^{3}

SU(2) gauge theory in the nonlinear gauge of Curci–Ferrari type is studied in the low-energy region. We give a classical solution that connects color electric charges. Its dual solution, which has a monopole configuration, is also presented. Due to gluon condensation subsequent to the ghost condensation, these classical fields become massive. The massive Lagrangian with the classical solution and that with the dual solution are derived. We show that these Lagrangians produce a linear potential between a quark and an antiquark. This is the mechanism of quark confinement, which is different from magnetic monopole condensation.

In the dual superconductor picture of quark confinement, magnetic monopoles are necessary (see, e.g., Ref. [

In Ref. [

In the previous paper [

In this paper, based on Ref. [

We review Ref. [

This Lagrangian requires gauge fixing, and an appropriate gauge-fixing term and a ghost term are necessary. The Lagrangian for these terms is written as

In Refs. [^{1}

In the presence of the VEV

Here, we used the notations

The gauge-fixing and ghost part becomes

We note that, if

When the classical field

First, we consider the magnetic potential

As an example,

To incorporate the current ^{2}

Then, performing this replacement and neglecting the components

Now we neglect

We choose the current
^{3}

The derivation of the linear potential will be discussed in

If we consider the magnetic monopole,

As we show in

Then Eq. (

Using Eqs. (

Even if

We solve Eq. (

Let us consider the color electric current

Then Eq. (

From this equation, we obtain

As in

Eq. (

Namely, the dual electric potential

Based on the dual relation in Eq. (^{4}

From the left-hand side (LHS) of Eq. (

We note that, because of Eq. (

The LHS of Eq. (

From the RHS of Eq. (

However, this equation is satisfied by the usual equation of motion

So, using Eq. (

We note that, if we multiply Eq. (

Since

Substituting

To derive the static potential between the electric charges

Then the first term in Eq. (

Next, for the space-like vector ^{5}

Instead of the field

By multiplying Eq. (

Since the current conservation

The first term and the second term yield the Yukawa potential and the linear potential, respectively.

The factor

In Eq. (

Next we consider another solution

Now we assume that Eq. (

If

Eq. (

This is the situation studied in

Thus we can conclude that the classical configuration that yields the quark confinement is the monopole solution of the dual gauge field

Zwanziger considered a local Lagrangian with two gauge fields

In the same way, we can derive the equivalent Lagrangian:

To study the quark confinement,

In this case, we can identify

In the present approach, we can identify

We make a comment. For the currents

In the previous papers [

In this paper, we considered the electric potential

From the Lagrangian

In the dual Ginzburg–Landau model of dual superconductor, the operator

We make some comments.

(1) Below the scale

(2) The operator

(3) Physical quantities should not depend on

Open Access funding: SCOAP

We employ the metric

From Eq. (

For simplicity, we use the notations

For a magnetic charge and an electric charge, we present monopole solutions and dual solutions in the massless case and the massive case.

In the massless case, we choose the magnetic potential, which describes a magnetic monopole, as

The corresponding dual magnetic potential

From Eq. (

We follow Zwanziger’s definition [

Namely,

In the massive case, Eq. (

The dual magnetic potential and its equation of motion change from Eq. (

Next we consider the color electric current

Using

Namely,

In the massive case, the potential

Instead of Eqs. (

First, we calculate the integral

The path

This function satisfies

When

We note that the damping behaviors

To see the meaning of the above infrared divergence, we choose

When

The relation between the length

From the ghost determinant

In the case of

This potential is analytically continued to the region

This potential coincides with the one obtained in Euclidean space, and leads to the condensation

In the case of

The one-loop diagrams in

The one-loop ghost diagrams. The dashed line is the ghost propagator

The series

In Ref. [

^{1} In Minkowski space, although we could not show

^{2} Equation (

^{3} Above the scale

^{4} This is Zwanziger’s dual field strength

^{5} The infrared behavior of Eq. (