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 $\alpha_{C}(x)$. 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], 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: or VQ Q (R) → σR . (1.1) 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.
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 [8,9,10], 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 [11]. 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 [11,12,13,14,15,16,17,18]. 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 [11,19]. 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 [20]. In our previous article [23], 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 Sect. 2, we briefly review the domain structure model. Angle parameter of the model is studied in Sect. 3. Then, in Sect. 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 Sect. 5. We summarize the main points of our study in Sect. 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 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 [1,11]: where the function α n C (x) represents the corresponding angle and 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.

Ansatz for the angle α C (x)
There are some functions to use as the ansatz for α C (x) [1,11,25]. 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: 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. 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. [11]. 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 word, 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. Now one should take two intervals for the square ansatz: 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 = 200. 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 [11] when the vacuum domains have been added to the model. In our previous papers [20,23], 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]: and the contribution of the vacuum domain added to the static potential is given by [11]: 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: 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 V 1 (R)/V 1/2 (R) and V 3/2 (R)/V 1/2 (R) induced 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, Fig. 7 shows potential ratios using the ansatz given in Eq. (3.2), for the choice of parameters f = 0.1, a = 0.05, and b = 4. The potential ratios V 1 (R)/V 1/2 (R) and V 3/2 (R)/V 1/2 (R) induced by center 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 Fig. 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. 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 Fig. 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.

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  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:    : Ratios of V j (R)/V 1/2 (R) induced by center vortex for adjoint and j = 3/2 representations using old ansatz for angle α C (x). The selected free parameters are f 1 = 0.1, a = 0.05, b = 4 [1]. The potential ratios start from the Casimir ratios but the ratios by this ansatz drop steeper than the ratios by square ansatz.
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     [1,20], respectively. Therefore, the potential ratios drop slower using square ansatz compared with the old ansatz. From Fig. 8, it is clear that at large distances, R ≥ 100, the static potentials induced by all domains agree with N -ality. As shown in Fig.  9, the potentials are screened at large distances where vacuum domain locates completely inside the Wilson loop. Therefore, to get the correct potentials at large distances, one has to use the center vortices and it is clear that the vacuum domains do not give the correct behavior. At large distances, a pair of gluon-anti gluon are popped out of the vacuum and combined with initial sources, and transform them into the lowest order representations in their class. Some examples 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)    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. In the following subsection, we study the 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. (2) For SU (2)  and also simultaneous creation of two oppositely oriented center vortices linked to a Wilson loop, produces a vacuum domain as the following:

Center domains in SU
We recall that combining the center vortices fluxes has been studied in Ref. [21], 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 is obtained. 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 [22]. 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 (1) max for the fundamental representation is equal to 2π. Figure 13 plots G r (α (n) ) versus x for a Wilson loop with R = 100 for the fundamental representation of SU (2) using square ansatz. The Wilson loop legs are located at x = 0 and x = 100. When the center vortex overlaps the minimal area of the Wilson loop, it affects the loop. The group factor interpolates smoothly from −1, when the vortex core is located entirely within the Wilson loop, to 1, when the core is entirely outside the loop. The interaction between center vortices is not considered.   is zero and in this case Re(G r ) = 1. Fig. 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  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. 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 Figs. 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. So 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 Fig.  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 z 1 z * 1 vacuum domain linked to different medium size loops (R < 20) is 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.
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 and 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 Fig. 4. It is obvious that both vacuum structures z 1 z * 1 and (z 1 ) 2 have contribution in the potential but it seems that z 1 z * 1 has a dominant role in increasing the Casimir regime. One can use the same arguments for SU (3) gauge group for explaining the potentials induced by the domains in Fig. 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 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 [11,19] and domain model [20]. The entire G(2) group can be covered by six SU (2) subgroups [16]. 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 [23]. We studied Re(G r ) for the G(2) gauge group. Using ansatz given in Eq. (3.4), Fig. 17 plots Re(G r (α (n) )) versus x for the 7 dimensional (fundamental) representation of G(2) gauge group for R = 100 . The timelike legs of the Wilson loop are located at x = 0 and x = 100. The group factor of the vacuum domain changes between 1 and the non trivial center elements of the SU (2) and SU (3) subgroups as the following: For SU (N ) gauge group, we have argued that vacuum domains may appear as a results of simultaneously creation of some center vortices linked to the Wilson loop. Therefore the foot print of center vortices has been observed in extremums of the vacuum domain group factor. Although G(2) gauge group does not have any center vortex but the extremums of the vacuum domain group factor have been related to the subgroups of G(2). One may argue that the G(2) vacuum is filled with center vortices of the subgroups. Simultaneous creation of three similarly oriented center vortices of the SU (3) subgroup linked to a Wilson loop may give a vacuum domain: vortex. One can do the same discussion in the vicinity of the right timelike leg of the Wilson loop.
As a result, in comparison 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 from center vortices. We discuss about the reason of these observations by studying the potentials and the group factor G r (α (n) ). The group factor plays an important role in the potential between quarks. According to vortex theory, the vacuum of QCD is filled with non trivial center vortices. One can construct vacuum domain by simultaneously creation of the center vortices linking the Wilson loop. We have discussed about the z 1 z * 1 and (z 1 ) 2 vacuum domains in the SU (2) gauge group. It seems that two oppositely center vortices in z 1 z * 1 vacuum domain attract each other and two similarly oriented vortices in (z 1 ) 2 vacuum domain repel each other. Therefore in SU (2) gauge group, the group factor of (z 1 ) 2 vacuum domain changes between 1 and non trivial center elements of the gauge group and the one of z 1 z * 1 vacuum domain changes between 1 and 0.9. Since the potential ratios start out at the ratios of the corresponding Casimirs, therefore for small size loops (R ≈ 1, 2) where the group factor is close to one, the potentials strongly agree with the Casimir scaling. In the intermediate regime, the potential ratios obtained from z 1 z * 1 vacuum domain drop slower than the one obtained from center vortices and the potential ratios obtained from (z 1 ) 2 vacuum domain drop faster than the one obtained from center vortices. Therefore the length of Casimir scaling regime increases by adding the contribution of the z 1 z * 1 vacuum domain to the potential induced by center vortices. On the other hand, by adding the contribution of the (z 1 ) 2 vacuum domain to the potential, the length of Casimir scaling regime decreases.
Comparison between the behavior of the group factor in SU (N ) gauge group with non trivial center elements, and G(2) gauge group with no non trivial center element is done, as well. In SU (N ) gauge groups, the group factor changes between 1 and center vortices of the group, but in G(2) gauge group it changes between 1 and center vortices of SU (2) and SU (3) subgroups. One can argue that the SU (2) and SU (3) subgroups have dominant roles in confinement regime in G(2) gauge group.