Initial Energy Density, Momentum and Flow in Heavy Ion Collisions Rainer Fries Texas A&M University & RIKEN BNL Heavy Ion Collisions at the LHC: Last Call

Initial Energy Density, Momentum and Flow in

Heavy Ion Collisions

Rainer FriesTexas A&M University & RIKEN BNL

Heavy Ion Collisions at the LHC: Last Call for Predictions CERN, May 25, 2007

LHC: Last Call 2 Rainer Fries

Outline

Space-time map of a high energy nucleus-nucleus collision.

Small time expansion in the McLerran-Venugopalan model

Energy density, momentum, flow

Matching to Hydrodynamics

Baryon Stopping

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?

PCM & clust. hadronization

NFD

NFD & hadronic TM

PCM & hadronic TM

CYM & LGT

string & hadronic TM

Initial stage< 1 fm/c

Equilibration, hydrodynamics


PCM & clust. hadronization

NFD

NFD & hadronic TM

PCM & hadronic TM

CYM & LGT

string & hadronic TM

Motivation

Possible 3 overlapping phases:1. Initial interaction: gluon saturation, classical fields

(clQCD), color glass2. Global evolution of the system + thermalization?

particle production? decoherence? instabilities?3. Equilibrium, hydrodynamics

What can we say about the global evolution of the system up to the point of equilibrium?

HydroNon-abeliandynamicsclQCD


Hydro + Initial Conditions

(Ideal?) hydro evolution of the plasma from initial conditions Energy momentum tensor for ideal hydro (+ viscous

corrections)

e, p, v, (nB, …) have initial values at = 0

Goal: measure EoS, viscosities, … Initial conditions enter as additional parameters

Constrain initial conditions: Hard scatterings, minijets (parton cascades) String or Regge based models; e.g. NeXus [Kodama et al.]

Color glass condensate [Hirano, Nara]

v,1 u pguupexT ,,0pl


A Simple Model Goal: estimate spatial distribution of energy and

momentum at some early time 0. (Ideal) 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

. Framework as general as possible w/o details of the dynamics Constrain Tpl

through Tf using energy momentum conservation

Use McLerran-Venugopalan model to compute F and Tf

pguupexT ,,0pl v,1 u

Color ChargesJ

Class. GluonField F

FieldTensor Tf

Plasma

Tensor Tpl

Hydro


The Starting Point: the MV 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 charge distributions McLerran-Venugopalan: Gaussian weight

JFD ,

x11 xJ

2

22

2exp

xxdOdO [McLerran, Venugopalan]


Color Glass: Two Nuclei

Gauge potential (light cone gauge): In sectors 1 and 2 single nucleus solutions Ai

1, Ai2.

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

ADDA

xAxAig

xA

xAxAxA

ii

iii

21

21

,2

,0

,0

22 zt

iA1iA2

[McLerran, Venugopalan][Kovner, McLerran, Weigert][Jalilian-Marian, Kovner, McLerran, Weigert]


Small Expansion

In the forward light cone: Perturbative solutions [Kovner, McLerran, Weigert]

Numerical solutions [Venugopalan et al; Lappi]

Analytic solution for small times? Solve equations in the forward light cone using

expansion in time : Get all orders in coupling g and sources !

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:

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


Note: order in coupled to order in the fields.

Expanding in powers of the boundary fields : Leading order terms can be resummed in

This reproduces the perturbative KMW result.

Perturbative Result

2,

2

2,

00 LO

10LO

kJAA

kJ

k

AA

ii kk

kk

ii AA 21 ,

In transverse Fourier space


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.

Corrections to transverse fields at order 3.

Gluon Near Field

jiij

ii

AAigF

AAigF

21210

210

,

,

E0

B0

0000)1( ,,22

FDFDe

F ijiji

☺

☺


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 times0E

0B


Gluon Near Field


Immediately after overlap: Strong longitudinal electric,

magnetic fields 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 initial longitudinal ‘pulse’: Main contribution to the energy momentum tensor

[RJF, Kapusta, Li]; [Lappi]; …

Particle production (Schwinger mechanism) [Kharzeev, Tuchin]; ...

Caveat: there might be structure on top (corrections from non-boost invariance, fluctuations)


Energy Momentum Tensor

Compute energy momentum tensor Tf.

Include random walk over charge distributions

E.g. energy density etc.

Initial value of the energy density:

Only diagonal contributions at order 0.

Energy and longitudinal momentum flow at order 1:

2200f 2

1BET

20

20

000f0 2

1BET

coshsinh2

1

sinhcosh2

1

031

001

iii

iii

T

T



0

0

0

0

)0(f

T

Initial structure: Longitudinal vacuum field Negative longitudinal pressure

General structure up to order 3 (rows 1 & 2 shown only)

Energy and momentum conservation:

..coshsinh16

coshsinh2

2cosh2sinh8

..4

sinhcosh16

sinhcosh2

..4

sinhcosh16

sinhcosh2

..sinhcosh16

sinhcosh2

2sinh2cosh84

113

10

12

222

32

02

10

2

011

31

01

113

10

12

0

2

0

f

ii

ii

T

0,3)( 0

1,2)( 03

f

4f

iOT

iOT



General structure up to order 3

Time hierarchy: O(0): Initial energy density, pressure O(1): Transverse ‘flow’ O(2): Decreasing energy density, build-up of other

components O(3): …

..coshsinh16

coshsinh2

2cosh2sinh8

..4

sinhcosh16

sinhcosh2

..4

sinhcosh16

sinhcosh2

..sinhcosh16

sinhcosh2

2sinh2cosh84

113

10

12

222

32

02

10

2

011

31

01

113

10

12

0

2

0

f

ii

ii

T



General structure up to order 3

Distinguish trivial and non-trivial contributions E.g. flow

Free streaming: flow = –gradient of energy density

Dynamic contribution:

..coshsinh16

coshsinh2

2cosh2sinh8

..4

sinhcosh16

sinhcosh2

..4

sinhcosh16

sinhcosh2

..sinhcosh16

sinhcosh2

2sinh2cosh84

113

10

12

222

32

02

10

2

011

31

01

113

10

12

0

2

0

f

ii

ii

T

00

free iiT

00000

shear ,, EBDBEDT jjijii


A Closer Look at Coefficients

So far just classical YM; add MV source modeling

E.g. consider initial energy density 0.

Contains correlators of 4 fields, e.g. Factorizes into two 2-point correlators:

2-point function Gk for nucleus k:

Analytic expression for Gk in the MV model is known. Caveat: logarithmically UV divergent for x 0! Naturally not seen in any numerical simulation so far.

0012 21

22

0 GGNNg

cc

21212

21 ~, AAAAAA

xAAxGN ik

ikkc 012

[T. Lappi]


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

Bending around


Estimating 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,ug gluons in cell u.

Can do discrete integrals easily

Size of the charges is = 1/Q0

Scale Q0 = UV cutoff !

uku

uk TR , bxx

guk

F

Aqukuk NCC

NN ,,, cell ofarea

,ukuk

Nb area density of charge

yxyx klc

kal

ak N2


Estimating the Boundary Fields

Field of the single nucleus k: Estimate non-linearities through screening on scale Rc ~

1/Qs

G = field profile for a single charge contains screening

Gives finite correlation function

Logarithmic singularity at x = y recovered for Q0

What about modes with kT > Q0? Use pQCD.

uu

iu

i

uuk

ik G

bxTgA bx

bxx

,

yx ii AA


Estimating Energy Density

Sum over contributions from all charges, recover continuum limit. Can be done analytically in simple situations In the following: center of head-on collision of very large

nuclei (RA >> Rc) with very slowly varying charge densities k (x) k.

E.g. initial energy density 0:

Depends logarithmically on ratio of scales = RcQ0.

2221

3

42.01ln c

sME N

[RJF, Kapusta, Li]



Here: central collision at RHIC Using parton distributions to

estimate parton area densities . [McLerran, Gyulassy]

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

scs RQ 22


Transverse Flow

Free-streaming part Pocket formula derived again for large nuclei and slowly

varying charge densities (center)

Transverse profile of the flow slope T0i

free/ for central collisions at RHIC:

221

30

free 42.01ln2

i

c

si

NT


Anisotropic Flow

Initial flow in the transverse plane:

Clear flow anisotropies for non-central collisions Caveat: this is flow of energy.

b = 8 fm

iT 0free

b = 0 fm

iT 0free


Coupling to the Plasma Phase

How to get an equilibrated (?) plasma? Difficult!

Use energy-momentum conservation to constrain the plasma phase Total energy momentum tensor of the system:

r(): interpolating function

Enforce

rTrTT 1plf

fT

plT

0 T


Coupling to the Plasma Phase

Here: instantaneous matching

I.e.

Leads to 4 equations to constrain Tpl. Ideal hydro has 5 unknowns: e, p, v

Matching to ideal hydro is only possible w/o ‘shear’ terms Tensor in this case:

0r

2coshsinhsinh2sinh

sinhcosh

sinhcosh

2sinhcoshcosh2cosh

21

2222

1221

21

fOO

OOT

pguupexT ,,0pl


The Plasma Phase

In general: need shear tensor for the plasma to match.

For central collisions (radial symmetry):

Non-vanishing shear tensor: Shear indeed related to pr = radial pressure

Need more information to close equations, e.g. equation of state

Recover boost invariance y = , but cut off at *

tanhv

v

22

z

rr

r

rr

pA

C

pA

CpApe

162

8

24

3

0

2

0

2

0

rC

B

A

ii

22 CprAC

Brz


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

[Mishustin, Kapusta]


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


Summary

Near-field in the MV model Expansion for small times Recursive solution known F, T: first 4 orders explicitly computed

Conclusions: Strong initial longitudinal fields Transverse energy flow exists naturally and might be

important Constraining initial conditions for hydro

Matching to plasma using energy & momentum conservation Natural emergence of shear contributions Estimates of energy densities Deceleration of charges baryon stopping


Backup


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

UUgi

A ik

ik 1


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



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


Energy Matching

Total energy content (soft plus pQCD) RHIC energy.

Documents

Initial Energy Density, Momentum and Flow in Heavy Ion Collisions Rainer Fries Texas A&M University & RIKEN BNL Heavy Ion Collisions at the LHC: Last Call