Contributions of the center vortices and vacuum domain in potentials between static sources

In this paper, we study the role of the domain structure of the Yang Mills vacuum. The Casimir scaling and N-ality are investigated in the potentials between static sources in various representations for SU(2) and SU(3) gauge groups based on the domain structure model using square ansatz for angle αC(x). We also discuss about the contributions of the vacuum domain and center vortices in the static potentials. As a result, the potentials obtained from vacuum domains agree with Casimir scaling better than the ones obtained from center vortices. The reasons of these observations are investigated by studying the behavior of the potentials obtained from vacuum domains and center vortices and the properties of the group factors. Then, the vacuum domains in SU(N) and G(2) gauge groups are compared and we argue that the G(2) vacuum is filled with center vortices of its subgroups.


Introduction
Mechanism of confinement is one of the challenging problems in QCD. Numerical methods (lattice gauge theory) and phenomenological models such as thick vortex model [1], gluon chain model [2] and dual superconductor models (for a review, see ref. [3]) are investigating quark confinement. Popular degrees of freedom which are responsible for the confinement are center vortices, Abelian monopoles, instantons and merons. In this article, center vortices are studied as the confining degrees of freedom.
There are numerical evidences in favor of the center vortex confinement mechanism (see refs. [4] and [5] for review). The vortex model proposed by 't Hooft in the late 1970's [6][7][8][9][10][11][12][13], interprets confinement based on condensation of thin vortices with fluxes quantized in terms of the center elements of the gauge group (center vortices). A center vortex is a color topological field configuration which is line like (in three dimensions) or surface like (in four dimensions). Each thin center vortex piercing the Wilson loop leads to a certain amount of disorder and its effect on the loop is to multiply the loop by a center element. One of the criteria for the color confinement is the area law for the large Wilson loops: Here C is a rectangular R × T loop in the x − t plane, Area(C) is the minimal surface spanned on the loop C, σ > 0 is the confining string tension, and VQ Q (R) is the static potential between sources at large distances R.

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Using thin center vortex model, one gets an area law for the potential of the quarks in the fundamental representation but not for the adjoint representation. According to the Monte Carlo data, confinement must be observed for color sources of higher representations [14][15][16][17], as well. Modifying the model by applying finite thicknesses to the thin vortices, the area law was observed for the Wilson loops of all representations [1].
Furthermore, by adding vacuum domains corresponding to the trivial center element z 0 = 1, confinement interval has been increased [18]. Vacuum domains carry quantized magnetic fluxes in terms of the trivial center element. This modified model has been called domain structure model in the literature. We study the characteristics of the vacuum domain and their effects on the potentials between color sources. G(2) gauge group is its own universal covering group and it has only one trivial center element. Therefore G(2) is an interesting laboratory, which attracts considerable attentions to examine the role of the trivial center element [18][19][20][21][22][23][24][25]. One expects that in G(2) gauge group the static potentials do not grow linearly over any certain range of distances because of the lack of nontrivial center element. However, the linear rise of the potentials for all representations at the intermediate distances was clearly observed in lattice gauge theory [18,26]. On the other hand, from the domain structure model, we have observed linear potentials between G(2) sources for different representations roughly proportional to the eigenvalue of the quadratic Casimir operator of the representation which is in agreement with the lattice results [27]. In our previous article [30], we have argued that the SU(2) and SU(3) subgroups of the G(2) gauge group may be responsible for the confinement potential of G (2). We applied the domain structure model to the G(2) gauge group and the thick vortex model to the SU (2) and SU(3) subgroups of G (2). We discussed about the reasons of observing linear potential in G(2) gauge group by comparing the potentials and extremums of the vortex profile Re(g r ) of the G(2) gauge group and its subgroups in the fundamental ({7}-dimensional) and adjoint ({14}-dimensional) representations.
The vacuum domain may play an important role not only in G(2) but also in SU(N ) gauge theories. In this paper, we study the role of the vacuum domain at intermediate regime for SU(N ) gauge groups and we discuss about the possibility of constructing a vacuum domain by center vortices. Then, the vacuum domain of G(2) is studied. By comparing with SU(2) and SU(3) groups, the role of these subgroups in observing confinement in G(2) is discussed. In section 2, we briefly review the domain structure model. Angle parameter of the model is studied in section 3. Then, in section 4, we obtain static potentials for various representations and their ratios for different kind of center domains for the SU(2) and SU(3) gauge groups. The interaction between the Wilson loops and vacuum domains and a comparison between SU(N ) and G(2) groups are discussed in section 5. We summarize the main points of our study in section 6. Finally, Cartan generators are constructed using tensor product and decomposition methods in the appendix.

Domain structure model of the Yang Mills vacuum
In this model, the vacuum is assumed to be filled with domain structures. In SU(N ) gauge group, there are N types of center domains including center vortices corresponding to the JHEP03(2015)016 nontrivial center elements of Z N subgroup enumerated by the value n = 1, . . . , N − 1 and the vacuum type corresponding to the z 0 = 1 center element (n = 0). For G(2) gauge group, there is of course only one center domain of vacuum type corresponding to z 0 = 1 which belongs to the trivial Z 1 subgroup. The probability that any given plaquette is pierced by an nth domain is equal to f n . Creation of a thick center domain linked to a Wilson loop in representation r has the effect of multiplying the Wilson loop by a group factor G r (α (n) ), i.e.
where the {H i i = 1, . . . , N −1} are the Cartan generators, angle α (n) shows the flux profile that depends on the location of the nth center domain with respect to the Wilson loop, and d r is the dimension of the representation r. If the core of the center domain is entirely enclosed by the loop, then where k is the N -ality of the representation r and if the core is entirely outside the minimal area of the loop, then the group factor is equal to 1. Phase factors of domains of type n and type N − n are complex conjugates of each other and they may be considered as the same type of domains but with magnetic flux pointing in opposite directions, so that The inter quark potential induced by the center domains is as the following [1,18]: where the function α n C (x) represents the corresponding angle and the amount it accepts, depends on both the Wilson contour C and the position of the vortex center x. In the next section, some reasonable ansatz for the angle α n C (x) are given.
3 Ansatz for the angle α C (x) There are some functions to use as the ansatz for α C (x) [1,18,32]. An appropriate ansatz must lead to a well-defined potential i.e. respecting linearity and Casimir scaling for the intermediate regime.
The Wilson contour C is a rectangular R × T with T ≫ R and left and right timelike legs of the loop are located at x = 0 and x = R, respectively. A few conditions that any ansatz must satisfy are as the followings: 1. If a center domain locates outside the minimal area of the Wilson loop, then α C (x) = 0.
2. If the minimal area of the Wilson loop is pierced by a center domain, then α C (x) = α max , where α max is obtained from the following maximum flux condition: exp(i α max · H r ) = e i2kπn/N I. 3. If R → 0, then the flux of the domain core must be zero inside the Wilson loop i.e. α c (x) → 0.
An ansatz introduced by Faber et al . [1] is: where a and b are free parameters and y(x) is The magnitude of y(x) shows the distance of the center of domain with respect to the nearest timelike leg of the Wilson loop. Figure 1 shows this old ansatz versus x for R = 100. Another ansatz was introduced by Greensite et al. [18]. Each domain, with cross section A d , is divided to subregions of area l 2 ≪ A d which l is a short correlation length. The color magnetic fluxes in subregions l 2 fluctuate randomly and almost independently. In other words, the color magnetic fluxes in neighboring regions of area l 2 are uncorrelated. The only constraint is that the total color magnetic fluxes of the subregions must correspond to an element of the gauge group center. The ansatz is introduced as the following: where A is the cross section of the center domain overlapping with the minimal area of the Wilson loop and µ is a free parameter. The cross section of a domain is a L d × L d square. Figure 2 schematically shows the interaction between the angle of square ansatz and the Wilson loop. One should take two intervals for the square ansatz: where The range of x i.e. − L d 2 ≤ x ≤ R + L d 2 has been restricted over all plaquettes within the minimal area of the Wilson loop, as well as plaquettes in the plane outside the perimeter of the loop which are located inside a distance L d 2 of the loop. Figure 3 shows this square ansatz versus x for R = 100. Center domains are located completely inside the Wilson loop at x = 0. The angle α(x) changes more drastically in the right plot where it is obtained by the center vortices (non zero α max ) compared with the left plot where the vacuum domains (α max = 0) are used. In the next section, we argue about the contribution of the vacuum domain to the potential between color sources at intermediate distances for the SU(N ) gauge theories.

Static potentials and Casimir scaling
The center vortex model [1] leads to linear regime for the static potential qualitatively in agreement with Casimir scaling hypothesis. The confinement regime has been increased [18] when the vacuum domains have been added to the model. In our previous papers [27,30], we have studied the role of the vacuum domain in G(2) gauge group which has one trivial center element, only. According to the center vortex theory, one does not expect confinement in a group without nontrivial center element. But using the domain model and from the numerical lattice calculations for the G(2) gauge group, the static potentials in different representations grow linearly at intermediate distances and the ratios of the linear regime slopes are roughly proportional to the Casimir ratios. Therefore, it is interesting to understand the role of the vacuum domain to the static potential in SU(N ) gauge theories. If one uses the square ansatz i.e. eq. (3.4), then the static potential induced by center vortices is as the following [1]:

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and the contribution of the vacuum domain added to the static potential is given by [18]: where f 0 is the probability that any given unit is pierced by a vacuum domain. Now, we obtain the static potential at different distances in SU(N ) gauge group (N = 2, 3), using the contributions of all domains, vacuum domain and center vortices, separately.

SU(2) case
First, we apply the model to the SU(2) gauge group. In SU(2) case, there is one nontrivial center element in addition to the trivial element Therefore, the static potential induced by all domains of SU(2) gauge group is obtained from eq. (2.4) where f 1 and f 0 are the probabilities that any given unit area is pierced by a center vortex and a vacuum domain, respectively. The free parameters L d , f 1 , f 0 , and L 2 d /(2µ) are chosen to be 100, 0.01, 0.03, and 4, respectively. We take the correlation length l = 1, therefore the static potentials are linear from the beginning (R = l). The square ansatz for the angles corresponding to the Cartan generator H 3 for the center vortex and the vacuum domain are:  Figure 6 plots the ratios for the linear regime using the vacuum domain, the center vortex, and all domains. These potential ratios start from the ratios of the corresponding Casimirs i.e.
In the range R ∈ [0, 20], the potential ratios by center vortices decrease slowly from 8/3 and 5 to about 2.34 and 3.65, respectively. In the same interval, the potential ratios V 1 (R)/V 1/2 (R) and V 3/2 (R)/V 1/2 (R) induced by vacuum domain drop very slowly from 8/3 and 5 to about 2.57 and 4.6 compared with the potential ratios induced by center vortices. On the other hand, figure 7 shows potential ratios using the ansatz given in eq.  vortices drop from 8/3 and 5 to about 2 and 2.5 in the range R ∈ [1,12], respectively [1]. So the potential ratios obtained from square ansatz drop slower than the ones by the old ansatz.
From figure 4, it is clear that at large distances, R ≥ 100, the static potentials induced by all domains agree with N -ality, as expected. Therefore, the main contribution to the potentials for large loops corresponds to center vortices and since the vacuum domain screens the potential at large distance, it dose not change the slope of the potential and N -ality. N -ality classifies the representations of a gauge group. At large distances, when the energy between two static sources is equal or greater than twice the gluon mass, a pair of gluon-anti gluon are popped out of the vacuum and combine with initial sources and transform them into the lowest order representations of their class. For examples In other words, static sources in representations {4}(j = 3/2) and {3}(j = 1) by combining with a gluon are transformed into the lowest order representation {2}(j = 1/2) and color singlet. Thus, the slope of representation {4} must be the same as the fundamental one and representation {3} must be screened. Screening is observed in figure 5, since vacuum domain locates completely inside the Wilson loop at large distances.
Therefore for SU(2) case, the fluctuations within a vacuum domain lead to a group disorder which agrees Casimir scaling stronger than center vortices while center vortex disorder leads to N -ality.  [1]. The potential ratios start from the Casimir ratios but the ratios by this ansatz drop steeper than the ratios by square ansatz.

SU(3) case
Next, we apply the model to the SU(3) gauge group. In this case, there are two nontrivial center elements in addition to the trivial center element (4.9) Since z 1 = (z 2 ) * , the vortex flux corresponding to z 1 is equivalent to an oppositely oriented vortex flux corresponding to z 2 . Therefore from eq. (2.4), the static potential induced by all domains in SU(3) gauge group is as the following: (4.10) where f 1 , f 2 , and f 0 are the probabilities that any given unit area is pierced by z 1 center vortex, z 2 center vortex, and the vacuum domain, respectively. As a result of eq. (2.3), The free parameters are chosen as the same as subsection 4.1. The square ansatz for angles of center vortex and vacuum domain are: (4.14) In these examples static sources in representations {6} and {8} are transformed into the lowest order representation {3} and color singlet, respectively. Therefore, the slope of {6} dimensional representation must be the same as the one for the fundamental representation and representation {8} must be screened. In summary, using the vacuum domain only, the intermediate potentials agree better with Casimir scaling compared with the case when center vortices are using. In addition, square ansatz for the group factor is a better choice if one wants to see the Casimir scaling.
From this section, we conclude that for SU (3), as well as SU(2) gauge group, the potential ratios induced by vacuum domain agree better with Casimir scaling compared with the potential ratios induced by center vortices. In the next section, we argue about the reasons of these observations by studying the behavior of the potentials induced by vacuum domains and center vortices and the properties of the group factor G r (α (n) ) for each case.    potentials between static sources and the behavior of the group factor G r (α (n) ) especially in SU(2) gauge group to investigate the contribution of the center domains.

Center domains in SU(2)
For SU(2) gauge group with the center group Z(2) = {z 0 = 1, z 1 = e πi }, simultaneous creation of two similarly oriented center vortices may give a vacuum domain and can be detected by a Wilson loop and also simultaneous creation of two oppositely oriented center vortices produces a vacuum domain and its effect on the Wilson loop is as the following: We recall that combining the center vortices fluxes has been studied in ref. [28], as well.
For SU(2) gauge group (z 1 ) 2 = z 1 z * 1 = 1. Therefore, if the loop is large enough to contain two vortices, the vacuum domain can be detected by the Wilson loop. However, we show that (z 1 ) 2 vacuum does not make a stable configuration and it is the z 1 z 1 * combination which is a stable configuration and makes the vacuum domain.
To understand the interaction between vortices, we study the potentials induced by vacuum domains and center vortices using the square ansatz. Figure 12 shows the static potentials of the fundamental representation, induced by vacuum domains corresponding to (z 1 ) 2 and z 1 z * 1 and center vortices. The potential energy induced by vacuum domains corresponding to two similarly oriented center vortices is larger than the twice of the potential energy induced by the center vortices. The extra positive energy may be interpreted as the interaction energy between center vortices constructing the vacuum domain. Therefore two vortices with the same flux orientations repel each other. On the other hand the potential energy induced by the vacuum domain corresponding to two oppositely oriented center vortices is less than the twice of the potential energy induced by the center vortices. Therefore an attraction occurs between two vortices with different flux orientations if they make a vacuum domain. Studying the group factors of the vacuum domains and the center vortices is also interesting [29]. The group factor for the fundamental representation of SU(2) is obtained from eq. (2.1) and the Cartan of the SU(2) gauge group: For the fundamental representation of SU(2) gauge group, when the center vortex is completely contained within the Wilson loop, Using the Cartan generator of SU(2), the maximum value of the angle α  max for the fundamental representation is zero and in this case Re(G r ) = 1. Figure 14 (right) plots Re(G r ) versus x for R = 100 for the z 1 z * 1 vacuum domain in the fundamental representation of SU (2). If the center of the vortex core is placed at x = 0 or x = 100, 50% of the maximum flux enters the Wilson loop and the value of the group factor is about 0.9. Since two oppositely oriented vortices of the z 1 z * 1 vacuum domain attract each other, the cores of two oppositely oriented vortices overlap each other and some part of the magnetic flux in each vortex is annihilated. Figure 15 (left) schematically represents the z 1 z * 1 vacuum domain. As a result, the magnetic flux of the center vortices does not conserve and we do not observe -1 (corresponding to the SU(2) center vortex) for the group factor of the vacuum domain when half flux of the vacuum domain locates inside the Wilson loop. Now, we discuss the effect of adding the contributions of the vacuum domains corresponding to (z 1 ) 2 and z 1 z * 1 to the potential induced by center vortices. According to figures 6 and 10, the potential ratios start out at the ratios of the corresponding Casimirs. Therefore for small size loops (R ≈ 1, 2) where α c is also small (α c ≈ 0), the potentials strongly agree with the Casimir scaling. As a result, for small size loops, the group factor is  close to one i.e. Re(G r ) ≈ 1. Therefore, if the group factors in medium size loops (R < 20) change very slowly, the potential ratios drop smoothly from Casimir ratios.
A comparison between group factors obtained from different domains for the two dimensional representation (j = 1/2) of the SU(2) gauge group is done by plotting figure 16 for the Casimir scaling regime. The value of the group factor obtained from center vortices changes smoothly from 1 to 0.75 for R < 20. In the same range of distances, the value of the group factor obtained from z 1 z * 1 vacuum domains changes from 1 to 0.92 where the changing rate is slower than the one obtained from center vortices. Therefore the magnetic flux of Group factor of (z 1 ) 2 vacuum domain R=2 R=5 R=10 R=20 Figure 16. Re(G r ) obtained from center domains versus x for different sizes of Wilson loops (different R) in the Casimir scaling regime. It is plotted for the two dimensional representation (j = 1/2) of the SU(2) gauge group. In this regime, the group factor of center vortices changes slowly from 1 to 0.75. For the same regime, the group factor of z 1 z * 1 vacuum domain changes from 1 to 0.92 which is very slower compared with the one obtained from center vortices. Also the group factor obtained from (z 1 ) 2 vacuum domains changes very fast from 1 to 0.2 compared with the one obtained from center vortices. Therefore as discussed in the text, by adding the contribution of the z 1 z * 1 vacuum domain in the potential obtained from center vortices, the length of Casimir scaling regime increases and by adding the contribution of the (z 1 ) 2 vacuum domain in the potential obtained from center vortices, the length of Casimir scaling regime decreases. approximately close to zero and their group factors is close to one. Since the group factor obtained from z 1 z * 1 vacuum domains changes slower than the one obtained from center vortices, therefore the potential ratios obtained from z 1 z * 1 vacuum domains violate from Casimir ratios slower than the one obtained from the center vortices. Also the value of group factor obtained from (z 1 ) 2 vacuum domains changes very fast from 1 to 0.2. Since the group factor obtained from (z 1 ) 2 vacuum domains changes faster than the one obtained from center vortices, therefore the potential ratios obtained from (z 1 ) 2 vacuum domains violate quickly from Casimir ratios.

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In summary, in the intermediate regime, the potential ratios obtained from z 1 z * 1 vacuum domain drop slower than the one induced by center vortices, and the potential ratios obtained from (z 1 ) 2 vacuum domain drop faster than the one obtained from center vortices. Therefore by 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 but by adding the contribution of the (z 1 ) 2 vacuum domain to the potential induced by center vortices, the length of Casimir scaling regime decreases. The above discussion can explain why the length of Casimir scaling is increased in figure 4. From the discussions and the fact that two vortices with the same orientation, z 1 and z 1 , 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. It should be noted that we do not add their contributions to the model. Figures 3 to 11 are plotted using center vortices and z 1 z * 1 contributions. However, if one adds the (z 1 ) 2 contributions in calculating potentials, the ratios of the slopes of the higher representation potentials to the fundamental representation potential would not change at large distances. This is because each of the z 1 vortices counts as a regular vortex. However, the Casimir ratios get worse as discussed. On the other hand, z 1 z * 1 contribution makes a stable configuration. They improve the Casimir scaling regime and since they screen the potential at large distances, they do not change the asymptotic slope of the potentials and therefore N -ality survives. Even though, the original model assumes stastical independence of vortex position, it should be noted that the interaction of vortices should be included if one wants to modify the model. Figure 17 shows the fundamental and adjoint representation potentials using center vortices only and center vortices plus z 1 z * 1 combinations. Adding z 1 z * 1 contributions, the general features of the potential at large and intermediate distances are survived and the length of Casimir scaling increase. One can use the same arguments for SU(3) gauge group for explaining the potentials induced by the domains in figure 8.

Comparison between SU(N) and G(2) gauge groups
As argued, for SU(N ) gauge groups which have non trivial center elements, the group factor of the vacuum domain changes between 1 and non trivial center elements of the gauge group. It is interesting to compare the behavior of the group factors of SU(N ) gauge group and G(2) gauge group which has only trivial center element z 0 = 1. For G(2) gauge group, a linear regime in agreement with Casimir scaling is observed from both lattice gauge theory [18,26] and domain model [27]. The entire G(2) group can be covered by six SU(2) subgroups [23]. Three of them, the non reducible ones, generate an SU(3) subgroup of G(2) which is seven dimensional and reducible. The representations of the remaining three SU(2) subgroups are seven dimensional, but they are reducible. The center elements of the SU(3) and SU(2) subgroups of G(2) in the fundamental representation are given by where I is the unit matrix, and z a ∈ {z 0 = 1, z 1 = e 2πi 3 , z 2 = e 4πi 3 } for the SU(3) subgroup center elements, and z a ∈ {z 0 = 1, z 1 = e πi } for the SU(2) subgroup center elements. We discussed the possible reasons of observing the confined potential at intermediate distances in our previous article [30]. We studied Re(G r ) for the G(2) gauge group. Using ansatz given in eq. vortex. One can do the same discussion in the vicinity of the right timelike leg of the Wilson loop.
As a result, compared with SU(N ) Yang-Mills theory where local extremums correspond to non trivial center elements of the gauge group, the local extremums for G (2) correspond to the non trivial center elements of the SU(2) and SU(3) subgroups. In other word, the vacuum domain in SU(N ) depends on the center vortices of the gauge groups and in G(2) depends on the center vortices of its subgroups.

Conclusions
Applying thick vortex model, which contains the vacuum domains, to the SU(2) and SU(3) gauge groups and using square ansatz for angle α C (x), we show that the static potentials of various representations grow linearly at intermediate distances and agree with N -ality at large distances. We compute Casimir ratios for SU(2) and SU(3) color sources at intermediate distances and we show that they are qualitatively in better agreement with Casimir ratios when using the square ansatz rather than the old ansatz for angle α C (x). We also study the contributions of the vacuum domain and center vortices to the static potentials. Our results for SU(2) and SU(3) gauge groups show that the potential ratios obtained from the vacuum domain agree better with Casimir scaling than the potential ratios obtained JHEP03(2015)016 The six independent states are: