PHYSICAL REVIEW D 97, 056015 (2018)

F4, E6 and G2 exceptional gauge groups in the vacuum domain structure model

† Amir Shahlaei* and Shahnoosh Rafibakhsh Department of Physics, Science and Research Branch, Islamic Azad University, Tehran 14665/678, Iran

(Received 4 December 2017; published 21 March 2018)

Using a vacuum domain structure model, we calculate trivial static potentials in various representations of F4, E6,andG2 exceptional groups by means of the unit center element. Due to the absence of the nontrivial center elements, the potential of every representation is screened at far distances. However, the linear part is observed at intermediate quark separations and is investigated by the decomposition of the exceptional group to its maximal subgroups. Comparing the group factor of the supergroup with the corresponding one obtained from the nontrivial center elements of SUð3Þ subgroup shows that SUð3Þ is not the direct cause of temporary confinement in any of the exceptional groups. However, the trivial potential obtained from the group decomposition into the SUð3Þ subgroup is the same as the potential of the supergroup itself. In addition, any regular or singular decomposition into the SUð2Þ subgroup that produces the Cartan generator with the same elements as h1, in any exceptional group, leads to the linear intermediate potential of the exceptional gauge groups. The other SUð2Þ decompositions with the Cartan generator different from h1 are still able to describe the linear potential if the number of SUð2Þ nontrivial center elements that emerge in the decompositions is the same. As a result, it is the center vortices quantized in terms of nontrivial center elements of the SUð2Þ subgroup that give rise to the intermediate confinement in the static potentials.

DOI: 10.1103/PhysRevD.97.056015

I. INTRODUCTION AND MOTIVATION gauge group center. A center vortex, which is topologically linked to a Wilson loop, changes the Wilson loop by a Quantum chromodynamics is the theory of strong group factor z : interactions. Quarks interact via gluons that are strong n force carriers and are attributed to the adjoint representation ð Þ → ð Þk ð Þ ð Þ of the SUð3Þ gauge group. The non-Abelian nature of W C zn W C ; 1 gluons causes QCD to be fundamentally a nonperturbative ¼ ð2πinÞ ¼ 1 2 … − 1 theory in the infrared sector. To understand the phenomena where zn exp N , n ; ; ;N , and k represents of the low energy regime, such as confinement, some the N-ality of the representation r. This property implies topological field configurations, such as center vortices a linear potential between static quarks, which means [1–15], are believed to play the key role in the nontrivial confinement. vacuum of QCD. They assign a criterion to confinement The thick center vortex model developed by Faber et al. [16] is aimed at studying the potentials of higher through the area-law falloff of the Wilson loop, which is ð Þ one of the most efficient order parameters for investigating representations of SU N gauge groups. In this model, the quark-antiquark static potential behaves differently in the large distance behavior of QCD. three regions. At short distances, the interaction is deter- In the center vortex model, confinement is the result mined by one-gluon exchange, which leads to a Coulomb- of the interaction between center vortices and the Wilson like potential [17–19]. At intermediate distances, the string loop. In fact, the Wilson loop running around the vortex tension of the linear potential is proportional to Casimir measures the vortex flux, which is quantized in terms of the scaling. In the asymptotic region, potentials of all repre- sentations with zero N-ality are screened. However, the *[email protected] potential of nonzero N-ality representations becomes par- † Corresponding author. allel to the one of the lowest representation with the same [email protected] N-ality [16,20–23]. In 2007, Greensite et al. [24] claimed that there is no Published by the American Physical Society under the terms of obvious reason to exclude the trivial center element from the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the model. In fact, even in G2 gauge theory, which only the author(s) and the published article’s title, journal citation, includes one trivial center element, one expects a linear and DOI. Funded by SCOAP3. potential from the breakdown of the perturbation theory to

2470-0010=2018=97(5)=056015(28) 056015-1 Published by the American Physical Society AMIR SHAHLAEI and SHAHNOOSH RAFIBAKHSH PHYS. REV. D 97, 056015 (2018) the onset of screening while all asymptotic string tensions where the group factor is described as are zero. Monte Carlo numerical lattice calculations for G2 1 [25–27] also confirm a confining potential, despite the G ½α⃗n ð Þ ¼ ½ ð α⃗n ⃗Þ ð Þ r C x Tr exp i C · H ; 3 absence of nontrivial center elements. In the new model, a dr vacuum domain is a closed tube of magnetic flux that, in which d depicts the dimension of the group represen- unlike a vortex, is quantized in units corresponding to the r tation; H , i ¼ 1; …;N− 1, are simultaneous diagonal gauge group trivial center. The string tensions are produced i generators of the group spanning the Cartan subalgebra; from random spatial variations of the color magnetic flux and n represents the type of center vortex. Vortices of type n quantized in terms of unity. But, what accounts for the and type N − n are complex conjugates of each other and intermediate linear potential in such gauge groups? To their magnetic fluxes are in the opposite directions. answer this question and by using the idea of domain Therefore, structures, Deldar et al. [28–30] showed that SUð2Þ and SUð3Þ subgroups of G2 have an important role in the G ½α⃗n ð Þ ¼ G½α⃗N−nð Þ ð Þ r C x r C x : 4 intermediate confinement of G2. In fact, they were moti- α⃗n ð Þ vated by the two works in Refs. [24,31]. Holland et al. [31] The function C x is the vortex profile ansatz. It depends investigated that a scalar Higgs field in the fundamental on the location of the vortex midpoint x, from the Wilson representation of G2 can break to SUð3Þ representations. loop, the shape of the contour C, and the vortex type n. So, one is able to interpolate between exceptional and Mathematically, there are various candidates that can ordinary confinement. Moreover, Greensite et al. [24] used simulate a well-defined potential, but all of them should the Abelian dominance idea to study SUð3Þ and SUð2Þ obey the following conditions: dominance in the G2 gauge theory. Therefore, it seems (1) As R → 0, then α → 0. interesting to investigate how confinement appears in a (2) When the vortex core lies entirely outside the theory with exceptional gauge groups in the framework of minimal planar area enclosed by the Wilson loop, the vacuum domain structure model. there is no interaction: In this paper, using the same method as in Refs. [28–30], ½ α⃗n ⃗¼I ⇒ α⃗n ¼ 0 ð Þ we present a general scheme to understand what kind of exp i C · H C : 5 group decompositions lead to the temporary confinement of the exceptional gauge groups in the vacuum domain struc- (3) Whenever the vortex core is completely inside the ture model. In the next section, the thick center vortex and planar area of the Wilson loop, vacuum domain structure models are discussed briefly. In exp½iα⃗n · H⃗¼z I ⇒ α⃗n ¼ α⃗n : ð6Þ Sec. III, some properties of exceptional groups are inves- C n C max tigated. We apply F4, E6,andG2 in the vacuum domain structure model and calculate the potentials in different Here, we have chosen the flux profile introduced in representations in Sec. IV. The decomposition of these gauge Ref. [16]: groups into their subgroups is investigated as well. α⃗n b α⃗n ðxÞ¼ max 1 − tanh ayðxÞþ ; ð7Þ C 2 R II. THICK CENTER VORTEX MODEL AND VACUUM DOMAIN STRUCTURES where a and b are free parameters of the model. The distance between the vortex midpoint and the nearest A center vortex is a closed tube of magnetic flux that is timelike leg of the Wilson loop is measured by yðxÞ: quantized in terms of the nontrivial center elements of the gauge group. It might be considered as linelike (in three x − R for jR − xj ≤ jxj dimensions) or as a surfacelike (in four dimensions) object. yðxÞ¼ : ð8Þ − j − j j j In a pure non-Abelian gauge theory, the random fluctua- x for R x > x tions in the number of center vortices that pierce the It seems changing the ansatz may have no effect on the minimal area of the Wilson loop give rise to the asymptotic G ½α⃗ð Þ string tension. In fact, a thin center vortex is capable of extremum points of the group factor r x [32], whereas α⃗n ðxÞ inducing the linear potential for the fundamental represen- an alteration of C is influential in the potential itself in a tation of the gauge group. Thickening the center vortices way that string tension ratios might be more or less in agreement with Casimir scaling [21,30]. leads to a bigger piercing area and these topological objects Now we are able to write the Wilson loop for SUðNÞ should be described by a profile function. Therefore, the gauge groups: gauge group centers in Eq. (1) should be replaced by a group factor, Y XN−1 n hWðCÞi ¼ 1 − f ð1 − G ½α⃗ ðxÞÞ hW0ðCÞi; ð9Þ ð Þ → G ½α⃗n ð Þ ð Þ n r C W C r C W C ; 2 x n¼1

056015-2 F4, E6 AND G2 EXCEPTIONAL GAUGE … PHYS. REV. D 97, 056015 (2018) where fn shows the probability that the midpoint of a center vortex is located at any plaquette in the plane of the Wilson loop. The probability of locating center vortices of any type at any two plaquettes is independent, which is an over- simplification of the model. hW0ðCÞi is the Wilson loop expectation value when no vortices are linked with the loop. It should be noted that in addition to the regions associated with the nontrivial center elements, the domains corre- sponding to unity center elements are also allowed in the vacuum. Therefore, the sum in Eq. (9) should contain n ¼ 0 as well. Using the fact that fn ¼ fN−n, the static potential between a color and an anticolor source induced by thick center vortices and vacuum domains is FIG. 1. Dynkin diagrams of G2, F4, and E6 exceptional groups. mX¼þ∞ XN−1 VðRÞ¼− ln 1 − f ð1 − ReG ½α⃗n ðx ÞÞ ; n r C m The sum of the ranks of the regular subgroups is equal to m¼−∞ n¼0 the rank of their supergroup. However, this is not true for ð Þ 10 the singular subgroups. It should be noted that if a factor ð1Þ where n ¼ 0 denotes a vacuum domain type vortex and U appears in a subgroup, it makes the subgroup as a n ¼ 1; …;N− 1 indicates the type of center vortex. nonsemisimple one. In this article, we briefly discuss how to derive the subgroups of F4 and use the same method for other III. SOME PROPERTIES OF exceptional groups. The extended Dynkin diagram is EXCEPTIONAL GROUPS structured by adding the most negative root (−γ) to the The ideas of symmetries and Lie exceptional groups set of simple roots (Fig. 2). Then, by omitting the original β have always been attractive in modern high energy physics. i roots, regular subgroups will emerge one by one. For β G2 is the simplest exceptional gauge group that confirms example, in Fig. 2, eliminating the root 2 leads to the ð3Þ ð3Þ the chance of having confinement without the center [31]. SU × SU subgroup of F4. Moreover, when the root β ð9Þ G2 gauge theory is a theoretical laboratory in which SUðNÞ 4 is omitted, the SO subgroup of F4 is obtained. It β subgroups are embedded. This provides us with an under- should be pointed out that omitting the root 3 gives off the ð2Þ ð4Þ standing not only about the exceptional G2 confinement but SU × SU subgroup. In some references, it has been also about the SUð3Þ confinement that happens in nature. claimed as a direct subgroup of F4 [35] and in some others ð9Þ In this section, we briefly explain some properties of the it is not [36,37]. However, it is a direct subgroup of SO . exceptional groups applied in this article, including their Therefore, it might be, at least, considered as an indirect subgroups and Dynkin diagrams. subgroup of F4. To achieve singular maximal subgroups of In general, there are five distinguishable exceptional exceptional groups, a determined method does not exist and groups named G2, F4, E6, E7, and E8. The subscripts each subgroup has to be extracted individually [34].All point out the ranks of the groups. The numbers of simple maximal subgroups of the F4 exceptional group have been roots and simultaneous diagonal generators of simple Lie presented in Table I. groups are equivalent to their rank. One may draw the whole root diagram by having simple roots and the angles between them in a simple Lie group. The angle between simple roots in a Dynkin diagram is always obtuse. Three, two, one or no lines between simple roots measure their mid angles, which are 150°, 135°, 120°, or 90°, respectively [33]. Figure 1 depicts Dynkin diagrams of the exceptional groups used in this research. Filled circles represent shorter roots and empty ones show longer roots in terms of their length. Using Dynkin diagrams, one is able to find the sub- groups of every lie group. There are three different sorts of maximal subgroups [34]:

(i) Regular maximal nonsemisimple subgroups, FIG. 2. Three different regular maximal subgroups of F4 (ii) Regular maximal semisimple subgroups, obtained from its extended Dynkin diagram by omitting the (iii) Singular (special) maximal subgroups. original simple roots one by one.

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5 6 7 8 9 10 TABLE I. Maximal subgroups of some exceptional groups h1 ¼ N1ðD5 þ D6 − D7 þ D8 − D9 − D10Þ; [37]. [R] and [S] represent regular and singular subgroups of each ¼ ð 3 þ 4 − 5 − 6 þ 10 − 11Þ group, respectively. h2 N2 D3 D4 D5 D6 D10 D11 ; ¼ N3 ð 2 − 2 3 − 4 þ 6 − 8 þ 9 E6 F4 G2 h3 2 D2 D3 D4 D6 D8 D9 ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ SU ×SU SU ×SU [R] SU [R] − D10 þ D11 − D12Þ; ×SUð3Þ [R] 10 11 12 SUð2Þ×SUð6Þ [R] SUð2Þ×Spð6Þ [R] SUð2Þ × SUð2Þ [R] N4 2 3 4 5 6 7 h4 ¼ ð−2D þ D − D þ D − D þ D SOð10Þ × Uð1Þ [R] SOð9Þ [R] SUð2Þ [S] 2 2 3 4 5 6 7 ð3Þ ð2Þ SU × G2 [S] SU × G2 [S] − D9 þ D12 − D13Þ; ð14Þ SUð3Þ [S] SUð2Þ [S] 9 12 13 Spð8Þ [S] G2 [S] where F4 [S] b ¼ − ð Þ Da Iab Iba¯ ; 15

and Iab are 26 × 26 matrices with the following matrix Based on the branching rules, an irreducible representa- elements: tion of a group can be decomposed into the irreps of its ð Þ ¼ δ δ ð Þ subgroup as follows [36]: Iab jk aj bk: 16

ð Þ¼⨁ ð Þ ð Þ Subscripts j and k take on the same values as a and b such R G miRi g ; 11 ∶−13 ≤ ≤ 13 i that a; b j, k with zero excluded. Using the standard normalization condition ð Þ ð Þ where R G is an irrep of the supergroup G and Ri g is the 1 Tr½h ;h ¼ δ ; ð17Þ irrep of the subgroup g. mi is the degeneracy of the a b 2 ab representation RiðgÞ in the decomposition of representation 36,37] RðGÞ ]. To be more precise, we consider one of the we calculate the normalization factors as follows: regular subgroups of F4: 1 N1 ¼ N2 ¼ pffiffiffi ; 2 6 F4 ⊃ SUð3Þ × SUð3Þ: 1 N3 ¼ N4 ¼ pffiffiffi : ð18Þ 2 3 Using the branching rules, one might write [36–38] To find the maximum amount of the domain structure ¯ ¯ 26 ¼ð8; 1Þ ⊕ ð3; 3Þ ⊕ ð3; 3Þ: ð12Þ flux, we use Eq. (6), using the fact that the F4 gauge group includes only one trivial center element, From Eq. (11), it is obvious that the first numbers in each exp½iα⃗· H⃗¼I; ð19Þ parenthesis could be considered as the degeneracy of the second representation emerging in the decomposition: and we find pffiffiffiffiffi ¯ 26 ¼ 8ð1Þþ3ð3Þþ3ð3Þ: ð13Þ αmax ¼ 2π 24; 1 pffiffiffiffiffi max α2 ¼ 6π 24; Therefore, an F4 “quark” is made up of three SUð3Þ quarks, pffiffiffiffiffi αmax ¼ 4π 48; three antiquarks, and one singlet. 3 pffiffiffiffiffi max α4 ¼ 2π 48: ð20Þ IV. CONFINEMENT WITHOUT A CENTER Now, one can calculate the static potential of Eq. (10) for A. F exceptional group 4 the fundamental representation of the F4 exceptional gauge The F4 exceptional group has rank four and contains group. This potential has been pictured in Fig. 3 for four Cartan generators. The diagonal generators for the R ∈ ½1; 100. The free parameters of the model are chosen fundamental 26-dimensional representation of F4 are to be a ¼ 0.05, b ¼ 4, and f ¼ 0.1 in every calculation of [39,40] this article.

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5 Fund. Rep. 26 × 26 ¼ 1 ⊕ 26 ⊕ 52 ⊕ 273 ⊕ 324: ð23Þ 4.5

4 As in SUðNÞ gauge groups, three F4 “quarks” can create a

3.5 baryon:

3 26 × 26 × 26 ¼ 1 ⊕ 5ð26Þ ⊕ 2ð52Þ ⊕ 4ð273Þ ⊕ 3ð324Þ 2.5 V(R) ⊕ 3ð1053Þ ⊕ 1274 ⊕ 2652 ⊕ 2ð4096Þ: 2 ð Þ 1.5 24 1 G ½α⃗n ð Þ Evidently, the function Re r C x looks predominant 0.5 in the potential formula in Eq. (10). It shows that the group 2πink 0 factor varies between 1 and expð Þ, corresponding to the 0 10 20 30 40 50 60 70 80 90 100 N R N-ality=k of the representation and the vortex type n.An unaffected Wilson loop that has not been pierced by any FIG. 3. The potential between two static sources in the vortex means ReG ½α⃗n ðxÞ ¼ 1. When the vortex is linked ∈ ½1 100 r C fundamental representation of F4 for R ; . Screening is to the Wilson loop, the group factor deviates from 1. Hence, clearly observed at large quark separations while the potential is to investigate what happens to the F4 potentials, one may linear at intermediate distances. The free parameters of the model study the behavior of the group factor. have been chosen to be a ¼ 0.05, b ¼ 4, and f ¼ 0.1. In Ref. [32], it has been proven that the third Cartan generator of the SUð4Þ gauge group, i.e., H3 ¼ In Fig. 3, the linear potential is demonstrably located in p1 ffiffi ½1 1 1 −3 2 6 diag ; ; ; , can produce the total potential indi- the approximate range of R ∈ ½2; 9. In addition, at large vidually. In the F4 exceptional group, one might use only distances where the vacuum domain is entirely located h1 or h2 Cartan generators or both of them together to find inside of the Wilson loop, a flat potential is induced. Hence, the same group factor and also the same potential as if we one can deduce that in groups without a nontrivial center, apply all four Cartan generators in our calculations. This static potentials of all representations behave like a SUðNÞ property will be helpful in the decomposition of the F4 representation with zero N-ality. representations into its subgroups. In fact, when the The adjoint representation of F4 is 52 dimensional. As a identical diagonal generators are constructed, the same “ ” consequence, like any gauge group, the bosonic gluons potentials as the F4 itself will be achieved. of the F4 exceptional group are made of the adjoint In Fig. 4, the real part of the group factor versus the representation. Thus, mathematically one can derive the location x of the vacuum domain midpoint has been plotted, way of screening of color sources in any representation for R ¼ 100 and in the range x ∈ ½−200; 300,by from tensor products of that representation with the adjoint one; i.e., when a singlet emerges, it means screening. So, 1 for the fundamental representation, one may write

0.8 26 × 52 ¼ 26 ⊕ 273 ⊕ 1053: ð21Þ

0.6 α] [

Therefore, the fundamental representation of F4 cannot be f screened just by one set of “gluons”. Energetically speak- Re G ing, color sources in the fundamental representation of F4 0.4 are not screened until the potential reaches that extent “ ” where four sets of gluons pop out of the vacuum: 0.2

all Cartan generators 4 times h1 and h2 only zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ 0 26 52 52 ¼ 1 ⊕ 46ð26Þ ⊕ 10ð52Þ ⊕ ð Þ -200 -100 0 100 200 300 × × × : 22 x

These tensor products have been calculated by the LieART FIG. 4. The real part of the group factor versus x, the location of the vacuum domain midpoint, for the fundamental representation project in Mathematica [41]. The numbers out of the of the F4 exceptional gauge group in the range x ∈ ½−200; 300, parentheses are the degeneracy of the representations being by applying h1 and h2 Cartan generators (stars) and all diagonal repeated in the tensor product. Therefore, four F4 “gluons” generators (solid line). It is clear that the two sets of data are the are able to screen an F4 “quark” to create a color singlet same. The distance R between color and anticolor sources is hybrid qGGGG. Moreover, two “quarks” form a singlet. equal to 100.

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1 in Eq. (25) is considered, its components are identical to the

0.8 ones for the Cartan generators h1 and h2 of Eq. (14).But 26 this is not the case with H8 . To examine the results coming 0.6 out of these two matrices, one is supposed to establish the same normalization condition as in Eq. (19): 0.4 α] [ f ð α26 26 þ α26 26Þ¼I ð Þ exp i max1 H3 i max2 H8 : 27

Re G 0.2

0 In this case, we have six distinctive equations and find pffiffiffi 26 -0.2 α ¼ 2π 6 max1 ; all Cartan generators pffiffiffi h3 and h4 only 26 -0.4 α ¼ 6π 2: ð28Þ -200 -100 0 100 200 300 max2 x Now, the potential of Eq. (10) could be calculated using FIG. 5. The same as Fig. 4 but in comparison with the group Eqs. (25) and (28). This potential is identical to the one in factor when only h3 or h4 is used. 26 Fig. 3, despite the difference between H8 and h1 or h2.To investigate this matter, one might manually estimate the considering all generators and also by utilizing only h1 and value of ReGr½α when the vacuum domain is completely h2. It is clear that both diagrams in Fig. 4 are identical and inside the Wilson loop: the group factor reaches the minimum amount of ≈0.076 at ¼ 0 ¼ 100 1 x and x . To confirm our conclusion, we have G1½α ¼ ð ½ ð α26 26ÞÞ ≈ 0 076 Re r SU3×SU3 ×Re Tr exp i max1 · H3 . plotted a similar diagram in Fig. 5 but by using only h3 or 26 h4 diagonal generators. In this figure, the group factor ð29Þ reaches the minimum amount equal to ≈−0.23, which is way less than the minimum amount of the F4 group factor. 1 It has been shown that [32] the group factor reaches the ReG2½α ¼ ×ReðTr½expðiα26 · H26ÞÞ ≈ 0.076: r SU3×SU3 26 max2 8 minimum points where 50% of the vortex maximum flux ð Þ enters the Wilson loop. These points are responsible for 30 the intermediate linear potential. We aim to show that the SUðNÞ subgroups of the exceptional groups might be the It is clear that both group factor functions earned by either α26 α26 reason for the appearance of the linear potential at inter- max1 or max2 reach the same amount, which is the mediate distances. It means that the minimum points of the minimum amount of the F4 group factor in Fig. 4 as well. group factor could be explained by the group decomposi- Consequently, based on the analogy of the group factor 26 26 tion into the subgroups. functions acquired by both H3 and H8 , we claim that, although the second matrix of SUð3Þ × SUð3Þ has different 1. SUð3Þ ×SUð3Þ decomposition components, it has the same effect as the Cartan h1 on the F4 group. Then, the trivial static potential of the F4 Using the decomposition in Eq. (13), we are able to exceptional group is similar to the potential gained by reconstruct Cartan generators of F4 with respect to its its SUð3Þ × SUð3Þ subgroup. Therefore, it seems that this SUð3Þ subgroup, decomposition could be generalized for higher representa- 8 times tions to find the corresponding potential. 1 zfflfflffl}|fflfflffl{ 26 pffiffiffi 3 3 3 3 3 3 The decomposition of the 52-dimensional adjoint rep- Ha ¼ diag½0; …; 0; λa; λa; λa; −ðλaÞ ; −ðλaÞ ; −ðλaÞ ; 6 resentation of the F4 is [36–38] ð25Þ 52 ¼ð8; 1Þ ⊕ ð1; 8Þ ⊕ ð6¯; 3Þ ⊕ ð6; 3¯Þ; where λ3, a ¼ 3, 8 are Cartan generators of SUð3Þ in the a 52 ¼ 8ð1Þþ1ð8Þþ6ð3Þþ6ð3¯Þ: ð31Þ fundamental representation:

1 F4 “ ” SUð3Þ λ3 ¼ ½1 −1 0 This shows that gluons are made of the usual 3 2 diag ; ; ; gluons (representation 8) and some additional gluons 1 consist of SUð3Þ quarks (representation 3) and antiquarks λ3 ¼ pffiffiffi diag½1; 1; −2: ð26Þ 3¯ 8 2 3 (representation ) and also eight singlets. It is clear that these representations have different trialities. Meanwhile, the matrices of Eq. (25) are normalized using Using Eq. (31), the Cartan generators of F4 in the adjoint 26 the normalization conditions in Eq. (17). If the matrix H3 representation might be reconstructed as follows:

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6 7 8 times times 26 1 zfflfflffl}|fflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ 52 pffiffiffiffiffi 8 3 3 52 Ha ¼ diag½0; …; 0; λa; λa; λa; 6 273 18 324 zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{6 times 5 −ðλ3Þ … −ðλ3Þ ð Þ a ; ; a ; 32 4 V(R) 3 3 where λa, a ¼ 3, 8, are the same generators as in Eq. (26) 8 and λa are simultaneous diagonal generators of the SUð3Þ gauge group in the adjoint eight-dimensional representa- 2 tion. Using Eq. (19), the maximum values of the vortex flux 1 for the adjoint representation of the F4 ⊃ SUð3Þ × SUð3Þ decomposition are 0 0 20 40 60 80 100 pffiffiffi R α52 ¼ 6π 2; max1 3 pffiffiffi 26 α52 ¼ 6π 6 ð Þ 52 max2 : 33 273 2.5 324

The potential between static sources in the fundamental 2 and adjoint representations of the F4 exceptional gauge group has been plotted in Fig. 6, along with the higher 1.5 ∈ ½1 100

representations in the range R ; . The decomposi- V(R) tion of the higher representations and the corresponding 1 Cartan generators have been presented in Appendix A.In Fig. 6, screening is observed for the potentials of every 0.5 representation at far distances. Since F4 does not own any 0 nontrivial center element, all representations act like ð Þ SU N representations with zero N-ality. Hence, screening -0.5 2 3 4 5 6 7 8 9 was anticipated. Another reason for this phenomenon is the R creation of gluons in the QCD vacuum that are able to screen the initial static color charges and produce a flat FIG. 6. Upper diagram: The potential between static color potential at high levels of energy: sources in the fundamental, adjoint, 273-dimensional, and 324- dimensional representations of the F4 exceptional gauge group in the range R ∈ ½1; 100. All potentials are screened at far distances, 52 52 ¼ 1 ⊕ 52 ⊕ 324 ⊕ × ; while linearity is evident at intermediate parts. Lower diagram: ∈ ½2 9 273 × 52 × 52 × 52 ¼ 1 ⊕ 15ð26Þ ⊕ ; The same as the upper diagram but in the range R ; . The slopes of the potentials have been given in the fourth row of 324 × 52 × 52 ¼ 1 ⊕ 26 ⊕ 3ð52Þ ⊕ : ð34Þ Table II. The potentials are in agreement with Casimir scaling qualitatively. Furthermore, in Fig. 6, there are linear parts at intermediate distance scales for all representations that are situated at the interval R ∈ ½2; 9, approximately. The linear potentials have been depicted in the lower diagram of Fig. 6. The slope of the linear potentials of different representations are TABLE II. Casimir numbers and Casimir ratios of different given in the fourth column of Table II, as well as the representations of the F4 exceptional group are presented in the second and third columns, respectively [42].a The slopes of the potential ratios (kr ) in the last column. It is observed that kF linear potentials of Fig. 6 and the potential ratios are given in the potential ratios are qualitatively in agreement with Casimir fourth and fifth columns, respectively. The numbers in paren- scaling (Cr ), which is presented in the third column of CF theses show the fit error. Table II. However, Casimir scaling has not been proved Rep. Casimir number Cr Potential slope kr numerically for F4. CF kF 2 Figure 7 presents the point-by-point ratio of the potential 26 3 1 0.252(7) 1 of each representation to the fundamental one in the range 52 1 1.5 0.331(7) 1.31(1) R ∈ ½1; 20. These ratios start up at the ratios of the 273 2 2 0.374(8) 1.48(1) 13 corresponding Casimirs. However, they abruptly decline 324 9 2.16 0.38(1) 1.5(1) at intermediate intervals. The inclination becomes more aIt should be noted that Casimir scaling of the representation pronounced as the dimension of the representations grows; 273 has not been reported in this reference.

056015-7 AMIR SHAHLAEI and SHAHNOOSH RAFIBAKHSH PHYS. REV. D 97, 056015 (2018) pffiffiffi 2.2 26− V /V α non ¼ 2π 6 52 26 max1 ; 2.1 V273/V26 pffiffiffi 26− V324/V26 α non ¼ 2π 2 ð Þ 2 max2 ; 37

1.9 and 1.8 pffiffiffi 52− 1.7 α non ¼ 6π 2; max1 pffiffiffi 1.6 α52−non ¼ 2π 6 ð Þ Potential ratios max2 ; 38 1.5

1.4 where the term “non” denotes a nontrivial center element. It

1.3 should be mentioned that an unusual normalization con- dition has been applied in Eq. (36). Therefore, neither the 1.2 0 5 10 15 20 G2 potentials nor the SUð3Þ ones are expected. However, as R the Cartan generators of Eq. (25) are taken, we expect the potentials obtained from Eqs. (37) and (38) to be parallel to FIG. 7. Potential ratios of the F4 representations to the fundamental one in the range R ∈ ½0; 20. These ratios start up the corresponding ones in Fig. 6, in some range of R.To at the values of the corresponding Casimir ratios presented in study this fact more accurately, we study the group factor Table II. function. The minimum points of the group factor function, which happen at the positions where half of the vortex flux enters e.g., the deviation from the exact Casimir scaling is more the Wilson loop, are responsible for the intermediate linear significant for representations 273 and 324. potential. Therefore, we compare the group factors of To investigate whether the linear potentials of the F4 different representations of F4 obtained from the trivial exceptional group are caused by the nontrivial center center element with the ones calculated from the decom- ð3Þ ð3Þ elements of the SU × SU subgroup or not, one position into the SUð3Þ subgroup. G ½α⃗ may plot the group factor function Re r with respect Figure 8 shows the real part of the group factor function ð3Þ to the nontrivial center elements of SU . Using the same for both fundamental and adjoint representations of the – method as in Refs. [28 30] and Eqs. (13) and (31), we are group F4 and the SUð3Þ subgroup using its nontrivial able to compose matrices containing center elements of center elements. The discrepancies in the minimum ð3Þ SU depending on the N-ality of each representation. amounts of the group factors in these figures are undeni- Thus, able. As a result, one might say that the center elements of the SUð3Þ subgroup are not the direct factors for the Z26 ¼ ½1 1 1 1 1 1 1 1 I I I SUð3Þ diag ; ; ; ; ; ; ; ;z 3×3;z 3×3;z 3×3; confinement of the F4 static potentials. The same reason is applicable for the higher representations of the F4 excep- z I3 3;z I3 3;z I3 3; × × × tional group. The calculations of the higher representations Z52 ¼ ½1 1 1 1 1 1 1 1 I I I SUð3Þ diag ; ; ; ; ; ; ; ; 8×8;z 3×3;z 3×3; have been presented in Appendix A. So far, we have shown that the decomposition of the F4 I I I I I z 3×3;z 3×3;z 3×3;z 3×3;z 3×3; representations into the SUð3Þ subgroup leads to the Cartan I I I I I ð Þ z 3×3;z 3×3;z 3×3;z 3×3;z 3×3 ; 35 generators that give the exact potential of F4, and Casimir scaling is also achieved. However, the SUð3Þ nontrivial center 2πi elements are not responsible for the linearity observed in the where z ¼ expð 3 Þ is the SUð3Þ nontrivial center element. We previously mentioned that z and z vortices potentials of the F4 representations. So, what accounts for the carry the same magnetic fluxes but in the opposite temporary confining potential? To answer this question, we directions. Interestingly, the numbers of z and z vortices study other subgroups of F4. Greensite et al. [24] found that ð3Þ that appear in the above decompositions of Eq. (35) are the SU and Z3-projected lattices are successful in reproducing the asymptotic string tension of G2 gauge theory. However, same. Therefore, one might conclude that the F4 vacuum domain consists of the SUð3Þ center vortices. For our no correlation between the gauge invariant Wilson loops and ð3Þ purpose, we use the normalization condition as follows: the SU and Z3-projected loops is observed. They conclude that the results of the SUð3Þ and Z3 projections in G2 gauge 26 52 theory are misleading. Therefore, they look for the smallest exp½iα⃗· H⃗ or ¼Z26 or 52I; ð36Þ SUð3Þ subgroup of G2,i.e.,SUð2Þ, and the Wilson loop imposing a “maximal SUð2Þ” gauge is calculated. It is observed that the 26 52 where H and H are the generators depicted in Eqs. (25) potential of the full G2 theory is approximately parallel to the and (32), respectively. Solving the corresponding equations one obtained from the SUð2Þ-only Wilson loop. However, results in SUð2Þ projection also appears to be problematic.

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1 F4 ⊃ SUð2Þ × Spð6Þ

0.8 26 ¼ð2; 6Þ ⊕ ð1; 14Þ; 52 ¼ð3; 1Þ ⊕ ð1; 21Þ ⊕ ð2; 140Þ: ð41Þ 0.6 ] α [

r In the next step, 0.4

Re G Spð6Þ ⊃ SUð2Þ × Spð4Þ 0.2 6 ¼ð2; 1Þ ⊕ ð1; 4Þ; 0 14 ¼ð1; 1Þ ⊕ ð1; 5Þ ⊕ ð2; 4Þ; Rep. 26 of F4 Rep. 26 of SU(3) × SU(3) 140 ¼ð1 4Þ ⊕ ð2 5Þ -0.2 ; ; ; -200 -100 0 100 200 300 x 21 ¼ð3; 1Þ ⊕ ð1; 10Þ ⊕ ð2; 4Þ: ð42Þ 1 Then, 0.8 Spð4Þ ⊃ SUð2Þ × SUð2Þ 0.6 4 ¼ð2 1Þ ⊕ ð1 2Þ

] ; ; ; α [ r 0.4 5 ¼ð1; 1Þ ⊕ ð2; 2Þ; Re G 10 ¼ð3; 1Þ ⊕ ð1; 3Þ ⊕ ð2; 2Þ: ð43Þ 0.2

Ultimately, for the F4 exceptional group, pure SUð2Þ 0 Rep. 52 of F4 subgroups are formed: Rep. 52 of SU(3) × SU(3) -0.2 -200 -100 0 100 200 300 26 ¼ð2 1Þ ⊕ ð2 1Þ ⊕ ð1 2Þ ⊕ ð2 1Þ ⊕ ð2 1Þ ⊕ ð1 2Þ x ; ; ; ; ; ; ⊕ ð1;1Þ ⊕ ð1; 1Þ ⊕ ð2; 2Þ ⊕ ð2;1Þ ⊕ ð1; 2Þ FIG. 8. The real part of the group factor versus the location x of the vacuum domain midpoint, for R ¼ 100 and in the range ⊕ ð2;1Þ ⊕ ð1; 2Þ; x ∈ ½−200; 300, for the fundamental and adjoint representation 52 ¼ð3; 1Þ ⊕ ð3; 1Þ ⊕ ð3; 1Þ ⊕ ð1; 3Þ ⊕ ð2; 2Þ ⊕ ð2;1Þ of F4 (solid lines) in comparison with the corresponding ones obtained from the SUð3Þ × SUð3Þ decomposition using its non- ⊕ ð1;2Þ ⊕ ð2; 1Þ ⊕ ð1; 2Þ ⊕ ð2;1Þ ⊕ ð1; 2Þ ⊕ ð1; 1Þ trivial center elements (dashed lines). The minimum values of the ⊕ ð2 2Þ ⊕ ð1 1Þ ⊕ ð2 2Þ ⊕ ð2 1Þ ⊕ ð1 2Þ ⊕ ð1 1Þ F4 group factor for the fundamental and adjoint representations ; ; ; ; ; ; −0 076 are 0.076 and . , respectively. It is clear that the minimum ⊕ ð2;2Þ ⊕ ð1; 1Þ ⊕ ð2; 2Þ: ð44Þ value of the group factors for the SUð3Þ × SUð3Þ decomposition is not identical to the corresponding ones for F4. The Cartan generators extracted out of these decomposi- tions are 2. SUð2Þ ×Spð6Þ subgroup 1 We try to achieve pure SUðNÞ subgroups of F4 out of 26 pffiffiffi 2 2 2 2 H ð2Þ ¼ diag½0; 0; 0; 0; σ3; 0; 0; 0; 0; σ3; 0; 0; σ3; σ3; this decomposition. One may focus on the Spð6Þ to branch SU 6 ð2Þ 2 2 it out to its SU subgroups, using the following process: 0; 0; σ3; 0; 0; σ3; 1 H52 ¼ pffiffiffi diag½0; 0; 0; 0; 0; 0; 0; 0; 0; σ3; σ2; σ2; 0; 0; Spð6Þ ⊃ SUð2Þ × Spð4ÞðRÞð39Þ SUð2Þ 3 2 3 3 3 2 2 2 2 2 2 2 2 σ3; 0; 0; σ3; 0; 0; σ3; 0; σ3; σ3; 0; σ3; σ3; 0; 0; σ3; and then 2 2 2 2 0; σ3; σ3; 0; σ3; σ3; ð45Þ

Spð4Þ ⊃ SUð2Þ × SUð2ÞðRÞ: ð40Þ 2 1 3 where σ3 ¼ 2 diag½1; −1 and σ3 ¼ diag½1; 0; −1 are diago- nal generators of the SUð2Þ gauge group in the fundamental Therefore, F4 branches to a pure SUð2Þ subgroup. For the and adjoint representations, respectively. Hence, with fundamental and adjoint representations of F4, one may respect to the SUð2Þ subgroup, we can reconstruct just have [36,37] one diagonal generator. It should be recalled that matrices of

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1 Eq. (45) are normalized with the normalization condition in Eq. (17). Considering the normalization coefficient, it is obvious that the components of these matrices are identical 0.8 26 52 to H3 and H3 in Eqs. (25) and (32), respectively. Therefore, one expects the potential between static color sources in the 0.6 ] α fundamental and adjoint representations to be the same as in [ f Fig. 6, using the maximum flux values below: pffiffiffi Re G 0.4 α26 ¼ 4π 6; max pffiffiffi α52 ¼ 12π 2 ð Þ max : 46 0.2

Rep. 26 of F4 The next step is to investigate if the nontrivial center Rep. 26 of SU(2) × SU(2) × SU(2) × SU(2) SUð2Þ iπ 0 element of SUð2Þ, i.e., z1 ¼ e , is responsible for the -200 -100 0 100 200 300 x confinement of F4. So, we calculate the maximum flux 1 values from Eq. (6). Similar to what we did for the SUð3Þ subgroup, we are going to develop matrices containing the nontrivial center 0.8 element of the SUð2Þ subgroup corresponding to the decomposition of Eq. (44): 0.6 ] α [ 26 r Z ¼ ½1 1 1 1 I 1 1 1 1 I 1 1 0.4 SUð2Þ diag ; ; ; ;z1 2×2; ; ; ; ;z1 2×2; ; ; Re G z1I2 2;z1I2 2; 1; 1;z1I2 2; 1; 1;z1I2 2; × × × × 0.2 Z52 ¼ ½1 1 1 1 1 1 1 1 1 I I I SUð2Þ diag ; ; ; ; ; ; ; ; ; 3×3;z1 2×2;z1 2×2; 1 1 I 1 1 I 1 1 I 1 I 0 ; ;z1 2×2; ; ;z1 2×2; ; ;z1 2×2; ;z1 2×2; Rep. 52 of F4 Rep. 52 of SU(2) × SU(2) × SU(2) × SU(2) I 1 I I 1 1 I 1 I -0.2 z1 2×2; ;z1 2×2;z1 2×2; ; ;z1 2×2; ;z1 2×2; -200 -100 0 100 200 300 I 1 I I ð Þ x z1 2×2; ;z1 2×2;z1 2×2 ; 47 I FIG. 9. The same as Fig. 8 but the dashed lines represent the where n×n are square identity matrices. Because the ð2Þ ð6Þ⊃ ð2Þ ð2Þ ð4Þ⊃ ¯ group factor for the SU ×SP SU ×SU ×SP representations 2 and 2 are the same in SUð2Þ, the vortices SUð2Þ×SUð2Þ×SUð2Þ×SUð2Þ decomposition. In each dia- z1 and z1 are the same in this gauge group. It is observed gram, the minimum values of the F4 group factor are the same that there are an even number of z1 vortices in the as the corresponding ones obtained from the SUð2Þ subgroup. decompositions of Eq. (47). Therefore, the vacuum domain These values are 0.076 and −0.076 for the fundamental and or the trivial vortex might be thought to have these center adjoint representations, respectively. vortices inside. The maximum flux condition of Eq. (6) could be written which occur at x ¼ 0 and x ¼ 100. Since the extremums at as follows: x ¼ 50 imply the complete interaction between vortices ð26Þ ð52Þ ð26Þ ð52Þ and the Wilson loop that results in a linear asymptotic exp½iα⃗· H⃗ or ¼Z or I; ð48Þ SUð2Þ SUð2Þ potential, it is the center element of the SUð2Þ subgroup that gives rise to the intermediate linear potential of F4. which leads to the amounts pffiffiffi The other chain of possible decomposition is α26−non ¼ 2π 6; max pffiffiffi α52−non ¼ 6π 2 ð Þ max : 49 F4 ⊃ SUð2Þ × Spð6ÞðRÞ To compare the extremums of the group factor function Spð6Þ ⊃ SUð2Þ × SUð2ÞðSÞ: ð50Þ of the F4 exceptional group with its SUð2Þ subgroup, the real part of this function has been plotted versus the location of the vortex midpoint for the fundamental and This decomposition generates matrices similar to h3 and h4. adjoint representations in Fig. 9. It is observed that the According to our previous discussion in Fig. 5, when h3 or group factors corresponding to this decomposition reach h4 is applied, the minimum points of the group factor the amounts 0.076 and −0.076 at x ¼ 50 for the funda- function are not identical to the ones for the F4 gauge mental and adjoint representations, respectively. These group. Using this fact, one might conclude that this amounts are identical to the corresponding ones for F4, decomposition is not responsible for the confinement in F4.

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3. SOð9Þ subgroup induced potentials by these matrices have different behaviors according to Fig. 5. Another regular maximal subgroup of F4 group is SOð9Þ. To extract a pure SUðNÞ subgroup out of this So far, we can conclude that, in order to determine the subgroup, one might choose the following decomposition subgroups whose Cartan decompositions result in a well- process: defined potential, one has to compare reconstructed Cartan matrices produced by means of the subgroups with the ones SOð9Þ ⊃ SUð2Þ × SUð4ÞðRÞ: ð51Þ of the main exceptional group itself. In the F4 case, the potential out of applying all of its Cartan generators is the We present three methods to decompose SUð4Þ into the same as the case where just h1 or h2 is used. Therefore, any SUð2Þ subgroups: regular or singular subgroup that is able to reconstruct the (i) A regular decomposition as follows: same diagonal matrices as one of these two generators produces the same potential as that of F4 itself. SUð4Þ ⊃ SUð2Þ × SUð2Þ × Uð1Þ: ð52Þ 4. SUð2Þ ×G2 subgroup ð1Þ There is a U factor in this decomposition that Ultimately, we are going to investigate the results of a makes it a nonsemisimple maximal subgroup. In direct singular maximal subgroup of the F4 exceptional fact, the Uð1Þ that appears in some branching rules is group, i.e., SUð2Þ × G2, because it shows a different a trivial Abelian Lie group composed by all 1 × 1 behavior. To make a pure SUðNÞ subgroup out of this matrices of eiϕ with real ϕ [43]. This factor is singular subgroup, one may choose to break G2 into its excluded in the branching rules [36,44]. In our case, SUð3Þ subgroup: we ignore it since it has no effect on our calculations. If we evade the Uð1Þ factor in Eq. (52), the same G2 ⊃ SUð3ÞðRÞ: ð56Þ results as in Eq. (44), where F4 has been decom- posed into its pure SUð2Þ subgroups, are achieved. It is obviously not a pure subgroup because it contains both Hence, this decomposition could be responsible for ð2Þ ð3Þ the intermediate linear potentials, as well. As the SU and SU subgroups. However, if one considers the ð2Þ results for the fundamental and adjoint representa- representations of SU as degeneracies of the irreducible ð3Þ tions are the same as in Fig. 9, we just present the representations of SU in the branching rules, the result ⊃ ð3Þ ð3Þ detailed calculations for the higher representations in will be the same as the F4 SU × SU decomposi- Appendix B. tion. We have argued that this decomposition is not (ii) SUð4Þ has a singular subgroup, responsible for the F4 confinement. A more challenging procedure is the following decom- SUð4Þ ⊃ Spð4ÞðSÞ; ð53Þ position: ⊃ ð2Þ ð Þ and it could be decomposed as follows: F4 SU × G2 S 26 ¼ð5; 1Þ ⊕ ð3; 7Þ; Spð4Þ ⊃ SUð2Þ × SUð2ÞðRÞ: ð54Þ 52 ¼ð3; 1Þ ⊕ ð1; 14Þ ⊕ ð5; 7Þ: ð57Þ The exact decompositions as in Eq. (44) and Cartan generators of Eq. (45) are obtained. Therefore, the G2 might be decomposed as same results are achieved. Consequently, singular ⊃ ð2Þ ð2ÞðÞ maximal subgroups are able to bring out the same G2 SU × SU R potentials as F4 as well. Furthermore, this decom- 7 ¼ð2; 2Þ ⊕ ð1; 3Þ; position could also describe the linear potential of 14 ¼ð3 1Þ ⊕ ð2 4Þ ⊕ ð1 3Þ ð Þ F4 correctly. ; ; ; : 58 (iii) Another chain of breaking to SUðNÞ subgroups could be a singular decomposition: Finally,

SUð4Þ ⊃ SUð2Þ × SUð2ÞðSÞ: ð55Þ 26 ¼ð5; 1Þ ⊕ ð2; 2Þ ⊕ ð1; 3Þ ⊕ ð2; 2Þ ⊕ ð1; 3Þ ⊕ ð2; 2Þ ⊕ ð1; 3Þ Although this decomposition seems to be similar to 52 ¼ð3 1Þ ⊕ ð3 1Þ ⊕ ð2 4Þ ⊕ ð1 3Þ ⊕ ð2 2Þ Eq. (52), due to the branching rules, representations ; ; ; ; ; decompose in a way that they produce exact ma- ⊕ ð1; 3Þ ⊕ ð2; 2Þ ⊕ ð1; 3Þ ⊕ ð2; 2Þ ⊕ ð1; 3Þ trices as h3 and h4 of Eq. (14) in the fundamental ⊕ ð2; 2Þ ⊕ ð1; 3Þ ⊕ ð2; 2Þ ⊕ ð1; 3Þ: ð Þ representation of F4. We previously learned that 59

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Using these decompositions, we can reproduce Cartan 1 generators in the fundamental and adjoint representations as follows: 0.8

0.6 26 1 2 2 3 2 2 3 H ¼ pffiffiffi diag½0; 0; 0; 0; 0; σ3; σ3; σ3; σ3; σ3; σ3; SUð2Þ×G2 ]

3 2 α [ f 2 2 3 0.4 σ3; σ3; σ3; Re G

52 1 4 4 3 2 2 0.2 H ¼ pffiffiffi diag½0; 0; 0; 0; 0; 0; σ3; σ3; σ3; σ3; σ3; SUð2Þ×G2 3 6 3 2 2 3 2 2 3 2 2 3 2 2 3 0 σ3; σ3; σ3; σ3; σ3; σ3; σ3; σ3; σ3; σ3; σ3; σ3; σ3: Rep. 26 of F4 ð Þ Rep. 26 of SU(2) × G 60 -0.2 2 -200 -100 0 100 200 300 x In these matrices, the normalization coefficients have been 1 2 3 4 computed from Eq. (17). σ3, σ3, and σ3 are Cartan ð2Þ generators of the SU gauge group in the fundamental, 0.8 adjoint and four-dimensional representations, respectively.

After an initial review, the elements of these matrices are 0.6 not fully the same as the corresponding ones in Eqs. (25) ] α [ and (32) or (45). Accordingly, the trivial static potentials, r 0.4 when we consider the trivial center element of the SUð2Þ Re G subgroup, are not identical to those of the F4 exceptional 0.2 group itself. We have investigated this subgroup in Ref. [45]. However, there is another aspect of this decom- 0 position that is appealing. ð2Þ Rep. 52 of F4 The center element matrices of the SU × G2 sub- Rep. 52 of SU(2) × G -0.2 2 group of F4 in the fundamental and adjoint representations -200 -100 0 100 200 300 are given by x ð2Þ 26 FIG. 10. The same as Fig. 9 but for the SU × G2 subgroup. Z ¼ diag½1; 1; 1; 1; 1;z1I2 2;z1I2 2; I3 3; SUð2Þ×G2 × × × In both figures, the minimum values of the F4 group factor are identical to the amount of the group factor corresponding to the z1I2 2;z1I2 2; I3 3; × × × SUð2Þ × G2 decomposition at x ¼ 50. Therefore, the slope of the z1I2 2;z1I2 2; I3 3; intermediate linear of F4 is the same as the asymptotic one for × × × ð2Þ 52 the SU × G2 subgroup. Z ¼ diag½1; 1; 1; 1; 1; 1;z1I4 4;z1I4 4; I3 3; SUð2Þ×G2 × × × I I I I I I I z1 2×2;z1 2×2; 3×3;z1 2×2;z1 2×2; 3×3;z1 2×2; decomposition of the representations to the SUð2Þ × G2 I I I I z1 2×2; 3×3;z1 2×2;z1 2×2; subgroup. When the vortex midpoint is located at x ¼ 50,it means that the vortex is completely inside the Wilson loop. I3 3;z1I2 2;z1I2 2; I3 3: ð61Þ × × × × The nonzero value of the group factor at this point results in Now, putting the above matrices in Eq. (6), the maximum a linear potential at large distances. As the value of the flux values are calculated as follows: group factor at this point is equal to the corresponding one for F4, the slope of this linear potential seems to be pffiffiffi α26-non ¼ 6π 2 identical to the intermediate linear potentials of F4. max ; pffiffiffi Therefore, one might say that the SUð2Þ × G2 ⊃ SUð2Þ × α52-non ¼ 6π 6 ð Þ ð2Þ ð2Þ max : 62 SU × SU subgroup of F4 is responsible for the intermediate confinement of this exceptional group. Figure 10 shows the group factor function versus the vortex An interesting point here is that Cartan generators of this midpoint x for the fundamental and adjoint representations, decomposition are different from h1 and h2.However,the respectively. As observed, the group factor has a totally minimum points of the F4 group factor can still be different behavior in comparison with Fig. 9. The minimum investigated correctly via this decomposition. The question points of the F4 representations occur at x ¼ 0 and is why this happens. In fact, the N-ality of the SUðNÞ x ¼ 100, where half of the vortex flux enters the Wilson representations or the center element matrix obtained from loop. These points are responsible for the intermediate the group decompositions has the predominant responsibil- linear potentials of F4. Now, we focus on the ity here. The representations could be classified by their

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N-ality. This means that the Wilson loop of the represen- It should be noted that, due to the similarity of these 27 tations with the same N-ality is affected by a vortex type n matrices, one can use only h1 to calculate the potential of in the same manner. To make it more clear, we investigate the fundamental representation of E6. The maximum flux center element matrices of the fundamental representation values for the domain structures, calculated from the in Eq. (47), obtained from the SUð2Þ × Spð6Þ subgroup, condition in Eq. (19), are ð2Þ and also in Eq. (61), by means of the SU × G2 pffiffiffiffiffi subgroup, which have different elements. In the former αmax ¼¼αmax ¼ 2π 24: ð65Þ ’ I ’ 1 6 one, there exist fourteen 1 s and six z1 2×2 s, while in the latter one, in Eq. (61), there exist five 1’s, six z1I2 2’s, and × On the other hand, if we include the nontrivial flux three I3 3’s. The number of z1I2 2 center elements is the × × condition in Eq. (6), the maximum amounts for the vortex same in both matrices, which corresponds to the funda- ¼ 1 I fluxes are mental representation with 2-ality . Elements 1 and 3×3 with zero 2-ality have no effect on the Wilson loop. So, the 4π pffiffiffi other elements of these two matrices do not affect the αnon ¼ αnon ¼∓ 6; max1 max4 3 Wilson loop. As a result, although the matrices of Eqs. (45) pffiffiffi αnon ¼ αnon ¼∓ 4π 6 and (60) have different elements and the potentials of these max3 max6 ; two generators behave differently, the number of center 8π pffiffiffi αnon ¼ αnon ¼∓ 6; ð66Þ vortices that emerge in both decompositions is the same. max2 max5 3 Thus, the group factor reaches the same minimum amount in both of them. Regarding 52-dimensional adjoint repre- where “non” indicates the answer pertaining to the non- sentation, the same process comes about. trivial center elements. Negative answers have been gained 2πi by type 1 vortices when Eq. (6) is equal to z1 ¼ expð 3 Þ B. E exceptional group and positive answers correspond to the type 2 vortices 6 −2πi [z2 ¼ expð 3 Þ]. Static potentials obtained by both of these E6 is the third exceptional group in terms of largeness. It maximum trivial and nontrivial flux values in Eqs. (65) and makes a 78-dimensional space with 78 generators and, (66) have been depicted in Fig. 11. As could have been similar to SUð3Þ, has Z3 as its group center [27]. Here, we predicted, at far distances, the potential obtained from the mostly focus on its trivial center to investigate the static trivial center element of E6 has been screened while the potential behavior. As the rank of E6 is six, it possesses six potential corresponding to the nontrivial center element is diagonal matrices that are shown as follows for the linear. This fact could be investigated by tensor products of fundamental representation [46]: the E6 “quark” and “gluons”: 27 h1 ¼ N1diag½−1; þ1; 0; 0; 0; 0; 0; 0; 0; −1; 0; 0; 0; −1; 27 × 78 ¼ 27 ⊕ 351 ⊕ 1728: ð67Þ þ 1; −1; −1; þ1; þ1; þ1; 0; −1; þ1; 0; 0; 0; 0; 27 h2 ¼ N2diag½0; −1; þ1; 0; 0; 0; 0; 0; −1; þ1; 0; −1; −1; It is seen that E6 “gluons” are not able to screen the potential of the E6 “quarks”. Similar to SUðNÞ gauge þ 1; 0; þ1; 0; 0; 0; −1; þ1; 0; −1; þ1; 0; 0; 0; 27 h3 ¼ N3diag½0; 0; −1; þ1; 0; 0; 0; −1; þ1; 0; −1; þ1; 0; 0; 18 0; −1; þ1; 0; −1; þ1; 0; 0; 0; −1; þ1; 0; 0; non-trivial 16 trivial 27 h ¼ N4diag½0; 0; 0; −1; þ1; 0; −1; þ1; 0; 0; 0; −1; þ1; 4 14 − 1 0 þ1 0 −1 þ1 0 0 0 0 0 −1 þ1 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; 12 27 ¼ ½0 0 0 0 −1 þ1 0 −1 −1 −1 þ1 þ1 0 h5 N5diag ; ; ; ; ; ; ; ; ; ; ; ; ; 10 V(R) þ 1; −1; 0; 0; þ1; 0; 0; 0; 0; 0; 0; 0; −1; þ1; 8 27 h6 ¼ N6diag½0; 0; 0; −1; −1; −1; þ1; þ1; 0; 0; þ1; 0; 0; 0; 6

0; 0; −1; 0; 0; −1; −1; þ1; þ1; þ1; 0; 0; 0: ð63Þ 4

2 These matrices are normalized and their normalization 0 coefficients could be calculated from Eq. (17): 0 10 20 30 40 50 60 70 80 90 100 R 1 N1 ¼¼N6 ¼ pffiffiffi : ð64Þ FIG. 11. Potential of the fundamental representation of E6 2 6 using trivial and nontrivial center elements for R ∈ ½1; 100.

056015-13 AMIR SHAHLAEI and SHAHNOOSH RAFIBAKHSH PHYS. REV. D 97, 056015 (2018) groups, one “quark” and one “antiquark” can join to create 1. SUð3Þ ×SUð3Þ ×SUð3Þ subgroup a meson, The fundamental representation decomposes as [36–38]

27 ¼ð3 3¯ 1Þ ⊕ ð1 3 3Þ ⊕ ð3¯ 1 3¯Þ ð Þ 27 × 27 ¼ 1 ⊕ 78 ⊕ 650; ð68Þ ; ; ; ; ; ; : 70 Thus, if we assume the first two representations in the and three “quarks” form a baryon, parentheses in Eq. (70), as degeneracies step by step, one might have 27 27 27 ¼ 1 ⊕ 2ð78Þ ⊕ 3ð650Þ ⊕ 2925 × × 27 ¼ 9ð1Þ ⊕ 3ð3Þ ⊕ 3ð3¯Þ: ð71Þ ⊕ 3003 ⊕ 25824: ð69Þ Although E6 has a nontrivial center element, the method of decomposing its representations leads to the SUð3Þ repre- “ ” Now, we aim to follow the same procedure applied for sentations with different trialities. Therefore, an E6 quark ð3Þ F4 to explain what actually accounts for the temporary could be decomposed into three SU quarks, three confinement in the trivial static potential of the E6 excep- antiquarks, and nine singlets. Now, two Cartan generators tional group. The main question is, which kinds of center with regard to this subgroup are reconstructed from vortices have filled the E6 QCD vacuum that give rise to the Eq. (71): intermediate confining potential obtained by the trivial 1 center element? In general, we have three candidates: H27 ¼ pffiffiffi diag½0; 0; 0; 0; 0; 0; 0; 0; 0; λ3; λ3; λ3; a 6 a a a (i) Nontrivial center elements of the E6 excep- 3 3 3 tional group; − ðλaÞ ; −ðλaÞ ; −ðλaÞ ; ð72Þ (ii) Nontrivial center elements of its SUð3Þ maximal subgroup; where a ¼ 3, 8. Now, one can consider the condition in (iii) Nontrivial center elements of its maximal SUð2Þ Eq. (19) and find subgroup. pffiffiffi To answer this question properly, the group factor α27 ¼ 2π 6; function using Eq. (63) for both trivial and nontrivial max1 pffiffiffi α27 ¼ 6π 2 ð Þ center elements of the E6 exceptional group have been max2 : 73 demonstrated in Fig. 12. Consequently, nontrivial center elements of E6 are not the direct reason of the intermediate The potential between the fundamental sources of the E6 linear part in the trivial potential of E6. Then, we go for using the six Cartan generators in Eq. (63) has been some of the maximal subgroups of E6 that have been presented in Fig. 13, which overlaps completely with the mentioned in Table I. data obtained from the above values in Eq. (73) and Cartan generators of Eq. (72). This fact is the result of the identical 1 5 Rep. 27 of E 0.8 6 Rep. 27 of SU(3) × SU(3) × SU(3)

0.6 4

0.4 ] α [ f 3 0.2 Re G 0 V(R) 2 -0.2

-0.4 1 non-trivial trivial -0.6 -200 -100 0 100 200 300 x 0 0 20 40 60 80 100 R FIG. 12. Solid line represents the group factor of the funda- mental representation of E6 versus the location x of the vacuum FIG. 13. The potential of the fundamental representation of E6 domain midpoint, using the trivial center element, for R ¼ 100 in using all Cartan generators (solid line) and the one corresponding the range x ∈ ½−200; 300. The dashed line shows the same to the SUð3Þ × SUð3Þ × SUð3Þ decomposition (stars) in the function versus the location x of the nontrivial center vortex. range R ∈ ½1; 100. The two sets of data are the same.

056015-14 F4, E6 AND G2 EXCEPTIONAL GAUGE … PHYS. REV. D 97, 056015 (2018)

27 27 27 6 components of H3 and h1 . Although H8 has different Rep.27 of E6 27 Rep. 78 of E matrix elements, it has the same effect as H3 on the E6 6 5 potentials. A similar discussion has been given in the F4 ⊃ SUð3Þ × SUð3Þ decomposition. Therefore, one might use 4 the SUð3Þ subgroup decomposition to find the E6 adjoint potential. 3 Reconstruction of Cartan generators in the adjoint V(R) representation of E6 with respect to its SUð3Þ subgroup is possible only when we want to calculate the trivial static 2 potential because they are identical. This method is not applicable for the potentials obtained by the nontrivial 1 center elements. 0 The adjoint representation might be decomposed as 0 10 20 30 40 50 60 70 80 90 100 follows: R 2.5 ¯ Rep. 27 of E6 78 ¼ð8; 1; 1Þ ⊕ ð1; 8; 1Þ ⊕ ð1; 1; 8Þ ⊕ ð3; 3; 3Þ Rep. 78 of E6 ⊕ ð3¯; 3¯; 3Þ¼16ð1Þ ⊕ 8 ⊕ 9ð3¯Þ ⊕ 9ð3Þ: ð74Þ 2

“ ” ð3Þ So, an E6 gluon has been decomposed into nine SU 1.5 quarks, nine antiquarks, one gluon, and 16 singlets. Hence, the Cartan generators structured from the SUð3Þ decom- V(R) position are 1

8 times 8 times 1 zfflfflffl}|fflfflffl{ zfflfflffl}|fflfflffl{ 0.5 H78 ¼ pffiffiffi diag½0; …; 0; 0; …; 0; λ8; a 2 6 a

9 times 9 times 0 zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ 3 4 5 6 7 8 9 10 3 3 3 3 R −ðλaÞ ; …; −ðλaÞ ; λa; …; λa: ð75Þ FIG. 14. Upper diagram: Trivial static potentials of the E6 Applying the maximum flux condition of Eq. (19) for these exceptional group for the fundamental and adjoint representations Cartan generators, we have in the range R ∈ ½1; 100. The potentials are screened at large pffiffiffi distances, which is due to the absence of the nontrivial center α78 ¼ 4π 6 element. At intermediate quark separations, the potentials are max1 ; pffiffiffi linear and have been presented in the lower diagram in the range α78 ¼ 12π 2 ð Þ ∈ ½3 10 max2 : 76 R ; . The ratio of the adjoint potential to the fundamental one is in agreement with Casimir scaling. The potential between static color sources using the trivial center element of the E6 exceptional gauge group for the 27 Z ¼ diag½1; 1; 1; 1; 1; 1; 1; 1; 1;zI3 3;zI3 3; fundamental and adjoint representations has been plotted in SUð3Þ × × I I I I Fig. 14. The screening is visible at large distances, while z 3×3;z 3×3;z 3×3;z 3×3 ; the intermediate parts are linear. The lower diagram shows 8 times 8 times the linear parts of the potentials in the interval R ∈ ½3; 10. zfflfflffl}|fflfflffl{ zfflfflffl}|fflfflffl{ Z78 ¼ ½0 … 0 0 … 0 I We have fitted our data to the equation VðRÞ¼aR þ b. SUð3Þ diag ; ; ; ; ; ; 8×8; The slopes of the potentials have been found to be 0.251(3) zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{9 times zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{9 times and 0.309(5) for the fundamental and adjoint representa- z I3 3; …;z I3 3;zI3 3; …;zI3 3: ð77Þ tions, respectively. Therefore, the ratio of the adjoint × × × × potential to the fundamental one in this range is 1.23(5). Similar to F4, the numbers of z and z vortices are the same In fact, the ratio of the adjoint potential to the fundamental in the above matrices. Using Eq. (6),wehave one starts from 1.384, which is the Casimir scaling of the pffiffiffi pffiffiffi adjoint representation [42]. But, similar to Fig. 7, the 27 27 α -non ¼ 2π 6; α -non ¼ 2π 2; adjoint potential ratio differs from the exact value of max1 pffiffiffi max2 pffiffiffi α78-non ¼ 4π 6 α78-non ¼ 4π 2 ð Þ Casimir scaling at intermediate distances. max1 ; max1 : 78 To find what accounts for the intermediate linear pote- ntial, one needs to construct a matrix that consists of SUð3Þ Using the above amounts, one might plot the group center elements with respect to Eqs. (70) and (74): factor for the fundamental and adjoint representations in

056015-15 AMIR SHAHLAEI and SHAHNOOSH RAFIBAKHSH PHYS. REV. D 97, 056015 (2018)

1 Then, one can choose the following maximal subgroup of SUð6Þ to decompose its representations: 0.8 SUð6Þ ⊃ SUð2Þ×SUð4Þ×Uð1ÞðRÞ 0.6 6 ¼ð2 1Þ ⊕ ð1 4Þ ] ; ; ; α [ f 15 ¼ð1 1Þ ⊕ ð2 4Þ ⊕ ð1 6Þ 0.4 ; ; ; ; Re G 20 ¼ð1;4Þ ⊕ ð1;4¯Þ ⊕ ð2;6Þ; 0.2 35 ¼ð1;1Þ ⊕ ð3;1Þ ⊕ ð1;15Þ ⊕ ð2;4Þ ⊕ ð2;4¯Þ; ð80Þ

0 Rep. 27 of E6 and for SUð4Þ, Rep. 27 of SU(3) × SU(3) × SU(3) -200 -100 0 100 200 300 x SUð4Þ ⊃ SUð2Þ × SUð2Þ × Uð1ÞðRÞ 1 4 ¼ð2; 1Þ ⊕ ð1; 2Þ;

0.8 6 ¼ð1; 1Þ ⊕ ð1; 1Þ ⊕ ð2; 2Þ; 15 ¼ð1; 1Þ ⊕ ð3; 1Þ ⊕ ð1; 3Þ ⊕ ð2; 2Þ ⊕ ð2; 2Þ: ð81Þ 0.6 ] α [

r Finally, we have

0.4 Re G 27 ¼ð2; 1Þ ⊕ ð2; 1Þ ⊕ ð1;2Þ ⊕ ð2; 1Þ ⊕ ð2; 1Þ ⊕ ð1;2Þ

0.2 ⊕ ð1;1Þ ⊕ ð2; 1Þ ⊕ ð1; 2Þ ⊕ ð2;1Þ ⊕ ð1; 2Þ ⊕ ð1; 1Þ ⊕ ð1;1Þ ⊕ ð2; 2Þ; 0 Rep. 78 of E6 Rep. 78 of SU(3) × SU(3) × SU(3) 78 ¼ð3; 1Þ ⊕ ð1; 1Þ ⊕ ð3;1Þ ⊕ ð1; 1Þ ⊕ ð3; 1Þ ⊕ ð1;3Þ -200 -100 0 100 200 300 x ⊕ ð2;2Þ ⊕ ð2; 2Þ ⊕ ð2; 1Þ ⊕ ð1;2Þ ⊕ ð2; 1Þ ⊕ ð1; 2Þ FIG. 15. The real part of the group factor function versus the ⊕ ð2;1Þ ⊕ ð1; 2Þ ⊕ ð2; 1Þ ⊕ ð1;2Þ ⊕ ð2; 1Þ ⊕ ð1; 2Þ location x of the vacuum domain midpoint, for R ¼ 100 and in ⊕ ð2;1Þ ⊕ ð1; 2Þ ⊕ ð1; 1Þ ⊕ ð1;1Þ ⊕ ð2; 2Þ ⊕ ð1; 1Þ the range x ∈ ½−200; 300, for the fundamental and adjoint representation of E6 (solid lines) in comparison with the same ⊕ ð1;1Þ ⊕ ð2; 2Þ ⊕ ð2; 1Þ ⊕ ð1;2Þ ⊕ ð2; 1Þ ⊕ ð1; 2Þ function versus the location x of the vortex midpoint obtained ⊕ ð1 1Þ ⊕ ð1 1Þ ⊕ ð2 2Þ ⊕ ð1 1Þ ⊕ ð1 1Þ ⊕ ð2 2Þ from the SUð3Þ × SUð3Þ × SUð3Þ decomposition (dashed lines). ; ; ; ; ; ; : ¼ 0 The minimum points of the E6 group factor that occur at x ð82Þ and x ¼ 100 reach the amounts 0.111 and −0.025 for the fundamental and adjoint representations, respectively, whereas −0 038 ð3Þ In Eqs. (80)–(82), the Uð1Þ factor has been ignored. these amounts are approximately 0 and . for the SU ð2Þ subgroup. Reconstruction of the Cartan generators for the SU subgroup of E6 and for the fundamental and adjoint Fig. 15 and compare them with the corresponding ones for representations using the decompositions in Eq. (82) is E6. In this figure, we can observe that the minimum values as follows: are not the same. Thus, nontrivial center elements of the 1 SUð3Þ subgroup are not in charge of the confining part in 27 ¼ pffiffiffi ½0 0 0 0 σ2 0 0 0 0 σ2 0 0 0 σ2 HSUð2Þ diag ; ; ; ; 3; ; ; ; ; 3; ; ; ; 3; the trivial potential of E6. The third possibility to inves- 6 2 2 2 tigate the linearity of the E6 trivial potentials in Fig. 14 is 0; 0; σ3; 0; 0; σ3; σ3; the nontrivial center element of the SUð2Þ subgroups. 35 times 20 times 1 zfflfflffl}|fflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ H78 ¼ pffiffiffi diag½0; …; 0; σ3; σ2; …; σ2: ð83Þ 2. SUð2Þ ×SUð6Þ subgroup SUð2Þ 2 6 3 3 3

Now, we decompose E6 into the SUð2Þ × SUð6Þ sub- It should be noted that, for the sake of simplicity, the group and have 78 components of the matrix HSUð2Þ are not in order because it 27 ¼ð2; 6¯Þ ⊕ ð1; 15Þ; does not have any effect on our calculations. To make a comparison with the potentials obtained from 78 ¼ð3; 1Þ ⊕ ð1; 35Þ ⊕ ð2; 20Þ: ð Þ 79 the trivial center element of the E6 exceptional group and

056015-16 F4, E6 AND G2 EXCEPTIONAL GAUGE … PHYS. REV. D 97, 056015 (2018)

1 its SUð2Þ subgroup, one can utilize Eq. (83) in Eq. (19) and find pffiffiffi 0.8 α27 ¼ 4π 6; max pffiffiffi 78 0.6

α ¼ 8π 6 ð Þ α] : 84 [ max f Re G Static potentials obtained by these maximum flux values 0.4 and Cartans of Eq. (83) are identical to the E6 potentials in

Fig. 14. These results were foreseeable due to the similarity 0.2 of the matrix components of Eq. (83) with the correspond-

Rep. 27 of E6 ing Cartan generators of E6 in Eqs. (72) and (75). Rep. 27 of SU(2) × SU(6) subgroup 0 Matrices made of center elements of the SUð2Þ subgroup -200 -100 0 100 200 300 considering Eq. (82) are x 1

27 Z ¼½1; 1; 1; 1;z1I2 2; 1; 1; 1; 1;z1; I2 2; 1; 1; 1; SUð2Þ × × 0.8 I 1 1 I 1 1 I I z1 2×2; ; ;z1 2×2; ; ;z1 2×2;z1 2×2 ; zfflfflffl}|fflfflffl{35 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{20 times 0.6 α]

78 [ Z ¼½1 … 1 I I … I ð Þ r SUð2Þ ; ; ; 3×3;z1 2×2; ;z1 2×2 : 85 0.4 Re G So, using Eq. (6), the flux maximum values are 0.2 pffiffiffi α27-non ¼ 2π 6; max 0 pffiffiffi Rep. 78 of E6 α78-non ¼ 4π 6: ð86Þ Rep. 78 of SU(2) × SU(6) subgroup max -200 -100 0 100 200 300 x Using these values, we are able to plot the group factor ð2Þ FIG. 16. The same as Fig. 15 but the dashed lines represent the function for the nontrivial center element of the SU group factor corresponding to the SUð2Þ × SUð6Þ subgroup of subgroup and compare the results with the same function E6. It is clear that in both diagrams, the minimum values are obtained by the trivial center element of the E6 exceptional identical. group or its SUð3Þ subgroup. Figure 16 depicts this comparison. It is clear that the minimum points are 4. G subgroup identical. So, one can conclude that the nontrivial center 2 element of the SUð2Þ subgroup is responsible for the In the F4 exceptional group case, its singular maximal linearity at intermediate distance scales. SUð2Þ × G2 subgroup has a property that could not produce the same potential as F4 itself, because its reconstructed Cartan matrices consist of different compo- 3. F subgroup 4 nents from h1 and h2 original Cartan generators of F4. There is another way to decompose E6 into its subgroup However, they are still able to produce the same linear part without having a Uð1Þ factor in the final result. For as some of the other SUð2Þ subgroups of F4. Here, for E6, instance, the decomposition chain described below can the branching G2 singular maximal subgroup into the satisfy our assumption and reconstruct matrices with the SUð2Þ × SUð2Þ subgroup attributes similarly to the same components as in Eq. (83): SUð2Þ × G2 subgroup of F4. According to the branching rules, the decomposition for the fundamental representation E6 ⊃ F4 ðSÞ of the E6 is as follows:

F4 ⊃ SUð2Þ × Spð6ÞðRÞ E6 ⊃ G2 ðSÞ Spð6Þ ⊃ SUð2Þ × Spð4ÞðRÞ 27 ¼ 27 Spð4Þ ⊃ SUð2Þ × SUð2ÞðRÞ: ð87Þ 78 ¼ 14 ⊕ 64: ð88Þ

Therefore, we expect the same result as the E6 ⊃ SUð2Þ × SUð6Þ decomposition. Then,

056015-17 AMIR SHAHLAEI and SHAHNOOSH RAFIBAKHSH PHYS. REV. D 97, 056015 (2018)

1 G2 ⊃ SUð2Þ × SUð2ÞðRÞ

27 ¼ð3; 3Þ ⊕ ð2; 4Þ ⊕ ð2; 2Þ ⊕ ð1; 5Þ ⊕ ð1; 1Þ; 0.8 14 ¼ð1; 3Þ ⊕ ð2; 4Þ ⊕ ð3; 1Þ; 0.6 64 ¼ð4; 2Þ ⊕ ð3; 5Þ ⊕ ð3; 3Þ ⊕ ð2; 6Þ ⊕ ð2; 4Þ ⊕ ð2; 2Þ α] [ f ⊕ ð1; 5Þ ⊕ ð1; 3Þ: ð89Þ 0.4 Re G

Therefore, 0.2 1 27 ¼ pffiffiffi ½σ3 σ3 σ3 σ4 σ4 σ2 σ2 σ5 0 0 HE6⊃G2 diag 3; 3; 3; 3; 3; 3; 3; 3; ; 3 6 Rep. 27 of E6 Rep. 27 of G2 subgroup 1 -0.2 H78 ¼ pffiffiffi diag½σ3; σ4; σ4; 0; 0; 0; σ2; σ2; σ2; σ2; σ5; -200 -100 0 100 200 300 E6⊃G2 6 6 3 3 3 3 3 3 3 3 x 1 5 5 3 3 3 6 6 4 4 2 2 5 3 σ3; σ3; σ3; σ3; σ3; σ3; σ3; σ3; σ3; σ3; σ3; σ3; σ3; ð90Þ 0.8 and 0.6 α] [ 27 r Z ¼ ½I I I I I 0.4 E6⊃G2 diag 3×3; 3×3; 3×3;z1 4×4;z1 4×4; Re G z1I2 2;z1I2 2; I5 5; 1; × × × 0.2 Z78 ¼ ½I I I 1 1 1 E6⊃G2 diag 3×3;z1 4×4;z1 4×4; ; ; ; 0 1I2 2 1I2 2 1I2 2 1I2 2 z × ;z × ;z × ;z × ; Rep. 78 of E6 I I I I I I Rep. 78 of G2 subgroup 5×5; 5×5; 5×5; 3×3; 3×3; 3×3; -0.2 -200 -100 0 100 200 300 I I I I x z1 6×6;z1 6×6;z1 4×4;z1 4×4; I I I I ð Þ FIG. 17. The same as Fig. 15 but the dashed lines represent the z1 2×2;z1 2×2; 5×5; 3×3 ; 91 group factor of the G2 subgroup. The flux condition of Eq. (19) gives pffiffiffi calculations in favor of the G2 exceptional gauge group α27-non ¼ 6π 6 max ; [25–27,31,47–50]. pffiffiffi – α78-non ¼ 12π 6: ð92Þ In Refs. [28 30], the static potentials of the G2 excep- SUð2Þ- max tional gauge group have been investigated, and the dom- inant role of nontrivial center elements of its SUð2Þ and Figure 17 shows the group factor obtained from this SUð3Þ subgroups on the intermediate confinement has been decomposition versus the location x of the vortex midpoint. studied. In this research, we are going to insert our It is observed that the amount of the group factor when the generalized method to calculate the potentials of higher vortex is completely inside the Wilson loop is equal to the representations of G2 as well. minimum values of the E6 group factor. Therefore, this To begin, one needs the original Cartan generators of the decomposition is able to describe E6 temporary confine- G2 exceptional group to simulate the static potential using ment. It should be pointed out that the numbers of center the vacuum domain structure model. The Cartan generators elements that emerge in the center element matrices of of G2 in the fundamental seven-dimensional representation Eqs. (85) and (90) are the same. So, an argument similar to are as follows [28,46]: that of the SUð2Þ × G2 subgroup of F4 could be applied here. 1 h7 ¼ pffiffiffi diag½þ1; −1; 0; 0; −1; þ1; 0; 1 2 2 C. G2 exceptional group 7 1 G2 is the simplest exceptional group with rank 2 and h2 ¼ pffiffiffi diag½þ1; þ1; −2; 0; −1; −1; þ2: ð93Þ likewise SUð3Þ. All of its representations are real and it is 2 6 its own universal covering group. Despite F4 and E6 exceptional groups that do not have numerical supports These matrices are normalized with the Eq. (17) condition. yet, pending future investigations, there are lattice Plotting the potential of Eq. (10) requires the group factor

056015-18 F4, E6 AND G2 EXCEPTIONAL GAUGE … PHYS. REV. D 97, 056015 (2018) in Eq. (3) and the flux profile in Eq. (7). In order to compute 14 ¼ 3 ⊕ 3¯ ⊕ 8; the maximum value of the flux profile, one has to apply the 27 ¼ 8 ⊕ 6 ⊕ 6¯ ⊕ 3 ⊕ 3¯ ⊕ 1; trivial flux condition in Eq. (19) and solve three indepen- dent equations: 64 ¼ 15 ⊕ 15¯ ⊕ 8 ⊕ 8 ⊕ 6 ⊕ 6¯ ⊕ 3 ⊕ 3¯; 77 ¼ 27 ⊕ 15 ⊕ 15¯ ⊕ 8 ⊕ 6 ⊕ 6¯; α7 α7 0 ¯ ¯ ¯ ¯ max1 max2 77 ¼ 15 ⊕ 15 ⊕ 10 ⊕ 10 ⊕ 8 ⊕ 6 ⊕ 6 ⊕ 3 ⊕ 3 ⊕ 1: exp pffiffiffi þ pffiffiffi ¼ I; 2 2 2 6 ð98Þ −α7 α7 exp pmaxffiffiffi 1 þ maxpffiffiffi2 ¼ I; 2 2 2 6 −α7 So, the corresponding Cartan generators are decomposed as exp pmaxffiffiffi 2 ¼ I; ð94Þ 6 follows: to find 1 H14 ¼ pffiffiffi diag½λ3; −ðλ3Þ; λ8; a 8 a a a pffiffiffi 7 α ¼ 2π 2; 27 1 8 6 6 3 3 max1 H ¼ pffiffiffiffiffi diag½λ ; λ ; −ðλ Þ ; λ ; −ðλ Þ ; 0; pffiffiffi a 18 a 6 a a a α7 ¼ 2π 6: ð95Þ max2 1 64 ¼ ½λ15 −ðλ15Þ λ8 λ8 λ6 −ðλ6Þ λ3 Ha 8 diag a ; a ; a; a; a; a ; a; 7 It can be easily shown that the first Cartan generatorphffiffiffi1 in − ðλ3Þ; αmax ¼ 4π 2 a Eq. (93), with the maximum flux value of 1 ,is 1 capable of producing the whole G2 potential individually, H77 ¼ pffiffiffiffiffiffiffiffi diag½λ27; λ15; −ðλ15Þ; λ8; λ6; −ðλ6Þ; 7 a 110 a a a a a a without using h2. In the next stage, we are going to calculate reconstructed Cartan generators in 7-, 14-, 27-, 770 1 15 15 10 10 8 0 H ¼ pffiffiffiffiffi diag½λ ; −ðλ Þ ; λ ; −ðλ Þ ; λ ; 64-, 77-, and 77 -dimensional representations from the a 2 22 a a a a a decomposition of G2 into its SUð3Þ subgroup. 6 6 3 3 λa; −ðλaÞ ; λa; −ðλaÞ ; 0; ð99Þ

1. SUð3Þ subgroup

The decomposition of the fundamental representation where the upper indices of the λa’s indicate the dimensions into the SUð3Þ subgroup is as follows [36,37]: of the SUð3Þ representations. The maximum flux values extracted from Eq. (19) are

G2 ⊃ SUð3ÞðRÞ pffiffiffi pffiffiffiffiffi 7 ¼ 3 ⊕ 3¯ ⊕ 1: ð96Þ α14 ¼ 2π 8; α14 ¼ 2π 24; max1 pffiffiffi max2 pffiffiffi α27 ¼ 6π 2; α27 ¼ 6π 6; The reconstructed Cartan generator with respect to this max1 max2 pffiffiffi α64 ¼ 16π α64 ¼ 16π 3 decomposition is 1 ; 2 ; max pffiffiffiffiffiffiffiffi max pffiffiffiffiffiffiffiffi α77 ¼ 2π 110; α77 ¼ 2π 330; max1 pffiffiffiffiffi max2 pffiffiffiffiffi 1 0 0 7 3 3 α77 ¼ 4π 22 α77 ¼ 4π 66 ð Þ H ¼ pffiffiffi diag½λ ; −ðλ Þ ; 0; ð97Þ max1 ; max2 : 100 a 2 a a with a ¼ 3, 8. It is clear that the decomposition of this The trivial static potentials of Eq. (10) for the funda- representation into the SUð3Þ subgroup results in the same mental, adjoint, 27-, 64-, 77-, and 770-dimensional repre- matrices as Eq. (93). Similar to the F4 and E6 exceptional sentations have been plotted in Fig. 18. Screening is groups, one is able to use SUð3Þ subgroups to reproduce observed for all representations, which is a consequence the group potentials. This matter enables us to calculate the of the adjoint “gluons” that pop out of the vacuum due to potentials of higher representations by taking the same high energies and screen the initial color sources. The procedure. The decomposition of the higher representations tensor products of all representations, when they create a of G2 into the SUð3Þ subgroup is [37] singlet, are an implication of this phenomenon:

056015-19 AMIR SHAHLAEI and SHAHNOOSH RAFIBAKHSH PHYS. REV. D 97, 056015 (2018)

10 7 14 TABLE III. The second column lists the Casimir numbers of 27 64 several representations of the G2 exceptional group [25–27,42]. 77 ’ 8 77 The Casimir ratios, slopes of the potentials obtained from the lower diagram of Fig. 18, and the potential ratios are given in the third, fourth, and fifth columns, respectively. The numbers in 6 the parentheses indicate the fit error.

V(R) Rep. Casimir numbers Cr Potential slope kr CF kF 4 1 7 2 1 0.329(8) 1 14 1 2 0.488(8) 1.48(1) 2 7 27 6 2.33 0.50(2) 1.52(1) 7 64 4 3.5 0.57(3) 1.74(3) 0 0 0 20 40 60 80 100 77 2 4 0.63(4) 1.91(4) R 5 4 77 2 5 0.68(5) 2.06(5) 7 14 27 3.5 64 77 77’ reach a plateau at R → ∞. To investigate the reason why 3 the potentials are linear at intermediate distances, we study

2.5 the effects of the subgroups of G2. The G2 exceptional group owns three direct maximal 2 V(R) subgroups, which are presented in Table I. Like those of the

1.5 F4 and E6 exceptional groups, the center elements of the SUð3Þ subgroup are not a direct cause of intermediate 1 confining potentials of G2 in several representations. Hence, we do not give a detailed calculation for this 0.5 subgroup, because the results are the same as those of 0 3 3.5 4 4.5 5 5.5 6 6.5 7 F4 and E6. So, we study the other subgroup of G2. It should R be mentioned that SUð2Þ × SUð2Þ is a regular subgroup. Hence, it proves that these features do not appear exclu- FIG. 18. Upper diagram: Static potentials of the G2 exceptional sively for singular maximal subgroups. group for the fundamental, adjoint, 27, 64, 77, and 770 repre- sentations in the range R ∈ ½1; 100. All potentials are screened at far distances and are linear at intermediate distance scales. Lower 2. SUð2Þ ×SUð2Þ subgroup diagram: Linear parts of the potentials in the range [3,7]. The slopes of the potentials are given in the fourth column of Table III. Using this subgroup, the fundamental and adjoint rep- resentations can be decomposed as follows [36,37]:

7 × 14 × 14 × 14 ¼ 1 ⊕ 10ð7Þ ⊕ 6ð14Þ ⊕ ;

14 × 14 ¼ 1 ⊕ 14 ⊕ 27 ⊕ ; 6 V14/V7 27 14 14 ¼ 1 ⊕ 3ð7Þ ⊕ 3ð14Þ ⊕ V27/V7 × × ; V /V 5 64 7 V77/V7 64 × 14 × 14 × 14 ¼ 2ð1Þ ⊕ 20ð7Þ ⊕ ; V77’/V7 77 × 14 × 14 ¼ 1 ⊕ 2ð7Þ ⊕ 4ð14Þ ⊕ ; 4 770 14 14 ¼ 1 ⊕ 3ð14Þ ⊕ 3ð27Þ ⊕ ð Þ × × : 101 3 Potential ratios The lower diagram of Fig. 18 shows the linear parts of the 2 potentials in the range R ∈ ½3; 7. The slopes of the linear potentials and the potential ratios (kr ) are listed in the fourth kF 1 and fifth columns of Table III, respectively. Comparing kr kF values with the values of Cr shows that potentials are in CF 0 qualitative agreement with Casimir scaling. The point-by- 0 5 10 15 20 point ratios of the potentials have been plotted in Fig. 19 in R the range R ∈ ½1; 20. The potential ratios start at accurate FIG. 19. Potential ratios in the different representations of the Casimir ratios. However, they plummet considerably at G2 exceptional group that launch at exact Casimir ratios but larger distances of R. In fact, the potential ratios almost deviate abruptly at farther distance ranges.

056015-20 F4, E6 AND G2 EXCEPTIONAL GAUGE … PHYS. REV. D 97, 056015 (2018)

1 G2 ⊃ SUð2Þ × SUð2ÞðRÞ 7 ¼ð2; 2Þ ⊕ ð1; 3Þ; 0.8

14 ¼ð1; 3Þ ⊕ ð3; 1Þ ⊕ ð2; 4Þ: ð102Þ 0.6

0.4 α]

Therefore, the reconstructed diagonal matrices for the [ f fundamental and adjoint representations of G2 with respect to the SUð2Þ × SUð2Þ subgroup are Re G 0.2

1 0 7 ¼ pffiffiffi ½σ2 σ2 σ3 HSUð2Þ diag 3; 3; 3 ; 6 -0.2

1 Rep. 7 of G 14 3 4 4 2 ¼ pffiffiffi ½σ 0 0 0 σ σ ð Þ Rep. 7 of SU(2) × SU(2) subgroup H ð2Þ diag 3; ; ; ; 3; 3 : 103 -0.4 SU 2 6 -200 -100 0 100 200 300 1 It is seen that the elements of these matrices are not 7 14 identical to H3 and H3 in Eqs. (96) and (98), respectively. 0.8 Therefore, the trivial potentials obtained from this decom- position are not the same as the original ones for G2. 0.6 Nevertheless, the center element matrix of the SUð2Þ α] [ r subgroup is 0.4 Re G Z7 ¼½ I I I SUð2Þ z1 2×2;z1 2×2; 3×3 ; 0.2 Z14 ¼½I 1 1 1 I I ð Þ SUð2Þ 3×3; ; ; ;z1 4×4;z1 4×4 : 104 0

Rep. 14 of G2 So, if one uses the nontrivial maximum flux condition in Rep. 14 of SU(2) × SU(2) subgroup -0.2 Eq. (6), the nontrivial maximum flux values are -200 -100 0 100 200 300

pffiffiffi FIG. 20. The real part of the group factor function versus the α7-non ¼ 2π 6 max -SUð2Þ ; location x of the vacuum domain midpoint, for R ¼ 100 and in pffiffiffiffiffi the range x ∈ ½−200; 300, for the fundamental and adjoint α14-non ¼ 2π 24: ð105Þ max -SUð2Þ representation of G2 (solid lines) in comparison with the same function versus the location x of the vortex midpoint obtained The group factor function of the fundamental and adjoint from the SUð2Þ × SUð2Þ decomposition (dashed lines). The representations obtained from this decomposition have minimum points of the G2 group factor, which occur at x ¼ 0 ¼ 100 −0 142 −0 143 been illustrated in Fig. 20, as well as the corresponding and x , reach the amounts . and . for the fundamental and adjoint representations, respectively. ones for G2. The detailed calculation for the higher representations has been given in Appendix C. In this figure, the minimum points of the G2 group factor, which G2 ⊃ SUð3Þ ¼ 0 ¼ 100 −0 142 occur at x and x , reach the values . and 7 ¼ 3 ⊕ 3¯ ⊕ 1 −0.143 for the fundamental and adjoint representations, ; respectively. The corresponding group factors of the 14 ¼ 3 ⊕ 3¯ ⊕ 8: ð106Þ SUð2Þ × SUð2Þ subgroup reach the same amounts at x ¼ 50. Therefore, similar to the F4 and E6 cases, the nontrivial center of the SUð2Þ subgroup induces temporary In the next step, confinement in the G2 exceptional group. Now, we go one step further and decompose the SUð3Þ subgroup into its SUð2Þ subgroup. This decomposition SUð3Þ ⊃ SUð2Þ × Uð1ÞðRÞ enables us to give a comprehensive conclusion from 3 ¼ 2 ⊕ 1; this work. 8 ¼ 3 ⊕ 2 ⊕ 2 ⊕ 1. ð107Þ 3. SUð3Þ ⊃ SUð2Þ ×Uð1Þ subgroup The decomposition of the fundamental and adjoint It should be recalled that the Uð1Þ factor has been ignored representations of G2 into the SUð3Þ subgroup are in these decompositions. Ultimately, one could have

056015-21 AMIR SHAHLAEI and SHAHNOOSH RAFIBAKHSH PHYS. REV. D 97, 056015 (2018) pffiffiffi 7 ¼ 2 ⊕ 1 ⊕ 2 ⊕ 1 ⊕ 1; α7-non ¼ 2π 2; max pffiffiffi 14 ¼ 2 ⊕ 1 ⊕ 2 ⊕ 1 ⊕ 3 ⊕ 2 ⊕ 2 ⊕ 1. ð108Þ α14-non ¼ 4π 2 ð Þ max : 111

Using the above decompositions, one is able to reconstruct The group factor function of the fundamental and adjoint the Cartan matrices for the fundamental and adjoint representations obtained from this decomposition have representations as follows: been illustrated in Fig. 21, as well as the corresponding ones for G2. It is observed that in each diagram the 1 7 pffiffiffi 2 2 minimum values of the two graphs are identical. Hence, H ð3Þ⊃ ð2Þ ¼ diag½σ3; 0; σ3; 0; 0; SU SU 2 the SUð2Þ gauge group has a dominant role in the linear 1 part of the trivial potentials of the exceptional gauge H14 ¼ pffiffiffi diag½σ2; 0; σ2; 0; σ3; σ2; σ2; 0: ð109Þ SUð3Þ⊃SUð2Þ 2 2 3 3 3 3 3 groups.

We are going to investigate the role of this decomposition in V. CONCLUSION the intermediate linear potentials of G2. So, the matrices of In this article, we have presented a generalized scenario the center elements are calculated as whereby the static potentials in different representations of 7 exceptional gauge groups could be calculated by means of Z ¼ diag½z1I2 2; 1;z1I2 2; 1; 1; SUð3Þ⊃SUð2Þ × × their unit center elements in the framework of the vacuum 14 Z ¼ ½ I 1 I 1 I I domain structure model. Although G2 and F4 exceptional SUð3Þ⊃SUð2Þ diag z1 2×2; ;z1 2×2; ; 3×3;z1 2×2; groups do not possess any nontrivial center elements and I 1 ð Þ z1 2×2; : 110 confinement is not expected, linear potential is observed for all representations at intermediate distances. This fact is Using the maximum flux condition in Eq. (6), we find also correct for the E6 exceptional gauge group, when one uses only the trivial center element in the calculation. 1 In addition, to calculate these types of trivial potentials, there is no need to use all the Cartan generators of the gauge 0.8 group. For example, concerning the G2, F4, and E6 exceptional groups, it seems adequate to consider only 7 26 27 0.6 their first Cartan generators, which are h1, h1 , and h1 in

α] their fundamental representations, respectively. On the [ f 0.4 other hand, if the Cartan generators reconstructed by

Re G the group decomposition into the maximal subgroups 0.2 have the same elements as h1, they are able to simulate the exact static potentials as the exceptional supergroups, 0 themselves. Thus, one is able to apply these subgroup decompositions to gain the static potential of the higher Rep. 7 of G2 Rep. 7 of the SU(2) subgroup -0.2 representations of the exceptional groups. Hence, Casimir -200 -100 0 100 200 300 x scaling of different representations of these groups is 1 observed. This method is not applicable for the potentials obtained by the nontrivial center elements of E6; i.e., the 0.8 potentials calculated by the nontrivial center elements in the thick center vortex model are not identical to the potentials 0.6 of their subgroups. Hence, it seems that this method is just valid when we use only the unit center element of the gauge α] [ r 0.4 groups to calculate the static potentials.

Re G To find the reason for the temporary confinement at

0.2 intermediate distances, we have turned to the center elements of the SUðNÞ subgroups by which their center

0 vortices indirectly produce the intermediate linear part in the supergroups. So, the group factor function ReGr½α⃗ðxÞ Rep. 14 of G2 Rep. 14 of the SU(2) subgroup has been investigated in different representations of the G2, -0.2 -200 -100 0 100 200 300 F4, and E6 exceptional gauge groups using the unit center x element only. Comparison of this function with the corre- FIG. 21. The same as Fig. 20 but the dashed lines represent the sponding one obtained from the nontrivial center elements group factor for the SUð3Þ ⊃ SUð2Þ × Uð1Þ decomposition. of the SUðNÞ subgroups shows that the center vortices of

056015-22 F4, E6 AND G2 EXCEPTIONAL GAUGE … PHYS. REV. D 97, 056015 (2018) the SUð3Þ subgroups in none of these exceptional groups confinement in different representations. In fact, if the could be responsible for the intermediate linear potential, number of center elements or center vortices in the matrix since the group factor functions reach different minimum of center elements obtained from two different decom- amounts. However, by means of the trivial center element positions is the same, the corresponding group factors reach of this subgroup, the same potential as the exceptional the same value when the vortex is located completely inside group itself is produced. the Wilson loop. The dominant role of the SUð2Þ subgroup Any regular or singular decomposition into the SUð2Þ in observing the temporary confinement obtained by the subgroup that produces a Cartan generator with the same unit center element is not exclusive to the exceptional elements as h1 gives rise to the linear intermediate parts in gauge groups. In the next work, we argue that this dominant the potentials of the supergroups. In fact, the extremums of role of the SUð2Þ subgroup is seen for the trivial potentials ReGr½α⃗ðxÞ, which occur at the points where 50% of the of the SUðNÞ gauge groups as well. We should mention vacuum domain flux enters the Wilson loop, are respon- that, due to the oversimplification of the model, the results sible for the intermediate linear potential. When the center that have been presented in this paper seem to be restricted element obtained from these SUð2Þ decompositions lies in the framework of the vacuum domain structure and thick entirely inside the Wilson loop, the corresponding group center vortex models. factor reaches a value that is equal to the extremum values of ReG ½α⃗ðxÞ for the given exceptional group. r APPENDIX A: F ⊃ SUð3Þ × SUð3Þ Furthermore, there are some subgroups such as SUð2Þ × 4 SUð2Þ for G2 and SUð2Þ × G2 for the F4 and G2 singular The decompositions of 273- and 324-dimensional irreps subgroup of E6 that produce different potentials from their of F4 to the irreps of the SUð3Þ × SUð3Þ subgroup are as supergroup. Yet, they are responsible for the temporary follows [36,37]:

273 ¼ð1; 1Þ ⊕ ð8; 1Þ ⊕ ð3; 3Þ ⊕ ð3¯; 3¯Þ ⊕ ð10; 1Þ ⊕ ð10¯ ; 1Þ ⊕ ð6; 3¯Þ ⊕ ð6¯; 3Þ ⊕ ð3; 6¯Þ ⊕ ð3¯; 6Þ ⊕ ð15; 3Þ ⊕ ð15¯ ; 3¯Þ ⊕ ð8; 8Þ; ðA1Þ

324 ¼ð1; 1Þ ⊕ ð8; 1Þ ⊕ ð1; 8Þ ⊕ ð3¯; 3¯Þ ⊕ ð3; 3Þ ⊕ ð6; 3¯Þ ⊕ ð6¯; 3Þ ⊕ ð27; 1Þ ⊕ ð6¯; 6¯Þ ⊕ ð6; 6Þ ⊕ ð15; 3Þ ⊕ ð15¯ ; 3¯Þ ⊕ ð8; 8Þ: ðA2Þ

Thus, the Cartan diagonal generators reconstructed by taking advantage of Eqs. (A1) and (A2) are

8times 10times 10times 1 zfflfflffl}|fflfflffl{ zfflfflffl}|fflfflffl{ zfflfflffl}|fflfflffl{ H273 ¼ pffiffiffiffiffidiag½0;0;…;0;λ3;λ3;λ3;−ðλ3Þ;−ðλ3Þ;−ðλ3Þ;0;…;0;0;…;0; a 3 14 a a a a a a zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{6 times zfflfflfflffl}|fflfflfflffl{6times zfflfflfflffl}|fflfflfflffl{15times zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{15times zfflfflfflffl}|fflfflfflffl{8times 3 3 3 3 6 6 6 6 6 6 3 3 3 3 8 8 −ðλaÞ ;…;−ðλaÞ ;;λa;…;λa;−ðλaÞ ;−ðλaÞ ;−ðλaÞ ;λa;λa;λa;λa;…;λa;;−ðλaÞ ;…;−ðλaÞ ;λa;…;λa; ðA3Þ

8 times 3 times 3 times 6 times 6 times 1 zfflfflffl}|fflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ H324 ¼ pffiffiffi diag½0; 0; …; 0; λ8; λ3; …; λ3; −ðλ3Þ; …; −ðλ3Þ; ðλ3Þ; …; −ðλ3Þ; λ3; …; λ3; a 9 2 a a a a a a a a a 6 6 15 15 8 zfflfflffl}|fflfflffl{27 times zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{times zfflfflfflfflffl}|fflfflfflfflffl{times zfflfflfflfflffl}|fflfflfflfflffl{times zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{times zfflfflfflfflffl}|fflfflfflfflffl{times 6 6 6 6 3 3 3 3 8 8 0; …; 0; −ðλaÞ ; …; −ðλaÞ ; λa; …; λa; λa; …; λa; −ðλaÞ ; …; −ðλaÞ ; λa; …; λa: ðA4Þ

Using the trivial flux condition in Eq. (19), the maximum flux values are calculated as follows: pffiffiffiffiffi pffiffiffiffiffi pffiffiffi pffiffiffi α273 ¼ 6π 14 α273 ¼ 6π 42 α324 ¼ 18π 2 α324 ¼ 18π 6 ð Þ max1 ; max2 ; max1 ; max2 : A5

The static potential calculated by means of the above equations has been given in Fig. 6. We can build up the center element matrices:

056015-23 AMIR SHAHLAEI and SHAHNOOSH RAFIBAKHSH PHYS. REV. D 97, 056015 (2018)

zfflfflffl}|fflfflffl{8 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{3 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{3 times zfflfflffl}|fflfflffl{10 times zfflfflffl}|fflfflffl{10 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{6 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{6 times Z273 ¼ ½1 1 … 1 I … I I … I 1 … 1 1 … 1 I … I I … I SUð3Þ diag ; ; ; ;zn 3×3; ;zn 3×3;zn 3×3; ;zn 3×3; ; ; ; ; ; ;zn 3×3; ;zn 3×3;zn 3×3; ;zn 3×3; zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{3 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{3 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{15 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{15 times zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{8 times I … I I … I I … I I … I I … I ð Þ zn 6×6; ;zn 6×6;zn 6×6; ;zn 6×6;zn 3×3; ;zn 3×3;zn 3×3; ;zn 3×3; 8×8; ; 8×8 ; A6

zfflfflffl}|fflfflffl{8 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{3 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{3 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{6 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{6 times Z324 ¼ ½1 1 … 1 I I … I I … I I … I I … I SUð3Þ diag ; ; ; ; 8×8zn 3×3; ;zn 3×3;zn 3×3; ;zn 3×3;zn 3×3; ;zn 3×3;zn 3×3; ;zn 3×3; zfflfflffl}|fflfflffl{27 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{6 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{6 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{15 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{15 times zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{8 times 1 … 1 I … I I … I I … I I … I I … I ð Þ ; ; ;zn 6×6; ;zn 6×6;zn 6×6; ;zn 6×6;zn 3×3; ;zn 3×3;zn 3×3; ;zn 3×3; 8×8; ; 8×8 : A7

Then, we use the nontrivial flux profile condition of Eq. (6) to estimate the maximum flux values for these representations: pffiffiffiffiffi pffiffiffiffiffi pffiffiffi pffiffiffi α273−non ¼ 6π 14 α273−non ¼ 2π 42 α324−non ¼ 18π 2 α324−non ¼ 6π 6 ð Þ max1 max2 max1 max2 : A8

Accordingly, using Eqs. (A2)–(A4) and (A8), the group factor functions could be plotted in Fig. 22.

⊃ ð Þ ⊃ ð Þ ð Þ APPENDIX B: F4 SO 9 SU 2 × SU 4 1

Representations 273 and 324 of the F4 exceptional group can be decomposed to the representations of the SOð9Þ 0.8 subgroup as follows: F4 ⊃ SOð9Þ 273 ¼ 9 ⊕ 16 ⊕ 36 ⊕ 84 ⊕ 128; 0.6 ] α

324 ¼ 1 ⊕ 9 ⊕ 16 ⊕ 44 ⊕ 126 ⊕ 128 ð Þ [ : B1 r 0.4

In the next step, Re G 0.2 SOð9Þ ⊃ SUð2Þ × SUð4Þ 9 ¼ð3; 1Þ ⊕ ð1; 6Þ; 0 Rep. 273 of F4 16 ¼ð2; 4Þ ⊕ ð2; 4¯Þ; Rep. 273 of SU(3) × SU(3) -0.2 -200 -100 0 100 200 300 36 ¼ð3; 1Þ ⊕ ð1; 15Þ ⊕ ð3; 6Þ; x 1 44 ¼ð1; 1Þ ⊕ ð5; 1Þ ⊕ ð3; 6Þ ⊕ ð1; 200Þ; 84 ¼ð1; 1Þ ⊕ ð1; 10Þ ⊕ ð1; 10¯ Þ ⊕ ð3; 6Þ ⊕ ð3; 15Þ; 0.8 126 ¼ð1; 6Þ ⊕ ð3; 10Þ ⊕ ð3; 10¯ Þ ⊕ ð1; 15Þ ⊕ ð3; 15Þ; 0.6 ¯ ¯ 128 ¼ð2; 4Þ ⊕ ð2; 4Þ ⊕ ð4; 4Þ ⊕ ð4; 4Þ ⊕ ð2; 20Þ ] α [ r ⊕ ð2; 20¯ Þ: ðB2Þ 0.4 Re G Now, we decompose SUð4Þ into its SUð2Þ subgroup: 0.2

SUð4Þ ⊃ SUð2Þ × SUð2Þ × Uð1Þ; 4 ¼ð2;1Þ ⊕ ð1;2Þ; 0

Rep. 324 of F4 6 ¼ð1;1Þ ⊕ ð1;1Þ ⊕ ð2;2Þ; Rep. 324 of SU(3) × SU(3) -0.2 -200 -100 0 100 200 300 10 ¼ð2;2Þ ⊕ ð3;1Þ ⊕ ð1;3Þ; x

15 ¼ð1;1Þ ⊕ ð2;2Þ ⊕ ð2;2Þ ⊕ ð3;1Þ ⊕ ð1;3Þ; FIG. 22. The same as Fig. 8 but for representations 273 and 20 ¼ð2;1Þ ⊕ ð2;1Þ ⊕ ð1;2Þ ⊕ ð1;2Þ ⊕ ð3;2Þ ⊕ ð2;3Þ; 324. The minimum values of the F4 group factor for representa- tions 273 and 324 are −0.025 and 0.037, respectively. It is seen 200 ¼ð1;1Þ ⊕ ð1;1Þ ⊕ ð1;1Þ ⊕ ð2;2Þ ⊕ ð2;2Þ ⊕ ð3;3Þ: that the minimum values of the group factors for the SUð3Þ × SUð3Þ decomposition are not identical to the corresponding ones ðB3Þ for F4.

056015-24 F4, E6 AND G2 EXCEPTIONAL GAUGE … PHYS. REV. D 97, 056015 (2018)

Ultimately we have

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{19 times zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{8 times zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{17 times zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{24 times 273 ¼ð1; 1Þ ⊕ ⊕ ð1; 1Þ ⊕ ð3; 1Þ ⊕ ⊕ ð3; 1Þ ⊕ ð2; 2Þ ⊕ ⊕ ð2; 2Þ ⊕ ð2; 1Þ ⊕ ⊕ ð2; 1Þ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{24 times zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{6 times zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{4 times zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{4 times ⊕ ð1; 2Þ ⊕ ⊕ ð1; 2Þ ⊕ ð1; 3Þ ⊕ ⊕ ð1; 3Þ ⊕ ð2; 3Þ ⊕ ⊕ ð2; 3Þ ⊕ ð3; 2Þ ⊕ ⊕ ð3; 2Þ ðB4Þ

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{18 times zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{11 times zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{21 times 324 ¼ 1 ⊕ ð1; 1Þ ⊕ ⊕ ð1; 1Þ ⊕ ð3; 1Þ ⊕ ⊕ ð3; 1Þ ⊕ ð2; 2Þ ⊕ ⊕ ð2; 2Þ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{24 times zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{24 times zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{10 times zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{4 times ⊕ ð2; 1Þ ⊕ ⊕ ð2; 1Þ ⊕ ð1; 2Þ ⊕ ⊕ ð1; 2Þ ⊕ ð1; 3Þ ⊕ ⊕ ð1; 3Þ ⊕ ð2; 3Þ ⊕ ⊕ ð2; 3Þ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{4 times ⊕ ð3; 2Þ ⊕ ⊕ ð3; 2Þ ⊕ ð3; 3Þ ⊕ ð5; 1ÞðB5Þ

1 and

0.8 91 times 70 times 14 times 1 zfflfflffl}|fflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ H273 ¼ pffiffiffiffiffi diag½0; …; 0; σ2; …; σ2; σ3; …; σ3; 0.6 SUð2Þ 3 14 3 3 3 3 α]

[ 78 21 r 105 times times times 0.4 zfflfflffl}|fflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ 324 1 2 2 3 3

Re G H ¼ pffiffiffi diag½0; …; 0; σ ; …; σ ; σ ; …; σ : ðB6Þ SUð2Þ 9 2 3 3 3 3 0.2 ð2Þ 0 The matrices made of the center elements of the SU Rep. 273 of F4 gauge group corresponding to the duality of its represen- Rep. 273 of SU(2) × SU(2) × SU(2) × U(1) tation with respect to Eqs. (B4) and (B5) are as follows: -0.2 -200 -100 0 100 200 300 x 1 zfflfflffl}|fflfflffl{91 times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{70times zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{14 times Z273 ¼ ½1 … 1 I … I I … I SUð2Þ diag ; ; ;z1 2×2; ;z1 2×2; 3×3; ; 3×3 ; 0.8 zfflfflffl}|fflfflffl{105times zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{78times zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{21 times Z324 ¼ ½1 … 1 I … I I … I 0.6 SUð2Þ diag ; ; ;z1 2×2; ;z1 2×2; 3×3; ; 3×3 : α]

[ ð Þ

r B7 0.4 Re G

0.2 The maximum flux values could be calculated from the nontrivial flux condition of Eq. (6): 0 pffiffiffiffiffi pffiffiffi Rep. 324 of F 273 324 4 α -non ¼ 6π 14; α -non ¼ 18π 2: ðB8Þ Rep. 324 of SU(2) × SU(2) × SU(2) × U(1) SUð2Þ- max SUð2Þ- max -0.2 -200 -100 0 100 200 300 x The group factor functions of these representations have FIG. 23. The real part of the group factor versus the location x been given in Fig. 23. of the vacuum domain midpoint, for R ¼ 100 and in the range x ∈ ½−200; 300, for representations 273 and 324 of F4 (solid lines) in comparison with the one obtained from the SOð9Þ ⊃ APPENDIX C: G2 ⊃ SUð2Þ × SUð2Þ SUð2Þ × SUð4Þ decomposition using its nontrivial center ele- ments (dashed lines). In each diagram, the minimum values are Decompositions of the G2 representations into the identical. SUð2Þ × SUð2Þ regular subgroup representations are

056015-25 AMIR SHAHLAEI and SHAHNOOSH RAFIBAKHSH PHYS. REV. D 97, 056015 (2018)

27 ¼ð3; 3Þ ⊕ ð2; 4Þ ⊕ ð2; 2Þ ⊕ ð1; 5Þ ⊕ ð1; 1Þ; 64 ¼ð4; 2Þ ⊕ ð3; 5Þ ⊕ ð3; 3Þ ⊕ ð2; 6Þ ⊕ ð2; 4Þ ⊕ ð2; 2Þ ⊕ ð1; 5Þ ⊕ ð1; 3Þ; 77 ¼ð5; 1Þ ⊕ ð4; 4Þ ⊕ ð3; 7Þ ⊕ ð3; 3Þ ⊕ ð2; 6Þ ⊕ ð2; 4Þ ⊕ ð1; 5Þ ⊕ ð1; 1Þ; 770 ¼ð4; 4Þ ⊕ ð3; 5Þ ⊕ ð3; 3Þ ⊕ ð3; 1Þ ⊕ ð2; 6Þ ⊕ ð2; 4Þ ⊕ ð2; 2Þ ⊕ ð1; 7Þ ⊕ ð1; 3Þ: ðC1Þ

The Cartan generators could be reconstructed as follows: 1 H27 ¼ pffiffiffi diag½σ3; σ3; σ3; σ4; σ4; σ2; σ2; σ5; 0; SUð2Þ 3 6 3 3 3 3 3 3 3 3 1 H64 ¼ pffiffiffi diag½σ2; σ2; σ2; σ2; σ5; σ5; σ5; σ3; σ3; σ3; σ6; σ6; σ4; σ4; σ2; σ2; σ5; σ3; SUð2Þ 8 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 H77 ¼ pffiffiffiffiffiffiffiffi diag½0; 0; 0; 0; 0; σ4; σ4; σ4; σ4; σ7; σ7; σ7; σ3; σ3; σ3; σ6; σ6; σ4; σ4; σ5; 0; SUð2Þ 330 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

0 1 H77 ¼ pffiffiffiffiffi diag½σ4; σ4; σ4; σ4; σ5; σ5; σ5; σ3; σ3; σ3; 0; 0; 0; σ6; σ6; σ4; σ4; σ2; σ2; σ7; σ3: ðC2Þ SUð2Þ 2 66 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

The center element matrices of the SUð2Þ × SUð2Þ subgroup are

Z27 ¼ ½I I I I I I I I 1 SUð2Þ diag 3×3; 3×3; 3×3;z1 4×4;z1 4×4;z1 2×2;z1 2×2; 5×5; ; Z64 ¼ ½ I I I I I I I I I I I I I I I I I I SUð2Þ diag z1 2×2;z1 2×2;z1 2×2;z1 2×2; 5×5; 5×5; 5×5; 3×3; 3×3; 3×3;z1 6×6;z1 6×6;z1 4×4;z1 4×4;z1 2×2;z1 2×2; 5×5; 3×3 ; Z77 ¼ ½1 1 1 1 1 I I I I I I I I I I I I I I I 1 SUð2Þ diag ; ; ; ; ;z1 4×4;z1 4×4;z1 4×4;z1 4×4; 7×7; 7×7; 7×7; 3×3; 3×3; 3×3;z1 6×6;z1 6×6;z1 4×4;z1 4×4; 5×5; ; Z770 ¼ ½ I I I I I I I I I I 1 1 SUð2Þ diag z1 4×4;z1 4×4;z1 4×4;z1 4×4; 5×5; 5×5; 5×5; 3×3; 3×3; 3×3; ; ; 1 I I I I I I I I ð Þ ;z1 6×6;z1 6×6;z1 4×4;z1 4×4;z1 2×2;z1 2×2; 7×7; 3×3 : C3

Using the nontrivial maximum flux condition of Eq. (6), we find pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffi α27-non ¼ 6π 6 α64-non ¼ 16π 3 α77-non ¼ 2π 330 α770-non ¼ 4π 66 ð Þ SUð2Þ- max ; SUð2Þ- max ; SUð2Þ- max ; SUð2Þ- max : C4

The group factor of representations 27, 64, 77, and 770 have been presented in Figs. 24–27.

1 1

0.8 0.8

0.6 0.6 α] α] [ [ r r 0.4 0.4 Re G Re G

0.2 0.2

0 0

Rep. 27 of G2 Rep. 64 of G2 Rep. 27 of SU(2) × SU(2) subgroup Rep. 64 of SU(2) × SU(2) subgroup -0.2 -0.2 -200 -100 0 100 200 300 -200 -100 0 100 200 300

FIG. 24. The same as Fig. 20 but for representation 27. The FIG. 25. The same as Fig. 20 but for representation 64. The extremum values of the F4 group factor at x ¼ 0 and x ¼ 100 are extremum values of the F4 group factor at x ¼ 0 and x ¼ 100 are approximately equal to 0.111. approximately equal to 0.

056015-26 F4, E6 AND G2 EXCEPTIONAL GAUGE … PHYS. REV. D 97, 056015 (2018)

1 1

0.8 0.8

0.6 0.6 α] α] [ [ r r 0.4 0.4 Re G Re G

0.2 0.2

0 0

Rep. 77 of G2 Rep. 77’ of G2 Rep. 77 of SU(2) × SU(2) subgroup Rep. 77’ of SU(2) × SU(2) subgroup -0.2 -0.2 -200 -100 0 100 200 300 -200 -100 0 100 200 300

FIG. 26. The same as Fig. 20 but for representation 77. The FIG. 27. The same as Fig. 20 but for representation 770. The extremum values of the F4 group factor at x ¼ 0 and x ¼ 100 are extremum values of the F4 group factor at x ¼ 0 and x ¼ 100 are approximately equal to 0.064. approximately equal to −0.038.

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