$, while $\langle P_{c}\rangle\neq 0$ in the vacuum with no monopole condensation $\langle\Phi\rangle=0$. Our results have been obtained at zero temperature in a model where taking real or imaginary mass of the monopoles both phases $\langle\Phi\rangle=0$ and $\langle\Phi\rangle\neq 0$ have been examined.

In the above discussion we have shown by using a toy model of classical monopole gas that the plasma phase at high temperature corresponds to the confinement phase, while the dipole phase at low temperature does to the deconfinement phase. It seems apparently that the model contradicts the feature of the gauge theories. But we can explain that the actual behavior of the monopole gas arising in the gauge theories is different from the model of the classical monopole gas. Namely, the monopoles condense in vacuum and a fraction of the monopoles is excited at low temperature. The magnetic Coulomb interaction is screened by the condensed monopoles so that the interaction range between the excited monopoles is short. The interaction is not effective. Because the interaction is not effective at the low temperature, the excited monopoles are almost free. They are in a plasma phase. Thus, as shown in the toy model, the extended Polyakov loop $\langle P_{c}\rangle$ vanishes. The confinement phase holds in low temperatures. As the temperature increases higher, the number of the excited monopoles increases more and the interaction between them becomes stronger. But the plasma phase still holds. Beyond the transition temperature between the confinement and the deconfinement phases, all the monopoles which condense in vacuum at low temperature are excited to form a dense monopole gas. The condensation disappears. Then, their magnetic interactions are strong because the mean distance of the monopoles is small. Probably, the interaction is sufficiently strong so that the monopoles would form dipoles. Thus, the monopole gas forms a dipole phase in which $\langle P_{c}\rangle\sim 1$. This feature is actual one we expect in the gauge theories. The recent studygyulassy of quark gluon plasma indicates that the proportion of the monopole density to the total density decreases as the temperature increases more in the deconfinement phase. The decrease of the proportion would be caused by the pair annihilation of the monopoles owing to the formation of the close dipoles.

Assuming spatial periodic boundary conditions in $4$ dimensional Minkowski space-time, we have proposed the extended Polyakov loop $P_{c}(A)$ as an order parameter of $Z_{N}$ symmetry; $P_{c}(A)=Tr(P\exp(i\int_{0}^{L}dx_{1}gA^{1}))$ where the path is taken along a spatial axis, e.g. $x_{1}$ axis. We have shown in the canonical formalism that the monopole condensation causes the vanishing of the extended Polyakov loop. Therefore, the $Z_{N}$ symmetry holds in the phase with the monopole condensation. We know that the monopole condensation leads to the quark confinement. Therefore, we expect that the extended Polyakov loop is an order parameter characterizing the confinement phase. Although we have defined the extended Polyakov loop in the Minkowski space-time, it can be also defined in the Euclidean space. It is interesting to see in lattice gauge theories with the condition $L>1/T$ that the extended Polyakov loop vanishes in sufficiently low temperature $T$ and does not vanish in sufficiently high temperature.

The author expresses thanks to Prof. K. Kondo in Chiba University and members of theory group in KEK for useful comments and discussions.