How string theory may contribute to ourunderstanding of Heavy Ion Collisions

Urs Achim WiedemannCERN PH-TH Department

24 April 2007

The Standard Model

!

LQCD + Lel"wk

Beyond theStandard Model

How does collectivityemerge from elementary

interactions ?

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

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

Content

1. Some Questions in Heavy Ion Physics

2. Some Answers from String Theory

3. To what extent do the answers addressthe questions?

Is the matter produced in heavy ioncollisions an ‘almost perfect fluid’?

Let’s explain the question…

Question 1:

Basic observation: Elliptic Flow,a Hallmark of a collective phenomenon

squeeze

bounce

!

dN

d"# 1+ 2v

2cos 2"( )[ ]

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

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

!

"

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

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

Why is the matter produced in heavy ioncollisions ‘almost black’ for hard partons?

Let’s explain the question…

Question 2:

Basic observation: suppression ofhigh pT hadron spectra

!

RAA (pT ,") =dN

AAdpTd"

ncoll dNNN

dpTd"

Centrality dependence:0-5% 70-90%

L large L small

Jet Quenching: Au+Au vs. d+Au● Final state suppression ● Initial state enhancement

partonicenergy loss

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…

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?

What is the fate of mesons in hot/densematter?

Question 3:

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

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?

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

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)

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

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"

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

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.

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

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?

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( )[ ]

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?

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

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

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; …

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

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

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 …