1. Heavy Ion Collisions in AdS5
Yuri Kovchegov
The Ohio State University
Based on the work done with
Javier Albacete, Shu Lin, and Anastasios Taliotis,
arXiv: [hep-th],
arXiv: [hep-th],
arXiv: [hep-ph]

2. Outline AdS/CFT techniques we use Bjorken hydrodynamics in AdS
Colliding shock waves in AdS:
Collisions at large coupling: complete nuclear stopping
Proton-nucleus collisions
All-orders resummation of rescatterings in one of the shock waves
Trapped surface and black hole production

3. AdS/CFT techniques

4. 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

5. 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.

6. 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

7. Thermal medium and Bjorken hydrodynamics in AdS

8. 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

9. 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.

10. 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 metric with e0 a constant.
In 4d gauge theory this gives Bjorken hydrodynamics: with

11. 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 rapidity-independent 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

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

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

14. 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

15. 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)

16. Model of heavy ion collisions in AdS
Imagine a collision of two shock waves in AdS:
We know the metric of both shock waves, and know that nothing happens before the collision.
Need to find a metric in the forward light cone! (cf. classical fields in CGC) ?
empty AdS5
1-graviton part
higher order
graviton exchanges

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

18. 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:

19. Shock waves collision: feature 1
Energy density at mid-rapidity grows with time!? This means longitudinal pressure is large and negative, and in some frames energy density is negative!
Energy-momentum tensor is very anisotropic:

20. Shock waves collision: feature 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 time the nucleus will run out of momentum and stop!

21. 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.

22. 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.

23. Proton-Nucleus Collisions

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

25. 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.

26. Physical Shocks
Summing all these graphs for the delta-function shock waves yields the transverse pressure:
Note the applicability region:

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

28. Physical Shocks: the Medium
Is this Bjorken hydro? Or a free-streaming medium?
Appears to be neither. At late times Not a free streaming medium.
For ideal hydrodynamics expect such that:
However, we get Not hydrodynamics either.

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

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

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

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

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

34. 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

35. 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
cf. Beuf et al ‘09,
Chesler & Yaffe, ‘09

36. 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!

37. 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 (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

38. 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.

39. 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 in the eikonal approximation.
Shock waves stop and probably lead to Landau-like rapidity-dependent hydrodynamics.
We performed a trapped-surface analysis showing that thermalization does happen in heavy ion collisions at strong coupling, and is much quicker than shock wave stopping.

40. Backup Slides

41. Thermalization problem

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

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

44. 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

45. 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

46. 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.

47. 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.

48. 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.

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

50. Rapidity-Independent Energy-Momentum Tensor
If then, as , one gets
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

51. 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 consider the expansion parameter in 4d is L2 t2. Also valid for early times only.
In the bulk the expansion is valid at small-z by the same token.

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

53. 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. +

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

55. Delta-prime shocks
Note that the energy momentum tensor becomes rapidity-dependent:
Thus we conclude that initially the matter distribution is rapidity-dependent. Hence at late times it will be rapidity-dependent too (causality). Can one get Bjorken hydro still? Probably not…

56. 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:

57. 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

58. Will this lead to Bjorken hydro?
Not clear at this point. But if yes, the transition may look like this:
(Yu.K., Taliotis ‘07)
Janik,
Peschanski
‘05
cf. Beuf et al ’09,
Chesler & Yaffe ‘09

59. Isotropization time
One can estimate this isotropization time from AdS/CFT (Yu.K, Taliotis ‘07) obtaining where 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 to in good agreement with hydrodynamics!

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