Early Time Evolution of High Energy Heavy Ion
Collisions
Rainer FriesTexas A&M University & RIKEN BNL
Talk at Quark Matter 2006, ShanghaiNovember 18, 2006
QM 2006 2 Rainer Fries
Outline
Motivation: space-time picture of the gluon field at early times
Small time expansion in the McLerran-Venugopalan model
Energy density
Flow
Matching to Hydrodynamics
In Collaboration with J. Kapusta and Y. Li
QM 2006 3 Rainer Fries
Motivation
RHIC: equilibrated parton matter after 1 fm/c or less. Hydrodynamic behavior
How do we get there? Pre-equilibrium phase: energy deposited between the
nuclei Rapid thermalization within less than 1 fm/c
PCM & clust. hadronization
NFD
NFD & hadronic TM
PCM & hadronic TM
CYM & LGT
string & hadronic TM
Initial stage< 1 fm/c
Equilibration, hydrodynamics
QM 2006 4 Rainer Fries
Motivation
RHIC: equilibrated parton matter after 1 fm/c or less. Hydrodynamic behavior
How do we get there? Pre-equilibrium phase: energy deposited between the
nuclei Rapid thermalization within less than 1 fm/c
Initial dynamics: color glass (clQCD) Later: Hydro How to connect color glass and
hydrodynamics? Compute spatial distribution of
energy and momentum at some early time 0. See also talk by T. Hirano.
Hydro
pQCD
clQCD
?
QM 2006 5 Rainer Fries
Plan of Action
Soft modes: hydro evolution from initial conditions
e, p, v, (nB) to be determined as functions of , x at = 0
Assume plasma at 0 created through decay of classical gluon field F with energy momentum tensor Tf
. Constrain Tpl
through Tf using energy momentum
conservation
Use McLerran-Venugopalan model to compute F and Tf
pguupexT ,,0pl v,1 u
Minijets
Color ChargesJ
Class. GluonField F
FieldTensor Tf
Plasma
Tensor Tpl
Hydro
QM 2006 6 Rainer Fries
Color Glass: Two Nuclei
Gauge potential (light cone gauge): In sectors 1 and 2 single nucleus solutions i
1, i2.
In sector 3 (forward light cone):
YM in forward direction: Set of non-linear differential
equations Boundary conditions at = 0
given by the fields of the single nuclei
xAA
xAxAii ,
,
0,,,1
0,,1
0,,1
2
33
jijii
ii
ii
FDADAigA
AAigAD
DD
xAxAig
xA
xAxAxA
ii
iii
21
21
,2
,0
,0
Kovner, McLerran, Weigert
22 zt
QM 2006 7 Rainer Fries
Small Expansion
In the forward light cone: Leading order perturbative solution (Kovner, McLerran, Weigert) Numerical solutions (Krasnitz, Venugopalan, Nara; Lappi)
Our idea: solve equations in the forward light cone using expansion in time : We only need it at small times anyway … Fields and potentials are regular for 0. Get all orders in coupling g and sources !
Solution can be given recursively!
xAxA
xAxA
in
n
ni
nn
n
0
0
,
,
YM equations
In the forward light cone
Infinite set of transverse differential equations
QM 2006 8 Rainer Fries
Solution can be found recursively to any order in !
0th order = boundary condititions:
All odd orders vanish
Even orders:
Note: order in coupled to order in the fields. Reproduces perturbative result (Kovner, McLerran,
Weigert)
Small Expansion
422
2
,,,1
,,2
1
nmlkm
ilk
nlk
jil
jk
in
nmlkm
il
ikn
ADAigFDn
A
ADDnn
A
xAxAig
xA
xAxAxA
ii
iii
210
210
,2
QM 2006 9 Rainer Fries
Field strength order by order: Longitudinal electric,
magnetic fields start with finite values.
Transverse E, B field start at order :
Corrections to longitudinal fields at order 2:
Gluon Near Field
jiij
ii
AAigF
AAigF
21210
210
,
,
Ez
Bz
0000)1( ,,22
FDFDe
F ijiji
21000
212
0002
,,41
,,41
FDDF
FDDF
ii
ii
QM 2006 10 Rainer Fries
Gluon Near Field
Before the collision: transverse fields in the nuclei E and B orthogonal
ii AxF 11
ii AxF 22
QM 2006 11 Rainer Fries
Gluon Near Field
Before the collision: transverse fields in the nuclei E and B orthogonal
Immediately after overlap: Strong longitudinal electric,
magnetic fields at early timeszE zB
QM 2006 12 Rainer Fries
Gluon Near Field
Before the collision: transverse fields in the nuclei E and B orthogonal
Immediately after overlap: Strong longitudinal electric and
magnetic field at early times
Transverse E, B fields start to build up linearly
iE
iB
QM 2006 13 Rainer Fries
Gluon Near Field
Reminiscent of color capacitor Longitudinal magnetic field of equal strength
Strong longitudinal pulse: recently renewed interest Topological charge (Venugopalan, Kharzeev; McLerran,
Lappi; …)
Main contribution to the energy momentum tensor (Fries, Kapusta, Li)
Particle production (Kharzeev and Tuchin, …)
QM 2006 14 Rainer Fries
Energy Density
Initial value :
Contains correlators of 4 fields Can be factorizes into two 2-point correlators (T. Lappi):
2-point function Gi for each nucleus i:
Analytic expression for Gi in the MV model is known. Caveat: logarithmically UV divergent for x 0!
Ergo: MV energy density has divergence for 0.
2200f 2
1BET
0012 21
22
0 GGNNg
cc
00
0f0 T
2121~ AAAA
xAAxGN iic 1112 01
QM 2006 15 Rainer Fries
Energy Momentum Tensor
Energy/momentum flow at order 1: In terms of the initial longitudinal
fields Ez and Bz.
No new non-abelian contributions
Corrections at order 2: E.g. for the energy density
sinh41
cosh41
2231
2201
zzii
zzii
BET
BET
22
2002 ,,
441
zi
zi
zzzz BAEAg
BBEET
Abelian correction Non-abelian correction
QM 2006 16 Rainer Fries
2coshsinhsinh2sinh
sinhcosh
sinhcosh
2sinhcoshcosh2cosh
021
220
22
1220
1
210
f
OO
OOT
Energy Momentum Tensor
General structure up to order 2:
02 iiv
QM 2006 17 Rainer Fries
Energy Momentum Tensor
General structure up to order 2:
2coshsinhsinh2sinh
sinhcosh
sinhcosh
2sinhcoshcosh2cosh
021
220
22
1220
1
210
f
OO
OOT
2
20
O
O
02 iiv
QM 2006 18 Rainer Fries
Compare Full Time Evolution
Compare with the time evolution in numerical solutions (T. Lappi)
The analytic solution discussed so far gives:Normalization Curvature
Curvature
Asymptotic behavior is known (Kovner, McLerran, Weigert)
T. Lappi
QM 2006 19 Rainer Fries
Modeling the Boundary Fields
Use discrete charge distributions
Coarse grained cells at positions bu in the nuclei.
Tk,u = SU(3) charge from Nk,uq quarks and antiquarks and Nk,u
g gluons in cell u.
Size of the charges is = 1/Q0, coarse graining scale Q0 = UV cutoff
Field of the single nucleus k:
Mean-field: linear field + screening on scale Rc = 1/Qs
G = field profile for a single charge contains screening
uku
uk TR , bxx
uu
iu
i
uuk
ik G
bxTgA bx
bxx
,
UUgi
A ik
ik 1
guk
F
Aqukuk NCC
NN ,,, cell ofarea
,ukuk
Nb area density of charge
QM 2006 20 Rainer Fries
Estimating Energy Density
Mean-field: just sum over contributions from all cells
Summation can be done analytically in simple situations
E.g. center of head-on collision of very large nuclei (RA >> Rc) with very slowly varying charge densities k (x) k.
Depends logarithmically on ratio of scales = Rc/.
2221
3
42.01ln c
sME N
RJF, J. Kapusta and Y. Li, nucl-th/0604054
QM 2006 21 Rainer Fries
Estimates for T
Here: central collision at RHIC Using parton distributions to
estimate parton area densities .
Cutoff dependence of Qs and 0
Qs independent of the UV cutoff.
E.g. for Q0 = 2.5 GeV: 0 260 GeV/fm3. Compare T. Lappi:
130 GeV/fm3 @ 0.1 fm/c
Transverse profile of 0: Screening effects: deviations
from nuclear thickness scaling
scs RQ 22
QM 2006 22 Rainer Fries
Transverse Flow
For large nucleus and slowly varying charge densities :
Initial flow of the field proportional to gradient of the source
Transverse profile of the flow slope i/ for central
collisions at RHIC:
221
301 42.01ln
2cosh
i
c
sii
NTv
QM 2006 23 Rainer Fries
Anisotropic Flow
Initial flow in the transverse plane:
Clear flow anisotropies for non-central collisions
b = 8 fm
i ib = 0 fm
QM 2006 24 Rainer Fries
Space-Time Picture
Finally: field has decayed into plasma at = 0
Energy is taken from deceleration of the nuclei in the color field.
Full energy momentum conservation:
fTf
QM 2006 25 Rainer Fries
Space-Time Picture
Deceleration: obtain positions * and rapidities y* of the baryons at = 0
For given initial beam rapidity y0 , mass area density m.
BRAHMS: dy = 2.0 0.4 Nucleon: 100 GeV 27 GeV We conclude:
aavayy 121coshcosh 00*
m
fa 0
(Kapusta, Mishustin)
20 GeV/fm 9f
QM 2006 26 Rainer Fries
Coupling to the Plasma Phase
How to relate field phase and plasma phase?
Use energy-momentum conservation to match: Instantaneous matching
0
0pl0f
T
TTT
2coshsinhsinh2sinh
sinhcosh
sinhcosh
2sinhcoshcosh2cosh
021
220
22
1220
1
210
f
OO
OOT
pguupexT ,,0pl
QM 2006 27 Rainer Fries
The Plasma Phase
Matching gives 4 equations for 5 variables
Complete with equation of state
E.g. for p = /3:
tanhv
cosh1
1 2
2
L
p
ppe
v
22 34 e
Bjorken: y = , but cut off at *
QM 2006 28 Rainer Fries
Summary
Near-field in the MV model Expansion for small times Recursive solution known
Fields and energy momentum tensor: first 3 orders Initially: strong longitudinal fields Estimates of energy density and flow
Relevance to RHIC: Deceleration of charges baryon stopping (BRAHMS) Matching to plasma using energy & momentum conservation
Outlook: Hydro! Soon. Connection with hard processes: get rid of the UV cutoff, jets
in strong color fields?
QM 2006 29 Rainer Fries
Backup
QM 2006 30 Rainer Fries
The McLerran-Venugopalan Model
Assume a large nucleus at very high energy: Lorentz contraction L ~ R/ 0 Boost invariance
Replace high energy nucleus by infinitely thin sheet of color charge Current on the light cone Solve Yang Mills equation
For an observable O: average over all charge distributions McLerran-Venugopalan: Gaussian weight
JFD ,
x11 xJ
2
22
2exp
sQx
xdOdO
QM 2006 31 Rainer Fries
Compare Full Time Evolution
Compare with the time evolution in numerical solutions (T. Lappi)
The analytic solution discussed so far gives:Normalization Curvature
Curvature
Asymptotic behavior is known (Kovner, McLerran, Weigert)
GeV
/fm
3
O(2 )
T. LappiInterpolation between near field and asymptotic behavior:
QM 2006 32 Rainer Fries
Role of Non-linearities
To calculate an observable O: Have to average over all possible charge distributions
We follow McLerran-Venugopalan: purely Gaussian weight
Resulting simplifications: e.g. 3-point functions vanish
Non-linearities: Boundary term is non-abelian (commutator of A1, A2) No further non-abelian terms in the energy-momentum
tensor before order 2.
2
22
2exp
sQx
xdOdO
QM 2006 33 Rainer Fries
Non-Linearities and Screening
Hence our model for field of a single nucleus: linearized ansatz, screening effects from non-linearities are modeled by hand.
Connection to the full solution:
Mean field approximation:
Or in other words: H depends on the density of charges and the coupling. This is modeled by our screening with Rc.
21
121
1 ,
42
1
,,#!3
,#!2 uu
uuuuuu
u
uu u
iu
iii
TTTg
TTig
T
Gbx
gUUgi
bxbx
Corrections introduce deviations from original color vector Tu
uuuu THTT bx
HGG 1
QM 2006 34 Rainer Fries
Compute Charge Fluctuations
Integrals discretized:
Finite but large number of integrals over SU(3)
Gaussian weight function for SU(Nc) random walk in a single cell u (Jeon, Venugopalan):
Here:
Define area density of color charges:
For 0 the only integral to evaluate is
v
vu
u TdTddd ,28
,18
21
ukc
uk
NTN
uk
cN e
NN
Tw ,2
,
/
4
,
guk
F
Aqukuk NCC
NN ,,,
cell ofarea
,ukuk
Nb
vuc
vvuuvuvu NNN
TTTTi ,2,1,2,1,2,12 1
,,Tr21
QM 2006 35 Rainer Fries
Estimating Energy Density
Mean-field: just sum over contributions from all cells E.g. energy density from longitudinal electric field
Summation can be done analytically in simple situations
E.g. center of head-on collision of very large nuclei (RA >> Rc) with very slowly varying charge densities k (x) k.
Depends logarithmically on ratio of scales = Rc/.
2221
3
42.01ln c
sME N
RJF, J. Kapusta and Y. Li, nucl-th/0604054
22
22
2
,,2,1
6
vu
vu
vuvu
vuvu
cE GG
xNN
Ng
bxbxbxbx
bbbbxx
QM 2006 36 Rainer Fries
Deceleration through Color Fields
Compare (in the McLerran-Venugopalan model): Fries, Kapusta & Li: f 260 GeV/fm3 @ = 0
Lappi: f 130 GeV/fm3 @ = 0.1 fm/c
Shortcomings: fields from charges on the light cone no recoil effects there are ambiguities in the MV model
Net-baryon number = good benchmark test
QM 2006 37 Rainer Fries
Color Charges and Currents
Charges propagating along the light cone, Lorentz contracted to very thin sheets ( currents J) Local charge fluctuations appear frozen (fluc >> 0)
Charge transfer by hard processes is instantaneous (hard << 0)
Solve classical EOM for gluon field
+-
1 2
’1’2+-
1 2
2 1
+-
1 2
’2 ’1
Charge fluctuations~ McLerran-Venugopalan model
(boost invariant)
Charge fluctuations+ charge transfer @ t=0
(boost invariant)
Charge fluctuations+ charge transfer with jets
(not boost invariant)
0, ; ,
JDJFD
I II III
QM 2006 38 Rainer Fries
Transverse Structure Solve expansion around = 0, simple transverse
structure Effective transverse size 1/ of charges, ~ Q0
During time , a charge feels only those charges with transverse distance < c
Discretize charge distribution, using grid of size a ~ 1/
Associate effective classical charge with ensemble of partons in each bin
Factorize SU(3) and x dependence
Solve EOM for two such charges colliding in opposite bins
a
)3( ; SUTTxQgxi
Bin in nucleus 1Bin in nucleus 2
Tube with field