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

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 086012 G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Magnetic field generated in heavy ion collisions ∼ m 2 π combined with: ◮ Axial Anomaly ⇒ C.M.E. (charge separation) C.M.W. (charge dependent v 2 ) ◮ Conformal Anomaly ⇒ photon v 2 (Ba¸ sar, Kharzeev, Skokov, arXiv:1206.1334) 0.16 0.14 0.12 0.1 v2 0.08 0.06 0.04 0.02 0 0 0.5 1 1.5 2 2.5 p ⊥ , GeV (talk by V. Skokov at xQCD, 08/22) G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Part I Instanton in a Magnetic Field G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Motivation & some lattice results Interplay between topology & magnetic field ◮ Chiral magnetic effect � J ∝ µ 5 � B ◮ What sources µ 5 ? 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¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Notation & conventions work in: ❘ 4 � 0 � � ✶ � α µ 0 chiral basis: γ µ = , γ 5 = α µ ¯ 0 0 − ✶ σ ) = α † α µ = ( ✶ , − i� σ ) , α µ = ( ✶ , i� ¯ µ � 0 � � � 0 α µ D µ D Dirac operator: / D = ≡ − D † α µ D µ ¯ 0 0 gauge field: A µ = A µ + a µ G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Notation & conventions � DD † � 0 � 2 ψ λ = ψ λ = λ 2 ψ λ � i/ diagonal form: D D † D 0 DD † = −D 2 χ = +1 : µ − F µν ¯ σ µν D † D = −D 2 χ = − 1 : µ − F µν σ µν ”supersymmetry:” for λ � = 0, DD † and D † D has identical spectra D . . . . . . D † χ = -1 χ = +1 G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Instanton & magnetic field DD † = −D 2 D † D = −D 2 µ − Bσ 3 , , µ − F µν σ µν − Bσ 3 Zero modes: Both spins, both chiralities � � � � F µν ˜ F µν ˜ Index thm: tr F µν = tr F µν G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Instanton & magnetic field DD † = −D 2 D † D = −D 2 µ − Bσ 3 , , µ − F µν σ µν − Bσ 3 Zero modes: Both spins, both chiralities � � � � F µν ˜ F µν ˜ Index thm: tr F µν = tr F µν ( F � = ˜ N + − N − � = − N − F ) G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Instanton & magnetic field DD † = −D 2 D † D = −D 2 µ − Bσ 3 , , µ − F µν σ µν − Bσ 3 Zero modes: Both spins, both chiralities � � � � F µν ˜ F µν ˜ Index thm: tr F µν = tr F µν ( F � = ˜ N + − N − � = − N − F ) Competition between instanton and magnetic field G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Instanton & magnetic field DD † = −D 2 D † D = −D 2 µ − Bσ 3 , , µ − F µν σ µν − Bσ 3 Zero modes: Both spins, both chiralities � � � � F µν ˜ F µν ˜ Index thm: tr F µν = tr F µν ( F � = ˜ N + − N − � = − N − F ) Competition between instanton and magnetic field ↓ ↓ try to align chiralities align spins G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Instanton zero mode: | ψ 0 | 2 = 64 ρ 2 ( x 2 + ρ 2 ) 3 192 ρ 4 Topological charge: q 5 ( x ) = ( x 2 + ρ 2 ) 4 G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Instanton zero mode: | ψ 0 | 2 = 64 ρ 2 ( x 2 + ρ 2 ) 3 192 ρ 4 Topological charge: q 5 ( x ) = ( x 2 + ρ 2 ) 4 B field zero mode: | ψ 0 | 2 ∝ f ( x 1 + ix 2 ) e − B | x 1 + ix 2 | 2 G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Instanton zero mode: | ψ 0 | 2 = 64 ρ 2 ( x 2 + ρ 2 ) 3 192 ρ 4 Topological charge: q 5 ( x ) = ( x 2 + ρ 2 ) 4 B field zero mode: | ψ 0 | 2 ∝ f ( x 1 + ix 2 ) e − B | x 1 + ix 2 | 2 B G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

✶ Large instanton limit 1 suppose: B << ρ √ G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

✶ Large instanton limit 1 suppose: B << ρ √ instanton is slowly varying → can do derivative expansion G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

✶ Large instanton limit 1 suppose: B << ρ √ instanton is slowly varying → can do derivative expansion η a µν x ν x 2 + ρ 2 ≈ 2 A a ρ 2 η a µ = 2 µν x ν + . . . G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Large instanton limit 1 suppose: B << ρ √ instanton is slowly varying → can do derivative expansion η a µν x ν x 2 + ρ 2 ≈ 2 A a ρ 2 η a µ = 2 µν x ν + . . . after appropriate gauge rotation & Lorentz transformation: 2 ( − x 2 , x 1 , − x 4 , x 3 ) τ 3 + B A µ = − F 2 ( − x 2 , x 1 , 0 , 0) ✶ 2 × 2 quasi-abelian, covariantly constant → soluble! G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Large instanton limit 2 ( − x 2 , x 1 , − x 4 , x 3 ) τ 3 + B A µ = − F 2 ( − x 2 , x 1 , 0 , 0) ✶ 2 × 2 � B − F � 0 F 12 = 0 B + F � − F 0 � F 34 = 0 F Landau problem with field strengths F 12 & F 34 G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Large instanton limit 2 ( − x 2 , x 1 , − x 4 , x 3 ) τ 3 + B A µ = − F 2 ( − x 2 , x 1 , 0 , 0) ✶ 2 × 2 � B − F � 0 F 12 = 0 B + F � − F 0 � F 34 = 0 F Landau problem with field strengths F 12 & F 34 G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Zero modes τ = − 1 , χ = − 1 , spin ↑ , n − = ( B + F ) F 2 π 2 π τ = +1 , χ = +1 , spin ↑ , n + = ( B − F ) F 2 π 2 π n + − n − = − F 2 n + + n − = B F , 2 π 2 2 π 2 F B+F x 1 x 3 x 2 x 4 -F B-F x 1 x 3 x 4 x 2 G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Dipole moments 2 � ¯ 3 = � ¯ 3 = 1 σ M σ E ψ Σ 12 ψ � , ψ Σ 34 ψ � G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Dipole moments 2 � ¯ 3 = � ¯ 3 = 1 σ M σ E ψ Σ 12 ψ � , ψ Σ 34 ψ � � σ 3 � � − σ 3 � 0 0 Σ 12 = , Σ 34 = 0 σ 3 0 σ 3 G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Dipole moments 2 � ¯ 3 = � ¯ 3 = 1 σ M σ E ψ Σ 12 ψ � , ψ Σ 34 ψ � � σ 3 � � − σ 3 � 0 0 Σ 12 = , Σ 34 = 0 σ 3 0 σ 3 m 2 m 2 � � � � m � ¯ ψ Σ 12 ψ � = tr 2 × 2 σ 3 + tr 2 × 2 σ 3 m 2 + DD † m 2 + D † D m 2 m 2 � � � � m � ¯ ψ Σ 34 ψ � = − tr 2 × 2 σ 3 + tr 2 × 2 σ 3 m 2 + DD † m 2 + D † D G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Dipole moments 2 � ¯ 3 = � ¯ σ M 3 = 1 σ E ψ Σ 12 ψ � , ψ Σ 34 ψ � � σ 3 � � − σ 3 � 0 0 Σ 12 = , Σ 34 = 0 σ 3 0 σ 3 BF m � ¯ ψ Σ 12 ψ � ≈ 2 π 2 F 2 m � ¯ ψ Σ 34 ψ � ≈ 2 π 2 � � F � ¯ ψ Σ 34 ψ ¯ ψ Σ 34 ψ � ≈ B 2 π 2 m 2 L 4 σ M 3 > σ E ◮ 3 G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Part II Sphaleron Rate in a Magnetic Field G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Sphaleron rate (basics) � g 2 Γ CS = (∆ Q 5 ) 2 a ( x ) g 2 � � d 4 x µν ˜ αβ ˜ 32 π 2 F a F µν 32 π 2 F a F αβ = a (0) V t dt = − c N 5 Γ CS Diffusion of topological charge: dN 5 T 3 ◮ CP odd effects in QCD (CME) ◮ Baryon number (B+L) violation in E.W. Weak coupling: Γ CS = κ g 4 T log(1 /g ) ( g 2 T ) 3 (B¨ odeker ’98) Strong coupling: Γ CS = ( g 2 N ) 2 256 π 3 T 4 (Son, Starinets ’02) G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Sphaleron rate with B field (holography) Gauge theory in magnetic field ⇔ Einstein-Maxwell theory Dynamics with magnetic field ⇔ Self-consistent solutions 2 r B >> T h r boundary AdS BTZ , T =T H AdS 5 R.G. flow IR UV CFT 1+1d CFT (temp=T) 3+1d CFT (N=4) G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Sphaleron rate with B field(holography) � � � � T^2 � � 1.6 1.5 1.4 1.3 1.2 1.1 � 5 10 15 20 T 2 ( g 2 N ) 2 � �  T 4 + 6 π 4 B 2 + O ( B 4 1 B << T 2 T 2 ) , 256 π 3    Γ CS = ( g 2 N ) 2 � �  B T 2 + 15 . 9 T 4 + O ( T 6 B >> T 2  √ B ) ,  3 π 5 384 G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

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 Γ CS for CME estimates G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field

Sphaleron rate with B field (holography) B >> T 2 ( g 2 N ) 2 B π × T 2 Γ CS = √ 384 3 π 4 G¨ ok¸ ce Ba¸ sar Instantons and Sphalerons in a Magnetic Field