1. 11 Department of Physics HIC with Dynamics┴ from Evolving Geometries in AdS arXiv: 1004.3500 [hep-th], Anastasios Taliotis Partial Extension of arXiv:0805.2927 [hep-th], arXiv:0902.3046 [hep-th], arXiv:0705.1234 [hep-ph] (published in JHEP and Phys. Rev. C) [ Albacete, Kovchegov, Taliotis]

2. 22 Outline Motivating strongly coupled dynamics in HIC AdS/CFT: What we need for this work State/set up the problem Attacking the problem using AdS/CFT Predictions/comparisons/conclusions/Summary Future work

3. 33 Motivating strongly coupled dynamics in HIC

4. 44 Notation/Facts Proper time: Rapidity: Saturation scale : The scale where density of partons becomes high. QGP CGCCGC  CGC describes matter distribution due to classical gluon fields and is rapidity- independent ( g<<1, early times).  Hydro is a necessary condition for thermalization. Bjorken Hydro describes successfully particle spectra and spectral flow. Is g??>>1 at late times?? Maybe; consistent with the small MFP implied by a hydro description.  No unified framework exists that describes both strongly & weakly coupled dynamics valid for times t >> 1/Q s Bj Hydro g<<1; valid up to times  ~ 1/Q S.

5. 55 Goal: Stress-Energy (SE) Tensor SE of the produced medium gives useful information. In particular, its form (as a function of space and time variables) allows to decide whether we could have thermalization i.e. it provides useful criteria for the (possible) formation of QGP. SE tensor will be the main object of this talk: we will see how it can be calculated by non perturbative methods in HIC.

6. 66 Most General Rapidity-Independent SE Tensor The most general rapidity-independent SE tensor for a collision of two transversely large nuclei is (at x 3 =0) which, due to gives  We will see three different regimes of p 3

7. 77 I.Early times : τQ s <<1 CGC II.Later times : τ>~1/Q s CGC III.Much later times:τQ s >>>1 Bjorken Hydrodynamics Classical gluon fields Pert. theory applies Describes RHIC data well (particle multiplicity dN/dn) Classical gluon fields Pert. Theory applies Energy is conserved Hydrodynamic description Does pert. Theory apply?? Describes data successfully (spectra dN/d 2 p T dn for K, ρ, n & elliptic flow) [Heınz et al] thermalization [Lappi ’06 Fukushima ’07: pQCD] [Talıotıs ’10: AdS/CFT] [Free streaming] 0 p(τ) Isotropization [Krasnitz, Nara,Venogopalan, Lappi, Kharzeev, Levin, Nardi]

8. 88 Bjorken Hydro & strongly coupled dynamics Deviations from the energy conservation are due to longitudinal pressure, P 3 which does work P 3 dV in the longitudinal direction modifying the energy density scaling with tau. If then, as, one gets.  It is suggested that neither classical nor quantum gluonic or fermionic fields can cause the transition from free streaming to Bjorken hydro within perturbation theory. [Kovchegov’05]  On the other hand Bjorken hydro describe simulations satisfactory.  Conclude that alternative methods are needed!

9. 99 AdS/CFT: What we need for this work

10. 10 Quantifying the Conjecture <exp z=0 ∫O φ 0 > CFT = Zs(φ |φ(z=0)= φo ) O is the CFT operator. Typically want <O 1 O 2 …O n > φ 0 =φ 0 (x 1,x 2,…,x d ) is the source of O in the CFT picture φ =φ (x 1,x 2,…,x d,z) is some field in string theory with B.C. φ (z=0)= φ 0

11. 11 Holographic renormalization Quantifying the Conjecture CFT = Zs(φ |φ(z=0)= φo ) Know the SE Tensor of Gauge theory is given by So g μν acts as a source => in order to calculate T μν from AdS/CFT must find the metric. Metric has its eq. of motion i.e. Einsteins equations. Example: [Witten ‘98]

12. 12  Energy-momentum tensor is dual to the metric in AdS. Using Fefferman-Graham coordinates one can write the metric as with z the 5 th dimension variable and the 4d metric.  Expand near the boundary (z=0) of the AdS space:  Using AdS/CFT can show:, and Holographic renormalization de Haro, Skenderis, Solodukhin ‘00

13. 13 State/set up the problem

14. 14 Initial T µν phenomenology AdS/CFT Dictionary Initial Geometry Dynamical Geometry Dynamical T µν (our result) EvolveEinstein's Eq. Strategy

15. 15 Field equations, AdS 5 shockwave; ∂g MN T μν  Eq. of Motion (units L=1) for g ΜΝ (x M = x ±, x 1, x 2, z) is generally given  AdS-shockwave with bulk matter: [Janik & Peschanski ’06] Then ~z 4 coef. implies ~ -δ μ + δ ν + µlog(r 1 ) δ(x + ) in QFT side Corresponding bulk tensor J MN :

16. 16 Single nucleus Single shockwave The picture in 4d is that matter moves ultrarelativistically along x - according to figure. Einstein's equations are satisfied trivially except (++) component; it satisfies a linear equation: □(z 4 t 1 )=J ++ This suggests may represent the shockwave metric as a single vertex: a graviton exchange between the source J ++ ( the nucleus living at z=0; the boundary of AdS ) and point X M in the bulk which gravitational field is measured.

17. 4D Picture of Collision 17

18. 18 Superposition of two shockwaves Non linearities of gravity ? Flat AdS Higher graviton ex. Due to non linearities One graviton ex.

19. Back-to-Back reactions for J MN In order to have a consistent expansion in µ 2 we must determine We use geodesic analysis Bulk source J ++ (J -- ) moves in the gravitational filed of the shock t 1 (t 2 ) Important: is conserved iff b≠0 19 Self corrections to J MN

20. 20 Calculation/results Step 1: Choose a gauge: Fefferman-Graham coordinates Step 2: Linearize field eq. expanding around 1/z 2 η MN (partial DE with w.r.t. x +,x -, z with non constant coef.). Step 3: Decouple the DE. In particular all components g (2) µν obey: □g (2) µν = A (2) µν (t 1 (x - ), t 2 (x + ),J) with box the d'Alembertian in AdS 5. Step 4: Solve them imposing (BC) causality-Determine the G R Step 5: Determine T μν by reading the z 4 coef. of g μν Side Remark: Gzz encodes tracelessness of Tµν Gzν encode conservation of Tµν

21. The Formula for T µν 21

22. Eccentricity-Momentum Anisotropy Momentum Anisotropy ε x = ε x (x) (left) and ε x = ε x (1/x) (right) for intermediate. 22 Agrees qualitatively with [Heinz,Kolb, Lappi,Venugopalan,Jas,Mrowczynski]

23. Conclusions Built perturbative expansion of dual geometry to determine T µν ; applies for sufficiently early times: µτ 3 <<1. T µν evolves according to causality in an intuitive way! There is a kinematical window where is invariant under. Our exact formula (when applicable) allows as to compute Spatial Eccentricity and Momentum Anisotropy. 23 [ Gubser ‘10 ]

24. When τ>>r 1,r 2 have ε~τ 2 log 2 τ-compare with ε~Q 2 s log 2 τ Despite J being localized, it still contributes to g µν and so to T µν not only on the light-cone but also inside. Impact parameter is required otherwise violate conservation of J MN and divergences of g µν. Not a surprise for classical field theories. Our technique has been applied to ordinary (4d) gravity and found similar behavior for g µν. A phenomenological model using the (boosted) Woods-Saxon profile: [Lappi, Fukushima] Taliotis’10 MS thesis.dept. of Mathematics, OSU [ Gubser,Yarom,Puf u ‘08] For τ> r 1, r 2 Note symmetry under when b=0; [Gubser’10]

25. Thank youThank you 25

26. Supporting slides 26

27. O(µ 2 ) Corrections to J µν 27 Remark: These corrections live on the forward light-cone as should!

28. 28 Field Equations