Initial Conditions from Shock Wave Collisions in AdS5
Yuri KovchegovThe Ohio State University
Based on the work done with Javier Albacete, Shu Lin, and Anastasios Taliotis,arXiv:0805.2927 [hep-th], arXiv:0902.3046 [hep-th], arXiv:0911.4707 [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
23
20 xx
30
30ln2
1
xx
xx
0x
3x
QGP
CGC
CGC (Color Glass Condensate) =classical gluon fields. The matter distribution due toclassical gluon fields is rapidity-independent.
QGP = Quark Gluon Plasma
xx
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)
z
y
x
t
p
p
pT
)(000
0)(00
00)(0
000)(
3
which, due to 0 T
gives
3p
d
d
0x
1x
2x
3x
3x
2x
1x
Color Glass at Very Early Times
In CGC at very early times
z
y
x
t
T
)(000
0)(00
00)(0
000)(
3p
d
d such that, since
1,1
log~ 2 SQ
0x
1x
2x
3x
we get, at the leading log level,
Energy-momentum tensor is
(Lappi ’06Fukushima ‘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:
d
d
such that
and
1
~
The total energy E~ e is conserved, as expected fornon-interacting particles.
z
y
x
t
p
pT
0000
0)(00
00)(0
000)(
0x
1x
2x
3x
SQ
1
Classical FieldsClassical Fields
CGC classical gluon field leads to energy density scaling as
1
~classical
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):
z
y
x
t
p
p
pT
)(000
0)(00
00)(0
000)(
p
d
d
such that
Using the ideal gas equation of state, , yieldsp3
3/4
1~
Bjorken, ‘83
The total energy E~ is not conserved, while the total entropy S is conserved.
0x
1x
2x
3x
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 4dworld
5d (super) gravitylives here in the AdS space
AdS5 space – a 5-dim space with a cosmological constant = -6/L2.(L is the radius of the AdS space.)
5th dimension
222
22 2 dzdxdxdxz
Lds
2
30 xxx
AdS/CFT Correspondence (Gauge-Gravity Duality)
Large-Nc, large g2 Nc N=4 SYM theory in our 4 space-timedimensions
Weakly coupledsupergravity in 5danti-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 5th dimension variable and the 4d metric.
Expand near the boundary of the AdS space:
For Minkowski world and with
Holographic renormalization
22
22 ),(~ dzdxdxzxgz
Lds
),(~ zxg
),(~ zxg
de Haro, Skenderis, Solodukhin ‘00
Bjorken Hydrodynamics in AdS
AdS Dual of a Static Thermal Medium
z
z=0
Our 4dworld
AdS5 black hole metric can be written as
5th dimension
2240
4240
4
240
4
2
22 )/1(
)/1(
)/1(dzxdzzdt
zz
zz
z
Lds
black hole horizonz0
Tz 2
0 with
Black hole in AdS5 ↔ Thermal medium in N=4 SYM theory.
AdS Dual of Bjorken Hydrodynamics
z=0
R3
black hole horizonz0
Janik, Peschanski ’05: to get Bjorken hydro dual need z0 =z0
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 e0 a constant. In 4d gauge theory this gives Bjorken hydrodynamics:
with
22220
2
0
2
0
2
22
3/4
4
3/4
4
3/4
4
11
1dzdxded
e
e
z
Lds z
z
z
z
y
x
t
p
p
pT
)(000
0)(00
00)(0
000)(
3/4
1~
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:0805.2927 [hep-th], arXiv:0902.3046 [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
)(~ xT
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^-):
22242
2
2
22 )(
22 dzdxdxzxT
Ndxdx
z
Lds
C
Janik, Peschanksi ‘05
Need the metric dual to a shock wave that solves Einstein equations:
06
2
12
gL
gRR
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:
22242
22 )(2 dzdxdxzxdxdxz
Lds
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 bothshock waves, and know thatnothing happens before the collision.
Need to find a metric in theforward light cone! (cf. classical fields in CGC)
24
22
224
12
222
2
22 )(
2)(
22 dxzxT
NdxzxT
Ndzdxdxdx
z
Lds
CC
empty AdS5 1-graviton part higher ordergraviton exchanges
?
Heavy ion collisions in AdS
24
22
224
12
222
2
22 )(
2)(
22 dxzxT
NdxzxT
Ndzdxdxdx
z
Lds
CC
empty AdS5 1-graviton part higher ordergraviton 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 considerthe 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.
)(~ xT
)(~ 2 xT
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
0 ttT
0)(
Janik, Peschanksi ‘05
Lowest Order Diagram
Simple dimensional analysis:
221~
Each graviton gives , hence get no rapidity dependence:e
tindependen Yee
Y
)(~~ 11
xT
)(~~ 22
xT Grumiller, Romatschke ‘08Albacete, 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)
0)('
221~
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!
2224)2/,( x
aaxaxT
3/1
1~
1~
Aax
)()()( xaxa
x
3/12 Ap pAa /~ 3/1
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 ithappens 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:
1p
2p
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.
2
1~p
xi
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.
2/5
)2/3(
2~
)(
1~
e
xxp
0
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.
2224)2/,( x
aaxaxT
Proton Stopping
We have the original shock wave:
We have the produced stuff:
Adding them together we see thatthe 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
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:
marginallytrappedsurface
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!
cfmpth /07.0~
)(
1~
1~
3/13/1
3/1
1~
1~
Aax
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 0.288 (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.
3/1~~ sentropyNGubser, 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.
1
~
3p0,
1~
1
dVp3
Positive longitudinal pressure and isotropization
1~
3p
d
d If then, as , one gets .03 p 1
1~
↔ 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
)(')(),(')( 222
211
xxtxxt
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:
0)(,0)( 21
xTdxxTdx
0)(,0)( 21
xTdxxTdx
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.
)(),( 222
211
xTxT
22
213 8)()()( pp
z
y
x
t
T
)(000
0)(00
00)(0
000)(
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 e0 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!
AGeVs /200
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