A Nearly Perfect Ink !?Theoretical Challenges from RHIC

Dublin - 29 July 2005

LATTICE 2005

Berndt Mueller (Duke University)

A perfect ink…

Is brilliantly dark and opaque Yet flows smoothly and easily

A painful challenge to fountain pen designers

A delightful challenge to physicists: Are the two requirements really

compatible?Hint:

1f

The road to the quark-gluon plasma

STAR

…Is hexagonal and 2.4 miles long

Two wealths

ChallengesC

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hallenge

sChallenge

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Challenges

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lleng

e

s

Challenge

s Challenges C

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A wealth of …A wealth of …

D A T A

Cornerstone results from RHIC

Anisotropic transverse flow

“Jet” suppression

Baryon/meson enhancement

Azimuthal anisotropy v2

py

)(tan,2cos 12

x

y

p

pv

Coordinate space: initial asymmetry

Momentum space: final asymmetry

pxx

y

Semiperipheral collisions

Signals early equilibration (teq 0.6 fm/c)

Jet quenching in Au+Au

PHENIX Data: Identified 0

d+Au

Au+Au

No quenching

Quenching!

Baryons vs. mesons

baryons mesons

RC

P

pT (GeV/c)

behaves like meson ?

(also -meson)

coll

Yield( | )( )

( )Yield( )

b AAR b

N b pp

Overview

QCD Thermodynamics What are the dynamical degrees of freedom? Is there a critical point, and where is it?

Thermalization How can it be so fast?

Transport in a thermal medium Viscosity, energy loss, collective modes

Hadronization Recombination vs. fragmentation

QCD phase diagram

B

Hadronicmatter

Critical endpoint

?

Plasma

Nuclei

Chiral symmetrybroken

Chiral symmetryrestored

Color superconductor

Neutron stars

T

1st order line ?

Quark-Gluon

RHIC

Space-time picture

eqPre-equil. phase

final2

( ) /( )

( ) /

( / )

T

T

A

dE dy

dV dy

dE dy

R

3( ) 4.5 GeV/fm

at 1 fm/c

in Au+Au (200 GeV)

Bjorken formula

Stages of a r.h.i. collision

Initial collision ≈ break-up of the coherent gluon field (“color glass condensate”)

Pre-equilibrium: the most puzzling stage

Equilibrium (T > Tc): hydrodynamic expansion in longitudinal and transverse directions

Hadronization: are there theoretically accessible domains in pT ?

Hadronic stage (T < Tc): Boltzmann transport of the hadronic resonance gas

Initial state: Gluon saturation

~ 1/Q2

2 2 1/3s

2 2 2s s

gluon density area ( , ) ( )

= 1A s s

A

xG x Q Q A x

R Q Q

2s ( , )Q x A

Details of space-time picture depend on gauge!

Gribov, Levin, Ryskin ’83

Blaizot, A. Mueller ’87

McLerran, Venugopalan ‘94

“Color glass condensate (CGC)”

CGC dynamicsInitial state occupation numbers ~ 1/s 1 → classical fields generated by random color sources on light cone:

2 20 2 2

1( ) exp [ ( )]a a

s

P P d x xg Q

Boost invariance → Hamiltonian gauge field dynamics in transverse

plane (x,). 2-dim lattice simulation shows rapid equipartitioning of

energy (eq ~ Qs-1). Krasnitz, Nara & Venugopalan; Lappi (hep-ph/0303076 )

( )a x ( )a x

( , )aiA x

Challenge: (3+1) dim. simulation without boost invariance

Saturation and dN/dKharzeev & Levin

Pseudorapidity = - ln tan

Rapidity y = ln tanh(E+pL)/(E-pL)

Assume nucleus is “black” for all gluons with kT Qs:

Qs(x) → Qs(y)

with x = Qse-|Y-y| .

Also predicts beam energy dependence of dN/dy.

Challenge:

How much entropy is produced by simple decoherence, how much during the subsequent full equilibration?

The(rmalization) mystery

Experiment demands: th 0.6 fm/c “Bottom up” scenario (Baier, A. Mueller, Schiff, Son):

“Hard” gluons with kT ~ Qs are released from the CGC; Released gluons collide and radiate thermal gluons;

Thermalization time th ~ [s13/5 Qs]-1 ≈ 2-3 fm/c

Perturbative dynamics among gluons does not lead to rapid thermalization.

Quasi-abelian instability (Mrowczynski; Arnold et al; Rebhan et al): Non-isotropic gluon distributions induce exponentially

growing field modes at soft scale k ~ gQs ; These coherent fields deflect and isotropize the “hard”

gluons.

After thermalization…

... matter is described by (relativistic) hydrodynamics !

Requires f L and small shear viscosity .

HTL pert. theory (nf=3):

Dimensionless quantity /s.

Classical transp. th.: ≈ 1.5Tf, s ≈ 4 → /s ≈ 0.4Tf.

3

4 1106.7

ln

T

g g (Baym…; Arnold,

Moore & Yaffe)

( )

0 with

( trace)P u

T

uT uu Pg

Azimuthal anisotropy v2

py

)(tan,2cos 12

x

y

p

pv

Coordinate space: initial asymmetry

Momentum space: final asymmetry

pxx

y

Semiperipheral collisions

Signals early equilibration (teq 0.6 fm/c)

– How small can it be?

0 00 0 0

41

3s T

T s s

D. TeaneyBoost invariant hydro with T00 ~ 1 requires /s ~ 0.1.

N=4 SUSY Yang-Mills theory (g1):

/s = 1/4(Kovtun, Son, Starinets).

Absolute lower bound on /s ?

/s = 1/4 implies f ≈ (5 T)-1 ≈ 0.3 d

Challenge: (3+1) dim. relativistic viscous fluid dynamics

QGP(T≈Tc) = sQGP

First attempts

3 ( )

ret

1( , ) ( , )

6

tt ik

ikd x d e dt T x t T x t

Nakamura & Sakai

SU(3) YM

Related (warm-up?) problem: EM conductivity ≈ q2f/2T.

nf = 3 QGP → / ≈ 20 T2.

Caveat: f(glue) f(quarks)

Method: spectral function repr. of G and Gret, fit of spectral fct. to G.

Challenge: Calculate /s for real QCD

QCD equation of state2

4

30T

20%

Challenge: Devise method for determining from data

Challenge: Identify the degrees of freedom as function of T

Is the (s)QGP a gaseous, liquid, or solid plasma ?

Challenge: QCD e.o.s. with light domain wall quarks

A possible method

24

23

30

2

45

T

s T

4 4

2 3 3

12150.96

128

s s

Eliminate T from and s :

Lower limit on requires lower limit on s and upper limit on .

BM & K. Rajagopal, hep-ph/0502174

Measuring and s

Entropy is related to produced particle number and is conserved in the expansion of the (nearly) ideal fluid: dN/dy → S → s = S/V.

Energy density is more difficult to determine: Energy contained in transverse degrees of freedom is not

conserved during hydrodynamical expansion. Focus in the past has been on obtaining a lower limit on ; here

we need an upper limit. New aspect at RHIC: parton energy loss. dE/dx is telling us

something important – but what exactly?

Entropy Two approaches:

1) Use inferred particle numbers at chemical freeze-out from statistical model fits of hadron yields;

2) Use measured hadron yields and HBT system size parameters as kinetic freeze-out (Pratt & Pal).

Method 2 is closer to data, but requires more assumptions.

Good news: results agree within errors: dS/dy = 5100 ± 400 for Au+Au (6% central, 200 GeV/NN)

→ s = (dS/dy)/(R20) = 33 ± 3 fm-3

Jet quenching in Au+Au

PHENIX Data: Identified 0

d+Au

Au+Au

No quenching

Quenching!

High-energy parton loses energy by

rescattering in dense, hot medium.q

q

Radiative energy loss:2/ TdE dx L k

“Jet quenching” = parton energy loss

q q

g

Scattering centers = color chargesL

2 2T

22

2ˆ

dq q dq

dqk

Density of scattering centers

Range of color forceScattering power of the QCD medium:

Energy loss at RHIC2ˆ 5 10 GeV /fmq Data suggest large energy loss parameter:

RHIC

Eskola, Honkanen, Salgado & Wiedemann

pT = 4.5–10 GeV

Dainese, Loizides, Paic

The Baier plot

Plotted against , is the same for a gas and for a perturbative QGP.

Suggests that is really a measure of the energy density.

Data suggest that may be larger than compatible with Baier plot.

Nonperturbat. calculation is needed.

Pion gas

QGP

Cold nuclear matter

q̂

q̂

q̂

RHIC data

sQGP

Challenge: Realistic calculation of gluon radiation in medium

Eikonal formalism

quark

x

x -

0

( ) ( ; ) ( )

( ; ) exp ( , )L

q x W x L q x

W x L i dx A x x

P

+Gluon radiation: x = 0

x

( )2 2

†

2

( ) 1 ( ,0) (0, ) ( , )2

Tr[w

( ; ) ( ; )]( , )

1ith

ik x ys Fg

A A

c

C x yN k dx dy e C x C y C x y

x y

W x L W y LC x y

N

Kovner, Wiedemann

Eikonal form. II†

2

2 †

0 0

2 2

Tr[ ( ; ) ( ; )( , )

1

1( ) ( ) ( , ) ( ) ( , )

2

1 1( ) ( ) ˆ

2 2i

i F

c

xLi

i

W x L W y LC x y

N

x y dx dx F x W x x F x W x x

x y L x yF F qL

Challenge: Compute F+i(x)F+i(0) for x2 = 0 on the lattice

Not unlike calculation of gluon structure function, maybe moments are calculable using euclidean techniques.

Where does Eloss go?

STAR p+p Au+Au

Lost energy of away-side jet is redistributed to rather large angles!

Trigger jetAway-side jet

Wakes in the QGP

Mach cone requires collective mode with (k) < k:

J. Ruppert and B. Müller, Phys. Lett. B 618 (2005) 123

Colorless sound? Colored sound = longitudinal gluons? Transverse gluons

Angular distribution depends on energy fraction in collective mode and propagation velocity

T. Renk & J. Ruppert

Baryons vs. mesons

baryons mesons

RC

P

pT (GeV/c)

behaves like meson ?

(also -meson)

coll

Yield( | )( )

( )Yield( )

b AAR b

N b pp

Hadronization mechanisms

qq

q

Baryon1

Meson

Fragmentation

q q

q q q

Baryon1

Meson

Recombination

M Q B Q2 3p p p p

Recombination wins…… for a thermal source

Baryons compete with mesons

Fragmentation dominates for a power-law tail

Quark number scaling of v2

In the recombination regime, meson and baryon v2 can be obtained from the quark v2 :

2 2 2 2v22

v3

v3v Btt

q tM q tp ppp

Hadronization RHIC data (Runs 4 and 5) will provide wealth of data on: Identified hadron spectra up to much higher pT (~10 GeV/c);

Elliptic flow v2 up to higher pT with particle ID;

Identified di-hadron correlations; Spectra and v2 for D-mesons….

D D

recombinationfragmentation

Challenge: Unified framework treating recombination as special case of QCD fragmentation “in medium”.

A. Majumder & X.N. Wang, nucl-th/0506040

Some other challenges

Where is the QCD critical point in (,T)?

What is the nature of the QGP in Tc < T < 2Tc ?

How well is QCD below Tc described by a weakly interacting resonance gas?

Thermal photon spectral function (m2,T).

Are there collective modes with (k) < k ?

Can lattice simulations help understand the dynamics of bulk (thermal) hadronization?

Don’t be afraid…

“Errors are the doors to discovery”

James Joyce

Back-up slides

Hard-soft dynamics

Nonabelian Vlasov equations generalizing “hard-thermal loop” effective theory.

Can be defined on (spatial 3-D) lattice with particles described as test charges or by multipole expansion (Hu & Müller, Moore, Bödeker, Rummukainen).

Poss. problem: short-distance lattice modes have wrong (k).

( ) ( ) ( ) (

ˆ(1, v) light-l

( )

ik

)

e

i i i

i

i

u

J x d Q u x x

D F gJ

k ~ gQs k ~ Qs

Challenge: Full (3+1) dim. simulation of hard-soft dynamics

Reco: Thermal quarks

2M

M3 3,

2B

B3 3, ,

( , ) ( , (1 ) ) ( )(2 )

' ( , ) ( , ' ) ( , (1 ') ) ( , ')(2 )

dN P uE d dxw R xP w R x P x

d P

dN P uE d dxdx w R xP w R x P w R x x P x x

d p

( , )w r p Quark distribution function at “freeze-out”

Relativistic formulation using hadron light-cone frame:

For a thermal distribution,

the hadron wavefunctions can be integrated out, eliminating the model dependence of predictions.

Remains exactly true even if higher Fock states are included!

( , ) exp( / )w r p p u T

Heavy quarks

Heavy quarks (c, b) provide a hard scale via their mass. Three ways to make use of this:

Color screening of (Q-Qbar) bound states;

Energy loss of “slow” heavy quarks;

D-, B-mesons as probes of collective flow.

RHIC program: c-quarks and J/;

LH”I”C program: b-quarks and .

RHIC data for J/ are forthcoming (Runs 4 & 5).

J/ suppression ?Vqq is screened at scale (gT)-1 heavy quark bound states dissolve above some Td.

Karsch et al.

Color singlet free energy

Quenched lattice simulations, with analytic continuation to real time, suggest Td 2Tc !

S. Datta et al. (PRD 69, 094507)

Challenge: Compute J/ spectral function in unquenched QCD

C-Cbar kinetics

J/

LHC

LHC

RHIC

RHIC

J/, can be ionized by thermal gluons. If resonances persist above Tc, J/ and can be formed by recombination in the medium: J/ may be enhanced at LHC!

Challenge: Multiple scattering theory of heavy quarks in a thermal medium

Analogous to the multiple scattering theory for high-pT partons, but using methods (NRQCD etc.) appropriate for heavy quarks.