Colliding Shock Waves in AdS 5

1. Colliding Shock Waves in AdS5 Yuri Kovchegov The Ohio State University Based on the work done with Javier Albacete and Anastasios Taliotis, arXiv:0805.2927 [hep-th], arXiv:0902.3046 [hep-th], arXiv:0705.1234 [hep-ph]

2. Outline • Problem of isotropization/thermalization in heavy ion collisions • AdS/CFT techniques • Bjorken hydrodynamics in AdS • Colliding shock waves in AdS: • Collisions at large coupling: complete nuclear stopping • Mimicking small-coupling effects: unphysical shock waves • Proton-nucleus collisions

3. Thermalization problem

4. Notations proper time rapidity QGP CGC valid up to times t ~ 1/QS The matter distribution due to classical gluon fields is rapidity-independent.

5. 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

6. 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

7. 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.

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

9. 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

10. If then, as , one gets . Rapidity-Independent Energy-Momentum Tensor Deviations from the scaling of energy density, like are due to longitudinal pressure , which does work in the longitudinal direction modifying the energy density scaling with tau. • Positive longitudinal pressure and isotropization ↔ deviations from

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

17. Shock wave in AdS Need the metric dual to a shock wave and solving 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

18. 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 (McLerran-Venugopalan model): the gluon field is given by 1-gluon exchange (Jalilian-Marian, Kovner, McLerran, Weigert ’96, Yu.K. ’96)

19. Bjorken Hydrodynamics in AdS

20. 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

21. 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?

22. Colliding shock waves in AdS Considered by Nastase; Shuryak, Sin, Zahed; Kajantie, Louko, Tahkokkalio; Grumiller, Romatschke; Gubser, Pufu, Yarom. I will follow J. Albacete, A. Taliotis, Yu.K. arXiv:0805.2927 [hep-th], arXiv:0902.3046 [hep-th]

23. McLerran-Venugopalan model 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

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

25. 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.

26. 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

27. Physical shock waves 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:

28. Physical shock waves: 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)

29. Physical shock waves: 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!

30. Physical shock waves • 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 hydrodynamics. • While Landau hydrodynamics is possible, it is Bjorken hydrodynamics which describes RHIC data rather well. Also baryon stopping data contradicts the conclusion of nuclear stopping at RHIC. • What do we do? We know that the initial stages of the collisions are weakly coupled (CGC)!

31. Unphysical shock waves • One can show that the conclusion about nuclear stopping holds for any energy-momentum tensor of the nuclei such that • To mimic weak coupling effects in the gravity dual we propose using unphysical shock waves with not positive-definite energy-momentum tensor:

32. Unphysical shock waves • Namely we take • This gives: • Almost like CGC at early times: • Energy density is now non-negative everywhere in the forward light cone! • The system may lead to Bjorken hydro. cf. Taliotis, Yu.K. ‘07

33. Will this lead to Bjorken hydro? • Not clear at this point. But if yes, the transition may look like this: (our work) Janik, Peschanski ‘05

34. Isotropization time • One can estimate this isotropization time from AdS/CFT (Yu.K, Taliotis ‘07) obtainingwhere e0 is the coefficient in Bjorken energy-scaling: • For central Au+Au collisions at RHIC at hydrodynamics requires e=15 GeV/fm3 at t=0.6 fm/c (Heinz, Kolb ‘03), giving e0=38 fm-8/3. This leads toin good agreement with hydrodynamics!

35. 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 using field theory? Landau hydro: results from strong coupling dynamics at all times in the collision. While possible, contradicts baryon stopping data at RHIC.

36. Proton-Nucleus Collisions

37. pA Setup • Consider pA collisions:

38. pA Setup • In terms of graviton exchanges need to resum diagrams like this: cf. gluon production in pA collisions in CGC!

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

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

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

42. 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.

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

44. Proton Stopping • What about the proton? Dueto our earlier result about shock wave stopping we should be able to see how it stops. • And we do:T++ goes to zero as x+ grows large!

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

46. More On Stopping: AA Case • Contour plot of transverse pressure for AA collisions. (Albacete,Yu.K., Taliotis, in preparation)

47. Conclusions • We have constructed graviton expansion for the collision of two shock waves in AdS, with the goal of obtaining energy-momentum tensor of the produced strongly-coupled matter in the gauge theory. • We have solved the pA scattering problem in AdS. • Real shock waves stop: Landau hydrodynamics. • Delta-prime shock waves don’t stop, but it is not clear what they lead to. Hopefully some form of ideal hydrodynamics. • Wherefore art thou Bjorken hydro?

48. Backup Slides

49. Delta-prime shocks • For delta-prime shock waves the result is surprising. The all-order eikonal answer for pA is given by LO+NLO terms: • That is, graviton exchange series terminates at NLO. +

50. Delta-prime shocks • The answer for transverse pressure iswith the shock waves • As p goes negative at late times, this is clearly not hydrodynamics and not free streaming.