How string theory may contribute to our understanding of ... · Elliptic flow vs. hydrodynamic...
Transcript of How string theory may contribute to our understanding of ... · Elliptic flow vs. hydrodynamic...
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How string theory may contribute to ourunderstanding of Heavy Ion Collisions
Urs Achim WiedemannCERN PH-TH Department
24 April 2007
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The Standard Model
!
LQCD + Lel"wk
Beyond theStandard Model
How does collectivityemerge from elementary
interactions ?
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The Standard Model
!
LQCD + Lel"wk
Beyond theStandard Model
How does collectivityemerge from elementary
interactions ?
The LHC Program
p+p @ 14 TeVATLASCMSLHCbALICE
Heavy Ions @ 5.5 TeVALICECMS
ATLAS
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The Standard Model
!
LQCD + Lel"wk
Beyond theStandard Model
How does collectivityemerge from elementary
interactions ?
String theory
String inspiredmodel building
Novel techniques forcalculating in thermal
QFTs
Topic of this talk
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Content
1. Some Questions in Heavy Ion Physics
2. Some Answers from String Theory
3. To what extent do the answers addressthe questions?
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Is the matter produced in heavy ioncollisions an ‘almost perfect fluid’?
Let’s explain the question…
Question 1:
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Basic observation: Elliptic Flow,a Hallmark of a collective phenomenon
squeeze
bounce
!
dN
d"# 1+ 2v
2cos 2"( )[ ]
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Elliptic flow vs. hydrodynamic simulations
PRC 72 (05) 014904200 GeV Au+Aumin-bias
Assumptions: - perfect (non-dissipative) fluid
- Bjorken boost invariance - ‘realistic’ equation of state - ‘realistic’ initial conditions - ‘realistic’ decoupling (freeze-out) !
Tµ" = # + p( ) uµ
u"$ p g
µ"
Results: - initial transverse pressure gradient - dependence of flow field elliptic flow
- size and pt-dependence of data accounted for by hydro (‘maximal’)
- characteristic mass dependence, since all particle species emerge from common flow field
!
"
!
uµ
!
v2(pT )
!
uµ!
v2
Strong claims at RHIC …
Equal energy density lines
Reactionplane
!
"
Kolb, Heinz;Teaney, Shuryak;Hirano, Nara;Huovinen
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Elliptic flow at RHIC - my view• Huge signal
• Rough agreementwith hydrodynamic modelsbased on perfect liquid.
• Open Questions:
- Can we constrain the size of dissipative properties such as viscosity? - Can dissipative properties of hot QCD matter be calculated from 1st principles?
- Are there independent tests of the manifestation and strength of collective flow?
!
dN
d"# 1+ 2v
2cos 2"( )[ ]
!
"
!
"
!
/2
!
0
!
0
!
"
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Dissipative Hydrodynamics• Do not neglect non-ideal components
• Region of validity? !
Ni
µ = niu
µ + n i
!
Tµ" = #uµ
u"$ p%
µ" + qµu" + q" uµ +&µ"
Spatio-temporal variations of macroscopic fluid should be smallif compared to microscopic reaction rates
!
" # n$ >> % = &µuµ Gradient expansion
• 2nd order viscous ‘Israel-Stewart’ hydrodynamics (shear viscosity only, neglect vorticity)
Entropy increase determined by viscosity:
!
T"µSµ # $µ%$
µ%2&
!
"#$%
µ$&
'D(%& +(µ' =) *µ
u' + 2"#(
%(µ,%
' )Equations of motion involverelaxation time and viscosity
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Constraints on viscosity from RHIC data
!
d(" s)
d"=
4
3#
" T
!
"
# T
1
s<<1
• very model-dependent (‘blast-wave-type’ study, not based on a hydrodynamic evolution)• indicate
• consistent with ideal fluid since Viscosity controls entropy increase
which is negligible if
D. Teaney, Phys. Rev. C68:034913, 2003
!
" s <<1
• In perturbation theory is parametrically large!
Is it at all conceivable that in a thermal QFT?
!
" s ~ 1 g2 log[1/g]
!
" s <<1
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Why is the matter produced in heavy ioncollisions ‘almost black’ for hard partons?
Let’s explain the question…
Question 2:
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Basic observation: suppression ofhigh pT hadron spectra
!
RAA (pT ,") =dN
AAdpTd"
ncoll dNNN
dpTd"
Centrality dependence:0-5% 70-90%
L large L small
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Jet Quenching: Au+Au vs. d+Au● Final state suppression ● Initial state enhancement
partonicenergy loss
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The medium-modified Final State Parton Shower
!
dI
d ln" dkT
=# sCR
(2$ )2" 2
2Re dy dy y
%
&0
%
& due'ikT u& e
'1
4d( ˆ q (( )u
2
y
%
&)
* +
,
- .
/0
0u.0
0sK(s = 0,y;u,y |")
Radiation offproduced parton
Target average determined bylight-like Wilson lines:
!
K s,y;u, y |"( ) = Drexp d#i"
2˙ r
2$
% &
'
( ) *
1
4ˆ q (#) r
2+ , -
. / 0 y
y
12
3 4 4
5
6 7 7 s= r(y )
u= r(y )
1
"89: 8 : : exp *1
4ˆ q Llongr
22
3 4 5
6 7
BDMPS transport coefficient
!
" Tr WA +(0)W
A(r)[ ]
tar
Reason for appearance of:
!
W xi( ) = P exp i dz
"TaAa
+xi,z"( )#[ ]
High energy scattering = phase rotation in target color field
!
"in
= # $i,x
i( )$i,xi{ }
% $i,x
i
!
"out
= # $i,x
i( ) % iW$
i&i
ri x
i( )( )$i,xi{ }
' &i,x
i
Baier, Dokshitzer, Mueller, Peigne, Schiff; Zakharov; Wiedemann…
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Quenching parameter from HI phenomenology
!
ˆ q =2
L2
" #"0( )" 0
L +" 0
$ ˆ q (") d" = 5 #15GeV
2
fm
Eskola, Honkanen, Salgado, Wiedemann Nucl Phys A747 (2005) 511
Can such a large value arise for a medium (in a thermal QFT),which displays a single momentum scale, T~300 MeV say?
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What is the fate of mesons in hot/densematter?
Question 3:
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Static quark-antiquark potential E(L)
!
W (C) =1
NTr P exp i A
C
"#
$ %
&
' (
)
* +
,
- .
!
W (C)T"exp #iL
timeE(L) # E
ren( )( )
• The T-dependence of E(L) has been studied in lattice QCD.
Bielefeld Group, hep-lat/0509001
time
transversedistance
L
Ltime
• Consider thermal expectation value of static Wilson loop in fundamental representation. For one can define a static quark- antiquark potential
!
Ltime
"#
• Lattice results support the argument of Matsui and Satz 1986 that quarkonium ‘melts’ due to color screening above the QCD phase transition.
Exponent of imaginary quantity
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J/Psi: a q-qbar pair moving through the mediumproduced in heavy ion collisions
Within reach in HICs at the LHC
• BUT: Connecting lattice QCD to heavy ion phenomenology requires model-dependent input
- formation time? - formation mechanism? - collective motion of ‘medium’
Can one get insight from 1st
principles of a thermal QFT onsome of these so-far model-dependent inputs?
- static potential for moving QQbar- spectral functions?
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What do these questions have in common?• The coupling constant is large (non-perturbative)
!
" ~ T 3 g2 log[1/g]
!
ˆ q RHIC
fitted>> ˆ q pert ~ g
2
Perturbatively:
!
" s Ttyp( ) =g2
4#~ 0.5 $ g
2Nc = % ~ 20
Large t’Hooft coupling is a better starting point:
• Problems involve real-time dynamics
- imaginary time formalism can be analytically continued but smallest Matsubara frequency is already , whereas hydrodynamic limit requires
!
2"T
!
" # 0
- lattice techniques are difficult to apply to - moving QQbar pairs - light-like Wilson lines - spectral functions
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AdS/CFT correspondenceN=4 SYM theory
!
gs," '= ls2
!
g2, Nc
1 gauge field4 Weyl fermions6 real scalars all in adjoint rep
Vanishing beta-functionTwo dim-less parameters
!
ds2 =
r2
R2"dt
2 + dr x 2( ) +
R2
r2
dr2 + R
2d#
5
2
Type IIB string theoryTwo dim-less parameters:String coupling and string length
Type IIB SUGRA lives in 10-dim space:
!
Z4D[J] = exp iS "cl[ ][ ]
!
g2
= 4" gs
!
" # g2Nc =
R2
$ '
For small, calculating correlation functions is a classical problem
!
" >>1, g2
!
limr"#
r
R2( )
$
%clr,x( ) = J(x)
Maldacena (1997)
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Wilson loops from AdS/CFT• Finite temperature N=4 SYM dual to AdS5 Schwarzschild black hole:
!
ds2 = " 1
2f dt
2 + r2
R2dx
1
2 + dx2
2 + dx3
2( ) + 1
fdr
2
!
f " r2
R21#
r0
4
r4( )
• Translation into field theoretic quantities:
Hawking temperature is QGP temperature
!
TH
=r0
" R2= T
String tension determines t’Hooft coupling
!
R2
" '= #
!
1 4"# '
!
" # gSYM2
N
Black holehorizon
• AdS/CFT - RecipeMaldacena (1998)Rey and Lee (1998)
!
WF(C)
T
" exp iS(C)( )
Our (3+1)-dim worldWilson loop Cin our world
horizon
!
r = r0
: area of stringworld sheet withboundary C.
!
S(C)
!
r = "
#$
Curvatureradius
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Quenching and Quarkonium dissociation
• Wilson loops for static and moving quark-antiquark potential and for quenching parameter are related by boosting the black-hole metric with rapidity (or velocity ).
!
"
!
v = tanh"
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Calculating the loop in boosted metric
• Parameterization of two-dimensional world-sheet bounded by C:
!
t = ", x1
=#
!
x2
= const., x3
= const.
Boundary condition:
!
y ± L
2( ) = ", r # r0y
!
x1
± L
2( ) = ± L
2
!
S(C) =1
4"# 'd$ d% det g#&' = (TLtime d$ L
0
l / 2
'
!
xµ = x
µ ",#( ), µ = t,1,2,3,r
Nambu-Goto action:
Our task: find catenary
!
r " r0y #( )
!
" = # /2for
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Finding S(C)
Not more difficult thanfinding the catenary
!
r = r(")
• Wilson loop can be considered as the space-time trajectory of a quark-antiquark pair.
• Key: open string connecting the quark pair can venture into the radial 5th dimension.
• Finding S(C): finding the shape of the string hanging from the spatial infinity of a black hole.
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Time-like vs. space-like world sheetTime-like world sheet:
!
cosh" < #
!
S(l) = "TLtime dyy4 # cosh2$
y4 #1( ) yc4 # q4( )1
%
&!
y'=1
qy4"1( ) y 4 " yc4( )
!
yc4
= cosh2" + q
2
e.o.m.
action
length
!
l = 2q2 dy1
y4 " yc
4( ) y 4 "1( )yc
#
$
Space-like world sheet:
!
cosh" > #
!
S(l) = i "TLtime dycosh
2# $ y 4
y4 $1( ) ym4 $ q4( )1
%
&!
y'2 =
1
q2y4"1( ) ym4 " q4( )
!
ym4
= cosh2" # q2
!
l = 2q2 dy1
y4 "1( ) ym4 " q4( )1
#
$
Boosted heavy quark-antiquark potential E(l)
Quenching parameter
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Velocity scaling of static potential
!
lmax
=2"# 3
4( )33/ 4# 1
4( )2
cosh1/ 2$
+O1
cosh5 / 2$
%
& '
(
) *
%
& '
(
) *
!
Lmax
=f ",#( )
$ T cosh"
f depends weakly on
!
",#
Consequences for pt-dependence of quarkonkium suppression at the LHC?
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Results for quenching parameter• The quenching parameter
- consider ordered limit:
!
"#$, %#$
- expand S(C) for small L:
!
ˆ q SYM =" 3 / 2
# 3
4( )# 5
4( )$T
3% 26.68 &SYM Nc T
3
!
ˆ q SYM
flow = cosh" f # sinh" f cos$( ) ˆ q SYM
• What if the medium is moving with rapidity at angle w.r.t. jet trajectory?
!
" f
!
"Liu, Rajagopal, Wiedemann, hep-ph/0612168
Liu, Rajagopal, Wiedemann,hep-ph/0605178
Herzog, Karch, Kovtun, Kozcaz, Yaffe hep-th/0605158S. Gubser, hep-th/0605182
!
WA
Clight" like( ) = exp "1
4ˆ q
L"
2L
2#
$ %
&
' (
= exp i2S Clight" like( )[ ]
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N=4 SYM Numerology
!
ˆ q SYM =" 3 / 2
# 3
4( )# 5
4( )$T
3% 26.68 &SYM Nc T
3
• If we relate N=4 SYM to QCD by fixing
!
Nc
= 3
!
"SYM
=1 2
!
ˆ q SYM = 4.4GeV
2
fm
!
ˆ q SYM =10.6GeV
2
fm
for T = 300 MeV
for T = 400 MeV
• In QGP of QCD, parton energy loss described perturbatively up to non-perturbative quenching parameter.
• We calculate quenching parameter in N=4 SYM (not necessarily a calculation of full energy loss of SYM)
This is close to values from experimental fits.
Is this comparison meaningful?
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Comment on: Is comparions meaningful?
• conformal
• no asmptotic freedom no confinement
• supersymmetric
• no chiral condensate
• no dynamical quarks, 6 scalar and 4 Weyl fermionic fields in adjoint representation
N=4 SYM theory
Physics nearvacuum and atvery high energyis very differentfrom that of QCD
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At finite temperature: Is comparions meaningful?
• conformal
• no asymptotic freedom no confinement
• supersymmetric (badly broken)
• no chiral condensate
• no dynamical quarks, 6 scalar and 4 Weyl fermionic fields in adjoint representation
N=4 SYM theory at finite T QCD at T ~ few x Tc
• near conformal (lattice)
• not intrinsic properties of QGP at strong coupling
• not present
• not present
• may be taken care of by proper normalization
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Quenching parameter for other theories
!
ˆ q CFT
ˆ q N= 4
=aCFT
aN= 4
=sCFT
sN= 4
• General CFTs with gravity dual: (large N and strong coupling)
a central charges entropy!
" =1Liu, Rajagopal, Wiedemann,hep-ph/0612168
• Near conformal theories: corrections small
!
ˆ q " 1# 3.121
3# vs
2( )( ) Buchel, hep-th/0605178
• Finite coupling and Nc corrections: hard Armesto, Edelstein, Mas hep-ph/0606245
• R-charge chemical potentials:Corrections small when chemical potentials small
Avramis, Sfetsos;Armesto, Edelstein, Mas;Lin, Matsuo; …
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What if quark is ‘dragged’ through N=4 SYM medium?Herzog, Karch, Kovtun, Kozcaz, Yaffe hep-th/0605158S. Gubser, hep-th/0605182
• Apply force to maintain momentum p of quark
!
˙ p = "µp + F = 0
Now, lost energy dissipates in hydrodynamic modes (Mach cone)
• Different from QCD quenching: since strongly coupled on all scales (divergence for v-> 1)
Fries, Gubser, Michalogiorgakis, Pufu,hep-th/0607022
• Range of validity of this picture set by Schwinger mechanism
!
dEq
dx="
2#T 2
v
1$ v2
!
Fcrit"M
2
#$ cosh% < &
Kinematic range does not overlap withthat of quenching calculation
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More on hydrodynamics
• A universal lower bound on viscosity from string theory
!
"
s>1
4#
!
1
4"1+135 #(3)
8 (2$)3 / 2+ ...
%
& '
(
) *
Strong coupling limitof N=4 SYMKovtun, Son, Starinets,hep-th/0309213
Arnold, Moore, Yaffe,JHEP 11 (2000) 001
!
" # g2Nc
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THE ENDof this talk
• Many chapters on comparing AdS/CFT to QCD written already
- Shear viscosity in QCD- diffusion constants- thermal spectral functions- quenching- drag
- quark-antiquark static potential- …
In thermal sector:Policastro, Son, Starinets, PRL 87, 081601 (2001)…
Policastro, Son, Starinets, hep-th/0205052, …
Teaney, hep-th/0602044, …
Liu, Rajagopal, Wiedemann hep-ph/0605178, PRL….
Liu, Rajagopal, Wiedemann hep-ph/0607062;PRL …
Herzog et al hep-th/0605158; Gubser hep-th/0605182Casalderrey-Solana, Teaney hep-ph/0605199
• ‘Comparison’ neither straightforward nor obviously far-fetched - rich testing ground for understanding non-perturbative properties of non-abelian thermal gauge field theories
• Many chapters to be written …