Monopoles, vortices and their correlations in SU(3) gauge group

Abstract

Topological defects such as monopoles, vortices and “chains” of the SU(3) gauge group are studied using its SU(2) subgroups. Two appropriate successive gauge transformations are applied to the subgroups to identify the chains of monopoles and vortices. Using the fact that the defects of the subgroups are not independent, the SU(3) defects and the Lagrangian are studied and compared with the Cho decomposition method obtained for monopoles. By comparing the results with the ones which are obtained directly for the SU(3) gauge group, the relations and the possible interactions between the defects of the subgroups are discussed.

1 Introduction

Since the time of discovering quarks, there have been so many attempts to describe the confinement problem but yet no universal, comprehensive and analytical description based on the basic principles of quantum chromodynamics to explain some more expected features of the confinement. Lattice calculations have given valuable clues to understand this phenomenon and to help proposing appropriate phenomenological models. Monopoles and vortices are among the rather successful candidates in describing the confinement problem, both via lattice calculations and phenomenological models. However, none of these candidates have been able to interpret all the characteristics of confinement like the Casimir scaling and N-altiy dependence. On the other hand, lattice results have shown some indications of field configurations of correlated vortices and monopoles, where center vortices have been observed to end at monopole world-lines. In fact, numerical simulations in the indirect maximal center gauge have found that monopole world lines lie on center vortex sheets and the magnetic flux of those monopole is concentrated along the sheet. In other words, the magnetic flux of monopoles at given time is concentrated in a tube-like structure, and the monopole “chain” is shorthand terminology for an ordering of monopoles and antimonopoles, in which half the magnetic flux from a monopole ends on the next antimonopole in the chain, while the other half ends on the previous antimonopole. Some phenomenological models have been presented to include both defects in the form of chains of monopoles and vortices. People hope that chains of monopoles and vortices may describe some more expected characteristics of the quark-antiquark potential in the confinement regime. Some of the papers in this regard are given in references [1,2,3,4,5,6,7,8,9]. We have also studied chains of monopoles and vortices and the corresponding field strength tensors and the Lagrangian for the SU(2) gauge group in ref. [10]. In this article, we generalize our previous article to contain the SU(3) gauge group and to study the possible defects and their interactions.

Basically, in many of the phenomenological models which try to describe quark confinement, one looks for magnetic topological objects or defects in the vacuum of QCD. There are a couple of ways to identify these magnetic defects. One of them is the field decomposition method proposed by Duan, Ge, Cho, Faddeev, Niemi and shabanov [11,12,13,14,15,16,17,18] where the Yang–Mills field is decomposed into other appropriate variables. As a result, topological defects such as monopoles appear in the theory. What is noticeable about the Cho decomposition method, is the fact that it allows a straightforward generalization of the SU(2) results to the SU(N) in a Weyl symmetric form. In Refs. [19, 20] using the Weyl symmetry, Abelian (Cho–Duan–Ge) decomposition of the SU(3) gauge field is discussed. Abelian gauge fixing is another method in which the formation of monopoles in the QCD ground state for SU(2) and SU(3) gauge groups are discussed [21].

The other clever idea to identify magnetic defects, is the use of appropriate gauge transformations to transform the color frames in such a way that they reveal monopoles, vortices, or both. Using this method, monopoles, vortices, and chains for the SU(2) gauge group are introduced [10, 22]. In this article, given the fact that the SU(3) group contains three SU(2) subgroups, we generalize this method to the SU(3) gauge group and identify monopoles, vortices and their correlations. Using SU(2) subgroups makes the task much easier than using directly the SU(3) gauge group. In addition, identifying SU(3) defects from the SU(2) subgroups, the relations between topological defects of the three SU(2) subgroups are discussed.

Here, we present an overview of our goals and the steps we go through. Using the appropriate gauge transformations for each of the three SU(2) subgroups, we obtain the local frames which contain defects such as monopoles, vortices or chains. Applying these gauge transformations, the transformed gauge fields look like the Cho decomposition fields for monopoles. In fact, we are dealing with Cho decomposition written based on local frames containing defects. We indicate the relationship between the magnetic defects obtained from the gauge transformations on the SU(2) subgroups and the results obtained by Cho decomposition, using directly the SU(3) gauge group. Then, we write the Lagrangian in two forms: in terms of the three SU(2) subgroups where we can learn how topological objects of each of the three SU(2) subgroups interact with each other; and by constructing the SU(3) Lagrangian from its SU(2) subgroups to study the interaction between topological objects in SU(3) group. These information may be used to write an effective Lagrangian to obtain the confinement potential using topological defects.

In Sect. 2 we review the above gauge transformation method for identifying monopoles, vortices and chains for the SU(2) gauge group. In Sect. 3 we first apply the appropriate gauge transformations to introduce the monopoles of each of the SU(2) subgroups. Then, using the results of each subgroup, we introduce the new local color frames and discuss the connection between the applied method and the Cho decomposition method. In addition, using the results of the Cho decomposition and the fact that the monopoles of the three SU(2) subgroups are not independent, we study the relations between the monopoles of three SU(2) subgroups and the monopoles of SU(3) group. Similar to the general procedure of Sect. 3, in Sect. 4 we identify the vortices for SU(3) gauge group. In Sect. 5, successively applying the two gauge transformations introduced in Sects. 3 and 4, we argue about the correlation between monopoles and vortices of the three SU(2) subgroups. We obtain local frames containing both monopole and vortex. The transformed gauge field is very similar to the Cho decomposition gauge fields for monopoles. Therefore, the results of the Cho decomposition can be used for finding out the relation between the correlated monopoles and vortices of the three SU(2) subgroups and the correlated monopoles and vortices of SU(3) group. At the end, we study the Lagrangian and discuss about the possible interactions between the defects. The conclusion and summary are given in Sect. 6.

2 Monopoles and center vortices as defects of the local color frames

By appropriate gauge transformations, we obtain the local color frames that contain the possible defects of the theory. As a result, the gluon fields are decomposed in such a way that the contribution of the magnetic defects appears explicitly in the final gauge field.

2.1 Monopole

In contrast to Dirac’s motivation that added monopoles to the Maxwell’s equation from the point of view of aesthetic, monopoles may appear in QCD as the effective agents to describe the confinement problem. Even though they do not exist in the standard form of the Lagrangian, it is possible to identify them by various methods like field decompositions and projections. Their existence has been confirmed by lattice calculations, as well. For example, as a result of the decomposition, one expects to observe the Wu–Yang monopoles which are point-like topological objects. There are some different ways to identify these defects. One of these methods is to use a nontrivial gauge transformation to construct the local color frame that contains monopoles.

By applying a nontrivial gauge transformation, the SU(2) Yang–Mills gauge field and the field strength tensor are changed as the following,

(1)

(2)

where $U\in SU(2)$ is a nontrivial topological mapping that is single-valued along any closed loop and the components of $\vec {T}$ are the generators of the SU(2) group.

To make a connection with the Cho decomposition, we introduce the frame $\hat{m}_a, a=1,2,3$, which is constructed by applying a rotation R(U) on the basis $\hat{e}_a$ of the color space,

$$\begin{aligned} UT_a U^{-1}=\hat{m}_a.\vec {T}\quad \text {or}, \quad \hat{m}_a=R(U)\hat{e}_a. \end{aligned}$$

(3)

This gauge transformation can be expressed in terms of Euler angles,

$$\begin{aligned} U=e^{-i\alpha T_3}e^{-i\beta T_2}e^{-i\gamma T_3} , \quad R(U)=e^{\alpha M_3}e^{\beta M_2}e^{\gamma M_3}, \end{aligned}$$

(4)

where $M_a$’s are the generators of SO(3) gauge group and $T_a$’s indicate half of the Pauli matrices.

In order for the frame $(\hat{m}_1,\hat{m}_2,\hat{m})$ to contain monopoles, we choose the parameters $\alpha$, $\beta$ and $\gamma$ such that $\alpha =\gamma =\varphi , \beta =\theta$, where $\theta$ and $\varphi$ are the polar and azimuthal angles. With this choice, the Abelian direction $\hat{m}$ takes the form of a hedgehog ($\hat{m}=\hat{r}$). The matrix form of the transformation U and the explicit form of the vectors $\hat{m}_a$ are calculated in [10].

Using the explicit forms of $T_a$’s in U, the gauge field of Eq. (1) is transformed as the following [10, 22],

$$\begin{aligned} \vec {A}_\mu ^U.\vec {T}=[\underbrace{(A_\mu ^3-C_\mu ^{(m)})}_{=A_\mu ^{(m)}}\hat{m}-\frac{1}{g}\hat{m}\times \partial _\mu \hat{m}+\underbrace{A_\mu ^1\hat{m}_1+A_\mu ^2\hat{m}_2}_{=\vec {X}_\mu ^{(m)}}].\vec {T}, \end{aligned}$$

(5)

where,

$$\begin{aligned} C_\mu ^{(m)}=-\frac{1}{g}\hat{m}_1.\partial _\mu \hat{m}_2. \end{aligned}$$

(6)

Using the gauge transformation U with the chosen parameters given after Eq. (4), the local color direction $\hat{m}$ would be equal to $\hat{r}$. Thus, choosing $A_\mu ^{(m)}=0$, $A_\mu ^a=0$ for $a=1,2$, a static Wu–Yang monopole [23] with a hedgehog form is obtained by Eq. (5).

We would like to compare the gauge field $\vec {A}_\mu ^U$ given by Eq. (5) with the gauge field obtained from the Cho decomposition. It is very interesting that the form of the resulted gauge fields of these two methods are very similar to each other. In Cho decomposition, by imposing the magnetic isometry to the gauge potential $\vec {A}_\mu$, the restricted potential is obtained,

$$\begin{aligned} D_\mu \hat{m}=(\partial _\mu +g \vec {A}_\mu \times )\hat{m}=0\Rightarrow \hat{A}_\mu =A_\mu ^{(m)}\hat{m}-\frac{1}{g}\hat{m}\times \partial _\mu \hat{m}. \end{aligned}$$

(7)

Choosing $A_\mu ^{(m)}=0$ and $\hat{m}=\hat{r}$, the restricted potential would describe the Wu–Yang monopole. By adding $\vec {X}_\mu ^{(m)}=A_\mu ^1\hat{m}_1+A_\mu ^2\hat{m}_2$ to the restricted potential, the extended Cho decomposition similar to what we have gotten in Eq. (5), is obtained. The major difference between the two methods is that in the Cho decomposition method the direction $\hat{m}$ is chosen to be $\hat{r}$ after the decomposition is done, while in the method used in this article, we use a specific nontrivial gauge transformation and as a result the form of a hedgehog for $\hat{m}$ is obtained, at the end.

2.2 Vortex

Center vortices are color magnetic defects which are localized on the closed two-dimensional surfaces (closed one-dimensional strings) in 4D (3D). Center vortices in SU(N) Yang–Mills theory are defined by the center of the gauge group. If a center vortex is linked to a Wilson loop, the Wilson loop is multiplied by an element proportional to the center of Z(N) group. Lattice calculations confirm that vortices are responsible for the color confinement [24] and removing them from the theory removes the linear behavior of the quark-antiquark potential.

In the continuum, thin center vortices are introduced by applying a gauge transformation V [25,26,27],

$$\begin{aligned} \vec {A}_\mu ^V\cdot \vec {T}=VA_\mu ^a T^aV^{-1}+\frac{i}{g}V\partial _\mu V^{-1}-I_\mu (\vartheta ). \end{aligned}$$

(8)

For simplicity we study the SU(2) group and choose $V\in SU(2)$. $I_\mu (\vartheta )$ represents the ideal vortex, and is localized on three-volume $\vartheta$ whose border gives the closed thin center vortex worldsheet. When crossing a three-volume $\vartheta$, the mapping V changes by a center element, $e^{\pm i\pi }V$. $I_\mu (\vartheta )$ is designed to cancel derivatives of the discontinuity of V at $\vartheta$ and keeping only the effect of the border of $\vartheta$, called the thin center vortices.

Using the gauge transformation V, a local basis $\hat{n}'_a$ in color space is introduced,

$$\begin{aligned} VT^aV^{-1}=\hat{n}'_a.\vec {T} , \quad \hat{n}'_a=R_{\hat{n}'}\hat{e}_a, \end{aligned}$$

(9)

and the frame dependent fields,

$$\begin{aligned} C_\mu ^{(v)a}=-\frac{1}{2g}\epsilon ^{abc}\hat{n}'_b\cdot \partial _\mu \hat{n}'_c, \end{aligned}$$

(10)

are defined so that they satisfy the following properties,

$$\begin{aligned} \hat{n}'_b\cdot \partial _\mu \hat{n}'_c=-g\epsilon ^{abc}C_\mu ^a ,\quad C_\mu ^aM^a=\frac{i}{g}R^{-1}\partial _\mu R. \end{aligned}$$

(11)

The matrices $M^a$, with elements $(M^a)^{bc}=-i\epsilon ^{abc}$ are the adjoint generators and $[M^a,M^b]=i\epsilon ^{abc}M^c$, $Tr(M^aM^b)=2\delta ^{ab}$. $\epsilon ^{abc}$ is the Levi–Civita symbol. For SU(2) gauge group, we have $V=e^{i\varphi T_3}$ or in adjoint representation $R_{\hat{n}'}=R_3(\varphi )$. Using the matrix form of V or its equivalent counterpart in the adjoint representation $R_3$, the components of $\hat{n}'_a,\ a=1,2,3$ are calculated [10],

$$\begin{aligned} \hat{n}'_1=\left( \begin{array}{c} \cos \varphi \\ -\sin \varphi \\ 0 \end{array}\right) ,\quad \hat{n}'_2=\left( \begin{array}{c} \sin \varphi \\ \cos \varphi \\ 0 \end{array}\right) ,\quad \hat{n}'_3=\left( \begin{array}{c} 0\\ 0\\ 1 \end{array}\right) . \end{aligned}$$

(12)

Using the components of $\hat{n}'_a$, the frame dependent fields are obtained,

$$\begin{aligned} C_\mu ^{(v)1}= & {} -\frac{1}{g}\hat{n}'_2\cdot \partial _\mu \hat{n}'_3=0, \nonumber \\ C_\mu ^{(v)2}= & {} -\frac{1}{g}\hat{n}'_3\cdot \partial _\mu \hat{n}'_1=0,\nonumber \\ C_\mu ^{(v)3}= & {} -\frac{1}{g}\hat{n}'_1\cdot \partial _\mu \hat{n}'_2=-\frac{1}{g}\partial _\mu \varphi . \end{aligned}$$

(13)

From the second equality of Eq. (11), one can simply show that $-C_\mu ^{(v)3}\hat{n}'_3\cdot \vec {T}=\frac{i}{g}V\partial _\mu V^{-1}$. However, unlike V, the adjoint representation R and the corresponding frame $\hat{n}'_a$ are always continuous, and as a result $C_\mu ^{(v)3}\hat{n}'_3$ contains no term concentrated on $\vartheta$ [27]. Therefore, one can rewrite the above equation as follows,

$$\begin{aligned} -C_\mu ^{(v)3}\hat{n}'_3\cdot \vec {T}=\frac{i}{g}V\partial _\mu V^{-1}-I_\mu (\vartheta ). \end{aligned}$$

(14)

Replacing $\frac{i}{g}V\partial _\mu V^{-1}$ from Eq. (14) in Eq. (8) and using the fact that $VA_\mu ^aT_aV^{-1}=A_\mu ^a\hat{n}'_a\cdot \vec {T}$, an equivalent representation for the thin configuration proposed in [25, 26] is obtained,

$$\begin{aligned} \vec {A}_\mu ^V\cdot \vec {T}=\left[ (A_\mu ^3-C_\mu ^{(v)3})\hat{n}'_3+A_\mu ^1\hat{n}'_1+A_\mu ^2\hat{n}'_2\right] \cdot \vec {T}. \end{aligned}$$

(15)

Using the definition of $C_\mu ^{(v)3}$ of Eq. (13),

$$\begin{aligned} \vec {A}_\mu ^V=(A_\mu ^3+\frac{1}{g}\partial _\mu \varphi )\hat{n}'_3+A_\mu ^1\hat{n}'_1+A_\mu ^2\hat{n}'_2. \end{aligned}$$

(16)

2.3 Chain

Since the scenarios written solely based on the monopoles or vortices have not been able to describe all the expected behaviors of the confining potential between a pair of quark and antiquark, it may be a smart idea to work with the configurations that include both of these objects. Chains are the configurations in which the vortex worldsheets have been attached to the monopole worldlines. Such configurations have been supported by lattice simulations [1, 28, 29].

It has been shown that by implying two successive gauge transformations U and V, one can observe a configuration that describes correlated monopoles and vortices or chains [22],

(17)

(18)

Using the above equations and the definition of Eq. (6), $C_\mu =-\frac{1}{g}\hat{n}_1.\partial _\mu \hat{n}_2$ which is correct for any magnetic defect, we have,

$$\begin{aligned} C_\mu ^{(n)}=C_\mu ^{(m)}+C_\mu ^{(v)}, \end{aligned}$$

(19)

where,

$$\begin{aligned} C_\mu ^{(m)}=-\frac{1}{g}\hat{m}_1.\partial _\mu \hat{m}_2,\ \ \ C_\mu ^{(v)}=\frac{1}{g}\partial _\mu \varphi . \end{aligned}$$

(20)

The magnetic potential $C_\mu ^{(n)}$ is a summation of two parts: $C_\mu ^{(m)}$ which represents the magnetic potential of the monopole, and $C_\mu ^{(v)}$ which represents the magnetic potential of the center vortex.

In [10] we applied the gauge transformations introduced in this section to obtain monopole, vortex, and chain for SU(2) gauge group and the corresponding Lagrangian. We have shown that the results obtained for monopoles are in agreement with what is obtained from Abelian Projection. We have also obtained a Lagrangian density for the correlated monopoles and vortices containing kinetic energy of the monopole, kinetic energy of the vortex, and the interaction between monopole and vortex.

Using the fact that SU(3) gauge group has three SU(2) subgroups, we generalize the results obtained for the SU(2) gauge group to the SU(3) gauge group to study the monopoles, vortices and their correlations for this group. Using SU(2) subgroups of SU(3) group, people have found magnetic monopoles by both Abelian gauge fixing [21] and field decomposition method [19] for the SU(3) gauge group. In this article, in addition to monopoles, we also study the vortex and the correlation between monopoles and vortices for the SU(3) gauge group. In addition, by comparing the results obtained with the help of SU(2) subgroups with the ones obtained directly from the Cho field decomposition for SU(3) monopoles, we discuss about the connection between SU(2) and SU(3) defects. For this purpose, in the next section, we first introduce and categorize SU(2) subgroups of SU(3) gauge group and then, by applying appropriate gauge transformations, we construct local color frames for each of the subgroups. Depending on the type of gauge transformation, magnetic defects like the monopole, vortex and chain are specified.

3 Identifying monopoles for SU(3) gauge group

One can identify the monopoles of the SU(3) gauge group directly as well as using its SU(2) subgroups. The second method is easier since the calculations are done for a lower color group and the results have already been reported in our earlier paper [10], as well. We follow the second method and study the SU(3) monopoles. Using the second method, we also study the relation between SU(2) subgroups defects and their SU(3) counterparts.

First, we use the appropriate gauge transformations for each of the three SU(2) subgroups. Then, we rewrite the results in terms of the local frames and find a connection between this method and the Cho decomposition. Using the results of the Cho decomposition for SU(3) group, we study the relation between the magnetic defects obtained in the three SU(2) subgroups with their counterparts in SU(3) gauge group. At the end, we write two Lagrangians with the help of both SU(3) and SU(2) subgroups and by comparison we learn how topological objects of the three SU(2) subgroups interact with each other.

We recall that monopoles appear as topological defects which are corresponding to the nontrivial second homotopy group $\Pi _2[SU(3)/U(1)^2]={\mathbb {Z}}^2$. It means that when SU(3) group is broken to its Cartan subgroup U$(1)^2$, two types of monopoles appear.

We first discuss about the SU(2) subgroups. SU(3) gauge group contains two Abelian directions. Suppose that $\hat{m}_i (i = 1, 2,\ldots , 8)$ represents a local orthonormal octet basis of SU(3). The Abelian directions are selected to be $\hat{m}_3=\hat{m}$ and $\hat{m}_8=\hat{m}'$. The space of SU(3) group is covered by three SU(2) subgroups. The corresponding local directions of the internal space can be grouped into the following three categories,

$$\begin{aligned}{} & {} (\hat{m}_1,\hat{m}_2,\hat{m}^1) , \quad \hat{m}^1=\hat{m},\nonumber \\{} & {} (\hat{m}_6,\hat{m}_7,\hat{m}^2) , \quad \hat{m}^2=-\frac{1}{2}\hat{m}+\frac{\sqrt{3}}{2}\hat{m},'\nonumber \\{} & {} (\hat{m}_4,-\hat{m}_5,\hat{m}^3) , \quad \hat{m}^3=-\frac{1}{2}\hat{m}-\frac{\sqrt{3}}{2}\hat{m}', \end{aligned}$$

(21)

where $\hat{m}^p(p=1,2,3)$ indicate the Abelian directions of SU(2) subgroups. As shown in the above equations, they are obtained from the Abelian directions $\hat{m}$ and $\hat{m}'$ of the SU(3) gauge group. In order to construct the above three local frames, we need to perform three nontrivial gauge transformations.

First subgroup: The generators of the first subgroup are $T_1, T_2$ and $T_3$, where $T_i=\frac{\lambda _i}{2}$ and $\lambda _i$s are Gell–Mann matrices given in Appendix A. The local color frame $(\hat{m}_1,\hat{m}_2,\hat{m}^1)$ is obtained by applying a nontrivial gauge transformation U,

$$\begin{aligned} U=e^{-i\alpha T_3}e^{-i\beta T_2}e^{-i\gamma T_3} . \end{aligned}$$

(22)

And,

$$\begin{aligned} UT_1U^{-1}= & {} \hat{m}_1\cdot \vec {T},\nonumber \\ UT_2U^{-1}= & {} \hat{m}_2\cdot \vec {T},\nonumber \\ UT_3U^{-1}= & {} \hat{m}^1\cdot \vec {T}. \end{aligned}$$

(23)

The frame $(\hat{m}_1,\hat{m}_2,\hat{m}^1)$ will contain a monopole, if one selects the parameters $\alpha$, $\beta$ and $\gamma$ such that $\alpha =\gamma =\varphi , \beta =\theta$, where $\theta$ and $\varphi$ are the polar and azimuthal angles. From Eq. (22) the resulted gauge transformation U is,

$$\begin{aligned} U=\left( \begin{array}{ccc} e^{-i\varphi }\cos \frac{\theta }{2}&{}-\sin \frac{\theta }{2}&{}0\\ \sin \frac{\theta }{2}&{}e^{i\varphi }\cos \frac{\theta }{2}&{}0\\ 0&{}0&{}1 \end{array} \right) , \end{aligned}$$

(24)

and the gauge field is transformed as the following,

$$\begin{aligned} \vec {A}_\mu ^U.\vec {T}= & {} U\left( \vec {A}_\mu .\vec {T}\right) U^{-1}+\frac{i}{g}U\partial _\mu U^{-1}u \nonumber \\= & {} U(X_\mu ^1 T_1+X_\mu ^2 T_2+A_\mu ^1T_3)U^{-1}+\frac{i}{g}U\partial _\mu U^{-1}.\nonumber \\ \end{aligned}$$

(25)

Calculating $\frac{i}{g}U\partial _\mu U^{-1}$ by the explicit form of Eq. (24) and using matrix forms of $T_1, T_2, T_3$, the transformed field is obtained,

$$\begin{aligned} \vec {A}_\mu ^U\cdot \vec {T}= & {} X_\mu ^1\underbrace{\big ((-\sin ^2\frac{\theta }{2}+\cos ^2\frac{\theta }{2}\cos 2\varphi )T_1+\cos ^2\frac{\theta }{2}\sin 2\varphi T_2-\sin \theta \cos \varphi T_3\big )}_{=\hat{m}_1\cdot \vec {T}} \nonumber \\{} & {} +X_\mu ^2\underbrace{\big (-\cos ^2\frac{\theta }{2}\sin 2\varphi T_1+(\sin ^2\frac{\theta }{2}+\cos ^2\frac{\theta }{2}\cos 2\varphi )T_2+\sin \theta \sin \varphi T_3\big )}_{=\hat{m}_2\cdot \vec {T}} \nonumber \\{} & {} +A_\mu ^1\underbrace{\big (\sin \theta \cos \varphi T_1+\sin \theta \sin \varphi T_2+\cos \theta T_3\big )}_{=\hat{m}^1\cdot \vec {T}} \nonumber \\{} & {} +\underbrace{\frac{1}{g}\big ((\sin \varphi \ \partial _\mu \theta -\cos \varphi \sin \theta \ \partial _\mu \varphi )T_1-(\cos \varphi \ \partial _\mu \theta +\sin \varphi \sin \theta \ \partial _\mu \varphi )T_2-(1+\cos \theta )\partial _\mu \varphi T_3\big )}_{=-(C_\mu ^{(m^1)}\hat{m}^1+\frac{1}{g}\hat{m}^1\times \partial _\mu \hat{m}^1)\cdot \vec {T}}, \end{aligned}$$

(26)

where,

$$\begin{aligned} C_\mu ^{(m^1)}=-\frac{1}{g}\hat{m}_1\cdot \partial _\mu \hat{m}_2=\frac{1}{g}(1+\cos \theta )\partial _\mu \varphi . \end{aligned}$$

(27)

The first three lines of Eq. (26) are regular. However, the last line is proportional to $\frac{1}{r}$ which indicates the contribution of the monopole and the term $C_\mu ^{(m^1)}$ diverges at $\theta =0$, representing a Dirac string. Since we are dealing with static objects, $\partial _\mu \varphi$ is replaced by $\vec {\nabla }\varphi$ and the last line of Eq. (26) can be written as the following,

$$\begin{aligned}{} & {} \frac{1}{g}\frac{1}{r}\Bigg ((\sin \varphi \ \hat{\theta }-\cos \varphi \ \hat{\varphi })T_1-(\cos \varphi \ \hat{\theta }+\sin \varphi \ \hat{\varphi })T_2 \nonumber \\{} & {} \quad -\frac{1+\cos \theta }{\sin \theta }\hat{\varphi }\ T_3\Bigg ). \end{aligned}$$

(28)

Next, we obtain the magnetic flux corresponding for each term of Eq. (28). These are the fluxes that would penetrate the area inside the closed contour $c(r,\theta )\equiv \{(r,\theta ,\varphi )|0\le \varphi <2\pi \}$,

$$\begin{aligned} \Phi ^{\text {flux}}= & {} \frac{1}{g}\int _0^{2\pi }r\sin \theta d\varphi \ \hat{\varphi }\cdot \frac{1}{r}(\sin \varphi \ \hat{\theta }-\cos \varphi \ \hat{\varphi })T_1\nonumber \\= & {} -\frac{1}{g}\int _0^{2\pi }\sin \theta \cos \varphi \ d\varphi \ T_1=0, \end{aligned}$$

(29)

$$\begin{aligned} \Phi ^{\text {flux}}= & {} -\frac{1}{g}\int _0^{2\pi }r\sin \theta d\varphi \ \hat{\varphi }\cdot \frac{1}{r}(\cos \varphi \ \hat{\theta }+\sin \varphi \ \hat{\varphi })T_2\nonumber \\= & {} -\frac{1}{g}\int _0^{2\pi }\sin \theta \sin \varphi \ d\varphi \ T_2=0, \end{aligned}$$

(30)

$$\begin{aligned} \Phi ^{\text {flux}}= & {} -\frac{1}{g}\int _0^{2\pi }r\sin \theta d\varphi \ \hat{\varphi }\cdot \frac{1+\cos \theta }{r\sin \theta }\hat{\varphi }T_3\nonumber \\= & {} -\frac{1}{g}(1+\cos \theta )\int _0^{2\pi }d\varphi \ T_3=-\frac{4\pi }{g}\frac{1+\cos \theta }{2} T_3. \end{aligned}$$

(31)

From the above equations, it is obvious that the magnetic flux of monopoles has no contribution in the color directions $T_1$ and $T_2$ and the total magnetic flux is obtained from the contribution of the monopole in the $T_3$th color direction. At $\theta =0$, the magnetic flux of a Dirac string that enters a monopole located at the origin $r=0$, is equal to $-\dfrac{4\pi }{g}T_3$.

Writing the transformed gauge field from Eq. (26) in terms of its components in the color space, the following expression for the gauge field is obtained for the first SU(2) subgroup,

$$\begin{aligned} \vec {A}_\mu ^U{=}(A_\mu ^1- C_\mu ^{(m^1)})\hat{m}^1-\frac{1}{g}\hat{m}^1\times \partial _\mu \hat{m}^1{+}X_\mu ^1 \hat{m}_1{+}X_\mu ^ 2\hat{m}_2. \nonumber \\ \end{aligned}$$

(32)

Identifying,

$$\begin{aligned} A_\mu ^{(m^1)}=A_\mu ^1- C_\mu ^{(m^1)} , \quad \vec {W}_\mu ^1= X_\mu ^ 1 \hat{m}_1+X_\mu ^ 2\hat{m}_2, \end{aligned}$$

(33)

we rewrite Eq. (32):

$$\begin{aligned} \vec {A}_\mu ^U=\underbrace{A_\mu ^{(m^1)}\hat{m}^1-\frac{1}{g}\hat{m}^1\times \partial _\mu \hat{m}^1}_{=\hat{A}_\mu ^{\left( m^1\right) }}+\vec {W}_\mu ^1, \end{aligned}$$

(34)

which is in the form of the extended Cho decomposition written in a frame that contains a monopole and the well-known string singularity.

The non-Abelian field strength tensor is as the following,

$$\begin{aligned} \vec {F}_{\mu \nu }=\partial _\mu \vec {A}_\nu ^U-\partial _\nu \vec {A}_\mu ^U+g\vec {A}_\mu ^U\times \vec {A}_\nu ^U. \end{aligned}$$

(35)

Using $\vec {A}_\mu ^U$ of Eq. (34), the non-Abelian field strength tensor of Eq. (35) is,

$$\begin{aligned} \vec {F}_{\mu \nu }^1=\underbrace{\left( F_{\mu \nu }^{( m^1)} +H_{\mu \nu }^{( m^1)}\right) \hat{m}^1}_{=\hat{F}_{\mu \nu }^1}+\vec {G}_{\mu \nu }^1+g\vec {W}_\mu ^1\times \vec {W}_\nu ^1, \end{aligned}$$

(36)

where,

$$\begin{aligned}{} & {} F_{\mu \nu }^{( m^1)}=\partial _\mu A_\nu ^{( m^1)}-\partial _\nu A_\mu ^{( m^1)},\nonumber \\{} & {} H_{\mu \nu }^{( m^1)}=\partial _\mu C_\nu ^{( m^1)}-\partial _\nu C_\mu ^{( m^1)},\nonumber \\{} & {} \vec {G}_{\mu \nu }^1=\hat{D}_\mu \vec {W}_\nu ^1-\hat{D}_\nu \vec {W}_\mu ^1, \quad \hat{D}_\mu =\partial _\mu +g\hat{A}_\mu ^{( m^1)}. \end{aligned}$$

(37)

It can be easily confirmed that $H_{\mu \nu }^{( m^1)}$ indicates the field strength of a magnetic monopole sitting at the origin along with a Dirac string at $\theta =0$ carrying a magnetic flux equal to $-\frac{4\pi }{g}T_3$ (see Fig. 1).

Fig. 1

Appearance of a monopole along with a Dirac string for the first subgroup

We do the same procedure for the two other SU(2) subgroups to construct the color frames that contain monopoles.

Second subgroup:The generators of the second subgroup are defined as $t_1=T_6, t_2=T_7$ and $t_3=-\frac{1}{2}T_3+\frac{\sqrt{3}}{2}T_8$. To construct the color frame $(\hat{m}_6,\hat{m}_7,\hat{m}^2)$, we apply a nontrivial gauge transformation $U'$ as the following,

$$\begin{aligned} U'=e^{-i\alpha t_3}e^{-i\beta t_2}e^{-i\gamma t_3} . \end{aligned}$$

(38)

And,

$$\begin{aligned} U't_1U'^{-1}= & {} \hat{m}_6\cdot \vec {t},\nonumber \\ U't_2U'^{-1}= & {} \hat{m}_7\cdot \vec {t},\nonumber \\ U't_3U'^{-1}= & {} \hat{m}^2\cdot \vec {t}. \end{aligned}$$

(39)

In order for the frame $(\hat{m}_6,\hat{m}_7,\hat{m}^2)$ to contain a monopole, we again choose the parameters $\alpha$, $\beta$ and $\gamma$ such that $\alpha =\gamma =\varphi , \beta =\theta$. Using the $t_i$ of this subgroup in Eq. (38), the gauge transformation $U'$ is obtained,

$$\begin{aligned} U'=\left( \begin{array}{ccc} 1&{}0&{}0\\ 0&{}e^{-i\varphi }\cos \frac{\theta }{2}&{}-\sin \frac{\theta }{2}\\ 0&{}\sin \frac{\theta }{2}&{}e^{i\varphi }\cos \frac{\theta }{2}\\ \end{array} \right) . \end{aligned}$$

(40)

It can be easily shown that the magnetic flux of monopole has no contribution in color directions $T_6$ and $T_7$ and the total magnetic flux is obtained from the contribution of monopole in $t_3$,

$$\begin{aligned} \Phi ^{\text {flux}}= & {} -\frac{1}{g}\int _0^{2\pi }r\sin \theta d\varphi \ \hat{\varphi }\nonumber \\{} & {} \cdot \frac{1+\cos \theta }{r\sin \theta }\hat{\varphi } \ \left( -\frac{1}{2}T_3+\frac{\sqrt{3}}{2}T_8\right) \nonumber \\= & {} -\frac{1}{g}(1+\cos \theta )\int _0^{2\pi }d\varphi \ \left( -\frac{1}{2}T_3+\frac{\sqrt{3}}{2}T_8\right) \nonumber \\= & {} -\frac{4\pi }{g}\frac{1+\cos \theta }{2} t_3. \end{aligned}$$

(41)

At $\theta =0$, the magnetic flux of a Dirac string that enters a monopole located at the origin $r=0$, is equal to $-\dfrac{4\pi }{g}t_3$.

Similar to the calculations of the first subgroup, here the transformed gauge field is,

$$\begin{aligned} \vec {A}_\mu ^{U'}=(A_\mu ^2- C_\mu ^{(m^2)})\hat{m}^2-\frac{1}{g}\hat{m}^2\times \partial _\mu \hat{m}^2+X_\mu ^ 6 \hat{m}_6+X_\mu ^ 7\hat{m}_7, \end{aligned}$$

(42)

where,

$$\begin{aligned} C_\mu ^{(m^2)}=-\frac{1}{g}\hat{m}_6\cdot \partial _\mu \hat{m}_7=\frac{1}{g}(1+\cos \theta )\partial _\mu \varphi . \end{aligned}$$

(43)

The components of $\hat{m}_6, \hat{m}_7$ and $\hat{m}^2$ are given in Appendix B. Identifying,

$$\begin{aligned} A_\mu ^{(m^2)}=A_\mu ^2- C_\mu ^{(m^2)} , \quad \vec {W}_\mu ^2= X_\mu ^6 \hat{m}_6+X_\mu ^ 7\hat{m}_7, \end{aligned}$$

(44)

we get,

$$\begin{aligned} \vec {A}_\mu ^{U'}=\underbrace{A_\mu ^{(m^2)}\hat{m}^2-\frac{1}{g}\hat{m}^2 \times \partial _\mu \hat{m}^2}_{=\hat{A}_\mu ^{(m^2)}}+\vec {W}_\mu ^2, \end{aligned}$$

(45)

which is again in the form of the extended Cho decomposition written in a frame which contains a monopole and the string singularity.

Also, similar to the calculations of the first subgroup, the non-Abelian field strength tensor is as the following,

$$\begin{aligned} \vec {F}_{\mu \nu }^2=\underbrace{\left( F_{\mu \nu }^{( m^2)} +H_{\mu \nu }^{( m^2)}\right) \hat{m}^2}_{=\hat{F}_{\mu \nu }^2}+\vec {G}_{\mu \nu }^2+g\vec {W}_\mu ^2\times \vec {W}_\nu ^2, \end{aligned}$$

(46)

where,

$$\begin{aligned}{} & {} F_{\mu \nu }^{( m^2)}=\partial _\mu A_\nu ^{( m^2)}-\partial _\nu A_\mu ^{( m^2)},\nonumber \\{} & {} H_{\mu \nu }^{( m^2)}=\partial _\mu C_\nu ^{( m^2)}-\partial _\nu C_\mu ^{( m^2)},\nonumber \\{} & {} \vec {G}_{\mu \nu }^2=\hat{D}_\mu \vec {W}_\nu ^2-\hat{D}_\nu \vec {W}_\mu ^2, \quad \hat{D}_\mu =\partial _\mu +g\hat{A}_\mu ^{( m^2)}, \end{aligned}$$

(47)

$H_{\mu \nu }^{( m^2)}$ indicates the field strength of a magnetic monopole sitting at the origin along with a Dirac string at $\theta =0$ carrying a magnetic flux equal to $-\frac{4\pi }{g}t_3$ (see Fig. 2).

Fig. 2

Appearance of a monopole along with a Dirac string for the second subgroup

Third subgroup:The generators of the third SU(2) subgroup are $t'_1=T_4, t'_2=-T_5$ and $t'_3=-\frac{1}{2}T_3-\frac{\sqrt{3}}{2}T_8$. The corresponding nontrivial gauge transformation which leads to a theory that contains the monopole is,

$$\begin{aligned} U''=\left( \begin{array}{ccc} e^{i\varphi }\cos \frac{\theta }{2}&{}0&{}\sin \frac{\theta }{2}\\ 0&{}1&{}0\\ -\sin \frac{\theta }{2}&{}0&{}e^{-i\varphi }\cos \frac{\theta }{2}\\ \end{array} \right) . \end{aligned}$$

(48)

The total magnetic flux is obtained only from the contribution of the monopole in the color direction $t_3^\prime$,

$$\begin{aligned} \Phi ^{\text {flux}}= & {} -\frac{1}{g}\int _0^{2\pi }r\sin \theta d\varphi \ \hat{\varphi }\nonumber \\{} & {} \cdot \frac{1+\cos \theta }{r\sin \theta }\hat{\varphi }\left( -\frac{1}{2}T_3-\frac{\sqrt{3}}{2}T_8\right) \nonumber \\= & {} -\frac{1}{g}(1+\cos \theta )\int _0^{2\pi }d\varphi \ \left( -\frac{1}{2}T_3-\frac{\sqrt{3}}{2}T_8\right) \nonumber \\= & {} -\frac{4\pi }{g}\frac{1+\cos \theta }{2}t_3^\prime , \end{aligned}$$

(49)

and the transformed gauge field is,

$$\begin{aligned} \vec {A}_\mu ^{U''}=(A_\mu ^3- C_\mu ^{(m^3)})\hat{m}^3-\frac{1}{g}\hat{m}^3\times \partial _\mu \hat{m}^3+X_\mu ^4 \hat{m}_4-X_\mu ^ 5\hat{m}_5, \end{aligned}$$

(50)

where,

$$\begin{aligned} C_\mu ^{(m^3)}=-\frac{1}{g}\hat{m}_4\cdot \partial _\mu (-\hat{m}_5)=\frac{1}{g}(1+\cos \theta )\partial _\mu \varphi . \end{aligned}$$

(51)

The components of $\hat{m}_4, -\hat{m}_5$ and $\hat{m}^3$ are given in Appendix B. Identifying,

$$\begin{aligned} A_\mu ^{(m^3)}=A_\mu ^3- C_\mu ^{(m^3)} , \quad \vec {W}_\mu ^3= X_\mu ^4 \hat{m}_4-X_\mu ^ 5\hat{m}_5, \end{aligned}$$

(52)

we obtain

$$\begin{aligned} \vec {A}_\mu ^{U''}=\underbrace{A_\mu ^{(m^3)}\hat{m}^3-\frac{1}{g}\hat{m}^3 \times \partial _\mu \hat{m}^3 }_{=\hat{A}_\mu ^{\left( m^3\right) }}+\vec {W}_\mu ^3, \end{aligned}$$

(53)

which is again in the form of the extended Cho decomposition written in a frame which contains a monopole and the string singularity.

The non-Abelian field strength is,

$$\begin{aligned} \vec {F}_{\mu \nu }^3=\underbrace{\left( F_{\mu \nu }^{( m^3)} +H_{\mu \nu }^{( m^3)}\right) \hat{m}^3}_{=\hat{F}_{\mu \nu }^3}+\vec {G}_{\mu \nu }^3+g\vec {W}_\mu ^3\times \vec {W}_\nu ^3, \end{aligned}$$

(54)

where,

$$\begin{aligned}{} & {} F_{\mu \nu }^{( m^3)}=\partial _\mu A_\nu ^{( m^3)}-\partial _\nu A_\mu ^{( m^3)},\nonumber \\{} & {} H_{\mu \nu }^{( m^3)}=\partial _\mu C_\nu ^{( m^3)}-\partial _\nu C_\mu ^{( m^3)},\nonumber \\{} & {} \vec {G}_{\mu \nu }^3=\hat{D}_\mu \vec {W}_\nu ^3-\hat{D}_\nu \vec {W}_\mu ^3, \quad \hat{D}_\mu =\partial _\mu +g\hat{A}_\mu ^{( m^3)}, \end{aligned}$$

(55)

$H_{\mu \nu }^{( m^3)}$ indicates the field strength of a magnetic monopole sitting at the origin along with a Dirac string at $\theta =0$ carrying a magnetic flux equal to $-\frac{4\pi }{g}t'_3$ (see Fig. 3).

Fig. 3

Appearance of a monopole along with a Dirac string for the third subgroup

It is clear from Eqs. (31), (41) and (49) that there are three magnetic monopoles with the following magnetic charges,

$$\begin{aligned} g_1= & {} -\frac{4\pi }{g}T_3,\nonumber \\ g_2= & {} -\frac{4\pi }{g}\left( -\frac{1}{2}T_3+\frac{\sqrt{3}}{2}T_8\right) =-\frac{4\pi }{g}t_3,\nonumber \\ g_3= & {} -\frac{4\pi }{g}\left( -\frac{1}{2}T_3-\frac{\sqrt{3}}{2}T_8\right) =-\frac{4\pi }{g}t_3^\prime . \end{aligned}$$

(56)

Given the fact that $g_1+g_2+g_3=0$, only two of the charges are independent. On the other hand, Eq. (56) can be written in the following form,

$$\begin{aligned} g_m=-\frac{4\pi }{g}(w_a\cdot T), \end{aligned}$$

(57)

where $w_a=1,2,3$ are the root vectors of the SU(3) gauge group and T indicates the vector $(T_3, T_8)$. The magnetic charges are proportional to the root vectors and only two of the root vectors are independent. As a result, two of these magnetic charges are independent. The root vectors are given in Appendix A.

From the above SU(2) subgroups discussions, the ultimate gauge field describing the monopoles of SU(3) gauge group can be written as the following,

$$\begin{aligned} \vec {A}_\mu =\sum _{p=1}^3 \left[ \dfrac{2}{3}(A_\mu ^{(m^p)}\hat{m}^p-\frac{1}{g}\hat{m}^p\times \partial _\mu \hat{m}^p)+\vec {W}_\mu ^p\right] . \end{aligned}$$

(58)

Therefore,

$$\begin{aligned} \vec {A}_\mu= & {} \dfrac{2}{3}\left\{ A_\mu ^{(m^1)}\hat{m}^1+A_\mu ^{(m^2)}\hat{m}^2+A_\mu ^{(m^3)}\hat{m}^3\right\} \nonumber \\{} & {} -\frac{2}{3}\frac{1}{g}\left\{ \hat{m}^1\times \partial _\mu \hat{m}^1+\hat{m}^2\times \partial _\mu \hat{m}^2+\hat{m}^3\times \partial _\mu \hat{m}^3\right\} \nonumber \\{} & {} +\sum _{p=1}^3\vec {W}_\mu ^p. \end{aligned}$$

(59)

Using Eq. (21) which shows the relations between SU(3) Abelian local color frames and its SU(2) subgroups counterparts, we rewrite the first two lines of Eq. (59) as follows,

$$\begin{aligned} \vec {A}_\mu= & {} \frac{2}{3} \left( A_\mu ^{(m^1)}-\frac{1}{2}A_\mu ^{(m^2)}-\frac{1}{2}A_\mu ^{(m^3)}\right) \hat{m}\nonumber \\{} & {} +\frac{2}{3} \frac{\sqrt{3}}{2} \left( A_\mu ^{(m^2)}-A_\mu ^{(m^3)}\right) \hat{m}^\prime \nonumber \\{} & {} -\frac{1}{g}\left( \hat{m}\times \partial _\mu \hat{m}+\hat{m}^\prime \times \partial _\mu \hat{m}^\prime \right) +\sum _{p=1}^3\vec {W}_\mu ^p. \end{aligned}$$

(60)

On the other hand, with the explicit definition of Cho decomposition for SU(3) gauge group in terms of local color frames $\hat{m}$ and $\hat{m}^\prime$, one may write the extended potential $\vec {A}_\mu$ as follows [13],

$$\begin{aligned} \vec {A}_\mu =\underbrace{A_\mu \hat{m}-\frac{1}{g} \hat{m}\times \partial _\mu \hat{m}}_{=\hat{A}_\mu }+\underbrace{A_\mu ^\prime \hat{m}^\prime -\frac{1}{g}\hat{m}^\prime \times \partial _\mu \hat{m}^\prime }_{=\hat{A}_\mu ^\prime }+\vec {X}_\mu , \end{aligned}$$

(61)

where $A_\mu =\hat{m}.\vec {A}_\mu$ and $A_\mu ^\prime =\hat{m}^\prime .\vec {A}_\mu ^\prime$ are the $T_3$-like and $T_8$-like Abelian components of the potential, respectively. $\vec {X}_\mu$ is the gauge covariant part of the potential, called the valence gluon.

Comparing Eqs. (60) and (61), we find the relation between Abelian components of the potential of the SU(3) gauge group with their counterparts for the three SU(2) subgroups,

$$\begin{aligned} \begin{aligned}&A_\mu =\frac{2}{3} \left( A_\mu ^{(m^1)}-\frac{1}{2}A_\mu ^{(m^2)}-\frac{1}{2}A_\mu ^{(m^3)}\right) ,\\&A_\mu ^\prime =\frac{1}{\sqrt{3}} \left( A_\mu ^{(m^2)}-A_\mu ^{(m^3)}\right) ,\\&\vec {X}_\mu =\sum _{p=1}^3\vec {W}_\mu ^p. \end{aligned} \end{aligned}$$

(62)

Also, using the results obtained from the three SU(2) subgroups, the ultimate field strength describing the SU(3) gauge group can be written as the following,

$$\begin{aligned} \vec {F}_{\mu \nu }=\sum _{p=1}^{3}\left[ \frac{2}{3}\hat{F}_{\mu \nu }^p+\vec {G}_{\mu \nu }^p\right] +g\sum _{p,q}\vec {W}_\mu ^p\times \vec {W}_\nu ^q. \end{aligned}$$

(63)

Therefore,

$$\begin{aligned} \vec {F}_{\mu \nu }= & {} \frac{2}{3}\left[ F_{\mu \nu }^{( m^1) } \hat{m}^1+F_{\mu \nu }^{( m^2) } \hat{m}^2+F_{\mu \nu }^{( m^3) } \hat{m}^3\right] \nonumber \\{} & {} +\frac{2}{3}\left[ H_{\mu \nu }^{( m^1) } \hat{m}^1+H_{\mu \nu }^{( m^2) } \hat{m}^2+H_{\mu \nu }^{( m^3) } \hat{m}^3\right] \nonumber \\{} & {} +\left[ \vec {G}_{\mu \nu }^1+\vec {G}_{\mu \nu }^2+\vec {G}_{\mu \nu }^3 \right] +g\sum _{p,q}\vec {W}_\mu ^p\times \vec {W}_\nu ^q. \end{aligned}$$

(64)

Once more, using Eq. (21), we rewrite the first two bracket of Eq. (64) as the following,

$$\begin{aligned} \vec {F}_{\mu \nu }= & {} \frac{2}{3}\left[ F_{\mu \nu }^{( m^1) } -\frac{1}{2}F_{\mu \nu }^{( m^2) } -\frac{1}{2}F_{\mu \nu }^{( m^3) }\right] \hat{m}\nonumber \\{} & {} +\frac{1}{\sqrt{3}}\left[ F_{\mu \nu }^{( m^2) }-F_{\mu \nu }^{( m^3) }\right] \hat{m}^\prime \nonumber \\{} & {} +\frac{2}{3}\left[ H_{\mu \nu }^{( m^1) } -\frac{1}{2}H_{\mu \nu }^{( m^2) }-\frac{1}{2}H_{\mu \nu }^{( m^3) }\right] \hat{m} \nonumber \\{} & {} +\frac{1}{\sqrt{3}}\left[ H_{\mu \nu }^{( m^2) }-H_{\mu \nu }^{( m^3) }\right] \hat{m}^\prime \nonumber \\{} & {} +\left[ \vec {G}_{\mu \nu }^1+\vec {G}_{\mu \nu }^2+\vec {G}_{\mu \nu }^3 \right] +g\sum _{p,q}\vec {W}_\mu ^p\times \vec {W}_\nu ^q. \end{aligned}$$

(65)

On the other hand, using the explicit definition of the Cho decomposition for SU(3) gauge group in terms of local frames $\hat{m}$ and $\hat{m}^\prime$, the extended filed strength tensor $\vec {F}_{\mu \nu }$ is written as the following,

$$\begin{aligned} \vec {F}_{\mu \nu }=F_{\mu \nu }\hat{m}+F_{\mu \nu }^\prime \hat{m}^\prime +H_{\mu \nu }\hat{m} +H_{\mu \nu }^\prime \hat{m}^\prime +\vec {K}_{\mu \nu }+g\vec {X}_\mu \times \vec {X}_\nu , \end{aligned}$$

(66)

where,

$$\begin{aligned} \begin{aligned}&F_{\mu \nu }=\partial _\mu A_\nu -\partial _\nu A_\mu , \quad F_{\mu \nu }^\prime =\partial _\mu A_\nu ^\prime -\partial _\nu A_\mu ^\prime \\&H_{\mu \nu }=\partial _\mu C_\nu -\partial _\nu C_\mu , \quad H_{\mu \nu }^\prime =\partial _\mu C_\nu ^\prime -\partial _\nu C_\mu ^\prime \\&\vec {K}_{\mu \nu }=D_\mu \vec {X}_\nu -D_\mu \vec {X}_\nu , \quad D_\mu =\partial _\mu +g\left( \hat{A}_\mu +\hat{A}_\mu ^\prime \right) . \end{aligned}\end{aligned}$$

(67)

$C_\mu$ and $C_\mu ^\prime$ are the magnetic potentials corresponding to the the magnetic field $H_{\mu \nu }$ and $H_{\mu \nu }^\prime$, directly defined in SU(3) gauge group. Comparing Eqs. (65)–(67), relations between physical quantities like field strength tensors of SU(3) gauge group and its SU(2) subgroups are obtained,

$$\begin{aligned}{} & {} F_{\mu \nu }=\frac{2}{3}\left[ F_{\mu \nu }^{( m^1) } -\frac{1}{2}F_{\mu \nu }^{( m^2) }-\frac{1}{2}F_{\mu \nu }^{( m^3) }\right] ,\nonumber \\{} & {} F_{\mu \nu }^\prime =\frac{1}{\sqrt{3}}\left[ F_{\mu \nu }^{( m^2) }-F_{\mu \nu }^{( m^3) }\right] ,\nonumber \\{} & {} H_{\mu \nu }=\frac{2}{3}\left[ H_{\mu \nu }^{( m^1) } -\frac{1}{2}H_{\mu \nu }^{( m^2) }-\frac{1}{2}H_{\mu \nu }^{( m^3) }\right] ,\nonumber \\{} & {} H_{\mu \nu }^\prime =\frac{1}{\sqrt{3}}\left[ H_{\mu \nu }^{( m^2) }-H_{\mu \nu }^{( m^3) }\right] ,\nonumber \\{} & {} \vec {K}_{\mu \nu }=\vec {G}_{\mu \nu }^1+\vec {G}_{\mu \nu }^2+\vec {G}_{\mu \nu }^3, \nonumber \\ {}{} & {} g\vec {X}_\mu \times \vec {X}_\nu =g\sum _{p,q}\vec {W}_\mu ^p\times \vec {W}_\nu ^q. \end{aligned}$$

(68)

Using the fact that $C_{\mu }^{( m^1) } +C_{\mu }^{( m^2) }+C_{\mu }^{( m^3) }=0$, only two of the SU(2) monopole magnetic potentials are independent. Replacing $H_{\mu \nu }$’s of Eqs. (37), (47) and (55) in $H_{\mu \nu }$ of Eqs. (68), we get,

$$\begin{aligned} H_{\mu \nu }= & {} \frac{2}{3}\left[ H_{\mu \nu }^{( m^1) } -\frac{1}{2}H_{\mu \nu }^{( m^2) }-\frac{1}{2}H_{\mu \nu }^{( m^3) }\right] \nonumber \\ H_{\mu \nu }= & {} \frac{2}{3}\left[ \left( \partial _\mu C_{\nu }^{( m^1) }-\partial _\nu C_{\mu }^{( m^1) }\right) \right. \nonumber \\{} & {} \left. -\frac{1}{2}\left\{ \partial _\mu \left( C_{\nu }^{( m^2) }+ C_{\nu }^{( m^3) }\right) -\partial _\nu \left( C_{\mu }^{( m^2) }+C_{\mu }^{( m^3) }\right) \right\} \right] \nonumber \\{} & {} \quad \rightarrow H_{\mu \nu }=H_{\mu \nu }^{(m^1)}\ \text {and} \ H_{\mu \nu }^\prime =\frac{1}{\sqrt{3}}\left[ H_{\mu \nu }^{( m^2) }-H_{\mu \nu }^{( m^3) }\right] . \end{aligned}$$

(69)

The above equations show the relations between the two monopole field strength tensors of the SU(3) gauge group in terms of their counterparts in the SU(2) subgroup.

Using Figs. 1, 2, 3 and Eq. (69), we end up to Fig. 4.

Fig. 4

Appearance of $T_3$-like monopole along with a Dirac string in SU(3) gauge group

In fact, Fig. 4 shows $H_{\mu \nu }$, one of the monopole field strength tensor of SU(3). It indicates a magnetic monopole sitting at the origin along with a Dirac string in $\theta = 0$ carrying a magnetic flux equal to $-\dfrac{4\pi }{g}T_3$. The interesting point is that it is obtained by SU(2) subgroups and not as a direct calculation of SU(3) gauge group.

In the same way, we get the other monopole field strength tensor of SU(3), $H_{\mu \nu }^\prime$. It indicates a magnetic monopole sitting at the origin along with a Dirac string in $\theta = 0$ carrying a magnetic flux equal $-\dfrac{4\pi }{g}T_8$ (Fig. 5).

Fig. 5

Appearance of $T_8$-like monopole along with a Dirac string in SU(3) gauge group

On the other hand, comparing Eqs. (37), (47), (55), (67) and (68), the relations between the monopole magnetic potential in SU(3) and its three SU(2) subgroups are as the following,

$$\begin{aligned} \begin{aligned}&C_{\mu }=\frac{2}{3}\left[ C_{\mu }^{( m^1) } -\frac{1}{2}C_{\mu }^{( m^2) }-\frac{1}{2}C_{\mu }^{( m^3) }\right] ,\\&C_{\mu }^\prime =\frac{1}{\sqrt{3}}\left[ C_{\mu }^{( m^2) }-C_{\mu }^{( m^3) }\right] . \end{aligned} \end{aligned}$$

(70)

However, since $C_{\mu }^{( m^1) } +C_{\mu }^{( m^2) }+C_{\mu }^{( m^3) }=0$, only two of the SU(2) monopole magnetic potentials are independent. Applying this fact to Eq. (70),

$$\begin{aligned} C_{\mu }=C_{\mu }^{( m^1) }, \quad C_{\mu }^\prime =\frac{1}{\sqrt{3}}\left[ C_{\mu }^{( m^2) }-C_{\mu }^{( m^3) }\right] . \end{aligned}$$

(71)

Therefore, we have found an interesting relation between the defined vector potentials of the two SU(3) monopoles and their counterparts of the SU(2) subgroups.

Finally, we discuss about the Lagrangian written directly for the SU(3) gauge group and compare it with the case where we use its SU(2) subgroups. Using field strength tensor of Eq. (64), the extended Lagrangian in terms of the three SU(2) subgroups is [20],

$$\begin{aligned} \mathcal {L}= & {} -\frac{1}{4} \vec {F}_{\mu \nu }.\vec {F}_{\mu \nu }\nonumber \\= & {} \sum _{p=1}^{3}\left\{ -\frac{1}{6} (\hat{F}_{\mu \nu }^p)^2-\frac{1}{4}(\vec {G}_{\mu \nu }^p)^2-\frac{g}{2}\hat{F}_{\mu \nu }^p.(\vec {W}_\mu ^p \times \vec {W}_\nu ^p)\right\} \nonumber \\{} & {} -\sum _{p,q} \frac{g^2}{4}(\vec {W}_\mu ^p \times \vec {W}_\nu ^q)^2 -\sum _{p,q,r}\frac{g}{2} \vec {G}_{\mu \nu }^p.(\vec {W}_\mu ^q \times \vec {W}_\nu ^r)\nonumber \\{} & {} -\sum _{p\ne q}\frac{g^2}{4}(\vec {W}_\mu ^p \times \vec {W}_\nu ^q).(\vec {W}_\mu ^q \times \vec {W}_\nu ^p)\nonumber \\{} & {} -\sum _{p\ne q}\frac{g^2}{4}(\vec {W}_\mu ^p \times \vec {W}_\nu ^p).(\vec {W}_\mu ^q \times \vec {W}_\nu ^q). \end{aligned}$$

(72)

We rewrite the term $-\frac{1}{4}\sum _{p=1}^3(\vec {G}_{\mu \nu }^p)^2$ using $\vec {G}_{\mu \nu }^p$’s defined in Eqs. (37), (47) and (55) in terms of $\vec {W}_\mu ^p$’s; and Eqs. (33), (44) and (52) which define $\vec {W}_\mu ^p$’s in terms of $X_\mu$’s.

$$\begin{aligned}{} & {} -\frac{1}{4}\sum _{p=1}^3(\vec {G}_{\mu \nu }^p)^2\nonumber \\{} & {} \quad =-\frac{1}{4}\bigg \{\Big [\big (\partial _\mu X_\nu ^1-g(A_\mu ^{(m^1)}+C_\mu ^{(m^1)})X_\nu ^2\big )-(\mu \leftrightarrow \nu )\Big ]^2\nonumber \\{} & {} \qquad +\Big [\big (\partial _\mu X_\nu ^2+g(A_\mu ^{(m^1)}+C_\mu ^{(m^1)})X_\nu ^1\big )-(\mu \leftrightarrow \nu )\Big ]^2\nonumber \\{} & {} \qquad +\Big [\big (\partial _\mu X_\nu ^6-g(A_\mu ^{(m^2)}+C_\mu ^{(m^2)})X_\nu ^7\big )-(\mu \leftrightarrow \nu )\Big ]^2\nonumber \\{} & {} \qquad +\Big [\big (\partial _\mu X_\nu ^7+g(A_\mu ^{(m^2)}+C_\mu ^{(m^2)})X_\nu ^6\big )-(\mu \leftrightarrow \nu )\Big ]^2\nonumber \\{} & {} \qquad +\Big [\big (\partial _\mu X_\nu ^4-g(A_\mu ^{(m^3)}+C_\mu ^{(m^3)})X_\nu ^5\big )-(\mu \leftrightarrow \nu )\Big ]^2\nonumber \\{} & {} \qquad +\Big [\big (\partial _\mu X_\nu ^5+g(A_\mu ^{(m^3)}+C_\mu ^{(m^3)})X_\nu ^4\big )-(\mu \leftrightarrow \nu )\Big ]^2\bigg \}.\nonumber \\ \end{aligned}$$

(73)

Using Eqs. (68), (70) and finding the field strengths of SU(2) versus the ones for SU(3), the Lagrangian can be rewritten for the SU(3) gauge group as follows,

$$\begin{aligned} \begin{aligned} \mathcal {L}&=-\frac{1}{4}F_{\mu \nu }^{\ 2}-\frac{1}{4}F_{\mu \nu }^{\prime \ 2}-\frac{1}{4}H_{\mu \nu }^{\ 2}-\frac{1}{4}H_{\mu \nu }^{\prime \ 2}\\&\quad -\frac{1}{2}\left| \left[ \partial _\mu +ig(A_\mu +C_\mu ) \right] R_\nu {-}\left[ \partial _\nu {+}ig(A_\nu +C_\nu ) \right] R_\mu \right| ^2\\&\quad -\frac{1}{2}\left| \left[ \partial _\mu -\frac{1}{2}ig(A_\mu +C_\mu )+\frac{\sqrt{3}}{2}ig(A_\mu ^\prime +C_\mu ^\prime ) \right] B_\nu \right. \\&\quad \left. -\left[ \partial _\nu -\frac{1}{2}ig(A_\nu +C_\nu )+\frac{\sqrt{3}}{2}ig(A_\nu ^\prime +C_\nu ^\prime ) \right] B_\mu \right| ^2\\&\quad -\frac{1}{2}\left| \left[ \partial _\mu -\frac{1}{2}ig(A_\mu +C_\mu )-\frac{\sqrt{3}}{2}ig(A_\mu ^\prime +C_\mu ^\prime ) \right] Y_\nu \right. \\&\quad \left. -\left[ \partial _\nu -\frac{1}{2}ig(A_\nu +C_\nu )-\frac{\sqrt{3}}{2}ig(A_\nu ^\prime +C_\nu ^\prime ) \right] Y_\mu \right| ^2\\&\quad +\text {other interactions}, \end{aligned} \end{aligned}$$

(74)

where,

$$\begin{aligned} R_\mu =\frac{X_\mu ^1+iX_\mu ^2}{\sqrt{2}},\quad B_\mu =\frac{X_\mu ^6+iX_\mu ^7}{\sqrt{2}},\quad Y_\mu =\frac{X_\mu ^4-iX_\mu ^5}{\sqrt{2}}. \end{aligned}$$

(75)

Since we are interested in studying the interaction of monopoles, we have just brought the terms that contain the monopole potentials of the second order. The rest of the Lagrangian is mentioned by “other interactions”. The above Lagrangian is equivalent to the Lagrangian obtained directly by Cho [13] who used the field decomposition method for SU(3) gauge group. Before proceeding to the interpretation of the Lagrangian components, we should point out the difference between the Cho Lagrangian and (74). Cho defined the dual magnetic potential $C_\mu ^*$ by $H_{\mu \nu }^*=\partial _\mu C_\nu ^*-\partial _\nu C_\mu ^*$. Where $C_\mu ^*$ indicates a timelike potential which has no string singularity. However, we need the Dirac string as we explain later. Therefore, we do not remove it at this point.

The first line of Eq. (74) shows the kinetic energy of the Abelian components and the topological defects. The second, the third and the fourth lines show the interaction of monopoles with each other. As it is clear from these sentences, each type of monopole can interact with its own type of monopole as well as with the other type. These interactions take place via the off-diagonal components of the gauge field. For example, the monopole corresponding to the potential $C_\mu$, interacts with the monopole corresponding to the potential $C'_\mu$ via $B_\mu$ and $Y_\mu$. In addition, $C_\mu$ can interact with monopoles of its own type via $R_\mu$, $B_\mu$ and $Y_\mu$ and $C'_\mu$ interacts with the monopole of its own type via $B_\mu$ and $Y_\mu$.

4 Identifying vortices for SU(3)

As mentioned in Sect. 2, to observe the effect of a vortex linking to a Wilson loop, a gauge transformation proportional to the center of the group should be applied. We use the SU(2) subgroups of SU(3) gauge group to identify the SU(3) vortices.

First subgroup: For this subgroup, the appropriate gauge transformation is,

$$\begin{aligned} V=e^{-i\xi T_3} . \end{aligned}$$

(76)

Recalling Eq. (9) for SU(2) gauge group, we have,

$$\begin{aligned} VT_1V^{-1}= & {} \hat{n}'_1\cdot \vec {T}, \nonumber \\ VT_2V^{-1}= & {} \hat{n}'_2\cdot \vec {T}, \nonumber \\ VT_3V^{-1}= & {} \hat{n}'^1\cdot \vec {T}. \end{aligned}$$

(77)

Choosing $\xi =-\varphi$ in Eq. (76), the gauge transformation V is,

$$\begin{aligned} V=\left( \begin{array}{ccc} e^{i\frac{\varphi }{2}}&{}0&{}0\\ 0&{}e^{-i\frac{\varphi }{2}}&{}0\\ 0&{}0&{}1 \end{array}\right) . \end{aligned}$$

(78)

The transformed gauge field is,

$$\begin{aligned}{} & {} \vec {A}_\mu ^V\cdot \vec {T}=V\vec {A}_\mu \cdot \vec {T}V^{-1}+\frac{i}{g}V\partial _\mu V^{-1}-\text {ideal vortex}\nonumber \\{} & {} \quad =V(X_\mu ^1T_1+X_\mu ^2T_2+A_\mu ^1T_3)V^{-1}+\frac{i}{g}V\partial _\mu V^{-1}\nonumber \\ {}{} & {} \qquad -\text {ideal vortex}. \end{aligned}$$

(79)

As mentioned in Sect. 2, the appearance of the last term is due to the discontinuity of gauge transformation [25, 26] and eventually will be removed.

Using V of Eq. (78) in the above equation,

$$\begin{aligned} \vec {A}_\mu ^V\cdot \vec {T}= & {} X_\mu ^1\underbrace{(\cos \varphi T_1-\sin \varphi T_2)}_{=\hat{n}'_1\cdot \vec {T}}\nonumber \\{} & {} +X_\mu ^2\underbrace{(\sin \varphi T_1+\cos \varphi T_2)}_{=\hat{n}'_2\cdot \vec {T}}+A_\mu ^1\underbrace{T_3}_{=\hat{n}'^1\cdot \vec {T}}\nonumber \\{} & {} +\underbrace{\frac{1}{g}\partial _\mu \varphi T_3}_{=-(C_\mu ^{(v^1)}\hat{n}'^1+\frac{1}{g}\hat{n}'^1\times \partial _\mu \hat{n}'^1)}, \end{aligned}$$

(80)

where,

$$\begin{aligned} C_\mu ^{(v^1)}=-\frac{1}{g}\hat{n}'_1\cdot \partial _\mu \hat{n}'_2=-\frac{1}{g}\partial _\mu \varphi , \end{aligned}$$

(81)

and $\frac{1}{g}\hat{n}'^1\times \partial _\mu \hat{n}'^1=0$. Thus, in terms of the local color frame, the first SU(2) subgroup gauge field is,

$$\begin{aligned} \vec {A}_\mu ^V =(A_\mu ^1-C_\mu ^{(v^1)})\hat{n}'^1+\underbrace{X_\mu ^1\hat{n}'_1+X_\mu ^2\hat{n}'_2}_{=\vec {W}_\mu ^1}. \end{aligned}$$

(82)

The vortex contribution is represented by the last term of Eq. (80) and its magnetic flux is obtained as the following,

$$\begin{aligned} \Phi ^{\text {flux}}=\frac{1}{g}\int _0^{2\pi }r\sin \theta d\varphi \ \hat{\varphi }\cdot \left( \frac{1}{r\sin \theta } \hat{\varphi }\ T_3\right) =\frac{2\pi }{g}T_3. \end{aligned}$$

(83)

Second subgroup: For this subgroup, the appropriate gauge transformation is obtained by its generators $t_i$,

$$\begin{aligned} V'=\left( \begin{array}{ccc} 1&{}0&{}0\\ 0&{}e^{i\frac{\varphi }{2}}&{}0\\ 0&{}0&{}e^{-i\frac{\varphi }{2}} \end{array}\right) . \end{aligned}$$

(84)

And the transformed gauge field is,

$$\begin{aligned} \vec {A}_\mu ^{V'}=(A_\mu ^2-C_\mu ^{(v^2)})\hat{n}'^2+\underbrace{X_\mu ^6\hat{n}'_6+X_\mu ^7\hat{n}'_7}_{=\vec {W}_\mu ^2}, \end{aligned}$$

(85)

where,

$$\begin{aligned} C_\mu ^{(v^2)}=-\frac{1}{g}\hat{n}'_6\cdot \partial _\mu \hat{n}'_7=-\frac{1}{g}\partial _\mu \varphi . \end{aligned}$$

(86)

The components of $\hat{n}'_6, \hat{n}'_7$ and $\hat{n}'^2$ are given in Appendix B. The magnetic flux corresponding to the singular term of the transformed gauge field is,

$$\begin{aligned} \Phi ^{\text {flux}}= & {} \frac{1}{g}\int _0^{2\pi }r\sin \theta d\varphi \ \hat{\varphi }\cdot \left( \frac{1}{r\sin \theta } \hat{\varphi }\ \left( -\frac{1}{2}T_3+\frac{\sqrt{3}}{2}T_8\right) \right) \nonumber \\= & {} \frac{2\pi }{g}\left( -\frac{1}{2}T_3+\frac{\sqrt{3}}{2}T_8\right) =\frac{2\pi }{g}t_3. \end{aligned}$$

(87)

Third subgroup: Using $t'_i$ for the third SU(2) subgroup, the appropriate gauge transformation and the gauge field is,

$$\begin{aligned} V''=\left( \begin{array}{ccc} e^{-i\frac{\varphi }{2}}&{}0&{}0\\ 0&{}1&{}0\\ 0&{}0&{}e^{i\frac{\varphi }{2}} \end{array}\right) . \end{aligned}$$

(88)

And,

$$\begin{aligned} \vec {A}_\mu ^{V''}=(A_\mu ^3-C_\mu ^{(v^3)})\hat{n}'^3+\underbrace{X_\mu ^4\hat{n}'_4-X_\mu ^5\hat{n}'_5}_{=\vec {W}_\mu ^3}, \end{aligned}$$

(89)

where,

$$\begin{aligned} C_\mu ^{(v^3)}=-\frac{1}{g}\hat{n}'_4\cdot \partial _\mu (-\hat{n}'_5)=-\frac{1}{g}\partial _\mu \varphi . \end{aligned}$$

(90)

The components of $\hat{n}'_4, -\hat{n}'_5$ and $\hat{n}'^3$ are given in Appendix B. The corresponding magnetic flux is,

$$\begin{aligned} \Phi ^{\text {flux}}= & {} \frac{1}{g}\int _0^{2\pi }r\sin \theta d\varphi \ \hat{\varphi }\cdot \left( \frac{1}{r\sin \theta } \hat{\varphi }\ \left( -\frac{1}{2}T_3-\frac{\sqrt{3}}{2}T_8\right) \right) \nonumber \\= & {} \frac{2\pi }{g}\left( -\frac{1}{2}T_3-\frac{\sqrt{3}}{2}T_8\right) =\frac{2\pi }{g}t_3^\prime . \end{aligned}$$

(91)

From Eqs. (83), (87) and (91), there exist three vortices with the following magnetic fluxes,

$$\begin{aligned} \Phi _1= & {} \frac{2\pi }{g}T_3, \nonumber \\ \Phi _2= & {} \frac{2\pi }{g}\left( -\frac{1}{2}T_3+\frac{\sqrt{3}}{2}T_8\right) =\frac{2\pi }{g}t_3, \nonumber \\ \Phi _3= & {} \frac{2\pi }{g}\left( -\frac{1}{2}T_3-\frac{\sqrt{3}}{2}T_8\right) =\frac{2\pi }{g}t_3^\prime . \end{aligned}$$

(92)

It is clear from Eq. (92) that $\Phi _1+\Phi _2+\Phi _3=0$. Therefore, only two of the vortices are independent. The above equations may be written in the compact following form,

$$\begin{aligned} \Phi _v=\frac{2\pi }{g}(w_a\cdot T), \end{aligned}$$

(93)

where $w_a=1,2,3$ are the root vectors of the SU(3) gauge group and T is the vector $(T_3, T_8)$. Similar to magnetic charges of monopoles, the magnetic fluxes of vortices are proportional to the root vectors and since two of the root vectors are independent, two of these vortices are independent.

We recall that the number of vortices of each SU(N) group is equal to the number of nontrivial center elements of that group. For example, for the SU(3) group, there are two nontrivial center elements $z_1=e^{i\frac{2}{3}\pi }$ and $z_2=e^{i\frac{4}{3}\pi }$. This means that we have two kinds of vortices for this group.

We would like to calculate the gauge fields, the corresponding field strength tensors and the Lagrangian density of the SU(3) vortices. From the SU(2) subgroups, the gauge field which contains the vortices of SU(3) gauge group can be written as the following,

$$\begin{aligned} \vec {A}_\mu =\sum _{p=1}^3 \Bigg [ \dfrac{2}{3}\underbrace{\big ((A_\mu ^{p}-C_\mu ^{(v^p)})\hat{n}'^p\big )}_{=\hat{A}_\mu ^p}+\vec {W}_\mu ^p\Bigg ]. \end{aligned}$$

(94)

Using the above gauge field, the field strength tensor and Lagrangian density are obtained as follows,

$$\begin{aligned} \begin{aligned} \vec {F}_{\mu \nu }&=\sum _{p=1}^3\dfrac{2}{3}\Big (\partial _\mu (A_\nu ^{p}-C_\nu ^{(v^p)}) -\partial _\nu (A_\mu ^{p}-C_\mu ^{(v^p)})\Big )\hat{n}'^p\\&\quad +\sum _{p=1}^3\big (\hat{D}_\mu ^p\vec {W}_\nu ^p-\hat{D}^p_\nu \vec {W}_\mu ^p\big )+g\sum _{p,q}\vec {W}_\mu ^p\times \vec {W}_\nu ^q, \end{aligned} \end{aligned}$$

(95)

where,

$$\begin{aligned} \hat{D}_\mu ^p=\partial _\mu +g\hat{A}_\mu ^p. \end{aligned}$$

(96)

Using Eq. (95), the Lagrangian in terms of three SU(2) subgroups is obtained,

$$\begin{aligned} \mathcal {L}= & {} -\frac{1}{4} \vec {F}_{\mu \nu }.\vec {F}_{\mu \nu }\nonumber \\= & {} \sum _{p=1}^{3}\left\{ -\frac{1}{6} (\hat{F}_{\mu \nu }^p)^2-\frac{1}{4}(\vec {G}_{\mu \nu }^p)^2-\frac{g}{2}\hat{F}_{\mu \nu }^p.(\vec {W}_\mu ^p \times \vec {W}_\nu ^p)\right\} \nonumber \\{} & {} -\sum _{p,q} \frac{g^2}{4}(\vec {W}_\mu ^p \times \vec {W}_\nu ^q)^2-\sum _{p,q,r}\frac{g}{2} \vec {G}_{\mu \nu }^p.(\vec {W}_\mu ^q \times \vec {W}_\nu ^r)\nonumber \\{} & {} -\sum _{p\ne q}\frac{g^2}{4}(\vec {W}_\mu ^p \times \vec {W}_\nu ^q).(\vec {W}_\mu ^q \times \vec {W}_\nu ^p)\nonumber \\{} & {} -\sum _{p\ne q}\frac{g^2}{4}(\vec {W}_\mu ^p \times \vec {W}_\nu ^p).(\vec {W}_\mu ^q \times \vec {W}_\nu ^q), \end{aligned}$$

(97)

where,

$$\begin{aligned} \begin{aligned} \hat{F}_{\mu \nu }^p&=\Big (\partial _\mu (A_\nu ^{p}-C_\nu ^{(v^p)})-\partial _\nu (A_\mu ^{p}-C_\mu ^{(v^p)})\Big )\hat{n}'^p,\\ \vec {G}_{\mu \nu }^p&=\big (\hat{D}_\mu ^p\vec {W}_\nu ^p-\hat{D}^p_\nu \vec {W}_\mu ^p\big ). \end{aligned} \end{aligned}$$

(98)

Now, we discuss about the interpretation of each term of the Lagrangian (97),

• $\sum _{p=1}^{3}-\frac{1}{6} (\hat{F}_{\mu \nu }^p)^2$:

  This term contains the kinetic energy of the vortices.

• $\sum _{p=1}^3-\frac{1}{4}(\vec {G}_{\mu \nu }^p)^2$:

  This expression indicates the interaction of the vortices with each other that happens via the off-diagonal components of the gauge field.

• $\sum _{p=1}^3-\frac{g}{2}\hat{F}_{\mu \nu }^p\cdot (\vec {W}_\mu ^p \times \vec {W}_\nu ^p)$,

• $-\sum _{p,q,r}\frac{g}{2} \vec {G}_{\mu \nu }^p.(\vec {W}_\mu ^q \times \vec {W}_\nu ^r)$:

  These terms show the interaction of the vortices with the off-diagonal components of the gauge field.

• $-\sum _{p,q} \frac{g^2}{4}(\vec {W}_\mu ^p \times \vec {W}_\nu ^q)^2 \times \vec {W}_\nu ^p)$,

• $-\sum _{p\ne q}\frac{g^2}{4}(\vec {W}_\mu ^p \times \vec {W}_\nu ^q).(\vec {W}_\mu ^q \times \vec {W}_\nu ^p)$,

• $-\sum _{p\ne q}\frac{g^2}{4}(\vec {W}_\mu ^p \times \vec {W}_\nu ^p).(\vec {W}_\mu ^q \times \vec {W}_\nu ^q)$:

  These terms contain the interaction of the off-diagonal components of the gauge field with each other.

5 Correlation of monopoles and vortices in SU(3) gauge group

Applying the two gauge transformations U and V of the previous sections in a successive form, VU, we get the configurations that include correlated monopoles and vortices. We use the SU(2) subgroups of SU(3) gauge group, in the same way as we have done for the monopoles and vortices. It can be easily shown that the vortex gauge transformation does not change the local color directions that have already been changed by the monopole gauge transformation, but would just add a phase factor to it. For example, for the first subgroup, the transformation V has been defined as the following,

$$\begin{aligned} V=e^{-i\xi \hat{m}^1\cdot \vec {T}}. \end{aligned}$$

(99)

Applying this transformation on a monopole gauge transformation U and by expanding the exponential factor, one gets,

$$\begin{aligned} \begin{aligned} VU&=e^{-i\xi \hat{m}^1\cdot \vec {T}}U=e^{-i\xi UT_3U^{-1}}U\\&=\left( 1-i\xi UT_3U^{-1}+\frac{(-i\xi )^2}{2!}(UT_3U^{-1})^2+\cdots \right) U\\&=U\left( 1-i\xi T_3+\frac{(-i\xi )^2}{2!} (T_3)^2+\cdots \right) \\&=U e^{-i\xi T_3}. \end{aligned} \end{aligned}$$

(100)

First subgroup: we define the color frame of the first subgroup as the following,

$$\begin{aligned} \begin{aligned} (VU)T_1(VU)^{-1}&=\hat{n}_1\cdot \vec {T},\\ (VU)T_2(VU)^{-1}&=\hat{n}_2\cdot \vec {T},\\ (VU)T_3(VU)^{-1}&=\hat{n}^1\cdot \vec {T}. \end{aligned} \end{aligned}$$

(101)

Choosing $\xi =-\varphi$ and using (100), the VU transformation matrix and the transformed gauge field are,

$$\begin{aligned} VU=\left( \begin{array}{ccc} \cos \frac{\theta }{2}e^{-i\frac{\varphi }{2}}&{}-\sin \frac{\theta }{2}e^{-i\frac{\varphi }{2}}&{}0\\ \sin \frac{\theta }{2}e^{i\frac{\varphi }{2}}&{}\cos \frac{\theta }{2}e^{i\frac{\varphi }{2}}&{}0\\ 0&{}0&{}1 \end{array}\right) , \end{aligned}$$

(102)

and,

$$\begin{aligned} \vec {A}_\mu ^{VU}\cdot \vec {T}= & {} (VU)\vec {A}_\mu \cdot \vec {T}(VU)^{-1}+\frac{i}{g}(VU)\partial _\mu (VU)^{-1}\nonumber \\= & {} (VU)(X_\mu ^1T_1+X_\mu ^2T_2+A_\mu ^1T_3)(VU)^{-1}\nonumber \\{} & {} \quad +\frac{i}{g}(VU)\partial _\mu (VU)^{-1}. \end{aligned}$$

(103)

Using the definition of VU from Eq. (102), the transformed gauge field is given,

$$\begin{aligned} \vec {A}_\mu ^{VU}\cdot \vec {T}= & {} X_\mu ^1\underbrace{(\cos \theta \cos \varphi T_1+\cos \theta \sin \varphi T_2-\sin \theta T_3)}_{=\hat{n}_1\cdot \vec {T}}\nonumber \\{} & {} +X_\mu ^2\underbrace{(-\sin \varphi T_1+\cos \varphi T_2)}_{=\hat{n}_2\cdot \vec {T}}\nonumber \\{} & {} +A_{\mu }^1\underbrace{(\sin \theta \cos \varphi T_1+\sin \theta \sin \varphi T_2+\cos \theta T_3)}_{=\hat{n}^1\cdot \vec {T}}\nonumber \\{} & {} +\underbrace{\frac{1}{g}(\sin \varphi \ \partial _\mu \theta \ T_1-\cos \varphi \ \partial _\mu \theta \ T_2-\partial _\mu \varphi \ T_3)}_{=-\big (C_\mu ^{(n^1)}\hat{n}^1+\frac{1}{g}\hat{n}^1\times \partial _\mu \hat{n}^1\big )\cdot \vec {T}}, \nonumber \\ \end{aligned}$$

(104)

where,

$$\begin{aligned} C_\mu ^{(n^1)}= & {} -\frac{1}{g}\hat{n}_1\cdot \partial _\mu \hat{n}_2 \nonumber \\= & {} \frac{1}{g}(1+\cos \theta )\partial _\mu \varphi -\frac{1}{g}\partial _\mu \varphi =\frac{1}{g}\cos \theta \partial _\mu \varphi \nonumber \\= & {} C_\mu ^{(m^1)}+C_\mu ^{(v^1)}, \end{aligned}$$

(105)

is the sum of magnetic potentials of a monopole and a vortex. In the following we discuss that the above vector field shows the vector field of a chain. Our reasoning is based on the flux we would get from this vector field and the contribution of the filed in the Lagrangian which causes the interaction between the monopoles and vortices. Using the above equation in the definition of vector field $\vec {A}_\mu ^{VU}$,

$$\begin{aligned} \vec {A}_\mu ^{VU}=\underbrace{(A_\mu ^1- C_\mu ^{(n^1)})}_{=A_\mu ^{(n^1)}}\hat{n}^1-\frac{1}{g}\hat{n}^1\times \partial _\mu \hat{n}^1+\underbrace{X_\mu ^1 \hat{n}_1+X_\mu ^2\hat{n}_2}_{=\vec {W}_\mu ^1}. \end{aligned}$$

(106)

Given the fact that the magnetic objects of our problem are static, the last line of Eq. (104) can be written as follows,

$$\begin{aligned} \frac{1}{g}\frac{1}{r}\left( \sin \varphi \ \hat{\theta }\ T_1-\cos \varphi \ \hat{\theta }\ T_2-\frac{1}{\sin \theta }\hat{\varphi }\ T_3\right) , \end{aligned}$$

(107)

where only the last term has a non-zero contribution to the magnetic flux,

$$\begin{aligned} \Phi ^{\text {flux}}&=-\int _0^{2\pi }(r\sin \theta \ d\varphi \ \hat{\varphi })\cdot \left( \frac{1}{g}\frac{1}{r\sin \theta }\hat{\varphi }\ T_3\right) \nonumber \\&=-\frac{2\pi }{g}T_3. \end{aligned}$$

(108)

The above magnetic flux is interpreted as the sum of the magnetic flux of a monopole attached to a Dirac string at $\theta =0$, $(-\frac{4\pi }{g}T_3)$, and the magnetic flux of a vortex, $(\frac{2\pi }{g}T_3)$.

Using $\vec {A}_\mu ^{VU}$ of Eq. (106), we get the non-Abelian field strength tensor,

$$\begin{aligned} \vec {F}_{\mu \nu }^1=\underbrace{(F_{\mu \nu }^{( n^1)} +H_{\mu \nu }^{(n^1)})\hat{n}^1}_{=\hat{F}_{\mu \nu }^1}+\vec {G}_{\mu \nu }^1+g\vec {W}_\mu ^1\times \vec {W}_\nu ^1, \end{aligned}$$

(109)

where,

$$\begin{aligned} F_{\mu \nu }^{( n^1)}&=\partial _\mu A_\nu ^{( n^1)}-\partial _\nu A_\mu ^{( n^1)}, \nonumber \\ H_{\mu \nu }^{( n^1)}&=\partial _\mu C_\nu ^{( n^1)}-\partial _\nu C_\mu ^{( n^1)}\nonumber \\&=\partial _\mu (C_\nu ^{( m^1)}+C_\nu ^{(v^1)})-\partial _\nu (C_\mu ^{( m^1)}+C_\mu ^{(v^1)}),\nonumber \\ \vec {G}_{\mu \nu }^1&=\hat{D}_\mu \vec {W}_\nu ^1-\hat{D}_\nu \vec {W}_\mu ^1, \quad \hat{D}_\mu =\partial _\mu +g\hat{A}_\mu ^{( n^1)}. \end{aligned}$$

(110)

It can be easily confirmed that $H_{\mu \nu }^{( n^1)}$ indicates the field strength of a magnetic monopole sitting at the origin along with a Dirac string at $\theta =0$ carrying a magnetic flux equal to $-\frac{4\pi }{g}T_3$; plus a line vortex carrying a magnetic flux equal to $\frac{2\pi }{g}T_3$ extending on the z-axis. The flux carried by the vortex is equal to the half of the flux of a Dirac string, so that the contribution of one Dirac string can be considered to be equivalent to the contribution of two vortices. At $\theta =0$, half of the Dirac string flux is canceled with the vortex flux located in the positive z-axis direction, and a line vortex carrying a magnetic flux equal to $\frac{-2\pi }{g}T_3$ remains in the positive z-axis. Finally, we have a monopole at the origin, which is connected to two line vortices, a line vortex carrying a magnetic flux equal to $\frac{-2\pi }{g}T_3$ on the positive z-axis and a line vortex carrying a magnetic flux equal to $\frac{2\pi }{g}T_3$ on the negative z-axis, which form a chain (see Fig. 7). We have reached to the definition of a chain defined by lattice people who claims that the magnetic flux of monopoles at given time is concentrated in a tube-like structure, and the chain is shorthand terminology for an ordering of monopoles and antimonopoles, in which half the magnetic flux from a monopole ends on the next antimonopole in the chain, while the other half ends on the previous antimonopole. In fact, applying the first gauge transformations shown in Eqs. (24), (40) and (48), we have found monopoles for the three subgroups in Eqs. (27), (43) and (51) that attached to their own Dirac strings at $\theta =0$. The appearance of Dirac strings confirms the existence of the antimonopoles at infinity. This is in agreement with figure (3) of the paper [30], which is shown in Fig. 6 in this article. Performing the second gauge transformation, the vortex appears along with the monopole which is attached to its Dirac sting. In other words, applying two successive gauge transformations like Eq. (100) gives us all the defects we need.

Fig. 6

Illustration of the connection between the Dirac string shown in plot (a); and the center vortex shown in plot (b) [30]. The interpretation of a chain represented in Fig. 7 is the same as this figure where two vortex lines enter a monopole

Fig. 7

Appearance of a chain of monopole and vortex for the first subgroup

Repeating the same procedure for the second and the third SU(2) subgroups, the gauge transformations and the gauge fields are obtained as follows.

Second subgroup: For this subgroup:

$$\begin{aligned} V'U'=\left( \begin{array}{ccc} 1&{}0&{}0\\ 0&{}\cos \frac{\theta }{2}e^{-i\frac{\varphi }{2}}&{}-\sin \frac{\theta }{2}e^{-i\frac{\varphi }{2}}\\ 0&{}\sin \frac{\theta }{2}e^{i\frac{\varphi }{2}}&{}\cos \frac{\theta }{2}e^{i\frac{\varphi }{2}}\\ \end{array}\right) . \end{aligned}$$

(111)

And,

$$\begin{aligned} \vec {A}_\mu ^{V'U'}=\underbrace{(A_\mu ^2- C_\mu ^{(n^2)})}_{=A_\mu ^{(n^2)}}\hat{n}^2-\frac{1}{g}\hat{n}^2\times \partial _\mu \hat{n}^2+\underbrace{X_\mu ^6 \hat{n}_6+X_\mu ^7\hat{n}_7}_{=\vec {W}_\mu ^2}, \end{aligned}$$

(112)

where,

$$\begin{aligned} C_\mu ^{(n^2)}= & {} -\frac{1}{g}\hat{n}_6\cdot \partial _\mu \hat{n}_7=\frac{1}{g}\cos \theta \partial _\mu \varphi \nonumber \\= & {} C_\mu ^{(m^2)}+C_\mu ^{(v^2)}, \end{aligned}$$

(113)

shows the sum of magnetic potentials of a monopole and a vortex. The components of $\hat{n}_6, \hat{n}_7$ and $\hat{n}^2$ are given in Appendix B. Using $\vec {A}_\mu ^{V'U'}$ of Eq. (112), we get the non-Abelian field strength tensor,

$$\begin{aligned} \vec {F}_{\mu \nu }^2=\underbrace{(F_{\mu \nu }^{( n^2)} +H_{\mu \nu }^{(n^2)})\hat{n}^2}_{=\hat{F}_{\mu \nu }^2}+\vec {G}_{\mu \nu }^2+g\vec {W}_\mu ^2\times \vec {W}_\nu ^2, \end{aligned}$$

(114)

where,

$$\begin{aligned} F_{\mu \nu }^{( n^2)}&=\partial _\mu A_\nu ^{( n^2)}-\partial _\nu A_\mu ^{( n^2)}, \nonumber \\ H_{\mu \nu }^{( n^2)}&=\partial _\mu C_\nu ^{( n^2)}-\partial _\nu C_\mu ^{( n^2)}\nonumber \\&=\partial _\mu (C_\nu ^{( m^2)}+C_\nu ^{(v^2)})-\partial _\nu (C_\mu ^{( m^2)}+C_\mu ^{(v^2)}),\nonumber \\ \vec {G}_{\mu \nu }^2&=\hat{D}_\mu \vec {W}_\nu ^2-\hat{D}_\nu \vec {W}_\mu ^2, \quad \hat{D}_\mu =\partial _\mu +g\hat{A}_\mu ^{( n^2)} . \end{aligned}$$

(115)

$H_{\mu \nu }^{( n^2)}$ indicates the field strength of a magnetic monopole sitting at the origin along with a Dirac string at $\theta =0$ carrying a magnetic flux equal to $-\frac{4\pi }{g}t_3$; plus a line vortex carrying a magnetic flux equal to $\frac{2\pi }{g}t_3$ extending on the z-axis. This combination makes a chain as explained for the first subgroup (see Fig. 8).

Fig. 8

Appearance of a chain of monopole and vortex for the second subgroup

Third subgroup: For this subgroup:

$$\begin{aligned} V''U''=\left( \begin{array}{ccc} \cos \frac{\theta }{2}e^{i\frac{\varphi }{2}}&{}0&{}\sin \frac{\theta }{2}e^{i\frac{\varphi }{2}}\\ 0&{}1&{}0\\ -\sin \frac{\theta }{2}e^{-i\frac{\varphi }{2}}&{}0&{}\cos \frac{\theta }{2}e^{-i\frac{\varphi }{2}}\\ \end{array}\right) . \end{aligned}$$

(116)

Calculations similar to what is done for the first subgroup leads to the following transformed gauge field,

$$\begin{aligned} \vec {A}_\mu ^{V''U''}=\underbrace{(A_\mu ^3- C_\mu ^{(n^3)})}_{A_\mu ^{(n^3)}}\hat{n}^3-\frac{1}{g}\hat{n}^3\times \partial _\mu \hat{n}^3+\underbrace{X_\mu ^4 \hat{n}_4-X_\mu ^ 5\hat{n}_5}_{\vec {W}_\mu ^3}, \end{aligned}$$

(117)

where,

$$\begin{aligned} \begin{aligned} C_\mu ^{(n^3)}&=-\frac{1}{g}\hat{n}_4\cdot \partial _\mu (-\hat{n}_5)=\frac{1}{g}\cos \theta \partial _\mu \varphi \\&=C_\mu ^{(m^3)}+C_\mu ^{(v^3)}, \end{aligned} \end{aligned}$$

(118)

indicates the sum of magnetic potentials of a monopole and a vortex. The components of $\hat{n}_4, -\hat{n}_5$ and $\hat{n}^3$ are given in Appendix B. Using $\vec {A}_\mu ^{V''U''}$ of Eq. (117), we get the non-Abelian field strength tensor,

$$\begin{aligned} \vec {F}_{\mu \nu }^3=\underbrace{(F_{\mu \nu }^{( n^3)} +H_{\mu \nu }^{(n^3)})\hat{n}^3}_{=\hat{F}_{\mu \nu }^3}+\vec {G}_{\mu \nu }^3+g\vec {W}_\mu ^3\times \vec {W}_\nu ^3, \end{aligned}$$

(119)

where,

$$\begin{aligned} F_{\mu \nu }^{( n^3)}= & {} \partial _\mu A_\nu ^{( n^3)}-\partial _\nu A_\mu ^{( n^3)},\nonumber \\ H_{\mu \nu }^{( n^3)}= & {} \partial _\mu C_\nu ^{( n^3)}-\partial _\nu C_\mu ^{( n^3)}\nonumber \\= & {} \partial _\mu (C_\nu ^{( m^3)}+C_\nu ^{(v^3)})-\partial _\nu (C_\mu ^{( m^3)}+C_\mu ^{(v^3)}),\nonumber \\ \vec {G}_{\mu \nu }^3= & {} \hat{D}_\mu \vec {W}_\nu ^3-\hat{D}_\nu \vec {W}_\mu ^3, \quad \hat{D}_\mu =\partial _\mu +g\hat{A}_\mu ^{( n^3)}.\nonumber \\ \end{aligned}$$

(120)

$H_{\mu \nu }^{( n^3)}$ indicates the field strength of a magnetic monopole sitting at the origin along with a Dirac string at $\theta =0$ carrying a magnetic flux equal to $-\frac{4\pi }{g}t'_3$; plus a line vortex carrying a magnetic flux equal to $\frac{2\pi }{g}t'_3$ extending on the z-axis, forming a chain (see Fig. 9).

Fig. 9

Appearance of a chain of monopole and vortex for the third subgroup

The corresponding magnetic fluxes for the second and the third subgroups are obtained as follows.

$$\begin{aligned} \Phi ^{\text {flux}}= & {} -\frac{1}{g}\int _0^{2\pi }(r\sin \theta \ d\varphi \ \hat{\varphi })\cdot \frac{1}{r\sin \theta }\hat{\varphi }\ t_3 \nonumber \\= & {} -\frac{2\pi }{g}t_3. \end{aligned}$$

(121)

The above magnetic flux is interpreted as the sum of the magnetic flux of a monopole attached to a Dirac string at $\theta =0$, $(-\frac{4\pi }{g}t_3)$, and the magnetic flux of a vortex, $(\frac{2\pi }{g}t_3)$.

For the third subgroup,

$$\begin{aligned} \Phi ^{\text {flux}}= & {} -\frac{1}{g}\int _0^{2\pi }(r\sin \theta \ d\varphi \ \hat{\varphi })\cdot \frac{1}{r\sin \theta }\hat{\varphi }\ t'_3\nonumber \\= & {} -\frac{2\pi }{g}\ t'_3. \end{aligned}$$

(122)

The above magnetic flux is interpreted as the sum of the magnetic flux of a monopole attached to a Dirac string at $\theta =0$, $(-\frac{4\pi }{g}t'_3)$, and the magnetic flux of a vortex, $(\frac{2\pi }{g}t'_3)$.

Finally, we reach to the point that we can gather all the information of the SU(2) subgroups to write a gauge field describing the chains in SU(3) group,

$$\begin{aligned} \vec {A}_\mu= & {} \sum _{p=1}^3 \frac{2}{3}(A_\mu ^{(n^p)}\hat{n}^p-\frac{1}{g}\hat{n}^p\times \partial _\mu \hat{n}^p)+\vec {W}_\mu ^p\nonumber \\= & {} \sum _{p=1}^3 \left( \frac{2}{3}\hat{A}_\mu ^p+\vec {W}_\mu ^p\right) \end{aligned}$$

(123)

where p changes from one to three which represents the three SU(2) subgroups. The SU(3) field strength tensor is obtained from the above gauge filed,

$$\begin{aligned} \vec {F}_{\mu \nu }=\sum _p\left( \frac{2}{3}\hat{F}_{\mu \nu }^p+\hat{D}_\mu ^p\vec {W}_\nu ^p-\hat{D}_\nu ^p\vec {W}_\mu ^p\right) +g\sum _{p,q}\vec {W}_\mu ^p\times \vec {W}_\nu ^q,\nonumber \\ \end{aligned}$$

(124)

where,

$$\begin{aligned} \hat{F}_{\mu \nu }^p=\big (\partial _\mu (A_\nu ^{(n^p)}+C_\nu ^{(n^p)})-\partial _\nu (A_\mu ^{(n^p)}+C_\mu ^{(n^p)})\big )\hat{n}^p, \end{aligned}$$

(125)

and

$$\begin{aligned} \hat{D}_\mu ^p=\partial _\mu +g\hat{A}_\mu ^p\times . \end{aligned}$$

(126)

By performing calculations similar to what was done for monopoles in Sect. 3, it is possible to rewrite the magnetic potentials and the field strength tensor of the chains in SU(3) gauge group in terms of their counterparts in SU(2) subgroups as the following. There exist two types of correlations of monopole and vortex, as shown in Fig. 10.

(127)

(128)

(129)

Fig. 10

Appearance of $T_3$-like chain and $T_8$-like chain in SU(3) gauge group

Using the above equations, one can argue that there are two types of correlated monopoles and vortices: $H_{\mu \nu }$ indicates the field strength of a magnetic monopole sitting at the origin attached to a vortex line carrying a flux equal to $-\frac{2\pi }{g}T_3$ at $\theta =0$; and a line vortex carrying a flux equal to $\frac{2\pi }{g}T_3$ at $\theta =\pi$ (see Fig. 10a) and $H'_{\mu \nu }$ indicates the field strength of a magnetic monopole sitting at the origin attached to a line vortex carrying a flux equal to $-\frac{2\pi }{g}T_8$ at $\theta =0$ and a line vortex carrying a flux equal to $\frac{2\pi }{g}T_8$ at $\theta =\pi$ (Fig. 10b)

Like monopoles, Lagrangian of chains may be written in two alternative ways: using magnetic potentials of SU(2) subgroups $C_\mu ^{(n^1)}, C_\mu ^{(n^2)}$ and $C_\mu ^{(n^3)}$, or SU(3) magnetic potentials $C_\mu$ and $C'_\mu$. The chain Lagrangian written with the SU(2) subgroups is similar to Eq. (72), the Lagrangian with monopole defects. The difference is that the $\hat{n}_a$ color frame is used for chains instead of $\hat{m}_a$ which is used for monopole defects; and the vortex potential $C_\mu ^{(v^a)}$ is added to the monopole potential $C_\mu ^{(m^a)}$. Since we are interested in studying the interactions between the monopoles and the vortices, we only look at those terms of the Lagrangian which are of the order of two with respect to $C_\mu ^{(n^a)}$. Those terms include the first and the second terms of the Lagrangian of the type (72) written for chains.

$$\begin{aligned} \mathcal {L}= & {} -\frac{1}{6}\sum _{p=1}^3\Big [\partial _\mu (A_\nu ^{(n^p)}+C_\nu ^{(m^p)}+C_\nu ^{(v^p)})\nonumber \\{} & {} -\partial _\nu (A_\mu ^{(n^p)}+C_\mu ^{(m^p)}+C_\mu ^{(v^p)})\Big ]^2\nonumber \\{} & {} -\frac{1}{4}\bigg \{\Big [\big (\partial _\mu X_\nu ^1-g(A_\mu ^{(n^1)}+C_\mu ^{(m^1)}+C_\mu ^{(v^1)})X_\nu ^2\big )-(\mu \leftrightarrow \nu )\Big ]^2\nonumber \\{} & {} +\Big [\big (\partial _\mu X_\nu ^2+g(A_\mu ^{(n^1)}+C_\mu ^{(m^1)}+C_\mu ^{(v^1)})X_\nu ^1\big )-(\mu \leftrightarrow \nu )\Big ]^2\nonumber \\{} & {} +\Big [\big (\partial _\mu X_\nu ^6-g(A_\mu ^{(n^2)}+C_\mu ^{(m^2)}+C_\mu ^{(v^2)})X_\nu ^7\big )-(\mu \leftrightarrow \nu )\Big ]^2\nonumber \\{} & {} +\Big [\big (\partial _\mu X_\nu ^7+g(A_\mu ^{(n^2)}+C_\mu ^{(m^2)}+C_\mu ^{(v^2)})X_\nu ^6\big )-(\mu \leftrightarrow \nu )\Big ]^2\nonumber \\{} & {} +\Big [\big (\partial _\mu X_\nu ^4-g(A_\mu ^{(n^3)}+C_\mu ^{(m^3)}+C_\mu ^{(v^3)})X_\nu ^5\big )-(\mu \leftrightarrow \nu )\Big ]^2\nonumber \\{} & {} +\Big [\big (\partial _\mu X_\nu ^5+g(A_\mu ^{(n^3)}+C_\mu ^{(m^3)}+C_\mu ^{(v^3)})X_\nu ^4\big )-(\mu \leftrightarrow \nu )\Big ]^2\bigg \}\nonumber \\{} & {} +\text {other interactions} \end{aligned}$$

(130)

The first term of the above Lagrangian contains the kinetic energy of the monopoles and vortices. It is clear from the Lagrangian that the monopoles of each subgroup interact with the monopoles and vortices of the same subgroup via the off-diagonal components of the gauge field. If we rewrite the SU(2) subgroups potentials $C_\mu ^{(n^1)}, C_\mu ^{(n^2)}$ and $C_\mu ^{(n^3)}$ in terms of their counterparts in SU(3) gauge group: $C_\mu$ and $C'_\mu$, and the same for $A_\mu ^{(n^1)}, A_\mu ^{(n^2)}$ and $A_\mu ^{(n^3)}$ in terms of $A_\mu$ and $A'_\mu$; we would get the following statements,

$$\begin{aligned} C_\mu ^{(n^1)}= & {} C_\mu ,\quad A_\mu ^{(n^1)}=A_\mu \nonumber \\ C_\mu ^{(n^2)}= & {} -\frac{1}{2}C_\mu +\frac{\sqrt{3}}{2}C'_\mu , \qquad A_\mu ^{(n^2)}=-\frac{1}{2}A_\mu +\frac{\sqrt{3}}{2}A'_\mu ,\nonumber \\ C_\mu ^{(n^3)}= & {} -\frac{1}{2}C_\mu -\frac{\sqrt{3}}{2}C'_\mu , \qquad A_\mu ^{(n^3)}=-\frac{1}{2}A_\mu -\frac{\sqrt{3}}{2}A'_\mu .\nonumber \\ \end{aligned}$$

(131)

The last three lines of the Lagrangian Eq. (130) contains the interaction between chains. We will use Eq. (131) to study the chains interactions.

The terms including the interaction between chains are as the following,

• $-\frac{1}{2}g^2C_\mu ^{(n^1)}C_\mu ^{(n^1)}\left( X_\nu ^2X_\nu ^2+X_\nu ^1X_\nu ^1\right)$

• $+\frac{1}{2}g^2C_\mu ^{(n^1)}C_\nu ^{(n^1)}\left( X_\mu ^2X_\nu ^2+X_\mu ^1X_\nu ^1\right)$

• $-\frac{1}{2}g^2C_\mu ^{(n^2)}C_\mu ^{(n^2)}\left( X_\nu ^7X_\nu ^7+X_\nu ^6X_\nu ^6\right)$

• $+\frac{1}{2}g^2C_\mu ^{(n^2)}C_\nu ^{(n^2)}\left( X_\mu ^7X_\nu ^7+X_\mu ^6X_\nu ^6\right)$

• $-\frac{1}{2}g^2C_\mu ^{(n^3)}C_\mu ^{(n^3)}\left( X_\nu ^5X_\nu ^5+X_\nu ^4X_\nu ^4\right)$

• $+\frac{1}{2}g^2C_\mu ^{(n^3)}C_\nu ^{(n^3)}\left( X_\mu ^5X_\nu ^5+X_\mu ^4X_\nu ^4\right)$

The above equations show that chains of each subgroup interact each other via the the off-diagonal components of their own subgroup. This happens because the Lagrangian is written for three independent SU(2) subgroups. However, to get the correct SU(3) Lagrangian, one should consider that only two of the SU(2) subgroup chains are independent and $C_{\mu }^{( n^1) } +C_{\mu }^{( n^2) }+C_{\mu }^{( n^3) }=0$. Therefore one of them can be written in terms of the other two. For example, one can choose $C_\mu ^{(n^1)}=-(C_{\mu }^{( n^2) }+C_{\mu }^{( n^3) })$ and then the first two lines of the above interactions can be rewritten as follows,

• $-\frac{1}{2}g^2C_\mu ^{(n^2)}C_\mu ^{(n^2)}\left( X_\nu ^2X_\nu ^2+X_\nu ^1X_\nu ^1\right)$

• $-\frac{1}{2}g^2C_\mu ^{(n^3)}C_\mu ^{(n^3)}\left( X_\nu ^2X_\nu ^2+X_\nu ^1X_\nu ^1\right)$

• $-g^2C_\mu ^{(n^2)}C_\mu ^{(n^3)}\left( X_\nu ^2X_\nu ^2+X_\nu ^1X_\nu ^1\right)$

• $+\frac{1}{2}g^2C_\mu ^{(n^2)}C_\nu ^{(n^2)}\left( X_\mu ^2X_\nu ^2+X_\mu ^1X_\nu ^1\right)$

• $+\frac{1}{2}g^2C_\mu ^{(n^3)}C_\nu ^{(n^3)}\left( X_\mu ^2X_\nu ^2+X_\mu ^1X_\nu ^1\right)$

• $+g^2C_\mu ^{(n^2)}C_\nu ^{(n^3)}\left( X_\mu ^2X_\nu ^2+X_\mu ^1X_\nu ^1\right)$

The third and sixth lines of the above equations show that the chains of the second and the third subgroups interact via the off-diagonal components of the first subgroup, as well as the off-diagonal components of their own subgroup. Therefore, if we choose $C_\mu ^{(n^2)}$ and $C_\mu ^{(n^3)}$ magnetic potentials to represent the independent chains, their interactions can be summarized as the following,

• $-\frac{1}{2}g^2C_\mu ^{(n^2)}C_\mu ^{(n^2)}\left( X_\nu ^7X_\nu ^7+X_\nu ^6X_\nu ^6+X_\nu ^2X_\nu ^2+X_\nu ^1X_\nu ^1\right)$

• $+\frac{1}{2}g^2C_\mu ^{(n^2)}C_\nu ^{(n^2)}\left( X_\mu ^7X_\nu ^7+X_\mu ^6X_\nu ^6+X_\mu ^2X_\nu ^2+X_\mu ^1X_\nu ^1\right)$

• $-\frac{1}{2}g^2C_\mu ^{(n^3)}C_\mu ^{(n^3)}\left( X_\nu ^5X_\nu ^5+X_\nu ^4X_\nu ^4+X_\nu ^2X_\nu ^2+X_\nu ^1X_\nu ^1\right)$

• $+\frac{1}{2}g^2C_\mu ^{(n^3)}C_\nu ^{(n^3)}\left( X_\mu ^5X_\nu ^5+X_\mu ^4X_\nu ^4+X_\mu ^2X_\nu ^2+X_\mu ^1X_\nu ^1\right)$

• $-g^2C_\mu ^{(n^2)}C_\mu ^{(n^3)}\left( X_\nu ^2X_\nu ^2+X_\nu ^1X_\nu ^1\right)$

• $+g^2C_\mu ^{(n^2)}C_\nu ^{(n^3)}\left( X_\mu ^2X_\nu ^2+X_\mu ^1X_\nu ^1\right)$

The interesting point is that the interaction between the chains of the second and the third subgroups are done via the off-diagonal components of the first subgroup while the interaction between two chains of the same subgroup is done via their own off-diagonal components as well as the first subgroup one’s.

Using Eq. (131) in Eq. (130), the Lagrangian density in terms of the SU(3) vector fields is as the following,

$$\begin{aligned} \begin{aligned} \mathcal {L}&=-\frac{1}{4}\Big [\partial _\mu (A_\nu +C_\nu )-\partial _\nu (A_\mu +C_\mu )\Big ]^2\\&\quad -\frac{1}{4}\Big [\partial _\mu (A'_\nu +C'_\nu )-\partial _\nu (A'_\mu +C'_\mu )\Big ]^2\\&\quad -\frac{1}{2}\left| \left[ \partial _\mu \!+\!ig(A_\mu \!+\!C_\mu ) \right] R_\nu -\left[ \partial _\nu +ig(A_\nu +C_\nu ) \right] R_\mu \right| ^2\\&\quad -\frac{1}{2}\left| \left[ \partial _\mu -\frac{1}{2}ig(A_\mu +C_\mu )+\frac{\sqrt{3}}{2}ig(A_\mu ^\prime +C_\mu ^\prime ) \right] B_\nu \right. \\&\quad \left. -\left[ \partial _\nu -\frac{1}{2}ig(A_\nu +C_\nu )+\frac{\sqrt{3}}{2}ig(A_\nu ^\prime +C_\nu ^\prime ) \right] B_\mu \right| ^2\\&\quad -\frac{1}{2}\left| \left[ \partial _\mu -\frac{1}{2}ig(A_\mu +C_\mu )-\frac{\sqrt{3}}{2}ig(A_\mu ^\prime +C_\mu ^\prime ) \right] Y_\nu \right. \\&\quad \left. -\left[ \partial _\nu -\frac{1}{2}ig(A_\nu +C_\nu )-\frac{\sqrt{3}}{2}ig(A_\nu ^\prime +C_\nu ^\prime ) \right] Y_\mu \right| ^2\\&\quad +\text {other interactions}. \end{aligned} \end{aligned}$$

(132)

Where,

$$\begin{aligned} R_\mu =\frac{X_\mu ^1+iX_\mu ^2}{\sqrt{2}},\quad B_\mu =\frac{X_\mu ^6+iX_\mu ^7}{\sqrt{2}},\quad Y_\mu =\frac{X_\mu ^4-iX_\mu ^5}{\sqrt{2}} \end{aligned}$$

(133)

The above Lagrangian describes the two different types of monopole and vortex correlations for the SU(3) gauge group. $C_\mu$ and $C'_\mu$ represent two types of chains in SU(3) gauge group and from the above Lagrangian it is understood that these chains interact with each other via the off-diagonal components of the gauge field. We hope to use these information to get an effective Lagrangian to study the condensation of vortices or chains for both SU(2) and SU(3) and the confinement.

6 Conclusions

Many proposals have been suggested to explain confinement which is one of the most challenging problems of QCD. Topological models containing magnetic defects, have been able to interpret some of the results given by lattice gauge theories. Monopoles and vortices are among the popular candidates that explain some expected characteristic of the quark-antiquark potential. However, none of them have predicted all the features of quark confinement. Chains of monopoles and vortices which somehow are observed in lattice calculations, attract people’s attention to work more on phenomenological models to compensate the shortcomings of other candidates. In Ref. [10], we discussed about the chains for the SU(2) gauge group by using two successive gauge transformations proposed in [22]. As a result of gauge transformations, we defined the local color frames which involve the monopole, the vortex or the chains. As an extension of our previous work, in this article we have studied the monopoles, vortices and especially chains in SU(3) using its SU(2) subgroups. Comparison is done with Cho decomposition method for SU(3) monopoles, and in order to get the correct SU(3) defects, interactions between their SU(2) counterparts are investigated. Using SU(2) subgroups not only makes the calculations easier but also reveals the possible relations between the SU(3) defects and their counterparts in the SU(2) subgroups. Correlations between monopoles and vortices are discussed for both groups. One of the task that can be done for the next step is finding an effective Lagrangian to obtain the confinement potential. It is reasonable to first investigate the condensation of vortices through an effective Lagrangian which is in progress.

Appendices

Appendix A: The SU(3) generators and root vectors

The eight Gell–Mann matrices are:

$$\begin{aligned} \lambda _1= & {} \left( \begin{array}{ccc} 0&{}1&{}0\\ 1&{}0&{}0\\ 0&{}0&{}0 \end{array}\right) , \ \ \lambda _2=\left( \begin{array}{ccc} 0&{}-i&{}0\\ i&{}0&{}0\\ 0&{}0&{}0 \end{array}\right) ,\\ \lambda _4= & {} \left( \begin{array}{ccc} 0&{}0&{}1\\ 0&{}0&{}0\\ 1&{}0&{}0 \end{array}\right) , \ \ \lambda _5=\left( \begin{array}{ccc} 0&{}0&{}-i\\ 0&{}0&{}0\\ i&{}0&{}0 \end{array}\right) ,\\ \lambda _6= & {} \left( \begin{array}{ccc} 0&{}0&{}0\\ 0&{}0&{}1\\ 0&{}1&{}0 \end{array}\right) , \ \ \lambda _7=\left( \begin{array}{ccc} 0&{}0&{}0\\ 0&{}0&{}-i\\ 0&{}i&{}0 \end{array}\right) ,\\ \lambda _3= & {} \left( \begin{array}{ccc} 1&{}0&{}0\\ 0&{}-1&{}0\\ 0&{}0&{}0 \end{array}\right) ,\ \ \lambda _8=\frac{1}{\sqrt{3}}\left( \begin{array}{ccc} 1&{}0&{}0\\ 0&{}1&{}0\\ 0&{}0&{}-2 \end{array}\right) . \end{aligned}$$

$T_a=\frac{\lambda _a}{2}$ are the generators of the SU(3) group. $T_3$ and $T_8$ are two diagonal generators.

The components of root vectors of SU(3) group are,

$$\begin{aligned} w_1=(1,0), \quad w_2=\left( -\frac{1}{2},-\frac{\sqrt{3}}{2}\right) , \quad w_3=\left( -\frac{1}{2},\frac{\sqrt{3}}{2}\right) . \end{aligned}$$

Appendix B: The components of the local color frames

The components of the local color frame containing monopoles for the second and the third subgroups are as follows.

$$\begin{aligned} \hat{m}_6\cdot \vec {t}= & {} \left( -\sin ^2\frac{\theta }{2}+\cos ^2\frac{\theta }{2}\cos 2\varphi \right) T_6+\cos ^2\frac{\theta }{2}\sin 2\varphi T_7\\{} & {} -\sin \theta \cos \varphi t_3,\\ \hat{m}_7\cdot \vec {t}= & {} -\cos ^2\frac{\theta }{2}\sin 2\varphi T_6+\left( \sin ^2\frac{\theta }{2}+\cos ^2\frac{\theta }{2}\cos 2\varphi \right) T_7\\{} & {} +\sin \theta \sin \varphi t_3,\\ \hat{m}^2\cdot \vec {t}= & {} \sin \theta \cos \varphi T_6+\sin \theta \sin \varphi T_7+\cos \theta t_3,\\ \hat{m}_4\cdot \vec {t}'= & {} \left( -\sin ^2\frac{\theta }{2}+\cos ^2\frac{\theta }{2}\cos 2\varphi \right) T_4\\{} & {} +\cos ^2\frac{\theta }{2}\sin 2\varphi (- T_5)-\sin \theta \cos \varphi t_3^\prime ,\\ -\hat{m}_5\cdot \vec {t}'= & {} -\cos ^2\frac{\theta }{2}\sin 2\varphi T_4\\{} & {} +\left( \sin ^2\frac{\theta }{2}+\cos ^2\frac{\theta }{2}\cos 2\varphi \right) (-T_5)+\sin \theta \sin \varphi t_3^\prime ,\\ \hat{m}^3\cdot \vec {t}'= & {} \sin \theta \cos \varphi T_4+\sin \theta \sin \varphi (-T_5)+\cos \theta t_3^\prime .\\ \end{aligned}$$

The components of the local color frame containing vortices for the second and the third subgroups are as follows.

$$\begin{aligned} \hat{n}'_6\cdot \vec {t}= & {} \cos \varphi T_6-\sin \varphi T_7,\\ \hat{n}'_7\cdot \vec {t}= & {} \sin \varphi T_6+\cos \varphi T_7,\\ \hat{n}'^2\cdot \vec {t}= & {} -\frac{1}{2}T_3+\frac{\sqrt{3}}{2}T_8,\\ \hat{n}'_4\cdot \vec {t}'= & {} \cos \varphi T_4-\sin \varphi (-T_5),\\ -\hat{n}'_5\cdot \vec {t}'= & {} \sin \varphi T_4+\cos \varphi (-T_5),\\ \hat{n}'^3\cdot \vec {t}'= & {} -\frac{1}{2}T_3-\frac{\sqrt{3}}{2}T_8. \end{aligned}$$

The components of the local color frame containing correlated monopoles and vortices for the second and the third subgroups are as follows.

$$\begin{aligned} \begin{aligned} \hat{n}_6\cdot \vec {t}&=\cos \theta \cos \varphi T_6+\cos \theta \sin \varphi T_7-\sin \theta t_3,\\ \hat{n}_7\cdot \vec {t}&=-\sin \varphi T_6+\cos \varphi T_7,\\ \hat{n}^2\cdot \vec {t}&=\sin \theta \cos \varphi T_6+\sin \theta \sin \varphi T_7+\cos \theta t_3,\\ \hat{n}_4\cdot \vec {t}'&=\cos \theta \cos \varphi T_4+\cos \theta \sin \varphi (-T_5)-\sin \theta t_3^\prime ,\\ -\hat{n}_5\cdot \vec {t}'&=-\sin \varphi T_4+\cos \varphi (-T_5),\\ \hat{n}^3\cdot \vec {t}'&=\sin \theta \cos \varphi T_4+\sin \theta \sin \varphi (-T_5)+\cos \theta t_3^\prime . \end{aligned} \end{aligned}$$