Monopole-center vortex chains in $SU(2)$ gauge theory

We study the relation between center vortex fluxes and monopole fluxes for $SU(2)$ gauge group in a model. This model is the same as thick center vortex model but we use monopole-antimonopole configurations instead of center vortices in the vacuum. Monopole-antimonopole configurations are line-like similar to center vortices. The group factor and potential for the fundamental representation of these configurations are studied and compared with those of center vortices. As a result, we obtain monopole-vortex chains which appear in lattice Monte Carlo simulations, as well. If these monopole-vortex chains intersect the large Wilson loop in two points, screening is observed for the potential in the fundamental representation at large distances and confinement is observed when one leg of the chain intersects the Wilson loop.


I. INTRODUCTION
The center vortices are color magnetic line-like (surface-like) objects in three (four) dimensions which are quantized in terms of center elements of the gauge group. Condensation of center vortices in the vacuum of QCD leads to quark confinement such that the color electric flux between quark and antiquark is compressed into tubes and a linear rising potential between static quarks is obtained. In the vortex picture, quark confinement emerges due to the interaction between center vortices and Wilson loops [1,2]. On the other hand, monopoles are playing the role of agents of confinement in the dual superconductor scenario [3,4]. Therefore, one may expect that there are some kind of relations between monopoles and center vortices. Monte Carlo simulations [5] indicate that a center vortex configuration after transforming to maximal Abelian gauge and then Abelian projection, appears in the form of the monopole-vortex chains in SU(2) gauge group. The idea of monopole-vortex chains has been studied by so many researchers [5][6][7][8][9].
In this article, monopole-vortex chains in SU(2) gauge group are investigated in a model. This model is the same as the thick center vortex model [10], but we use monopoleantimonopole flux instead of the center vortex flux. The motivation is to see if by this simple model we can observe the idea of monopole-vortex chains which has already been confirmed by lattice calculations, as well as some other phenomenological models. In this model, monopole-antimonopole configurations which are line-like and similar to center vortices are assumed to exist in the vacuum. Studying the group factors of the monopole-antimonopole configurations and center vortices, we understand that the monopole-antimonopole configurations are constructed of two center vortices. Increasing the thickness of the center vortex core increases the energy of the center vortex and therefore the energy of the vacuum which is made of these vortices. As a result, the potential energy between static quark-antiquark increases. This fact can be confirmed by this model, as well. This is a trivial fact from the physical point of view that the condensation of vortices leads to quark confinement.
Classically, it is very similar to the Aharanov-Bohm effect where increasing the thickness of the magnetic flux and therefore the magnetic energy of the system, changes the interference pattern. Using this simple model, we have calculated the potentials induced by the monopole-antimonopole configurations and center vortices. Comparing these potentials, we observe that the monopole-antimonopole configurations leads to a larger static quark-antiquark potential compared with the case when we use two center vortices in the model. We interpret this extra energy as a repulsive energy between two center vortices constructing the monopole-antimonopole configurations and then we discuss that the monopole-antimonopole configurations can deform to the monopole-vortex chains as confirmed by lattice calculations and other phenomenological models.
In section II, the formation of monopoles which is related to the Abelian gauge fixing method is reviewed in SU(2) gauge group. A model with structures of center vortices and monopole-antimonopole configurations are studied in sections III and IV. Then in section V, we study the group factors and potentials of these structures to argue monopole-vortex chains. Finally, we summarize the main points of our study in section VI.

II. ABELIAN GAUGE FIXING AND MAGNETIC MONOPOLE CHARGES
By Abelian gauge fixing, magnetic monopoles are produced in a non Abelian gauge theory. Specific points in the space where the Abelian gauge fixing becomes undetermined are sources of magnetic monopoles. In the following the formation of the magnetic charge by Abelian gauge fixing method is discussed [11].
In order to reduce a non Abelian gauge theory into an Abelian gauge theory, the gluon field under a gauge transformation can not be diagonalized. In fact, the gluon field A µ has four components and only one of them can be aligned simultaneously. Therefore, a scalar field is used to fix a gauge. One can consider a scalar field Φ (x) in the adjoint representation of SU(N) as the following: where T a are the N 2 − 1 generators of the SU (N) gauge group. A gauge which diagonalizes the matrix Φ (x) is called Abelian gauge. Now, we consider the SU(2) gauge group. A gauge transformation Ω (x) can diagonalize the field Φ (x): where The eigenvalues λ (x) of the matrix Φ (x) are degenerated when λ = 0 and therefore three components Φ a=1,2,3 ( r) are zero at specific points r = r 0 : In the vicinity of the point r = r 0 we can express Φ ( r) in terms of a Taylor expansion: where C ab = ∂Φa ∂x b r= r 0 . Therefore the field Φ ( r) has the hedgehog shape in the vicinity of the point r = r 0 . One can define another coordinate system where the point r 0 is placed at the origin. In this coordinate system, the field Φ ( r ′ ) has the form: Dropping the prime on x ′ and using the spherical coordinates for the vector r, one get to The gauge transformation Ω which diagonalizes the hedgehog field Φ is Therefore the hedgehog field Φ is diagonalized as the following: The gluon field transforms under the same gauge transformation: One can obtain: Thus, the gluon field under the gauge transformation of Eq. (8) can be separated into a regular part A R and a singular part: The singular part has the form of a gauge field in the vicinity of a magnetic monopole with magnetic charge equal to: To summarize, we observe that in the vicinity of the points where the eigenvalues of the matrix Φ (x) are degenerate, the singular part of the gluon field in the Abelian gauge behaves like a monopole with magnetic charge g = − 4π e T 3 .

III. A MODEL OF VACUUM STRUCTURE
In this model [10], where The {H i i = 1, .., N − 1} are the Cartan generators, S is a random element of SU(N) gauge group and angle α n C (x) shows the flux profile which depends on the Wilson loop size and the location x of the center vortex with respect to the Wilson contour C. The random group orientations associated with S are uncorrelated, and should be averaged. The averaged contribution of G over orientations in the group manifold specified by S is where G r ( α n C (x)) is called the group factor and I dr is the d r × d r unit matrix. In SU(N) case, the group factor of the fundamental representation interpolates smoothly from e i2πn N , if the core of the center vortex is located completely inside the Wilson loop, to 1, if the core is completely exterior. The Wilson loop C is assumed as a rectangular R × T loop in the x − t plane with T ≫ R where the left and the right time-like legs of the Wilson loop are located at x = 0 and x = R. In other words, the two static charges are located at these points.
A desired ansatz for angle α n C (x) must lead to a well-defined potential i.e. linearity and Casimir scaling at the intermediate distances. Any reasonable ansatz for the angle α n C (x) must satisfy the following conditions: 1. α n C (x) = 0 when a center vortex locates far outside the Wilson loop.
2. α n C (x) = α n max when a center vortex locates deep inside a large Wilson loop. The maximum value of the angle α n max is obtained from the following maximum flux condition: where k is the N-ality of representation r.
An ansatz for the flux profile which would meet these conditions is assumed as the following [10] where n indicates the center vortex type, a, b are free parameters of the model, α n i(max) corresponding to Eq. (17) indicates the maximum value of the flux profile and R is the distance between two static charges. y(x), the nearest distance of x from the timelike side of the loop, is The flux of Eq. (18) is one of the many examples that can give the appropriate potential.
Some other examples were discussed in Ref. [13].
For SU(2) gauge group, when the vortex core is entirely contained within the Wilson loop, using Eq. (17), we get where H 3 is Cartan generator and z 1 I = e πi I is the center element of SU(2) gauge group.
Therefore, the maximum value of the angle α 1 max for the fundamental representation is equal to 2π. Thus, the ansatz of the flux profile given in Eq. (17) for SU(2) is obtained as the following probabilities of piercing the plaquettes in the Wilson loop by center vortices are uncorrelated.
Assuming that an nth center vortex appears in any given plaquette with the probability f n , the expectation value of the Wilson loop is obtained: where < W 0 (C) > denotes T r U...U which no vortex pierces the Wilson loop.
One of the criteria for the color confinement is the area law for the Wilson loop i .e.
Here A(C) is the minimal surface spanned on the Wilson loop C and σ > 0 is the confining string tension. Using Eq. (24) into Eq. (22), the string tension is obtained as the following One gets the static potential induced by center vortices between static color charges in representation r at distance R as the following where the center of vortex cores pierces the middle of plaquettes i.e. x = (n + 1 2 )a (n ∈ (−∞, ∞)) where a is the lattice spacing. We use a = 1 throughout this paper. Although R takes only integer values in the lattice formulation, but the figures related to V r (R) are platted over the continuous interval.
For SU(2) gauge group, the static potential induced by z 1 center vortices at f 1 ≪ 1 and small distances between static charges (small R) where α 1 (x) ≪ 2π is obtained as the where spin index j shows the representations in SU(2) gauge theory. According to Eq. (27), the static potential is proportional to the eigenvalue of the quadratic Casimir operator i.e.
V j (R) ∼ j(j + 1) in agreement with the Casimir scaling effect observed in lattice simulations [14]. The Casimir proportionality of the static potential induced by center vortices can be generalized from SU(2) to SU(N). For observing the property of Casimir scaling in the potentials at intermediate regime, the probability f n should be far smaller than 1. Therefore, the probability f n is chosen 0.1 as a desired value in the calculations.
The Casimir scaling is not found at intermediate distances for any choice of the free parameters related to the ansatz in Eq. (18). But it is observed for a large region of the parameter space. As an example, the extent of Casimir scaling region at intermediate distances can be changed by any factor F by setting a → a/F, b → bF . The thickness of the center vortex would be on the order 1/a for the ansatz given in Eq. (17). Therefore, choosing F > 1 as an integer value, increases the thickness of the center vortex and the Casimir scaling region while F < 1, decreases these quantities.
In the next section, we investigate the effect of a monopole flux on a Wilson loop. . Therefore, the total magnetic flux of a monopole crossing the surface S is equal the magnetic charge g as the following [11,15] where V is the volume enclosed by the surface S. For SU(2) gauge group, g is the monopole charge in Eq. (13). If we attribute a thickness for these configurations as what is done for the thick center vortices, the effect of a monopole-antimonopole configuration on a Wilson loop is to multiply the loop by a group element the same as the one in Eq. (16). If a monopole-antimonopole configuration is entirely contained within the loop, then where eg satisfies the charge quantization condition eg = 2nπ.
For SU(2) gauge group, corresponding to Eq. (13), the magnetic charge of the monopole is the Cartan generator. When a monopole-antimonopole pair is entirely contained within the Wilson loop, using SU(2) magnetic charge into Eq. (17), we get where index n = 0 is related to the monopole-antimonopole configurations. The sign in the exponent is not important since the direction of the configuration which pierces the Wilson loop is not important. Therefore the maximum value of the angle α 0 max for the fundamental representation is equal to 4π. Therefore the ansatz of the flux profile given in Eq. (17) for the monopole-antimonopole configurations of SU(2) gauge theory is obtained as the following The potential induced by monopole-antimonopole configurations is the same as the one induced by center vortices represented in Eq. (26). For SU(2) gauge group, the potential induced by monopole-antimonopole configurations for the fundamental representation is obtained as the following In the next section, we study the group factors and the potentials for the center vortices and the monopole-antimonopole configurations.

V. SU(2) AND VACUUM STRUCTURES
To study the center vortices and monopole-antimonopole configurations in the vacuum for SU(2) gauge group, we discuss the interaction between the Wilson loop and these con-figurations. First, the group factors of these configurations which have an important role in producing the potentials of Eq. (26) [16,18] are studied and the relation between these configurations is discussed. Then, with calculating the potentials induced by these configurations, interactions inside these configurations are studied.

A. Interaction between the Wilson loop and center vortices
First, we calculate the group factor of the center vortices in SU(2) gauge group. The group factor for the fundamental representation of SU(2) is obtained from Eq. (16) where H 3 is the Cartan generator of SU(2) gauge group. According to Eq. (20), the maximum value of the angle α 1 max for the fundamental representation is equal to 2π. Using ansatz given in Eq.

B. Interaction between the Wilson loop and monopole fluxes
Next, we calculate the group factor of the monopole-antimonopole configurations in SU(2) gauge group. Using Eq. (16), the group factor for the fundamental representation is obtained as the following According to Eq. (30), the maximum value of the angle α 0 max for the fundamental representation is equal to 4π. Using ansatz given in Eq. these center vortices is investigated, in details. Before that we study another approach, explained in ref. [12], for obtaining the relation between center vortex and monopole fluxes.

The contribution of this Abelian configuration on the Wilson loop is
Comparing Eq. (36) with the contribution of an Abelian field configuration to the Wilson loop which is W = e iqΦ (q means units of the electric charge and q = 1 for the fundamental representation) [8], the flux of this Abelian configuration is equal to π.
Therefore, the flux of this Abelian configuration corresponding to the half of the magnetic charge g, is the same as one center vortex on the Wilson loop.
In the next subsection the interaction between center vortices inside the monopoleantimonopole configuration is studied.

C. Monopole-vortex chains
In the previous sections, we have shown that the flux between a monopole-antimonopole pair is constructed from the fluxes of two vortices. To understand the interaction between two center vortices inside the monopole-antimonopole configuration, we study the potentials In addition, the dual superconductor picture of quark confinement was proposed by Nambu in 1970's [17]. Ginzburg-Landau theory defines two parameters: the superconducting coherence length ξ and the London magnetic field penetration depth λ.
As an interesting possibility, the repel of two center vortices may mean the Type-II superconductor of the QCD vacuum, that is, the Ginzburg-Landau parameter κ = λ/ξ of the QCD vacuum is larger than 1/ √ 2.
We would like to mention that the interaction between vortices has been studied by the domain model (the modified thick center vortex model) in ref. [18], as well. In that article, based on "energetics" we have shown that two vortices with the same flux orientations inside (z 1 ) 2 vacuum domains repel each other. While two vortices with opposite flux orientations inside z 1 z * 1 vacuum domains attract each other. The group factors analysis of (z 1 ) 2 and z 1 z * 1 vacuum domains agree with this article. Since two similarly oriented vortices inside (z 1 ) 2 vacuum domain repel each other, we conclude that they do not make a stable configuration and one should consider each of them as a single vortex in the model. On the other hand, since two vortices with the opposite orientation inside z 1 z * 1 vacuum domain attract each other, we conclude that they make a stable configuration. Adding the contribution of the z 1 z * 1 vacuum domain to the potential obtained from center vortices, the length of the Casimir scaling regime increases [18]. The results of this paper is in agreement with our previous paper.
To summarize, in this article, we obtain a chain of monopole-vortex. The magnetic flux coming from a monopole inside the chain is squeezed into vortices of finite thickness and a non-orientable closed loop is formed. The non-orientable closed loop means that two vortex lines inside the loop have different orientations of magnetic fluxes.   [5,19] as shown in Fig. 9. Therefore a center vortex upon Abelian projection would appear in the form of monopole-vortex chains. Indeed Abelian monopoles and center vortices correlate with each other. Figure 10a shows some monopole-vortex chains in SU (2) gauge group [19]. In addition, the monopole-vortex junctions called as nexuses are studied in ref. [20]. Some solutions to the equations of motion obtained from the low-energy effective FIG. 10: a) Some monopole-vortex chains in SU (2) gauge group shown in ref. [19]. b) The monopole-vortex chain shown in ref. [7]. Therefore, the monopole-vortex chain obtained in this article agrees with the results of lattice gauge theory and chain models.
energy functional E of QCD [7] are studied. Several thick vortices meet at a monopole-like center (nexus), with finite action and non-singular field strengths. In SU(N) gauge group each nexus is the source of N center vortices. Figure 10b shows monopole-vortex chain obtained by Cornwall for SU(2) gauge group [7]. In ref. [6], examples of the monopole-vortex chains are also plotted using the method of ref. [7].
Therefore the monopole-vortex chain in the vacuum obtained from the model agrees with the results of lattice gauge theory and chain models. The resulting monopole-vortex chains agree with the lattice calculations and phenomenological models. In general, these monopole-vortex chains should be observed in 3 dimensions.
In the model, the Wilson loop, which is a rectangular R ×T loop in the x−t plane, probably intersects with one of the legs of the chain at a time. Many random piercings of the Wilson loop by these legs and then averaging those random piercings leads to the confinement.

VII. ACKNOWLEDGMENTS
We are grateful to the Iran National Science Foundation (INSF) and the research council of the University of Tehran for supporting this study. [