NEW ANOMALY INDUCED SECOND ORDER TRANSPORT F. PEA-BENTEZ IFT - - - PowerPoint PPT Presentation

NEW ANOMALY INDUCED SECOND ORDER TRANSPORT

• F. PEÑA-BENÍTEZ

IFT - UAM/CSIC

BASED ON: 1304.5529 WITH EUGENIO MEGÍAS

OUTLINE

ANOMALIES AND HYDRODYNAMICS STRONGLY COUPLED MODEL FLUID GRAVITY CORRESPONDENCE RESULTS SUMMARY, ACTUAL AND FUTURE DIRECTIONS

ANOMALIES AND HYDRODYNAMICS

rµJµ = 3cA 4 µνρλFµνFρλ + cm 4 µνρλRα

βµνRβ αρλ

TA TB TC Aµ Aν Aρ

TA hµν hλβ Aρ

(NON)CONSERVED CURRENT

ANOMALIES AND HYDRODYNAMICS

rµJµ = 3cA 4 µνρλFµνFρλ + cm 4 µνρλRα

βµνRβ αρλ

[NEIMAN AND OZ, ‘10] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10]

Jµ

(1)ano = ξωµ + ξBBµ

Jµ = jµ + jµ

ano

⇠B = cAµ − n ✏ + p ✓1 2cAµ2 + cmT 2 ◆

⇠ = 1 2cAµ2 + cmT 2 − n ✏ + p ✓1 3cAµ3 + 2cmµT 2 ◆

[ERDMENGER, ET AL, ’08] [BANERJEE, ET AL, ’08] [YAROM, JENSEN, LOGANAYAGAM,10]

ANOMALIES AND HYDRODYNAMICS

rµJµ = 3cA 4 µνρλFµνFρλ + cm 4 µνρλRα

βµνRβ αρλ

[NEIMAN AND OZ, ‘10] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10]

Jµ

(1)ano = ξωµ + ξBBµ

Jµ = jµ + jµ

ano

⇠B = cAµ − n ✏ + p ✓1 2cAµ2 + cmT 2 ◆

⇠ = 1 2cAµ2 + cmT 2 − n ✏ + p ✓1 3cAµ3 + 2cmµT 2 ◆

[ERDMENGER, ET AL, ’08] [BANERJEE, ET AL, ’08] [YAROM, JENSEN, LOGANAYAGAM,10]

ANOMALIES AND HYDRODYNAMICS

rµJµ = 3cA 4 µνρλFµνFρλ + cm 4 µνρλRα

βµνRβ αρλ

[NEIMAN AND OZ, ‘10] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10]

Jµ

(1)ano = ξωµ + ξBBµ

Jµ = jµ + jµ

ano

⇠B = cAµ − n ✏ + p ✓1 2cAµ2 + cmT 2 ◆

⇠ = 1 2cAµ2 + cmT 2 − n ✏ + p ✓1 3cAµ3 + 2cmµT 2 ◆

[ERDMENGER, ET AL, ’08] [BANERJEE, ET AL, ’08] [YAROM, JENSEN, LOGANAYAGAM,10]

PARITY AND TIME REVERSAL PROPERTIES

ANOMALIES AND HYDRODYNAMICS

rµJµ = 3cA 4 µνρλFµνFρλ + cm 4 µνρλRα

βµνRβ αρλ

[NEIMAN AND OZ, ‘10] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10]

Jµ

(1)ano = ξωµ + ξBBµ

Jµ = jµ + jµ

ano

⇠B = cAµ − n ✏ + p ✓1 2cAµ2 + cmT 2 ◆

⇠ = 1 2cAµ2 + cmT 2 − n ✏ + p ✓1 3cAµ3 + 2cmµT 2 ◆

[ERDMENGER, ET AL, ’08] [BANERJEE, ET AL, ’08] [YAROM, JENSEN, LOGANAYAGAM,10]

~ j = B ~ B

PARITY AND TIME REVERSAL PROPERTIES

ANOMALIES AND HYDRODYNAMICS

rµJµ = 3cA 4 µνρλFµνFρλ + cm 4 µνρλRα

βµνRβ αρλ

[NEIMAN AND OZ, ‘10] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10]

Jµ

(1)ano = ξωµ + ξBBµ

Jµ = jµ + jµ

ano

⇠B = cAµ − n ✏ + p ✓1 2cAµ2 + cmT 2 ◆

⇠ = 1 2cAµ2 + cmT 2 − n ✏ + p ✓1 3cAµ3 + 2cmµT 2 ◆

[ERDMENGER, ET AL, ’08] [BANERJEE, ET AL, ’08] [YAROM, JENSEN, LOGANAYAGAM,10]

P-even T-odd

~ j = B ~ B

PARITY AND TIME REVERSAL PROPERTIES

ANOMALIES AND HYDRODYNAMICS

rµJµ = 3cA 4 µνρλFµνFρλ + cm 4 µνρλRα

βµνRβ αρλ

[NEIMAN AND OZ, ‘10] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10]

Jµ

(1)ano = ξωµ + ξBBµ

Jµ = jµ + jµ

ano

⇠B = cAµ − n ✏ + p ✓1 2cAµ2 + cmT 2 ◆

⇠ = 1 2cAµ2 + cmT 2 − n ✏ + p ✓1 3cAµ3 + 2cmµT 2 ◆

[ERDMENGER, ET AL, ’08] [BANERJEE, ET AL, ’08] [YAROM, JENSEN, LOGANAYAGAM,10]

P-odd T-odd P-even T-odd

~ j = B ~ B

PARITY AND TIME REVERSAL PROPERTIES

ANOMALIES AND HYDRODYNAMICS

rµJµ = 3cA 4 µνρλFµνFρλ + cm 4 µνρλRα

βµνRβ αρλ

[NEIMAN AND OZ, ‘10] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10]

Jµ

(1)ano = ξωµ + ξBBµ

Jµ = jµ + jµ

ano

⇠B = cAµ − n ✏ + p ✓1 2cAµ2 + cmT 2 ◆

⇠ = 1 2cAµ2 + cmT 2 − n ✏ + p ✓1 3cAµ3 + 2cmµT 2 ◆

[ERDMENGER, ET AL, ’08] [BANERJEE, ET AL, ’08] [YAROM, JENSEN, LOGANAYAGAM,10]

P-odd T-odd P-even T-odd P-odd T-even

~ j = B ~ B

PARITY AND TIME REVERSAL PROPERTIES

ANOMALIES AND HYDRODYNAMICS

rµJµ = 3cA 4 µνρλFµνFρλ + cm 4 µνρλRα

βµνRβ αρλ

[NEIMAN AND OZ, ‘10] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10]

Jµ

(1)ano = ξωµ + ξBBµ

Jµ = jµ + jµ

ano

⇠B = cAµ − n ✏ + p ✓1 2cAµ2 + cmT 2 ◆

⇠ = 1 2cAµ2 + cmT 2 − n ✏ + p ✓1 3cAµ3 + 2cmµT 2 ◆

[ERDMENGER, ET AL, ’08] [BANERJEE, ET AL, ’08] [YAROM, JENSEN, LOGANAYAGAM,10]

SECOND ORDER ANOMALOUS CONSTITUTIVE RELATIONS

τ µν

(2) = a=15

X

a=1

ΛaT (a)µν

jµ

(2) = a=10

X

a=1

ξaJ (a)µ AT SECOND ORDER IN A CONFORMAL FLUID WE HAVE MANY CONTRIBUTIONS

SECOND ORDER ANOMALOUS CONSTITUTIVE RELATIONS

jµ

(2)ano = a=5

X

a=1

˜ ξa ˜ J (a)µ

τ µν

(2) = a=15

X

a=1

ΛaT (a)µν τ µν

(2)ano = a=8

X

a=1

˜ Λa ˜ T (a)µν

jµ

(2) = a=10

X

a=1

ξaJ (a)µ AT SECOND ORDER IN A CONFORMAL FLUID WE HAVE MANY CONTRIBUTIONS

SECOND ORDER ANOMALOUS CONSTITUTIVE RELATIONS

jµ

(2)ano = a=5

X

a=1

˜ ξa ˜ J (a)µ

τ µν

(2) = a=15

X

a=1

ΛaT (a)µν τ µν

(2)ano = a=8

X

a=1

˜ Λa ˜ T (a)µν

jµ

(2) = a=10

X

a=1

ξaJ (a)µ AT SECOND ORDER IN A CONFORMAL FLUID WE HAVE MANY CONTRIBUTIONS I WILL FOCUS ON ANOMALOUS CONTRIBUTION WITH SECOND DERIVATIVE TERMS

SECOND ORDER ANOMALOUS CONSTITUTIVE RELATIONS

jµ

(2)ano = a=5

X

a=1

˜ ξa ˜ J (a)µ

˜ T (1)µν = Πµν

αβDαωβ

˜ T (4)µν = Πµν

αβDαBβ

˜ J (5)µ = µναβuνDαEβ

τ µν

(2) = a=15

X

a=1

ΛaT (a)µν τ µν

(2)ano = a=8

X

a=1

˜ Λa ˜ T (a)µν

jµ

(2) = a=10

X

a=1

ξaJ (a)µ AT SECOND ORDER IN A CONFORMAL FLUID WE HAVE MANY CONTRIBUTIONS I WILL FOCUS ON ANOMALOUS CONTRIBUTION WITH SECOND DERIVATIVE TERMS

SECOND ORDER ANOMALOUS CONSTITUTIVE RELATIONS

˜ J (5)µ = µναβuνDαEβ r ⇥ ~ E

P-even T-even

THE CONDUCTIVITIES ASOCIATED TO THE ANOMALOUS SOURCES ARE DISSIPATIVE AND PARITY VIOLATING!

SECOND ORDER ANOMALOUS CONSTITUTIVE RELATIONS

˜ J (5)µ = µναβuνDαEβ r ⇥ ~ E

P-even T-even

THE CONDUCTIVITIES ASOCIATED TO THE ANOMALOUS SOURCES ARE DISSIPATIVE AND PARITY VIOLATING! PARITY AND TIME REVERSAL AGAIN!

S = SEHM + SCS + SGH + SCSK

STRONGLY COUPLED MODEL

SCS = Z √−g✏MNP QRAM h 3 FNP FQR + RA

BNP RB AQR

i

S = SEHM + SCS + SGH + SCSK

UNDER A GAUGE TRANSFORMATION

STRONGLY COUPLED MODEL

SCS = Z √−g✏MNP QRAM h 3 FNP FQR + RA

BNP RB AQR

i

S = Z

∂

√ −h ⇠✏µνρλ ⇣ 3 FµνFρλ + Rα

βµνRβ αρλ

⌘

S = SEHM + SCS + SGH + SCSK

UNDER A GAUGE TRANSFORMATION

STRONGLY COUPLED MODEL

SCS = Z √−g✏MNP QRAM h 3 FNP FQR + RA

BNP RB AQR

i

S = Z

∂

√ −h ⇠✏µνρλ ⇣ 3 FµνFρλ + Rα

βµνRβ αρλ

⌘

−λ = 1 4cm

−1 3κ = 1 4cA

c(⇥) = ⇤(⇥) r+ + O() W1(⇥) = 1 + O() W2(⇥) = 1 + O() W3(⇥) = 1 − f(⇥) + O() W4µ(⇥) = O() W5µν(⇥) = O() aµ(⇥) = O()

BOUNDARY BACKGROUND GAUGE FIELD COUNTS THE NUMBER OF TRANSVERSE DERIVATIVES EPSILON ZERO MEANS NO X DEPENDENCE AND THE ANSATZ BECOME IN THE BOOSTED CHARGED BLACK HOLE SOLUTION

FLUID/GRAVITY ANSATZ

uµ(x)uµ(x) = −1

ds2 = −2W1(ρ)uµdxµ dr2 + rAνdxν + r2  W2(ρ)ηµν + W3(ρ)uµuν + 2W4σ(ρ) r+ P σ

µ uν + W5µν(ρ)

r2

+

dxµdxν

• A =

⇣ a(b)

ν

+ aµ(ρ)P µ

ν + r+c(ρ)uν

⌘ dxν

Jµ ∼ lim

✏→0

⇣ r2

+a(¯ 2,✏) µ

+ Jct

µ

⌘ Tµ⌫ ∼ lim

✏→0

⇣ 4r2

+W (¯ 4,✏) 5µ⌫ + T ct µ⌫

⌘

A = ⇣ a(b)

ν

+ aµ(ρ)P µ

ν + r+c(ρ)uν

⌘ dxν

ds2 = −2W1(ρ)uµdxµ dr2 + rAνdxν + r2  W2(ρ)ηµν + W3(ρ)uµuν + 2W4σ(ρ) r+ P σ

µ uν + W5µν(ρ)

r2

+

dxµdxν

• FLUID/GRAVITY

HOLOGRAPHIC ONE POINT FUNCTIONS

SOME OF THE COEFFICIENTS IN THE TENSOR AND VECTOR SECTOR

˜ Λ1 = −2η˜ lω

˜ T (1)µν = Πµν

αβDαωβ

ANOMALY IN THE ENERGY MOMENTUM TENSOR

SOME OF THE COEFFICIENTS IN THE TENSOR AND VECTOR SECTOR

˜ Λ1 = −2η˜ lω

˜ T (1)µν = Πµν

αβDαωβ

˜ lω = 2κµ3 p + 211π2µλ p ✓ 3r2

+ − 2µ2 − πTµ2

r+ ◆

ANOMALY IN THE ENERGY MOMENTUM TENSOR

SOME OF THE COEFFICIENTS IN THE TENSOR AND VECTOR SECTOR

˜ Λ1 = −2η˜ lω ˜ Λ2 = −2η˜ lB

˜ T (1)µν = Πµν

αβDαωβ

˜ T (4)µν = Πµν

αβDαBβ

˜ lω = 2κµ3 p + 211π2µλ p ✓ 3r2

+ − 2µ2 − πTµ2

r+ ◆

ANOMALY IN THE ENERGY MOMENTUM TENSOR

SOME OF THE COEFFICIENTS IN THE TENSOR AND VECTOR SECTOR

˜ Λ1 = −2η˜ lω ˜ Λ2 = −2η˜ lB

˜ T (1)µν = Πµν

αβDαωβ

˜ T (4)µν = Πµν

αβDαBβ

˜ lB = κµ2 + 23λπ2T 2 p ˜ lω = 2κµ3 p + 211π2µλ p ✓ 3r2

+ − 2µ2 − πTµ2

r+ ◆

ANOMALY IN THE ENERGY MOMENTUM TENSOR

SOME OF THE COEFFICIENTS IN THE TENSOR AND VECTOR SECTOR

˜ Λ1 = −2η˜ lω

˜ T (1)µν = Πµν

αβDαωβ

˜ lω = 2κµ3 p + 211π2µλ p ✓ 3r2

+ − 2µ2 − πTµ2

r+ ◆ ω = i η 4pk2 ⌥ Ck3 + O(k4)

ANOMALY IN THE ENERGY MOMENTUM TENSOR

SOME OF THE COEFFICIENTS IN THE TENSOR AND VECTOR SECTOR

˜ Λ1 = −2η˜ lω

˜ T (1)µν = Πµν

αβDαωβ

˜ lω = 2κµ3 p + 211π2µλ p ✓ 3r2

+ − 2µ2 − πTµ2

r+ ◆ SHEAR WAVES DISPERSION RELATION

[KHARZEEV, YEE]

ω = i η 4pk2 ⌥ Ck3 + O(k4)

ANOMALY IN THE ENERGY MOMENTUM TENSOR

SOME OF THE COEFFICIENTS IN THE TENSOR AND VECTOR SECTOR

˜ Λ1 = −2η˜ lω

˜ T (1)µν = Πµν

αβDαωβ

˜ lω = 2κµ3 p + 211π2µλ p ✓ 3r2

+ − 2µ2 − πTµ2

r+ ◆ ω = −i η 4pk2 ⇣ 1 ± ˜ lωk + O(k2) ⌘ SHEAR WAVES DISPERSION RELATION

[KHARZEEV, YEE]

ANOMALY IN THE ENERGY MOMENTUM TENSOR

SOME OF THE COEFFICIENTS IN THE TENSOR AND VECTOR SECTOR

˜ ξ5 = σ˜ lE ˜ J (5)µ = µναβuνDαEβ p ˜ lE = −23r3

+¯

µ π (κ log 2 − +2λ(1 + 2 log 2)) + O(¯ µ3)

ANOMALY IN THE CHARGE CURRENT

SOME OF THE COEFFICIENTS IN THE TENSOR AND VECTOR SECTOR

˜ ξ5 = σ˜ lE ˜ J (5)µ = µναβuνDαEβ p ˜ lE = −23r3

+¯

µ π (κ log 2 − +2λ(1 + 2 log 2)) + O(¯ µ3) ~ J = ⇠B ~ B − ˜ ⇠5 @ ~ B @t + . . .

ANOMALY IN THE CHARGE CURRENT

SUMMARY & OUTLOOK

TA hµν hλβ Aρ

# MEASURABLE IN THE LAB IN HYDRODYNAMICAL REGIMES?? # HIGH TEMPERATURES ENHANCE ANOMALY INDUCE CURRENTS # THE MIXED ANOMALY CONTRIBUTES NON TRIVIALLY IN HYDRODYNAMICAL TRANSPORT COEFF., SO IT HAS TO BE CONSIDERED! # ANOMALIES SHOW QUANTUM EFFECTS AT MACROSCOPICAL LEVEL # ANOMALIES INDUCE A CHARACTERISTIC LENGTH IN WHICH HELICITY +/- CAN BE DISTINGUISHED # KUBO FORMULAS IN TERM OF TWO POINT FUNCTIONS CAN BE WRITTEN FOR THE COEFFICIENTS I WAVE SHOWN.

THANKS...