^{3}

In this paper, we show the existence of magnetic monopoles in the pure

In high-energy physics, quark confinement is a long-standing problem to be solved in the framework of quantum chromodynamics (QCD). The dual superconductivity picture [

Nevertheless, we know [

Recently, it has been shown that the gauge-invariant mass term of the Yang–Mills field can be introduced by combining the change of variables and a

This fact provides a natural understanding of the gauge field decomposition in the Yang–Mills theory called the

According to the gauge-independent BEH mechanism, the color field ^{1}

This way of introducing the color field will facilitate understanding of the role of the color field itself. The remaining field

Consequently, we can introduce a gauge-invariant mass term

To obtain the pure Yang–Mills theory from the complementary gauge–scalar model, we must solve an issue. The naively extended Yang–Mills theory written in terms of the field variables

Such color field configurations satisfying the reduction condition are obtained from the field equations of the complementary gauge–scalar model, since it is shown [

We show in this paper that magnetic monopoles do exist in the pure

It should be remarked that, within the framework of the reformulated Yang–Mills theory, the configurations of the color field

This paper is organized as follows. In

In this section, we review the procedure [

In what follows, we introduce, respectively, the inner and exterior products for the Lie-algebra-valued fields by

To begin with, we construct a composite vector boson field

The outline to obtain the massive Yang–Mills theory from the “complementary” gauge–scalar model. The double-lined arrow stands for our approach in this paper. The dotted box shows the approach in Refs. [

Moreover, the kinetic term of the scalar field is identical to the mass term of the vector field

By using the definition of the massive vector field

Then, we regard a set of field variables

In the gauge–scalar model,

Moreover, notice that the degrees of freedom of the original gauge field ^{2}

The reduction condition indeed eliminates the two extra degrees of freedom introduced by the radially fixed scalar field into the Yang–Mills theory, since

Following the Faddeev–Popov procedure, we insert unity into the functional integral to incorporate the reduction condition:

The Jacobian

The obtained massive Yang–Mills theory indeed has the same degrees of freedom as the usual Yang–Mills theory because the massive vector boson

It should be remarked that the solutions of the field equations of the gauge–scalar model satisfy the reduction condition automatically. (But the converse is not true.) The field equations besides Eq. (

To eliminate the Lagrange multiplier field

The field equations (

By applying the covariant derivative

Moreover, by taking the exterior product of Eq. (

Hence, the simultaneous solutions of the coupled field equations (

The relation between the solutions of the field equations of the gauge–scalar model and the reduction condition.

In this section, we shall summarize the essence of the Georgi–Glashow model. We introduce the Georgi–Glashow model by the Lagrangian density:

We take the standard static and spherically symmetric ansatz for the ’t Hooft–Polyakov monopole with a unit magnetic charge [

In the spherically symmetric case, we can rewrite the Lagrangian

By rescaling

The field equations are obtained as

We assume asymptotic behavior for small

By substituting these series expansions into the field equations (

In this section we examine the existence of the static and stable configuration in the massive Yang–Mills theory. For this purpose, we follow the scaling argument due to Derrick [

In the gauge–adjoint scalar model with a radial-fixing constraint, the static energy

By rescaling the spatial variable

Then the scaled energy

For the massive Yang–Mills theory (

We find that

This result should be compared with the pure (massless) Yang–Mills theory and the

Notice that for the pure (massless) Yang–Mills theory only the term of the gauge field exists and hence there is no stationary point under the scaling, which implies the non-existence of the static and stable soliton solutions in the

Because of the constraint (^{3}

The profile functions

We redefine the Lagrangian density

The equations for the profile functions

Equation (

By substituting Eq. (

Thus we can determine the Lagrange multiplier field

First, we examine the asymptotic behavior of

For small

Here, one finds that

In order to obtain the asymptotic behavior of

The linear differential equation (

Here the first two terms of Eq. (

Under the boundary conditions

The Taylor expansion of the solution (

Thus, under the boundary conditions

For large

In a similar way to the above, we can determine the coefficients

(Top) The solution

From this numerical solution, we can calculate the static energy or the rest mass of a magnetic monopole

This result also shows that the obtained solution

We define the energy density

(Left panel) The energy density

Based on Eq. (

The Yang–Mills monopole mass,

We shall separate the gauge field

In the present ansatz, by using the normalized scalar field

In what follows, we adopt the polar coordinate system

The behavior of

We perform a singular gauge transformation, which makes

As mentioned before,

Thus,

One can find that the field

For the Yang–Mills magnetic monopole obtained in the massive Yang–Mills theory, we do not need to introduce artificial regularization by hand to remedy the short-distance (or ultraviolet) singularity and instability of the Wu–Yang magnetic monopole in the pure massless Yang–Mills theory as worked out in Refs. [

We examine the magnetic charge

The magnetic charge

The magnetic charge density

From the definition of

In the top panel of

The short-distance behavior of (top) the magnetic charge density

In order to investigate the behavior of the chromo-magnetic field

Then,

For

We find that in the radially fixed case of Yang–Mills theory, the chromo-magnetic field diverges at the origin due to its logarithmic behavior:

See the panels in the second line of

It should be noticed that we have included the volume element

In this paper we have constructed the magnetic monopole

In the long-distance region, we observed that the Yang–Mills magnetic monopole configuration

By using the Yang–Mills magnetic monopole found in this paper, we can show quark confinement in the 3D Yang–Mills theory in the same way as the 3D Georgi–Glashow model shown by Polyakov [

The authors would like to express their sincere thanks to the referee for giving valuable comments to revise the manuscript. S.N. would like to thank Nakamura Sekizen-kai for a scholarship. R.M. was supported by a Grant-in-Aid for Japan Society for the Promotion of Science (JSPS) Research Fellow Grant Number 17J04780. M.W. was supported by the Ministry of Education, Culture, Sports, Science and Technology, Japan 501100001691 (MEXT scholarship). K.-I.K. was supported by a Grant-in-Aid for Scientific Research, JSPS KAKENHI Grant Number (C) No. 15K05042.

Open Access funding: SCOAP

^{1}The “complementarity” originates from the confinement–Higgs complementarity in the gauge–scalar model that says that there is no phase transition between the two phases, confinement and Higgs, which are analytically connected in the phase diagram [

^{2}The massive vector field

^{3}It should be noted that in this setup the reduction condition (