department of physics hic from ads/cft anastasios taliotis
DESCRIPTION
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 ). 1. Outline. - PowerPoint PPT PresentationTRANSCRIPT
11
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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OutlineMotivating 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
33
Motivating strongly coupled dynamics in HIC
44
Notation/FactsProper time:
Rapidity:
Saturation scale : The scale where density of partons becomes high.
23
20 xx
12ln
x0 x3x0 x3
1
2ln
xx
0x
3x
QGP
CGCCGC describes matter distribution due to classical gluon fields and is rapidity-independent ( 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.
No unified framework exists that describes both strongly & weakly coupled dynamics
valid for times t >> 1/Qs
Bj Hydro
g<<1; valid up to times ~ 1/QS.
sQ
JFD
55
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.
66
Introduction to AdS/CFT
77
Type IIB superstring N =4 SYM SU(Nc)
Q. gravity & fields Q. strings
Clas. fields & part. Clas. Strings
1/ cN
=> (Ignore QM / small ) => Large Nc
=> (Ignore extended objects/small ) => Large λ
1/sl
L
5L Radius of S
4 4/ 's s cL l t Hooft g N
2YM sg g
(10)pl
L
4 4 (10)(10)/ ~ 1/p cL l N G
(10)G
Scales & ParametersScales & Parameters
lp(10) /L 1
ls / L 1 sg
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Quantifying the Conjecture
<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
[Witten ‘98]
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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
• Know the SE Tensor of Gauge theory is given by
• 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 >.
Example:de Haro, Skenderis, Solodukhin ‘00
gg
S
gT |1
2
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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 (z=0) of the AdS space:
Using AdS/CFT can show: , and
Holographic renormalization
22 2 2
52( , )
Lds g x z dx dx dz L d
z
),(~ zxg
),(~ zxg
...,),(
lim2
4402
2
coefzeiz
zxgNT
z
c
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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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Initial Tµνphenomenology
AdS/CFTDictionary
Initial Geometry
Dynamical Geometry Dynamical Tµν
(our result)
EvolveEinst. Eq.
Strategy
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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
; empty space reduces
Empty & “Flat” AdS space:
implies Tμν=0 in QFT
side
Empty but not flat AdS-shockwave: [Janik & Peschanski ’06]
Then ~z4 coef. implies <Tμν (xμ)>= δμ - δν - < Tμν (x-)> in QFT side
JgRgR 6
2
1 04 gR
]2[1 22
22 dzdxdxdx
zds
...,),(
lim2
4402
2
coefzeiz
zxgNT
z
c
])()(2
2[1 2224
2
2
2
2 dzdxdxzxTN
dxdxz
dsc
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Single nucleus Single shockwave
Choose T-- (x-) a localized function along x- but not
along ┴ plane. So take
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.
T ~ (x )
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Superposition of two shockwavesNon linearities of gravity
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22
224
12
222
2
22 )(
2)(
22 dxzxT
NdxzxT
Ndzdxdxdx
z
Lds
CC
?
Flat AdS
Higher graviton ex.
Due to non linearities One graviton ex.
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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, μ1
2μ2, μ1μ22, …
• So reconstruct by expanding around the flat AdS:
flat AdS, single shockwave(s), higher gravitons
...),(),(),(),()2()1()0(
zxgzxgzxgzxg MNMNMNMN
MNg)0(
MNg)1( )2( j
MNg
2020
Insight from Dim. Analysis, symmetries, kinematics & conservation
Tracelessness + conservation Tμν(x+, x-) provide 3 equations. Also have x+ x- symmetry. Expect:
)(~~ 11
xT
oijij
ooo hThThThT ~,~,~,~
For the case Ti =μi δ(x) shock-waves [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.
Y
)(~~ 22
xT
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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: . Encodes Tμν.
• δ profiles: Get corrections:
T+ -~T┴┴ ~ ho ~μ1 μ2τ2 and T- - ~ μ1 μ2(x+)2
• Step profiles: Here δ’s are smeared;
• At the nucleus will run out of momentum and stop!
)()()( xaxa
x
)()(~)( 2102
xtxth
2224)2/,( x
aaxaxT
[Grumiller, Romatschke ’08][Albacete, Kovchegov, Taliotis’08]
ax
1
~
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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 ithappens in AA collisions using field theory or AdS/CFT?
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II. pA: State/set up the problem
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pA collisions
11 pt 22 pt
]23[
xxzzzz hhhz
})(2
1]2
2
11
2
7[{)( 2
622
2
zxxxxzzzz hxtzhhhz
zxtz
Eq. for transverse component:
Diagrammatic Representation
Scalar Propagator
Multiple graviton ex.
vertex ~ t2
)()(16 215 xtxtz
Initial Condition vertex
)()(16 215 xtxtz
cf. gluon production in pAcollisions in CGC!
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Eikonal Approximation &Diagrams Resummation
•Nucleus is Lorentz-contracted and so 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
2/1~ pxi
2828
Calculation (δ profiles)• Particular profiles:
• Diagram ressumation (all orders in μ2) in the forward LC yields:
• Recalling the duality mapping:
• Finally recalling ho;ei encodes <Tμν> through
yields to the results:
),(;| 4
xxhg eioz
eioijij
eioeioeio hThThThT ;;;; ~,~,~,~
4|2
2
2 zc g
NT
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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
• Not (any other) Ideal Hydrodynamics eitherIndeed, from and considering μ=ν=+ deduce that T++ >0; however T++ is found strictly negative!
• Proton stopping in pA also For AA, it was found earlier thatwith estimation stopping time estimated by . Same result recovered here by considering the total T++and expanding to O(μ2;x-=α/2):
2/5
)2/3(
2~
)(
1~
e
xxp
T (x a,x a /2) ~a 22x2
ax 2/1
(Landau Hydro??)
Ttot Tin
Tprod
Nc2
2 21a1{1 (
1
182(x)2 x
1)}, 0 x a1
3131
Proton Stopping(Landau Hydro??)
T++
X+
)0( x
3232
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]
1 2 2 21,2 ( ) ( )r x b x
2 1 1 2 1 1( ) ( ) ( ) ( )I IIT r r r A r r r A
2 2 1( ) ( ) { }IIIr r r A b b
•Causality separates evolution in a very intuitive way!
•General form of SE tensor: For given proper time τ it has the form
Snapshot of the collision at given proper time τ
Eccentricity-Momentum Anisotropy
Momentum Anisotropy εx= εx (x≡τ/b) (left) and εx = εx(1/x) (right) for intermediate x≡τ/b .
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Agrees qualitatively with [Heinz,Kolb, Lappi,Venugopalan,Jas,Mrowczynski]
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
Tµν is invariant under .
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
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),,,( 210
lim bxxTb
[Gubser ‘10]
[Lappi, Fukushima]
3636
Thank you