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:

f
1

The road to the quark-gluon plasma
…Is hexagonal and 2.4 miles long
STAR
Two wealths
A wealth of …
Challenges
A wealth of …
Challenges
Challenges
DATA
Challenges
Cornerstone results from RHIC
 Anisotropic
 “Jet”
transverse flow
suppression
 Baryon/meson
enhancement
Azimuthal anisotropy v2
Semiperipheral collisions
Coordinate space:
initial asymmetry
Momentum space:
final asymmetry
y
py
x
v 2  cos2 ,   tan 1(
py
px
)
Signals early equilibration (teq  0.6 fm/c)
Jet quenching in Au+Au
PHENIX Data: Identified p0
No quenching
d+Au
Quenching!
Au+Au
Yield(b | AA)
R(b) 
N coll (b)Yield( pp)
baryons
RCP
Baryons vs. mesons
pT (GeV/c)
mesons
f behaves like meson ?
(also -meson)
Overview

QCD Thermodynamics



Thermalization


How can it be so fast?
Transport in a thermal medium


What are the dynamical degrees of freedom?
Is there a critical point, and where is it?
Viscosity, energy loss, collective modes
Hadronization

Recombination vs. fragmentation
QCD phase diagram
RHIC
T
Critical
endpoint ?
QuarkGluon
Plasma
Chiral symmetry
restored
Hadronic
matter
1st order
line ?
Chiral symmetry
broken
Nuclei
Color
superconductor
Neutron stars
B
Space-time picture
Bjorken formula
dET (t ) / dy
 (t ) 
dV (t ) / dy
(dET / dy )final

p RA2t
Pre-equil. phase
teq
 (t )  4.5 GeV/fm3
at t  1 fm/c
in Au+Au (200 GeV)
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
Gribov, Levin, Ryskin ’83
~ 1/Q2
Blaizot, A. Mueller ’87
McLerran, Venugopalan ‘94
 Qs2 ( x, A)
“Color glass condensate (CGC)”
xGA ( x, Qs2 )  s (Qs2 )
gluon density  area =

2
p RA
Qs2
A1/ 3 x  
1
2
Qs
Details of space-time picture depend on gauge!
CGC dynamics
Initial state occupation numbers
~ 1/s  1 → classical fields
generated by random color
sources on light cone:


1
P(  a )  P0 exp   2 2  d 2 x [  a ( x )]2 
 g Qs

 a ( x )
 a ( x )
Aia ( x ,t )
Boost invariance → Hamiltonian gauge field dynamics in transverse
plane (x,t). 2-dim lattice simulation shows rapid equipartitioning of
energy (teq ~ Qs-1). Krasnitz, Nara & Venugopalan; Lappi (hep-ph/0303076 )
Challenge: (3+1) dim. simulation without boost invariance
Saturation and dN/d
Kharzeev & Levin
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:
Pseudorapidity  = - ln tan q
Rapidity y = ln tanh(E+pL)/(E-pL)
How much entropy is produced
by simple decoherence, how
much during the subsequent
full equilibration?
The(rmalization) mystery

Experiment demands: tth  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 tth ~ [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 .
 T

 0 with
T   (  P)u  u  Pg    (  u   u   trace)
HTL pert. theory (nf=3):
T3
  106.7 4
g ln g 1
Dimensionless quantity /s.
(Baym…; Arnold,
Moore & Yaffe)
Classical transp. th.:  ≈ 1.5Tf, s ≈ 4 → /s ≈ 0.4Tf.
Azimuthal anisotropy v2
Semiperipheral collisions
Coordinate space:
initial asymmetry
Momentum space:
final asymmetry
y
py
x
v 2  cos2 ,   tan 1(
py
px
)
Signals early equilibration (teq  0.6 fm/c)
 – How small can it be?
D. Teaney
Boost invariant hydro with T0t0 ~ 1
requires /s ~ 0.1.
s
t0

4  

3T0t 0 s s
T0t 0
1
N=4 SUSY Yang-Mills theory (g1):
/s = 1/4p (Kovtun, Son, Starinets).
Absolute lower bound on /s ?
QGP(T≈Tc) = sQGP
/s = 1/4p implies f ≈ (5 T)-1 ≈ 0.3 d
Challenge: (3+1) dim. relativistic viscous fluid dynamics
First attempts
Challenge: Calculate /s for real QCD
t
Nakamura & Sakai
t
1
    d 3 x  dt e (t t )  dt  Tik ( x, t )T ik ( x, t )
6


Method: spectral function repr. of
Gb and Gret, fit of spectral fct. to Gb.
Related (warm-up?) problem:
EM conductivity  ≈ q2f/2T.
SU(3) YM
nf = 3 QGP → / ≈ 20 T2.
Caveat: f(glue) 
f(quarks)
ret
QCD equation of state

p
2
30
T
Challenge: QCD e.o.s. with light domain wall quarks
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 ?
4
A possible method
BM & K. Rajagopal, hep-ph/0502174
Eliminate T from  and s :
 
p2
T4
30
2
2p 3
s 
T
45
1215 s 4
s4

 0.96 3
2
3
128p 

Lower limit on  requires lower limit on s
and upper limit on .
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)
2)
Use inferred particle numbers at chemical freeze-out from
statistical model fits of hadron yields;
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)/(pR2t0) = 33 ± 3 fm-3
Jet quenching in Au+Au
PHENIX Data: Identified p0
No quenching
d+Au
Quenching!
Au+Au
“Jet quenching” = parton energy loss
High-energy parton loses energy by
rescattering in dense, hot medium.
q
q
Radiative energy loss:
dE / dx
 L kT 2
L
Scattering centers = color charges
q
q
Density of
scattering centers
g
d
2
2
qˆ    q dq
  kT 
2
dq

2
Scattering power of
the QCD medium:
2
Range of color force
Energy loss at RHIC
Data suggest large energy loss parameter:
Eskola, Honkanen, Salgado
& Wiedemann
qˆ  5 10 GeV2 /fm
Dainese, Loizides, Paic
RHIC
pT = 4.5–10 GeV
The Baier plot




Plotted against , q̂ is the
same for a p gas and for
a perturbative QGP.
Suggests that q̂ is really a
measure of the energy
density.
Data suggest that q̂ may
be larger than compatible
with Baier plot.
Nonperturbat. calculation
is needed.
RHIC data
sQGP
QGP
Pion gas
Cold nuclear matter
Challenge: Realistic calculation of gluon radiation in medium
Eikonal formalism
Kovner, Wiedemann
x
quark

q ( x )  Wb ( x ; L) qb ( x )
x-
L


W ( x ; L)  P exp i  dx A ( x , x ) 
 0

b
Gluon radiation:
N g (k ) 
+
 s CF
ik ( x
dx
dy
e


2p 
  y )
x
x = 0
x  y
1  C ( x , 0)  C (0, y )  C ( x , y ) 
2 2 
x y
with C ( x , y ) 
Tr[WA† ( x ; L)WA ( y ; L)]
N c2  1
Eikonal form. II
C ( x , y ) 
Tr[W † ( x ; L)W ( y ; L)
N c2  1
L
x

1
2
 ( x  y )  dx  dx F  i ( x )W ( x , x ) Fi  ( x )W † ( x , x )
2
0
0
1
1
( x  y ) 2 L F  i Fi   F  ( x  y ) 2 Lqˆ
2
2
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
Away-side jet
p+p
Au+Au
Trigger jet
Lost energy of away-side jet is redistributed to rather large angles!
Wakes in the QGP
J. Ruppert and B. Müller, Phys. Lett. B 618 (2005) 123
Angular distribution depends
on energy fraction in collective
mode and propagation velocity
Mach cone requires
collective mode with w(k) < k:
 Colorless sound
? Colored sound = longitudinal gluons
? Transverse gluons
T. Renk & J. Ruppert
Yield(b | AA)
R(b) 
N coll (b)Yield( pp)
baryons
RCP
Baryons vs. mesons
pT (GeV/c)
mesons
f behaves like meson ?
(also -meson)
Hadronization mechanisms
q
q q
q
q
qq q
Recombination
Fragmentation
Baryon
Meson
1
Baryon
1
Meson
pM  2 pQ
pB  3 pQ
Recombination wins…
… for a thermal source
Fragmentation
dominates for a
power-law tail
Baryons compete
with mesons
Quark number scaling of v2
In the recombination regime, meson and baryon v2 can be obtained
from the quark v2 :
 pt 
v  pt   2 v  
 2
M
2
q
2
 pt 
v  pt   3v  
 3
B
2
q
2
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….
Challenge: Unified framework treating recombination
as special case of QCD fragmentation “in medium”.
A. Majumder & X.N. Wang,
nucl-th/0506040
D
fragmentation
D
recombination
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 w(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
k ~ gQs
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).
k ~ Qs
D F   gJ 
J  ( x)    dt Qi (t )ui (t )  ( x  xi (t ))
i
ˆ light-like
ui  (1, v)
Poss. problem: short-distance
lattice modes have wrong w(k).
Challenge: Full (3+1) dim. simulation of hard-soft dynamics
Reco: Thermal quarks
Relativistic formulation using hadron light-cone frame:
w (r , p) 
Quark distribution function at “freeze-out”
E
2
dN M
P u



d

dxw
(
R
,
xP
)
w
(
R
,
(1

x
)
P
)
f
(
x
)


b
M
d 3P 
(2p )3  , b 
E
2
dN B
P u




d

dxdx
'
w
(
R
,
xP
)
w
(
R
,
x
'
P
)
w
(
R
,
(1

x

x
')
P
)
f
(
x
,
x
')


b

B
d3 p 
(2p )3  , b , 
For a thermal distribution,
w(r , p) exp( p  u / T )
the hadron wavefunctions can be integrated out, eliminating the
model dependence of predictions.
Remains exactly true even if higher Fock states are included!
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/Y;
LH”I”C program: b-quarks and .
 RHIC data for J/Y are forthcoming (Runs 4 & 5).
J/Y suppression ?
Vqq is screened at scale (gT)-1 
heavy quark bound states
dissolve above some Td.
Color singlet free energy
Quenched lattice simulations,
with analytic continuation to real
time, suggest Td  2Tc !
S. Datta et al. (PRD 69, 094507)
Karsch et al.
Challenge: Compute J/Y spectral function in unquenched QCD
C-Cbar kinetics
J/Y
RHIC
LHC
J/Y,  can be ionized by thermal
gluons. If resonances persist above
Tc, J/Y and  can be formed by
recombination in the medium: J/Y
may be enhanced at LHC!
Challenge: Multiple scattering theory
of heavy quarks in a thermal medium
RHIC

LHC
Analogous to the multiple scattering
theory for high-pT partons, but using
methods (NRQCD etc.) appropriate
for heavy quarks.