QCD in a strong magnetic field Part I: magnetic field and anomaly - PowerPoint PPT Presentation

QCD in a strong magnetic field Part I: magnetic field and anomaly Yoshimasa Hidaka (RIKEN)

Introduction

450,000G Orders of magnitude for magnetic fields wikipedia Typical magnet 50G Neodymium magnet 12,500G (strongest permanent magnet) Strongest continuous magnetic field produced in a laboratory ～ 10 13 G Magnetars Heavy ion collisions ～ 10 4 MeV 2 ～ 10 17 G The early Universe ～ 10 22 G (Electroweak transition)

450,000G Orders of magnitude for magnetic fields wikipedia Typical magnet 50G Neodymium magnet 12,500G (strongest permanent magnet) Strongest continuous magnetic field produced in a laboratory ～ 10 13 G Magnetars Heavy ion collisions ～ 10 4 MeV 2 ～ 10 17 G The early Universe ～ 10 22 G (Electroweak transition)

Strong magnetic field in heavy ion collisions

Strong magnetic field in heavy ion collisions B √ Strong magnetic field eB ∼ 100MeV ∼ 10 17 − 10 18 Gauss Kharzeev, McLerran, Warringa (2008)

Magnetic field in heavy ion collisions Kharzeev, McLerran, Warringa (’08) 10 5 b = 4 fm b = 8 fm b = 12 fm 10 4 eB (MeV 2 ) 10 3 10 2 10 1 10 0 0 0.5 1 1.5 2 2.5 3 τ (fm) Strong magnetic filed is the QCD scale.

Nonlinear e ff ects Synchrotron radiation B e − γ e − Real photon decays into dileptons e − B γ e +

Nonlinear e ff ects Synchrotron radiation B e − γ e − Real photon decays into dileptons e − B γ e + in heavy ion collisions, Tuchin (’11) (’12)

Nonlinear e ff ects Synchrotron radiation B e − γ e − Real photon decays into dileptons e − B γ e + in heavy ion collisions, Tuchin (’11) (’12) Vacuum birefringence Hattori, Itakura (’12) → Hattori’s talk

Chiral magnetic e ff ect q f B i N c X J i V = µ A 2 π 2 f Chiral separation e ff ect q f B i N c X J i µ A = 2 π 2 f related to chiral anomaly

Bali, Bruckmann, Endrodi, Fodor, Katz, Schafer, JHEP 1202 (2012) 044 Magnetic catalysis

Relativistic fields in a magnetic field

Symmetry of QCD in a strong constant magnetic field Internal symmetry SU (2) L × SU (2) R × U (1) B × U (1) A

Symmetry of QCD in a strong constant magnetic field Internal symmetry SU (2) L × SU (2) R × U (1) B × U (1) A U (1) I 3 ,V × U (1) I 3 ,A × U (1) B × U (1) A B

Symmetry of QCD in a strong constant magnetic field Internal symmetry SU (2) L × SU (2) R × U (1) B × U (1) A U (1) I 3 ,V × U (1) I 3 ,A × U (1) B × U (1) A B U (1) I 3 ,V × U (1) B = U (1) em × U (1) B SSB, anomaly

Symmetry of QCD in a strong constant magnetic field Internal symmetry SU (2) L × SU (2) R × U (1) B × U (1) A U (1) I 3 ,V × U (1) I 3 ,A × U (1) B × U (1) A B U (1) I 3 ,V × U (1) B = U (1) em × U (1) B SSB, anomaly Lorentz symmetry SO (3 , 1) → SO (1 , 1) t,z × SO (2) x,y

Symmetry of QCD in a strong constant magnetic field Internal symmetry SU (2) L × SU (2) R × U (1) B × U (1) A U (1) I 3 ,V × U (1) I 3 ,A × U (1) B × U (1) A B U (1) I 3 ,V × U (1) B = U (1) em × U (1) B SSB, anomaly Lorentz symmetry SO (3 , 1) → SO (1 , 1) t,z × SO (2) x,y Discrete symmetry C, CP , and T are broken.

Lorentz force Classical equation of motion Charged scalar particle in a magnetic field H ¨ x = e ( E + ˙ x × B ) , p p 2 + m 2 H = B

Lorentz force Classical equation of motion (Landau) quantization Closed orbital motion in the transverse plane Charged scalar particle in a magnetic field H ¨ x = e ( E + ˙ x × B ) , p p 2 + m 2 H = B

Charged scalar particle Klein-Gordon equation in a magnetic field ( − D 2 µ − m 2 ) φ ( x ) = 0 D µ = ∂ µ + ieA µ [ D x , D y ] = − ieB B = (0 , 0 , B )

Charged scalar particle Klein-Gordon equation in a magnetic field ( − D 2 µ − m 2 ) φ ( x ) = 0 D µ = ∂ µ + ieA µ [ D x , D y ] = − ieB B = (0 , 0 , B ) ( − D 2 x − D 2 φ ( x ) = e − i ω t + ip z z ϕ ( x, y ) y ) ϕ ( x, y ) = λϕ ( x, y ) λ = ω 2 − p 2 z − m 2

Charged scalar particle Klein-Gordon equation in a magnetic field ( − D 2 µ − m 2 ) φ ( x ) = 0 D µ = ∂ µ + ieA µ [ D x , D y ] = − ieB B = (0 , 0 , B ) ( − D 2 x − D 2 φ ( x ) = e − i ω t + ip z z ϕ ( x, y ) y ) ϕ ( x, y ) = λϕ ( x, y ) λ = ω 2 − p 2 z − m 2 Introducing 1 − 1 [ X, P ] = i X = iD y , P ≡ iD x , √ √ eB eB

Charged scalar particle Klein-Gordon equation in a magnetic field ( − D 2 µ − m 2 ) φ ( x ) = 0 D µ = ∂ µ + ieA µ [ D x , D y ] = − ieB B = (0 , 0 , B ) ( − D 2 x − D 2 φ ( x ) = e − i ω t + ip z z ϕ ( x, y ) y ) ϕ ( x, y ) = λϕ ( x, y ) λ = ω 2 − p 2 z − m 2 Introducing 1 − 1 [ X, P ] = i X = iD y , P ≡ iD x , √ √ eB eB y ) ϕ ( x, y ) = eB ( X 2 + P 2 ) ϕ ( x, y ) ( − D 2 x − D 2 looks like a harmonic oscillator.

(cf. ) Degeneracy Magnetic translation R x = x − iD y R y = y + iD x [ R x , R y ] = − i eB eB eB [ D 2 x + D 2 y , R x,y ] = 0 (R x , R y ) corresponds to the center of motion. S = eB 2 π Z dR x dR y Z dxdp = eBV ⊥ [ x, p ] = i ~ N = eB 2 π ~ 2 π 2 π

1 1 a † = a = 2( X + iP ) 2( X − iP ) √ √ r r eB eB 2 ( R x − iR y ) b † = B b = 2 ( R x + iR y ) Continuum Discrete

Energy: 1 1 a † = a = 2( X + iP ) 2( X − iP ) √ √ r r eB eB 2 ( R x − iR y ) b † = B b = 2 ( R x + iR y ) Continuum ✓ ◆ a † a + 1 − D 2 x − D 2 y = 2 eB Discrete 2 p E n = eB (2 n + 1) + p 2 z + m 2 Wave function: | n, l i = ( a † ) n ( b † ) l p p | 0 , 0 i n ! l !

Angular momentum (symmetric gauge) Energy: 1 1 a † = a = 2( X + iP ) 2( X − iP ) √ √ r r eB eB 2 ( R x − iR y ) b † = B b = 2 ( R x + iR y ) Continuum ✓ ◆ a † a + 1 − D 2 x − D 2 y = 2 eB Discrete 2 p E n = eB (2 n + 1) + p 2 z + m 2 Wave function: | n, l i = ( a † ) n ( b † ) l p p | 0 , 0 i n ! l ! L z = i ( xp y − yp x ) = b † b − a † a

Weyl Fermion σ µ = (1 , − σ i ) i σ µ D µ ψ L = 0 ✓ i ∂ 0 − i ∂ z √ ✓ ◆ 2 eBa † ◆ ∂ 0 − ∂ z − D x + iD y i ψ L = ψ L = 0 i √ ∂ 0 + ∂ z − D x − iD y i ∂ 0 + i ∂ z − i 2 eBa

Weyl Fermion σ µ = (1 , − σ i ) i σ µ D µ ψ L = 0 ✓ i ∂ 0 − i ∂ z √ ✓ ◆ 2 eBa † ◆ ∂ 0 − ∂ z − D x + iD y i ψ L = ψ L = 0 i √ ∂ 0 + ∂ z − D x − iD y i ∂ 0 + i ∂ z − i 2 eBa p 2 eBn + p 2 Positive energy solution: E n = z p E n � p z ! 1 p 2 E n | u L ( n, p z , l ) i = | n, l i √ − i 2 eB √ E n − p z a

Higher modes can move up and down directions. related to the chiral magnetic effect. LLL has spin up, and down moving Weyl Fermion σ µ = (1 , − σ i ) i σ µ D µ ψ L = 0 ✓ i ∂ 0 − i ∂ z √ ✓ ◆ 2 eBa † ◆ ∂ 0 − ∂ z − D x + iD y i ψ L = ψ L = 0 i √ ∂ 0 + ∂ z − D x − iD y i ∂ 0 + i ∂ z − i 2 eBa p 2 eBn + p 2 Positive energy solution: E n = z p E n � p z ! 1 p 2 E n | u L ( n, p z , l ) i = | n, l i √ − i 2 eB √ E n − p z a ✓ 1 ◆ E 0 = | p z | | u L (0 , p z , l ) i = θ ( � p z ) | 0 , l i 0

B = 0 E p z

B 6 = 0 E · · ·· · · n = 4 n = 3 n = 2 n = 1 p z

LLL B 6 = 0 E · · ·· · · n = 4 n = 3 n = 2 n = 1 p z

Free Dirac particle in a magnetic field E 2 m 2

Free Dirac particle in a magnetic field Orbital quantization E 2 m 2

Free Dirac particle in a magnetic field Orbital Zeeman e ff ect (spin 1/2) quantization up down E 2 m 2

Free Dirac particle in a magnetic field Orbital Zeeman e ff ect (spin 1/2) quantization up down E 2 m 2

Free Dirac particle in a magnetic field Orbital Zeeman e ff ect (spin 1/2) quantization up down E 2 m 2 Lowest Landau Level (LLL)

Free vector particle in a magnetic field Zeeman e ff ect (spin 1) up zero down E 2 m 2 Lowest Landau Level (LLL)

Scalar boson becomes heavier. Weyl and Dirac Fermions LLL: zeromode. Vector boson has instability if eB > m 2

Chiral magnetic and separation effects

Chiral magnetic and separation effects LLL E · · ·· · · n = 4 n = 3 n = 2 n = 1 p z

LLL At finite density, average of current cannot vanish. Chiral magnetic and separation effects E · · ·· · · n = 4 µ n = 3 n = 2 n = 1 p z

Chiral magnetic and separation effects Current average L ⌘ 1 Z J z d 3 x h J z L ( x ) i V Z dp z = N c p z X ( f q ( n, l, p z ) − f ¯ q ( n, l, p z )) 2 π V ⊥ E n n ≥ 1 ,l Z dp z + N c X 2 π ( − θ ( − p z ) f q (0 , l, p z ) + θ ( p z ) f ¯ q (0 , l, p z )) V ⊥ l

Current average Chiral magnetic and separation effects L ⌘ 1 Z J z d 3 x h J z L ( x ) i V Z dp z = N c p z X ( f q ( n, l, p z ) − f ¯ q ( n, l, p z )) 2 π V ⊥ E n n ≥ 1 ,l Z dp z + N c X 2 π ( − θ ( − p z ) f q (0 , l, p z ) + θ ( p z ) f ¯ q (0 , l, p z )) V ⊥ l Higher orders are cancelled out. Only LLL contributes to J L.