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

▶
qcd
SMART_READER_LITE
LIVE PREVIEW

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

Feb 16, 2023 588 likes •1.29k views

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)

slide-1
SLIDE 1

QCD

in a strong magnetic field

Yoshimasa Hidaka (RIKEN)

Part I: magnetic field and anomaly

slide-2
SLIDE 2

Introduction

slide-3
SLIDE 3

Orders of magnitude for magnetic fields

Neodymium magnet Typical magnet

50G 12,500G

(strongest permanent magnet)

Strongest continuous magnetic field

produced in a laboratory

450,000G

Magnetars Heavy ion collisions The early Universe

(Electroweak transition)

～104MeV2～1017 G ～1022 G ～1013 G

wikipedia

slide-4
SLIDE 4

Orders of magnitude for magnetic fields

Neodymium magnet Typical magnet

50G 12,500G

(strongest permanent magnet)

Strongest continuous magnetic field

produced in a laboratory

450,000G

Magnetars Heavy ion collisions The early Universe

(Electroweak transition)

～104MeV2～1017 G ～1022 G ～1013 G

wikipedia

slide-5
SLIDE 5

Strong magnetic field in heavy ion collisions

slide-6
SLIDE 6

Strong magnetic field in heavy ion collisions

B

Strong magnetic field

Kharzeev, McLerran, Warringa (2008)

√ eB ∼ 100MeV ∼ 1017 − 1018Gauss

slide-7
SLIDE 7

Kharzeev, McLerran, Warringa (’08)

Magnetic field in heavy ion collisions

b = 12 fm b = 8 fm b = 4 fm τ(fm) eB (MeV2) 3 2.5 2 1.5 1 0.5 105 104 103 102 101 100

Strong magnetic filed is the QCD scale.

slide-8
SLIDE 8

Synchrotron radiation Real photon decays into dileptons

B

γ

e−

e+

B

γ

e− e−

Nonlinear effects

slide-9
SLIDE 9

Synchrotron radiation Real photon decays into dileptons

B

γ

e−

e+

B

γ

e− e−

Nonlinear effects

in heavy ion collisions, Tuchin (’11) (’12)

slide-10
SLIDE 10

Synchrotron radiation Real photon decays into dileptons

B

γ

e−

e+

B

γ

e− e−

Nonlinear effects

in heavy ion collisions, Tuchin (’11) (’12)

Vacuum birefringence Hattori, Itakura (’12) →Hattori’s talk

slide-11
SLIDE 11

Ji

V =

X

f

qfBiNc 2π2 µA

Ji

A =

X

f

qfBiNc 2π2 µ

Chiral magnetic effect Chiral separation effect related to chiral anomaly

slide-12
SLIDE 12

Magnetic catalysis

Bali, Bruckmann, Endrodi, Fodor, Katz, Schafer, JHEP 1202 (2012) 044
slide-13
SLIDE 13

Relativistic fields in a magnetic field

slide-14
SLIDE 14

Symmetry of QCD in a strong constant magnetic field

Internal symmetry

SU(2)L × SU(2)R × U(1)B × U(1)A

slide-15
SLIDE 15

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)I3,V × U(1)I3,A × U(1)B × U(1)A B

slide-16
SLIDE 16

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)I3,V × U(1)I3,A × U(1)B × U(1)A B

SSB, anomaly

U(1)I3,V × U(1)B = U(1)em × U(1)B

slide-17
SLIDE 17

Symmetry of QCD in a strong constant magnetic field

Internal symmetry

SU(2)L × SU(2)R × U(1)B × U(1)A SO(3, 1) → SO(1, 1)t,z × SO(2)x,y

Lorentz symmetry

U(1)I3,V × U(1)I3,A × U(1)B × U(1)A B

SSB, anomaly

U(1)I3,V × U(1)B = U(1)em × U(1)B

slide-18
SLIDE 18

Symmetry of QCD in a strong constant magnetic field

Internal symmetry

SU(2)L × SU(2)R × U(1)B × U(1)A SO(3, 1) → SO(1, 1)t,z × SO(2)x,y

Lorentz symmetry

U(1)I3,V × U(1)I3,A × U(1)B × U(1)A B

SSB, anomaly

U(1)I3,V × U(1)B = U(1)em × U(1)B

Discrete symmetry C, CP , and T are broken.

slide-19
SLIDE 19

Charged scalar particle in a magnetic field

Classical equation of motion

H ¨ x = e(E + ˙ x × B),

H = p p2 + m2

Lorentz force

B

slide-20
SLIDE 20

Charged scalar particle in a magnetic field

Classical equation of motion

H ¨ x = e(E + ˙ x × B),

H = p p2 + m2

Lorentz force

B

(Landau) quantization Closed orbital motion in the transverse plane

slide-21
SLIDE 21

Charged scalar particle

(−D2

µ − m2)φ(x) = 0

Dµ = ∂µ + ieAµ

B = (0, 0, B)

[Dx, Dy] = −ieB

Klein-Gordon equation in a magnetic field

slide-22
SLIDE 22

Charged scalar particle

(−D2

µ − m2)φ(x) = 0

Dµ = ∂µ + ieAµ

B = (0, 0, B)

φ(x) = e−iωt+ipzzϕ(x, y) (−D2

x − D2 y)ϕ(x, y) = λϕ(x, y)

λ = ω2 − p2

z − m2

[Dx, Dy] = −ieB

Klein-Gordon equation in a magnetic field

slide-23
SLIDE 23

Charged scalar particle

(−D2

µ − m2)φ(x) = 0

Dµ = ∂µ + ieAµ

B = (0, 0, B)

φ(x) = e−iωt+ipzzϕ(x, y) (−D2

x − D2 y)ϕ(x, y) = λϕ(x, y)

λ = ω2 − p2

z − m2

[X, P] = i

Introducing

X = 1 √ eB iDy, P ≡ −1 √ eB iDx,

[Dx, Dy] = −ieB

Klein-Gordon equation in a magnetic field

slide-24
SLIDE 24

Charged scalar particle

(−D2

µ − m2)φ(x) = 0

Dµ = ∂µ + ieAµ

B = (0, 0, B)

φ(x) = e−iωt+ipzzϕ(x, y) (−D2

x − D2 y)ϕ(x, y) = λϕ(x, y)

λ = ω2 − p2

z − m2

[X, P] = i

Introducing

X = 1 √ eB iDy, P ≡ −1 √ eB iDx,

[Dx, Dy] = −ieB

Klein-Gordon equation in a magnetic field

looks like a harmonic oscillator.

(−D2

x − D2 y)ϕ(x, y) = eB(X2 + P 2)ϕ(x, y)

slide-25
SLIDE 25

Rx = x − iDy eB Ry = y + iDx eB

[D2

x + D2 y, Rx,y] = 0

[Rx, Ry] = − i eB

Degeneracy

N = eB Z dRxdRy 2π = eBV⊥ 2π

Z dxdp 2π~

[x, p] = i~

(cf. ) Magnetic translation

S = eB 2π

(Rx, Ry) corresponds to the center of motion.

slide-26
SLIDE 26

a = 1 √ 2(X + iP) a† = 1 √ 2(X − iP) b = r eB 2 (Rx − iRy) b† = r eB 2 (Rx + iRy)

Continuum

Discrete

B

slide-27
SLIDE 27

a = 1 √ 2(X + iP) a† = 1 √ 2(X − iP) b = r eB 2 (Rx − iRy) b† = r eB 2 (Rx + iRy)

Continuum

Discrete

B

−D2

x − D2 y = 2eB

✓ a†a + 1 2 ◆

Energy: Wave function: |n, li = (a†)n

p n! (b†)l p l! |0, 0i En = p eB(2n + 1) + p2

z + m2

slide-28
SLIDE 28

a = 1 √ 2(X + iP) a† = 1 √ 2(X − iP) b = r eB 2 (Rx − iRy) b† = r eB 2 (Rx + iRy)

Continuum

Discrete

B

Lz = i(xpy − ypx) = b†b − a†a

Angular momentum (symmetric gauge)

−D2

x − D2 y = 2eB

✓ a†a + 1 2 ◆

Energy: Wave function: |n, li = (a†)n

p n! (b†)l p l! |0, 0i En = p eB(2n + 1) + p2

z + m2

slide-29
SLIDE 29

σµ = (1, −σi)

iσµDµψL = 0

i ✓ ∂0 − ∂z −Dx + iDy −Dx − iDy ∂0 + ∂z ◆ ψL = ✓ i∂0 − i∂z i √ 2eBa† −i √ 2eBa i∂0 + i∂z ◆ ψL = 0

Weyl Fermion

slide-30
SLIDE 30

σµ = (1, −σi)

iσµDµψL = 0

i ✓ ∂0 − ∂z −Dx + iDy −Dx − iDy ∂0 + ∂z ◆ ψL = ✓ i∂0 − i∂z i √ 2eBa† −i √ 2eBa i∂0 + i∂z ◆ ψL = 0

Weyl Fermion

Positive energy solution: En =

p 2eBn + p2

z

|uL(n, pz, l)i = 1 p2En pEn pz

−i √ 2eB √En−pz a

! |n, li

slide-31
SLIDE 31

σµ = (1, −σi)

iσµDµψL = 0

i ✓ ∂0 − ∂z −Dx + iDy −Dx − iDy ∂0 + ∂z ◆ ψL = ✓ i∂0 − i∂z i √ 2eBa† −i √ 2eBa i∂0 + i∂z ◆ ψL = 0

Weyl Fermion

LLL has spin up, and down moving

Higher modes can move up and down directions. |uL(0, pz, l)i = θ(pz) ✓1 ◆ |0, li

related to the chiral magnetic effect.

E0 = |pz|

Positive energy solution: En =

p 2eBn + p2

z

|uL(n, pz, l)i = 1 p2En pEn pz

−i √ 2eB √En−pz a

! |n, li

slide-32
SLIDE 32

E pz

B = 0

slide-33
SLIDE 33

E

· · ·· · ·

n = 1

n = 3 n = 4 n = 2 pz

B 6= 0

slide-34
SLIDE 34

E

· · ·· · ·

n = 1

n = 3 n = 4 n = 2 pz

B 6= 0

LLL

slide-35
SLIDE 35

E2

Free Dirac particle in a magnetic field

m2

slide-36
SLIDE 36

Orbital quantization

E2

Free Dirac particle in a magnetic field

m2

slide-37
SLIDE 37

Orbital quantization Zeeman effect (spin 1/2) up down

E2

Free Dirac particle in a magnetic field

m2

slide-38
SLIDE 38

Orbital quantization Zeeman effect (spin 1/2) up down

E2

Free Dirac particle in a magnetic field

m2

slide-39
SLIDE 39

Orbital quantization Zeeman effect (spin 1/2) up down Lowest Landau Level (LLL)

E2

Free Dirac particle in a magnetic field

m2

slide-40
SLIDE 40

Zeeman effect (spin 1) up zero

E2

Free vector particle in a magnetic field

m2

down Lowest Landau Level (LLL)

slide-41
SLIDE 41

Scalar boson

becomes heavier.

has instability if

Weyl and Dirac Fermions Vector boson

eB > m2

LLL: zeromode.

slide-42
SLIDE 42

Chiral magnetic and separation effects

slide-43
SLIDE 43

E

· · ·· · ·

LLL

n = 1

n = 3 n = 4 n = 2 pz

Chiral magnetic and separation effects

slide-44
SLIDE 44

E

· · ·· · ·

LLL

n = 1

n = 3 n = 4 n = 2 pz µ

At finite density, average of current cannot vanish.

Chiral magnetic and separation effects

slide-45
SLIDE 45

Chiral magnetic and separation effects

Jz

L ⌘ 1

V Z d3xhJz

L(x)i

Current average

= Nc V⊥ X

n≥1,l

Z dpz 2π pz En (fq(n, l, pz) − f¯

q(n, l, pz))

+Nc V⊥ X

l

Z dpz 2π (−θ(−pz)fq(0, l, pz) + θ(pz)f¯

q(0, l, pz))

slide-46
SLIDE 46

Chiral magnetic and separation effects

Jz

L ⌘ 1

V Z d3xhJz

L(x)i

Current average

Higher orders are cancelled out. Only LLL contributes to JL.

= Nc V⊥ X

n≥1,l

Z dpz 2π pz En (fq(n, l, pz) − f¯

q(n, l, pz))

+Nc V⊥ X

l

Z dpz 2π (−θ(−pz)fq(0, l, pz) + θ(pz)f¯

q(0, l, pz))

slide-47
SLIDE 47

Chiral magnetic and separation effects

If the system is thermal equilibrium Jz

L =

fq = 1 e(|pz|−µL)/T + 1

f¯

q =

1 e(|pz|+µL)/T + 1

linear in B, independent of T

Nc V⊥ X

l

Z dpz 2π (−θ(−pz)fq(0, l, pz) + θ(pz)f¯

q(0, l, pz))

= −NceB 2π Z ∞ dpz 2π ✓ 1 e(pz−µL)/T + 1 − 1 e(pz−µL)/T + 1 ◆ = −NceB 2π 1 2π T ⇣ ln(1 + eµL/T ) − ln(1 + e−µL/T ) ⌘ = −NceB 2π 1 2π µL

slide-48
SLIDE 48

Chiral magnetic effect Chiral separation effect

Fukushima, Kharzeev, Warringa (’08)

Son, Zhitnitsky(’04)

Metlitski, Zhitnitsky(’05)

~ JV = ~ JR + ~ JL = Nc 4⇡2 (µR − µL)e ~ B = Nc 2⇡2 µAe ~ B ~ JA = ~ JR − ~ JL = Nc 4⇡2 (µR + µL)e ~ B = Nc 2⇡2 µV e ~ B

slide-49
SLIDE 49

nA = jz

V

nV = jz

A

In 1+1d, Using chemical potential,

Chiral magnetic effect

Chiral separation effect

(Jµ

A = −✏µνJV ν)

LLL projected theory

nV = NceB 2π µV π nA = NceB 2π µA π ~ JV = Nc 2⇡2 µAe ~ B ~ JA = Nc 2⇡2 µV e ~ B

slide-50
SLIDE 50

Perturbative vs Holographic

Yee(’09), Rubakov(’10), Rebhan, Schmitt, Stricker(’09), Lifshytz, Lippert(’09), Gorsky, Kopnin, Zayakin(’10)

Holographic models

j = σB

Chiral magnetic conductivity

Yee JHEP11 (2009) 08

10 20 30 40 50 0.5 1.0 5 10 15 20 25 30 Ω T 0.5 0.5 1.0 Σ Σ0

Sakai-Sugimoto model

Kharzeev, Warringa, Phys. Rev. D80, 034028(2009)

Perturbative

µ/T = 0.1

slide-51
SLIDE 51

Chiral magnetic effect

  • n the Lattice
[Fukushima, Kharzeev, Warringa]

Yamamoto(’11)

  • Phys. Rev. D 84, 114504 (2011)
slide-52
SLIDE 52

Yannis Burnier, Dmitri E. Kharzeev, Jinfeng Liao, Ho-Ung Yee (’11)

Chiral magnetic wave

Kharzeev, Yee(’10)

~ JV = 1 2⇡2 µAe ~ B ~ JA = 1 2⇡2 µV e ~ B

slide-53
SLIDE 53

Yannis Burnier, Dmitri E. Kharzeev, Jinfeng Liao, Ho-Ung Yee (’11)

Chiral magnetic wave

Kharzeev, Yee(’10)

~ JV = 1 2⇡2 µAe ~ B ~ JA = 1 2⇡2 µV e ~ B ✓ ∂0 ⌥ NceBα 2π2 ∂z DL∂2

z

◆ j0

L,R = 0

gapless collective mode in chiral symmetric phase

0.0 0.5 1.0 1.5 2.0 eB GeV 2 0.0 0.2 0.4 0.6 0.8 1.0 Kharzeev, Yee, Phys. Rev. D83, 085007 (2011)

Sakai-Sugimoto model velocity

T = 150MeV T = 200MeV

T = 250MeV

slide-54
SLIDE 54

Chiral magnetic waves in Heavy Ion Collisions

±

Observed A

  • 0.1
  • 0.05

0.05 0.1

(%)

2

v

3.6 3.65 3.7 3.75

  • π

+

π

STAR Preliminary

±

A

  • 0.04
  • 0.02

0.02 0.04

) (%)

+

π (

2

) - v

  • π

(

2

v

  • 0.1

0.1 0.2

Au + Au 200 GeV: 30-40% < 0.5 GeV/c

T

0.15 < p

STAR Preliminary

  • G. Wang et al [STAR Coll], arxiv:1210.5498

v2

± = v2  qe

ρe A± A± = N+ − N− N+ + N−

slide-55
SLIDE 55

Gapless mode couples to photon Photon becomes massive

1 2 3 4 5 6 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 t

(b) electric field (c) quark number density

  • N. Tanji, Ann. Phys. 325:2018 (2010)
  • A. Iwazaki, Phys.Rev.C80 052202 (2009)

ω =

  • k2

+ m2 γ,

Electric field

Particle number

Ez = E0 cos(ωt − k · x),

1.0

0.5

n/nmax

LLL approximation Schwinger mechanism + Maxwell Eq. From anomaly equation

slide-56
SLIDE 56

Chiral kinetic theory

Son, Yamamoto (’12), 1210.8158, Stephanov, Yin (’12)

Action with Berry phase

S = Z dt( ˙ x · p + ˙ x · A(x) − ˙ p · A(p) − H(x, p))

slide-57
SLIDE 57

Chiral kinetic theory

Son, Yamamoto (’12), 1210.8158, Stephanov, Yin (’12)

Action with Berry phase

˙ x = v + ˙ p × Ω

Eom

˙ p = ˙ x × B + F

v = ∂H/∂p F = −∂H/∂x B = r × A Ω = rp × A S = Z dt( ˙ x · p + ˙ x · A(x) − ˙ p · A(p) − H(x, p))

slide-58
SLIDE 58

Chiral kinetic theory

Son, Yamamoto (’12), 1210.8158, Stephanov, Yin (’12)

Action with Berry phase

˙ x = v + ˙ p × Ω

Eom

˙ p = ˙ x × B + F

v = ∂H/∂p F = −∂H/∂x B = r × A Ω = rp × A S = Z dt( ˙ x · p + ˙ x · A(x) − ˙ p · A(p) − H(x, p))

G = (1 + B · Ω)

√ G ˙ x = v + F × Ω + B(v · Ω) √ G ˙ p = F + v × B + Ω(F · B)

slide-59
SLIDE 59

Chiral kinetic theory

Son, Yamamoto (’12), 1210.8158, Stephanov, Yin (’12)

Action with Berry phase

˙ x = v + ˙ p × Ω

Eom

˙ p = ˙ x × B + F

v = ∂H/∂p F = −∂H/∂x B = r × A Ω = rp × A S = Z dt( ˙ x · p + ˙ x · A(x) − ˙ p · A(p) − H(x, p))

G = (1 + B · Ω)

√ G ˙ x = v + F × Ω + B(v · Ω) √ G ˙ p = F + v × B + Ω(F · B)

Chiral magnetic effect

j = Z

p

√ Gf ˙ x = Z fv + F × Z

p

fΩ + B Z

p

f(v · Ω)

slide-60
SLIDE 60

Chiral vortical effect

Son, Surówka(’09)

Hydrodynamics

Jµ = nuµ + νµ

∂µT µν = F νλJλ

∂µJµ = CEµBµ

T µν = (✏ + P)uµuν + Pgµν + ⌧ µν

slide-61
SLIDE 61

Chiral vortical effect

Son, Surówka(’09)

Hydrodynamics

Jµ = nuµ + νµ

∂µT µν = F νλJλ

∂µJµ = CEµBµ

T µν = (✏ + P)uµuν + Pgµν + ⌧ µν

ν = −σTP µν∂ν(µ/T) + σEµ + ξBBµ + ξωµ

sµ = suµ − µ T + Dωµ + DBBµ

!µ = 1 2✏µνρσuν@ρuσ

slide-62
SLIDE 62

Chiral vortical effect

Son, Surówka(’09)

Hydrodynamics

Jµ = nuµ + νµ

∂µT µν = F νλJλ

∂µJµ = CEµBµ

T µν = (✏ + P)uµuν + Pgµν + ⌧ µν

∂µsµ ≥ 0

⇠ = C ✓ µ2 − 2 3 nµ3 ✏ + P ◆

⇠B = C ✓ µ − 1 2 nµ2 ✏ + P ◆

ν = −σTP µν∂ν(µ/T) + σEµ + ξBBµ + ξωµ

sµ = suµ − µ T + Dωµ + DBBµ

!µ = 1 2✏µνρσuν@ρuσ

slide-63
SLIDE 63

Chiral vortical effect

Linear response theory Perturbative and Holographic

Landsteiner, Megıas, Pena-Benitez (’11)

⇠ = lim

kn→0

X

ij

✏ijn i 2kn hJi

AT 0jiω=0

Amado, Landsteiner, Pena-Benitez (’11)

ξ = µ2 4π2 + T 2 12

slide-64
SLIDE 64

related to gravitational anomaly(?)

Chiral vortical effect

Linear response theory Perturbative and Holographic

Landsteiner, Megıas, Pena-Benitez (’11)

⇠ = lim

kn→0

X

ij

✏ijn i 2kn hJi

AT 0jiω=0

Amado, Landsteiner, Pena-Benitez (’11)

ξ = µ2 4π2 + T 2 12

slide-65
SLIDE 65

related to gravitational anomaly(?)

Chiral vortical effect

Linear response theory Perturbative and Holographic

Landsteiner, Megıas, Pena-Benitez (’11)

⇠ = lim

kn→0

X

ij

✏ijn i 2kn hJi

AT 0jiω=0

Amado, Landsteiner, Pena-Benitez (’11)

ξ = µ2 4π2 + T 2 12

  • ne-loop exact

Golkar, Son (’12)

slide-66
SLIDE 66

Summary Magnetic field + Chiral fermion

= magnetic (separation) effects

Chiral vortical effect

slide-67
SLIDE 67

Summary Magnetic field + Chiral fermion

= magnetic (separation) effects

Chiral vortical effect

Lorentz force Coriolis force

slide-68
SLIDE 68

Generating functional for Fluid

Banerjee, Bhattacharya, Bhattacharyya, Jain, Minwalla, Sharma (’12)

Jensen, Kaminski, Kovtun, Meyer, Ritz, Yarom (’12)

Gravitational and gauge background with a timelike killing vector ds2 = −e2σ(dt + aidxi)2 + gijdxidxj

A = A0dx0 + Aidxi

T = e−σT0 + · · · µ = e−σ(µ0 + A) + · · ·

Local temperature Local chemical potential

slide-69
SLIDE 69

Wanom = C 2 ✓Z A0 3T0 AdA + A2 6T0 Ada ◆

Next leading order (First derivative)

Leading order

W0 = Z d3x√g3 eσ T0 P(T0e−σ, e−σA0) T µν = −2T0 δW δgµν

Jµ = T0 δW δAµ

= nuµ = (✏ + P) + Pgµν