Resurgence and Hydrodynamics in Gauss-Bonnet Holography Ben Meiring - PowerPoint PPT Presentation

Resurgence and Hydrodynamics in Gauss-Bonnet Holography Ben Meiring & Jorge Casalderrey-Solana (Oxford) ben.meiring@physics.ox.ac.uk Jan, 2017 1

Hydrodynamics in 3+1 Dimensions The equation of motion for Hydrodynamics is the conservation equation ∇ µ T µν = 0 (1) where T µν = T µν ( ǫ, P, u µ ) with ǫ the energy density, P the Pressure, and u µ the fluid velocity. 2

Hydrodynamics in 3+1 Dimensions The equation of motion for Hydrodynamics is the conservation equation ∇ µ T µν = 0 (1) where T µν = T µν ( ǫ, P, u µ ) with ǫ the energy density, P the Pressure, and u µ the fluid velocity. For a perfect fluid ideal = ( ǫ + P ) u µ u ν − Pη µν . T µν (2) 2

Hydrodynamics in 3+1 Dimensions The equation of motion for Hydrodynamics is the conservation equation ∇ µ T µν = 0 (1) where T µν = T µν ( ǫ, P, u µ ) with ǫ the energy density, P the Pressure, and u µ the fluid velocity. For a perfect fluid ideal = ( ǫ + P ) u µ u ν − Pη µν . T µν (2) For a non-ideal fluid, we include every possible tensor combination of ∂ µ , u µ and η µν with co-efficients c i . 2

Hydrodynamics in 3+1 Dimensions The equation of motion for Hydrodynamics is the conservation equation ∇ µ T µν = 0 (1) where T µν = T µν ( ǫ, P, u µ ) with ǫ the energy density, P the Pressure, and u µ the fluid velocity. For a perfect fluid ideal = ( ǫ + P ) u µ u ν − Pη µν . T µν (2) For a non-ideal fluid, we include every possible tensor combination of ∂ µ , u µ and η µν with co-efficients c i . T µν = T µν ideal + c 1 ∂ µ u ν + c 2 ∂ ν u µ + c 3 η µν ∂ α u α + c 4 u µ u ν ∂ α u α + ... (3) Generally symmetries of the theory can be used to constrain these co-efficients c i . 2

Hydrodynamics in 3+1 Dimensions For example, in a conformal theory all these co-efficients (at order ∂ µ u ν ) are contrained to η the shear viscousity T µν = T µν ideal − ησ µν + ... (4) where σ µν = ( ∂ µ u ν + ∂ ν u µ − 2 3 ( u µ u ν + η µν ) ∂ α u α ) . 3

Hydrodynamics in 3+1 Dimensions For example, in a conformal theory all these co-efficients (at order ∂ µ u ν ) are contrained to η the shear viscousity T µν = T µν ideal − ησ µν + ... (4) where σ µν = ( ∂ µ u ν + ∂ ν u µ − 2 3 ( u µ u ν + η µν ) ∂ α u α ) . In general we can include all derivatives of u µ T µν = T µν ideal + O ( ∼ ∂ µ u ν ) + O ( ∼ ( ∂ µ u ν ) 2 ) + ... (5) ◮ This series is known as the Gradient Expansion and orders itself in ∂ µ u ν << 1 when u µ is slowly varying. 3

Hydrodynamics in 3+1 Dimensions For example, in a conformal theory all these co-efficients (at order ∂ µ u ν ) are contrained to η the shear viscousity T µν = T µν ideal − ησ µν + ... (4) where σ µν = ( ∂ µ u ν + ∂ ν u µ − 2 3 ( u µ u ν + η µν ) ∂ α u α ) . In general we can include all derivatives of u µ T µν = T µν ideal + O ( ∼ ∂ µ u ν ) + O ( ∼ ( ∂ µ u ν ) 2 ) + ... (5) ◮ This series is known as the Gradient Expansion and orders itself in ∂ µ u ν << 1 when u µ is slowly varying. ◮ The co-efficients c i are known as transport co-efficients and uniquely specify our theory. 3

Bjorken Flow There is a phenomologically relevant model for Heavy Ion collisions known as Bjorken Flow. 4

Bjorken Flow There is a phenomologically relevant model for Heavy Ion collisions known as Bjorken Flow. y y x z Longitudinal Plane Transverse Plane Figure: Head-on and Side profiles for a Lead-Lead collision. The overlapping region results in an energy density that evolves longitudinally according to hydrodynamics. 4

Bjorken Flow ◮ This energy density T 00 = ǫ is a function of only the proper time, and the form is known to all orders: ǫ ( τ ) = τ − 4 / 3 ( ǫ 0 + ǫ 1 τ − 2 / 3 + ǫ 2 τ − 4 / 3 + ... ) (6) 5

Bjorken Flow ◮ This energy density T 00 = ǫ is a function of only the proper time, and the form is known to all orders: ǫ ( τ ) = τ − 4 / 3 ( ǫ 0 + ǫ 1 τ − 2 / 3 + ǫ 2 τ − 4 / 3 + ... ) (6) ◮ Each new factor of τ − 2 / 3 comes from exactly each new order of ∂ µ u ν in the gradient expansion, and the transport co-efficients are related to each ǫ i . 5

Bjorken Flow ◮ This energy density T 00 = ǫ is a function of only the proper time, and the form is known to all orders: ǫ ( τ ) = τ − 4 / 3 ( ǫ 0 + ǫ 1 τ − 2 / 3 + ǫ 2 τ − 4 / 3 + ... ) (6) ◮ Each new factor of τ − 2 / 3 comes from exactly each new order of ∂ µ u ν in the gradient expansion, and the transport co-efficients are related to each ǫ i . ◮ To gain some understanding of this evolving system analytically, we need a way to calculate the energy co-efficients for a QCD-like theory at Strong Coupling. 5

Bjorken Flow ◮ This energy density T 00 = ǫ is a function of only the proper time, and the form is known to all orders: ǫ ( τ ) = τ − 4 / 3 ( ǫ 0 + ǫ 1 τ − 2 / 3 + ǫ 2 τ − 4 / 3 + ... ) (6) ◮ Each new factor of τ − 2 / 3 comes from exactly each new order of ∂ µ u ν in the gradient expansion, and the transport co-efficients are related to each ǫ i . ◮ To gain some understanding of this evolving system analytically, we need a way to calculate the energy co-efficients for a QCD-like theory at Strong Coupling. ◮ N = 4 SYM (a QCD-like theory) can be re-written at infinite coupling as a gravitational theory. 5

The Fluid-Gravity correspondence We can perform classical gravity calculations to find strongly coupled QFT results. Hydrodynamical QFT Black Hole Geometry Figure: Some Gauge theories and Gravity theories are conjectured to be the same theory under a field redefinition. 6

The Fluid-Gravity correspondence We can perform classical gravity calculations to find strongly coupled QFT results. Hydrodynamical QFT Black Hole Geometry Figure: Some Gauge theories and Gravity theories are conjectured to be the same theory under a field redefinition. 7

The Fluid-Gravity Correspondence The geometry that is dual to Bjorken Flow Hydrodynamics in N = 4 SYM at infinite coupling is given by ds 2 = − r 2 A ( r, τ ) dτ 2 +2 dτdr +( rτ +1) 2 e B ( r,τ ) dy 2 + r 2 e C ( r,τ ) dx 2 ⊥ (7) where r is the radial distance towards the Black Hole, and τ is the proper time. 8

The Fluid-Gravity Correspondence The geometry that is dual to Bjorken Flow Hydrodynamics in N = 4 SYM at infinite coupling is given by ds 2 = − r 2 A ( r, τ ) dτ 2 +2 dτdr +( rτ +1) 2 e B ( r,τ ) dy 2 + r 2 e C ( r,τ ) dx 2 ⊥ (7) where r is the radial distance towards the Black Hole, and τ is the proper time. A , B and C are defined by: � 4 τ − 2 3 i A i ( r − 1 τ − 1 / 3 ) , � 1 A ( τ, r ) = � A 0 = 1 − rτ 1 / 3 i =0 τ − 2 3 i B i ( r − 1 τ − 1 / 3 ) , B ( τ, r ) = � B 0 = 0 i =0 τ − 2 3 i C i ( r − 1 τ − 1 / 3 ) , C ( τ, r ) = � C 0 = 0 . i =0 (Kinoshita, Mukohyama & Nakamura [arXiv:0807.3797v2]) 8

The Fluid-Gravity Correspondence The geometry that is dual to Bjorken Flow Hydrodynamics in N = 4 SYM at infinite coupling is given by ds 2 = − r 2 A ( r, τ ) dτ 2 +2 dτdr +( rτ +1) 2 e B ( r,τ ) dy 2 + r 2 e C ( r,τ ) dx 2 ⊥ (8) where r is the radial distance towards the Black Hole, and τ is the proper time. This looks a little like a space with a blackhole a horizon sinking into the radial direction. Boundary Theory r = ∞ r = τ − 1 3 Black Hole Figure: Schematic cartoon of the Geometry. 9

Resurgence We can calculate ǫ i to large orders from this solution. 3 + ǫ 2 τ − 4 / 3 + ... ) ǫ ( τ ) = τ − 4 / 3 ( ǫ 0 + ǫ 1 τ − 2 (9) 10

Resurgence We can calculate ǫ i to large orders from this solution. 3 + ǫ 2 τ − 4 / 3 + ... ) ǫ ( τ ) = τ − 4 / 3 ( ǫ 0 + ǫ 1 τ − 2 (9) But after some finite order, the co-efficients start to contribute more and more! Figure: Energy density co-efficients ǫ 1 /n as a function of order n. n Note that ( n !) 1 /n ∼ n for large n . [arXiv:1302.0697v2] 10

Resurgence Using the identity: ∞ � e − uτ 2 / 3 � � n � � τ − 2 u n n ! = du (10) 3 τ 2 / 3 0 11

Resurgence Using the identity: ∞ � e − uτ 2 / 3 � � n � � τ − 2 u n n ! = du (10) 3 τ 2 / 3 0 we can write our diverging series 3 + ǫ 2 τ − 4 / 3 + ... ) ǫ ( τ ) = τ − 4 / 3 ( ǫ 0 + ǫ 1 τ − 2 (11) 11

Resurgence Using the identity: ∞ � e − uτ 2 / 3 � � n � � τ − 2 u n n ! = du (10) 3 τ 2 / 3 0 we can write our diverging series 3 + ǫ 2 τ − 4 / 3 + ... ) ǫ ( τ ) = τ − 4 / 3 ( ǫ 0 + ǫ 1 τ − 2 (11) as an integral of a converging series ∞ � � � ǫ 0 e − uτ 2 / 3 � 2! u 2 + ǫ 1 3! u 3 + ǫ 2 4! u 4 + ... � ǫ ( τ ) = du (12) τ 2 / 3 0 11

Resurgence This convergent series is called the Borel Sum ζ ( u ) = ǫ 0 2! u 2 + ǫ 1 3! u 3 + ǫ 2 4! u 4 + ... (13) 12

Resurgence This convergent series is called the Borel Sum ζ ( u ) = ǫ 0 2! u 2 + ǫ 1 3! u 3 + ǫ 2 4! u 4 + ... (13) If we plot ζ ( u ) in the complex plane we can examine the pole structure. Figure: Poles of the ζ ( u ) series containing non-perturbative information. [arXiv:1302.0697v2] 12

Resurgence To evaluate ∞ � � e − uτ 2 / 3 � ǫ ( τ ) = ζ ( u ) (14) du τ 2 / 3 0 we take the leading pole contribution. 13