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. S =

• d4xL,

L = − 1 4g2

1

FµνF µν − 1 4g2

2

Ga

µνGµνa + Tr(DµΦ)†DµΦ − (V1 + V2) ,

where DµΦ = (∂µ − iAµσ0/2 − iCa

µσa/2)Φ

Potential V1 = λ1 8 (Tr Φ†Φ − 2ξ)2 , V2 = λ2 8 (Tr Φ†σaΦ)2 Properties of this theory:

◮ scalar sector of a supersymmetric theory ◮ possesses many localized solutions (strings, etc.)

Spontaneous symmetry breaking

Let Φ =

• φ1

ψ1 φ2 ψ2

• with this notation:

V1 = λ1 8 (φ†φ + ψ†ψ − 2ξ)2 , V2 = λ2 8

• (φ†φ − ψ†φ)2 + 4|ψ†φ|2

, 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 → VΦV † Color-flavor locking: gauge and color symmetry both broken spontaneously, SU(2)CF remains unbroken Topology permits vortex solutions

Some vortex solutions

An (n1, n2) vortex: Φ = φ1(r)ein1ϑ ψ2(r)ein2ϑ

• ,

Aϑ = a(r) C3

ϑ = c3(r)

with real radial functions Radial equation for φ1, ψ2, a, c3 solved numerically Further solutions generated: orientational normal modes Φ → VΦV † , V ∈ SU(2) explicitly: Φ = χ+✶ + χ−naσa , ca = na˜ c3 where χ± = (φ1D + φ2D)/2.

(Shifman etal., Auzzi etal.)

Some vortex solutions

An (n1, n2) vortex: Φ = φ1(r)ein1ϑ ψ2(r)ein2ϑ

• ,

Aϑ = a(r) C3

ϑ = c3(r)

with real radial functions Radial equation for φ1, ψ2, a, c3 solved numerically Further solutions generated: orientational normal modes Φ → VΦV † , V ∈ SU(2) explicitly: Φ = χ+✶ + χ−naσa , ca = na˜ c3 where χ± = (φ1D + φ2D)/2.

(Shifman etal., Auzzi etal.)

BPS

Energy as sum-of-squares if λi = g2

i (fixed by SUSY):

EBPS = 1 4g2

1

• Fik ± g2

1

2 ǫik(Tr Φ†σ0Φ − 2ξ) 2 + 1 4g2

2

• Ga

ik ± g2 2

2 ǫik Tr Φ†σaΦ 2 + 1 2 Tr(DiΦ ± iǫikDkΦ)†(DiΦ ± iǫimDmΦ) ± ξ 4Fikǫik ∓ ǫik∂i Tr(Φ†DkΦ) . minimal energy: all squares vanish: Fik = ∓g2

1

2 ǫik(Tr Φ†σ0Φ − 2ξ) , Ga

ik = ∓g2 2

2 ǫik Tr Φ†σaΦ , DiΦ = ∓iǫikDkΦ , Energy: EBPS =

• d2xEBPs = 2πξϕ

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)−1Φ0(x + iy)

Φ0 holomorphic given, zeros of its determinant: position of vortices

◮ S = S1S2, Ωi = SiS† i , Ω1 = exp(ψ) ◮ one equation for Ω = SS†, reduced to a holomorphic splitting

problem ∂¯

z(Ω2∂zΩ−1 2 ) = −g2 2

4

• Φ0Φ†

0Ω−1 2

− ✶ N Tr Φ0Φ†

0Ω−1 2

• e−ψ ,

∂¯

z∂zψ = − g2 1

4N

• Tr(Φ0Φ†

0Ω−1 2 )e−ψ − Nξ

• .

Also for more general gauge groups

Adding twist

Straight string: translation invariance along axis z: Φ(xµ) = Φ(xi) exp i 2Mωαxα

• ,

Aµ(xν) = (Ai(xj), Aα(xj)) , Ca

µ(xν) = (Ca i (xj), Ca α(xj)) ,

Decoupling

ω2 = −ωαωα = 0 ensures that the equations for Φ(xi), Ca

i , Ai are unchanged (i = 1, 2)

Aα = ωαA , Ca

α = ωαCa

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: Φ(xµ) = Φ(xi) exp i 2Mωαxα

• ,

Aµ(xν) = (Ai(xj), Aα(xj)) , Ca

µ(xν) = (Ca i (xj), Ca α(xj)) ,

Decoupling

ω2 = −ωαωα = 0 ensures that the equations for Φ(xi), Ca

i , Ai are unchanged (i = 1, 2)

Aα = ωαA , Ca

α = ωαCa

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: D2

i

1 g2

1

Aσ0 + 1 g2

2

Caσa

• = −1

2

• Φ(ΦM − CΦ)† + (ΦM − CΦ)Φ†

= −ΦMΦ† + 1 2{C, ΦΦ†} , Physical quantities, like momentum in string axis direction and energy: E = EBPS +

• d2xQ ,

where EBPS = 2πξϕ (no. flux quanta) and T03 =

• d2xQ

where Q is a current: ω0Q = ω0 4 Tr

• (ΦM − CΦ)MΦ† + ΦM(ΦM − CΦ)†

, Angular momentum: T0ϑ = 1 g2

1

F0ϑFϑr + 1 g2

2

Ga

0rGa ϑr + Tr D0Φ†DϑΦ + Tr DϑΦ†D0Φ ,

Gauss constraint

Gauss constraint for out-of-plane fields: D2

i

1 g2

1

Aσ0 + 1 g2

2

Caσa

• = −1

2

• Φ(ΦM − CΦ)† + (ΦM − CΦ)Φ†

= −ΦMΦ† + 1 2{C, ΦΦ†} , Physical quantities, like momentum in string axis direction and energy: E = EBPS +

• d2xQ ,

where EBPS = 2πξϕ (no. flux quanta) and T03 =

• d2xQ

where Q is a current: ω0Q = ω0 4 Tr

• (ΦM − CΦ)MΦ† + ΦM(ΦM − CΦ)†

, Angular momentum: T0ϑ = 1 g2

1

F0ϑFϑr + 1 g2

2

Ga

0rGa ϑr + Tr D0Φ†DϑΦ + Tr DϑΦ†D0Φ ,

An Ansatz for rotationally symmetric solutions in the plane Φ(xi) = φ1(r) ψ1(r)eiNϑ φ2(r) ψ2(r)eiNϑ

• ,

Aϑ = a(r) , Ca

ϑ = ca(r) .

Minimal Ansatz:φi, ψi real. c2 = 0: consistency condition Diagonalizable: elementary or (n1, n2) vortex, VΦDV † Coincident composite vortices (Shifman, Auzzi): N = −1, flux 2

(Auzzi, Shifman, Yung)

Parameter α: angle of (φ1(∞), φ2(∞)) and (1, 0)

Small α

◮ φ1, ψ2, a, c3 of unit magnitude ◮ c1, φ2, ψ1 small

An Ansatz for rotationally symmetric solutions in the plane Φ(xi) = φ1(r) ψ1(r)eiNϑ φ2(r) ψ2(r)eiNϑ

• ,

Aϑ = a(r) , Ca

ϑ = ca(r) .

Minimal Ansatz:φi, ψi real. c2 = 0: consistency condition Diagonalizable: elementary or (n1, n2) vortex, VΦDV † Coincident composite vortices (Shifman, Auzzi): N = −1, flux 2

(Auzzi, Shifman, Yung)

Parameter α: angle of (φ1(∞), φ2(∞)) and (1, 0)

Small α

◮ φ1, ψ2, a, c3 of unit magnitude ◮ c1, φ2, ψ1 small

An Ansatz for rotationally symmetric solutions in the plane Φ(xi) = φ1(r) ψ1(r)eiNϑ φ2(r) ψ2(r)eiNϑ

• ,

Aϑ = a(r) , Ca

ϑ = ca(r) .

Minimal Ansatz:φi, ψi real. c2 = 0: consistency condition Diagonalizable: elementary or (n1, n2) vortex, VΦDV † Coincident composite vortices (Shifman, Auzzi): N = −1, flux 2

(Auzzi, Shifman, Yung)

Parameter α: angle of (φ1(∞), φ2(∞)) and (1, 0)

Small α

◮ φ1, ψ2, a, c3 of unit magnitude ◮ c1, φ2, ψ1 small

An Ansatz for rotationally symmetric solutions in the plane Φ(xi) = φ1(r) ψ1(r)eiNϑ φ2(r) ψ2(r)eiNϑ

• ,

Aϑ = a(r) , Ca

ϑ = ca(r) .

Minimal Ansatz:φi, ψi real. c2 = 0: consistency condition Diagonalizable: elementary or (n1, n2) vortex, VΦDV † Coincident composite vortices (Shifman, Auzzi): N = −1, flux 2

(Auzzi, Shifman, Yung)

Parameter α: angle of (φ1(∞), φ2(∞)) and (1, 0)

Small α

◮ φ1, ψ2, a, c3 of unit magnitude ◮ c1, φ2, ψ1 small

Perturbative framework

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 2 4 6 8 10 r φ1(r) φ1(r) α-1 ψ1(r)α-1 ψ2(r)

• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

0.5 1 2 4 6 8 10 r a(r) c1(r)α-1 c3(r)

Expansion in α: φ1 = φ(0)

1

+ α2φ(2)

1

+ . . . , φ2 = αφ(1)

2

+ . . . , ψ1 = αψ(1)

1

+ . . . , ψ2 = ψ(0)

2

+ α2ψ(2)

2

+ . . . , and a = a(0) + α2a(2) + . . . , c3 = c(0)

3

+ α2c(2)

3

+ . . . , c1 = αc(1)

1

+ . . . .

Twisting the elementary vortex

Reminder: elementary vortex: Φ = VΦDV †: Φ = χ+✶ + χ−naσa , ca = na˜ c3 where χ± = (φ1D + φ2D)/2. Ansatz for out-of-plane fields: let M = m0σ0 + maσa, A = m0 , Ca = (mn)na + ˜ maΣ , ˜ ma = ma − (mn)na , yielding one equation 1 r (rΣ′)′ − c2

3

r 2 Σ = g2

2

• χ2

+(Σ − 1) + χ2 −(Σ + 1)

• .

Note: m0 and parallel part to na: pure gauge ω0Q =

• m2 − (nm)2

Σ(χ2

− − χ2 +) + (χ2 − + χ2 +)

• ,

(see also Collie, Eto etal.)

Twisting the elementary vortex

Reminder: elementary vortex: Φ = VΦDV †: Φ = χ+✶ + χ−naσa , ca = na˜ c3 where χ± = (φ1D + φ2D)/2. Ansatz for out-of-plane fields: let M = m0σ0 + maσa, A = m0 , Ca = (mn)na + ˜ maΣ , ˜ ma = ma − (mn)na , yielding one equation 1 r (rΣ′)′ − c2

3

r 2 Σ = g2

2

• χ2

+(Σ − 1) + χ2 −(Σ + 1)

• .

Note: m0 and parallel part to na: pure gauge ω0Q =

• m2 − (nm)2

Σ(χ2

− − χ2 +) + (χ2 − + χ2 +)

• ,

(see also Collie, Eto etal.)

Twisting the elementary vortex

Reminder: elementary vortex: Φ = VΦDV †: Φ = χ+✶ + χ−naσa , ca = na˜ c3 where χ± = (φ1D + φ2D)/2. Ansatz for out-of-plane fields: let M = m0σ0 + maσa, A = m0 , Ca = (mn)na + ˜ maΣ , ˜ ma = ma − (mn)na , yielding one equation 1 r (rΣ′)′ − c2

3

r 2 Σ = g2

2

• χ2

+(Σ − 1) + χ2 −(Σ + 1)

• .

Note: m0 and parallel part to na: pure gauge ω0Q =

• m2 − (nm)2

Σ(χ2

− − χ2 +) + (χ2 − + χ2 +)

• ,

(see also Collie, Eto etal.)

Twisting the coincident composite vortex

Gauss constraints: ∇2A = −g2

1J0 3 ,

D2

i Ca = −g2 2Ja 3 ,

Two cases considered: m0 = −m3 = ω/2 and m1 = ω Simplest case: m0 = −m3 = ω/2 A = A(r), Ca = Ca(r), C2 = 0 Q = −ω 2

• (A − ω)(ψ2

1 + ψ2 2) + 2C1ψ1ψ2 + C3(ψ2 1 − ψ2 2)

• .

◮ For small α: energy difference very small

E = EBPS +

• d2xQ ,

EBPS = 4πξ

◮ α = 0: bgr. agrees with elementary, m0, m3 twist pure gauge

Twisting the coincident composite vortex

Gauss constraints: ∇2A = −g2

1J0 3 ,

D2

i Ca = −g2 2Ja 3 ,

Two cases considered: m0 = −m3 = ω/2 and m1 = ω Simplest case: m0 = −m3 = ω/2 A = A(r), Ca = Ca(r), C2 = 0 Q = −ω 2

• (A − ω)(ψ2

1 + ψ2 2) + 2C1ψ1ψ2 + C3(ψ2 1 − ψ2 2)

• .

◮ For small α: energy difference very small

E = EBPS +

• d2xQ ,

EBPS = 4πξ

◮ α = 0: bgr. agrees with elementary, m0, m3 twist pure gauge

Twisting the coincident composite vortex

The other case: m1 = ω A = 0, C1,3 = ˆ C1,3 cos ϑ, C2 = ˆ C2 sin ϑ Q ω = 1 2

• ω(φ2

1 + φ2 2 + ψ2 1 + ψ2 2) − (φ1ψ2 + φ2ψ1)ˆ

C1 +(φ1ψ2 − φ2ψ1)ˆ C2 − (φ1ψ1 − φ2ψ2)ˆ C3 ,

◮ energy difference O(1) (for small α too)

E = EBPS +

• d2xQ ,

EBPS = 4πξ

◮ α = 0: bgr. agrees with elementary, m0, m3 twist pure gauge,

(m1)2 + (m2)2 in Q

Symmetries

Rotational symmetry of the static solution Φ(r, ϑ) = exp

• i N

2 ϕ

• Φ(r, ϑ − ϕ) exp
• −i N

2 ϕσ3

• ,

i.e., rotation compensated by internal symmetries Twisted solution: Φ(xµ) = Φ(xi) exp i 2ωαxαM

• Also rotationally symmetric if [M, σ3] = 0

m0, m3-twisted vortex more symmetric, lower energy

Stability

◮ static BPS solution: minimal energy ◮ twisted vortex carries conserved charge

it is also localised to the core

Numerical solutions: m0 = −m3 = ω/2

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 2 4 6 8 10 r φ1(r) φ1(r) α-1 ψ1(r)α-1 ψ2(r)

• 3
• 2.5
• 2
• 1.5
• 1
• 0.5

0.5 1 2 4 6 8 10 r a(r) c1(r)α-1 c3(r) 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 1 2 3 4 5 6 7 8 r C1(r)/ω+sin(α)/2

• J1/ω
• 0.0007
• 0.0006
• 0.0005
• 0.0004
• 0.0003
• 0.0002
• 0.0001

0.0001 0.0002 0.0003 0.0004 1 2 3 4 5 6 7 8 r C0(r)/ω-1/2 C3(r)/ω+cos(α)/2

• J0/ω
• J3/ω

Numerical solutions: m0 = −m3 = ω/2 and m1 = ω

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 1 2 3 4 5 6 7 8 r C1(r)/ω+sin(α)/2

• J1/ω
• 0.0007
• 0.0006
• 0.0005
• 0.0004
• 0.0003
• 0.0002
• 0.0001

0.0001 0.0002 0.0003 0.0004 1 2 3 4 5 6 7 8 r C0(r)/ω-1/2 C3(r)/ω+cos(α)/2

• J0/ω
• J3/ω
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 r C1(r)/ ω -cos α C2(r)/ω + 1

• J1/ω
• J2/ω

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 1 2 3 4 5 6 7 8 r C3(r)/ω + sin α

• J3/ω

Energy and angular momentum density

• 5e-05

5e-05 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 1 2 3 4 5 6 7 8 r Q/ω2 T0ϑ/ω

◮ ω−2

d2xQ = 0.0124

◮ ω−2

d2xT0ϑ = −0.00117

• 0.05

0.05 0.1 0.15 0.2 0.25 1 2 3 4 5 6 7 8 r Q/ω2 T0ϑ/ω/cosθ

◮ ω−2

d2xQ = 12.4

◮ No net angular momentum

Note: Energy of static (BPS) vortex 2πξϕ, here ϕ = 2 A remark: m1 case: constraint for A = 0, consistent, reduces Gauss constraints to quadrature

Conclusions

◮ Static vortices well known

◮ in the Abelian Higgs model ◮ in non-Abelian gauge theories ◮ BPS case (SUSY)

◮ With multiple fields, twisted strings also possible ◮ Known for the elementary vortex ◮ Interesting properties for the composite coincident vortex

◮ Energy difference sometimes very small ◮ Momentum in the direction of the string axis ◮ Rotating and no net angular momentum case

THANK YOU FOR YOUR ATTENTION!

References

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(vortex solutions, moduli; also references therein)

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the Higgs phase: the moduli matrix approach J. Phys. A: Math.

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(moduli matrix approach; see also references therein)

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085404

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