Instantons and Sphalerons in a Magnetic Field

Transcription

1 Stony Brook University 06/27/2012 GB, G.Dunne & D. Kharzeev, arxiv: , PRD GB, D. Kharzeev, arxiv: , PRD

2 Outline Motivation & some lattice results General facts on Dirac operator Large instanton limit & dipole moments Sphaleron rate at strong coupling

3 Motivation & some lattice results Interplay between topology & magnetic field Chiral magnetic effect J = q B µ 5 2π π What sources µ 5? Instanton + magnetic field Lattice results sphalerons, η domains, etc.. ITEP group (electric & dipole moments) T. Blum et al. (zero modes B) A. Yamamoto (C.M. conductivity) (Polikarpov et al. 09)

4 Notation & conventions work in: R 4 (T 4 ) chiral basis: γ µ = ( 0 ) αµ ᾱ µ 0, γ 5 = ( ) Dirac operator: /D = α µ = (1, i σ), ᾱ µ = (1, i σ) = α µ ( ) 0 αµ D µ ᾱ µ D µ 0 ( 0 ) D D 0 gauge field: A µ = A µ + a µ

5 Notation & conventions diagonal form: ( i /D ) 2 ψλ = ( ) DD 0 0 D ψ D λ = λ 2 ψ λ χ = +1 : DD = Dµ 2 F µν σ µν χ = 1 : D D = Dµ 2 F µν σ µν supersymmetry: for λ 0, DD and D D has identical spectra. D. χ = -1 D χ = +1

6 magnetic field: DD = D D = D 2 µ Bσ 3 Zero modes: Positive spin, both chiralities

7 magnetic field: DD = D D = D 2 µ Bσ 3 Zero modes: Positive spin, both chiralities BPST instanton: DD = D 2 µ, D D = D 2 µ F µν σ µν Zero modes: Both spins, definite chirality Index theorem: N + N = N = 1 32π 2 d 4 x F a µν F a µν

8 Instanton & magnetic field DD = D 2 µ Bσ 3,, D D = D 2 µ F µν σ µν Bσ 3

9 Instanton & magnetic field DD = D 2 µ Bσ 3,, D D = D 2 µ F µν σ µν Bσ 3 Zero modes: Both spins, both chiralities ) ) Index thm: tr (F µν Fµν = tr (F µν Fµν + (dim) f µν fµν

10 Instanton & magnetic field DD = D 2 µ Bσ 3,, D D = D 2 µ F µν σ µν Bσ 3 Zero modes: Both spins, both chiralities ) ) Index thm: tr (F µν Fµν = tr (F µν Fµν

11 Instanton & magnetic field DD = D 2 µ Bσ 3,, D D = D 2 µ F µν σ µν Bσ 3 Zero modes: Both spins, both chiralities ) ) Index thm: tr (F µν Fµν = tr (F µν Fµν N + N N (F F)

12 Instanton & magnetic field DD = D 2 µ Bσ 3,, D D = D 2 µ F µν σ µν Bσ 3 Zero modes: Both spins, both chiralities ) ) Index thm: tr (F µν Fµν = tr (F µν Fµν N + N N (F F) Competition between instanton and magnetic field

13 Instanton & magnetic field DD = D 2 µ Bσ 3,, D D = D 2 µ F µν σ µν Bσ 3 Zero modes: Both spins, both chiralities ) ) Index thm: tr (F µν Fµν = tr (F µν Fµν N + N N (F F) Competition between instanton and magnetic field try to align chiralities align spins

14 Instanton zero mode: ψ 0 2 = Topological charge: q 5 (x) = 64 ρ2 (x 2 +ρ 2 ) ρ4 (x 2 +ρ 2 ) 4

15 Instanton zero mode: ψ 0 2 = Topological charge: q 5 (x) = 64 ρ2 (x 2 +ρ 2 ) ρ4 (x 2 +ρ 2 ) 4 B field zero mode: ψ 0 2 f(x 1 + ix 2 )e B x 1+ix 2 2

16 Instanton zero mode: ψ 0 2 = Topological charge: q 5 (x) = 64 ρ2 (x 2 +ρ 2 ) ρ4 (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

17 Large instanton limit suppose: 1 B << ρ

18 Large instanton limit suppose: 1 B << ρ instanton is slowly varying can do derivative expansion

19 Large instanton limit suppose: 1 B << ρ instanton is slowly varying can do derivative expansion A a µ = 2 ηa µν x ν x 2 +ρ 2 2 ρ 2 η a µνx ν +...

20 Large instanton limit suppose: 1 B << ρ instanton is slowly varying can do derivative expansion A a µ = 2 ηa µν x ν x 2 +ρ 2 2 ρ 2 η a µνx ν +... after appropriate gauge rotation & Lorentz transformation: A µ = F 2 ( x 2, x 1, x 4, x 3 )τ 3 + B 2 ( x 2, x 1, 0, 0)12 2 quasi-abelian, covariantly constant soluble!

21 Large instanton limit A µ = F 2 ( x 2, x 1, x 4, x 3 )τ 3 + B 2 ( x 2, x 1, 0, 0)12 2 F 12 = ( B F 0 ) 0 B + F F 34 = ( ) F 0 0 F Landau problem with field strengths F 12 & F 34

22 Large instanton limit A µ = F 2 ( x 2, x 1, x 4, x 3 )τ 3 + B 2 ( x 2, x 1, 0, 0)12 2 F 12 = ( B F 0 ) 0 B + F F 34 = ( ) F 0 0 F Landau problem with field strengths F 12 & F 34 Topological charge: 1 32π 2 F a µν F a µν = F 2 2π 2

23 Zero modes τ = 1, χ = 1, spin, n = (B+F ) 2π τ = +1, χ = +1, spin, n + = (B F ) 2π F 2π F 2π n + + n = B F 2π 2, n + n = F 2 2π 2 F B+F x 3 x 1 x 4 x 2 -F B-F x 3 x 1 x 4 x 2

24 Zero modes B < F χ = +1 : n + = 0 χ = 1 : n = { (B+F ) F 2π ( B+F ) 2π 2π, (τ 3 = 1, spin ) F 2π, (τ 3 = +1, spin ) n + + n = n = F 2 2π 2, n + n = n = F 2 2π 2 n n = BF 2π 2

25 Quantization on T 4 twisted boundary conditions fractional Pontryagin index ( t Hooft 81) zero twist quasi-abelian, covariantly constant, SD solutions (van Baal 96) M: Instanton flux N:Magnetic flux Zero modes: B > F index: N + N = 2M 2 = F 2 L 4 2π 2 total number: N + + N = 2MN = B F L4 2π 2 B < F index: N + N = 2M 2 = F 2 L 4 2π 2 total number: N + + N = 2M 2 = F 2 L 4 2π 2

26 Dipole moments σ M i = 1 2 ɛ ijk ψσ jk ψ, σ E i = ψσ i4 ψ

27 Dipole moments σ M 3 = 1 2 ψσ 12 ψ, σ E 3 = ψσ 34 ψ

28 Dipole moments σ3 M = 1 2 ψσ 12 ψ, σ3 E = ψσ 34 ψ ( ) ( ) σ3 0 σ3 0 Σ 12 =, Σ 0 σ 34 = 3 0 σ 3

29 Dipole moments σ3 M = 1 2 ψσ 12 ψ, σ3 E = ψσ 34 ψ ( ) ( ) σ3 0 σ3 0 Σ 12 =, Σ 0 σ 34 = 3 0 σ 3 ( m ψσ m 2 12 ψ = tr 2 2 σ 3 m 2 + DD ( m ψσ m 2 34 ψ = tr 2 2 σ 3 m 2 + DD ) + tr 2 2 (σ 3 m 2 m 2 + D D ) m + tr 2 2 (σ 2 3 ) m 2 + D D )

30 Dipole moments σ3 M = 1 2 ψσ 12 ψ, σ3 E = ψσ 34 ψ ( ) ( ) σ3 0 σ3 0 Σ 12 =, Σ 0 σ 34 = 3 0 σ 3 ( m ψσ m 2 ) ( m 2 ) 12 ψ tr 2 2 m 2 + DD + tr 2 2 m 2 + D D ( m ψσ m 2 ) ( m 2 ) 34 ψ tr 2 2 m 2 + DD + tr 2 2 m 2 + D D

31 Dipole moments σ3 M = 1 2 ψσ 12 ψ, σ3 E = ψσ 34 ψ ( ) ( ) σ3 0 σ3 0 Σ 12 =, Σ 0 σ 34 = 3 0 σ 3 ( ) ( ) m ψσ B F F 12 ψ 2π 2π ( ) ( m ψσ B F F 34 ψ 2π 2π ( B + F + 2π ( B + F ) + 2π ) ( ) F 2π ) ( ) F 2π

32 Dipole moments σ3 M = 1 2 ψσ 12 ψ, σ3 E = ψσ 34 ψ ( ) ( ) σ3 0 σ3 0 Σ 12 =, Σ 0 σ 34 = 3 0 σ 3 m ψσ 12 ψ BF 2π 2 m ψσ 34 ψ F 2 2π 2

33 Dipole moments σ3 M = 1 2 ψσ 12 ψ, σ3 E = ψσ 34 ψ ( ) ( ) σ3 0 σ3 0 Σ 12 =, Σ 0 σ 34 = 3 0 σ 3 m ψσ 12 ψ BF 2π 2 m ψσ 34 ψ F 2 2π 2 σ M 3 > σ E 3

34 Dipole moments σ3 M = 1 2 ψσ 12 ψ, σ3 E = ψσ 34 ψ ( ) ( ) σ3 0 σ3 0 Σ 12 =, Σ 0 σ 34 = 3 0 σ 3 m ψσ 12 ψ BF 2π 2 m ψσ 34 ψ F 2 2π 2 σ M 3 > σ E 3 ψσ 34 ψ ψσ 34 ψ ( F 2π 2 m 2 L 4 ) B

35 Sphaleron rate (basics) Γ CS = ( Q 5) 2 V t = g d 4 2 x 32π 2 F µν a µν F a (x) g2 32π 2 F αβ a αβ F a (0) Diffusion of topological charge: dn 5 dt CP odd effects in QCD (CME) = c N 5 Γ CS T 3 Baryon number (B+L) violation in E.W. Weak coupling: Γ CS = κ g 4 T log(1/g) (g 2 T ) 3 (Bödeker 98) Strong coupling: Γ CS = (g2 N) 2 256π 3 T 4 (Son, Starinets 02)

36 Sphaleron rate (holography) 5-d Einstein-Maxwell-(Chern Simons): S = 1 16πG 5 d 5 x g(r+f MN F MN 12 l 2 ) πG 5 A F F F = Bdx 1 dx 2 solutions of the form: (D Hoker, Kraus 08-11) ds 2 = U(r)dt 2 + dr2 U(r) + e2v (r) (dx dx 2 2) + e 2W (r) dx 2 3 with: U(r h ) = 0, e 2V (r), e 2W (r) r r 2 (AdS 5 ), AdS CFT r=0 AdS 3 IR B, T=0 R.G. flow 1+1d CFT (magnetic catalysis) 3+1d CFT (N=4) UV r boundary AdS 5

37 Sphaleron rate (holography) 5-d Einstein-Maxwell-(Chern Simons): S = 1 16πG 5 d 5 x g(r+f MN F MN 12 l 2 ) πG 5 A F F F = Bdx 1 dx 2 solutions of the form: (D Hoker, Kraus 08-11) ds 2 = U(r)dt 2 + dr2 U(r) + e2v (r) (dx dx 2 2) + e 2W (r) dx 2 3 with: U(r h ) = 0, e 2V (r), e 2W (r) r r 2 (AdS 5 ), AdS CFT r h BTZ, T =T IR H B >> T R.G. flow 1+1d CFT (temp=t) 3+1d CFT (N=4) 2 UV r boundary AdS 5

38 Sphaleron rate (holography) 5-d Einstein-Maxwell-(Chern Simons): S = 1 16πG 5 d 5 x g(r+f MN F MN 12 l 2 ) πG 5 A F F F = Bdx 1 dx 2 solutions of the form: (D Hoker, Kraus 08-11) ds 2 = U(r)dt 2 + dr2 U(r) + e2v (r) (dx dx 2 2) + e 2W (r) dx 2 3 with: U(r h ) = 0, e 2V (r), e 2W (r) r r 2 (AdS 5 ), AdS r h some BH, T =T IR H B, T R.G. flow UV r boundary AdS 5 CFT some CFT (temp=t) 3+1d CFT (N=4)

39 Sphaleron rate (holography) T^ ( ) (g 2 N) 2 T B 2 + O( B4 ), B << T 2 256π 3 6π 4 T 2 Γ CS = ) (g 2 N) 2 B T T 4 + O( T 6 B ), B >> T π 5 ( T

40 Sphaleron rate (holography) B >> T Γ CS = (g2 N) π 4 B π T 2

41 Sphaleron rate (holography) B >> T Landau level density Γ CS = (g2 N) π 4 B π T 2 zero modes are magnetically confined in lowest Landau levels

42 Sphaleron rate (holography) B >> T Landau level density Γ CS = (g2 N) π 4 B π T 2 diffusion scale in 1+1d zero modes are magnetically confined in lowest Landau levels strong interaction back-reaction of B into the sphaleron spherical symmetry axial symmetry

43 Conclusions & speculations Instanton + magnetic field has a rich structure Electric and magnetic dipole moments Zero modes play a crucial role 1 st order derivative expansion captures some lattice results Confinement? (instantons with nonzero holonomy) 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? (Diamagnetic response of zero modes?)