Initial Conditions from Shock Wave Collisions in AdS 5 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]

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

Thermalization problem

Timeline of a Heavy Ion Collision (particle production)

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

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 we get, at the leading log level, Energy-momentum tensor is (Lappi ’06 Fukushima ‘07)

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  is conserved, as expected for non-interacting particles.

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

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~  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 AdS/CFT would help accomplish this goal.

AdS/CFT techniques

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

AdS/CFT Correspondence (Gauge-Gravity Duality) Large-N c, large  g 2 N c 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.

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 of the AdS space:  For Minkowski world and with Holographic renormalization de Haro, Skenderis, Solodukhin ‘00

Bjorken Hydrodynamics in AdS

AdS Dual of a Static Thermal Medium z z=0 Our 4d world AdS 5 black hole metric can be written as 5 th dimension black hole horizon z0z0 with Black hole in AdS 5 ↔ Thermal medium in N=4 SYM theory.

AdS Dual of Bjorken Hydrodynamics z=0 R3R3 black hole horizon z0z0 Janik, Peschanski ’05: to get Bjorken hydro dual need z 0 =z 0  Black hole recedes into the bulk: medium in 4d expands and cools off.

Asymptotic geometry  Janik and Peschanski ’05 showed that in the rapidity- independent case the geometry of AdS space at late proper times  is given by the following metric with e 0 a constant.  In 4d gauge theory this gives Bjorken hydrodynamics: with

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

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.

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

Shock wave in AdS 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 Need the metric dual to a shock wave that solves Einstein equations:

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)

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 AdS 5 1-graviton parthigher order graviton exchanges ?

Heavy ion collisions in AdS empty AdS 5 1-graviton parthigher order graviton exchanges

Expansion Parameter  Depends on the exact form of the energy- momentum tensor of the colliding shock waves.  For the parameter in 4d is   : the expansion is good for early times  only.  For that we will also consider the expansion parameter in 4d is  2  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 t .  This gives (in rapidity-independent case) along with Janik, Peschanksi ‘05

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

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. ( 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 rapidity- dependent 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

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

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.

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+X+

Colliding shock waves: trapped surface analysis see also Gubser, Pufu, Yarom ’08,’09; Lin, Shuryak ’09. Yu.K., Lin ‘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 ‘pre- horizon’, whose appearance indicates that gravitational collapse is inevitable.  Pioneered in AdS 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 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!

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/d in AdS and the amount of baryon stopping to make a more comprehensive comparison. Gubser, Pufu, Yarom, ‘08

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 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.

Backup Slides

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 If then, as, one gets. ↔ deviations from

 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. 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… Delta-prime shocks

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:  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

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

Isotropization time  One can estimate this isotropization time from AdS/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 =15 GeV/fm3 at =0.6 fm/c (Heinz, Kolb ‘03), giving  0 =38 fm-8/3. This leads to in good agreement with hydrodynamics!