Applications of string theory to the very hot and the very cold - PowerPoint PPT Presentation

Applications of string theory to the very hot and the very cold Steve Gubser Princeton University 15th European Workshop on String Theory, ETH Z¨ urich Based on work with C. Herzog, A. Nellore, S. Pufu, F. Rocha, and A. Yarom September 9, 2009

ptContents 1 The very hot: heavy-ion collisions 3 1.1 Equation of state and bulk viscosity . . . . . . . . . . . . . . . . . 4 1.2 Drag force on heavy quarks . . . . . . . . . . . . . . . . . . . . . 8 1.3 Stochastic forces and the Einstein relation . . . . . . . . . . . . . 13 1.4 The worldsheet horizon . . . . . . . . . . . . . . . . . . . . . . . 15 2 The very cold: superconductors and superfluids 17 2.1 The basics of superconducting black holes . . . . . . . . . . . . . 17 2.2 A candidate ground state . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Embedding in string theory . . . . . . . . . . . . . . . . . . . . . 22 2.4 A critical velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Conclusions 30

Gubser, Applications to the hot and the cold, 9-9-09 3 The very hot: heavy-ion collisions 1. The very hot: heavy-ion collisions T peak ≈ 300 MeV for central RHIC collisions, about 200 , 000 times hotter than the core of the sun, and about 1 . 7 times bigger than T c ≈ 180 MeV where QCD deconfines. First natural question: What is the equation of state? Lattice gives pretty reliable answers (except T c is hard to pin down in MeV ). ǫ/ǫ free = 0 . 88 ↔ λ SY M = 5 . 5 ǫ/ǫ free = 0 . 77 ↔ λ SY M = 6 π [Bazavov et al. 2009]

Gubser, Applications to the hot and the cold, 9-9-09 4 Equation of state and bulk viscosity 1.1. Equation of state and bulk viscosity Authors of [Kharzeev and Tuchin 2008; Karsch et al. 2008] suggest a way to trans- late EOS into a prediction for bulk viscosity: 1 � T 5 ∂ ǫ − 3 p � ζ = − 16 ǫ vac + (quark terms) . (1) 9 ω 0 ∂T T 4 (1) comes out of a low-energy theorem (“sum rule”) for θ ≡ T µ µ : � � T ∂ � G E (0 ,� d 4 x � θ ( x ) θ (0) � = ∂T − 4 � θ (0) � + (quark terms) , 0) = (2) plus observation that � θ (0) � = ǫ − 3 p + 4 ǫ vac , plus (crucially) the assumption of a low-frequency parametrization ω 2 0) = 9 ζω ρ ( ω,� 0 ω 0 ∼ 1 GeV (3) ω 2 π 0 + ω 2 for the spectral measure of the two-point function of T µ µ . Because (3) is ad hoc , it seems worthwhile to obtain ζ using strongly coupled meth- ods and compare with (1).

Gubser, Applications to the hot and the cold, 9-9-09 5 Equation of state and bulk viscosity The results [Gubser and Nellore 2008; Gubser et al. 2008ab]: ζ rises near T c , but not so much as (1) predicts. 2 c s • Type I: smooth cross- � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 0.30 over: quasi-realistic. � � � 0.25 � QPM � � • Type II: nearly second � lattice, 2 � 1 flavors � � 0.20 � order, c 2 s → 0 at T c . lattice, pure glue � � 0.15 Type I BH � • Type III: No BH below � Type II BH � 0.10 � � � T c , like [Gursoy et al. Type III BH � � 0.05 2008b]. 4.0 T � T c 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 • Sharper behavior of c 2 s gives Ζ � s sharper ζ/s . � Type I BH, �� 3.94 � Type II BH, �� 3.99 1.000 • Large ζ at T c is hard to ar- 0.500 � Type II BH, �� 3 lattice, pure glue range with a reasonably re- sum rule, 2 � 1 0.100 alistic EOS. � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 0.050 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • Poses a challenge for “soft � � � � � � � � � � � � � � � � � � � � � � � � � 0.010 � � � � � � � � statistical hadronization” � 0.005 � � � � � � proposal of [Karsch et al. � 4.0 T � T c 0.001 1.0 1.5 2.0 2.5 3.0 3.5 2008].

Gubser, Applications to the hot and the cold, 9-9-09 6 Equation of state and bulk viscosity The method: Reproduce the lattice EOS using 1 � R − 1 � 2( ∂φ ) 2 − V ( φ ) L = . (4) 2 κ 2 5 V ( φ ) can be adjusted to match dependence of s ≡ dp c 2 (5) speed of sound: dǫ 5 to get desired ǫ/T 4 at some high scale (say 3 GeV ). A quasi- on T . Then adjust κ 2 realistic EOS comes from V ( φ ) = − 12 cosh γφ + bφ 2 γ = 0 . 606 , b = 2 . 057 . (6) L 2 Authors of [Gursoy and Kiritsis 2008; Gursoy et al. 2008ab] took same starting √ 2 point (4) further: an appropriate V ( φ ) , with V ∼ − φ 2 e 3 φ , gives a Hawking- Page transition to confinement; logarithmic RG in UV; glueball with m 2 ∼ n , as in linear confinement; and favorable comparison with thermodynamic and transport quantities [Gursoy et al. 2009ab].

Gubser, Applications to the hot and the cold, 9-9-09 7 Equation of state and bulk viscosity Once conformal invariance is broken, we can investigate bulk viscosity [Gubser et al. 2008ba], following a number of earlier works, e.g. [Parnachev and Starinets 2005; Buchel 2005 2007]: ζ = 1 1 � d 3 x dt e iωt θ ( t ) � [ T µ x ) , T ν 9 lim ω Im µ ( t, � ν (0 , 0)] � . (7) ω → 0 3,1 R t,x Shear viscosity relates to absorption probability for h / ϕ h z an h 12 graviton. Bulk vis- 12 ii cosity relates to absorption of a mixture of the h ii gravi- ton and the scalar φ . ii / ϕ η ∼ p absorb 12 ζ ∼ p absorb horizon z = z H + e 2 B ( r ) dr 2 ds 2 = e 2 A ( r ) � − h ( r ) dt 2 + d� x 2 � φ = φ ( r ) . (8) h ( r ) In a gauge where δφ = 0 , let’s set h 11 = e − 2 A δg 11 = e − 2 A δg 22 = e − 2 A δg 33 . Then 3 A ′ − 4 A ′ + 3 B ′ − h ′ − e − 2 A +2 B 6 hA ′ − h ′ B ′ h ′ � − 1 � � � ω 2 + h ′′ h ′ 11 = 11 + h 11 h h 2 h (9)

Gubser, Applications to the hot and the cold, 9-9-09 8 Drag force on heavy quarks 1.2. Drag force on heavy quarks The results: [Herzog et al. 2006; Gubser 2006a] v Quark can’t slow down T mn q because m = ∞ R 3,1 h mn AdS −Schwarzschild fundamental string 5 Horizon is “sticky” because of gravitational redshift: prevents string from moving. horizon √ dp dt = − π λ 1 − v 2 = − p v 2 m Q T 2 √ √ τ Q = SY M 2 τ Q πT 2 λ SY M τ charm ≈ 2 fm τ bottom ≈ 6 fm if T QCD = 250 MeV

Gubser, Applications to the hot and the cold, 9-9-09 9 Drag force on heavy quarks The method: Consider a more general problem of embedding a string in a warped background [Herzog 2006; Gursoy et al. 2009b; Gubser and Yarom 2009]:  τ + ζ ( r )  ds 2 = − e 2 A ( r ) h ( r ) dt 2 vτ + vζ ( r ) + ξ ( r )   X µ ( τ, r ) = 0  , x 2 + dr 2 (10)   + e 2 A ( r ) d�   0  h ( r ) r Using classical equations of motion and a gauge choice for ζ , find � vξ ′ ξ ′ ( r ) = − π ξ h − v 2 ζ ′ ( r ) = h − v 2 , (11) he 4 A / (2 πα ′ ) 2 − π 2 he A ξ where π ξ = ∂ L string /∂ξ ′ . To make ξ ′ ( r ) everywhere real, we must choose � h ( r ∗ ) e 2 A ( r ∗ ) h ( r ∗ ) = v 2 . (12) π ξ = − where 2 πα ′ F drag can be argued to be precisely ( π ξ , 0 , 0) .

Gubser, Applications to the hot and the cold, 9-9-09 10 Drag force on heavy quarks A recent study shows that these equilibration times are at least roughly consistent with R AA of non-photon electrons: F drag = − γ T 2 � p � 1.8 m Q =0.3 � ± 1.6 (a) b=3.1fm, c+b->e =1.0 � γ ≈ 2 based on AdS/CFT =3.0 � 1.4 PHENIX(0-10%) STAR(0-5%) 1.2 Colored triples show 1 different freezeout AA R assumptions 0.8 0.6 0.4 Analysis should work for p T > ∼ 3 GeV . 0.2 [Akamatsu et al. 2008] 0 0 1 2 3 4 5 6 7 8 9 10 p [GeV] T To get this γ ≈ 2 , have to match SYM and QCD at fixed energy density, and also set λ ≡ g 2 Y M N = 5 . 5 to approximately match the static q - ¯ q force calculated from the lattice [Gubser 2006c].

Gubser, Applications to the hot and the cold, 9-9-09 11 Drag force on heavy quarks A bit more detail on why g 2 Y M N ≈ 5 . 5 based on matching string theory to lattice q - ¯ q potential: • Lattice people define an effective coupling: q ( r, T ) ≡ 3 4 r 2 ∂F q ¯ q α q ¯ ∂r . (13) • Analogous quantity in string theory receives contributions from two configura- tions: q r q q r q 3,1 x 3,1 R R y AdS 5 −Schwarzschild AdS 5 −Schwarzschild fundamental massless exchange string horizon horizon • Simplest approximation to U-curve contribution is zero temperature result: 3 π 2 α SYM ( T =0) ≡ 3 4 r 2 ∂V q ¯ q � g 2 ∂r = Y M N Γ(1 / 4) 4 . (14) T � = 0 results in a bit of Debye screening.