Initial Conditions from Shock Wave Collisions in Ad

Initial Conditions from Shock Wave Collisions in Ad. S 5 Yuri Kovchegov The Ohio State University Based on the work done with Javier Albacete, Shu Lin, and Anastasios Taliotis, ar. Xiv: 0805. 2927 [hep-th], ar. Xiv: 0902. 3046 [hep-th], ar. Xiv: 0911. 4707 [hep-ph]

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

Thermalization problem

Timeline of a Heavy Ion Collision (particle production)

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

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 x 3 =0) which, due to gives

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

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 Ø The total energy E~ e t is conserved, as expected for non-interacting particles.

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

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 Ø The total energy E~ e t is not conserved, while the total entropy S is conserved.

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 Ad. S/CFT would help accomplish this goal.

Ad. S/CFT techniques

Ad. S/CFT Approach z=0 Our 4 d world 5 d (super) gravity lives here in the Ad. S space 5 th dimension z Ad. S 5 space – a 5 -dim space with a cosmological constant L= -6/L 2. (L is the radius of the Ad. S space. )

Ad. S/CFT Correspondence (Gauge-Gravity Duality) Large-Nc, large l=g 2 Nc N=4 SYM theory in our 4 space-time dimensions Weakly coupled supergravity in 5 d anti-de Sitter space! Ø Can solve Einstein equations of supergravity in 5 d to learn about energy-momentum tensor in our 4 d world in the limit of strong coupling! Ø Can calculate Wilson loops by extremizing string configurations. Ø Can calculate e. v. ’s of operators, correlators, etc.

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

Bjorken Hydrodynamics in Ad. S

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

Ad. S Dual of Bjorken Hydrodynamics Janik, Peschanski ’ 05: to get Bjorken hydro dual need z 0 =z 0(t). z=0 R 3 z 0 black hole horizon Black hole recedes into the bulk: medium in 4 d expands and cools off.

Asymptotic geometry ¡ ¡ Janik and Peschanski ’ 05 showed that in the rapidityindependent case the geometry of Ad. S space at late proper times t is given by the following metric with e 0 a constant. In 4 d gauge theory this gives Bjorken hydrodynamics: with

Bjorken hydrodynamics in Ad. S ¡ 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 Ad. S the problem of thermalization = problem of black hole production in the bulk

Colliding shock waves in Ad. S J. Albacete, A. Taliotis, Yu. K. ar. Xiv: 0805. 2927 [hep-th], ar. Xiv: 0902. 3046 [hep-th] see also Nastase; Shuryak, Sin, Zahed; Kajantie, Louko, Tahkokkalio; Grumiller, Romatschke.

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

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

Diagrammatic interpretation The metric of a shock wave in Ad. S corresponding to the ultrarelativistic nucleus in 4 d can be represented as a graviton exchange between the boundary of the Ad. S 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, Mc. Lerran, Weigert ’ 96, Yu. K. ’ 96)

Model of heavy ion collisions in Ad. S ¡ ¡ ¡ Imagine a collision of two shock waves in Ad. S: 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 Ad. S 5 1 -graviton part ? higher order graviton exchanges

Heavy ion collisions in Ad. S empty Ad. S 5 1 -graviton part higher order graviton exchanges

Expansion Parameter ¡ Depends on the exact form of the energymomentum tensor of the colliding shock waves. ¡ For the parameter in 4 d is m t 3 : the expansion is good for early times t only. ¡ For that we will also consider the expansion parameter in 4 d is L 2 t 2. Also valid for early times only. ¡ In the bulk the expansion is valid at small-z by the same token.

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

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:

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)

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 time the nucleus will run out of momentum and stop!

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 rapiditydependent hydrodynamics also describes RHIC data rather well. ¡ However baryon stopping data contradicts the conclusion of nuclear stopping at RHIC.

Landau vs Bjorken Landau hydro: results from strong coupling dynamics (at all times) in the collision. While possible, contradicts baryon stopping data at RHIC. 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?

Proton-Nucleus Collisions

p. A Setup ¡ Solving the full AA problem is hard. To gain intuition need to start somewhere. Consider p. A collisions:

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

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

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

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

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.

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

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.

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!

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

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

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 ‘prehorizon’, whose appearance indicates that gravitational collapse is inevitable. Pioneered in Ad. S by Gubser, Pufu, Yarom ’ 08: marginally trapped surface

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

Black Hole Production ¡ Using trapped surface analysis one can estimate 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!

Black Hole Production ¡ Estimating the produced entropy by calculating the area of the trapped surface one gets the energyscaling of particle multiplicity: Gubser, Pufu, Yarom, ‘ 08 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 d. N/dh in Ad. S and the amount of baryon stopping to make a more comprehensive comparison.

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.

Conclusions ¡ We have constructed graviton expansion for the collision of two shock waves in Ad. S, with the goal of obtaining energy-momentum tensor of the produced strongly-coupled matter in the gauge theory. ¡ We have solved the p. A scattering problem in Ad. S 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.

Backup Slides

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

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

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.

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…

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:

Unphysical shock waves ¡ Namely we take ¡ This gives: cf. Taliotis, Yu. K. ‘ 07 ¡ 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. ¡

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

Isotropization time ¡ One can estimate this isotropization time from Ad. S/CFT (Yu. K, Taliotis ‘ 07) obtaining where e 0 is the coefficient in Bjorken energy-scaling: ¡ For central Au+Au collisions at RHIC at hydrodynamics requires e=15 Ge. V/fm 3 at t=0. 6 fm/c (Heinz, Kolb ‘ 03), giving e 0=38 fm-8/3. This leads to in good agreement with hydrodynamics!