Initial Conditions from Shock Wave Collisions in AdS 5

1. Initial Conditions from Shock Wave Collisions in AdS5 Yuri Kovchegov The Ohio State University Based on the work done with Javier Albacete, Shu Lin, and Anastasios Taliotis, arXiv:0805.2927 [hep-th], arXiv:0902.3046 [hep-th], arXiv:0911.4707 [hep-ph]

2. Outline • Problem of isotropization/thermalization in heavy ion collisions • AdS/CFT techniques we use • Bjorken hydrodynamics in AdS • Colliding shock waves in AdS: • Collisions at large coupling: complete nuclear stopping • Proton-nucleus collisions • Trapped surface and black hole production

3. Thermalization problem

4. Timeline of a Heavy Ion Collision (particle production)

5. Notations proper time rapidity QGP CGC CGC (Color Glass Condensate) = classical gluon fields. The matter distribution due to classical gluon fields is rapidity-independent. QGP = Quark Gluon Plasma

6. Most General Rapidity-Independent Energy-Momentum Tensor The most general rapidity-independent energy-momentum tensor for a high energy collision of two very large nuclei is (at x3 =0) which, due to gives

7. Color Glass at Very Early Times (Lappi ’06 Fukushima ‘07) In CGC at very early times we get, at the leading log level, such that, since Energy-momentum tensor is

8. Color Glass at Later Times: “Free Streaming” At late times classical CGC gives free streaming, which is characterized by the following energy-momentum tensor: such that and • The total energy E~ e t is conserved, as expected for • non-interacting particles.

9. Classical Fields from numerical simulations by Krasnitz, Nara, Venugopalan ‘01 • CGC classical gluon field leads to energy density scaling as

10. Much Later Times: Bjorken Hydrodynamics In the case of ideal hydrodynamics, the energy-momentum tensor is symmetric in all three spatial directions (isotropization): such that Using the ideal gas equation of state, , yields Bjorken, ‘83 • The total energy E~ e t is not conserved, while the total entropy S is conserved.

11. The Problem • Can one show in an analytic calculation that the energy-momentum tensor of the medium produced in heavy ion collisions is isotropic over a parametrically long time? • That is, can one start from a collision of two nuclei and obtain Bjorken hydrodynamics? • Even in some idealized scenario? Like ultrarelativistic nuclei of infinite transverse extent? • Let us proceed assuming that strong-coupling dynamics from AdS/CFT would help accomplish this goal.

12. AdS/CFT techniques

13. AdS/CFT Approach z=0 Our 4d world 5d (super) gravity lives here in the AdS space 5th dimension AdS5 space – a 5-dim space with a cosmological constant L= -6/L2. (L is the radius of the AdS space.) z

14. AdS/CFT Correspondence (Gauge-Gravity Duality) Large-Nc, large l=g2 Nc N=4 SYM theory in our 4 space-time dimensions Weakly coupled supergravity in 5d anti-de Sitter space! • Can solve Einstein equations of supergravity in 5d to learn about energy-momentum tensor in our 4d world in the limit of strong coupling! • Can calculate Wilson loops by extremizing string configurations. • Can calculate e.v.’s of operators, correlators, etc.

15. Holographic renormalization de Haro, Skenderis, Solodukhin ‘00 • Energy-momentum tensor is dual to the metric in AdS. Using Fefferman-Graham coordinates one can write the metric as with z the 5th dimension variable and the 4d metric. • Expand near the boundary of the AdS space: • For Minkowski world and with

16. Bjorken Hydrodynamics in AdS

17. AdS Dual of a Static Thermal Medium Black hole in AdS5 ↔ Thermal medium in N=4 SYM theory. z=0 Our 4d world 5th dimension black hole horizon z0 z AdS5 black hole metric can be written as with

18. AdS Dual of Bjorken Hydrodynamics Janik, Peschanski ’05: to get Bjorken hydro dual need z0 =z0(t). z=0 R3 black hole horizon z0 Black hole recedes into the bulk: medium in 4d expands and cools off.

19. Asymptotic geometry • Janik and Peschanski ’05 showed that in the rapidity-independent case the geometry of AdS space at late proper times t is given by the following metricwith e0 a constant. • In 4d gauge theory this gives Bjorken hydrodynamics: with

20. Bjorken hydrodynamics in AdS • Looks like a proof of thermalization at large coupling. • It almost is: however, one needs to first understand what initial conditions lead to this Bjorken hydrodynamics. • Is it a weakly- or strongly-coupled heavy ion collision which leads to such asymptotics? If yes, is the initial energy-momentum tensor similar to that in CGC? Or does one need some pre-cooked isotropic initial conditions to obtain Janik and Peschanski’s late-time asymptotics? • In AdS the problem of thermalization = problem of black hole production in the bulk

21. Colliding shock waves in AdS J. Albacete, A. Taliotis, Yu.K. arXiv:0805.2927 [hep-th], arXiv:0902.3046 [hep-th] see also Nastase; Shuryak, Sin, Zahed; Kajantie, Louko, Tahkokkalio; Grumiller, Romatschke.

22. Single Nucleus in AdS/CFT An ultrarelativistic nucleus is a shock wave in 4d with the energy-momentum tensor

23. Shock wave in AdS Need the metric dual to a shock wave that solves Einstein equations: The metric of a shock wave in AdS corresponding to the ultrarelativistic nucleus in 4d is (note that T_ _ can be any function of x^-): Janik, Peschanksi ‘05

24. Diagrammatic interpretation The metric of a shock wave in AdS corresponding to the ultrarelativistic nucleus in 4d can be represented as a graviton exchange between the boundary of the AdS space and the bulk: cf. classical Yang-Mills field of a single ultrarelativistic nucleus in CGC in covariant gauge: given by 1-gluon exchange (Jalilian-Marian, Kovner, McLerran, Weigert ’96, Yu.K. ’96)

25. Model of heavy ion collisions in AdS • Imagine a collision of two shock waves in AdS: • We know the metric of bothshock waves, and know thatnothing happens before the collision. • Need to find a metric in theforward light cone! (cf. classical fields in CGC) ? empty AdS5 1-graviton part higher order graviton exchanges

26. Heavy ion collisions in AdS empty AdS5 1-graviton part higher order graviton exchanges

27. Expansion Parameter • Depends on the exact form of the energy-momentum tensor of the colliding shock waves. • For the parameter in 4d is m t3 :the expansion is good for early times t only. • For that we will also considerthe expansion parameter in 4d is L2t2. Also valid for early times only. • In the bulk the expansion is valid at small-z by the same token.

28. What to expect • There is one important constraint of non-negativity of energy density. It can be derived by requiring thatfor any time-like tm. • This gives (in rapidity-independent case)along with Janik, Peschanksi ‘05

29. Lowest Order Diagram Simple dimensional analysis: The same result comes out of detailed calculations. Grumiller, Romatschke ‘08 Albacete, Taliotis, Yu.K. ‘08 Each graviton gives , hence get no rapidity dependence:

30. Shock waves collision: problem 1 • Energy density at mid-rapidity grows with time!? This violates condition. This means in some frames energy density at some rapidity is negative! • I do not know of a good explanation: it may be due to some Casimir-like forces between the receding nuclei. (see e.g. work by Kajantie, Tahkokkalio, Louko ‘08)

31. Shock waves collision: problem 2 • Delta-functions are unwieldy. We will smear the shock wave:with and . (L is the typical transverse momentum scale in the shock.) • Look at the energy-momentum tensor of a nucleus after collision: • Looks like by the light-cone timethe nucleus will run out of momentum and stop!

32. Shock waves at lowest order • We conclude that describing the whole collision in the strong coupling framework leads to nuclei stopping shortly after the collision. • This would not lead to Bjorken hydrodynamics. It is very likely to lead to Landau-like rapidity-dependent hydrodynamics. This is fine, as rapidity-dependent hydrodynamics also describes RHIC data rather well. • However baryon stopping data contradicts the conclusion of nuclear stopping at RHIC.

33. Landau vs Bjorken Bjorken hydro: describes RHIC data well. The picture of nuclei going through each other almost without stopping agrees with our perturbative/CGC understanding of collisions. Can we show that it happens in AA collisions? Landau hydro: results from strong coupling dynamics (at all times) in the collision. While possible, contradicts baryon stopping data at RHIC.

34. Proton-Nucleus Collisions

35. pA Setup • Solving the full AA problem is hard. To gain intuition need to start somewhere. Consider pA collisions:

36. pA Setup • In terms of graviton exchanges need to resum diagrams like this: In QCD pA with gluons cf. A. Mueller, Yu.K., ’98; B. Kopeliovich, A. Tarasov and A. Schafer, ’98; A. Dumitru, L. McLerran, ‘01.

37. Eikonal Approximation • Note that the nucleus is Lorentz-contracted. Hence all and are small.

38. Physical Shocks • Summing all these graphs forthe delta-function shock wavesyields the transverse pressure: • Note the applicability region:

39. Physical Shocks • The full energy-momentum tensor can be easily constructed too. In the forward light cone we get:

40. Physical Shocks: the Medium • Is this Bjorken hydro? Or a free-streaming medium? • Appears to be neither. At late timesNot a free streaming medium. • For ideal hydrodynamics expectsuch that: • However, we getNot hydrodynamics either.

41. Physical Shocks: the Medium • Most likely this is an artifact of the approximation, this is a “virtual” medium on its way to thermalization.

42. Proton Stopping • What about the proton? If our earlier conclusion about shock wave stopping based onis right, we should be able to see how it stops.

43. Proton Stopping • We have the original shock wave: • We have the produced stuff: • Adding them together we see thatthe shock wave is cancelled:T++ goes to zero as x+ grows large!

44. Proton Stopping • We get complete proton stopping (arbitrary units): T++ of the proton X+

45. Colliding shock waves: trapped surface analysis Yu.K., Lin ‘09 see also Gubser, Pufu, Yarom ’08,’09; Lin, Shuryak ’09.

46. Trapped Surface: Shock Waves with Sources • To determine whether the black hole is produced and to estimate the generated entropy use the trick invented by Penrose – find a ‘trapped surface’, which is a ‘pre-horizon’, whose appearance indicates that gravitational collapse is inevitable. • Pioneered in AdS by Gubser, Pufu, Yarom ’08: marginally trapped surface

47. Trapped Surface: Shock Waves without Sources • Sources in the bulk are sometimes hard to interpret in gauge theory. However, if one gets rid of sources by sending them off to IR the trapped surface remains: Yu.K., Shu Lin, ‘09

48. Black Hole Production • Using trapped surface analysis one can estimate the thermalization time (Yu.K., Lin ’09; see also Grumiller, Romatschke ’08) • This is parametrically shorter than the time of shock wave stopping: • (Part of) the system thermalizes before shock waves stop!

49. Black Hole Production • Estimating the produced entropy by calculating the area of the trapped surface one gets the energy-scaling of particle multiplicity:where s is the cms energy. • The power of 1/3 is not too far from the phenomenologically preferred 0.288 (HERA) and 0.2 (RHIC). • However, one has to understand dN/dh in AdS and the amount of baryon stopping to make a more comprehensive comparison. Gubser, Pufu, Yarom, ‘08

50. Black Hole Production • It appears that the black hole is at z= ∞ with a horizon at finite z, independent of transverse coordinates, similar to Janik and Peschanski case. • In our case we have rapidity-dependence. • We conclude that thermalization does happen in heavy ion collisions at strong coupling. • We expect that it happens before the shock waves stop.