gluon fields at early times and initial conditions for hydrodynamics rainer fries university of...

Gluon Fields at Early Times and Initial

Conditions for Hydrodynamics

Rainer FriesUniversity of Minnesota

2006 RHIC/AGS Users’ MeetingJune 7, 2006

with Joe Kapusta, Yang Li

Gluon Fields at Early Times 2 Rainer Fries

Introduction

Initial phase of a high energy nuclear collision? Interactions between partons. Energy deposited between the nuclei. Equilibration, entropy production.

Plasma at time > 0.5 … 1 fm/c. Hydrodynamic evolution

PCM & clust. hadronization

NFD

NFD & hadronic TM

PCM & hadronic TM

CYM & LGT

string & hadronic TM

Initial stage< 1 fm/c

Equilibration, hydrodynamics


Introduction

Initial phase of a high energy nuclear collision? Plasma at time > 0.5 …1 fm/c. Path to equilibrium ??

Hydro evolution of the plasma from initial conditions

, p, v, (nB, …) to be determined as functions of , x at = 0

Goal: measure EoS, viscosities, … Initial conditions add more parameters

pguupxT ,,0pl v,1 u


Introduction

Initial phase of a high energy nuclear collision? Plasma at time > 0.5 …1 fm/c. Path to equilibrium ?? Hydro evolution of the plasma from initial

conditions Goal: measure EoS, viscosities Constrain initial conditions:

Hard scatterings, minijets (parton cascades) String based models NeXus, HIJING Color glass + hydro (Hirano, Nara)


Color Glass Large nuclei at very large energy: color glass state

Saturation Gluon density sets a scale

High density limit of QCD

Large number of gluons in the wave function: classical description of the gluon field

3/12

22 ~

,A

RQxG

QA

sss


Color Glass + Phenomenology Results galore from CGC

Kharzeev, Levin, Nardi ; Kovchegov, Tuchin Krasnitz and Venugopalan, Lappi

Our mission: Try to understand some of the features analytically Make contact with phenomenology, hydro Produce numerical estimates

Our approach to deal with this very complex system: Use simple setup: McLerran-Venugopalan Model (for now …) Ask the right questions: just calculate energy momentum

tensor Use controlled approximations: e.g. small time expansion If not possible, make reasonable model assumptions


Outline

Minijets

Color ChargesJ

Class. GluonField F

FieldTensor Tf

Plasma

Tensor Tpl

Hydro


The McLerran-Venugopalan Model

Assume a large nucleus at very high energy: Lorentz contraction in longitudinal direction L ~ R/ 0 No longitudinal length scale in the problem boost

invariance

Replace high energy nucleus by infinitely thin sheet of color charge Current on the light cone Solve Yang Mills equations

JFD , 0, JD

x xJ


Color Glass: Single Nucleus Gluon field of single nucleus is transverse

F+ = 0 Fi = 0 Fi+ = (x)i(x) Fij = 0

Transverse field

Field created by charge fluctuations: Nucleus is overall color neutral.

Charge takes random walk in SU(3) space.

2

22

2exp

sQx

xdOdO

x iii eUUUgi

1

Longitudinal electric field Ez

Longitudinal magnetic field Bz


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

xA

xxAii ,

,

3

0,,,1

0,,1

0,,1

23

3

33

jijii

ii

ii

FDDig

igD

DD

xxig

x

xxx

ii

iii

21

213

,2

,0

,0


Small Expansion

Idea: solve equations in the forward light cone using expansion in time : We only believe color glass at small times anyway … Fields and potentials are regular for 0. Get all orders in g!

Solution can be given recursively!

xx

xx

in

n

ni

nn

n

30

3

0

,

,

YM equations

In the forward light cone

Infinite set of transverse differential equations


Small Expansion

Idea: solve equations in the forward light cone using expansion in time : 0th order in :

All odd orders vanish:

2nd order

Arbitrary order in can be written down. Note: order in coupled to order in the fields.

xxig

x

xxx

ii

iii

210

2103

,2

0

0

12

123

x

x

n

in

jiji

ii

FD

DD

00223

00022

,4

1

,,8

1

RJF, J. Kapusta and Y. Li, nucl-th/0604054


Gluon Near Field

Structure of the field strength tensor

Longitudinal electric, magnetic fields start with finite values.

For 0 : longitudinal fields = color capacitor?

Strong longitudinal pulse (re)discovered recently. Fries, Kapusta and Li, QM 2005; Kharzeev and Tuchin;

Lappi and McLerran, hep-ph/0602189

jiij

i

ii

igF

F

igF

21210

0

210

,

0

,

Ez

Bz


Gluon Near Field

Structure of the field strength tensor

Longitudinal electric, magnetic fields start with finite values.

Transverse E & B fields start at order O()

jiij

i

ii

igF

F

igF

21210

0

210

,

0

,

Ez

Bz

0

,,2

0

211

00001

1

F

FDFDx

F

F

ijiji


Input Fields

Use discrete charge distribution and coarse graining

Assume distribution of quarks & gluons at positions bu in the nuclei. e.g. charge distribution for nucleus 1 Tk,u = SU(3) matrices R = profile function of a single charge

Write field of these charges in nucleus 1 as

G = field profile for a single charge In a weak field or abelian limit, this would be the exact

solution, e.g. for 2-D Coulomb for point charges:

uu

u TR ,11 bxx

uu

iu

i

uu

i Gbx

Tg bxbx

x

,11

x

xGxxR

2

1 2


Coarse Graining & Screening

Coarse graining Transverse resolution of the gluon field ~ 1/Qs

Gluon modes with k > Qs: hard processes

Use finite transverse size ~ 1/Qs for R.

Screening: remove infrared singularity with cutoff Rc.

Impose screening by hand

Then

Rc should depend on the density of charges and should in addition be smaller than 1/QCD.

This screening should be provided self-consistently by the non-linearities in the YM equations.

1112

12 ii

c gi

gR

cc

x

Rx

KRe

xG 1

/

21

22


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


Charge Fluctuations

We have to evaluate

Use discretization: finite but large number of integrals over SU(3)

Gaussian weight function for SU(Nc) random walk (Jeon & Venugopalan):

N = number of color charges in the cell around bu, calculated from the number of quarks, antiquarks and gluons.

v

vu

u TdTddd ,28

,18

21

NTNcN

ceNN

Tw /4

2

gF

Aqq N

CC

NNN


Energy Density

Color structure of the longitudinal field:

Energy density

SU(3) random walk for the scalar appearing in :

It’s really fluctuations: energy ~ N1N2 , field ~ N1N2

vuc

vvuuvuvu NNN

TTTTi ,2,1,2,1,2,12 1

,,Tr21

vuvu

TTig ,2,13

,

,function real0 field Long.

2210

2

0 21

21

FF ME


Estimating Energy Density

Energy density created in the center of a head-on collision (x = 0) of large nuclei (RA >> Rc)

Only depends on ratio of scales = Rc/. Use approx. constant number density of charges 1, 2

(quarks+antiquarks+9/4 gluons)

Numerical value for Qs = 1 GeV, Rc = 1 fm at RHIC: 450 GeV/fm3. Remember: this is for 0. Scheme for charge density: partons in the wave function

minus hard processes.

2221

3

42.01ln c

sME N RJF, J. Kapusta and Y. Li,

nucl-th/0604054


Going into the Forward Light Cone

Next coefficient in the energy density, order 2 , is negative.

expansion takes us to 1/Qs

Match small expansion and large asymptotic behavior. Asymptotics: weak fields at large (Kovner, McLerran and

Weigert)

GeV

/fm

3

O(2 )


Going into the Forward Light Cone

Compare to the full result

Numerical result by McLerran & Lappi

GeV

/fm

3

Preliminary

O(2 )


Energy Momentum Tensor Early time structure of the energy momentum:

Hierarchy of terms:

Energy and momentum conservation:

2coshsinhsinh2cosh

sinhcosh

sinhcosh

2coshcoshcosh2cosh

21

22

11

21

f

CABBC

BDAEB

BEDAB

CBBCA

T

2,

,

1,

OxC

Ox

OxA

B

2f 0

OT


Matching of the E P Tensors Thermalization?

Independent of the mechanism: energy and momentum have to be conserved!

= local energy density, p = pressure

Interpolate between the field and the plasma phase E.g. rapid thermalization around = 0 :

pguupxT ,,0pl v,1 u

0

0pl0f

T

TTT


The Plasma Phase

Matching gives 4 equations for 5 variables

Complete set of equations e.g. by applying equation of state

E.g. for p = /3:

tanh

cosh

1

1 2

2

Lv

pCA

pCACAp

Bv

B

22 34 BCACA

Bjorken: y = , but cut off at some value*


Initial Conditions for the QGP Flow starts to build up linearly with time: System starts to flow before thermalization.

210 ~~ iii TB f

Preliminary


3D Space-Time Picture

Force acting on the light cone charges Deceleration of the nuclei; Trajectory for each bin of mass m: start at beam rapidity

y0 (Kapusta & Mishustin)

Obtain positions * and rapidities y* of the baryons at = 0

Eventually: baryon number distribution

Finally: decay into plasma at = 0

fTf

1sinhcosh2

0

2

0

t

yz

y 20am


Summary

Problem: how to understand the initial energy and momentum tensor of the plasma from early gluon fields.

Introduce small time expansion in the MV model. Estimate initial energy density and its decay with

time using a model with discrete, screened charges.

Calculate the full energy momentum tensor and match to the plasma phase using energy and momentum conservation.


Backup


Color Glass: Single Nucleus

Current for one nucleus: Current (in + direction): Transverse distribution of charge: (x)

Solve Yang-Mills equations

Gluon field of single nucleus is transverse F+ = 0 Fi = 0 Fi+ = (x)i(x) Fij = 0 where No longitudinal electric or magnetic field in the nuclei. Transverse electric and magnetic fields are orthogonal

to each other.

But what is the color distribution (x)?

x xJ

x ii

JFD , 0,

JD


Thermalization ?

Experimental results indicate thermalization of partons at time scales 0 < 1fm/c

Strong longitudinal fields: pair production

Numerical work by Lappi: Dirac equation in background field Quark-antiquark pairs produced copiously Ng / Nq ~ 4/Nf after short time, close to chemical

equilibrium

Once thermalization is reached: hydrodynamic evolution Energy momentum tensor of the quark gluon plasma

pguupxT ,,0pl v,1 u


More Flow

This can lead to radial flow early in the plasma phase…

… and to elliptic flowb = 8 fm

gluon fields at early times and initial conditions for hydrodynamics rainer fries university of...

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