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 r µ J µ = 3 c A 4 � µ νρλ F µ ν F ρλ + c m β µ ν R β 4 � µ νρλ R α αρλ T B A µ h µ ν A ρ A ρ T A T A h λβ A ν T C (NON)CONSERVED CURRENT

ANOMALIES AND HYDRODYNAMICS r µ J µ = 3 c A 4 � µ νρλ F µ ν F ρλ + c m β µ ν R β 4 � µ νρλ R α αρλ J µ = j µ + j µ (1) ano = ξω µ + ξ B B µ J µ ano ✓ 1 ◆ n 2 c A µ 2 + c m T 2 ⇠ B = c A µ − ✏ + p ✓ 1 ◆ ⇠ = 1 n 2 c A µ 2 + c m T 2 − 3 c A µ 3 + 2 c m µT 2 ✏ + p [ERDMENGER, ET AL, ’08] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10] [BANERJEE, ET AL, ’08] [NEIMAN AND OZ, ‘10] [YAROM, JENSEN, LOGANAYAGAM,10]

ANOMALIES AND HYDRODYNAMICS r µ J µ = 3 c A 4 � µ νρλ F µ ν F ρλ + c m β µ ν R β 4 � µ νρλ R α αρλ J µ = j µ + j µ (1) ano = ξω µ + ξ B B µ J µ ano ✓ 1 ◆ n 2 c A µ 2 + c m T 2 ⇠ B = c A µ − ✏ + p ✓ 1 ◆ ⇠ = 1 n 2 c A µ 2 + c m T 2 − 3 c A µ 3 + 2 c m µT 2 ✏ + p [ERDMENGER, ET AL, ’08] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10] [BANERJEE, ET AL, ’08] [NEIMAN AND OZ, ‘10] [YAROM, JENSEN, LOGANAYAGAM,10]

ANOMALIES AND HYDRODYNAMICS PARITY AND TIME REVERSAL PROPERTIES r µ J µ = 3 c A 4 � µ νρλ F µ ν F ρλ + c m β µ ν R β 4 � µ νρλ R α αρλ J µ = j µ + j µ (1) ano = ξω µ + ξ B B µ J µ ano ✓ 1 ◆ n 2 c A µ 2 + c m T 2 ⇠ B = c A µ − ✏ + p ✓ 1 ◆ ⇠ = 1 n 2 c A µ 2 + c m T 2 − 3 c A µ 3 + 2 c m µT 2 ✏ + p [ERDMENGER, ET AL, ’08] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10] [BANERJEE, ET AL, ’08] [NEIMAN AND OZ, ‘10] [YAROM, JENSEN, LOGANAYAGAM,10]

ANOMALIES AND HYDRODYNAMICS PARITY AND TIME REVERSAL PROPERTIES j = � B ~ ~ B r µ J µ = 3 c A 4 � µ νρλ F µ ν F ρλ + c m β µ ν R β 4 � µ νρλ R α αρλ J µ = j µ + j µ (1) ano = ξω µ + ξ B B µ J µ ano ✓ 1 ◆ n 2 c A µ 2 + c m T 2 ⇠ B = c A µ − ✏ + p ✓ 1 ◆ ⇠ = 1 n 2 c A µ 2 + c m T 2 − 3 c A µ 3 + 2 c m µT 2 ✏ + p [ERDMENGER, ET AL, ’08] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10] [BANERJEE, ET AL, ’08] [NEIMAN AND OZ, ‘10] [YAROM, JENSEN, LOGANAYAGAM,10]

ANOMALIES AND HYDRODYNAMICS PARITY AND TIME REVERSAL PROPERTIES j = � B ~ ~ B r µ J µ = 3 c A 4 � µ νρλ F µ ν F ρλ + c m β µ ν R β 4 � µ νρλ R α αρλ J µ = j µ + j µ (1) ano = ξω µ + ξ B B µ J µ ano P-even T-odd ✓ 1 ◆ n 2 c A µ 2 + c m T 2 ⇠ B = c A µ − ✏ + p ✓ 1 ◆ ⇠ = 1 n 2 c A µ 2 + c m T 2 − 3 c A µ 3 + 2 c m µT 2 ✏ + p [ERDMENGER, ET AL, ’08] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10] [BANERJEE, ET AL, ’08] [NEIMAN AND OZ, ‘10] [YAROM, JENSEN, LOGANAYAGAM,10]

ANOMALIES AND HYDRODYNAMICS PARITY AND TIME REVERSAL PROPERTIES j = � B ~ ~ B r µ J µ = 3 c A 4 � µ νρλ F µ ν F ρλ + c m β µ ν R β 4 � µ νρλ R α αρλ J µ = j µ + j µ (1) ano = ξω µ + ξ B B µ J µ ano P-even P-odd T-odd T-odd ✓ 1 ◆ n 2 c A µ 2 + c m T 2 ⇠ B = c A µ − ✏ + p ✓ 1 ◆ ⇠ = 1 n 2 c A µ 2 + c m T 2 − 3 c A µ 3 + 2 c m µT 2 ✏ + p [ERDMENGER, ET AL, ’08] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10] [BANERJEE, ET AL, ’08] [NEIMAN AND OZ, ‘10] [YAROM, JENSEN, LOGANAYAGAM,10]

ANOMALIES AND HYDRODYNAMICS PARITY AND TIME REVERSAL PROPERTIES j = � B ~ ~ B r µ J µ = 3 c A 4 � µ νρλ F µ ν F ρλ + c m β µ ν R β 4 � µ νρλ R α αρλ J µ = j µ + j µ (1) ano = ξω µ + ξ B B µ J µ ano P-even P-odd T-odd T-odd ✓ 1 ◆ n 2 c A µ 2 + c m T 2 ⇠ B = c A µ − P-odd ✏ + p T-even ✓ 1 ◆ ⇠ = 1 n 2 c A µ 2 + c m T 2 − 3 c A µ 3 + 2 c m µT 2 ✏ + p [ERDMENGER, ET AL, ’08] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10] [BANERJEE, ET AL, ’08] [NEIMAN AND OZ, ‘10] [YAROM, JENSEN, LOGANAYAGAM,10]

ANOMALIES AND HYDRODYNAMICS r µ J µ = 3 c A 4 � µ νρλ F µ ν F ρλ + c m β µ ν R β 4 � µ νρλ R α αρλ J µ = j µ + j µ (1) ano = ξω µ + ξ B B µ J µ ano ✓ 1 ◆ n 2 c A µ 2 + c m T 2 ⇠ B = c A µ − ✏ + p ✓ 1 ◆ ⇠ = 1 n 2 c A µ 2 + c m T 2 − 3 c A µ 3 + 2 c m µT 2 ✏ + p [ERDMENGER, ET AL, ’08] [SON, SUROWKA, ‘09] [LANDSTEINER, MEGÍAS, PB,10] [BANERJEE, ET AL, ’08] [NEIMAN AND OZ, ‘10] [YAROM, JENSEN, LOGANAYAGAM,10]

SECOND ORDER ANOMALOUS CONSTITUTIVE RELATIONS AT SECOND ORDER IN A CONFORMAL FLUID WE HAVE MANY CONTRIBUTIONS a =15 a =10 X Λ a T ( a ) µ ν X ξ a J ( a ) µ τ µ ν j µ (2) = (2) = a =1 a =1

SECOND ORDER ANOMALOUS CONSTITUTIVE RELATIONS AT SECOND ORDER IN A CONFORMAL FLUID WE HAVE MANY CONTRIBUTIONS a =15 a =10 X Λ a T ( a ) µ ν X ξ a J ( a ) µ τ µ ν j µ (2) = (2) = a =1 a =1 a =8 a =5 X T ( a ) µ ν Λ a ˜ ˜ X ξ a ˜ ˜ J ( a ) µ τ µ ν j µ (2) ano = (2) ano = a =1 a =1

SECOND ORDER ANOMALOUS CONSTITUTIVE RELATIONS AT SECOND ORDER IN A CONFORMAL FLUID WE HAVE MANY CONTRIBUTIONS a =15 a =10 X Λ a T ( a ) µ ν X ξ a J ( a ) µ τ µ ν j µ (2) = (2) = a =1 a =1 a =8 a =5 X T ( a ) µ ν Λ a ˜ ˜ X ξ a ˜ ˜ J ( a ) µ τ µ ν j µ (2) ano = (2) ano = a =1 a =1 I WILL FOCUS ON ANOMALOUS CONTRIBUTION WITH SECOND DERIVATIVE TERMS

SECOND ORDER ANOMALOUS CONSTITUTIVE RELATIONS AT SECOND ORDER IN A CONFORMAL FLUID WE HAVE MANY CONTRIBUTIONS a =15 a =10 X Λ a T ( a ) µ ν X ξ a J ( a ) µ τ µ ν j µ (2) = (2) = a =1 a =1 a =8 a =5 X T ( a ) µ ν Λ a ˜ ˜ X ξ a ˜ ˜ J ( a ) µ τ µ ν j µ (2) ano = (2) ano = a =1 a =1 I WILL FOCUS ON ANOMALOUS CONTRIBUTION WITH SECOND DERIVATIVE TERMS T (1) µ ν = Π µ ν ˜ αβ D α ω β J (5) µ = � µ ναβ u ν D α E β ˜ T (4) µ ν = Π µ ν ˜ αβ D α B β

SECOND ORDER ANOMALOUS CONSTITUTIVE RELATIONS J (5) µ = � µ ναβ u ν D α E β ˜ r ⇥ ~ E THE CONDUCTIVITIES ASOCIATED TO THE ANOMALOUS SOURCES ARE DISSIPATIVE AND PARITY VIOLATING! P-even T-even

SECOND ORDER ANOMALOUS CONSTITUTIVE RELATIONS PARITY AND TIME REVERSAL AGAIN! J (5) µ = � µ ναβ u ν D α E β ˜ r ⇥ ~ E THE CONDUCTIVITIES ASOCIATED TO THE ANOMALOUS SOURCES ARE DISSIPATIVE AND PARITY VIOLATING! P-even T-even

STRONGLY COUPLED MODEL S = S EHM + S CS + S GH + S CSK Z √− g ✏ MNP QR A M h  i 3 F NP F QR + � R A BNP R B S CS = AQR

STRONGLY COUPLED MODEL S = S EHM + S CS + S GH + S CSK Z √− g ✏ MNP QR A M h  i 3 F NP F QR + � R A BNP R B S CS = AQR UNDER A GAUGE TRANSFORMATION Z − h ⇠✏ µ νρλ ⇣  √ ⌘ β µ ν R β 3 F µ ν F ρλ + � R α � S = αρλ ∂

STRONGLY COUPLED MODEL S = S EHM + S CS + S GH + S CSK Z √− g ✏ MNP QR A M h  i 3 F NP F QR + � R A BNP R B S CS = AQR UNDER A GAUGE TRANSFORMATION Z − h ⇠✏ µ νρλ ⇣  √ ⌘ β µ ν R β 3 F µ ν F ρλ + � R α � S = αρλ ∂ − λ = 1 − 1 3 κ = 1 4 c m 4 c A

FLUID/GRAVITY ANSATZ  W 2 ( ρ ) η µ ν + W 3 ( ρ ) u µ u ν + 2 W 4 σ ( ρ ) µ u ν + W 5 µ ν ( ρ ) � ds 2 = − 2 W 1 ( ρ ) u µ dx µ � dr 2 + r A ν dx ν � + r 2 P σ dx µ dx ν r 2 r + + c ( ⇥ ) = ⇤ ( ⇥ ) ⇣ ⌘ a ( b ) + a µ ( ρ ) P µ A = ν + r + c ( ρ ) u ν dx ν + O ( � ) ν r + W 1 ( ⇥ ) = 1 + O ( � ) EPSILON ZERO MEANS NO X DEPENDENCE AND W 2 ( ⇥ ) = 1 + O ( � ) BOUNDARY BACKGROUND THE ANSATZ BECOME GAUGE FIELD IN THE BOOSTED W 3 ( ⇥ ) = 1 − f ( ⇥ ) + O ( � ) CHARGED BLACK HOLE SOLUTION W 4 µ ( ⇥ ) = O ( � ) W 5 µ ν ( ⇥ ) = O ( � ) a µ ( ⇥ ) = O ( � ) u µ ( x ) u µ ( x ) = − 1 COUNTS THE NUMBER OF TRANSVERSE DERIVATIVES

FLUID/GRAVITY  W 2 ( ρ ) η µ ν + W 3 ( ρ ) u µ u ν + 2 W 4 σ ( ρ ) µ u ν + W 5 µ ν ( ρ ) � ds 2 = − 2 W 1 ( ρ ) u µ dx µ � dr 2 + r A ν dx ν � + r 2 P σ dx µ dx ν r 2 r + + ⇣ ⌘ a ( b ) + a µ ( ρ ) P µ A = ν + r + c ( ρ ) u ν dx ν ν HOLOGRAPHIC ONE POINT FUNCTIONS ⇣ ⌘ + a (¯ r 2 2 , ✏ ) + J ct J µ lim ∼ µ µ ✏ → 0 + W (¯ ⇣ ⌘ 4 , ✏ ) 4 r 2 5 µ ⌫ + T ct T µ ⌫ lim ∼ µ ⌫ ✏ → 0

SOME OF THE COEFFICIENTS IN THE TENSOR AND VECTOR SECTOR ANOMALY IN THE ENERGY MOMENTUM TENSOR T (1) µ ν = Π µ ν Λ 1 = − 2 η ˜ ˜ ˜ αβ D α ω β l ω

SOME OF THE COEFFICIENTS IN THE TENSOR AND VECTOR SECTOR ANOMALY IN THE ENERGY MOMENTUM TENSOR T (1) µ ν = Π µ ν Λ 1 = − 2 η ˜ ˜ ˜ αβ D α ω β l ω l ω = 2 κ µ 3 + 2 11 π 2 µ λ + − 2 µ 2 − π Tµ 2 ✓ ◆ ˜ 3 r 2 p p r +

SOME OF THE COEFFICIENTS IN THE TENSOR AND VECTOR SECTOR ANOMALY IN THE ENERGY MOMENTUM TENSOR T (1) µ ν = Π µ ν Λ 1 = − 2 η ˜ ˜ ˜ αβ D α ω β l ω l ω = 2 κ µ 3 + 2 11 π 2 µ λ + − 2 µ 2 − π Tµ 2 ✓ ◆ ˜ 3 r 2 p p r + T (4) µ ν = Π µ ν ˜ αβ D α B β Λ 2 = − 2 η ˜ ˜ l B