Instantons and Sphalerons in a Magnetic Field G ok ce Ba sar - - PowerPoint PPT Presentation

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Instantons and Sphalerons in a Magnetic Field G ok ce Ba sar Stony Brook University 08/17/2012 Quark Matter 2012, Washington D.C. GB, G.Dunne & D. Kharzeev , arXiv:1112.0532, PRD 85 045026 GB, D. Kharzeev, arXiv:1202.2161, PRD 85

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SLIDE 1

Instantons and Sphalerons in a Magnetic Field

G¨

  • k¸

ce Ba¸ sar

Stony Brook University

08/17/2012 Quark Matter 2012, Washington D.C.

GB, G.Dunne & D. Kharzeev , arXiv:1112.0532, PRD 85 045026 GB, D. Kharzeev, arXiv:1202.2161, PRD 85 086012

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 2

Magnetic field generated in heavy ion collisions ∼ m2

π

combined with:

◮ Axial Anomaly ⇒ C.M.E. (charge separation)

C.M.W. (charge dependent v2)

◮ Conformal Anomaly ⇒ photon v2

(Ba¸ sar, Kharzeev, Skokov, arXiv:1206.1334)

0.5 1 1.5 2 2.5 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 p⊥, GeV v2

(talk by V. Skokov at xQCD, 08/22)

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 3

Part I Instanton in a Magnetic Field

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 4

Motivation & some lattice results

Interplay between topology & magnetic field

◮ Chiral magnetic effect

J ∝ µ5 B

◮ What sources µ5?

sphalerons, η domains, etc..

◮ Instanton + magnetic field ◮ Lattice results

◮ ITEP group (electric & dipole moments) ◮ T. Blum et al. (zero modes ∝ B) ◮ A. Yamamoto (C.M. conductivity) (Polikarpov et al. ’09) G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 5

Notation & conventions

work in: ❘4 chiral basis: γµ = αµ ¯ αµ

  • ,

γ5 = ✶ −✶

  • αµ = (✶, −i

σ) , ¯ αµ = (✶, i σ) = α†

µ

Dirac operator: / D =

  • αµDµ

¯ αµDµ

  • ≡

D −D†

  • gauge field:

Aµ = Aµ + aµ

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 6

Notation & conventions

diagonal form:

  • i/

D 2 ψλ = DD† D†D

  • ψλ = λ2ψλ

χ = +1 : DD† = −D2

µ − Fµν¯

σµν χ = −1 : D†D = −D2

µ − Fµνσµν

”supersymmetry:” for λ = 0, DD† and D†D has identical spectra . . . . . . D D†

χ = +1 χ = -1

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 7

Instanton & magnetic field

DD† = −D2

µ − Bσ3,

, D†D = −D2

µ − Fµνσµν − Bσ3

Zero modes: Both spins, both chiralities Index thm: tr

  • Fµν ˜

Fµν

  • = tr
  • Fµν ˜

Fµν

  • G¨
  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 8

Instanton & magnetic field

DD† = −D2

µ − Bσ3,

, D†D = −D2

µ − Fµνσµν − Bσ3

Zero modes: Both spins, both chiralities Index thm: tr

  • Fµν ˜

Fµν

  • = tr
  • Fµν ˜

Fµν

  • N+ − N− = −N−

(F = ˜ F)

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 9

Instanton & magnetic field

DD† = −D2

µ − Bσ3,

, D†D = −D2

µ − Fµνσµν − Bσ3

Zero modes: Both spins, both chiralities Index thm: tr

  • Fµν ˜

Fµν

  • = tr
  • Fµν ˜

Fµν

  • N+ − N− = −N−

(F = ˜ F) Competition between instanton and magnetic field

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 10

Instanton & magnetic field

DD† = −D2

µ − Bσ3,

, D†D = −D2

µ − Fµνσµν − Bσ3

Zero modes: Both spins, both chiralities Index thm: tr

  • Fµν ˜

Fµν

  • = tr
  • Fµν ˜

Fµν

  • N+ − N− = −N−

(F = ˜ F) Competition between instanton and magnetic field ↓ ↓ try to align chiralities align spins

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 11

Instanton zero mode: |ψ0|2 =

64 ρ2 (x2+ρ2)3

Topological charge: q5(x) =

192 ρ4 (x2+ρ2)4

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 12

Instanton zero mode: |ψ0|2 =

64 ρ2 (x2+ρ2)3

Topological charge: q5(x) =

192 ρ4 (x2+ρ2)4

B field zero mode: |ψ0|2 ∝ f(x1 + ix2)e−B|x1+ix2|2

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 13

Instanton zero mode: |ψ0|2 =

64 ρ2 (x2+ρ2)3

Topological charge: q5(x) =

192 ρ4 (x2+ρ2)4

B field zero mode: |ψ0|2 ∝ f(x1 + ix2)e−B|x1+ix2|2

B

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 14

Large instanton limit

suppose:

1 √ B << ρ

✶

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 15

Large instanton limit

suppose:

1 √ B << ρ

instanton is slowly varying → can do derivative expansion ✶

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 16

Large instanton limit

suppose:

1 √ B << ρ

instanton is slowly varying → can do derivative expansion Aa

µ = 2 ηa

µν xν

x2+ρ2 ≈ 2 ρ2 ηa µνxν + . . .

✶

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 17

Large instanton limit

suppose:

1 √ B << ρ

instanton is slowly varying → can do derivative expansion Aa

µ = 2 ηa

µν xν

x2+ρ2 ≈ 2 ρ2 ηa µνxν + . . .

after appropriate gauge rotation & Lorentz transformation: Aµ = −F

2 (−x2, x1, −x4, x3)τ 3 + B 2 (−x2, x1, 0, 0)✶2×2

quasi-abelian, covariantly constant → soluble!

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 18

Large instanton limit

Aµ = −F

2 (−x2, x1, −x4, x3)τ 3 + B 2 (−x2, x1, 0, 0)✶2×2

F12 = B − F B + F

  • F34 =

−F F

  • Landau problem with field strengths F12 & F34

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 19

Large instanton limit

Aµ = −F

2 (−x2, x1, −x4, x3)τ 3 + B 2 (−x2, x1, 0, 0)✶2×2

F12 = B − F B + F

  • F34 =

−F F

  • Landau problem with field strengths F12 & F34

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 20

Zero modes

τ = −1 , χ = −1 , spin ↑ , n− = (B+F)

2π F 2π

τ = +1 , χ = +1 , spin ↑ , n+ = (B−F)

2π F 2π

n+ + n− = B F

2π2

, n+ − n− = − F 2

2π2 F x4 x3

  • F

x4 x3 B+F x1 x2 B-F x1 x2

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 21

Dipole moments

σM

3 = 1 2 ¯

ψΣ12ψ , σE

3 = ¯

ψΣ34ψ

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 22

Dipole moments

σM

3 = 1 2 ¯

ψΣ12ψ , σE

3 = ¯

ψΣ34ψ Σ12 = σ3 σ3

  • ,

Σ34 = −σ3 σ3

  • G¨
  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 23

Dipole moments

σM

3 = 1 2 ¯

ψΣ12ψ , σE

3 = ¯

ψΣ34ψ Σ12 = σ3 σ3

  • ,

Σ34 = −σ3 σ3

  • m ¯

ψΣ12ψ = tr2×2

  • σ3

m2 m2 + DD†

  • + tr2×2
  • σ3

m2 m2 + D†D

  • m ¯

ψΣ34ψ = −tr2×2

  • σ3

m2 m2 + DD†

  • + tr2×2
  • σ3

m2 m2 + D†D

  • G¨
  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 24

Dipole moments

σM

3 = 1 2 ¯

ψΣ12ψ , σE

3 = ¯

ψΣ34ψ Σ12 = σ3 σ3

  • ,

Σ34 = −σ3 σ3

  • m ¯

ψΣ12ψ ≈ BF 2π2 m ¯ ψΣ34ψ ≈ F 2 2π2 ¯ ψΣ34ψ ¯ ψΣ34ψ ≈

  • F

2π2m2L4

  • B

◮

σM

3 > σE 3

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 25

Part II Sphaleron Rate in a Magnetic Field

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 26

Sphaleron rate (basics)

ΓCS = (∆Q5)2 V t =

  • d4x

g2 32π2 F a

µν ˜

F µν

a (x) g2

32π2 F a

αβ ˜

F αβ

a (0)

  • Diffusion of topological charge: dN5

dt = −c N5 ΓCS T 3 ◮ CP odd effects in QCD (CME) ◮ Baryon number (B+L) violation in E.W.

Weak coupling: ΓCS = κ g4T log(1/g) (g2T)3 (B¨

  • deker ’98)

Strong coupling: ΓCS = (g2N)2

256π3 T 4

(Son, Starinets ’02)

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 27

Sphaleron rate with B field (holography)

Gauge theory in magnetic field ⇔ Einstein-Maxwell theory Dynamics with magnetic field ⇔ Self-consistent solutions

r r AdS CFT IR UV 1+1d CFT (temp=T) 3+1d CFT (N=4) AdS5

boundary

B >> T

R.G. flow

BTZ , T =T

h H 2

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 28

Sphaleron rate with B field(holography)

5 10 15 20

  • T2

1.1 1.2 1.3 1.4 1.5 1.6

  • T^2

ΓCS =       

(g2N)2 256π3

  • T 4 +

1 6π4 B2 + O(B4 T 2 )

  • ,

B << T 2

(g2N)2 384 √ 3π5

  • B T 2 + 15.9 T 4 + O(T 6

B )

  • ,

B >> T 2

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 29

Sphaleron rate with B field (holography) B ∼ T 2

◮ for B = T 2 the effect is ∼ %0.2 ◮ it is safe to ignore the effects of B field on ΓCS

for CME estimates

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 30

Sphaleron rate with B field (holography) B >> T 2 ΓCS =

(g2N)2 384 √ 3 π4 B π × T 2

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 31

Sphaleron rate with B field (holography) B >> T 2

Landau level density ↑

ΓCS =

(g2N)2 384 √ 3 π4 B π × T 2

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 32

Sphaleron rate with B field (holography) B >> T 2

Landau level density ↑

ΓCS =

(g2N)2 384 √ 3 π4 B π × T 2

↓

diffusion scale in 1+1d

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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SLIDE 33

Conclusions

◮ Anomalies + magnetic field have a rich structure ◮ Electric and magnetic dipole moments ◮ Zero modes play a crucial role ◮ 1st order derivative expansion captures some lattice results ◮ Confinement ? (instantons with nonzero holonomy), CSB ? ◮ At strong coupling:

◮ Magnetic field always increases the sphaleron rate ◮ Back-reaction of magnetic field into non-abelian sector ◮ Strong magnetic field leads to dimensional reduction

◮ Weak coupling?

G¨

  • k¸

ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field