Transcript Slide 1
Department of Physics
HIC from AdS/CFT
Anastasios Taliotis
Work done in collaboration with Javier Albacete
and Yuri Kovchegov, arXiv:0805.2927 [hep-th],
arXiv:0902.3046 [hep-th], arXiv:0705.1234 [hep-ph],
arXiv:1004.3500 [hep-ph]
(published in JHEP and Phys. Rev. C)
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Outline
Motivating strongly coupled dynamics
Introduction to AdS/CFT
I. AA: State/set up the problem
Attacking the problem using AdS/CFT
Predictions/comparisons/conclusions/Summary
II. pA: State/set up the problem
Predictions/Conclusions
III. Transverse Dynamics-a quick look
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Motivating strongly coupled
dynamics in HIC
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Notation/Facts
2
2
x
x
Proper time:
0
3
x0
1 x0 x3 1 x
ln
Rapidity: ln
2 x 0 x 3 2 x
Saturation scale Qs: The scale where
density of partons becomes high.
valid for times t >> 1/Qs
Bj QGP
Hydro
g<<1; valid up to times ~ 1/QS.
CGC describes matter distribution due
CGC
to classical gluon fields and is rapidityindependent ( g<<1, early times).
Hydro is a necessary condition for
thermalization. Bjorken Hydro describes
successfully particle spectra and spectral flow. Is
g??>>1 at late times?? Maybe; consistent with
the small MFP implied
by
a hydro description.
D F J
No unified framework exists that describes both
strongly & weakly coupled dynamics
x3
4
Goal: Stress-Energy (SE) Tensor
• SE of the produced medium gives useful information.
• In particular, its form (as a function of space and time
variables) allows to decide whether we could have
thermalization i.e. it provides useful criteria for the
(possible) formation of QGP.
• SE tensor will be the main object of this talk: we will
see how it can be calculated by non perturbative
methods in HIC.
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Introduction to AdS/CFT
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Scales & Parameters
N =4 SYM SU(Nc)
Type IIB superstring
l p (10)
L Radius of S 5
Q. gravity & fields
Q. strings
L
1/ Nc
Clas. fields & part.
Clas. Strings
L / l t ' Hooft gs Nc
4
4
s
ls
L
4
L /l
4
p (10)
N c ~ 1/ G
(10)
1/
2
gYM
gs
lp(10) /L 1=> (Ignore QM / small G (10) ) => Large Nc
ls / L 1 => (Ignore extended objects/small g s ) => Large λ
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Quantifying the Conjecture
[Witten ‘98]
<exp z=0∫O φ0>CFT = Zs(φ|φ(z=0)= φo)
O is the CFT operator. Typically want <O1 O2…On>
φ0 =φ0 (x1,x2,… ,xd) is the source of O in the CFT picture
φ =φ (x1,x2,… ,xd ,z) is some field in string theory with B.C.
φ (z=0)= φ0
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How to use the correspondence
• Take functional derivatives on both sides. LHS gives
correlation functions. RHS is the machine that
computes them (at any value of coupling!!).
• Must write fields φ (that act as source in the CFT) as
a convolution with a boundary to bulk propagator:
φ (xμ,z)= ∫dxν' φ0 (xμ’)Δ(xμ – xμ’,z)
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• φ (xμ,z) being a field of string theory must obey some
equation of motion; say □ φ=0. Then Δ(xμ – xμ’,z) is
specified solving
□ Δ=δ(xμ – xμ’) δ(z)
Note:
• Usually approximate string theory by SUGRA and hence Zs
by a single point (saddle point); we approximated the large
coupling gauge problem with a point of string theory!! Once
we know Zs, we are done; can compute anything in CFT.
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Holographic renormalization
de Haro, Skenderis, Solodukhin ‘00
Example:
• Know the SE Tensor of Gauge theory is given by
T
S
1
2
g g
|g
• So gμν acts as a source => in order to calculate Tμν from
AdS/CFT must find the metric. Metric has its eq. of motion
i.e. Einsteins equations.
• Then by varying the Zs w.r.t. the metric at the boundary
(once at z=0) can obtain < Tμν >.
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Holographic renormalization
Energy-momentum tensor is dual to the metric in AdS. Using
Fefferman-Graham coordinates one can write the metric as
2
L
ds 2 2 g ( x, z ) dx dx dz 2 L2 d 5
z
~ ( x, z ) the 4d metric.
with z the 5th dimension variable and g
~
Expand g ( x, z ) near the boundary (z=0) of the AdS space:
Using AdS/CFT can show:
2
c
2 z 0
N
T
lim
2
,
g ( x , z )
z4
, i.e. z 4 coef .
and
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I. AA: State/set up the
problem
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Rmrks:
• Deal with N=4 SYM theory
• Coupling is tuned large and remains large at all times
• Forget previous results of pQCD
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Strategy
Initial Tµν
phenomenology
Initial Geometry
Evolve
Einst.
Eq.
AdS/CFT
Dictionary
Dynamical
Geometry
Dynamical Tµν
(our result)
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Field equations, AdS5 & examples
gμν Tμν
Eq. of Motion (units L=1) for gΜΝ(xM = x±, x1, x2, z) is generally given
R
1
g R 6g
2
J
Empty & “Flat” AdS space:
; empty space reduces R
4 g 0
1
2
ds 2 [2dx dx dx dz 2 ]
z
2
g ( x , z )
N c2
4
T 2 lim
,
i
.
e
.
z
coef . implies Tμν=0 in QFT side
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2 z 0
z
Empty but not flat AdS-shockwave:
[Janik & Peschanski ’06]
2
1
2
2
ds 2 [2dx dx 2 T ( x ) z 4 (dx ) 2 dx dz 2 ]
z
Nc
2
Then ~z4 coef. implies
<Tμν (xμ)>= δμ - δν - < Tμν (x-)> in QFT side
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Single nucleus
Single shockwave
Choose T-- (x-) a localized function along x- but not
along ┴ plane. So take
T ~ (x )
μ is associated with the energy carried by nucleus ([μ]=3).
May represent the shockwave metric as a
single vertex: a graviton exchange between
the source (the nucleus living at z=0; the
boundary of AdS) and point X in the bulk which
gravitational field is measured.
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Superposition of two shockwaves
Non linearities of gravity
2
L
ds2 2
z
?
2
2
2
2
2
2
4
2
4
2
z dx
2 dx dx dx dz 2 T1 ( x ) z dx 2 T2 ( x ) Higher
graviton
NC
NC
ex.
Flat AdS
One graviton ex.
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Due to non linearities
Built up a perturbative approach
• Motivation: Knowing gMN in the forward light cone we automatically
know Tμν of QFT after the ion’s collision just read it from ∂gMN (~z4
coefficient).
• Know that Ti ~μi (i=1,2). Higher graviton exchanges; i.e. corrections to
gMN should come with extra powers of μ1 and μ2: μ1μ2, μ12μ2, μ1μ22, …
• So reconstruct by expanding around the flat AdS:
( 0)
(1)
( 2)
g MN ( x , z ) g MN ( x , z ) g MN ( x , z ) g MN ( x , z ) ...
(0)
g MN
flat AdS, (1)
g MN
single shockwave(s),
( j 2)
g MN
higher gravitons
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Insight from Dim. Analysis, symmetries,
kinematics & conservation
Tracelessness + conservation Tμν(x+, x-) provide 3
equations. Also have x+ x- symmetry. Expect:
T
~
ho , T ~
ho , T ~ ho , T ij ~ ij ho
~ T1 ~ 1 ( x )
Y
~ T2 ~ 2 ( x )
For the case Ti =μi δ(x) shockwaves [μi]=3 and as energy
density has [ε]=4 then we expect
that the first correction to ε must be
ε~ μ1 μ2 x+ x- i.e.rapidity
independent as diagram suggests.
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Calculation/results
• Step 1.: Linearize field eq. expanding around ημν
(partial DE with w.r.t. x+,x-, z with non constant coef.).
• Step 2.: Decouple the DE. In particular g(2)┴┴=h(x+,x-, z )
obeys: □h=8/3 z6 t1(x-) t2 (x+) with box the d'Alembertian in AdS5.
• Step 3.: Solve them imposing (BC) causality. Find:
h= z4 ho(t1(x-) , t2(x+)) + z6 h1(t1(x-) , t2(x+)).
• Step 4.: Use rest components of field eq. in order to determine rest
components of gμν.
• Step 5.: Determine Tμν by reading the z4 coef. of gμν
Conclude: Tμν has precisely the form we suspected for any t1, t2:
Tμν is encoded in a single coefficient!
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Particular sources (nucleus profile)
• Only need ho:
•
( )2 h0 ~ t1 ( x )t2 ( x ) . Encodes Tμν.
δ profiles: Get corrections:
[Grumiller, Romatschke ’08]
[Albacete, Kovchegov, Taliotis’08]
T+ -~T┴┴ ~ ho ~μ1 μ2τ2 and T- - ~ μ1 μ2(x+)2
• Step profiles: Here δ’s are smeared; ( x ) ( x ) (a x )
a
T ( x a, x a / 2) 4 2 2 x 2
a
• At x ~
1
a
the nucleus will run out of momentum and stop!
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Conclusions/comparisons/summary
• Constructed graviton expansion for the collision of two
shock waves in AdS. Goal is obtain SE tensor of the
produced strongly-coupled matter in the gauge theory. Can
go to any finite order. Lower order hold for early times.
• LO agress with [Grumiller, Romatschke ‘08]. NLO and NNLO
corrections have been also performed.
• They confirm: Tμν is encoded in a single coefficient h0(x+,x-).
Also come with alternate sign.
• Likely nucleus stops. A more detailed calculation (all order
ressumation in A) in pA [Albacete, Taliotis, Yu.K. ‘09] confirms it.
• Possibly have Landau hydro. However its Bjorken hydro
that describes (quite well) RHIC data.
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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 using 24
field theory or AdS/CFT?
II. pA: State/set up the
problem
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Diagrammatic
Representation
pA collisions
16z 5t1 ( x )t2 ( x )
Initial Condition vertex
cf. gluon production in pA
collisions in CGC!
1
t p
1
Scalar Propagator
Multiple graviton ex.
vertex ~ t2
t2 p2
Eq. for transverse component:
3
z [ hz hzz 2hx x ] 16z 5t1 ( x )t2 ( x )
z
2
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1
1 6
z t 2 ( x ) {z z [
hz hzz 2hx x ] z t2 ( x )hx x z }
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2z
2
2
2
Eikonal Approximation &
Diagrams Resummation
•Nucleus is Lorentz-contracted and so xi ~ 1/ p2 are
small; hence ∂+ is large compared to ∂- and ∂z.
•This allows to sub the vertices and propagators with
effectives and simplify problem. For more see [Kovchegov,
Albacete, Taliotis’09].
•Apprxn applies for
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Calculation (δ profiles)
• Particular profiles:
• Diagram ressumation (all orders in μ2) in the forward LC yields:
g|z 4 ho;ei ( x , x )
• Recalling the duality mapping:
T
N c2
g | z 4
2
2
• Finally recalling ho;ei encodes <Tμν> through
T
~
ho;ei , T ~
ho;ei , T ~ ho;ei , T ij ~ ij ho;ei
yields to the results:
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Results
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Conclusions
• Not Bjorken hydro
Indeed instead of T┴ ┴=p ~1/τ4/3 it is found that
1
p~
2
(x )
x
~
e (3 / 2)
5/ 2
• Not (any other) Ideal Hydrodynamics either
Indeed, from
and considering μ=ν=+ deduce
that T++ >0; however T++ is found strictly negative!
• Proton stopping in pA also
(Landau Hydro??)
2 2
For AA, it was found earlier that T (x a, x a/2) ~ 2 x
a
with estimation stopping time estimated by x 1/ 2a . Same result
recovered here by considering the total T++and expanding to O(μ2;x-=α/2):
tot
T
in
T
prod
T
N 2
c
2
1
2 a1
{1 (
1
1 82 (x )2 x
1)}, 0 x a1
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Proton Stopping
(Landau Hydro??)
T++
( x 0)
X+
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Future Work
• Use CGC as initial condition in order to evolve
the metric to later times! Ambiguities Many
initial metrics give same initial condition. Choose
the simplest?
• Include transverse dynamics? Very hard but…
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Recent Work
arXiv: 1004.3500v1[hep-th] - [Taliotis]
Snapshot of the
collision at given
proper time τ
r1,2 ( x1 b) 2 ( x 2 ) 2
•Causality separates evolution in a very intuitive way!
•General form of SE tensor: For given proper time τ it has the form
T (r2 r1 ) (r1 ) A (r2 r1 ) ( r1 ) A
( r2 ) (r2 r1 ) A
I
II
III
{b b}
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Eccentricity-Momentum Anisotropy
Momentum Anisotropy εx= εx (x≡τ/b) (left) and εx = εx(1/x) (right) for
intermediate x≡τ/b .
Agrees qualitatively with [Heinz,Kolb,
Lappi,Venugopalan,Jas,Mrowczynski]
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Conclusions
Built perturbative expansion of dual geometry to determine Tµν ;
applies for sufficiently early times: µτ3<<1.
Tµν evolves according to causality in an intuitive way! Also
T
(
,x
,x
,b
).
[Gubser ‘10]
Tµν is invariant under lim
1
2
b
0
Our exact formula (when applicable) allows as to compute Spatial
Eccentricity and Momentum Anisotropy .
When τ>>r1 ,r2 have ε~τ2 log 2 τ-compare with ε~Qs2log 2τ
[Lappi, Fukushima]
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Thank you
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