Colliding Shock Waves in AdS5

1
Colliding Shock Waves in AdS5
Yuri Kovchegov The Ohio State University Based
on the work done with Javier Albacete and
Anastasios Taliotis, arXiv0805.2927 hep-th,
arXiv0902.3046 hep-th, arXiv0705.1234
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.
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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
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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
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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.

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Classical Fields
from numerical simulations by Krasnitz, Nara,
Venugopalan 01

• CGC classical gluon field leads to energy
  density scaling as

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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

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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
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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.

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AdS/CFT techniques
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AdS/CFT Approach
z0
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
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AdS/CFT Correspondence (Gauge-Gravity Duality)
Large-Nc, large lg2 Nc N4 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.

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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

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Single Nucleus in AdS/CFT

• An ultrarelativistic nucleus is a shock wave
  in 4d with the energy-momentum tensor

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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
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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-Maria
n, Kovner, McLerran, Weigert 96, Yu.K. 96)
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Bjorken Hydrodynamics in AdS
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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

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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 Peschanskis late-time asymptotics?

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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.
arXiv0805.2927 hep-th, arXiv0902.3046
hep-th
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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
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Heavy ion collisions in AdS
empty AdS5
1-graviton part
higher order graviton exchanges
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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 L2
  t2. Also valid for early times only.
• In the bulk the expansion is valid at small-z by
  the same token.

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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
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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
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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)

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Physical shock waves problem 2

• Delta-functions are unwieldy. We will smear the
  shock wavewith 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!

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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)!

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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

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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
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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
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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 AuAu collisions at RHIC at
  hydrodynamics requires e15 GeV/fm3 at
  t0.6 fm/c (Heinz, Kolb 03), giving e038
  fm-8/3. This leads toin good agreement with
  hydrodynamics!

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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.
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Proton-Nucleus Collisions
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pA Setup

• Consider pA collisions

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pA Setup

• In terms of graviton exchanges need to resum
  diagrams like this

cf. gluon production in pA collisions in CGC!
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Eikonal Approximation

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

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Physical Shocks

• Summing all these graphs for the delta-function
  shock wavesyields the transverse pressure
• Note the applicability region

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Physical Shocks

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

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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.

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Physical Shocks the Medium

• Most likely this is an artifact of the
  approximation, this is a virtual medium on its
  way to thermalization.

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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 doT goes to zero as x grows large!

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Proton Stopping

• We get complete proton stopping (arbitrary units)

T of the proton
X
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More On Stopping AA Case

• Contour plot of transverse pressure for AA
  collisions.
• (Albacete,Yu.K., Taliotis, in preparation)

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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 dont stop, but it is not
  clear what they lead to. Hopefully some form of
  ideal hydrodynamics.
• Wherefore art thou Bjorken hydro?

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Backup Slides
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Delta-prime shocks

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

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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.

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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