Holographic thermalization patterns Stefan Stricker TU Vienna - PowerPoint PPT Presentation

Holographic thermalization patterns Stefan Stricker TU Vienna EPS-HEP, Stockholm July 22 2013 R. Baier, SS, O. Taanila, A. Vuorinen, 1207.116 (PRD) D. Steineder, SS, A. Vuorinen, 1209.0291 (PRL), 1304.3404 (JHEP) S. Stricker, arxive:1307.2736 INSTITUTE for THEORETICAL PHYSICS Vienna University of Technology Saturday, 20 July, 13

Motivation Quark gluon plasma Produced in heavy collisions at RHIC and LHC Behaves as a strongly coupled liquid Thermalization process not well understood: Goals Gain insight into the thermalization process Modification of production rates of photons Modification of energy momentum tensor correlators Which modes thermalize first: top-down or bottom-up ? Dependence on coupling strength Strategy SYM where strong and weak coupling regimes are accessible AdS/CFT: strongly coupled N=4 SYM is dual to classical SUGRA in AdS space Saturday, 20 July, 13

Outline Photons in N=4 SYM plasma Motivation Equilibrium properties at infinite and finite coupling Out of equilibrium at } thermalization scenarios Out of equilibrium at Plasma constituents Properties of energy momentum tensor correlators Saturday, 20 July, 13

Thermalization scenarios Bottom up scenario At weak coupling Scattering processes In the early stages many soft gluons are emitted which then thermalize the system ( Baier et al (2001) ) Supported by classical Yang-Mills simulations (Berges et al (2013)) Driven by instabilities Instabilities isotropize the momentum distributions more rapidly than scattering processes ( Kurkela, Moore (2011) ) Top down scenario At strong coupling UV modes thermalize first In AdS calculations, follows naturally from causality Saturday, 20 July, 13

Photon emission in heavy ion collisions Photons are emitted at all stages of the collision Initial hard scattering processes: quark anti-quark annihilation: on-shell photon or virtual photon → dilepton pair Strongly coupled out of equilibrium phase: no quasiparticle picture Additional (uninteresting) emissions from charged hadron decays Saturday, 20 July, 13

Probing the plasma Probing the plasma Once produced photons stream through the plasma almost unaltered Provide observational window in the thermalization process of the plasma Quantity of interest µ = − 2Im( Π ret ) µ µ ( k 0 ) χ µ Spectral density : Number of emitted photons Fluctuation dissipation theorem η µ ν Π < µ ν ( ω ) = − 2 n B ( ω )Im( Π ret ) µ µ ( ω ) = n B ( ω ) χ ( ω ) Production rate k 0 d Γ γ α µ ν ( ω = k 0 ) 4 π 2 η µ ν Π < d 3 k = Saturday, 20 July, 13

Photon emission in equilibrium SYM plasma Hassanain, Schvellinger (2012) Huot et al (2006) Perturbative result Increasing the coupling: slope at k=0 decreases, hydro peak broadens and moves right Strong coupling result Decreasing coupling from : peak sharpens and moves left λ = ∞ Saturday, 20 July, 13

Equilibrium summary Equilibrium picture in SYM fairly complete How does photon/dilepton production get modified out of equilibrium Can one access thermalization at finite coupling? Saturday, 20 July, 13

The falling shell setup AdS AdS-bh Danielsson, Keski-Vakkuri, Kruczenski (1999) r = 0 r s r = ∞ r h Outside and inside spacetime ds 2 = ( π TL ) 2 L 2 u = r 2 f ( u ) dt 2 + dx 2 + dy 2 + dz 2 � h � 4 u 2 f ( u ) du 2 metric: + r 2 u ⇢ f + ( u ) = 1 − u 2 , for u > 1 f ( u ) = , f − ( u ) = 1 , for u < 1 Saturday, 20 July, 13

The falling shell setup AdS AdS-bh Danielsson, Keski-Vakkuri, Kruczenski (1999) r = 0 r s r = ∞ r h Outside and inside spacetime ds 2 = ( π TL ) 2 L 2 u = r 2 f ( u ) dt 2 + dx 2 + dy 2 + dz 2 � h � 4 u 2 f ( u ) du 2 metric: + r 2 u ⇢ f + ( u ) = 1 − u 2 , for u > 1 f ( u ) = , f − ( u ) = 1 , for u < 1 Thermalization from geometric probes: Entanglement entropy and Wilson loop: always top down thermalization Saturday, 20 July, 13

The falling shell setup AdS AdS-bh Danielsson, Keski-Vakkuri, Kruczenski (1999) r = 0 r s r = ∞ r h Outside and inside spacetime ds 2 = ( π TL ) 2 L 2 u = r 2 f ( u ) dt 2 + dx 2 + dy 2 + dz 2 � h � 4 u 2 f ( u ) du 2 metric: + r 2 u ⇢ f + ( u ) = 1 − u 2 , for u > 1 f ( u ) = , f − ( u ) = 1 , for u < 1 Thermalization from geometric probes: Entanglement entropy and Wilson loop: always top down thermalization Saturday, 20 July, 13

The falling shell setup AdS AdS-bh Danielsson, Keski-Vakkuri, Kruczenski (1999) E in E − E out r = 0 r = ∞ r s r h Outside and inside spacetime ds 2 = ( π TL ) 2 L 2 u = r 2 f ( u ) dt 2 + dx 2 + dy 2 + dz 2 � h � 4 u 2 f ( u ) du 2 metric: + r 2 u ⇢ f + ( u ) = 1 − u 2 , for u > 1 f ( u ) = , f − ( u ) = 1 , for u < 1 Outside solution E + = c + E in + c − E out Saturday, 20 July, 13

Holographic Green’s functions Off-equilibrium correlators offer a useful window to thermalization: Probe how different energy scales approach equilibrium Related to measurable quantites, e.g. production rates Some computational details Solve classical EoM for bulk electric field E inside and outside the shell p E � ( ω � ) | u s = f m E + ( ω + ) | u s , Matching conditions: E 0 f m E 0 � ( ω � ) | u s = + ( ω + ) | u s . Use conventional methods to obtain retarded correlator E 0 1 + c � out Π ( ω , q ) = − N 2 c T 2 E ( u, Q ) = − N 2 c T 2 E 0 ( u, Q ) c + E 0 lim in Π therm 1 + c � E out 8 8 u ! 0 c + E in Behaviour of crucial for out of equilibrium dynamics c − /c + Quasistatic approximation: Energy scale of interest >> characteristic time scale of shell’s motion Saturday, 20 July, 13

Photon spectral density 1.0 virtuality 0.8 ω 2 − ˆ q 2 v = ˆ c=0 0.6 ω 2 ˆ Μ Χ Μ c=0.8 2 T Ω N c parametrize 0.4 q = c ˆ ω 0.2 c=1 0.0 0 5 10 15 20 25 Ω ê T spectral density for r s /r h =1.1 for different virtualities Out of equilibrium effect: oscillations around thermal value As the shell approaches the horizon equilibrium is reached Saturday, 20 July, 13

Relative deviation of spectral density 0.10 Relative deviation from thermal equilibrium 0.05 R 0.00 ω ) = χ (ˆ ω ) − χ th (ˆ ω ) R (ˆ χ th (ˆ ω ) - 0.05 - 0.10 20 40 60 80 w ê T relative deviation R for r s =1.1 and c=1, 0.8, 0 Top down thermalization: highly energetic modes are closer to equ. value Highly virtual field modes thermalize first ✓ ◆ 1 + f 1 ( u s ) R ≈ 1 2 χ (ˆ ω ) ≈ ˆ , 3 ω ˆ ˆ ω ω Saturday, 20 July, 13

Photon production rate at infinite coupling 1.5 1.0 d G ê dk 0 100 a H p N c L 2 T 3 0.5 0.0 0 1 2 3 4 5 6 7 w ê T photon production rate for r s /r h =1.1, 1.01, 1.001 Enhancement of production rate Hydro peak broadens and moves right Apparently no dramatic observable signature in off-equilibrium photon production Saturday, 20 July, 13

Photon production rate at infinite coupling 1.5 1.0 d G ê dk 0 100 a H p N c L 2 T 3 0.5 0.0 0 1 2 3 4 5 6 7 w ê T photon production rate for r s /r h =1.1, 1.01, 1.001 Enhancement of production rate Hydro peak broadens and moves right Apparently no dramatic observable signature in off-equilibrium photon production Combining the two allows to study thermalization at finite coupling! Saturday, 20 July, 13

Finite coupling corrections Key relation in AdS/CFT: Go beyond : add terms to SUGRA action, i.e. first non trivial terms in a small curvature expansion Leading order corrections: Gubser et al; Pawelczyk, Theisen (1998) Improved type IIB SUGRA action: γ ≡ 1 bcdef + 1 8 ζ (3) λ − 3 ⇣ ⌘ abcmn F + mn abfmn F + mn T abcdef = i r a F + F + � 3 F + , 2 def dec 16 Paulos (2008) Leads to -corrected metric EoM for different fields Saturday, 20 July, 13

Quasinormal modes infinite coupling ` Re w - 4 - 2 0 2 4 0 - 1 à à - 2 ` æ æ Im w ω n | q =0 = n ( ± 1 − i ) - 3 ì ì - 4 ò ò Strong coupling equivalent to quasiparticle picture at weak coupling Characterize the response of the system to inf. perturbation Appear as poles in the retarded correlator: First indication of top down thermalization at strong coupling Saturday, 20 July, 13

Quasinormal modes finite coupling ` Re w 00 1 2 3 4 5 6 7 - 1 Ê Ê Ê Ê Ê Ê l =1000 Ï ‡ - 2 ‡‡ ‡ ‡ ‡ ` l =2000 Im w Ú Ï Ï l =3000 ÏÏ Ï - 3 Ú Ú l = • l =5000 Ú - 4 Ú Ù l =10000 Effect of decreasing coupling: Imaginary part decreases Outside the limit, response of the plasma appears to change, moving towards λ = ∞ a quasiparticle picture Larger impact on higher energetic modes Convergence of strong coupling expansion not guaranteed when shift is of What happens if we take system further away from equilibrium? Saturday, 20 July, 13

Photon production rate at intermediate coupling 0.020 0.015 d G ê dk 0 a N c 2 T 3 0.010 0.005 0.000 0 2 4 6 8 w ê T emission rate for photons r s /r h =1.01 and Behaviour qualitatively similar to equilibrium case: in particular the result is much less sensitive to finite coupling corrections than QNM spectrum Saturday, 20 July, 13