PHYSICAL REVIEW D 100, 114504 (2019)

Domain structures and potentials in SU(3)

Seyed Mohsen Hosseini Nejad * Faculty of Physics, Semnan University, P.O. Box 35131-19111, Semnan, Iran

(Received 25 July 2019; published 6 December 2019)

We analyze the static potentials for various representations in SU(3) Yang-Mills theory within the framework of the domain model of center vortices. The influence of vortex interactions is investigated on the static potentials. We show that, by ad hoc choosing the probability weights of the different vortex configurations contributing to the static potential, a phenomenologically satisfactory result for the different representations can be achieved. In particular including vacuum domains, a way to effectively parametrize vortex interactions, is crucial in obtaining an (almost) everywhere convex potential when interpolating between the short distances and the asymptotic regimes.

DOI: 10.1103/PhysRevD.100.114504

I. INTRODUCTION In this paper, this artifact is studied through analyzing Understanding quark confinement and the dynamical vortex interactions. We represent various forms of the center mechanism behind it is a big challenge in QCD. The vortex picture of confinement for static quark potentials in interaction between static quark sources at small separa- different representations of SU(3). In the thick center vortex – tions is dominated by one- exchange and the potential model, we investigate the Yang Mills vacuum of the SU(3) is Coulomb-like. At intermediate distances, quark confine- gauge theory including two types of center vortices. In some ment arises referring to the color electric flux-tube for- literatures, two types of vortices may be regarded as the same mation and linear potentials. In this range of distances, the type of vortex but with magnetic flux pointing in opposite string tensions for different representations are qualitatively directions. Without this constraint, we study the behavior of in agreement with Casimir scaling [1–3]. At asymptotic these center vortices on static potentials in this analytical distances, the string tensions depend only on the N-ality of model. Although interactions of both types of center vortices the representations [4]. In addition, the quark potential must with sufficiently large Wilson loops are the same, their interactions with medium size Wilson loops are different. be everywhere convex and without concavity [5]. Numerical N simulations [6–11] and infrared models [12–19] have indi- Besides, Casimir scaling and -ality regimes for some cated that center vortices [20–25] which are quantized representations do not connect smoothly and some kind of unexpected concavity occurs in the model which explicitly magnetic flux tubes could account for the quark confinement disagrees with lattice results. In Refs. [13,33,34],adomain via the area law of the . Furthermore, numerical structure is assumed in the vacuum for G(2) and SU N simulations have shown that the center vortices could also ð Þ gauge theories. The total magnetic flux through each domain account for spontaneous chiral breaking [26–32]. corresponds to a center element of ZðNÞ subgroup. In the The thick center vortex model [12,13] is a phenomeno- framework of the domain model of center vortices, we logical model trying to understand the in analyze the domain structures with a fixed vortex profile terms of the interaction of the Wilson loops with the center for removing concavity and improving Casimir scaling vortices. However, the potentials induced by center vortices especially for higher representations of the SU(3) gauge for some representations show unphysical concavity when group. Interactions between two types of vortices are interpolating between the short distances and the asymp- discussed using a dual analogy to the type II superconduc- totic regimes. For removing the concavity, in Ref. [16], the tivity where it seems that two vortices repel each other while vortex profile is allowed to fluctuate. the vortex-antivortex interaction is attractive. Moreover, we argue that the same interactions may be confirmed by the model. We show that interactions between two types of *[email protected] center vortices may deform them to the configurations with the lowest magnitude of center fluxes where there are Published by the American Physical Society under the terms of appeared vortices of type one as well as vacuum domains the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to on the vacuum. We show that ad hoc choosing the probability the author(s) and the published article’s title, journal citation, weights of the domain structures is crucial in obtaining an and DOI. Funded by SCOAP3. (almost) everywhere convex potential.

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n n ⃗ In Sec. II, we analyze the static potentials in various where the function GrðαCðxÞÞ ¼ 1=drTr exp½iα⃗CH, dr is representations and their ratios induced by center vortices the dimension of the representation, and fHig is the set of n in SU(3) gauge theory within the framework of the thick generators from the Cartan subalgebra. The function αCðxÞ center vortex model. We investigate the contributions of denotes the vortex profile and this angle depends on both the center vortices and the vortex interactions in the potentials Wilson loop C and the position of the vortex center x. If the in Sec. III. Then, in Sec. IV the confinement mechanism in center vortex is all contained within the Wilson loop the background of center vortices would be reformulated n ⃗ kr exp½iα⃗CH¼ðznÞ I where kr is the N-ality of representa- for removing concavity and improving the Casimir scaling. tion r. Using this constraint, the maximum value of the angle We summarize the main points of our study in Sec. V. αn max could be calculated. If the center vortex is outside the n ⃗ loop exp½iα⃗CH¼I and therefore it has no effect on the II. STATIC POTENTIALS INDUCED BY TWO loop. The quark potential induced by the center vortices is as TYPES OF SU(3) CENTER VORTICES follows [12]: N Any non-Abelian SUð Þ gauge theory of confinement X XN−1 should explain some features of the confining force which n VrðRÞ¼− ln 1 − fn½1 − ReGrðα⃗CðxÞÞ ; ð2:3Þ can be verified in lattice simulations. If one neglects x n¼1 dynamical in the vacuum in the first approximation, where the parameter fn determines the probability that any the static quark potential of nonperturbative regime has given plaquette is pierced by an nth center vortex. An ansatz distinct behavior in two ranges of interquark distances. At n for the angle α⃗C was introduced by Greensite et al. [13]. Each intermediate distances, from the onset of the confinement to A L L the onset of color screening, the quark potential is expected center vortex with square cross section v ¼ v × v to be linearly rising and the string tension of the quark contains small independently fluctuating subregions of area l2 ≪ A l potential for the representation r is approximately propor- v which is a short correlation length. The only tional to Cr, the eigenvalue of the quadratic Casimir constraint is that the total magnetic fluxes of the subregions Cr must correspond to a center element of the gauge group. This operator for the representation r, i.e., σr ≈ σF where F CF square ansatz is as follows: denotes the fundamental representation [1–3]. When the A A A2 A 2 energy between quarks suffices, a gluon pair is created in α⃗n x α⃗n x v − αn ; : Cð Þ · Cð Þ¼ 2 þ max ð2 4Þ the vacuum and Casimir scaling breaks down and is 2μ Av Av Av N replaced by an -ality dependent law [4]. Therefore, at A asymptotic distances, the quark potential depends on the where is the cross section of the center vortex overlapping with the minimal area of the Wilson loop and μ is a free N-ality kr of the representation, i.e., σr ¼ σðkrÞ. The string parameter. tension σðkrÞ corresponds to the lowest dimensional N N k Now, we apply the model to the SU(3) gauge group with representation of SUð Þ with -ality r. In addition, the Z 3 lattice results [5] show that the static quark potential must center ð Þ. The homotopy group be everywhere convex, i.e., Π1½SUð3Þ=Zð3Þ ¼ Zð3Þ; ð2:5Þ dV d2V implies that the SU(3) gauge theory has center vortices > 0 and ≤ 0: ð2:1Þ dr dr2 corresponding to the nontrivial center elements. In SU(3) case, there are two types of center vortices corresponding to Therefore, it is crucial to obtain a convex potential without the nontrivial center elements z1 ¼ expði2π=3Þ and any concavity when interpolating between the short dis- z2 ¼ expði4π=3Þ. In some literatures, vortices of type z1 tances and the asymptotic regimes. and type z2 have phase factors which could be considered Any model of the quark confinement should be able to complex conjugates of one another (z1 ¼ z ) and therefore explain these features for the potentials between static 2 two vortices may be regarded as the same type of vortex but quarks. The thick center vortex model has been fairly with magnetic flux pointing in opposite directions. Without successful in describing the mechanism of confinement in this constraint, vortex fluxes of two types of center vortices QCD [12]. However there are still some shortcomings are different and we analyze the behavior of these center within the model which is at the focus of this article. In vortices on static potentials. Using Eq. (2.3), the static this model, the vacuum is assumed to be filled with center potential induced by center vortices in SUð3Þ gauge group is vortices. In SUðNÞ gauge group, there are N − 1 types as follows: of center vortices corresponding to the nontrivial center elements of zn ¼ expði2πn=NÞ ∈ ZðNÞ enumerated by the LvX=2þR value n ¼ 1; …;N− 1. The effect of a thick center vortex on 1 VrðRÞ¼− ln½ð1 − f1 − f2Þþf1ReGrðα ðxÞÞ a planar Wilson loop is to multiply the loop by a group factor C x¼−Lv=2 n 2 WrðCÞ → GrðαCðxÞÞWrðCÞ; ð2:2Þ þ f2ReGrðαCðxÞÞ; ð2:6Þ

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C C f6g 2 5; f8g 2 25; C ¼ . C ¼ . f3g f3g C C f10g 4 5; f15sg 7: : C ¼ . C ¼ ð2 8Þ f3g f3g

In the range R ∈ ½0; 20, the potential ratios for the various FIG. 1. The Young diagrams for the lowest representations of representations drop slowly from Casimir ratios. However, SU(3). The N-ality k of the representations are shown below the s the deviations from the exact Casimir scaling are much diagrams. The label means the representation is symmetric. greater for higher representations. At large distances, the static potentials agree with N-ality where f1, f2 are the probabilities that any given plaquette where can bind to the initial sources and string is pierced by z1 and z2 center vortices, respectively. The tensions of the representations are reduced to the lowest- square ansatz given in Eq. (2.4) for the angles corresponding dimensional representation with the same N-ality. In to the center vortices for all representations are particular, zero N-ality representations are screened. For example, an adjoint charge combining with a gluon can A A A2 4π A 2 form a color-singlet, ½f8g ⊗ f8g¼f1g ⊕ …. More α1 x 2 v − ffiffiffi ; ð Cð ÞÞ ¼ 2 þ p dynamical gluons might be required for screening of higher 2μ Av Av 3 Av representations with zero N-ality. Nonzero N-ality repre- A A A2 8π A 2 α2 x 2 v − ffiffiffi : : sentations through combining with gluons are transformed ð Cð ÞÞ ¼ 2 þ p ð2 7Þ 2μ Av Av 3 Av into the lowest order representations. For example, a tensor product of ½f6g ⊗ f8g¼f3¯g ⊕ … shows that the slope 2 The free parameters Lv, f1, f2, and Lv=ð2μÞ are chosen to be of the potential for the representation f6g must be the same 100, 0.01, 0.01, and 4, respectively. The correlation length is as the one for the representation f3g. taken l ¼ 1 and therefore the potentials are linear from the As a result, the model leads to Casimir scaling at the beginning (R ¼ l). Now, we study the static potentials of the intermediate distances and exhibits N-ality at the asymp- lowest representations in SUð3Þ gauge theory. Figure 1 totic regimes, in agreement with lattice calculations. But shows the Young diagrams as well as N-ality k of the these two regimes for several representations do not representations. connect smoothly and some kind of unexpected concavity Figure 2(a) plots the static potentials VrðRÞ induced by occurs in the model which explicitly disagrees with lattice two types of center vortices for these representations in the results. range R ∈ ½0; 100. At intermediate distances, the potentials In the next section, for reducing the concavity of some are linear in the range R ∈ ½0; 20. The potential ratios representations, we argue about the behavior of two types V R =V R r frgð Þ f3gð Þ for the various representation are shown of center vortices on the vacuum through analyzing their in Fig. 2(b). These ratios start out at the Casimir ratios: effects on the Wilson loops.

5 8 r=6 {3} r=8 {6} 4 6 r=10 {8} r=15s {10} (R)

3 {15s} {3} 4 V(R) 2 (R)/V {r}

V 2 1

0 0 0 20 40 60 80 100 0 5 10 15 20 (a)R (b) R

FIG. 2. (a) The static potentials using both types of center vortices for various representations of SUð3Þ. The concavity is appeared for V R =V R several representations. (b) Potential ratios frgð Þ f3gð Þ at the intermediate distances. The ratios start from the Casimir ratios and violate slowly from the Casimir ratios in this regime. However, the deviations from the exact Casimir scaling are much greater for higher 2 representations. The free parameters are Lv ¼ 100, f1 ¼ f2 ¼ 0.01, and Lv=ð2μÞ¼4.

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III. CENTER VORTEX CONTRIBUTIONS The behavior of a group factor gives some information IN THE POTENTIALS about the details of its potential. The functions of the group factors for the lowest representations of SU(3) can be found For analyzing the concavity of the potentials in SU(3) in the Appendix. Now, we analyze the group factors gauge group, we study the potentials induced by two types corresponding to two types of center vortices for the of center vortices in more details. As shown in Fig. 2(a), the adjoint representation close to the concavity regime (about concavity is appeared for several representations such as R ¼ 60). The timelike legs of the Wilson loop are located at the adjoint representation. Figure 3 depicts the potentials x ¼ 0 and x ¼ 60. When the center vortex overlaps the induced by two types of center vortices individually for the minimal area of the Wilson loop, it affects the Wilson loop. adjoint representation. The concavity is appeared in the As shown in Fig. 4(a), z1 vortex group factor changes static potential induced by center vortices of type two while smoothly with a minimum value around any timelike leg there is no this artifact in the potential obtained by center (x ¼ 0, 60) while for the z2 vortex group factor a wavy vortices of type one. character with equal large sizes of maxima and minima is observed the neighborhood of any these regimes.

1.4 Therefore, the large fluctuations of the group factor around 1.2 any timelike leg lead to the concavity behavior in the potentials. As shown in Fig. 4(b), at large distances 1 (R ¼ 100) governed with the N-ality, the group factors for both types of center vortices in the adjoint representa- 0.8 tion interpolate from 1, when the vortex core is located

V(R) 0.6 entirely within the Wilson loop, to 1, when the core is entirely outside the loop. As shown in Fig. 4, a fluctuation 0.4 {8}, z with a minimum is appeared around each timelike leg for 1 the z1 vortex group factor in the adjoint representation 0.2 {8}, z 2 while two of these fluctuations occur around each timelike z z 0 leg for the 2 vortex group factor. Since 2 vortices are 2 0 20 40 60 80 100 characterized by the center element z2 ¼ z , there is R 1 periodicity in the z2 vortex group factor and its potential z FIG. 3. The static potentials induced by two types of center compared with those of the 1 vortex. vortices individually for the adjoint representation. The concavity Furthermore, Fig. 5(a) depicts the group factors corre- is observed for the potential induced by z2 center vortices while sponding to two types of center vortices for the medium there is no this artifact in the potential obtained by z1 center size Wilson loop with R ¼ 15 for the fundamental repre- vortices. The free parameters are Lv ¼ 100, f1 ¼ f2 ¼ 0.01, and sentation and the ones for the large size loop with R ¼ 100 L2= 2μ 4 v ð Þ¼ . are plotted in Fig. 5(b). As shown, the z1 vortex group

R=60, z R=100, z 1 1.2 1.2 1 R=60, z R=100, z 2 2 1 1 ) ) α α ( ( 0.8 0.8 {8} {8} 0.6 0.6 Re G Re G 0.4 0.4

0.2 0.2

0 0 −50 0 50 100 −50 0 50 100 150 (a)x (b) x G α x R 60 FIG. 4. (a) The group factors Re f8gð Þ of the two types of center vortices vs corresponding to Fig. 3 at ¼ , close to the concavity regime. The fluctuations of the z2 vortex group factor with equal large sizes of maxima and minima around any time-like leg lead to concavity behavior in the potentials. (b) The same as (a) but for the large size Wilson loop with R ¼ 100. In agreement with the color screening in the large regime for the adjoint representation, the group factors for both types of center vortices when the vortex core 2 is located entirely within the Wilson loop is equal to 1. The free parameters are Lv ¼ 100 and Lv=ð2μÞ¼4.

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1 1 R=15, z R=100, z 1 1 R=15, z R=100, z 2 2 ) 0.5 ) 0.5 α α ( ( {3} {3} Re G Re G 0 0

−0.5 −0.5 −40 −20 0 20 40 60 −50 0 50 100 150 (a)x (b) x

G α x FIG. 5. (a) The group factors Re f3gð Þ of the two types of center vortices vs for the fundamental representation in the intermediate distance with R ¼ 15. (b) the same as (a) but for the asymptotic distance with R ¼ 100. The group factors for both types of center vortices when the vortex core is located entirely within the Wilson loop is equal to −0.5. Decreasing the size of the loop, the minimum value of the z1 vortex group factor is increased and becomes close to trivial value while the one of the z2 vortex group factor is close to 2 center vortex value (−0.5). The free parameters are Lv ¼ 100 and Lv=ð2μÞ¼4. factor in the fundamental representation changes smoothly break down somewhat the Casimir scaling at intermediate around each timelike leg and therefore one could expect distances. Figure 6 plots the potential ratios induced by 2 that the group factor of z2 ¼ z1 changes smoothly the center vortices for the range R ∈ ½0; 20. As shown, the neighborhood of each timelike leg. contributions of two types of the center vortices are At large distances, the group factors for both types of compared. For various representations, the potential ratios center vortices in the fundamental representation interpo- obtained from the z1 vortices which start from the Casimir late from −0.5, when the vortex core is located entirely ratios drop slower than those induced by both types of within the Wilson loop, to 1, when the core is entirely vortices. outside the loop. As shown in Fig. 5(a), decreasing the size For detailed analysis of two types of vortices, we note to of the Wilson loop (R ¼ 15), the minimum value of the z1 the interactions between vortices. The QCD vacuum could vortex group factor is increased while the one of the z2 be described in terms of a Landau-Ginzburg model of a vortex group factor is close to center vortex value (−0.5). dual superconductor where it follows the electric flux The value of the z2 vortex group factor for the medium size tube formation and confinement of the electric charge. Wilson loops is about center vortex value which is related A dual superconductor is like type II superconductors but to N-ality regimes. Therefore, we expect that the z2 vortices the roles of the electric and magnetic fields, and electric

7

2.4 6 5 (R) (R) 2.2 {3} {3} 4 2 r=6, z 1 3 r=10, z (R)/V (R)/V 1 r=6, z and z {r} {r} 1.8 1 2 r=10, z and z 1 2 V V r=8, z 2 1 r=15s, z 1 1.6 r=8, z and z 1 1 2 r=15s, z and z 1 2 1.4 0 0 5 10 15 20 0 5 10 15 20 (a)R (b) R V R =V R FIG. 6. The potential ratios of frgð Þ f3gð Þ induced by center vortices for the various representations. Upper curve of any representation shows the contribution of the z1 vortices which violates more slowly from the Casimir ratio compared with the contribution of the z1 vortices plus the z2 vortices. For all representations, the ratios induced by the z1 vortices agree better with Casimir 2 scaling compared with the ratios induced by z2 vortices. The free parameters are Lv ¼ 100, f1 ¼ f2 ¼ 0.01, and Lv=ð2μÞ¼4.

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4 The next step, the confinement mechanism in the back- ground of center vortices would be reformulated for remov- ing this concavity and we discuss two types of SU(3) center 3 vortices in more details.

2 IV. VACUUM DOMAINS AND REMOVING THE V(R) {3} CONCAVITY OF THE POTENTIALS {6} The QCD vacuum is a dual analogy to the type II 1 {8} {10} superconductivity. As argued, it seems that two vortices {15s} repel each other while the vortex-antivortex interaction is attractive. On the one hand, two z1 vortices within the z2 0 0 20 40 60 80 100 vortex repel each other and one could observe them as the R single vortices. On the other hand, in addition to two types of center vortices, there are their antivortices corresponding z FIG. 7. The static potentials using 1 center vortices for the to complex conjugates of center elements on the vacuum. various representations of SUð3Þ. Although the concavity is Therefore, z2 and z1 vortex configurations attract each removed somewhat this artifact stays for higher presentations 2 other and they would merge forming z2z1 ¼ z1z1 ¼ z1z0 especially 15s . The free parameters are Lv 100, f1 0.01, f g ¼ ¼ 2 where z1z is equal to the identity element z0 1.In and Lv=ð2μÞ¼4. 1 ¼ Refs. [13,33,34], vacuum domains corresponding to the identity element are also allowed in the model. The Yang– and magnetic charges, have been interchanged [35,36]. Mills vacuum has a domain structure where there are Properties of superconductors are often described in terms domains of the center-vortex type and of the vacuum type. ξ of the superconducting coherence length and the London Therefore, we observe that attractions between z2 and z λ 1 magnetic penetration depth . The Ginzburg-Landau vortices are forming z1 vortices as well as vacuum domains. parameter κ ¼ λ=ξ of the type-II superconductor is larger z z pffiffiffi One could apply the same argument for 2 and 1 vortex than 1= 2. In the type II superconductors, there are configurations. The domain structures can be readily vortices as the magnetic flux lines as well as the magnetic generalized to SU(4) and beyond. For example, in SU fluxes pointing in opposite directions of vortices (anti- (4), there are nontrivial center elements z1 ¼ expðiπ=2Þ, 2 3 vortices). The interaction between vortices is repulsive z2 ¼ z1, and z3 ¼ z1. The attractions between vortices and while the vortex-antivortex interaction is attractive [37,38]. antivortices in SU(4) may form center vortices as well as Furthermore, one may find the same interactions between vacuum domains. Using Eq. (2.3), the static potential vortices in the model which is discussed in the next section. induced by center vortices as well as the vacuum domains is Now, using these results, the vacuum is argued in SU(3) case, filled with z2 vortices as well as z1 vortices. Such z2 LvX=2þR XN−1 2 V R − 1 − f 1 − G α⃗n x ; vortices are characterized by the center element z2 ¼ z1. rð Þ¼ ln n½ Re rð Cð ÞÞ x −L =2 n 0 The z2 vortex is constructed of two z1 vortices with the ¼ v ¼ z same flux orientations and therefore these 1 vortices ð4:1Þ according to the interactions in the type-II superconductor repel each other. One may conclude that z2 vortices do not where the contribution of the vacuum domains (n ¼ 0)is make a stable configuration and one should consider each added. If a vacuum domain is all contained within the Wilson z z 0 ⃗ of 1 vortices within the 2 vortices as a single vortex in the loop exp½iα⃗CH¼z0I. The total magnetic flux through a model. In addition, only vortices with the smallest magni- vacuum domain is zero value and therefore the square ansatz tude of center flux have substantial probability [12]. In fact, given in Eq. (2.4) for the angle of vacuum domain for all this probability for the z2 vortex should be less than the one representations is for the z1 vortex. In previous section, we considered the f A A A2 general case that all possible n are included and therefore α0 x 2 v − : : ð Cð ÞÞ ¼ 2 ð4 2Þ the concavity is appeared for several representations. 2μ Av Av Now, we assume only z1 vortices in the vacuum. Figure 7 shows the potentials for the various representations in the Now, to understand the interactions between two types range R ∈ ½0; 100. of SU(3) center vortices, we study the static potentials in Although, using only z1 vortices, two Casimir scaling and the fundamental representation. Figure 8 shows the static N-ality distances are smoothly connected for some repre- potentials induced by vortex configurations in the funda- sentations, the concavity occurs for higher representations mental representation at large distances where the ansatz especially f15sg. Indeed, this concavity is observed inde- of the vortex profile has no role in the potentials. Each pendent of the ansatz for the angle [16]. configuration is appeared in the plane of the Wilson loop

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8 7 2 * (z ) and z {15s}, f =0.01 f =0.05 7 1 1 6 1 0 z and z z*=I 1 1 1 6 5 f =0.03 0 5 4 f =0.01

V(R) 0 4 V(R) 3

3 2 f =0 0 2 1

1 0 100 120 140 160 180 200 0 50 100 150 200 R R

FIG. 8. The static potential induced by z1 and z0 vortex 15s FIG. 9. The static potential for the representation f g. On the configurations is compared with the one induced by z1 and z2 vacuum, there are z1 vortices with the fixed probability f1 ¼ 0.01 vortex configurations in the fundamental representation at large while the probability f0 of vacuum domains is gradually z z distances. It seems that for minimizing the energy 2 and 1 attract increased. The concavity could almost be removed by appearing z z each other and deform to 1 and 0 vortex configurations. For the vacuum domains, a way to effectively parametrize vortex each configuration, the free parameters are fn ¼ 0.01, Lv ¼ 100, L 100 L2= 2μ 4 2 interactions. The free parameters are v ¼ and v ð Þ¼ . and Lv=ð2μÞ¼4. with the probability fn ¼ 0.01. We assume that there are z1 Therefore, in SU(3) case using Eq. (4.1), the static z1 vortex as well as z1 antivortex on the vacuum. Adding z1 potential induced by vortices and vacuum domains is vortex to the vortex configurations may lead either to z2 ¼ 2 Lv=2þR z and z vortex configurations or to z1 and z0 ¼ z z1 X 1 1 1 V R − 1 − f − 2f f G α0 x vortex configurations. In other words, it is interesting to rð Þ¼ ln½ð 0 1Þþ 0Re rð Cð ÞÞ x −L =2 observe that this z1 vortex is attracted by which one of the ¼ v z z 1 initial vortices, 1 vortex or 1 antivortex. One expects that þ 2f1ReGrðαCðxÞÞ; ð4:3Þ the ensemble of the vortex configurations leads to a z minimum energy. The potential energy induced by 1 where the vortices of type n ¼ 2 is substituted with those of and z2 vortex configurations is more than the one induced type n ¼ 1 and vacuum domains. f0, f1 are the proba- by z1 and z0 vortex configurations. It seems that for z z bilities that any given plaquette is pierced by vacuum minimizing the energy 2 and 1 attract each other and z z z domains and 1 vortices, respectively. In Fig. 9, the static deform to 1 and 0 vortex configurations. The extra 15s z z potential for the representation f g which has shown the negative energy of the potential induced by 1 and 0 z z worst concavity is plotted. On the vacuum, there are 1 vortex configurations compared with the one induced by 1 f 0 01 z vortices with the fixed probability 1 ¼ . but the and 2 vortex configurations may be interpreted as an f probability 0 of vacuum domains is gradually increased attraction energy between z1 and z vortices and repulsion 1 from zero to 0.05. As shown, the concavity could almost be between two z1 vortices. Therefore, two magnetic vortex removed by appearing the vacuum domains in the vacuum. fluxes with the same orientation may repel each other 27 while those with the opposite orientation attract each other. The concavity for the higher representation f g is also It seems that the model also confirms the interactions eliminated. Therefore, the satisfactory result can be between vortices in the type II superconductivity. In achieved by ad-hoc choosing the probability weights of addition, we studied the interaction between vortices in the different vortex configurations. In particular including the model based on energetics in Refs. [34,39] approving vacuum domains, a way to effectively parametrize vortex the same results for the interactions. It seems that the interactions, is crucial in obtaining an (almost) everywhere attractions between two types of center vortices produce convex potential. vortices of type n ¼ 1 and vacuum domains. It is possible To check the details of the static potential in the that only vortices with the smallest magnitude of center representation f15sg, we analyze its group factors. flux have substantial probability to find the midpoints of Figure 10(a) depicts the group factors corresponding to them at any given location [12]. It seems that z2 vortices, z1 vortices and vacuum domains for the medium size which its magnitude of center flux is twice the one of z1 Wilson loop close to concavity regime (about R ¼ 70) for vortices, interacting with z1 vortices are decomposed to the the representation f15sg and those for the large size configurations with the lowest magnitude of center fluxes, loop are plotted in Fig. 10(b). As shown in Fig. 10(a), i.e., z1 vortices and vacuum domains. for the z1 vortex group factor, a wavy character with a large

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1.2 1.5 R=100, z R=70, z 1 1 1 R=100, Vac. Dom. R=70, Vac. Dom. 1 0.8 ) ) α α ( 0.6 ( 0.5 {15s} {15s} 0.4 Re G 0.2 Re G 0

0 −0.5 −0.2 −50 0 50 100 −50 0 50 100 150 (a)x (b) x G α z x R 70 FIG. 10. (a) The group factors Re f15sgð Þ of the 1 vortices and vacuum domains vs at ¼ corresponding to Fig. 9, close to the concavity regime. The fluctuations of the z1 vortex group factor with equal large sizes of maxima and minima around any timelike leg lead to the concavity behavior in the potentials. The vacuum domain group factor changes smoothly close to the trivial value 1 around any time-like leg removing the concavity in the potential. (b) The same as (a) but for the large size Wilson loop with R ¼ 100. Since the N-ality of the representation f15sg is the same as the one of the fundamental representation, the z1 vortex group factor like the one of the fundamental representation interpolates from −0.5, when the vortex core is located entirely within the Wilson loop, to 1, when the core is entirely outside the loop. Also, in the same interval, when the core of vacuum domain is located entirely within the loop, the group factor 2 reaches to the trivial value 1. The free parameters are Lv ¼ 100 and Lv=ð2μÞ¼4. amplitude appears around to any timelike leg (x ¼ 0, 70). representation, interpolates from −0.5, when the vortex The same behavior occurs for the z2 vortex group factor in core is located entirely within the Wilson loop, to 1, when the adjoint representation and therefore the concavity is the core is entirely outside the loop. Also, in the same appeared in its potential. Increasing large fluctuations of the interval, when the core of the vacuum domain is located group factor leads to the concavity behavior in the entirely within the loop, the group factor reaches to the potentials. But, the vacuum domain group factor changes trivial value 1. Besides, as show in Ref. [34], the vacuum smoothly (small fluctuations) close to trivial value 1 around domains could enhance the Casimir scaling at the inter- any timelike leg. Therefore the concavity of the potential mediate distances. could be removed by including the vacuum domain As a result, small fluctuations of the group factor close contribution to the potential. As shown in Fig. 10(b), the to the trivial value, which occur because of the inter- group factor for z1 vortices in the representation f15sg at actions between center vortices, could remove concavity large distances (R ¼ 100), like the one of the fundamental in the static potentials and also improve the Casimir

10 r=6 8 {3} {6} r=8 8 {8} r=10 r=15s 6 {10} (R) 6 {15s} {3}

V(R) 4

(R)/V 4 {r} V 2 2

0 0 0 50 100 150 200 0 5 10 15 20 (a)R (b) R

FIG. 11. (a) The static potentials using both z1 center vortices and vacuum domains for the various representations of SUð3Þ. The N V R =V R Casimir scaling and -ality regimes connect naturally to each other without almost any concavity. (b) Potential ratios frgð Þ f3gð Þ at the intermediate distances. These potentials agree with Casimir scaling better than those obtained from both types of vortices, 2 especially for the higher representations. The free parameters are Lv ¼ 100, f0 ¼ 0.05, 2f1 ¼ 0.01, and Lv=ð2μÞ¼4.

114504-8 DOMAIN STRUCTURES AND QUARK POTENTIALS IN SU(3) … PHYS. REV. D 100, 114504 (2019) scaling at the intermediate regime. But the large fluctua- In addition, the potential ratios induced by z1 vortices tions of the group factor could create the concavity in the starting out at the Casimir ratios at intermediate distances static potentials and break down the Casimir scaling at the drop slower than those of z2 vortices. Analyzing the intermediate regime. interactions between two types of center vortices, the In Fig. 11(a), the potentials VrðRÞ induced by z1 confinement mechanism of center vortices is reformulated vortices and vacuum domains for the various representa- for removing the concavity of the potentials and also tions for the range R ∈ ½0; 200 are plotted and the improving the Casimir scaling at the intermediate regimes. potential ratios are shown in Fig. 11(b).Therefore,the The QCD vacuum is a dual analogy to the type II satisfactory potentials for the different representations superconductivity where it seems that two vortices repel can be achieved by ad hoc choosing the probability each other while the vortex-antivortex interaction is weights of the different vortex configurations. In particu- attractive. We show that the model may also confirm lar including the vacuum domains, a way to effectively the same interactions between vortices based on ener- parametrize vortex interactions, is crucial in obtaining an getics. On the one hand, z2 vortices are characterized by 2 (almost) everywhere convex potential when interpolating the center element z2 ¼ z1 and two z1 vortices within the between the short distances and the asymptotic regimes. z2 vortex may repel each other and one could observe In addition, the potential ratios starting out at the Casimir them as the single vortices. However using only z1 ratios at intermediate distances drop very slowly from the vortices, this concavity would still remain for some higher exact Casimir scaling for all representations, especially for representations. On the other hand, in addition to two the higher representations. Therefore, the convex poten- types of center vortices, there are their antivortices on tials in agreement with Casimir scaling at intermediate the vacuum. We show, like superconductivity, that z2 regimes with a fixed vortex profile could be obtained, if and z1 vortex configurations may attract each other and one includes the contribution of vortex interactions in the therefore they would merge forming z2z1 ¼ z1z0 where z0 static potentials. is equal to the identity element. We observe that attrac- d 2 Furthermore, when the properties of vortices in ¼ tions between z2 and z vortices are forming z1 vortices as Z N 1 dimensions and ð Þ models were being worked out, it well as vacuum domains. Therefore, z2 vortices, which its was found that a real-space group approach magnitude of center flux is twice the one of z1 vortices, Z N to the ð Þ models reproduced the correct change in critical within the interactions with the z1 vortices may be N 4 behavior at ¼ if vacancies were included. In fact, it was decomposed to the configurations with the lowest mag- found that the vacancy fugacity mimicked the vortex nitude of center fluxes. As a result, the vacuum in stead of fugacity, and was a relevant variable in the disordered z1 and z2 vortices is filled with z1 vortices and vacuum phase. In the framework of the real-space renormalization domains. We show that by ad hoc choosing the probability – group approach [40 42], analyzing the convexity could be weights of the different vortex configurations, satisfactory interesting and we will focus on this idea in the future result for the static potentials can be achieved. In par- works. ticular including the vacuum domains, a way to effectively parametrize vortex interactions, is crucial in obtaining the V. CONCLUSION convex potentials in agreement with Casimir scaling at The static potentials in various representations depend intermediate regimes. on basic properties. At the intermediate regime, the potentials are governed by Casimir scaling while this ACKNOWLEDGMENTS feature breaks down in the asymptotic regime and is replaced by the N-ality dependent law. These two regimes The author would like to thank the Vice-Chancellors of should be connected smoothly to each other without Semnan University for their great support. any concavity. In this paper, we analyze the static potentials in SU(3) Yang-Mills theory within the frame- APPENDIX: GROUP FACTORS OF THE work of the domain model of center vortices where there REPRESENTATIONS are two types of vortices. The two types of vortices may r be regarded as the same type of vortex with magnetic flux The Cartan generators for the representation within the pointing in opposite directions. Without this constraint, group factors of the static potential given in Eq. (4.1) can be we study the behavior of these center vortices on static calculated using the tensor method. One can obtain the real potentials. The interactions of both types of center part of the group factors for all center domains in several vortices with large size Wilson loops are the same but representations as: their interactions with the medium size Wilson loops 1 αn αn are different. The potentials induced by both vortex G αn 2 ffiffiffi ffiffiffi ; Re f3gð Þ¼ cos p þ cos p ðA1Þ types show concave behavior for several representations. 3 2 3 3

114504-9 SEYED MOHSEN HOSSEINI NEJAD PHYS. REV. D 100, 114504 (2019) n n n n 1 α α 2α ReG 6 ðα Þ¼ 2 cos pffiffiffi þ 3 cos pffiffiffi þ cos pffiffiffi ; ðA2Þ f g 6 2 3 3 3 n n 1 3α ReG 8 ðα Þ¼ 4 þ 4 cos pffiffiffi ; ðA3Þ f g 8 2 3 n n n 1 3α 6α ReG 10 ðα Þ¼ 3 þ 6 cos pffiffiffi þ cos pffiffiffi ; ðA4Þ f g 10 2 3 2 3 n n n n n n 1 α α 2α 5α 4α ReG 15s ðα Þ¼ 4 cos pffiffiffi þ 3 cos pffiffiffi þ 5 cos pffiffiffi þ 2 cos pffiffiffi þ cos pffiffiffi : ðA5Þ f g 15 2 3 3 3 2 3 3

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