Introducing vortices in the continuum using direct and indirect methods

Inspired by direct and indirect maximal center gauge methods which confirm the existence of vortices in lattice calculations and by using the connection formalism, we show that under some appropriate gauge transformations vortices and chains appear in the QCD vacuum of the continuum limit. In the direct method, by applying center gauge transformation and \textquotedblleft center projection,\textquotedblright QCD is reduced to a gauge theory including vortices, which corresponds to the non-trivial first homotopy group $\Pi_1\left( \text{SO}(3)\right) =Z_2.$ On the other hand, using the indirect method, in addition to the center gauge transformation and \textquotedblleft center projection,\textquotedblright an initial step called Abelian gauge transformation and then Abelian projection are applied. Therefore, instead of single vortices, chains that contain monopoles and vortices appear in the theory.

Inspired by direct and indirect maximal center gauge methods which confirm the existence of vortices in lattice calculations and by using the connection formalism, we show that under some appropriate gauge transformations vortices and chains appear in the QCD vacuum of the continuum limit. In the direct method, by applying center gauge transformation and "center projection,"QCD is reduced to a gauge theory including vortices, which corresponds to the non-trivial first homotopy group Π1 (SO(3)) = Z2. On the other hand, using the indirect method, in addition to the center gauge transformation and "center projection,"an initial step called Abelian gauge transformation and then Abelian projection are applied. Therefore, instead of single vortices, chains that contain monopoles and vortices appear in the theory.

I. INTRODUCTION
Quantum chromodynamics is the non-Abelian gauge theory of the strong interaction which describes the hadrons in terms of quarks and gluons. There are many books that discuss QCD; for instance, see Refs. [1,2]. However, quarks have not been observed as isolated particles in the real world. Only hadrons (mesons and baryons) are observed as some color singlet combinations. This experimental fact reflects the confinement mechanism as one of the most controversial unsolved issues in particle physics in the low-energy regime or large distances [3,4]. During the past decades, many ideas have been proposed to approach this problem. There are many articles about this subject; for instance, see Refs. [5][6][7][8][9][10][11].
The area law of the Wilson loop average is a wellknown gauge-invariant criterion in studying quark confinement. It leads to a linear potential between a pair of static quark-antiquark. To study the linear part of the confinement potential, the quenched approximation is used where the dynamical quarks are removed for the infrared regime [4]. In fact, one can obtain some collective modes from gluons [12] which are associated with some topological degrees of freedom of the QCD vacuum, and as a result it is assumed that the QCD vacuum is filled with the topological objects obtained from these collective modes. Magnetic monopoles and center vortices are among the main candidates for describing confinement and each has its own fans.
For the non-perturbative description, people use lattice QCD simulations and phenomenological models to look * zahra.asmaee@ut.ac.ir † sdeldar@ut.ac.ir ‡ mkiamari@ipm.ir for the confinement and topological objects. The results of the phenomenological models must be in agreement with the results of the lattice QCD, though. In fact, lattice QCD can be served as a laboratory for confirming the correctness or incorrectness of the phenomenological models.
In the absence of matter fields, some various mechanisms of confinement have been suggested to extract the topological degrees of freedom of pure Yang-Mills theory. One of those mechanisms is the picture of the dual superconductor and appearance of Abelian monopoles. It was proposed by Nambu [13], Mandelstam [14], 't Hooft [15] and Polyakov [16] in the 1970s. The idea is that the QCD vacuum can behave like a dual superconductor and it is filled with magnetic monopoles. Just as the Meissner effect leads to the condensation of the Cooper pairs as electrically charged objects in an ordinary superconductor, the magnetic monopoles are condensed in a dual superconductor and squeeze the chromoelectric flux between the quark-antiquark pair inside a tube. Therefore, confinement of electric fields is obtained as a result of the condensation of magnetic monopoles in this picture [12,17,18].
The second possible mechanism is given by the center vortex model [19][20][21][22][23][24][25]. Historically, vortex-like structures were introduced in superconductors in 1959. Even though they were not observed at that time, they were recognized a few years later by Abrikosov [26]. It was proposed in various forms by 't Hooft [27][28][29][30], Nielsen and Olesen [31], Ambjorn and Olesen [32], Mack and Petkova [33,34], and Cornwall [35] in the late 1970s with a field theoretical approach. The idea is that the QCD vacuum is filled with closed magnetic vortices, and it is assumed that the vortices are condensed in the QCD vacuum. If a Wilson loop is linked to a vortex in an SU(N ) gauge group, the Wilson loop obtains a phase difference equal to e i2πn/N (n = 0 to N − 1) corresponding to the type of arXiv:2111.10636v3 [hep-lat] 20 May 2022 the vortex. As a result, some disorders are created in the lattice which eventually lead to an area law fall-off and confinement.
Vortices are defined by the center of the SU(N ) gauge group and there exist (N − 1) distinct vortices, which are called non-Abelian Z N vortices. The simplest vortices are defined by the Z 2 gauge group and they have the topology of tubes (in three Euclidean dimensions) or surfaces of finite thickness (in four dimensions) carrying some welldefined magnetic fluxes [5,27,28,[31][32][33][34][35].
Lattice calculations show that Z N vortices produce full string tensions as the Yang-Mills vacuum does. This is an encouraging motivation to study confinement via center vortices. If the center vortices are removed from the lattice, the string tension also disappears [19,21,23,[36][37][38].
The vortex condensation picture relies upon center gauge fixing and center projection. After performing center projection in lattice QCD, the full QCD with SU(N ) gauge symmetry is reduced to a gauge theory with a Z(N ) gauge symmetry. These vortices are called projection vortices (or p-vortices). Unlike monopoles, the modified models of vortices like thick center vortices can qualitatively explain the Casimir scaling dependence for all representations [20].
To study the confinement problem by center vortices, one first has to discuss the existence of vortices in the continuum limit. The most common methods of identifying vortices in the lattice simulation are direct maximal center gauge (DMCG) [21] and indirect maximal center gauge (IMCG) [19]. Inspired by these two methods of identifying vortices in lattice calculations and by the help of the connection formalism [12], we discuss the appearance of vortices in the continuum limit of QCD.
We review DMCG and IMCG methods in lattice QCD in Sec. II. In Sec. III, motivated by the methods proposed in lattice calculations, we introduce the vortices in the continuum by direct method for SU(N ) gauge group. As an example, by applying an appropriate gauge transformation in the SU(2) gauge group and using the results of Sec. III, we show in Sec. IV that under the center gauge transformation the vortex and anti-vortex can appear in the theory. Then, we remove the term that represents the anti-vortex. The theory has an SO(3) symmetry containing the vortex, which corresponds to the non-trivial first homotopy group of Π 1 (SO(3)) = Z 2 . Removal of the contribution of the anti-vortex is called "center projection" in our paper. In Sec. V, we introduce thin vortices in the continuum by the indirect method for SU(N ) gauge group. Sec. VI is brought in two subsections. In Sec. VI A, applying an Abelian gauge transformation for SU(2) gauge theory, we show that the QCD vacuum is filled with monopoles and anti-monopoles. It is shown that after Abelian projection the monopole appears in the vacuum and the gauge group symmetry is reduced from SU(2) to U(1) and we have a monopole vacuum. Then, in Sec. VI B we show that under a center gauge transformation on the monopole vacuum the vortex and anti-vortex appear in the gauge theory along with the monopole. After applying a "center projection" we have a gauge theory that contains chains including monopoles and vortices.

II. DMCG AND IMCG IN LATTICE QCD
There are some methods to identify vortices in lattice calculations and by using appropriate gauge transformations, which are in agreement with the vortex condensation picture [19-21, 23, 24].
In lattice QCD, the action is expressed in terms of link variables on which the gluon fields are defined. The idea is that under an appropriate gauge transformation the link variables U µ (x) get as close as possible to the center gauge group; center (SU(N )) = Z N . Then, after a projection, a smaller set of degrees of freedom remains. This job is usually done via two methods in lattice QCD calculation. In the following, we briefly review both methods.

A. Direct maximal center gauge method
This method was proposed by Del Debbio et al. [21], who tried to maximize the following quantity by determining the gauge transformation G(x) ∈ SU(N ): shows the gauge transformation of the link variables U µ (x) and as a result of the above maximization, U G µ becomes as close as possible to the center elements.μ is a unit vector along the µ direction. Then, by performing the center projection, one replaces the transformed link variable U G µ (x) by the closest associated center elements of the group Z N . As an example, the center projection is defined for the SU(2) gauge group as where 1 represents a 2 × 2 unit matrix. P-vortices identified by the DMCG method, are related to the nonperturbative degrees of freedom; see Ref. [4] for more details.

B. Indirect maximal center gauge method
The indirect maximal center gauge method was originally examined in lattice QCD for the SU(2) gauge group [19]. In general, for an SU(N ) gauge group, the procedure is as the follows. For the first step, a non-Abelian gauge configuration is fixed under Abelian gauge fixing, and after Abelian projection, the SU(N ) gauge symmetry is reduced to [U(1)] N −1 . In the second step, the remaining [U(1)] N −1 configuration is fixed under center gauge fixing such that the transformed gauge fields become as close as possible to the center elements. Finally, by performing a center projection, one gets the center elements. For example, for the SU(2) gauge group, where θ(x, µ) parametrizes the links. In general, in both methods, identification of the vortices is done by using gauge fixing and then projection.

III. DIRECT METHOD OF INTRODUCING VORTICES IN THE CONTINUUM
In this section, inspired by DMCG method in lattice QCD which confirms the existence of vortices in the infrared regime, we show that vortices appear in the theory by a singular gauge transformation. In this procedure, we use the connection formalism, which is applied for the singular gauge transformation. We would like to mention that we do not find a continuum formula for maximal center gauge transformation, Eq.(1), which has already been done in Ref. [39]. Instead, motivated by the fact that vortices exist in the infrared part of the theory, as shown by the lattice calculations, we use the connection formalism to make explicit the vortices, somehow similar to the procedure done in Ref. [12] for monopoles.

A. Link variable and transformed gluon field
In lattice gauge theory, color confinement can be studied by quenched approximation where only gluon fields exist in the theory. Gluons are defined on link variables as follows: Under a gauge transformation G(x) ∈ SU(N ), the link variables are transformed as Using Eq.(4) in the above equation, Since the lattice spacing a is small enough, we can use the exponential expansion of e iagAµ(x) , and using the Taylor expansion for G † (x +μ), Thus, in the continuum limit where a → 0, the gluon field is transformed as (8) where A G µ (x) ∈ SU(N ). In terms of group generators, T c are generators of the SU(N ) group, and c is the color index.
Since we are interested in observing topological defects from the gluon fields, we have to use an appropriate gauge transformation. Thin vortices appear as topological defects after center gauge transformations and some subsequent efforts.
Equation (9) can be used to study the vortices if G(x) ≡ N (x) is defined as a center gauge transformation, To study the contribution of thin vortices in the continuum, we first recall that in lattice QCD calculations when the Wilson loop links to the vortex it receives a phase difference equal to e i2πn/N associated with the non-trivial center element contribution Z(k) [18], C indicates an open circle from x − to x + , where x indicates the location of the intersection of C and the hypersurface Σ. is an infinitesimal quantity so that in the limit where → 0, C = C.
We use Eq. (12) to obtain the appropriate gauge transformation N (x), which gives the non-trivial center elements indicating the existence of vortices in the last line of Eq. (12).
From Fig. 1(a), an ideal vortex is defined on (D − 1)-dimensional hypersurface Σ, while the thin vortex is defined on (D−2)-dimensional boundary S = ∂Σ [18,39]. Piercing the hypersurface by the Wilson loop results in a discontinuity Z(k). The center vortex in D = 2, 3, and 4 is defined as a string, surface, and volume, respectively.
The relation between an ideal vortex and a thin vortex is as follows [18,39]: In fact, intersecting the hypersurface Σ of an ideal vortex with a Wilson loop C gives a phase to the Wilson loop proportional to a center group element. The boundary ∂Σ indicates the location of the thin vortex, which is gauge equivalent to the ideal vortex. Thus, the ideal vortex field is not unique and can be gauge transformed to a thin vortex field which has the support only on the boundary ∂Σ [18].
Eq. (10), On the other hand, in analogy to the lattice calculation where the thin vortex links to the Wilson loop [ Fig.1(b)], one can define a gauge field in the coset space by removing the ideal vortex [39] We recall that A N µ . T is still singular on ∂Σ. The gauge field configuration A N µ . T induces the same behavior for arbitrary Wilson loop as the configuration A N µ . T does. In other words, for x / ∈ hypersurface, we only see the boundary of the vortex, called the thin vortex field. In fact, by this choice of x, we have removed the hypersurface Σ from the space-time. So, the contribution of the ideal vortex defined on the hypersurface vortex would be zero;

B. Field strength tensor and connection formalism
In this subsection, we discuss the connection formalism, which has already been used in some references, for instance, Refs. [12,40]. In fact, we generalize the connection formalism, previously applied to the Abelian gauge transformation, to the center gauge transformation.
The Yang-Mills Lagrangian has an SU(N ) symmetry and is given by where the SU(N ) non-Abelian field strength tensor called and for a regular system can be written as whereD µ is the covariant-derivative operator. But topological defects appear as a result of singular gauge transformation. To observe these defects explicitly, we rewrite the Yang-Mills gauge theory in terms of the covariant-derivative operatorD µ and the ordinary derivative operator∂ µ , Using (20) For regular systems, the first term on the right-hand side of Eq.(20) is zero, so we have Eq. (18). But this term is not zero for singular systems. Therefore, where F µν is the SU(N ) non-Abelian field strength tensor, and Eq. (21) is applied when the singularity exists in the system. As a result of singular gauge transformation, topological defects like monopoles and vortices appear in the theory.
We study the behavior of the non-Abelian field strength tensor under singular gauge transformations.
In general, if G(x) ∈ SU(N ) represents a regular gauge transformation, the field strength tensor is transformed On the other hand, for a singular system where F µν is defined by Eq.(21), one gets (25) This is a noticeable result. The last term of Eq. (25) shows the difference between this equation and Eq. (22).
The advantage of using the connection formalism technique is that the gauge theory will remain gauge invariant after the singular gauge transformation. Equation (25) is valid for both the Abelian and center gauge transformations. It has already been discussed for the Abelian gauge transformation [12,40] and we intend to use it for the center gauge transformation, as well.
If one uses Eq.(25) without applying any projection, a full QCD will be obtained at the end. We discuss how we perform "center projection" in Sec. IV.

IV. DIRECT METHOD FOR INTRODUCING VORTICES IN SU(2) GAUGE GROUP
The formation of center vortices in the QCD vacuum relies upon two steps: center gauge transformation and "center projection". Using the results of Sec. III, we discuss these two steps for the SU(2) gauge group.
Step 1: Center gauge transformation In general, a 2 × 2 gauge transformation G(x) ∈ SU(2) is written in terms of three Euler angles α,β,γ, where T c 's are generators of the SU(2) group and σ c 's are Pauli matrices. The center gauge transformation G(x) ≡ N (x) ∈ SU(2) is continuous everywhere except at the hypersurface of the vortex. Therefore, the Euler angles are selected in a way that the constraint of Eq.(12) is satisfied. There are different choices for the angles. One can choose α = γ = ϕ 2 and β = 0, It can be shown that where (−1 2×2 ) represents the non-trivial contribution of the Z(2) gauge group. Thus, the contribution of an ideal vortex is observed at ϕ = 0. On the other hand, outside the hypersurface, the contribution of the thin vortex is represented by a pure gauge shown in Eq. (16), The spatial component of thin vortex is Equation (30) represents the gauge field associated with the thin vortex in cylindrical coordinates. The thin vortex is observed at ρ = 0 [39] in the third direction of color space. Under a center gauge transformation, the gluon field is defined by Eq. (15), where the first term on the right-hand side is regular and the second term indicates a topological defect. Replacing Eqs. (27) and (29) in Eq.(31), one obtains The magnetic vortex flux Φ flux is The total contribution of the magnetic flux is in the third direction in color space.
Using Eq.(25) of Sec. III B for the transformed field strength, With the help of Eq.(32), we rewrite the first term of Eq.(34) which is linear in terms of A N µ . T , The first three sets of brackets of Eq.
The first three brackets of Eq.(36) represent interactions between gluon fields and are regular. The fourth and the fifth brackets indicate interactions between the thin vortex and off-diagonal gluon fields but with an opposite sign compared with their counterparts in Eq. (35). Back to Eq.(34), the last term can be rewritten with the help of Eq. (27), . T in the above equation indicates the field strength of an anti-thin vortex carrying a magnetic flux equal to Φ flux = + 2π g T 3 .
Step 2: "Center projection" Adding F linear µν , F bilinear µν , and F singular µν together, one finds out that the similar terms with opposite signs cancel each other in F bilinear µν and F linear µν . On the other hand, the anti-thin vortex field strength tensor contribution represented by F singular µν is canceled by the thin vortex field strength tensor contribution brought in the last term of F linear µν , and finally, one is left with a full QCD field strength tensor.
In fact, with the above parametrization, one can argue that the vacuum is filled with thin vortices and anti-thin vortices, (see Fig.(2)).
To have only the contribution of the thin vortices, we remove the F singular µν term in Fig.(2). We call this procedure "center projection" and the "center projected field strength tensor" is defined as follows: All the terms in Eq. (38) are regular except the last term, which represents the field strength of a thin vortex field.
Therefore, the gauge symmetry is SO(3), and thin vortices appear as the topological defects corresponding to the non-trivial first homotopy group Π 1 (SO(3)) = Z 2 .

FIG. 2: Appearance of a vortex and an anti-vortex
under center gauge transformation.

V. INDIRECT METHOD OF INTRODUCING VORTICES IN THE CONTINUUM
In this section, motivated by the IMCG method in lattice QCD which confirms the existence of vortices, we study vortices in the continuum. As mentioned in Sec. II B, in the indirect method, in addition to the center gauge transformation and center projection [19], an initial step including Abelian gauge transformation and then Abelian projection is done.
Two successive gauge transformations are performed such that the first one is an Abelian gauge transformation M (x) ∈ SU(N ) and the second one is a center gauge transformation N (x) ∈ SU(N ). The transformation of link variables is or In the last equality, we used Eq. (4). Similar to what we have done in Sec. III A, we can use the exponential expansion and by using the Taylor expansion for M † (x + µ) and N † (x +μ), For the continuum limit where a → 0, N (x) indicates a center gauge transformation, and the contribution of vortices must be obtained from this gauge transformation. Therefore, similar to the Sec. III A and Eq.(15), the gauge field in the coset space is written as This is somehow similar to an Abelian gauge fixing plus center gauge fixing of IMCG method in lattice calculations. However, an intermediate step including Abelian projection must be applied, and we discuss it for the SU(2) gauge group in Sec. VI.

VI. INDIRECT METHOD OF INTRODUCING VORTICES IN SU(2) GAUGE GROUP
In this section, we apply the procedure explained in the previous section to the SU(2) gauge group, and we show that, unlike Sec. IV where we have gotten vortices as a result of center gauge transformation and a "center projection", we get a chain of vortices and monopole by the indirect method.
We discuss this section in two subsections. In Sec. VI A, we study the Abelian gauge transformation and Abelian projection, which leads to the emergence of the monopole.
In Sec. VI B, in addition to steps 1˚and 2 in Sec. VI A, a center gauge transformation followed by a "center projection" which leads to the appearance of chains containing vortices and monopoles, is discussed.

A. Abelian gauge tTransformations and Abelian projection: Monopole
The appearance of monopoles relies upon Abelian gauge transformation followed by an Abelian projection, which is discussed in this subsection for the SU(2) gauge group.
Lattice studies show that within a good approximation the string tension between a pair of quark and anti-quark is described by Abelian variables of the maximal Abelian gauge transformation. Therefore, in the continuum limit, the idea of the Abelian gauge transformation is to repress the contribution of the off-diagonal components of the gauge fields so that the contribution of diagonal compo-nents is dominant in the low-energy regime.
We perform a local rotation in color space called an Abelian gauge transformation. As a result, magnetic monopoles can be extracted from the diagonal part of the non-Abelian gauge field.
In this paper, we choose γ(x) = −ϕ, According to Sec. III A, we define G(x) ≡ M (x) as an Abelian gauge transformation; then the transformation of gluon field is given by Eq. (9), (45) The first term on the right-hand side of Eq.(45) is regular under Abelian gauge transformation M (x), but the second term is singular. Replacing Eq. (44) in Eq. (45), where the singularity appears in the inhomogeneous term of the above equation defined by A singular coordinates and is given by where A singular (θ, ϕ) = A c singular (θ, ϕ)T c . It is observed from Eq.(47) that there exists a magnetic monopole as a point defect at the origin, r = 0 along with a Dirac string at θ = π. The magnetic flux Φ flux (θ) of the inhomogeneous term is given by It is observed that the total contribution of the magnetic flux is located along the third direction of the color space. At θ = π, the magnetic flux of a Dirac string that enters a monopole located at the origin r = 0 is equal to 4π g T 3 .
We have discussed in Sec. III B that under a gauge transformation, the field strength tensor is obtained from Eq. (25). We rewrite it as follows: Equation (49) can be calculated using Eqs. (44) and (46). Since the two color directions T 1 and T 2 have no contribution in the magnetic flux, we will suppress these nondiagonal components of the gauge fields in the "Abelian projection" step. It can be easily confirmed that the first term of the Abelian sector F linear If we add the contents of the above three terms, the similar terms with opposite signs are canceled, and a field strength tensor which gives a full QCD is obtained [12].
Step 2: Abelian projection The sum of the two terms F linear µν + F singular µν represents a gauge configuration that only contains a monopole at r = 0. However, it is exactly canceled by the anti-monopole arisen from F bilinear µν . Thus, one can claim that the vacuum is filled with monopoles and anti-monopoles.
From Fig.(3), it is observed that, in order to have only the contribution of the monopole, we discard the term F bilinear µν so that [12] From Eq. (46), the gauge field is changed to As a result, F bilinear µν , which gives the anti-monopole contribution, is equal to zero, and the remaining part F linear µν + F singular µν describes an Abelian projected QCD, which contains a monopole at r = 0 and is called the monopole vacuum.

B. Center gauge transformation and "Center
Projection": Chain As mentioned before, the vortex recognition by the indirect method is done in four steps. We have discussed the first two steps in Sec. VI A and we explain the final two steps in the following.
Step 3: Center gauge transformation; We have shown that under two successive gauge transformations the gluon field is changed by Eq. (43). After Abelian projection, the bracket in Eq.(43) should be replaced by Eq.(51), For x / ∈ hypersurface, the thin vortex is defined in Eq. (16).
In fact, Eq.(53) expresses that a center gauge transformation is applied on a monopole vacuum. Using the center gauge transformation defined by Eq. (27) and the Abelian projected field defined by Eq.(52), the above equation is changed to The first three terms on the right-hand side of Eq.(54) are regular, and the last term is defined in the spherical indicates a defect representing a monopole located at the origin r = 0 along with the two vortices at θ = 0, π. This singular term remarkably represents a chain containing monopole and vortices. In fact, the magnetic potential of the chain defined by E µ ≡ − 1 g cosθ∂ µ ϕT 3 , can be interpreted as the sum of two terms: a magnetic potential of a monopole along with a Dirac string defined by The magnetic flux Φ flux (θ) passing through a closed contour C(r, θ) is defined as There is also a magnetic vortex whose flux is equal to − 2π g T 3 at θ = π. It is located in the negative direction of the z axis and exits from the magnetic monopole placed at r = 0. In fact, the sum of the two fluxes Φ Dirac string + Φ line vortex represents the contribution of a vortex equal to + 2π g T 3 , which enters the magnetic monopole sitting at the origin, r = 0. As a result, the magnetic flux of the monopole is obtained as the sum of the absolute values of the fluxes of the two vortices entering into it.
Equation (25) is used to obtain the field strength tensor of the transformation, We use Eq.(56) to study some various topological defects. A full QCD is obtained if one uses Eq.(56) without applying any projection. In this section, we discuss the possible resulting defects after Abelian and "center projections".
Looking at the last line of Eq.(56), we define the first term by F linear T and rewrite it using Eq.(54), The last line of the above equation contains some defects as explained in the following: The first term of Eq.(58) represents the field strength of a magnetic monopole located at r = 0, the second term indicates the field strength of a Dirac string at θ = π, and the third term represents the field strength of a thin vortex field that extended on the z axis. (See Fig.(4).) It is clear that the second term of Eq.
where In fact, the contribution of the vortex and the Dirac string appearing in F linear µν is exactly canceled by the contribution of the anti-vortex and the anti-Dirac string in F singular µν . As a result, a monopole vacuum is obtained unless we remove some of the singularities from the theory.
Step 4: "Center projection" As explained in Sec. IV, a "center projection" is done by removing F singular µν defined in Eq.(59). This means that "center projection" is obtained by F linear µν + F bilinear µν . On the other hand, we have shown that F bilinear µν is zero, and as a result, the center projected field strength tensor is as follows: Therefore, only a monopole attached to a Dirac string and a vortex remain. We can interpret these configuration as a chain as shown in Fig.(6).
FIG. 6: Appearance of a chain after Abelian gauge transformation, Abelian projection, center gauge transformation, and "center projection".
The first plot on the left-hand side of Fig.(6) represents a monopole at r = 0 plus a Dirac string located at θ = π carrying a magnetic flux equal to 4π g T 3 . The second plot on the left-hand side indicates a vortex carrying a magnetic flux equal to − 2π g T 3 extending on the z axis. Combining these two plots, a chain shown on the right-hand side of Fig.(6) is obtained. A chain contains a monopole at r = 0 and two vortices entering it. The flux of the vortex sitting at θ = 0 is equal to − 2π g T 3 , and the flux of the vortex sitting at θ = π is equal to + 2π g T 3 .
The latter vortex is obtained as a result of combining the flux of the Dirac string sitting in the negative z direction and the first vortex located in the z direction. Our arguments about the chains of monopoles and vortices are in agreement with the work of Del Debbio et al. [24], which is done by lattice QCD (see Fig.(7)), and also in agreement with the results of Reinhardt and Engelhardt [41], in which two vortex enter a monopole (see Fig.(8)). We end this section by discussing the possible advantages of using chains. As we mentioned at the beginning of the article, in both the dual superconductor model and the center vortex model, monopoles and vortices can explain some aspects of the color confinement like the linear potential between a quark and anti-quark. However, none of these models nor the associated defects is able to describe all the expected features of the confining potential between color sources.
At intermediate distances, a well-defined linear con-  [41]. The interpretation of a chain represented in Fig. (6) is the same as this figure where two vortex lines enter a monopole.
fining potential is expected, V R (r) ∼ σ R r, in which σ R is the string tension of representation R. The confining potential should agree with the Casimir scaling at intermediate distances. It means that the string tension of the potential between a quark and an anti-quark in representation R, σ R , is approximately proportional to the quadratic Casimir operator C R of representation R, i.e., where F indicates the fundamental representation and σ F shows the string tension of the fundamental representation. C F denotes the eigenvalue of the Casimir operator of representation F . We recall that the dependence of the potential slope to the Casimir scaling applies only for the intermediate distances and it is valid and exact for the large N limit [42,43]. In addition, at large distances, the k-string tension depends on the N -ality of the representation; it is equal to the fundamental representation string tension for the non-zero N-ality representations and zero for the zero N -ality representations [44]. Proportionality with Casimir scaling for the intermediate distances and the N -ality dependence of the potentials at large distances are confirmed by lattice calculations for the fundamental and a variety of higher representations [45][46][47]. Therefore, any phenomenological model which tries to describe the potential between static color sources is expected to interpret these two features. Vortex based models have been able to explain the N -ality dependence. However, to get the Casimir scaling for all representations, the models have been modified by defining a thickness for the vortex [24,48]. On the other hand, lattice results confirm the existence of chains of monopoles and vortices [25,49] that may explain the agreement of the potentials with Casimir scaling for higher representations. In this article, we have followed this approach to study the existence of chains of monopoles and vortices for the continuum.
We recall that an Abelian-projected theory gives the N -ality dependence (after all, it can still contain vortices), but it does not give the Casimir scaling dependence at intermediate distances [23,50] In this paper, motivated by direct and indirect methods of identifying vortices in lattice QCD, we have shown the existence of chains of monopoles and vortices for the continuum.

VII. CONCLUSIONS
Motivated by lattice QCD, which discusses the vortex contribution in color confinement, we have tried to introduce vortices in the continuum.
In the absence of matter fields, we work in the quenched approximation where dynamical quarks are removed from the theory. Therefore, the theory includes only the gluon fields, in this limit.
In recent years, the identification of vortices in lattice QCD has seen significant progress. Therefore, one expects to observe the same physics in the continuum limit when one uses the lattice results for the limit where a → 0.
Inspired by direct and indirect maximal center gauge methods which have studied vortices in lattice calculations and by using connection formalism technique, we have tried to recognize the vortices in the continuum. We have introduced the thin vortices from the gluon fields via both direct and indirect methods for the SU(N ) gauge group. We also get some help from the techniques proposed by Engelhardt and Reinhardt.
For an example, from the direct method, we have shown that under center gauge transformation the QCD vacuum of the SU(2) gauge group is filled with the vortices and anti-vortices. Then, applying a "center projection" we reach a theory that contains the thin vortex. The theory has an SO(3) symmetry containing the vortex, which corresponds to the non-trivial first homotopy group of Π 1 (SO(3)) = Z 2 .
Then, using the indirect method, we have shown that under Abelian gauge transformation for the SU(2) case the gauge theory would contain monopoles along with the Dirac strings and anti-Dirac strings as well as antimonopoles. Then, applying Abelian projection and removing the anti-monopole contribution, we end up with a theory that includes only the monopoles. In other words, SU(2) gauge symmetry is reduced to a U(1) gauge symmetry, and monopoles appear as the topological defects corresponding to the non-trivial second homotopy group Π 2 (SU(2)/U(1)) = Z.
Next, we have done a center gauge transformation on the Abelian vacuum. As a result, we get the monopole along with the Dirac string, the vortex, and anti-Dirac string, and the anti-vortex. Eventually, by applying "center projection", we end up with a theory that contains chains including monopole and two vortices. In subsection VI A, we express that the field strength is changed by Eq. (49) when Abelian gauge transformation is applied. One can also show that, where, + 1 g sinθ (∂ µ θ∂ ν ϕ − ∂ µ ϕ∂ ν θ) represents the field strength of a magnetic monopole at r = 0 and 1 g (1 − cosθ) [∂ µ , ∂ ν ] ϕ indicates the field strength of a Dirac string at θ = π. The Abelian sector F bilinear µν . T is also obtained by Eq. (46), And the above singular term shows the field strength of an anti-Dirac string at θ = π.