Twisted non-Abelian vortices rpd Lukcs, in collaboration with Pter - PowerPoint PPT Presentation

Twisted non-Abelian vortices Árpád Lukács, in collaboration with Péter Forgács, Fidel A. Schaposnik Wigner RCP RMKI, Budapest, Hungary Non-Perturbative Methods in Quantum Field Theory 8–10. October 2014

Outline Introduction Dimensional reduction: twist Some solutions Static vortex solutions Twisting the elementary vortex Twisting the coincident composite vortex Conclusions

Motivation Localised solutions ◮ Classical solutions ◮ Important in Quantum Theory as well ◮ Non-perturbative Vortices and strings ◮ A vortex is a 2D solution ◮ Can be the cross section of a string ◮ Strings may play an important role in confinement ◮ non-Abelian vortices have nice mathematical properties too

Theory considered Bosonic sector of N = 2 supersymmetric SU ( 2 ) × U ( 1 ) gauge theory, SU ( 2 ) flavor symmetry. d 4 x L , � S = L = − 1 1 F µν F µν − µν G µν a + Tr ( D µ Φ) † D µ Φ − ( V 1 + V 2 ) , G a 4 g 2 4 g 2 1 2 where D µ Φ = ( ∂ µ − iA µ σ 0 / 2 − iC a µ σ a / 2 )Φ Potential V 1 = λ 1 V 2 = λ 2 8 ( Tr Φ † Φ − 2 ξ ) 2 , 8 ( Tr Φ † σ a Φ) 2 Properties of this theory: ◮ scalar sector of a supersymmetric theory ◮ possesses many localized solutions (strings, etc.)

Spontaneous symmetry breaking Let � � φ 1 ψ 1 Φ = φ 2 ψ 2 with this notation: V 1 = λ 1 V 2 = λ 2 8 ( φ † φ + ψ † ψ − 2 ξ ) 2 , ( φ † φ − ψ † φ ) 2 + 4 | ψ † φ | 2 � � , 8 i.e., vacuum: both φ, ψ normalized to ξ and orthogonal Symmetry breaking pattern: U ( 1 ) × SU ( 2 ) × SU ( 2 ) global → SU ( 2 ) CF where SU ( 2 ) CF preserves the VEV, E.g., choosing � Φ � = ξ ✶ : SU ( 2 ) CF acts as Φ → V Φ V † Color-flavor locking: gauge and color symmetry both broken spontaneously, SU ( 2 ) CF remains unbroken Topology permits vortex solutions

Some vortex solutions An ( n 1 , n 2 ) vortex: A ϑ = a ( r ) � φ 1 ( r ) e in 1 ϑ � Φ = , ψ 2 ( r ) e in 2 ϑ C 3 ϑ = c 3 ( r ) with real radial functions Radial equation for φ 1 , ψ 2 , a , c 3 solved numerically Further solutions generated: orientational normal modes Φ → V Φ V † , V ∈ SU ( 2 ) explicitly: Φ = χ + ✶ + χ − n a σ a , c a = n a ˜ c 3 where χ ± = ( φ 1 D + φ 2 D ) / 2. (Shifman etal., Auzzi etal.)

Some vortex solutions An ( n 1 , n 2 ) vortex: A ϑ = a ( r ) � φ 1 ( r ) e in 1 ϑ � Φ = , ψ 2 ( r ) e in 2 ϑ C 3 ϑ = c 3 ( r ) with real radial functions Radial equation for φ 1 , ψ 2 , a , c 3 solved numerically Further solutions generated: orientational normal modes Φ → V Φ V † , V ∈ SU ( 2 ) explicitly: Φ = χ + ✶ + χ − n a σ a , c a = n a ˜ c 3 where χ ± = ( φ 1 D + φ 2 D ) / 2. (Shifman etal., Auzzi etal.)

BPS Energy as sum-of-squares if λ i = g 2 i (fixed by SUSY): � 2 � 2 F ik ± g 2 ik ± g 2 1 1 � � 2 ǫ ik ( Tr Φ † σ 0 Φ − 2 ξ ) 1 2 G a 2 ǫ ik Tr Φ † σ a Φ E BPS = + 4 g 2 4 g 2 1 2 + 1 2 Tr ( D i Φ ± i ǫ ik D k Φ) † ( D i Φ ± i ǫ im D m Φ) ± ξ 4 F ik ǫ ik ∓ ǫ ik ∂ i Tr (Φ † D k Φ) . minimal energy: all squares vanish: F ik = ∓ g 2 2 ǫ ik ( Tr Φ † σ 0 Φ − 2 ξ ) , 1 ik = ∓ g 2 2 G a 2 ǫ ik Tr Φ † σ a Φ , D i Φ = ∓ i ǫ ik D k Φ , Energy: � d 2 x E BPs = 2 πξϕ E BPS = where ϕ is the number of flux quanta

BPS multi-vortices Multi vortex solutions also possible ◮ No interaction between vortices ◮ Moduli: positions, orientations Moduli matrix approach (Eto etal.): ◮ Φ = S ( x + iy , x − iy ) − 1 Φ 0 ( x + iy ) Φ 0 holomorphic given, zeros of its determinant: position of vortices ◮ S = S 1 S 2 , Ω i = S i S † i , Ω 1 = exp ( ψ ) ◮ one equation for Ω = SS † , reduced to a holomorphic splitting problem 2 ) = − g 2 � � − ✶ e − ψ , z (Ω 2 ∂ z Ω − 1 2 0 Ω − 1 0 Ω − 1 Φ 0 Φ † N Tr Φ 0 Φ † ∂ ¯ 2 2 4 z ∂ z ψ = − g 2 � 2 ) e − ψ − N ξ � 1 0 Ω − 1 Tr (Φ 0 Φ † ∂ ¯ . 4 N Also for more general gauge groups

Adding twist Straight string: translation invariance along axis z : � i � Φ( x µ ) = Φ( x i ) exp 2 M ω α x α , A µ ( x ν ) = ( A i ( x j ) , A α ( x j )) , C a µ ( x ν ) = ( C a i ( x j ) , C a α ( x j )) , Decoupling ω 2 = − ω α ω α = 0 ensures that the equations for Φ( x i ) , C a i , A i are unchanged ( i = 1 , 2) C a α = ω α C a A α = ω α A , The out-of-plane components satisfy a Gauss-constraint Solutions equivalent to solving mass deformed theory ◮ adjoint scalars: out-of-plane gauge field components ◮ mass matrix – twisting matrix (Collie, Eto etal., Gorsky etal.)

Adding twist Straight string: translation invariance along axis z : � i � Φ( x µ ) = Φ( x i ) exp 2 M ω α x α , A µ ( x ν ) = ( A i ( x j ) , A α ( x j )) , C a µ ( x ν ) = ( C a i ( x j ) , C a α ( x j )) , Decoupling ω 2 = − ω α ω α = 0 ensures that the equations for Φ( x i ) , C a i , A i are unchanged ( i = 1 , 2) C a α = ω α C a A α = ω α A , The out-of-plane components satisfy a Gauss-constraint Solutions equivalent to solving mass deformed theory ◮ adjoint scalars: out-of-plane gauge field components ◮ mass matrix – twisting matrix (Collie, Eto etal., Gorsky etal.)

Gauss constraint Gauss constraint for out-of-plane fields: � 1 A σ 0 + 1 = − 1 � Φ(Φ M − C Φ) † + (Φ M − C Φ)Φ † � D 2 C a σ a � i g 2 g 2 2 1 2 = − Φ M Φ † + 1 2 { C , ΦΦ † } , Physical quantities, like momentum in string axis direction and energy: � d 2 x Q , E = E BPS + where E BPS = 2 πξϕ (no. flux quanta) and � d 2 x Q T 03 = where Q is a current: ω 0 Q = ω 0 (Φ M − C Φ) M Φ † + Φ M (Φ M − C Φ) † � � 4 Tr , Angular momentum: T 0 ϑ = 1 F 0 ϑ F ϑ r + 1 G a 0 r G a ϑ r + Tr D 0 Φ † D ϑ Φ + Tr D ϑ Φ † D 0 Φ , g 2 g 2 1 2

Gauss constraint Gauss constraint for out-of-plane fields: � 1 A σ 0 + 1 = − 1 � Φ(Φ M − C Φ) † + (Φ M − C Φ)Φ † � D 2 C a σ a � i g 2 g 2 2 1 2 = − Φ M Φ † + 1 2 { C , ΦΦ † } , Physical quantities, like momentum in string axis direction and energy: � d 2 x Q , E = E BPS + where E BPS = 2 πξϕ (no. flux quanta) and � d 2 x Q T 03 = where Q is a current: ω 0 Q = ω 0 (Φ M − C Φ) M Φ † + Φ M (Φ M − C Φ) † � � 4 Tr , Angular momentum: T 0 ϑ = 1 F 0 ϑ F ϑ r + 1 G a 0 r G a ϑ r + Tr D 0 Φ † D ϑ Φ + Tr D ϑ Φ † D 0 Φ , g 2 g 2 1 2

An Ansatz for rotationally symmetric solutions in the plane � φ 1 ( r ) ψ 1 ( r ) e iN ϑ � Φ( x i ) = , ψ 2 ( r ) e iN ϑ φ 2 ( r ) A ϑ = a ( r ) , C a ϑ = c a ( r ) . Minimal Ansatz: φ i , ψ i real. c 2 = 0: consistency condition Diagonalizable: elementary or ( n 1 , n 2 ) vortex, V Φ D V † Coincident composite vortices (Shifman, Auzzi): N = − 1, flux 2 (Auzzi, Shifman, Yung) Parameter α : angle of ( φ 1 ( ∞ ) , φ 2 ( ∞ )) and ( 1 , 0 ) Small α ◮ φ 1 , ψ 2 , a , c 3 of unit magnitude ◮ c 1 , φ 2 , ψ 1 small

An Ansatz for rotationally symmetric solutions in the plane � φ 1 ( r ) ψ 1 ( r ) e iN ϑ � Φ( x i ) = , ψ 2 ( r ) e iN ϑ φ 2 ( r ) A ϑ = a ( r ) , C a ϑ = c a ( r ) . Minimal Ansatz: φ i , ψ i real. c 2 = 0: consistency condition Diagonalizable: elementary or ( n 1 , n 2 ) vortex, V Φ D V † Coincident composite vortices (Shifman, Auzzi): N = − 1, flux 2 (Auzzi, Shifman, Yung) Parameter α : angle of ( φ 1 ( ∞ ) , φ 2 ( ∞ )) and ( 1 , 0 ) Small α ◮ φ 1 , ψ 2 , a , c 3 of unit magnitude ◮ c 1 , φ 2 , ψ 1 small

An Ansatz for rotationally symmetric solutions in the plane � φ 1 ( r ) ψ 1 ( r ) e iN ϑ � Φ( x i ) = , ψ 2 ( r ) e iN ϑ φ 2 ( r ) A ϑ = a ( r ) , C a ϑ = c a ( r ) . Minimal Ansatz: φ i , ψ i real. c 2 = 0: consistency condition Diagonalizable: elementary or ( n 1 , n 2 ) vortex, V Φ D V † Coincident composite vortices (Shifman, Auzzi): N = − 1, flux 2 (Auzzi, Shifman, Yung) Parameter α : angle of ( φ 1 ( ∞ ) , φ 2 ( ∞ )) and ( 1 , 0 ) Small α ◮ φ 1 , ψ 2 , a , c 3 of unit magnitude ◮ c 1 , φ 2 , ψ 1 small

An Ansatz for rotationally symmetric solutions in the plane � φ 1 ( r ) ψ 1 ( r ) e iN ϑ � Φ( x i ) = , ψ 2 ( r ) e iN ϑ φ 2 ( r ) A ϑ = a ( r ) , C a ϑ = c a ( r ) . Minimal Ansatz: φ i , ψ i real. c 2 = 0: consistency condition Diagonalizable: elementary or ( n 1 , n 2 ) vortex, V Φ D V † Coincident composite vortices (Shifman, Auzzi): N = − 1, flux 2 (Auzzi, Shifman, Yung) Parameter α : angle of ( φ 1 ( ∞ ) , φ 2 ( ∞ )) and ( 1 , 0 ) Small α ◮ φ 1 , ψ 2 , a , c 3 of unit magnitude ◮ c 1 , φ 2 , ψ 1 small

Perturbative framework 1 1 0.8 0.5 0.6 0 0.4 -0.5 φ 1 (r) φ 1 (r) α -1 0.2 -1 ψ 1 (r) α -1 ψ 2 (r) 0 -1.5 -0.2 -2 a(r) c 1 (r) α -1 -0.4 -2.5 c 3 (r) -0.6 -3 0 2 4 6 8 10 0 2 4 6 8 10 r r Expansion in α : φ 1 = φ ( 0 ) + α 2 φ ( 2 ) ψ 1 = αψ ( 1 ) + . . . , + . . . , 1 1 1 φ 2 = αφ ( 1 ) ψ 2 = ψ ( 0 ) + α 2 ψ ( 2 ) + . . . , + . . . , 2 2 2 and a = a ( 0 ) + α 2 a ( 2 ) + . . . , c 1 = α c ( 1 ) + . . . . 1 c 3 = c ( 0 ) + α 2 c ( 2 ) + . . . , 3 3