Solitons in the SU(3) Faddeev-Niemi Model

Solitons in the SU(3) Faddeev-Niemi Model Yuki Amari Tokyo University of Science amari.yuki.ph@gmail.com Based on arxiv:1805,10008 with PRD 97, 065012 (2018) In collaboration with Nobuyuki Sawado (TUS) Quarks-2018 @Valday Russia, 29 May 2018

Contents 1. Background & motivation 2. Hopfions in the SU(2) Faddeev-Niemi model 3. 2D instantons in the F 2 nonlinear σ-model 4. Hopfions in the SU(3) Faddeev-Niemi model 5. Summary & Outlook

Knots as particles Atoms are knotted vortex tubes in ether. Lord Kelvin

Knots as particles Atoms are knotted vortex tubes in ether. Lord Kelvin The mass spectrum of glueballs R. V. Buniy, T. W. Kephart, A model of glueballs PLB 576, 127 C. J. Morningstar and M. J. Peardon PRD 60, 034509 (1999)

SU(2) Yang-Mills fields in the IR limit L. D. Faddeev & A. J. Niemi,PRL 82, 1624 (1999) QCD in UV limit asymptotic freedom QCD in IR limit color confinement Very different phases! In the IR limit with monopole condensation, some other order parameter could play a central role rather than the gauge field A μ. Cho-Faddeev-Niemi (CFN) decomposition A a μ = C μ n a + ε abc μ n b n c + ρ μ n a + σε abc μ n b n c C μ : Abelian gauge field, n = n 1, n 2, n 3 with n = 1, ρ, σ: scalar field If n = Ԧr r, C μ = ρ = σ = 0, it gives the Wu-Yang monopole.

Effective model in confinement phase In the confinement phase φ = 0 where φ = ρ + iσ, we get SU(2) Faddeev-Niemi model S eff = නd 4 x Λ 2 μ n 2 + μ n ν n 2 For static configurations, n defines a mapping S 3 S 2. Preimage of the Hopf map OPS = S 2 Phys. sp. = R 3 ~S 3 π 3 S 2 = Z Knot solitons

CFN decomposition of SU(3) gauge fields Decomposition L. D. Faddeev and A. J. Niemi, PLB 449, 214 (1999) A μ = C μ a n a + i n a, μ n a + ρ ab n a, μ n b + iσ ab n a, μ n b n a = Uh a U U SU 3 SU(3) Faddeev-Niemi model 2 S eff = නd 4 x M 2 Tr μ n a μ n a + 1 e 2 F μν a F a μν a=1 F μν a = i 2 Tr n a μ n b, ν n b OPS= SU 3 /U 1 2 = F 2 π 3 F 2 = Z Knot solitons Cf. S 2 = SU(2)/U(1)

Contents 1. Background & motivation 2. Hopfions in the SU(2) Faddeev-Niemi model 3. 2D instantons in the F 2 nonlinear σ-model 4. Hopfions in the SU(3) Faddeev-Niemi model 5. Summary & Outlook

The SU(2) Faddeev-Niemi model J. Gladikowski & M. Hellmund, PRD 56, 5194 (1997) The static energy L. D. Faddeev & A. J. Niemi, Nature 387, 58 (1997) E = නd 3 x M 2 i n 2 + 1 e 2 in j n 2 The Hopfions P. Sutcliffe, Proc. Roy. Soc. Lond. A 463, 3001 (2007) R. A. Battye & P. M. Putcliffe, PRL 81, 4798 (1998)

The SU(2) Faddeev-Niemi model J. Gladikowski & M. Hellmund, PRD 56, 5194 (1997) The static energy L. D. Faddeev & A. J. Niemi, Nature 387, 58 (1997) E = නd 3 x M 2 i n 2 + 1 e 2 in j n 2 Stereographic projection S 2 CP 1 n = 1 Δ u + u, i u u, u 2 1 Δ = 1 + u 2 At the end of my talk, we see the e.o.m. of the SU 3 model reduces to that of the SU 2 case with the stereographic projection.

Contents 1. Background & motivation 2. Hopfions in the SU(2) Faddeev-Niemi model 3. 2D instantons in the F 2 nonlinear σ-model 4. Hopfions in the SU(3) Faddeev-Niemi model 5. Summary & Outlook

The action density The F 2 nonlinear σ-model S = Z A μ Z B 2 + ZB μ Z C 2 + ZC μ Z A 2

The action density The F 2 nonlinear σ-model S = Z A μ Z B 2 + ZB μ Z C 2 + ZC μ Z A 2 D μ (a) μ Z a μ Z a Copies of the CP 2 NLσ-model A. D'Adda et.al., NPB 146, 63 (1978) S = න d 2 x D μ A Z A 2 + Dμ C Z C 2 ZC μ Z A 2

The action density The F 2 nonlinear σ-model S = Z A μ Z B 2 + ZB μ Z C 2 + ZC μ Z A 2 D μ (a) μ Z a μ Z a Copies of the CP 2 NLσ-model A. D'Adda et.al., NPB 146, 63 (1978) S = න d 2 x D A 2 μ Z A + C 2 Dμ Z C 2 ZC μ Z A 2π N A + N C න d 2 x Z 2 C μ Z A Winding number of a map S 2 CP 2 N a = i 2π නd2 x ε μν D μ a Z a Dν (a) Za N A + N B + N C = 0

Parametrization Isomorphism SU 3 /U 1 2 SL 3, C /B + 1 0 0 u 1 1 0 u 2 u 3 1 = c 1, c 2, c 3 SL 3, C 6 DOF = dim F 2 Gram-Schmidt orthogonalization process U = Z Α, Z Β, Z C SU 3

Parametrization Isomorphism SU 3 /U 1 2 SL 3, C /B + 1 0 0 u 1 1 0 u 2 u 3 1 = c 1, c 2, c 3 SL 3, C 6 DOF = dim F 2 Gram-Schmidt orthogonalization process U = Z Α, Z Β, Z C SU 3 Z A = 1 Δ 1 1 u 1 u 2 Z B = 1 Δ 1 Δ 2 u 1 u 2 u 3 1 u 1 u 2 u 3 + u 2 2 u 1 u 2 + u 3 + u 3 u 1 2 Z C = 1 u 1 u 3 u 2 u 3 Δ 2 1 Δ 1 = 1 + u 1 2 + u 2 2 Δ 2 = 1 + u 3 2 + u 1 u 3 u 2 2

The 2D instantons The BPS-like bound S 2π N A + N C න d 2 x Z 2 C μ Z A satisfied if and only if u j = u j x 1 ± ix 2 for j = 1,2,3.

The 2D instantons The BPS-like bound S 2π N A + N C න d 2 x Z 2 C μ Z A satisfied if and only if u j = u j x 1 ± ix 2 for j = 1,2,3. Stationary points in the bound correspond to the instantons. π N A + N C න d 2 x Z 2 C μ Z A 0 u 1 = u 3 = 0 u 2 = u 2 x 1 ± ix 2 H. T. Ueda, Y. Akagi & N. Shannon, PRA. 93, 021606(R) (2016) Embedding type

Contents 1. Background & motivation 2. Hopfions in the SU(2) Faddeev-Niemi model 3. 2D instantons in the F 2 nonlinear σ-model 4. Hopfions in the SU(3) Faddeev-Niemi model 5. Summary & Outlook

L = The SU(3) Faddeev-Niemi model The Lagrangian 2 a=1 M 2 Tr μ n a μ n a F μν a = i 2 Tr n a μ n b, ν n b 1 e 2 F μν a F a μν For simplicity, we set M = e = 1. n a = Uh a U U = Z Α, Z Β, Z C SU 3 The Cartan-Weyl basis h 1 = λ 3 / 2, h 2 = λ 8 / 2 e 1 = 0 1 0 0 0 0 0 0 0, e 2 = 0 0 0 0 0 0 1 0 0, e 3 = 0 0 0 0 0 1 0 0 0 e p = e p T

Equations of motion Cartan decomposition U μ U = ia μ a h a + ij μ p e p The EL eq. is equivalent to conservation of the Noether current K μ associated to U gu, g SU(3). μ K μ = 0 K μ = UB μ U μ B μ + U μ U, B μ = 0 The equations of motion μ J p μ ig p μν J pν B μ = i + ir pμ J p μ ig p μν J p ν +G pμν J p 1 μ J p+1 ν = 0 p J μ p iα a p F μν a J pν e p p = 1,2,3 R μ p = α a p A μ a, G p μν = α a p F μν a α a p = Tr e p h a, e p

Trivial embedding The embedding can be realized if two of the scalars vanish. u 1 = u 3 = 0, u 2 = u U = 1 Δ 1 0 u 0 Δ 0 u 0 1 Δ = 1 + u 2

Trivial embedding The embedding can be realized if two of the scalars vanish. u 1 = u 3 = 0, u 2 = u U = 1 Δ 1 0 u 0 Δ 0 u 0 1 Δ = 1 + u 2 We have J μ 1 = J μ 3 = 0, J μ 2 = i Δ μu The e.o.m for p = 1, 3 are automatically satisfied and for p = 2 reduces to μ μ u ig μν ν u + ir μ μ log Δ μ u ig μν ν u = 0 R μ = i Δ u μ u u μ u G μν = 2i Δ 2 μu ν u μ u ν u

Genuine solution Our strategy (i) We impose J μ 2 u 3 μ u 1 μ u 2 = 0. u 2 = f u 1, u 3 = f u 1 (ii)el eq. for p = 1, 3 are proportional to each other. 1 + u 1 2 + u 2 2 / 1 + u 3 2 + u 1 u 3 u 2 2 = const.

Genuine solution Our strategy (i) We impose J μ 2 u 3 μ u 1 μ u 2 = 0. (ii)el eq. for p = 1, 3 are proportional to each other. u 1 = u 3 = u 2 = u 2 u 2 = f u 1, u 3 = f u 1 1 + u 1 2 + u 2 2 / 1 + u 3 2 + u 1 u 3 u 2 2 = const. 2u, U = 1 Δ 1 2u u 2 2u 1 u 2 2u u 2 2u 1 The e.o.m for p = 2 is automatically satisfied and for p = 1, 3 reduce to μ μ u ig μν ν u + ir μ μ log Δ μ u ig μν ν u = 0

Summary We confirmed the existence of Hopfions in the SU 3 Faddeev-Niemi model. The model is an effective model of the SU(3) pure YM theory. We found an ansatz of nonembedding type which the e.o.m. reduces to the SU 2 Faddeev-Niemi model. Outlook Stability of the Hopfions Implication of the constraint Estimation of mass spectrum of glueballs

Reformulation Cartan decomposition of the Murer-Cartan form U μ U = ia μ a h a + ij μ p e p p = ±1, ±2, ±3. Under the gauge transformation U U exp iθ a h a, A μ a A μ a + μ θ a, The energy functional 3 E = නd 3 x q=1 J μ p J μ p e iθa α a p J q i J q i 1 4 J q [ij q j] J q+1 q+1 [i J 2 j] J [i q J j] q J i q J j q J j q J i q The energy represents the gauge fixing functional for a nonlinear maximal Abelian gauge.

χ 1 0 0 χ 2 χ 4 0 χ 3 χ 5 1 χ 1 χ 4 Parametrization = c 1, c 2, c 3 χ 1 0, χ 4 0 10 dof Gramm-Schmit orthogonalization process χ 2 /χ 1 = u 1, χ 3 /χ 1 = u 2, χ 5 /χ 4 = u 3 U = Z A e iθ 1, Z Β e iθ 4, Z C e i θ 1+θ 4 θ i = arg χ i 8 dof = dim SU(3) W = Z Α, Z Β, Z C 6 dof = dim F 2 Z A = 1 Δ 1 1 u 1 u 2 Z B = 1 Δ 1 Δ 2 u 1 u 2 u 3 1 u 1 u 2 u 3 + u 2 2 u 1 u 2 + u 3 + u 3 u 1 2 Z C = 1 u 1 u 3 u 2 u 3 Δ 2 1 Δ 1 = 1 + u 1 2 + u 2 2 Δ 2 = 1 + u 3 2 + u 1 u 3 u 2 2

The Hopf invariant M. Kisielowski, J. Phys. A 49, 17, 175206 (2016) Exact sequence 0 π 3 SU 3 π 3 SU 3 /U 1 2 0 Isotropy π 3 F 2 π 3 SU 3 Hopf invariant =Winding number of U: S 3 SU(3) =CS term for the SU 3 pure gauge R H = 1 24π 2 නTr U du 3 = 1 16π 2 න Tr R dr + 2 Tr R R R 3 The Hopf invariant H = 1 8π 2 නd3 x ε ijk A i a F jk a R = U du + 1 8π 2 නd3 x ε ijk Im J i 1 J j 2 J k 3

CFN decomposition of SU(3) gauge fields There are two options of the decomposition: SB pattern Order parameter space Homotopy group SU 3 U 1 2 F 2 = SU 3 /U 1 2 π 3 F 2 = Z SU 3 U 2 CP 2 = SU 3 /U 2 π 3 CP 2 = 0 Decomposition L. D. Faddeev and A. J. Niemi, PLB 449, 214 (1999) A μ = C μ a n a + i n a, μ n a + ρ ab n a, μ n b + iσ ab n a, μ n b SU(3) Faddeev-Niemi model 2 S eff = නd 4 x M 2 Tr μ n a μ n a + 1 e 2 F μν a F a μν a=1 n a = Uh a U U SU 3 F μν a = i 2 Tr n a μ n b, ν n b

Effective models of the SU(2) YM theory (1) Coulomb phase The original Yang-Mills action at low energy (2) Higgs phase φ 0 φ = ρ + iσ S eff = නd 4 x G 2 μν + D μ φ 2 + 1 φ 2 2 G μν = μ C ν ν C μ, D μ = μ + ic μ Abelian Higgs model (3) Confinement phase φ = 0 S eff = නd 4 x Λ 2 μ n 2 + μ n ν n 2 SU(2) Faddeev-Niemi model

The SU(2) Faddeev-Niemi model The static energy J. Gladikowski & M. Hellmund, PRD 56, 5194 (1997) L. D. Faddeev & A. J. Niemi, Nature 387, 58 (1997) E = නd 3 x M 2 i n 2 + 1 e 2 in j n 2 n const. as x n R 3 ~S 3 S 2 π 3 S 2 = Z The top. charge can be defined. Hopf map Derrick s theorem Scaling x λx E = E 2 + E 4 E n; λ = λ 1 E 2 + λe 4 λ E n; 1 = E 2 + E 4 = 0 λ 2 E n; 1 = 2E 2 > 0 If E 4 is absent, there are no stable solitons.

The SU(2) Faddeev-Niemi model J. Gladikowski & M. Hellmund, PRD 56, 5194 (1997) The static energy L. D. Faddeev & A. J. Niemi, Nature 387, 58 (1997) E = නd 3 x M 2 i n 2 + 1 e 2 in j n 2 Stereographic projection S 2 CP 1 n = 1 Δ u + u, i u u, u 2 1 Δ = 1 + u 2 The complex scalar u is convenient to solve the EL eq.. But, the topological charge cannot be defined by n nor u. We need to introduce Z = Z 0, Z 1 T where u = Z 1 /Z 0. Hopf invariant π 3 S 2 H = 1 4π නd3 x A da = Z A = iz dz

Applications in condensed matter physics There are many suggestions that knot solitons appear in condensed matter physics; Triplet superconductor Nematic liquid crystal E. Babaev, PRL 88, 177002 (2002) P. J. Ackerman & I. I. Smalyukh, PRX 7, 011006 (2017) SU(3) analogue Color superconductor Spin-nematic described by the SU 3 Heisenberg model. OPS of the SU 3 AFH model on the Cubic lattice = F 2 Continuum limit The SU(3) Faddeev-Niemi (like) model