A new scheme for color confinement and violation of the non-Abelian Bianchi identities
aa r X i v : . [ h e p - l a t ] M a y A new scheme for color confinement and violation of the non-Abelian Bianchiidentities
Tsuneo Suzuki ∗ Kanazawa University, Kanazawa 920-1192, Japan
Katsuya Ishiguro
Library and Information Technology, Kochi University, Kochi 780-8520, Japan
Vitaly Bornyakov
NRC ”Kurchatov Institute” -IHEP, 142281, Protvino, RussiaSchool of Biomedicine, Far Eastern Federal University, Vladivostok 690950, Russia (Dated: September 25, 2018)A new scheme for color confinement in QCD due to violation of the non-Abelian Bianchi identitiesproposed earlier is revised. The violation of the non-Abelian Bianchi identities (VNABI) J µ is equalto Abelian-like monopole currents k µ defined by the violation of the Abelian-like Bianchi identities.Although VNABI is an adjoint operator satisfying the covariant conservation law D µ J µ = 0, itsatisfies, at the same time, the Abelian-like conservation law ∂ µ J µ = 0. There are N − SU ( N ) QCD. The charge of each component of VNABI is assumed to satisfy theDirac quantization condition. Each color component of the non-Abelian electric field E a is squeezedby the corresponding color component of the solenoidal current J aµ . Then only the color singletsalone can survive as a physical state and non-Abelian color confinement is realized. This confinementpicture is completely new in comparison with the previously studied monopole confinement scenariobased on an Abelian projection after some partial gauge-fixing, where Abelian neutral states cansurvive as physical.To check if the scenario is realized in nature, numerical studies are done in the framework of lat-tice field theory by adopting pure SU (2) gauge theory for simplicity. Considering J µ ( x ) = k µ ( x ) inthe continuum formulation, we adopt an Abelian-like definition of a monopole following DeGrand-Toussaint as a lattice version of VNABI, since the Dirac quantization condition of the magneticcharge is satisfied on lattice partially. To reduce severe lattice artifacts, we introduce various tech-niques of smoothing the thermalized vacuum. Smooth gauge fixings such as the maximal centergauge (MCG), block-spin transformations of Abelian-like monopoles and extraction of physically im-portant infrared long monopole loops are adopted. We also employ the tree-level tadpole improvedgauge action of SU (2) gluodynamics. With these various improvements, we measure the densityof lattice VNABI: ρ ( a ( β ) , n ) = P µ,s n pP a ( k aµ ( s n )) / (4 √ V n b ), where k aµ ( s n ) is an n blockedmonopole in the color direction a , n is the number of blocking steps, V n = V /n ( b = na ( β )) is thelattice volume (spacing) of the blocked lattice. Beautiful and convincing scaling behaviors are seenwhen we plot the density ρ ( a ( β ) , n ) versus b = na ( β ). A single universal curve ρ ( b ) is found from n = 1 to n = 12, which suggests that ρ ( a ( β ) , n ) is a function of b = na ( β ) alone. The universalcurve seems independent of a gauge fixing procedure used to smooth the lattice vacuum since thescaling is obtained in all gauges adopted. The scaling, if it exists also for n → ∞ , shows that thelattice definition of VNABI has the continuum limit and the new confinement scenario is realized. PACS numbers: 12.38.AW,14.80.Hv
I. INTRODUCTION
Color confinement in quantum chromodynamics(QCD) is still an important unsolved problem [1].As a picture of color confinement, ’t Hooft [2] andMandelstam [3] conjectured that the QCD vacuum isa kind of a magnetic superconducting state caused bycondensation of magnetic monopoles and an effect dualto the Meissner effect works to confine color charges.However, in contrast to SUSY QCD [4] or Georgi- ∗ e-mail:suzuki04@staff.kanazawa-u.ac.jp Glashow model [5, 6] with scalar fields, to find color mag-netic monopoles which condense is not straightforward inQCD.An interesting idea to realize this conjecture is toproject QCD to the Abelian maximal torus group by apartial (but singular) gauge fixing [7]. In SU (3) QCD,the maximal torus group is Abelian U (1) . Then colormagnetic monopoles appear as a topological object. Con-densation of the monopoles causes the dual Meissner ef-fect [8–10].Numerically, an Abelian projection in non-local gaugessuch as the maximally Abelian (MA) gauge [11–13] hasbeen found to support the Abelian confinement scenariobeautifully [14–20]. Also the Abelian dominance and thedual Meissner effect are observed clearly in local unitarygauges such as F
12 and Polyakov (PL) gauges [21].However, although numerically interesting, the idea ofAbelian projection[7] is theoretically very unsatisfactory.1) In non-perturabative QCD, any gauge-fixing is notnecessary at all. There are infinite ways of such a partialgauge-fixing and whether the ’t Hooft scheme is gaugeindependent or not is not known. 2) After an Abelianprojection, only one (in SU (2)) or two (in SU (3)) gluonsare photon-like with respect to the residual U (1) or U (1) symmetry and the other gluons are massive charged mat-ter fields. Such an asymmetry among gluons is unnatu-ral. 3) How to construct Abelian monopole operators ina gauge-independent way in terms of original gluon fieldsis not clear at all.In this paper, we propose a new theoretical schemefor color confinement based on the dual Meissner effectwhich is free from the above problems. The idea wasfirst expressed by one of the authors (T.S.) in Ref.[22]and was extended in Ref.[23]. However, the proofs of theDirac quantization condition of g am in SU (2) and SU (3)shown in Refs.[22, 23] are incorrect. Without knowingthe explicit form of the gauge-field configuration corre-sponding to VNABI, it is impossible to prove the Diracquantization condition theoretically. Since the authorsexpect that VNABI play an important role in color con-finement, the Dirac quantization conditions for g am in SU (2) and SU (3) are assumed. Also the simultaneousdiagonalization of VNABI J µ for all µ can not be provedfrom the Coleman-Mandula theorem[24] and Lorentz in-variance contrary to the assertion in Ref.[23]. When thesimultaneous diagonalization of J µ for all µ is assumed,the condensation of J µ and electric color invariance ofthe confinement vacuum can be compatible.Then to check if the above scheme is realized in na-ture, we study the proposal in the framework of thenon-Abelian lattice gauge theory. For simplicity weadopt pure SU (2) lattice gauge theory. First considering J µ ( x ) = k µ ( x ) in the continuum, we define VNABI onlattice as an Abelian-like monopole following DeGrand-Toussaint[25]. Then as a most important point to beclarified, we are going to study if the lattice VNABI hasthe non-trivial continuum limit, namely if the scaling ofthe density exists.The lattice monopoles exist as a closed loop due tothe current conservation law. As shown later explicitly,monopole closed loops are contaminated by lattice arti-facts. Hence it is absolutely necessary to introduce vari-ous techniques avoiding such large lattice artifacts in or-der to analyse especially such a quantity as the monopoledensity, since all lattice artifacts contribute positively tothe density. We introduce various techniques of smooth-ing the thermalized vacuum. Smooth gauge fixings suchas the maximal center gauge (MCG)[26, 27], block-spintransformations of Abelian-like monopoles and extrac-tion of physically important infrared long monopoles aretaken into account. We also employ the tree-level tadpoleimproved gauge action. II. A NEW CONFINEMENT SCHEME BASEDON VNABIA. Equivalence of J µ and k µ First of all, we prove that the Jacobi identities of co-variant derivatives lead us to conclusion that violation ofthe non-Abelian Bianchi identities (VNABI) J µ is noth-ing but an Abelian-like monopole k µ defined by viola-tion of the Abelian-like Bianchi identities without gauge-fixing. Define a covariant derivative operator D µ = ∂ µ − igA µ . The Jacobi identities are expressed as ǫ µνρσ [ D ν , [ D ρ , D σ ]] = 0 . (1)By direct calculations, one gets[ D ρ , D σ ] = [ ∂ ρ − igA ρ , ∂ σ − igA σ ]= − ig ( ∂ ρ A σ − ∂ σ A ρ − ig [ A ρ , A σ ]) + [ ∂ ρ , ∂ σ ]= − igG ρσ + [ ∂ ρ , ∂ σ ] , where the second commutator term of the partial deriva-tive operators can not be discarded, since gauge fieldsmay contain a line singularity. Actually, it is the ori-gin of the violation of the non-Abelian Bianchi identities(VNABI) as shown in the following. The non-AbelianBianchi identities and the Abelian-like Bianchi identitiesare, respectively: D ν G ∗ µν = 0 and ∂ ν f ∗ µν = 0. The re-lation [ D ν , G ρσ ] = D ν G ρσ and the Jacobi identities (1)lead us to D ν G ∗ µν = 12 ǫ µνρσ D ν G ρσ = − i g ǫ µνρσ [ D ν , [ ∂ ρ , ∂ σ ]]= 12 ǫ µνρσ [ ∂ ρ , ∂ σ ] A ν = ∂ ν f ∗ µν , (2)where f µν is defined as f µν = ∂ µ A ν − ∂ ν A µ = ( ∂ µ A aν − ∂ ν A aµ ) σ a /
2. Namely Eq.(2) shows that the violation ofthe non-Abelian Bianchi identities is equivalent to thatof the Abelian-like Bianchi identities.Denote the violation of the non-Abelian Bianchi iden-tities as J µ : J µ = 12 J aµ σ a = D ν G ∗ µν . (3)Eq.(3) is gauge covariant and therefore a non-zero J µ is agauge-invariant property. An Abelian-like monopole k µ without any gauge-fixing is defined as the violation of theAbelian-like Bianchi identities: k µ = 12 k aµ σ a = ∂ ν f ∗ µν = 12 ǫ µνρσ ∂ ν f ρσ . (4)Eq.(2) shows that J µ = k µ . (5)Several comments are in order.1. Eq.(5) can be considered as a special case of theimportant relation derived by Bonati et al.[28] inthe framework of an Abelian projection to a simplecase without any Abelian projection. Actually it ispossible to prove directly without the help of theJacobi identities J aµ − k aµ = Tr σ a D ν G ∗ µν − ∂ ν f ∗ aµν = − ig Tr σ a [ A ν , G ∗ µν ] − igǫ µνρσ Tr σ a [ ∂ ν A ρ , A σ ]= 0 .
2. VNABI J µ transforms as an adjoint operator, sothat does the Abelian-like monopole current k µ .This can be proved also directly. Consider a regulargauge transformation A ′ µ = V A µ V † − ig ∂ µ V V † . Then k ′ µ = ǫ µνρσ ∂ ν ∂ ρ A ′ σ = ǫ µνρσ ∂ ν ∂ ρ ( V A σ V † − ig ∂ σ V V † )= V ( ǫ µνρσ ∂ ν ∂ ρ A σ ) V † = V k µ V † . (6)3. The above equivalence shows VNABI is essentiallyAbelian-like. It was already argued that singulari-ties of gauge fields corresponding to VNABI mustbe Abelian[29], although the reasoning is different.4. The covariant conservation law D µ J µ = 0 is provedas follows[28]: D µ J µ = D µ D ν G ∗ νµ = ig G νµ , G ∗ νµ ]= ig ǫ νµρσ [ G νµ , G ρσ ] = 0 , (7)where ∂ µ ∂ ν G ∗ µν = 0 (8)is used. The Abelian-like monopole satisfies theAbelian-like conservation law ∂ µ k µ = ∂ µ ∂ ν f ∗ µν = 0 (9)due to the antisymmetric property of the Abelian-like field strength[30]. Hence VNABI satisfies alsothe same Abelian-like conservation law ∂ µ J µ = 0 . (10) Both Eqs.(7) and (10) are compatible, since thedifference between both quantities[ A µ , J µ ] = 12 ǫ µνρσ [ A µ , ∂ ν f ρσ ]= ǫ µνρσ [ A µ , ∂ ν ∂ ρ A σ ]= − ǫ µνρσ ∂ ν ∂ µ [ A ρ , A σ ]= ig ( ∂ µ ∂ ν G ∗ µν − ∂ µ ∂ ν f ∗ µν )= 0 , where (8) and (9) are used. Hence the Abelian-likeconservation relation (10) is also gauge-covariant.5. The Abelian-like conservation relation (10) givesus three conserved magnetic charges in the case ofcolor SU (2) and N − SU ( N ). But these are kinematical relations com-ing from the derivative with respect to the diver-gence of an antisymmetric tensor [30]. The numberof conserved charges is different from that of theAbelian projection scenario [7], where only N − SU ( N ). B. Proposal of the vacuum in the confinementphase
Now we propose a new mechanism of color confine-ment in which VNABI J µ play an important role in thevacuum. For the scenario to be realized, we make twoassumptions concerning the property of VNABI.1. If VNABI are important physically, they must sat-ify the Dirac quantization condition between thegauge coupling g and the magnetic charge g am for a = 1 , , SU (2) and a = 1 ∼ SU (3).Since we do not know theoretically the property ofVNABI, we have to assume the Dirac qunatizationconditions: gg am = 4 πn a , where n a is an integer.2. The vacuum in the color confinement phase shouldbe electric color invariant. Since VNABI transformas an adjoint operator, we have to extract electriccolor invariant but magnetically charged quantityfrom VNABI. One possible way it to assume thatVNABI satisfy[ J µ ( x ) , J ν = µ ] = 0which make it possible to diagonalize VNABI J µ simultaneously for all µ . At present, the authors donot know if the second assumption is the only wayto have the magnetically charged but electricallyneutral vacuum in the confinement phase. TABLE I:
Comparison between the ’tHooft Abelian projection studies and the present work in SU (2) QCD. ˆ φ ′ = V † p σ V p ,where V p is a partial gauge-fixing matrix of an Abelian projection. ( u c , d c ) is a color-doublet quark pair. MAmeans maximally Abelian.The ’tHooft Abelian projection scheme This work and Refs.[32, 33]Previous works[11–21] Reference [28]Origin of k µ A singular gauge transformation k µ = Tr J µ ˆ φ ′ k aµ = J aµ No. of conserved k µ A aµ One photon A µ with k µ + 2 massive A ± µ Three gluons A aµ with k aµ Flux squeezing One electric field E µ Three electric fields E aµ Number of physical mesons 2 Abelian neutrals, ¯ u c u c and ¯ d c d c u c u c + ¯ d c d c Expected confining vacuum Condensation of Abelian monopoles Condensation of color-invariant λ µ [9]Privileged gauge choice A singular gauge MA gauge No need of gauge-fixing Using the above assumption, VNABI can be diagonal-ized by a unitary matrix V d ( x ) as follows: V d ( x ) J µ ( x ) V † d ( x ) = λ µ ( x ) σ , where λ µ ( x ) is the eigenvalue of J µ ( x ) and is then colorinvariant but magnetically charged. Then one getsΦ( x ) ≡ V † d ( x ) σ V d ( x ) (11) J µ ( x ) = 12 λ µ ( x )Φ( x ) , (12) X a ( J aµ ( x )) = X a ( k aµ ( x )) = ( λ µ ( x )) . (13)Namely the color electrically charged part and the mag-netically charged part are separated out. From (12) and(10), one gets ∂ µ J µ ( x ) = 12 ( ∂ µ λ µ ( x )Φ( x ) + λ µ ( x ) ∂ µ Φ( x ))= 0 . (14)Since Φ( x ) = 1, ∂ µ λ µ ( x ) = − λ µ ( x )(Φ( x ) ∂ µ Φ( x ) + ∂ µ Φ( x )Φ( x ))= 0 . Hence the eigenvalue λ µ itself satisfies the Abelian con-servation rule.Furthermore, when use is made of (6), it is possible toprove that 12 ǫ µνρσ ∂ ν f ′ µν ( x ) = λ µ ( x ) σ , (15)where f ′ µν ( x ) = ∂ µ A ′ ν ( x ) − ∂ ν A ′ µ ( x ) A ′ µ = V d A µ V † d − ig ∂ µ V d V † d , ≡ A ′ aµ σ a . Namely, 12 ǫ µνρσ ∂ ν f ′ , ρσ ( x )( x ) = 0 (16)12 ǫ µνρσ ∂ ν f ′ ρσ ( x )( x ) = λ µ ( x ) . (17)The singularity appears only in the diagonal componentof the gauge field A ′ µ .It is very interesting to see that f ′ µν ( x ) is actually thegauge invariant ’tHooft tensor[5]: f ′ µν ( x ) = Tr Φ( x ) G µν ( x ) + i g Tr Φ( x ) D µ Φ( x ) D ν Φ( x ) , in which the field Φ( x ) (11) plays a role of the scalar Higgsfield in Ref.[5]. To be noted is that the field Φ( x ) (11) isdetermined uniquely by VNABI itself in the gluodynam-ics without any Higgs field. In this sense, our schemecan be regarded as a special Abelian projection scenariowith the partial gauge-fixing condition where J µ ( x ) arediagonalized. The condensation of the gauge-invariantmagnetic currents λ µ does not give rise to a spontaneousbreaking of the color electric symmetry. Condensation ofthe color invariant magnetic currents λ µ may be a keymechanism of the physical confining vacuum[9, 10].The main difference between our new scheme and pre-vious Abelian projection schemes is that in the formerthere exist N − N − N − N − N − k ABµ ( x ) = Tr { J µ ( x )Φ AB ( x ) } , (18)where k ABµ ( x ) is an Abelian monopole, Φ AB ( x ) = V † AB ( x ) σ V AB ( x ) and V AB ( x ) is a partial gauge-fixingmatrix in some Abelian projection like the MA gauge.Making use of Eq.(12), we get k ABµ ( x ) = λ µ ( x ) ˜Φ ( x ) , (19)where ˜Φ( x ) = V AB ( x ) V † d ( x ) σ V † AB ( x ) V d ( x )= ˜Φ a ( x ) σ a . The relation (18) is important, since existence of anAbelian monopole in any Abelian projection scheme isguaranteed by that of VNABI J µ in the continuum limit.Hence if in any special gauge such as MA gauge, Abelianmonopoles remain non-vanishing in the continuum assuggested by many numerical data [14–20], VNABI alsoremain non-vanishing in the continuum. III. LATTICE NUMERICAL STUDY OF THECONTINUUM LIMITA. Definition of VNABI on lattice
Let us try to define VNABI on lattice. In the previ-ous section, VNABI J µ ( x ) is shown to be equivalent inthe continuum limit to the violation of the Abelian-likeBianchi identities J µ ( x ) = k µ ( x ).On lattice, we have to define a quantity which leadsus to the above VNABI in the continuum limit. Thereare two possible definitions which lead us to the aboveVNABI in the naive continuum limit. One is a quantitykeeping the adjoint transformation property under thelattice SU (2) gauge transformation V ( s ): U ( s, µ ) ′ = V ( s ) U ( s, µ ) V † ( s + µ ) . Here U ( s, µ ) is a lattice gauge link field. Such a quantitywas proposed in Ref[31]: J µ ( s ) ≡ (cid:0) U ( s, ν ) U µν ( s + ν ) U † ( s, ν ) − U µν ( s ) (cid:1) ,U µν ( s ) ≡ U ( s, µ ) U ( s + µ, ν ) U † ( s + ν, µ ) U † ( s, ν )where U µν ( s ) is a plaquette variable corresponding to thenon-Abelian field strength. This transforms as an adjointoperator: J ′ µ ( s ) = V ( s ) J µ ( s ) V † ( s ) (20)and satisfies the covariant conservation law X µ D Lµ J µ ( s ) = X µ (cid:0) U ( s + µ, µ ) J µ ( s ) U † ( s, µ ) − J µ ( s ) (cid:1) = 0 . However it does not satisfy the Abelian conservation law: X µ (cid:0) J µ ( s + µ ) − J µ ( s ) (cid:1) = 0 . (21) Moreover it does not have a property corresponding tothe Dirac quantization condition satisfied by the contin-uum VNABI, as we assumed. The last point is very un-satisfactory, since the topological property as a monopoleis essential.Hence we adopt here the second possibility whichcan reflect partially the topological property satisfied byVNABI. That is, we define VNABI on lattice as theAbelian-like monopole[32, 33] following DeGrand andToussaint[25]. First we define Abelian link and plaquettevariables: θ aµ ( s ) = arctan( U aµ ( s ) /U µ ( s )) ( | θ aµ ( s ) | < π ) (22) θ aµν ( s ) ≡ ∂ µ θ aν ( s ) − ∂ ν θ aµ ( s ) , (23)where ∂ ν ( ∂ ′ ν ) is a forward (backward) difference. Thenthe plaquette variable can be decomposed as follows: θ aµν ( s ) = ¯ θ aµν ( s ) + 2 πn aµν ( s ) ( | ¯ θ aµν | < π ) , (24)where n aµν ( s ) is an integer corresponding to the numberof the Dirac string. Then VNABI as Abelian monopolesis defined by k aµ ( s ) = − (1 / ǫ µαβγ ∂ α ¯ θ aβγ ( s + ˆ µ )= (1 / ǫ µαβγ ∂ α n aβγ ( s + ˆ µ ) J µ ( s ) ≡ k aµ ( s ) σ a . (25)This definition (25) of VNABI satisfies the Abelian con-servation condition (21) and takes an integer value whichcorresponds to the magnetic charge obeying the Diracquantization condition. The eigenvalue λ µ is definedfrom (13) as ( λ µ ( s )) = X a ( k aµ ( s )) . (26)However Eq.(25) does not satisfy the transformationproperty (20) on the lattice. We will demonstrate thatthis property is recovered in the continuum limit by show-ing the gauge invariance of the monopole density or thesquared monopole density (26) in the scaling limit. TABLE II:
A typical example of monopole loop distribu-tions (Loop length (L) vs Loop number (No.))for various gauges in one thermalized vacuum on24 lattice at β = 3 . I and L denote the color compo-nent and the loop length of the monopole loop,respectively.NGF I=1 MCG I=1 DLCG I=1L No L No L No4 154 4 166 4 1646 20 6 64 6 668 7 8 30 8 2810 2 10 13 10 1514 1 12 11 12 1016 1 14 4 14 3407824 1 16 5 16 618 1 18 222 2 20 124 2 22 128 1 24 230 1 26 332 1 30 134 2 36 136 1 44 144 1 48 146 1 54 148 1 58 158 1 124 1124 1 1106 12254 1 1448 1AWL I=1 MAU1 I=1 MAU1 I=3L No L No L No4 142 4 73 4 1906 66 6 32 6 808 36 8 13 8 2210 8 10 11 10 1512 7 12 6 12 214 3 14 3 14 316 3 16 2 16 118 1 18 3 18 320 1 20 2 20 322 3 22 1 24 126 3 30 2 36 128 1 34 2 42 130 2 58 1 60 132 1 148 1 66 134 1 5188 1 146 140 1 318 146 1 722 158 1120 1308 11866 1 B. Simulation details
1. Tadpole improved gauge action
First of all, we adopt the tree level improved action ofthe form [34] for simplicity in SU (2) gluodynamics: S = β imp X pl S pl − β imp u X rt S rt (27)where S pl and S rt denote plaquette and 1 × S pl,rt = 12 Tr(1 − U pl,rt ) , (28)the parameter u is the input tadpole improvement fac-tor taken here equal to the fourth root of the averageplaquette P = h t rU pl i . In our simulations we have notincluded one–loop corrections to the coefficients, for thesake of simplicity.The lattices adopted are 48 for β = 3 . ∼ . for β = 3 . ∼ .
9. The latter was taken mainlyfor studying finite-size effects. The simulations with theaction (27) have been performed with parameters givenin Table V in AppendixA following similarly the methodas adopted in Ref.[35].
2. The non-Abelian string tension
In order to fix the physical lattice scale we need tocompute one physical dimensionful observable the valueof which is known. For this purpose we choose the stringtension σ . The string tension for the action (27) wascomputed long ago in [35, 36] but we improve this mea-surement according to present standards. We use thehypercubic blocking (HYP) invented by the authors ofRef. [37–40] to reduce the statistical errors. After onestep of HYP, APE smearing [41] were applied to thespace-like links. The spatial smearing is made, as usu-ally, in order to variationally improve the overlap witha mesonic flux tube state. The results of the measuredstring tensions are listed also in Table V in AppendixA.
3. Introduction of smooth gauge-fixings
Monopole loops in the thermalized vacuum producedin the above improved action (27) still contain largeamount of lattice artifacts. Hence we here adopt a gauge-fixing technique smoothing the vacuum, although anygauge-fixing is not necessary in principle in the contin-uum limit[42]:1. Maximal center gauge (MCG).The first gauge is the maximal center gauge[26, 27]which is usually discussed in the framework of thecenter vortex idea. We adopt the so-called direct
FIG. 1: b = na ( β ) in unit of 1 / √ σ versus β b = n a ( β ) β b=na( β ) versus β n=1n=2n=3n=4n=6n=8n=12 TABLE III:
The n = 4 blocked monopole loop distribution(Loop length (L) vs Loop number (No.)) invarious gauges on 6 reduced lattice volume at β = 3 . maximal center gauge which requires maximizationof the quantity R = X s,µ (Tr U ( s, µ )) (29)with respect to local gauge transformations. Thecondition (29) fixes the gauge up to Z (2) gaugetransformation and can be considered as the Lan-dau gauge for the adjoint representation. In oursimulations, we choose simulated annealing algo-rithm as the gauge-fixing method which is knownto be powerful for finding the global maximum. Fordetails, see the reference[43].2. Direct Laplacian center gauge (DLCG).The second is the Laplacian center gauge[44] whichis also discussed in connection to center vortex idea.Here we adopt the so-called direct Laplacian centergauge (DLCG). Firstly, we require maximization of the quantity R M = X s,µ Tr (cid:2) M T ( s ) U A ( s, µ ) M ( s, µ ) (cid:3) (30)where U A ( s, µ ) denotes the adjoint representationof U ( s, µ ) and M ( s, µ ) is a real-valued 3 × SU (2) gauge theory which satisfies the constraint1 V X s X j M Tij ( s ) M jk ( s ) = δ ik (31)with V lattice volume. Matrix field M ( s ) whichleads to a global maximum of R M is composedof the three lowest eigenfunctions of a latticeLaplacian operator. Secondly, to determine thecorresponding gauge transformation, we construct SO (3) matrix-valued field which is the closest to M ( s ) and satisfies the corresponding Laplaciancondition by local gauge transformation. Finally,the SO (3) matrix-valued field is mapped to anSU(2) matrix-valued field which is used to thegauge transformation for the original lattice gaugefield in fundamental representation. After that,DLCG maximizes the quantity (29) with respectto solving a lattice Laplacian equation.3. Maximal Abelian Wilson loop gauge (AWL).Another example of a smooth gauge is introduced.It is the maximal Abelian Wilson loop gauge(AWL) in which R = X s,µ = ν X a ( cos ( θ aµν ( s )) (32)is maximaized. Here θ aµν ( s ) have been introducedin eq. (24). Since cos ( θ aµν ( s )) are 1 × × U (1) Landau gauge (MAU1).The fourth is the combination of the maximalAbelian gauge (MAG) and the U (1) Landaugauge[12, 13]. Namely we first perform the max-imal Abelian gauge fixing and then with respect tothe remaining U (1) symmetry the Landau gaugefixing is done. This case breaks the global SU (2)color symmetry contrary to the previous three cases(MCG, DLCG and AWL) but nevertheless we con-sider this case since the vacuum is smoothed fairlywell. MAG is the gauge which maximizes R = X s, ˆ µ Tr (cid:16) σ U ( s, µ ) σ U † ( s, µ ) (cid:17) (33)with respect to local gauge transformations. Thenthere remains U (1) symmetry to which the Lan-dau gauge fixing is applied, i.e., P s,µ cosθ µ ( s ) ismaximized[46].
4. Extraction of infrared monopole loops
An additional improvement is obtained when we ex-tract important long monopole clusters only from totalmonopole loop distribution. Let us see a typical exampleof monopole loop distributions in each gauge in compar-ison with that without any gauge fixing starting from athermalized vacuum at β = 3 . lattice. They areshown in Table II. One can find almost all monopole loopsare connected and total loop lengths are very large whenno gauge fixing (NGF) is applied as shown in the NGFcase. On the other hand, monopole loop lengths becomemuch shorter in all smooth gauges discussed here. Also itis found that only one or few loops are long enough andothers are very short as observed similarly in old papersin MAG. The long monopole clusters are called as in-frared monopoles and they are the key ingredient givingconfinement as shown in the old papers[47]. It is im-portant that in addition to MAU1, all other three MCG,DLCG and AWL cases also have similar behaviors. Sincesmall separate monopole loops can be regarded as latticeartifacts, we extract only infrared monopoles alone. Al-though there observed only one infrared monopole loopin almost all cases, there are some vacua (especially forlarge beta) having two or three separate long loops whichcan be seen as infrared one, since they have much longerlength than other shorter ones. We here define as in-frared monopoles as all loops having loop lengths longerthan 10% of the longest one. The cutoff value is not socritical. Actually the definition of infrared loops itselfhas an ambiguity, since even in the longest loop, we cannot separate out some short artifact loops attached ac-cidentally to the real infrared long loop. But such anambiguity gives us numerically only small effects as seenfrom the studies of different cutoff values.
5. Blockspin transformation
Block-spin transformation and the renormalization-group method is known as the powerful tool to studythe continuum limit. We introduce the blockspin trans-formation with respect to Abelian-like monopoles. Theidea was first introduced by Ivanenko et al.[48] and ap-plied in obtaining an infrared effective monopole actionin Ref.[49]. The n blocked monopole has a total magneticcharge inside the n cube and is defined on a blocked re-duced lattice with the spacing b = na , a being the spacingof the original lattice. The respective magnetic currents FIG. 2:
The VNABI (Abelian-like monopoles) densityversus a ( β ) in MCG on 48 . Top: total density;bottom: infrared density. n in the legend means n -step blocked monopoles. M onopo l e d e n s it y a( β ) MCG monopole density monopole2 monopole3 monopole4 monopole6 monopole8 monopole12 monopole 0 0.5 1 1.5 2 2.5 3 3.5 4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 M onopo l e d e n s it y a( β ) MCG IFmonopole density IFmonopole2 IFmonopole3 IFmonopole4 IFmonopole6 IFmonopole8 IFmonopole12 IFmonopole are defined as k ( n ) µ ( s n ) = 12 ǫ µνρσ ∂ ν n ( n ) ρσ ( s n + ˆ µ )= n − X i,j,l =0 k µ ( ns n +( n − µ + i ˆ ν + j ˆ ρ + l ˆ σ ) , (34) n ( n ) ρσ ( s n ) = n − X i,j =0 n ρσ ( ns n + i ˆ ρ + j ˆ σ ) , FIG. 3:
The VNABI (Abelian-like monopoles) density versus b = na ( β ) in MCG on 48 . Top: total density; bottom:infrared density. M onopo l e d e n s it y (cid:13)(cid:0)(cid:1)(cid:2)(cid:3) β (cid:17) MCG monopole density monopole2 monopole3 monopole4 monopole6 monopole8 monopole12 monopole 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 M onopo l e d e n s it y (cid:4)(cid:5)(cid:6)(cid:7)(cid:8) β (cid:9) MCG IFmonopole density IFmonopole2 IFmonopole3 IFmonopole4 IFmonopole6 IFmonopole8 IFmonopole12 IFmonopole where s n is a site number on the reduced lattice. Forexample, k (2) µ ( s ) = X i,j,l =0 k µ (2 s + ˆ µ + i ˆ ν + j ˆ ρ + l ˆ σ ) ,k (4) µ ( s ) = X i,j,l =0 k µ (4 s + 3ˆ µ + i ˆ ν + j ˆ ρ + l ˆ σ )= X i,j,l =0 k (2) µ (2 s + ˆ µ + i ˆ ν + j ˆ ρ + l ˆ σ ) . These equations show that the relation between k (4) µ ( s )and k (2) µ ( s ) is similar to that between k (2) µ ( s ) and k µ ( s )and hence one can see the above equation (34) corre-sponds to the usual block-spin transformation. After the0 FIG. 4:
The fit of the infrared VNABI (Abelian-like monopoles) density data in MCG on 48 lattice to Eq.(36). M onopo l e d e n s it y (cid:10)(cid:11)(cid:12)(cid:14)(cid:15) β (cid:16) MCG IFmonopole density log( ρ (b))=0.5302-1.4756b+0.1304b MCG 1 IFmonopoleMCG 2 IFmonopoleMCG 3 IFmonopoleMCG 4 IFmonopoleMCG 6 IFmonopoleMCG 8 IFmonopoleMCG 12 IFmonopole ρ (b) FIG. 5:
The VNABI (Abelian-like monopole) density at b = 0 . , . , . , . n in MCG on 48 . The data usedare derived by a linear interpolation of two nearest data below and above for the corresponding b and n . As anexample, see the original data at b = 1 . M onopo l e d e n s it y n MCG monopole density for different n b=0.5b=1.0b=1.5b=2.0
TABLE IV:
IF monopole density ρ IF around b = 1 . n in MCG case on 48 . n β b = na ( β ) db ρ IF error3 3.0 1.1184 0.0012 3.94E-01 1.42E-033 3.1 0.9465 0.0024 4.82E-01 4.06E-034 3.2 1.052 0.0016 3.99E-01 1.40E-024 3.3 0.866 0.0008 5.32E-01 2.37E-036 3.4 1.092 0.0012 3.93E-01 2.80E-036 3.5 0.9318 0.0024 4.64E-01 7.44E-038 3.6 1.0712 0.0072 3.77E-01 9.20E-038 3.7 0.9064 0.0008 4.75E-01 3.78E-0312 3.8 1.1412 0.0012 3.70E-01 4.43E-0312 3.9 0.9948 0.0024 4.56E-01 8.36E-03 block-spin transformation, the number of short lattice ar-tifact loops decreases while loops having larger magneticcharges appear. We show an example of the loop lengthand loop number distribution of the four step ( n = 4 )blocked monopoles in TableIII with respect to the sameoriginal vacuum as in TableII. For reference, we show therelation between the spacing of the blocked lattice and β in Fig.1. In Fig.1 and in what follows we present spacings a and b in units of 1 / √ σ . C. Numerical results
Now let us show the simulation results with respectto VNABI (Abelian-like monopole ) densities. Since1
FIG. 6:
The VNABI (Abelian-like monopoles) densityversus b = na ( β ) in AWL on 48 . Top: totaldensity; bottom: infrared density. M onopo l e d e n s it y (cid:18)(cid:19)(cid:20)(cid:21)(cid:22) β (cid:23) AWL monopole density monopole2 monopole3 monopole4 monopole6 monopole8 monopole12 monopole 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 M onopo l e d e n s it y (cid:24)(cid:25)(cid:26)(cid:27)(cid:28) β (cid:29) AWL IFmonopole density IFmonopole2 IFmonopole3 IFmonopole4 IFmonopole6 IFmonopole8 IFmonopole12 IFmonopole monopoles are three-dimensional objects, the density isdefined as follows: ρ = P µ,s n qP a ( k aµ ( s n )) √ V n b , (35)where V n = V /n is the 4 dimensional volume of the re-duced lattice, b = na ( β ) is the spacing of the reducedlattice after n -step blockspin transformation. s n is thesite on the reduced lattice and the superscript a denotesa color component. Note that P a ( k aµ ) is gauge-invariantin the continuum limit. Although the global color invari-ance is exact except in MAU1 gauge, the average of thedensity of each color component of | k aµ | is not equal tothe average of the above ρ , since two or three coloredmonopoles can run on the same dual links. In general,the density ρ is a function of two variables β and n . FIG. 7:
The VNABI (Abelian-like monopoles) densityversus b = na ( β ) in DLCG on 24 . M onopo l e d e n s it y (cid:30)(cid:31) !" β DLCG monopole density monopole2 monopole3 monopole4 monopole6 monopole
1. Scaling
For the purpose of studying the continuum limit, it isusual to analyse scaling behaviors. First of all, let usshow the data of MCG case in Fig.2. In this Figure andin what follows we present the monopole density ρ inunits of σ . . When the scaling exists for both the stringtension and the monopole density, we expect ρ → constas a ( β ) → V → ∞ , since a ( β ) is measured inunit of the string tension. In the case of total monopoledensity such a behavior is not seen yet. When infraredmonopoles alone and blocked monopoles are considered,the behavior becomes flatter as seen from Fig.2. But stillthis scaling is not conclusive. We need to study larger β regions on larger lattice volumes. These features are verymuch similar in other smooth gauges as AWL, DLCG andMAU1 and so their data are not shown here.
2. Scaling under the block-spin transformations
It is very interesting to see that more beautifuland clear scaling behaviors are observed when we plot ρ ( a ( β ) , n ) versus b = na ( β ). As one can see from the fig-ures shown below for various smooth gauges consideredin this work, one can see a universal function ρ ( b ) for β = 3 . ∼ . β = 3 . ∼ .
7) and n = 1 , , , , , , n = 1 , , , ,
6) on 48 (24 ) lattice. Namely ρ ( a ( β ) , n ) is a function of b = na ( β ) alone. Thus we observe clearindication of the continuum ( a ( β ) →
0) limit for the lat-tice VNABI studied in this work.
3. MCG case
First we show the case of MCG gauge-fixed vacua indetails. As can be seen from Fig.3, data for ρ ( a ( β ) , n ) can2 FIG. 8:
The VNABI (Abelian-like monopoles) densityversus b = na ( β ) for k and k components inMAU1 on 48 . Top: total density; bottom:infrared density. M onopo l e d e n s it y $%&’( β ) MAU1 k and k monopole density k monopolek monopole M onopo l e d e n s it y *+,-. β / MAU1 k and k IFmonopole density k IFmonopolek IFmonopole be expressed by a function of one argument b = na ( β )alone. There is a very beautiful scaling behavior for therange of β = 3 . ∼ . n = 1 , , , , , ,
12. Whenwe are restricted to long infrared monopoles alone, thedensity becomes substantially reduced for small b < . b region as shown in Fig.3. The violation of scaling forsmall b region is mainly due to the ambiguity of extract-ing infrared monopoles. When we restrict ourselves tothe data for b ≥ .
5, the scaling function ρ ( b ) is obtainedusing the χ fit to a simple function as shown in Fig.4: ρ ( b ) = exp( a + a b + a b ) , (36) a = 0 . , a = − . , a = 0 . . But the fit is not good enough, since χ /N dof = 12 .
56 for N dof = 44. Here we show the function (36) only for thepurpose of illustration, since we have not found a simplebut better fit.To see in more details, let us consider the data pointsat b = 0 . , . , . , . n . Especially the data at b = 1 . β from 3 . ≤ β ≤ . n at b = 1 . , . , . b = 0 .
4. AWL case
Very similar behaviors are seen in the AWL gauge case.Again beautiful scaling behaviors for the range of β =3 . ∼ . n = 1 , , , , , ,
12 are seen in Fig.6. Butin the case of infrared monopoles shown in Fig.6, a scalingviolation is observed for small b region.
5. DLCG case
Since the DLCG gauge-fixing needs much time forlarger lattice, we evaluate monopole density only on 24 lattice. As seen from Fig.7, a scaling behavior is found,although small deviations exist for small b region.
6. MAU1 case
Now we discuss the case of MAU1 gauge. In this gauge,the global isospin symmetry is broken. Hence let us firstevaluate the monopole density in each color direction.Namely ρ a = P µ,s n | k aµ ( s n )) | V n b . (37)As expected we find ρ ∼ ρ = ρ , so that we show ρ and ρ . The results are shown in Fig.8. Here thescaling is seen clearly with respect to the off-diagonal k currents, but the violation is seen for the diagonal k currents especially at small b region. Similar behaviorsare found when we are restricted to infrared monopoles.However when we evaluate the monopole density (35),we can observe similar beautiful scaling behaviors as inMCG and AWL cases. They are shown in Fig.9. D. Gauge dependence
Since P a ( k aµ ) should be gauge-invariant according toour derivation in section II, we compare the data in differ-ent smooth gauges. Look at Fig.10, which show the com-parison of the data in four gauges (MCG, AWL, DLCGand MAU1). One can see that data obtained in thesefour different gauges are in good agreement with eachother providing strong indication of gauge independence. This is the main result of this work.
Note that in MAU1gauge, the global color invariance is broken and usuallyoff-diagonal color components of gauge fields are said tohave large lattice artifacts. However here we performedadditional U1 Landau gauge-fixing with respect to the re-maining U (1) symmetry after MA fixing, which seems to3 FIG. 9:
The VNABI (Abelian-like monopoles) density (35) versus b = na ( β ) in MAU1 on 48 . Top: total density; bottom:infrared density. M onopo l e d e n s it y β MAU1 monopole density monopole2 monopole3 monopole4 monopole6 monopole8 monopole12 monopole M onopo l e d e n s it y β ; MAU1 IFmonopole density IFmonopole2 IFmonopole3 IFmonopole4 IFmonopole6 IFmonopole8 IFmonopole12 IFmonopole make the vacua smooth enough as those in MCG gaugecase. The fact that the scaling functions ρ ( b ) obtainedin MCG gauge can reproduce other three smooth-gaugedata seems to show that it is near to the smallest den-sity corresponding to the continuum limit without largelattice artifact effects. In other non-smooth gauges orwithout any gauge-fixing (NGF), ρ does not satisfy thescaling and actually becomes much larger. This is dueto our inability to suppress lattice artifacts in the non-smooth gauges or without gauge-fixing. E. Volume dependence in MCG case
The volume dependence is also studied when the twodata on 48 and 24 lattices in MCG are plotted for thesame β region (3 . ≤ β ≤ .
6) and the blocking steps(1 ≤ n ≤
6) as shown in Fig.11. We found sizable finitevolume effects for β = 3 . L = 24 becomes La < . / √ σ . Vol-ume dependence for (3 . ≤ β ≤ .
6) is very small as seenfrom Fig.11.4
FIG. 10:
Comparison of the VNABI (Abelian-like monopoles) densities versus b = na ( β ) in MCG, AWL, DLCG and MAU1cases. DLCG data only are on 24 lattice. Here ρ ( b ) is a scaling function (36) determined from the Chi-Square fitto the IF monopole density data in MCG. Top: total density; bottom: infrared density. M onopo l e d e n s it y <=>?@ β A Monopole densty in smooth gaugeslog( ρ (b))=0.5302-1.4756b+0.1304b MCG monopoleAWL monopoleMAU1 monopoleDLCG monopole ρ (b) M onopo l e d e n s it y BCDEF β G IFmonopole densty in smooth gaugeslog( ρ (b))=0.5302-1.4756b+0.1304b MCG IFmonopoleAWL IFmonopoleMAU1 IFmonopoleDLCG IFmonopole ρ (b) F. Gauge action dependence
Let us in short check how the gauge action adoptedhere improves the density ρ behavior by comparing thedata in the tadpole improved action with those in thesimple Wilson gauge action. It is shown in Fig.12. Thedensity in the Wilson action is higher especially for b ≤ . IV. CONCLUSIONS
In conclusion, we have proposed a new color confine-ment scheme which is summarized as follows:1. VNABI is equal to the Abelian-like monopole com-ing from the violation of the Abelian-like Bianchiidentities.2. VNABI satisfies the Abelian-like conservation lawas well as the covariant one. Hence there are N − SU ( N ).5 FIG. 11:
Volume dependence of VNABI (Abelian-likemonopole) density in the case of MCG in 48 and24 tadpole improved gauge action. The data for3 . ≤ β ≤ . ≤ n ≤ M onopo l e d e n s it y HIJKL β M Volume dependenceMCG monopole density
N24 1 monopoleN24 2 monopoleN24 3 monopoleN24 4 monopoleN24 6 monopoleN48 1 monopoleN48 2 monopoleN48 3 monopoleN48 4 monopoleN48 6 monopole FIG. 12:
Gauge action dependence of VNABI(Abelian-like monopole) densities in the case ofDLCG in 24 tadpole improved and Wilson gaugeactions, The data for 3 . ≤ β ≤ . ≤ n ≤ M onopo l e d e n s it y NOPQR β S DLCG monopole density : improved action2 : improved action3 : improved action4 : improved action6 : improved action1 : Wilson action2 : Wilson action3 : Wilson action4 : Wilson action6 : Wilson action
3. All magnetic charges are assumed to satisfy theDirac quantization condition.4. VNABI can be defined on lattice as lattice Abelian-like monopoles. Previous numerical results suggestthat the dual Meissner effect due to condensation ofVNABI must be the color confinement mechanismof QCD. The role of Abelian monopoles is playedby VNABI. This must be a new scheme for colorconfinement in QCD.5. VNABI are assumed to satisfy [ J µ , J ν = µ ] = 0 lead-ing to the simultaneous diagonalization for all µ .6. Condensation of the color invariant magnetic cur-rents λ µ which are the eigenvalue of VNABI J µ may be a key mechanism of the physical confining vacuum.Then to check if the new confinement scenario iscorrect in the continuum limit, densities of VNABIdefined on lattice were studied extensively in thiswork. Since VNABI is equivalent to Abelian-likemonopoles in the continuum, VNABI on lattice is definedas lattice Abelian-like monopoles following DeGrand-Toussaint[25]. This definition even on lattice keeps par-tially the topological property of VNABI satisfied in thecontinuum.In the thermalized vacuum, there are plenty of lat-tice artifact monopoles which contribute equally to thedensity, so that we have adopted various improvementtechniques reducing the lattice artifacts. One of them isto adopt the tadpole improved gauge action. The secondis to introduce various gauges smoothing the vacuum,although gauge-fixing is not necessary at all in the con-tinuum. We have considered here four smooth gauges,MCG, DLCG, AWL and MAU1. The third is to performa blockspin renormalization group study.With these improvement techniques, we have been ableto get very beautiful results. First of all, in MCG,AWL and MAU1 gauges, clear scaling behaviors are ob-served up to the 12-step blockspin transformations for β = 3 . ∼ .
9. Namely the density ρ ( a ( β ) , n ) is a func-tion of b = na ( β ) alone, i.e. ρ ( b ). If such scaling be-haviors are seen for n → ∞ , the obtained curve depend-ing on b = na ( β ) alone corresponds to the continuumlimit a ( β ) →
0. It is just the renormalized trajectory.The second beautiful result is the gauge independenceof the measured densities at least with respect to MCG,AWL and MAU1 smooth gauges on 48 and DLCG on24 adopted here. The gauge independence is the prop-erty expected in the continuum limit, since the observedquantity ρ in (35) is gauge invariant in the continuum.These beautiful results suggest that the lattice VNABIadopted here has the continuum limit and hence the newconfinement scenario can be studied on lattice with theuse of the lattice VNABI.Let us note that monopole dominance and the dualMeissner effect due to VNABI as Abelian monopoles wereshown partially without any smooth gauge fixing with theuse of random gauge transformations in Ref.[32, 33], al-though scaling behaviors were not studied enough. Moreextensive studies of these effects and derivation of in-frared effective VNABI action using block-spin transfor-mation in these smooth gauges discussed here and its ap-plication to analytical studies of non-perturbative quan-tities will appear in near future. Acknowledgments
The numerical simulations of this work were done usingcomputer clusters HPC and SX-ACE at Reserach Centerfor Nuclear Physics (RCNP) of Osaka University and the6supercomputer at ITEP, Moscow. The authors wouldlike to thank RCNP for their support of computer facili-ties. Work of VB was supported by Russian Foundationfor Basic Research (RFBR) grant 16-02-01146. One ofthe authors (T.S.) would like to thank Prof. T. Kugoand Prof. H. Tamura for pointing him the errors in theoriginal paper and fruitful discussions.
Appendix A: Tadpole improved action
The parameter u has been iterated over a series ofMonte Carlo runs in order to match the fourth root ofthe average plaquette P . The values of u are shown inTable V. TABLE V:
Details of the simulations with improved action β imp L N conf u < P > / √ σa Appendix B: The maximal Abelian Wilson loopgauge
In the maximal Abelian Wilson loop gauge (AWL), R = X s,µ = ν X a ( cos ( θ aµν ( s )) (B1)is maximized. Here θ aµν ( s ) is defined in Eq.(23).Since the gauge transformation property of the Abelianlink fields is not simple, to do the gauge-fixing efficientlyis not easy. Hence we adopt a gauge fixing iterationmethod of a minimal gauge transformation starting fromthe already-known smooth gauge configurations such asthose in the maximal center gauge (MCG) or the directLaplacian center gauge (DLCG) where the quantity R in(B1) is known to be already large. At the site s , the minimal gauge transformation is writ-ten as U ′ ( s, µ ) = e i~α ( s ) · ~σ U ( s, µ )= (1 + i~α ( s ) · ~σ ) U ( s, µ ) + O (( ~α ) ) . Hence in case of the minimal gauge transformation, weget U ′ ( s, µ ) = U ( s, µ ) − ~α ( s ) · ~U ( s, µ ) ~U ′ ( s, µ ) = ~U ( s, µ ) + U ( s, µ ) ~α ( s ) − ~α × ~U ( s, µ ) . Then an Abelian link field (22) is transformed as θ ′ aµ ( s ) = θ aµ ( s ) + δ aµ ( s ) ,δ aµ ( s ) = α a ( s )+ 1( U ( s, µ )) + ( U a ( s, µ )) × (cid:0) U a ( s, µ ) X b = a α b ( s ) U b ( s, µ ) − ǫ abc U ( s, µ ) U c ( s, µ ) (cid:1) . The function R is changed as follows: R ′ = X a,µ = ν,s cos ( θ a ′ µν ( s ))= X a,µ = ν,s cos ( θ aµν ( s ) + δ aµ ( s ) − δ aν ( s ))= R − X a,µ = ν,s ( δ aµ ( s ) − δ aν ( s )) sin ( θ aµν ( s ))= R − X b α b ( s ) A b ( s )7 A b ( s ) = 2 X a = b X µ = ν ( U b ( s, µ ) − ǫ bca U c ( s, µ )) × U ( s, µ ) sin ( θ aµν ( s )) U ( s, µ )) + ( U a ( s, µ )) . Hence if we choose α b ( s ) = − cA b ( s ) ( c > , we get R ′ = R + c X b ( A b ( s )) ≥ R. The maximum value of R is 3 .
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