Monopole-center vortex chains in SU(2) gauge theory
aa r X i v : . [ h e p - ph ] O c t Monopole-center vortex chains in SU (2) gauge theory Seyed Mohsen Hosseini Nejad ∗ and Sedigheh Deldar † Department of Physics, University of Tehran,P.O. Box 14395/547, Tehran 1439955961, Iran
Abstract
We study the relation between center vortex fluxes and monopole fluxes for SU (2) gauge groupin a model. This model is the same as the thick center vortex model but we use monopole-antimonopole configurations instead of center vortices in the vacuum. Comparing the group factorfor the fundamental representation obtained by monopole-antimonopole configurations with the oneobtained by the center vortices, we conclude that the flux between the monopole-antimonopole canbe split into two center vortex fluxes. Studying the potentials induced by monopole-antimonopoleconfigurations and center vortices, we obtain monopole-vortex chains which appear in lattice MonteCarlo simulations, as well. We show that two similarly oriented center vortices inside the monopole-antimonopole configuration repel each other and make a monopole-vortex chain. PACS numbers: 11.15.Ha, 12.38.Aw, 12.38.Lg, 12.39.Pn ∗ [email protected] † [email protected] . INTRODUCTION The center vortices are color magnetic line-like (surface-like) objects in three (four) di-mensions which are quantized in terms of center elements of the gauge group. Condensationof center vortices in the vacuum of QCD leads to quark confinement such that the colorelectric flux between quark and antiquark is compressed into tubes and a linear rising po-tential between static quarks is obtained. In the vortex picture, quark confinement emergesdue to the interaction between center vortices and Wilson loops [1, 2]. On the other hand,monopoles are playing the role of agents of confinement in the dual superconductor scenario[3, 4]. Therefore, one may expect that there are some kind of relations between monopolesand center vortices. Monte Carlo simulations [5] indicate that a center vortex configurationafter transforming to maximal Abelian gauge and then Abelian projection, appears in theform of the monopole-vortex chains in SU (2) gauge group. The idea of monopole-vortexchains has been studied by so many researchers [5–9].In this article, monopole-vortex chains in SU (2) gauge group are investigated in a model.This model is the same as the thick center vortex model [10], but we use monopole-antimonopole flux instead of the center vortex flux. The motivation is to see if by this simplemodel we can observe the idea of monopole-vortex chains which has already been confirmedby lattice calculations, as well as some other phenomenological models. In this model,monopole-antimonopole configurations which are line-like and similar to center vortices areassumed to exist in the vacuum. Studying the group factors of the monopole-antimonopoleconfigurations and center vortices, we understand that the monopole-antimonopole configu-rations are constructed of two center vortices. Increasing the thickness of the center vortexcore increases the energy of the center vortex and therefore the energy of the vacuum whichis made of these vortices. As a result, the potential energy between static quark-antiquarkincreases. This fact can be confirmed by this model, as well. This is a trivial fact fromthe physical point of view that the condensation of vortices leads to quark confinement.Classically, it is very similar to the Aharanov-Bohm effect where increasing the thicknessof the magnetic flux and therefore the magnetic energy of the system, changes the inter-ference pattern. Using this simple model, we have calculated the potentials induced bythe monopole-antimonopole configurations and center vortices. Comparing these potentials,we observe that the monopole-antimonopole configurations leads to a larger static quark-2ntiquark potential compared with the case when we use two center vortices in the model. Weinterpret this extra energy as a repulsive energy between two center vortices constructing themonopole-antimonopole configurations and then we discuss that the monopole-antimonopoleconfigurations can deform to the monopole-vortex chains as confirmed by lattice calculationsand other phenomenological models.In section II, the formation of monopoles which is related to the Abelian gauge fixingmethod is reviewed in SU (2) gauge group. A model with structures of center vortices andmonopole-antimonopole configurations are studied in sections III and IV. Then in sectionV, we study the group factors and potentials of these structures to argue monopole-vortexchains. Finally, we summarize the main points of our study in section VI. II. ABELIAN GAUGE FIXING AND MAGNETIC MONOPOLE CHARGES
By Abelian gauge fixing, magnetic monopoles are produced in a non Abelian gaugetheory. Specific points in the space where the Abelian gauge fixing becomes undeterminedare sources of magnetic monopoles. In the following the formation of the magnetic chargeby Abelian gauge fixing method is discussed [11].In order to reduce a non Abelian gauge theory into an Abelian gauge theory, the gluonfield under a gauge transformation can not be diagonalized. In fact, the gluon field A µ hasfour components and only one of them can be aligned simultaneously. Therefore, a scalarfield is used to fix a gauge. One can consider a scalar field Φ ( x ) in the adjoint representationof SU ( N ) as the following: Φ ( x ) = Φ a ( x ) T a (1)where T a are the N − SU ( N ) gauge group. A gauge which diagonalizesthe matrix Φ ( x ) is called Abelian gauge.Now, we consider the SU (2) gauge group. A gauge transformation Ω ( x ) can diagonalizethe field Φ ( x ): Φ = Φ a T a → ΩΦΩ † = λT = λ − λ , (2)where λ = q Φ + Φ + Φ . (3)3he eigenvalues λ ( x ) of the matrix Φ ( x ) are degenerated when λ = 0 and therefore threecomponents Φ a =1 , , ( ~r ) are zero at specific points ~r = ~r :Φ ( ~r ) = 0 Φ ( ~r ) = 0 Φ ( ~r ) = 0 (4)In the vicinity of the point ~r = ~r we can express Φ ( ~r ) in terms of a Taylor expansion:Φ ( ~r ) = Φ a ( ~r ) T a = T a C ab ( x b − x b ) , (5)where C ab = ∂ Φ a ∂x b (cid:12)(cid:12)(cid:12) ~r = ~r . Therefore the field Φ ( ~r ) has the hedgehog shape in the vicinity ofthe point ~r = ~r . One can define another coordinate system where the point ~r is placed atthe origin. In this coordinate system, the field Φ ( ~r ′ ) has the form:Φ ( ~r ′ ) = x ′ a T a x ′ a = C ab ( x b − x b ) . (6)Dropping the prime on x ′ and using the spherical coordinates for the vector ~r , one get toΦ ( ~r ) = x a T a = r cos θ e − iϕ sin θe iϕ sin θ − cos θ . (7)The gauge transformation Ω which diagonalizes the hedgehog field Φ isΩ ( θ, ϕ ) = e iϕ cos θ sin θ − sin θ e − iϕ cos θ . (8)Therefore the hedgehog field Φ is diagonalized as the following:ΩΦΩ † = r − = rT . (9)The gluon field transforms under the same gauge transformation: ~A = ~A a T a → Ω (cid:18) ~A + 1 ie ~ ∇ (cid:19) Ω † . (10)One can obtain:1 ie Ω ~ ∇ Ω † = 1 e (cid:18) − ~e θ T e iϕ − ~e ϕ θr sin θ T + ~e ϕ r (cos ϕT − sin ϕT ) (cid:19) . (11)Thus, the gluon field under the gauge transformation of Eq. (8) can be separated into aregular part ~A R and a singular part: ~A = ~A a T a = ~A Ra T a − e ~n ϕ θr sin θ T . (12)4he singular part has the form of a gauge field in the vicinity of a magnetic monopole withmagnetic charge equal to: g = − πe T . (13)To summarize, we observe that in the vicinity of the points where the eigenvalues ofthe matrix Φ ( x ) are degenerate, the singular part of the gluon field in the Abelian gaugebehaves like a monopole with magnetic charge g = − πe T . III. A MODEL OF VACUUM STRUCTURE
In this model [10], the Yang Mills vacuum is dominated by center vortices which have afinite thickness (a core). In SU ( N ) gauge group, there are N − z n = e i πnN enumerated by the value n = 1 , ..., N −
1. The effect of piercing a Wilson loop by a thick center vortex is assumed tobe represented by insertion of a group element G in the link product as the following W ( C ) = T r (cid:2)
U...U (cid:3) −→ T r (cid:2)
U...G...U (cid:3) , (14)where G ( ~α nC ( x ) , S ) = S exp h i~α nC ( x ) · ~ H i S † . (15)The {H i i = 1 , .., N − } are the Cartan generators, S is a random element of SU ( N ) gaugegroup and angle ~α nC ( x ) shows the flux profile which depends on the Wilson loop size and thelocation x of the center vortex with respect to the Wilson contour C . The random grouporientations associated with S are uncorrelated, and should be averaged. The averagedcontribution of G over orientations in the group manifold specified by S is G ( ~α nC ( x )) = Z dS S exp h i~α nC ( x ) · ~ H i S † == 1 d r T r (cid:16) exp h i~α nC ( x ) · ~ H i(cid:17) I d r ≡ G r ( ~α nC ( x )) I d r , (16)where G r ( ~α nC ( x )) is called the group factor and I d r is the d r × d r unit matrix. In SU ( N )case, the group factor of the fundamental representation interpolates smoothly from e i πnN , ifthe core of the center vortex is located completely inside the Wilson loop, to 1, if the core iscompletely exterior. The Wilson loop C is assumed as a rectangular R × T loop in the x − t T ≫ R where the left and the right time-like legs of the Wilson loop are locatedat x = 0 and x = R . In other words, the two static charges are located at these points.A desired ansatz for angle ~α nC ( x ) must lead to a well-defined potential i.e. linearity andCasimir scaling at the intermediate distances. Any reasonable ansatz for the angle ~α nC ( x )must satisfy the following conditions:1. ~α nC ( x ) = 0 when a center vortex locates far outside the Wilson loop.2. ~α nC ( x ) = ~α nmax when a center vortex locates deep inside a large Wilson loop. The maxi-mum value of the angle ~α nmax is obtained from the following maximum flux condition:exp( i~α nmax · ~ H r ) = exp( iα ni ( max ) H ir ) = e i kπn/N I, (17)where k is the N -ality of representation r .3. ~α nC ( x ) = 0 as R → α ni ( x ) = α ni ( max ) − tanh( ay ( x ) + bR )] , (18)where n indicates the center vortex type, a, b are free parameters of the model, α ni ( max ) corresponding to Eq. (17) indicates the maximum value of the flux profile and R is thedistance between two static charges. y ( x ), the nearest distance of x from the timelike sideof the loop, is y ( x ) = x − R for | R − x | ≤ | x |− x for | R − x | > | x | (19)The flux of Eq. (18) is one of the many examples that can give the appropriate potential.Some other examples were discussed in Ref. [13].For SU (2) gauge group, when the vortex core is entirely contained within the Wilsonloop, using Eq. (17), we get exp[ iα max H ] = z I, (20)where H is Cartan generator and z I = e πi I is the center element of SU (2) gauge group.Therefore, the maximum value of the angle α max for the fundamental representation is equal6o 2 π . Thus, the ansatz of the flux profile given in Eq. (17) for SU (2) is obtained as thefollowing α ( x ) = π [1 − tanh( ay ( x ) + bR )] . (21)Figure 1a schematically shows the interaction of center vortices with an R × T Wilson loopusing the ansatz for the flux profile of center vortices which is given in Eq. (18). The ansatzof the flux profile in Eq. (21) is plotted in Fig. 1b. In this model, as an assumption, the a) b) −50 0 50 100 15001234567 x α ( x ) FIG. 1: a) The figure schematically shows the interaction between the SU (2) center vortices withthe ansatz of the flux profile given in Eq. (21) and the Wilson loop as a rectangular R × T loopin the x − t plane as well as some parameters of the ansatz. The effect of center vortex on theloop is assumed by insertion of a group element G in the link product of the Wilson loop whichinterpolates smoothly between G = + I at α = 0, when the center vortex locates far outside theWilson loop, and G = − I at α max = 2 π , when the center vortex completely locates inside theWilson loop. b) The angle α versus x corresponds to the ansatz for SU (2) gauge theory. Theleft and right time-like legs of the Wilson loop are located at x = 0 and x = R = 100. The freeparameters a and b are chosen to be 0 .
05 and 4, respectively. probabilities of piercing the plaquettes in the Wilson loop by center vortices are uncorrelated.Assuming that an n th center vortex appears in any given plaquette with the probability f n ,the expectation value of the Wilson loop is obtained: < W ( C ) > = Y x ( − N − X n =1 f n (1 − Re G r [ ~α nC ( x )]) ) < W ( C ) >, (22)7here f n = f N − n and G r [ ~α n ( x )]) = G ∗ r [ ~α N − n ( x )]) . (23) < W ( C ) > denotes T r (cid:2)
U...U (cid:3) which no vortex pierces the Wilson loop.One of the criteria for the color confinement is the area law for the Wilson loop i . e .< W ( C ) > = exp (cid:0) − σA ( C ) (cid:1) < W ( C ) > . (24)Here A ( C ) is the minimal surface spanned on the Wilson loop C and σ > σ = − A X x ln ( − N − X n =1 f n (1 − Re G r [ ~α nC ( x )]) ) . (25)One gets the static potential induced by center vortices between static color charges inrepresentation r at distance R as the following V r ( R ) = − ∞ X x = −∞ ln ( − N − X n =1 f n (1 − Re G r [ ~α nC ( x )]) ) , (26)where the center of vortex cores pierces the middle of plaquettes i.e. x = ( n + ) a ( n ∈ ( −∞ , ∞ )) where a is the lattice spacing. We use a = 1 throughout this paper. Although R takes only integer values in the lattice formulation, but the figures related to V r ( R ) areplatted over the continuous interval.For SU (2) gauge group, the static potential induced by z center vortices at f ≪ R ) where α ( x ) ≪ π is obtained as thefollowing V j ( R ) = ( f ∞ X x = −∞ α ( x ) ) j ( j + 1) , (27)where spin index j shows the representations in SU (2) gauge theory. According to Eq. (27),the static potential is proportional to the eigenvalue of the quadratic Casimir operator i.e.V j ( R ) ∼ j ( j + 1) in agreement with the Casimir scaling effect observed in lattice simulations[14]. The Casimir proportionality of the static potential induced by center vortices canbe generalized from SU (2) to SU ( N ). For observing the property of Casimir scaling in thepotentials at intermediate regime, the probability f n should be far smaller than 1. Therefore,the probability f n is chosen 0 . F by setting a → a/F, b → bF . The thickness ofthe center vortex would be on the order 1 /a for the ansatz given in Eq. (17). Therefore,choosing F >
F <
1, decreases these quantities.In the next section, we investigate the effect of a monopole flux on a Wilson loop.
IV. MONOPOLE-ANTIMONOPOLE CONFIGURATIONS
Now, we consider monopole-antimonopole pairs as the Abelian configurations in the vac-uum. We are assuming that the magnetic fields between monopole and antimonopole areinitially localized in a tube as plotted in Fig. 2. We use these configurations in the thickcenter vortex model instead of center vortices. The monopole-antimonopole configurationsare line-like similar to the center vortices. The effect of piercing a Wilson loop by themonopole-antimonopole configuration is represented by insertion of a phase e ie R S ~B.d~s [8] inthe link product where e is the color electric charge and R S ~B.d~s is the total magnetic fluxof the monopole. The magnetic field of a monopole with topological charge g obeys theMaxwell equation ~ ∇ . ~B = gδ ( ~r ). Therefore, the total magnetic flux of a monopole crossingthe surface S is equal the magnetic charge g as the following [11, 15]Φ m = Z S ~B.d~s = Z V d r ~ ∇ . ~B = Z V d r gδ ( ~r ) = g, (28)where V is the volume enclosed by the surface S . For SU (2) gauge group, g is the monopolecharge in Eq. (13). If we attribute a thickness for these configurations as what is done forthe thick center vortices, the effect of a monopole-antimonopole configuration on a Wilsonloop is to multiply the loop by a group element the same as the one in Eq. (16). If amonopole-antimonopole configuration is entirely contained within the loop, thenexp h i~α n · ~ H i = e ieg , (29)where eg satisfies the charge quantization condition eg = 2 nπ .For SU (2) gauge group, corresponding to Eq. (13), the magnetic charge of the monopole is g = − πe H and the one for antimonopole is g = + πe H where H = diag (cid:0) , − (cid:1) represents9 IG. 2: A schematic view of the monopole-antimonopole configuration which is initially consideredto be localized. This configuration is line-like, the same as the center vortices. The arrows on thelines show the direction of the magnetic field. the Cartan generator. When a monopole-antimonopole pair is entirely contained within theWilson loop, using SU (2) magnetic charge into Eq. (17), we getexp[ iα max H ] = e ± i π H = e i π I, (30)where index n = 0 is related to the monopole-antimonopole configurations. The sign in theexponent is not important since the direction of the configuration which pierces the Wilsonloop is not important. Therefore the maximum value of the angle α max for the fundamentalrepresentation is equal to 4 π . Therefore the ansatz of the flux profile given in Eq. (17) for themonopole-antimonopole configurations of SU (2) gauge theory is obtained as the following α ( x ) = 2 π [1 − tanh( ay ( x ) + bR )] . (31)The potential induced by monopole-antimonopole configurations is the same as the oneinduced by center vortices represented in Eq. (26). For SU (2) gauge group, the potentialinduced by monopole-antimonopole configurations for the fundamental representation isobtained as the following V f ( R ) = − ∞ X x = −∞ ln { (1 − f ) + f G f [ α ( x )] } . (32)In the next section, we study the group factors and the potentials for the center vorticesand the monopole-antimonopole configurations. V. SU(2) AND VACUUM STRUCTURES
To study the center vortices and monopole-antimonopole configurations in the vacuumfor SU (2) gauge group, we discuss the interaction between the Wilson loop and these con-10gurations. First, the group factors of these configurations which have an important rolein producing the potentials of Eq. (26) [16, 18] are studied and the relation between theseconfigurations is discussed. Then, with calculating the potentials induced by these configu-rations, interactions inside these configurations are studied. A. Interaction between the Wilson loop and center vortices
First, we calculate the group factor of the center vortices in SU (2) gauge group. Thegroup factor for the fundamental representation of SU (2) is obtained from Eq. (16) G j =1 / = 12 j + 1 Tr exp[ iα H ] = cos ( α , (33)where H is the Cartan generator of SU (2) gauge group. According to Eq. (20), themaximum value of the angle α max for the fundamental representation is equal to 2 π . Usingansatz given in Eq. (21), Fig. 3a shows G r ( α n ) obtained from center vortices versus x fora fundamental representation Wilson loop with R = 80. The legs of the Wilson loop arelocated at x = 0 and x = 80. The free parameters a and b are chosen to be 0 .
05 and4, respectively. The group factor interpolates smoothly from −
1, when the vortex core islocated completely inside the Wilson loop, to 1, when the core is entirely outside the loop.Figure 4a shows G r ( α n ) obtained from center vortices versus x for small sizes of Wilson loops(small R ). For small size of the Wilson loop, center vortices are partially located inside theWilson loop and the maximum flux is not center vortex flux. Therefore the minimum ofthe group factor of center vortices increases with decreasing the size of the Wilson loop andapproaches to 1. B. Interaction between the Wilson loop and monopole fluxes
Next, we calculate the group factor of the monopole-antimonopole configurations in SU (2)gauge group. Using Eq. (16), the group factor for the fundamental representation is obtainedas the following G f = cos ( α , (34)According to Eq. (30), the maximum value of the angle α max for the fundamental represen-tation is equal to 4 π . Using ansatz given in Eq. (31), Fig. 3b plots G r ( α n ) obtained from11he monopole-antimonopole configurations versus x for a Wilson loop of the size R = 80 forthe fundamental representation. The Wilson loop legs are located at x = 0 and x = 80. Thefree parameters a and b are chosen to be 0 . /F and 4 F , respectively. When the monopole-antimonopole configuration overlaps the minimal area of the Wilson loop, it affects the loop.For F = 1, the value of the group factor is 1, when the monopole-antimonopole configurationcore is located completely inside or outside the Wilson loop. For F >
1, the thickness ofcenter vortices is increased and the maximum value of the flux profile α is less than 4 π .Therefore, when the center of vortex core is located in the middle of the Wilson loop withthe size R = 80, G r ( α ) becomes less than 1. Increasing the size of the Wilson loop, themaximum value of the group factor reaches to 1.When the center of monopole-antimonopole configuration is placed at x = 0 or x = 80,half of the maximum flux enters the Wilson loop. The group factor interpolates smoothlyfrom 1, when the monopole-antimonopole configuration core is entirely outside the loop, to −
1, when the half of the core is located inside the Wilson loop. As shown in Fig. 3b, theresults are the same by changing the free parameters. This behavior of the group factor issimilar to the group factor of the center vortex which changes smoothly between 1, when thecenter vortex is located completely outside the Wilson loop and −
1, when the center vortexis completely inside the loop. Therefore, half of the monopole-antimonopole flux is equal tothe vortex flux i . e . the monopole-antimonopole configuration is constructed from two centervortices. Figure 4b shows G r ( α n ) obtained from monopole-antimonopole configuration versus x for small sizes of Wilson loops. For small R , monopole-antimonopole configurations arepartially located inside the Wilson loop and the maximum flux is not equal to the totalflux of the monopole-antimonopole configuration. Assuming the monopole-antimonopoleconfiguration is constructed from center vortices, we observe that the value of − R is equalto 13. Decreasing the size of the Wilson loop, the minimum of the group factor of themonopole-antimonopole configuration increases and deviates from − ) −100 −50 0 40 80 150 200−1−0.8−0.6−0.4−0.200.20.40.60.81 x R e [ g r ( α ) ] Group factor of center vortex b) −100 −50 0 50 100 150 200−1−0.8−0.6−0.4−0.200.20.40.60.81 x R e [ g r ( α ) ] Group factor of monopole−antimonopole configuration F=1F=2
FIG. 3: a) Re( G r ) obtained from center vortices versus x for the fundamental representation ofthe SU (2) gauge group for R = 80. The free parameters a and b are chosen to be 0 .
05 and 4,respectively. When center vortices locate completely inside the Wilson loop, the value of the groupfactor is − a and b are chosen to be 0 . /F and 4 F , respectively. When half of the core ofthe monopole-antimonopole configuration locates inside the Wilson loop (at x = 0 or x = 80),the flux inside the loop is equivalent to the center vortex flux. Therefore the fluxes of the centervortices inside the monopole-antimonopole configuration do not have an overlap. It seems thattwo similarly oriented vortices repel each other. As shown, by changing the free parameters (forexample, varying F = 1 to F = 2), the results are the same and half of the core of the monopole-antimonopole configuration is equivalent to the center vortex flux. By varying F = 1 to F = 2,the thickness of center vortices is increased and becomes more than the size of the Wilson loop( R = 80). Therefore, when the center of vortex core is located in the middle of the Wilson loop( x = 40), G r ( α ) becomes less than 1. Increasing the size of the Wilson loop, the maximum valueof the group factor reaches to 1. We show in Fig. 5 that by changing F = 1 to F = 2 the staticpotentials are just scaled up. these center vortices is investigated, in details. Before that we study another approach,explained in ref. [12], for obtaining the relation between center vortex and monopole fluxes.Using fractional flux of a monopole, the flux of a center vortex is constructed in SU (2)gauge group. Substituting H from Eq. (13) in Eq. (20) and α max = 2 π , we get [12]exp [ i π H ] = exp h − ie g i = z I. (35)13 ) −100 −50 0 50 10000.10.20.30.40.50.60.70.80.91 x R e [ g r ( α ) ] Group factor of center vortex R=7R=10R=13 b) −100 −50 0 50 100−1−0.8−0.6−0.4−0.200.20.40.60.81 x R e [ g r ( α ) ] Group factor of monopole−antimonopole configuration R=7R=10R=13
FIG. 4: a) Re( G r ) obtained from center vortices versus x for the fundamental representation ofthe SU (2) gauge group for small sizes of Wilson loops (small R ). The free parameters a and b arechosen to be 0 .
05 and 4, respectively. Decreasing the size of the Wilson loop, the minimum value ofthe group factor increases and approaches to 1. b) the same as a) but obtained from the monopole-antimonopole configurations. Assuming the monopole-antimonopole configuration is constructedfrom center vortices, we observe that the value of − R is equal to 13. Decreasing the size of the Wilsonloop, the minimum of the group factor of the monopole-antimonopole configuration increases anddeviates from − According to Eq. (28), g is equal to the total magnetic flux of a monopole. Thereforethe effect of a center vortex on the Wilson loop is the same as the effect of an Abelianconfiguration corresponding to the half of the matrix flux g on the Wilson loop.Now, we obtain the flux of this Abelian configuration and compare it with the flux ofcenter vortex which is equal to Φ v = π [19].The contribution of this Abelian configuration on the Wilson loop is W = G f = 1 d f Tr (cid:16) exp h − ie g i(cid:17) = 12 Tr e − iπ e iπ = e iπ . (36)Comparing Eq. (36) with the contribution of an Abelian field configuration to the Wilsonloop which is W = e iq Φ ( q means units of the electric charge and q = 1 for the fundamentalrepresentation) [8], the flux of this Abelian configuration is equal to π .Therefore, the flux of this Abelian configuration corresponding to the half of the magnetic14harge g , is the same as one center vortex on the Wilson loop.In the next subsection the interaction between center vortices inside the monopole-antimonopole configuration is studied. C. Monopole-vortex chains
In the previous sections, we have shown that the flux between a monopole-antimonopolepair is constructed from the fluxes of two vortices. To understand the interaction betweentwo center vortices inside the monopole-antimonopole configuration, we study the potentialsinduced by the center vortices and the monopole-antimonopole configurations using the“center vortex model”. Using Eqs. (26) and (32), Fig. 5 shows the static potential of thefundamental representation at intermediate distances induced by monopole-antimonopoleconfigurations compared with the one induced by the center vortices. The potential energyinduced by monopole-antimonopole configurations is larger than the twice of the potentialenergy induced by the center vortices. The free parameters a , b and f n ( n = 0 ,
1) are chosento be 0 . /F , 4 F and 0 .
1, respectively. As shown in Fig. 5, the result do not change byvarying the factor F related to the free parameters. Using two center vortices in the modelwithout any interaction, potential at small distances is obtained to be equal to the casewhen we use one center vortex with a thickness of twice the original one. We recall thatincreasing the thickness of the center vortex core would increase the energy of the centervortex and therefore the energy of the vacuum which is made of these vortices. As a result,the potential energy between static quark-antiquark increases. This is shown in figure 5. Onthe other hand, if there is no interaction between the vortices of the monopole-antimonopolepair, the induced potential for the small distances is expected to be equal to the inducedpotential by the two non interacting vortices. However, as shown in figure 5, the inducedpotentials are not equal. The extra energy obtained for the induced potential betweenthe quark-antiquark using monopole-antimonopole configurations, can be interpreted as arepulsion energy between the two center vortices constructing the configuration. Therefore,two vortices with the same flux orientations inside the monopole-antimonopole configurationrepel each other.The interaction between the constructing vortices of monopole-antimonopole pair can beobserved by the small or intermediate size Wilson loops. For large enough Wilson loops,15wo center vortices, constructing the monopole-antimonopole configuration, are located com-pletely inside the Wilson loop. Thus, the effect of two center vortices ( z ) on the large Wilsonloops is trivial ( z = I ). The flat potential at large distances in Fig. 6, shows this trivialbehavior. Therefore, interaction between two vortices can not be observed for R greaterthan the vortex core size. The vortex core size is about 20 with the free parameters we usedin the model. We recall that the lengths are dimensionless in the model. Thus, interactionbetween vortices of the monopole-antimonopole configuration is possible for R less than 20.Using the monopole-antimonopole configuration of the vacuum, we only show that thereis a repulsion between two center vortices within the monopole-antimonopole configurationand they construct a monopole-vortex chain. However, these monopole-vortex chains shouldbe observed in 3 dimensions. In the model, since the Wilson loop is a rectangular R × T loop in the x − t plane, it probably intersects with one of the legs of the chain at a time.Many random piercings of the Wilson loop by these legs and then averaging those randompiercings leads to the confinement. In fact, the repulsion deforms the localized flux andonly one of the vortices would intersect the Wilson loop as confirmed by the chain models[5–9]. These cases are shown in Fig. 6. In ref. [9], Reinhardt and et al . explained that themonopole-antimonopole flux splits into two equal portions of center vortex fluxes shown inFig. 7. The Wilson loop which intersects one of these center vortices leads to confinementfor the static sources.In addition, the dual superconductor picture of quark confinement was proposed byNambu in 1970’s [17]. Ginzburg-Landau theory defines two parameters: the superconductingcoherence length ξ and the London magnetic field penetration depth λ .As an interesting possibility, the repel of two center vortices may mean the Type-IIsuperconductor of the QCD vacuum, that is, the Ginzburg-Landau parameter κ = λ/ξ ofthe QCD vacuum is larger than 1 / √ z ) vacuum domains repel each other. While two vortices with opposite flux orientationsinside z z ∗ vacuum domains attract each other. The group factors analysis of ( z ) and z z ∗ vacuum domains agree with this article. Since two similarly oriented vortices inside ( z ) vacuum domain repel each other, we conclude that they do not make a stable configuration16nd one should consider each of them as a single vortex in the model. On the other hand,since two vortices with the opposite orientation inside z z ∗ vacuum domain attract eachother, we conclude that they make a stable configuration. Adding the contribution of the z z ∗ vacuum domain to the potential obtained from center vortices, the length of the Casimirscaling regime increases [18]. The results of this paper is in agreement with our previouspaper.To summarize, in this article, we obtain a chain of monopole-vortex. The magnetic fluxcoming from a monopole inside the chain is squeezed into vortices of finite thickness and anon-orientable closed loop is formed. The non-orientable closed loop means that two vortexlines inside the loop have different orientations of magnetic fluxes. Figure 8 schematicallyshows the interaction between center vortices inside the monopole-antimonopole configura-tion. a) V ( R ) Potentials, a=0.05,b=4 cen. vor. (twice)mon. −antimono. con. b)
11 12 13 14 15 1611.522.533.544.555.5 R V ( R ) Potentials, a=0.025,b=8 cen. vor. (twice)mon. −antimono. con.
FIG. 5: a) The potential energy V mon ( R ) induced by monopole-antimonopole configurations andthe twice of the potential V vor ( R ) obtained by the center vortices. The free parameters a and b are chosen to be 0 . /F and 4 F where F = 1 and the probability f n ( n = 0 ,
1) is chosen 0 . V mon ( R ) / V vor ( R ) is about 1 .
5. The extra positive potential energy of staticpotential induced by monopole-antimonopole configurations compared with the twice of the staticpotential obtained from center vortices shows that two similarly oriented center vortices inside themonopole-antimonopole configuration repel each other and make a monopole-vortex chain. b) thesame as a) but for F = 2. The potential ratios V mon ( R ) / V vor ( R ), obtained from a) and b), are thesame within the errors. Therefore, varying the free parameters do not change the physical results.
20 40 60 80 1000510152025 R V ( R ) Potentials induced by Monopole−vortex chains induced by Monopole−vortex chainsinduced by Monopole−antimonopole pairs
FIG. 6: The potential energy induced by the monopole-vortex chains. The free parameters a , b and f n are chosen to be 0 .
05, 4 and 0 .
1. If the monopole-vortex chains intersect in two points withthe large Wilson loop, the static sources are screened at large distances. On the other hand, if oneleg of the monopole-vortex chain intersects the large Wilson loop, confinement is observed for thefundamental representation.a) b)FIG. 7: a) The monopole-antimonopole pair in SU (2) gauge group. These configurationscontribute unity to the Wilson loop C . b) Assuming that the magnetic flux of the monopole-antimonopole configuration is split into two center vortex fluxes, one leg of this chain contributes − C . Therefore these chains lead to the confinement for the static sources [9]. Our understanding of monopole-antimonopole flux and monopole vortex chain is also inagreement with other research about this topic as comes in the following. According to theMonte Carlo simulations, after Abelian projection almost all monopoles are sitting on topof the vortices [5, 19] as shown in Fig. 9. Therefore a center vortex upon Abelian projectionwould appear in the form of monopole-vortex chains. Indeed Abelian monopoles and centervortices correlate with each other. Figure 10a shows some monopole-vortex chains in SU (2)gauge group [19]. In addition, the monopole-vortex junctions called as nexuses are studiedin ref. [20]. Some solutions to the equations of motion obtained from the low-energy effective18 IG. 8: A schematic view of a monopole-vortex chain obtained from the monopole-antimonopoleconfiguration. Center vortices of the monopole-antimonopole configuration repel each other andmake a monopole-vortex chain. The arrows on the vortex lines show the direction of the magneticfield of the vortex.FIG. 9: monopoles priced by P-vortices. Almost all monopoles (about 93%) are priced by one P-vortex (middle panel). Only very small fractions of monopoles either are not pierced at all (about3%)(left panel), or are pierced by more than one line (about 4%)(right panel) [5].a) + + - +- + - +-+- +- + - b)FIG. 10: a) Some monopole-vortex chains in SU (2) gauge group shown in ref. [19]. b) Themonopole-vortex chain shown in ref. [7]. Therefore, the monopole-vortex chain obtained in thisarticle agrees with the results of lattice gauge theory and chain models. energy functional E of QCD [7] are studied. Several thick vortices meet at a monopole-likecenter (nexus), with finite action and non-singular field strengths. In SU ( N ) gauge groupeach nexus is the source of N center vortices. Figure 10b shows monopole-vortex chainobtained by Cornwall for SU (2) gauge group [7]. In ref. [6], examples of the monopole-19ortex chains are also plotted using the method of ref. [7].Therefore the monopole-vortex chain in the vacuum obtained from the model agrees withthe results of lattice gauge theory and chain models. VI. CONCLUSION
The formation of the monopole-vortex chains which are observed in lattice simulationis studied in a model. This model is similar to the thick center vortex model but insteadof center vortices in the model we use monopole-antimonopole configurations which areline-like the same as center vortices. Comparing group factors of monopole-antimonopoleconfigurations and center vortices, we observe that the flux of the monopole-antimonopoleconfiguration is constructed from two center vortex fluxes. Calculating the induced quark-antiquark potential from two non interacting vortices and monopole-antimonopole config-urations and comparing the plots, we observe that the potential energy induced by themonopole-antimonopole configurations is larger than the twice of the one induced by thetwo non interacting center vortices configurations. The extra positive energy is interpreted asthe repulsive energy between the vortices inside the monopole-antimonopole configuration.The resulting monopole-vortex chains agree with the lattice calculations and phenomenolog-ical models. In general, these monopole-vortex chains should be observed in 3 dimensions.In the model, the Wilson loop, which is a rectangular R × T loop in the x − t plane, probablyintersects with one of the legs of the chain at a time. Many random piercings of the Wilsonloop by these legs and then averaging those random piercings leads to the confinement. VII. ACKNOWLEDGMENTS
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