early time evolution of high energy heavy ion collisions

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


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


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

?


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


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


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


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


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


Gluon Near Field

Before the collision: transverse fields in the nuclei E and B orthogonal

ii AxF 11

ii AxF 22


Gluon Near Field


Immediately after overlap: Strong longitudinal electric,

magnetic fields at early timeszE zB


Gluon Near Field


Immediately after overlap: Strong longitudinal electric and

magnetic field at early times

Transverse E, B fields start to build up linearly

iE

iB


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, …)


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


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


2coshsinhsinh2sinh

sinhcosh

sinhcosh

2sinhcoshcosh2cosh

021

220

22

1220

1

210

f

OO

OOT


General structure up to order 2:

02 iiv



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


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


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


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


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


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


Anisotropic Flow

Initial flow in the transverse plane:

Clear flow anisotropies for non-central collisions

b = 8 fm

i ib = 0 fm


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


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


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


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 *


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?


Backup


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


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:


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


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


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


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


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


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


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

early time evolution of high energy heavy ion collisions

Documents

b fields

transverse fields

longitudinal fields

magnetic fields

forward light coneinfinite

fieldfield strength

energy momentum tensor

b orthogonalimmediately