, 5 January QCD. Wilson. (1) Wilson loop operator QCD [1,2] [3] [4][5] U(2) [2-7]

Transcription

1 , 5 January QCD ( ) Wilson (1) Wilson loop operator QCD [1,2] (2) QCD [3] (3) QCD [4][5] (4) SU(3) U(2) [6][7] [1] [2-7] 1

2 References SU(3) [1] K.-I. Kondo and Y. Taira, Prog. Theor. Phys. 104 (2000) [2] K.-I. Kondo, Phys. Rev. D77, (2008). [3] K.-I. Kondo, T. Shinohara and T. Murakami, Prog. Theor. Phys. 120, 1 50 (2008). [4] K.-I. Kondo, A. Shibata, T. Shinohara, T. Murakami, S. Kato, and S. Ito, Phys. Lett. B669, (2008). [5] A. Shibata, K.-I. Kondo and T. Shinohra, Phys. Lett. B691, (2010). [6] K.-I. Kondo, A. Shibata, T. Shinohara and S. Kato, Phys. Rev. D83, (2011). [7] A. Shibata, K.-I. Kondo, S. Kato and T. Shinohara, Phys. Rev. D87, (2013). 2

3 Contents 1. Quark Confinement for pure Yang-Mills theory a gauge invariant Wilson criterion area law of the Wilson loop average =a linear potential 2. Dual superconductivity picture for quark confinement: magnetic monopoles and dual Meissner effect 3. Wilson loop operator and the inherent magnetic monopole (through a non-abelian Stokes theorem) 4. Reformulating SU(N) Yang-Mills theory based on change of variables 5. Lattice reformulation and Numerical results for SU(3) case 6. Conclusion and discussion 3

4 Chapter 1: Quark Confinement for pure Yang-Mills theory: (a gauge invariant) Wilson criterion area law of the Wilson loop average =a linear potential 4

5 Wilson loop and static quark potential For a closed path C, we define the Wilson loop operator W C [A ] for the non-abelian Yang-Mills field A µ (x) by W C [A ] :=tr [ { P exp ig C }] dx µ A µ (x) /tr(1), A µ (x) = Aµ A (x)t A. (1) where P denotes the path-ordering prescription. In Yang-Mills theory, we consider a vacuum expectation value or the Wilson loop average W (C) of the Wilson loop operator W C [A ] for a closed loop C: W (C) = W C [A ] YM. (2) For a closed path C of side lengths T and R, the Wilson loop average W (C) is related to a static quark-antiquark potential V Q Q(R) as Therefore, V Q Q(R) is defined by W (C) exp[ T V Q Q(R)], (T R 1). (3) V Q Q(R) = lim T 1 T ln W (C). (4) 5

6 The static quark-antiquark potential V Q Q(R) obtained in this way is obviously gauge invariant, since the Wilson loop operator is gauge invariant. A naive picture of quark confinement is obtained, if V Q Q(R) as R. (5) It is shown 1 that the Wilson loop average W (C) for a closed loop C behaves W (C) = exp[ σ S C L + ], (6) where S is the minimal area of the (2-dimensional) surface Σ spanned by a loop C (i.e., C is a boundary of Σ, C = Σ) and L is the perimeter length of the loop C. If σ 0, the Wilson loop is said to obey the area law. If σ = 0, C 0, the Wilson loop is said to obey the perimeter law. 1 K. Osterwalder and R. Schrader, Commun. Math. Phys 42, 281 (1975); 31, 83 (1973). 6

7 The Wilson criterion for quark confinement 2 is : quark is confined if the Wilson loop average obeys the area law. the area law is equal to the linear potential: which indeed satisfies V Q Q(R) as R. V Q Q(R) = σr, (7) It is proved that the V Q Q(R) defined from the Wilson loop average does not increase faster than σr with a constant σ. On the lattice, a stronger result is proved 3 that V (R) is bounded from the above by the linear function in R, i.e., V Q Q(R) CR. The Wilson loop is efficient only when a dynamical pair creation does not occur in increasing the distance R. In the Wilson loop operator, Q Q is introduced as an external source which is assumed to be sufficiently or infinitely heavy so that we do not need to consider its own dynamics (This is equivalent to neglecting the kinetic term of quarks). 2 K.G. Wilson, Phys. Rev. D10, (1974). 3 B. Simon and L.G. Yaffe, Phys. Lett. 115B, 145 (1982). 7

8 Simulation result of the static quark potential Static quark-antiquark potential V Q Q(R)= Coulomb+Linear: V Q Q(R) = α R + σr + c σ: string tension [mass 2 ], α: dimensionless [mass 0 ], c: [mass 1 ], σ 0 confinement V Q Q(R) as R σ = 0 deconfinement V Q Q(R) < as R [V(r)-V(r 0 )] r β = 6.0 β = 6.2 β = 6.4 Cornell ƒ ƒ\ƒ r/r 0 Figure 1: The static quark-antiquark potential V (r) as a function of the distance r in SU(3) Yang-Mills theory, which is obtained by numerical simulations in the framework of lattice gauge theory. Note that the potential is normalized so that V (r 0 ) = 0 and β = 2N/g 2 YM for SU(N). See G.S. Bali,[hep-ph/ ], Phys.Rept.343, 1 (2001). 8

9 Chapter 2: Dual superconductivity picture for quark confinement: magnetic monopoles and dual Meissner effect 9

10 Introduction Dual superconductor picture for Yang-Mills vacuum [Nambu (1974), t Hooft (1975), Mandelstam (1976),Polyakov (1975,1977)...] The key ingredients of the hypothesis: QCD vacuum=dual superconductor are * Dual Meissner effect In the dual superconductor, chromoelectric flux is squeezed into tubes. [ In the ordinary superconductor, magnetic flux is squeezed into tubes] dual dual superconductor * condensation of chromomagnetic monopoles [ electric charges condense into Cooper pairs ] superconductor We must answer the following questions: * How to introduce magnetic monopoles in the YM gauge theory without scalar fields? cf. t Hooft-Polyakov magnetic monopole * How to define the duality in the non-abelian gauge theory? * How to preserve the original (non-abelian) gauge invariance? 10

11 The basic ingredient for dual superconductivity is the existence of magnetic monopoles to be condensed and the realization of the dual Meissner effect. [Nambu 1974, Mandelstam 1976, t Hooft 1978] Dirac magnetic monopole in U(1) Maxwell theory is realized when and only when there are singularities in A, leading to the violation of the Bianchi identity for U(1) gauge theory. As far as A is non-singular, the magnetic-monopole current vanishes identically: k = δ F = df = dda 0, (1) t Hooft-Polyakov magnetic monopole in the presence of adjoint Higgs fields, obtained by the spontaneous symmetry breaking of the original gauge symmetry G to a subgroup H: G H A basic question: How magnetic monopoles can be defined in Yang-Mills theory without adjoint scalar fields or QCD? 1. t Hooft Abelian projection, Abelian magnetic monopole is obtained by the Abelian projection due to the explicit breaking of the original gauge symmetry G to the maximal torus subgroup H, G H, e.g., SU(2) U(1), SU(3) U(1) U(1) magnetic monopole=gauge-fixing defect [ t Hooft, Nucl. Phys. B190, 455 (1981)] 11

12 It is possible to realize magnetic monopoles in the pure non-abelian gauge theory, without explicitly introducing singularities in the gauge connection, without introducing the adjoint Higgs field, and without breaking the original gauge symmetry. 2. Cho and Duan & Ge (field decomposition, new variables) [Cho, Phys. Rev. D21, 1080 (1980)] [Duan and Ge, Sinica Sci., 11, 1072(1979)] The second method has been fully developed in the last decade: Path integral formulation is completed (action and measure for new field variables). The relevant lattice gauge formulation are available for numerical simulations. In particular, The gauge-invariance of the magnetic monopole is guaranteed from the beginning by construction. The direct relevance of the magnetic monopole to the Wilson loop and Abelian dominance in the operator level are manifest via a non-abelian Stokes theorem. For SU(2) gauge group, the second method have already reproduced all essential results obtained so far by the first method, Infrared Abelian dominance in the string tension (Wilson loop average) magnetic monopole dominance in the string tension (Wilson loop average) 12

13 The first method (Abelian projection) is reproduced as a special limit of the second method (Cho and Duan & Ge). In fact, the first method is nothing but a gauge-fixed version of the second method. The purpose of this talk is to reconsider the dual superconductivity in SU(3) Yang- Mills theory as a mechanism for quark confinement. We focus on how the SU(3) case is different from SU(2) one. For SU(3) gauge group, we ask what kind of magnetic monopoles are defined and how they are responsible for quark confinement in SU(3) case. We obtain the exact relationship between the Wilson loop and the magnetic monopole. Consequently, (1) the magnetic monopole are inherent in the Wilson loop operator and can be defined in a gauge-invariant way. (2) the Wilson loop operator plays the role of a probe for detecting magnetic monopole. 13

14 Three steps of our strategy: Step1) We give a definition of a gauge-invariant magnetic monopole inherent in the Wilson loop operator even in pure SU(N) Yang-Mills theory without adjoint scalar fields. a non-abelian Stokes theorem for the Wilson loop operator (a path-integral representation of the Wilson loop operator using the coherent state for the Lie group) [Diakonov & Petrov, 1989,...],[Kondo,1998],[Kondo & Taira, 2000],[Kondo, 2008] Step2) We give an optimal description of the magnetic monopole derived in 1). a novel reformulation of Yang-Mills theory using new field variables [Cho, 1980],[Duan & Ge, 1979],[Faddeev & Niemi, 1999],[Shabanov, 1999],[Kondo, Shinohara & Murakami, 2005,2008] Step3) For SU(3), we confirm the infrared dominance of the restricted variables (extraction of the infrared-dominant variable) and the non-abelian U(2) magnetic monopole dominance for quark confinement (in the string tension). cf. [infrared Abelian dominance and magnetic monopole dominance in MA gauge] a lattice version of the reformulation of the Yang-Mills theory and numerical simulations on a lattice [Kato et al.,, 2006] [Ito et al., 2007][Shibata et al., 2007] [Kondo, Shibata, Shinohara, Murakami, Kato & Ito, 2008] [Shibata, Kondo & Shinohara,2010] [Shibata et al, 2013] 14

15 Chapter 3: Wilson loop operator and the inherent magnetic monopole (through a non-abelian Stokes theorem) SU(2)[Diakonov and Petrov, Phys. Lett. B224, 131 (1989)] SU(2)[Kondo, hep-th/ , Phys.Rev.D58, (1998)] SU(3)[Kondo & Taira, hep-th/ , Mod.Phys.Lett. A15, 367 (2000)] SU(N)[Kondo & Taira, hep-th/ , Prog.Theor.Phys. 104, 1189 (2000)] SU(N)[Kondo, hep-th/ ] SU(N)[Kondo, arxiv: , Phys.Rev.D77, (2008)] 15

16 SU(2) Wilson loop and magnetic monopole Non-Abelian Stokes theorem for the Wilson loop The Wilson loop operator for SU(2) Yang-Mills connection W C [A ] :=tr [ P exp {ig YM C }] dx µ A µ (x) /tr(1), A µ (x) = Aµ A (x)σ A /2 A non-abelian Stokes theorem for the Wilson loop operator in the J representation of SU(2): J = 1/2, 1, 3/2, 2, [Diakonov & Petrov, 1989] W C [A ] := [dµ(n)] Σ exp {ijg YM Σ: Σ=C } ds µν F µν, no path-ordering n(x) = n A (x)σ A /2 := U (x)(σ 3 /2)U(x), U(x) SU(2) (A, B, C {1, 2, 3}) F µν (x) := µ [A A ν (x)n A (x)] ν [A A µ (x)n A (x)] g 1 ϵ ABC n A (x) µ n B (x) ν n C (x), and [dµ(n)] Σ is the product measure of an invariant measure on SU(2)/U(1) over Σ: [dµ(n)] Σ := x Σ dµ(n(x)), dµ(n(x)) = 2J + 1 4π δ(na (x)n A (x) 1)d 3 n(x). 16

17 The geometric and topological meaning of the Wilson loop operator [Kondo, arxiv: , Phys.Rev.D77, (2008)] [Kondo, hep-th/ ] W C [A ] = dµ Σ (n) exp { ijg YM (Ξ Σ, k) + ijg YM (N Σ, j) }, C = Σ k := δ F = df, Ξ Σ := δ 1 Θ Σ (D-3)-forms j := δf, N Σ := δ 1 Θ Σ 1-forms (D-indep.) Θ µν Σ (x) = d 2 S µν (x(σ))δ D (x x(σ)) Σ 1 (k, Ξ Σ ) := d D xk µ 1 µ D 3 (x)ξ µ 1 µ D 3 Σ (x), (j, N Σ ) := (D 3)! d D xj µ (x)n µ Σ (x) k and j are gauge invariant and conserved currents, δk = 0 = δj. The magnetic monopole is a topological object of co-dimension 3. D=3: 0-dimensional point defect point-like magnetic monopole (cf. Wu-Yang type) D=4: 1-dimensional line defect magnetic monopole loop (closed loop) The Wilson loop operator knows the (gauge-invariant) magnetic monopole! We do not need to use the Abelian projection[ t Hooft,1981] (partial gauge fixing) to define magnetic monopoles in Yang-Mills theory! 17

18 Magnetic charge quantization: The non-vanishing magnetic charge is obtained without introducing Dirac singularities in c µ : (x) = Aµ A (x)n A (x). The magnetic charge obeys the quantization condition of Dirac type. n(x) := n1 (x) sin β(x) cos α(x) n 2 (x) = sin β(x) sin α(x) SU(2)/U(1) = Sint, 2 n 3 (x) cos β(x) The magnetic charge g m is nothing but a number of times Sint 2 from Sphys 2 to S2 int. [Π 2(SU(2)/U(1)) = Π 2 (S 2 ) = Z] is wrapped by a mapping g m := = V S 2 phy =g 1 YM d 3 xk 0 = V d 3 x i ( 1 2 ϵijk F jk ) dσ jk g 1 YM n ( jn k n) = g 1 YM S 2 int S 2 phy sin βdβdα = 4πg 1 n (n = 0, ±1, ) YM (β, α) dσ jk sin β (x j, x k ) where (β,α) (x µ,x ν ) is the Jacobian: (xµ, x ν ) S 2 phy (β, α) S2 int SU(2)/U(1) and S 2 int is a surface of a unit sphere with area 4π. 18

19 For D = 3, k(x) = 1 2 ϵjkl l f jk (x) = ρ m (x) denotes the magnetic charge density at x, and Ξ Σ (x) = Ω Σ (x)/(4π) agrees with the (normalized) solid angle at the point x subtended by the surface Σ bounding the Wilson loop C. The magnetic part reads W m A := exp { ijg YM (Ξ Σ, k) } = exp {ijg YM d 3 xρ m (x) Ω } Σ(x) 4π The magnetic charge q m obeys the Dirac quantization condition: q m := d 3 xρ m (x) = 4πg 1 n (n Z) YM [Proof] The non-abelian Stokes theorem does not depend on the surface Σ chosen for spanning the surface bounded by the loop C, See the next page. [Kondo, arxiv , Phys.Rev.D77, (2008)] 19

20 2πn = 1 2 g d 3 xρ YM m (x) Ω Σ 1 (x) 1 4π 2 g d 3 xρ YM m (x) Ω Σ 2 (x) 4π = 1 2 g d 3 xρ YM m (x) Ω Σ 1 (x) Ω Σ2 (x) 4π = 1 2 g d 3 xρ m (x) = 1 2 g q YM m, (1) - Figure 2: Quantization of the magnetic charge. The difference between the surface integrals of f over two surfaces Σ 1 and Σ 2 with the same boundary C is equal to the surface integral Σ f of f over one closed surface Σ = Σ 1 + Σ 2. Here the direction of the normal vector to the surface Σ 2 must be consistent with the direction of the line integral over C. The magnetic charge Q m is non-zero if the magnetic monopole exists inside Σ, otherwise it is zero. 20

21 For the ensemble of point-like magnetic charges: k(x) = n qmδ a (3) (x z a ) a=1 { n } { n } = W m A = exp ij g YM 4π a=1 q a mω Σ (z a ) = exp ij a=1 n a Ω Σ (z a ), n a Z The magnetic monopoles in the neighborhood of the Wilson surface Σ (Ω Σ (z a ) = ±2π) contribute to the Wilson loop W m A = n a=1 exp(±i2πjn a ) = { n a=1 ( 1)n a (J = 1/2, 3/2, ) = 1 (J = 1, 2, ) = N-ality dependence of the asymptotic string tension [Kondo, arxiv: , J.Phys.G35, (2008)] 21

22 For D = 4, the magnetic part reads using Ω µ Σ (x) is the 4-dim. solid angle W m A = exp {ijg YM } d 4 xω µ Σ (x)kµ (x) Suppose the existence of the ensemble of magnetic monopole loops C a, k µ (x) = n a=1 = W m A = exp q a m { C a ijg YM dy µ aδ (4) (x x a ), n a=1 q a ml(σ, C a) } q a m = 4πg 1 YM n a = exp { 4πJi } n n a L(Σ, C a), n a Z where L(Σ, C ) is the linking number between the surface Σ and the curve C. L(Σ, C ) := a=1 C dy µ (τ)ξ µ Σ (y(τ)) where the curve C is identified with the trajectory of a magnetic monopole and the surface Σ with the world sheet of a hadron (meson) string for a quark-antiquark pair. 22

23 Coherent state and maximal stability group H: maximal stability subgroup h H h Λ = Λ e iϕ(h) (2) for a reference state Λ of a given representation of a Lie group G. g = ξh G, ξ G/ H, h H. (3) g, Λ := g Λ = ξh Λ = ξ Λ e iϕ(h) = ξ, Λ e iϕ(h). (4) The coherent state is constructed as ξ, Λ = ξ Λ G/ H. (5) G = SU(2), H = U(1) = H, G = SU(3), H = U(1) U(1), U(2), G = SU(4), H = U(1) U(1) U(1), U(1) U(2), SU(2) U(2), U(3), G = SU(N), H H = U(1) N 1,..., U(N 1), (6) 23

24 {H j, E α, E α }: Cartan basis of the Lie algebra G with the generators {T A } Let Λ be the highest-weight state, i.e., H j Λ = Λ j Λ, E α Λ = 0 (α R + ), (7) where R + (R ) is a subsystem of positive (negative) roots. Then the coherent state is given by 4 ξ, Λ = ξ Λ = exp β R ( η β E β η β E β) Λ, η β C, (8) E α : (off-diagonal) shift-up operators, E α = E α : (off-diagonal) shift-down operators E α Λ = 0 (some α R ); E β Λ = Λ + β (some β R ); For the fundamental representation, the highest-weight state is represented by Λ = (1,, 0, 0) T = 1. 0 or Λ = (1,, 0, 0), (9) 0 4 [Feng, Gilmore and Zhang, Rev. Mod. PHys. 62, 867 (1990)] 24

25 G=SU(2): Fundamental representation H 1 = 1 2 σ 3 = 1 2 ( ) 1 0, E := 1 2 σ 1+i 1 2 σ 2 = ( ) 0 1, E 0 0 := 1 2 σ 1 i 1 2 σ 2 = 1 2 σ 3 Λ = 1 ( ) 1 2 σ 3 = 1 Λ highest, 0 2 ( ) ( ( ) ( ( ) ( ( ) ( E + =, E 0 0) = ; E 0 1) + =, E 1 0) =, 1 0) 1 2 σ 3 Λ = 1 ( ) 0 2 σ 3 = 1 Λ lowest, 1 2 ( ) 0 0, α α H1 - α Λ H1 Figure 3: Root diagram and weight diagram of the fundamental representation of SU(2) where Λ is the highest weight of the fundamental representation. 25

26 G=SU(3): H 1 = 1 2 λ 3 = , H 2 = 1 2 λ 8 = [1, 0] : 1 2 λ = , 2 2 λ = , Λ = ( 1 3 2, ) := ν 1 [ 1, 1] : 1 2 λ = , 2 2 λ = , Λ = ( 1 3 2, ) := ν 2 [0, 1] : 1 2 λ = , 2 λ = 1 0 0, Λ = (0, 1 ) := ν 3 (10) Figure 4: The weight diagram and root vectors to define the coherent state in the fundamental representations [1, 0] = 3, [0, 1] = 3 of SU(3) where Λ = h 1 = ν 1 := ( 1 2, 1 2 ) is the 3 highest weight of the fundamental representation, and the other weights are ν 2 := ( 1 2, 1 ν 3 := (0, 1 3 ). 2 3 ) and 26

27 G=SU(3) H2 α(3) α(2) H1 - α(1) α(1) - α(2) α(3) Figure 5: The root diagram of SU(3), where positive root vectors are given by α (1) = (1, 0), α (2) = ( 1 2, 3 2 ), and α(3) = ( 1 2, 3 2 ). The two simple roots are given by α1 := α = α (1), and α 2 := β = α (3). G=SU(3) has the positive roots given by ( ) α (1) = (1, 0), α (2) 1 3 = 2,, α (3) = 2 ( 1 2, ) 3, (11) 2 where the simple roots are α 1 := α (1), α 2 := α (3). (12) 27

28 G=SU(3): H 1 = 1 2 λ 3, H 2 = 1 2 λ 8, For Λ = ν 1 E ±α (1) := 1 2 (λ 1 ± iλ 2 ) = E ±α (2) := 1 2 (λ 4 ± iλ 5 ) = E ±α (3) := 1 2 (λ 6 ± iλ 7 ) = ( ( ) ), ( ( , ( ) ) ) ( 0 0 0, , E ±α (1) Λ = 0, ν 2,, E ±α (2) Λ = 0, ν 3, ), E ±α (3) Λ = 0, 0, (13) Figure 6: The weight diagram and root vectors to define the coherent state in the fundamental representations [1, 0] = 3, [0, 1] = 3 of SU(3) where Λ = h 1 = ν 1 := ( 1 2, 1 2 ) is the 3 highest weight of the fundamental representation, and the other weights are ν 2 := ( 1 2, 1 ν 3 := (0, 1 3 ). 2 3 ) and 28

29 The maximal stability subgroup H is defined by h H h Λ = Λ e iϕ(h), (14) for a reference state Λ of a given representation of a Lie group G. Every representation R of SU(3) with the Dynkin index [m,n] belongs to (I) or (II): (I) [Maximal] m 0 and n 0 = H = H = U(1) U(1). maximal torus e.g., adjoint rep.[1,1], {H 1, H 2 } u(1) + u(1), (II) [Minimal] m = 0 or n = 0 = H = U(2). when the weight vector Λ is orthogonal to some of the root vectors, e.g., fundamental rep. [1,0], {H 1, H 2, E β, E β } u(2), where Λ β, β. H2 H2 Λ H2 ν2 - α(1) ν1 Λ - α(2) - α(3) α(2) - α(2) H1 H1 - α(1) H1 ν3 - α(2) α(3) 29

30 In the minimal case (I), the coset G/ H is given by whereas in the maximal case (II), SU(3)/U(2) = SU(3)/(SU(2) U(1)) = CP 2, (15) SU(3)/(U(1) U(1)) = F 2. (16) CP n is the complex projective space and F n is the flag space. Therefore, the fundamental representations of SU(3) belong to the minimal case (I), and hence the maximal stability group is U(2), rather than the maximal torus group U(1) U(1). [Kondo & Taira, hep-th/ , Mod.Phys.Lett. A15, 367 (2000)] [Kondo & Taira, hep-th/ , Prog.Theor.Phys. 104, 1189 (2000)] - α(1) H2 Λ - α(3) - α(2) H1 Figure 7: The weight diagram and root vectors required to define the coherent state in the adjoint representation [1, 1] = 8 of SU(3), where Λ = ( 1 2, 3 2 ) is the highest weight of the adjoint representation. 30

31 SU(N) Wilson loop operator and magnetic monopole The SU(N) Wilson loop operator in a given representation specified by Λ : W C [A ] :=tr [ P exp {ig YM C }] dx µ A µ (x) /tr(1), A µ (x) = A A µ (x)λ A /2 su(n) can be cast into an equivalent form without the path-ordering P W C [A ] = [dµ(g)] C exp [ig YM where g YM is the Yang-Mills coupling constant, C ] A, A := A µ (x)dx µ, (1) [dµ(g)] C := x C dµ(g(x)), dµ(g) : an invariant measure on G = SU(N) (2) A := A µ (x)dx µ, A µ (x) = tr{ρ[g(x) A µ (x)g(x) + ig 1 YM g(x) µ g(x)]}, (3) ρ := Λ Λ, g(x) G = SU(N). (4) Λ : a reference state (highest weight state of the rep.) making a rep. of the Wilson loop we consider. tr(ρ) = Λ Λ = 1 follows from the normalization of Λ. 31

32 We follow the standard procedures of deriving the path-integral formula: 1. Divide the path L into N infinitesimal segments s ] P exp [ ig YM dsa (s) = P s 0 N 1 n=0 where ϵ := (s s 0 )/N and s n := s 0 + nϵ. We set s = s N. exp[ ig YM ϵa (s n )], (5) 2. Insert a complete set of states at each partition point: 1 = g n, Λ dµ(g n ) g n, Λ, g n := g(s n ) (n = 1,, N 1), (6) where the state is normalized g n, Λ g n, Λ = Take the limit N and ϵ 0 appropriately such that Nϵ = s s 0 is fixed. 4. Replace the trace of the operator O with the integral: N 1 tr(o) = dµ(g 0 ) g 0, Λ O g 0, Λ. (7) 32

33 { s ]} N 1 tr P exp [ ig YM dsa (s) s 0 N 1 N 1 = lim dµ(g n ) g n, Λ exp[ ig N,ϵ 0 YM ϵa (s n )] g n+1, Λ. (8) n=0 n=0 For taking the limit ϵ 0 in the final step, it is sufficient to retain the O(ϵ) terms. Therefore, we obtain g n, Λ exp[ ig YM ϵa (s n )] g n+1, Λ = Λ g(x n ) exp[ ig YM ϵa (s n )]g(x n+1 ) Λ = Λ exp[ ig YM ϵa g (s n )] Λ = Λ [1 ig YM ϵa g (s n ) + O(ϵ 2 )] Λ = Λ Λ iϵg YM Λ A g (s n ) Λ + O(ϵ 2 ) = exp[ iϵg YM Λ A g (s n ) Λ ] + O(ϵ 2 ), (9) where where we have used the normalization condition, Λ Λ = 1. 33

34 Finally, we obtain a path-integral representation of the Wilson loop operator: W C [A ] = [dµ(g)] C exp ( ig YM where we have introduced the one-form A g by C ) A g, [dµ(g)] C := dµ(g(x)), (10) x C A g (x) =A g µ(x)dx µ := Λ A g µ (x) Λ dx µ, (11) A g µ (x) :=g(x) A µ (x)g(x) + ig 1 YM g(x) µ g(x). (12) and [dµ(g)] C is the product measure of dµ(g n ) along the loop C: [dµ(g)] C := N 1 lim dµ(g n ), (13) N,ϵ 0 n=0 It should be remarked that the path-ordering has disappeared in the path-integral representation. We call this result the pre-non-abelian Stokes theorem (pre-nast). 34

35 Now the argument of the exponential is Abelian quantity. Therefore, we can apply the (usual) Stokes theorem: C= S ω = S dω, (14) to replace the line lintegral to the surface integral. Thus we obtain a version of the non-abelian Stokes theorem (NAST): W C [A ] := where the F is the two-form defined by [dµ(g)] Σ exp [ ig YM Σ: Σ=C ] F g, (15) F g = da g = 1 2 F µν(x)dx g µ dx ν (16) Here we have replaced the integration measure on the loop C by the integration measure on the surface Σ: [dµ(g)] Σ := dµ(g(x)), (17) x Σ: Σ=C by inserting additional integral measures, dµ(g(x)) = 1. 35

36 For the highest-weight state Λ of a representation R of a group G, we define a matrix ρ with the matrix element ρ ab by Then the trace of ρ has a unity: ρ := Λ Λ, ρ ab := Λ a Λ b = λ a λ b. (18) tr(ρ) = ρ aa = Λ a Λ a = λ a λ a = 1, (19) since the highest-weight state is normalized Λ Λ = 1. Moreover, the matrix element Λ O Λ of an arbitrary matrix O is written in the trace form: since Λ O Λ = λ b O baλ a = ρ ab O ba = tr(ρo). The one-form A g = A g µ(x)dx µ is defined by Λ O Λ = tr(ρo), (20) A g µ(x) :=tr{ρ[g(x) A µ (x)g(x) + ig 1 YM g(x) µ g(x)]} =tr{g(x)ρg(x) A µ (x)} + tr{ρig 1 YM g(x) µ g(x)}, (21) 36

37 where we have defined the traceless field ñ(x) (the normalized and traceless field is n(x)) [ ñ(x) := g(x) ρ 1 ] g(x). (22) tr(1) The field strength F g is calculated to be F g µν(x) := µ A g ν(x) ν A g µ(x) = µ tr{g(x)ρg (x)a ν (x)} ν tr{g(x)ρg (x)a µ (x)} the field strength F g is rewritten as + ig YM tr{g(x)ρg (x)[ig 1 YM g(x) µg (x), ig 1 YM g(x) νg (x)]} + ig 1 YM tr{ρg (x)[ µ, ν ]g(x)}. (23) F g µν = µ tr{ña ν } ν tr{ña µ } + ig YM tr{ñ[ω µ, Ω ν ]} + ig 1 YM tr{ρg [ µ, ν ]g}, (24) where we have defined Ω µ (x) := ig 1 YM g(x) µg (x). (25) 37

38 Then it is rewritten into the surface-integral form using a usual Stokes theorem: W C [A ] = dµ Σ (g) exp [ig YM Σ: Σ=C ] F, (26) where dµ Σ (g) := x Σ dµ(g(x)), the resulting two-form F := da = 1 2 F µν(x)dx µ dx ν is given by F µν (x) = 2(N 1)/N[G µν (x) + ig 1 YM tr{ρg(x) [ µ, ν ]g(x)}], (27) with the field strength G µν defined by G µν (x) := µ tr{n(x)a ν (x)} ν tr{n(x)a µ (x)} + 2(N 1) ig 1 N tr{n(x)[ YM µn(x), ν n(x)]}, (28) by introducing the so-called color field n(x) := N/[2(N 1)]g(x) [ρ 1/tr(1)] g(x), (29) which is normalized (n(x) n(x) = 1) and traceless (tr{n(x)} = 0). 38

39 Finally, the Wilson loop operator in the fundamental rep. of SU(N) reads [Kondo, arxiv: , Phys.Rev.D77, (2008)] [Kondo, hep-th/ ] W C [A ] = dµ Σ (g) exp { ig YM (Ξ Σ, k) + ig YM (N Σ, j) }, C = Σ k := δ F = df, Ξ Σ := δ 1 Θ Σ (D-3)-forms j := δf, N Σ := δ 1 Θ Σ 1-forms (D-indep.) Θ µν Σ (x) = d 2 S µν (x(σ))δ D (x x(σ)) Σ 1 (Ξ Σ, k) := d D xk µ 1 µ D 3 (x)ξ µ 1 µ D 3 Σ (x), (N Σ, j) := (D 3)! d D xj µ (x)n µ Σ (x) k,j: magnetic current k and electric current j which are conserved, δk = 0 = δj. := dδ + δd: the D-dimensional Laplacian, Θ: the vorticity tensor, which has the support on the surface Σ (with the surface element ds µν (x(σ))) whose boundary is the loop C. The last part ig 1 YM tr{ρg(x) [ µ, ν ]g(x)} in F corresponds to the Dirac string, which is not gauge invariant and does not contribute to the Wilson loop (unless the gauge is fixed). The gauge-invariant magnetic monopole k is inherent in the Wilson loop operator. 39

40 SU(2) case: For the fundamental rep. of SU(2), the highest-weight state Λ yields ( ( ( ) Λ =, ρ := Λ Λ = (1, 0) =, ρ 0) 0) = σ 3 2, (30) = n(x) =g(x) σ 3 2 g(x) SU(2)/U(1) S 2 CP 1. (31) Then the gauge invariant magnetic monopole current (D-3)-form k = 1 2 δ f is obtained from f µν = µ 2tr{nA ν } ν 2tr{nA µ } + ig 1 YM 2tr{n[ µn, ν n]} =2tr { nf µν + ig 1 YM n[d µn, D ν n] }. (32) The existence of magnetic monopole is suggested from a nontrivial Homotopy class of the map n : S 2 SU(2)/U(1) π 2 (SU(2)/U(1)) = π 1 (U(1)) = Z. (33) Magnetic charge obeys the quantization condition a la Dirac: Q m := d 3 xk 0 = 4πg 1 l, l Z. YM (34) 40

41 cf. the Abelian magnetic monopole of t Hooft Polyakov type: n A ˆϕ A (x)/ ˆϕ(x). (35) For SU(2), if we choose a special gauge in which the color field is uniform: n(x) = (n 1 (x), n 2 (x), n 3 (x)) = (0, 0, 1), (36) then f µν = µ Aν 3 ν Aµ 3. (37) The Wilson loop operator reduces to the Abelian-projected form: W C [A ] = exp [ig YM Σ: Σ=C where the two-form F := da = 1 2 F µν(x)dx µ dx ν is defined by ] F, (38) F µν (x) = G µν (x) = 1 2 f µν(x). (39) cf. Abelian projection, See reviews by Chernodub & Polikarpov, Greensite,... 41

42 SU(3) case: For the fundamental rep. of SU(3), the highest-weight sate Λ yields Λ = 1 0, ρ := Λ Λ = , ρ = , (40) = n(x) = g(x) g(x) SU(3)/U(2) CP 2 (41) Then the gauge invariant magnetic monopole current (D-3)-form k = δ f is given by G µν := µ tr{na ν } ν tr{na µ } ig 1 YM tr{n[ µn, ν n]}. (42) Homotopy class of the map n : S 2 SU(3)/U(2) π 2 (SU(3)/[SU(2) U(1)]) = π 1 (SU(2) U(1)) = π 1 (U(1)) = Z. (43) Magnetic charge obeys the quantization condition: Q m := d 3 xk 0 = 2π 3g 1 l, l Z. YM (44) 42

43 The target space of the color field is specified by the maximal stability group H: n(x) = g(x)diag.(λ 1, λ 2, λ 3 )g(x) G/ H, (45) The gauge-invariant magnetic monopoles inherent in the SU(3) Wilson loop operator for the fundamental rep. are non-abelian U(2) magnetic monopole in the sense of Goddard Nuyts Olive Weinberg. c.f. Abelian projection method=the partial gauge fixing from an original gauge group G to the maximal torus subgroup H: G = SU(3) H = U(1) U(1) (46) π 2 (SU(3)/[U(1) U(1)]) = π 1 (U(1) U(1)) = Z 2. (47) = two kinds of Abelian U(1) magnetic monopoles for any rep. No such difference for SU(2). For any rep. of SU(2), magnetic monopole is U(1), since H = H = U(1). (48) - α α H1 - α Λ H1 43

44 Chapter 4: Reformulating SU(N) Yang-Mills theory based on change of variables References: original and maximal option for SU(N): Y.M. Cho, Phys. Rev. Lett. 44, 1115(1980). L. Faddeev and A.J. Niemi, Phys.Lett.B 449, 214(1999). Phys.Lett.B 464, 90(1999). minimal option for SU(3) and all the other possible options for SU(N), N > 3: K.-I. Kondo, T. Shinohara and T. Murakami, Prog. Theor. Phys. 120, 1(2008). 44

45 Reformulating Yang-Mills theory using new variables We can construct the decomposition: A µ (x) = V µ (x) + X µ (x), (1) such that (a) V µ alone reproduces the Wilson loop operator: W C [A ] = W C [V ], (2) and that (b) F µν [V ] := µ V ν ν V µ ig YM [V µ, V ν ] in n direction agrees with G µν : G µν (x) = tr{n(x)f µν [V ](x)}. (3) For this purpose, we impose the defining equations: (I) n(x) is a covariant constant in the background V µ (x): 0 = D µ [V ]n(x) := µ n(x) ig YM [V µ (x), n(x)], = (b) (4) (II) X µ (x) does not have the H-commutative part: ( X µ (x) H := 1 2 N 1 ) [n, [n, ]] X µ (x) = 0. = (a) (5) N 45

46 By solving the defining equations, such fields V µ (x) and X µ (x) are determined uniquely: X µ = ig 1 2(N 1) [n, D YM µ [A ]n] L ie(g/ N H), V µ =C µ + B µ L ie(g), C µ = A µ B µ = ig 1 YM At the same time, the color field 2(N 1) [n, [n, A µ ]] L ie( N H), 2(N 1) [n, µ n] L ie(g/ N H). (6) n(x) L ie(g/ H) must be obtained by solving the reduction condition χ = 0 for a given A, e.g., χ[a, n] := [n, D µ [A ]D µ [A ]n] L ie(g/ H). (7) 46

47 For the gauge transformations of the original field A µ (x) A µ (x) A µ(x) := U(x)A µ (x)u(x) 1 + ig 1 YM U(x) µu(x) 1. (8) new fields obey the gauge transformation: if the color field obeys X µ (x) X µ(x) := U(x)X µ (x)u(x) 1, V µ (x) V µ(x) := U(x)V µ (x)u(x) 1 + ig 1 YM U(x) µu(x) 1, (9) n(x) n (x) := U(x)n(x)U(x) 1. (10) The gauge invariance of G µν (x) and hence the magnetic current k follow from these transformation rules. 47

48 We consider the Wilson loop average W (C) := W C [A ] YM = Z 1 YM DA e S YM[A ] W C [A ]. (11) original YM = reformulated YM field variables Aµ A L (G) action S YM [A ] = n β, Cν k, Xν b = S YM [n, C, X ] integration measure DAµ A Wilson loop operator W C [A ] = Dn β DCν k DXν b χ) red FP [n, c, X ] = dµ Σ (g) exp { ig YM (k, Ξ Σ ) + ig YM (j, N Σ ) } Thus, we have arrived at the Wilson loop average in the reformulated YM theory: W C [A ] YM = Z 1 YM dµ Σ (g) Dn β DC k ν DX b ν Jδ( χ) red FPe S YM [n,c,x ] exp { ig YM (k, Ξ Σ ) + ig YM (j, N Σ ) }, (12) where χ = 0 is the reduction condition written in terms of the new variables: χ := χ[n, C, X ] := D µ [V ]X µ, (13) 48

49 and red FP is the Faddeev-Popov determinant associated with the reduction condition: red FP := det ( ) δχ δθ χ=0 = det ( ) δχ δn θ. (14) χ=0 which is obtained by the BRST method as red FP [n, c, X ] = det{ D µ[v + X ]D µ [V X ]}. The Jacobian J is very simple, irrespective of the choice of the reduction condition: J = 1. (15) [Kondo, Shinohara & Murakami, arxiv: , Prog.Theor.Phys. 120, 1 50 (2008)] 49

50 Chapter 5: Lattice reformulation and Numerical results for SU(3) case Lattice reformulation and Numerical simulations SU(2) Ito, Kato, Kondo, Murakami, Shibata and Shinohara, Phys.Lett.B 645, 67(2007). Shibata, Kato, Kondo, Murakami, Shinohara and Ito, Phys. Lett. B653, 101 (2007). Lattice reformulation SU(N) Kondo, Shibata, Shinohara, Murakami, Kato and Ito, Phys. Lett. B669, 107(2008) Shibata, Kondo and Shinohara, Phys. Lett. B691, 91(2010) Lattice reformulation and Numerical simulations SU(3) Kondo, Shibata, Shinohara and Kato, Phys. Rev. D83, (2011) Shibata, Kondo, Shinohara and Kato, in preparation 50

51 Lattice reformulation and numerical simulations Lattice reformulation [Kondo, Shibata, Shinohara, Murakami, Kato and Ito, Phys. Lett. B669, 107(2008)] [Shibata, Kondo and Shinohara, Phys. Lett. B691, 91(2010) ] V(R)/a U : T=6 V : T=6 Mono: fit U fit V sigma_u fit mono = sigma_v = sigma_m = sigma_ref = sigma_u/sigma_ref = sigma_v/sigma_ref = sigma_m/sigma_ref = Figure 8: quark-antiquark potential: (from above to below) full SU(3) potential V f (r), restricted part V a (r) and magnetic monopole part V m (r) at β = 5.7 on 16 4 lattice W C [A ] YM V f (r) full SU(3) quark-antiquark potential, W C [V ] YM V a (r) restricted field part R/a infrared V dominance in the string tension (85 90%), e ig YM (k,ξ Σ) YM V m (r) magnetic monopole part U(2) magnetic monopole dominance in the string tension (75%), 51

52 We study the color flux produced by a quark-antiquark pair, see the left-panel of Fig.9. y tr WLU p L tr W U p q q Y X T position x L distance y distance y x W Figure 9: (Left) The setup for measuring the color flux produced by a quark antiquark pair. (Right) The connected correlator between a plaquette P and the Wilson loop W. In order to explore the color flux in the gauge invariant way, we use the connected correlator ρ W of the Wilson line (see the right panel of Fig.9), ρ W := tr ( UP L W L ) tr (W ) 1 N tr (U P ) tr (W ) tr (W ), (1) 52

53 In the naive continuum limit, ρ W reduces to the field strength: ( ε 0 tr gϵ ρ W gϵ 2 2 F µν L W L ) F µν q q := + O(ϵ 4 ), (2) tr (W ) β Thus, the color filed strength produced by a q q pair is given by F µν = 2N ρ W Ex Ex X Y X Y Figure 10: Measurement of chromo-electric flux in the SU(3) Yang-Mills theory. (Left) E x from the original field A. (Right) E x from the restricted field V. These are numerical evidences supporting non-abelian dual superconductivity due to non-abelian magnetic monopoles as a mechanism for quark confinement in SU(3) Yang-Mills theory. 53

54 To obtain correlation functions of field variables, we need to fix the gauge and we have adopted the Landau gauge. Fig.11 shows two-point correlation functions of color field, indicating the global SU(3) color symmetry preservation, no specific direction in color space: n A (0)n B (r) = δ AB D(r) lattice β= lattice β= correlation correlation distance (r/a) Figure 11: Color field correlators n A (0)n B (r) (Left) A = B, (Right) A B (A, B = 1,, 8) measured at β = 6.2 on 24 4 lattice, using 500 configurations under the Landau gauge. We have also checked that one-point functions vanish, distance (r/a) n A (x) = ±

55 10 1 correlation <OO> beta=6.20 <AA> <VV> <XX> 0.1 g 2 a 2 <OO> distance (r/a) Figure 12: Field correlators as functions of r (from above to below) Vµ A (0)Vµ A (r), Aµ A (0)Aµ A (r), and Xµ A (0)Xµ A (r). Fig. 14 shows correlators of new fields V, X, and original fields A, indicating the infrared dominance of restricted correlation functions Vµ A (0)Vµ A (r) in the sense that the variable V is dominant in the long distance, while the correlator Xµ A (0)Xµ A (r) of SU(3)/U(2) variable X decreases quickly. For X, at least, we can introduce a gauge-invariant mass term 1 2 M 2 X X A µ X A µ, since X transforms like an adjoint matter field under the gauge transformation. The naively estimated mass of X is M X = σ phys = 1.1 GeV. This value should be compared with the result in MA gauge. 55

56 magnetic-monopole loops for SU(3) Z X Y Figure 13: The magnetic-monopole loops on the 4 dimensional lattice where the 3-dimensional plot is obtained by projecting the 4-dimensional dual lattice space to the 3-dimensional one, i.e., (x, y, z, t) (x, y, z). 56

57 Instantons to magnetic monopole via reduction SU(2) By solving the reduction condition, a circular loop of monopole current k has been obtained from the Jackiw-Nohl-Rebbi two-instanton configuration: µ b 0 =(20.100, 0.100, 0.100, 0.100) µ b 1 =(-9.900, , 0.100, 0.100) x 3 µ b 2 =(-9.900, , 0.100, 0.100) x x e e e e e e e e e e e+00 Figure 14: JNR two-instanton and the associated circular magnetic current k x,µ. Fukui, Kondo, Shibata and Shinohara, arxiv: [hep-th], Phys.Rev. D82, (2010). Jackiw-Nohl-Rebbi two-instanton as a source of magnetic monopole loop. 57

58 Conclusion and discussion 1) We have defined a gauge-invariant magnetic monopole k inherent in the Wilson loop operator by using a non-abelian Stokes theorem for the Wilson loop operator, even in SU(N) Yang-Mills theory without adjoint scalar fields. For quarks in the fundamental representation, H = U(N) for G = SU(N) G=SU(2) Abelian magnetic monopole SU(2)/U(1) G=SU(3) non-abelian magnetic monopole SU(3)/U(2) 2) We have constructed a new reformulation of Yang-Mills theory using new field variables, which gives an optimal description of the magnetic monopole derived in 1). The reformulation allows options discriminated by the maximal stability group H. For G = SU(3), two options are possible: The minimal option H = U(2) gives an optimized description of quark confinement through the Wilson loop in the fundamental representation. The maximal option, H = H = U(1) U(1), the new theory reduces to a manifestly gauge-independent reformulation of the conventional Abelian projection in the maximal Abelian gauge. 58

59 The idea of using new variables is originally due to Cho, and Faddeev & Niemi, where N 1 color fields n (j) (j = 1,..., N 1) are introduced. However, our reformulation in the minimal option is new for SU(N), N 3: we introduce only a single color field n for any N, which is enough for reformulating the quantum Yang-Mills theory to describe confinement of the fundamental quark. By constructing a lattice version of the reformulation of the SU(N) Yang-Mills theory and performing numerical simulations on a lattice, 3) For SU(3), we have confirmed the infrared dominance of the restricted variables V and the non-abelian magnetic monopole dominance for quark confinement (in the string tension), cf. [infrared Abelian dominance and magnetic monopole dominance in MA gauge] 4) We have shown the evidence of the dual Meissner effect caused by non-abelian magnetic monopoles in SU(3) Yang-Mills theory: the tube-shaped flux of the chromoelectric field originating from the restricted field including the non-abelian magnetic monopoles. To confirm the non-abelian dual superconductivity picture in SU(3) Yang-Mills theory, we plan to do further checkes, e.g., determination of the type of dual superconductor, measurement of the penetraing depth, induced magnetc current around color flux due to magnetic monopole condensations, and so on. 59

60 Questions: breaking of the dual gauge symmetry, generation of dual gluon mass relationship between magnetic monopoles and vortex. We can define a gauge-invariant vortex which ends on the non-abelian magnetic monopole. relationship between magnetic monopoles and instantons or merons. SU(2) case Kondo, Fukui, Shibata and Shinohara, arxiv: [hep-th], Phys.Rev.D78, (2008) Fukui, Kondo, Shibata and Shinohara, arxiv: [hep-th], Phys.Rev.D82, (2010) Extension to finite temperature case: magnetic monopole liquid (T c < T < 2T c ), magnetic monopole gas (T > 2T c ), proposed by [Chernodub & Zakharov,2007] Skyrme-Fadeev-Niemi model as an low-energy effective theory, a non-linear sigma model written in terms of n CP N 1 = SU(N)/U(N 1) Large N analysis 60