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Colliding Shock Waves in AdS5

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Title: 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.
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
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
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
14
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.

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-Maria
n, 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 Peschanskis 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.
arXiv0805.2927 hep-th, arXiv0902.3046
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 L2
    t2. 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 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!

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 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!

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 for the 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 doT 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 dont 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 LONLO 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.

51
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
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