early time evolution of high energy nuclear collisions

Early Time Evolution of High Energy Nuclear

Collisions

Rainer FriesTexas A&M University & RIKEN BNL

Early Time Dynamics in Heavy Ion CollisionsMcGill University, Montreal, July 18, 2007

With J. Kapusta and Y. Li

ETD-HIC 2 Rainer Fries

Motivation

ETD Questions

ETD IdeasQGP

HydroclQCDCGC

pQCD

How much kinetic energy is lost in the collision of two nuclei with a total kinetic energy of 40 TeV? How long does it take to decelerate them?

How is this energy stored initially?Does it turn into a thermalized plasma?How and when would that happen?

Pheno.Models


PCM & clust. hadronization

NFD

NFD & hadronic TM

PCM & hadronic TM

CYM & LGT

string & hadronic TM

Motivation

Assume 3 overlapping phases:1. Initial interaction: energy deposited between the nuclei;

gluon saturation, classical fields (clQCD), color glass2. Pre-equilibrium / Glasma: decoherence? thermalization?

particle production? instabilities?3. Equilibrium (?): (ideal ?) hydrodynamics

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

HydroNon-abeliandynamicsclQCD


Outline

Goal: space-time map of a high energy nucleus-nucleus collision.

Small time expansion of YM; McLerran-Venugopalan model

Energy density, momentum, flow

Matching to Hydrodynamics

Baryon Stopping


Hydro + Initial Conditions

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 = 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 ,,pl


Hydro + Initial Conditions

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

Assume plasma at 0 created through decay of gluon field F with energy momentum tensor Tf

. Even w/o detailed knowledge of non-abelian dynamics:

constraints from energy & momentum conservation for Tpl

Tf !

Need gluon field F and Tf at small times.

Estimate using classical Yang-Mills theory

v,1 u pguupexT ,,pl


Classical Color Capacitor

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 classical Yang Mills equation

McLerran-Venugopalan model: For an observable O: average over charge distributions 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

kJAA

kJk

AA

ii00

LO

10LO

,

2,

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]; ...

Caveats: Instability from quantum fluctuations? [Fukushima, Gelis,

McLerran]

Corrections from violations of boost invariance?


Energy Momentum Tensor

Compute energy momentum tensor Tf.

Initial value of the energy density:

Only diagonal contributions at order 0:

Longitudinal vacuum field

Negative longitudinal pressure maximal anisotropy transv. long. Leads to the deceleration of the nuclei

Positive transverse pressure transverse expansion

20

20

000f0 2

1BET

0

0

0

0

)0(f

T



Energy and longitudinal momentum flow at order 1:

Distinguish hydro-like contributions and non-trivial dynamic contributions

Free streaming: flow = –gradient of transverse pressure

Dynamic contribution: additional stress

coshsinh2

1

sinhcosh2

1

31

01

iii

iii

T

T

0 ii

0000 ,, BEDEBD jjiji



Order O( 2): first correction to energy density etc.

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


McLerran Venugopalan Model

So far just classical YM; add color random walk.

E.g. consider initial energy density 0.

Correlator of 4 fields, 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! Not seen in previous numerical simulations on a lattice. McLerran-Venugopalan does not describe UV limit correctly;

use pQCD

00~~ 2121210 GGAAAA lkjiklijklij

xAAxG ik

ikk 0

[T. Lappi]


Estimating Energy Density

Initial energy density in the MV model

Q0: UV cutoff

k2: charge density in nucleus k from

Compatible with estimate using screened abelian boundary fields modulo exact form of logarithmic term. [RJF, Kapusta, Li (2006)]

2

2022

221

26

0 ln8

1

QNNg cc

yxgyx kak

ak

222


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


Transverse Flow @ O(1)

Free-streaming part in the MV model.

Dynamic contribution vanishes!

2

2022

221

26

ln8

1

QNNg icci

0i


Anisotropic Flow

Sketch of 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?

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

Analytic structure of Tf as function of

With etc…

Matching to ideal hydro only possible w/o ‘stress’ terms

0r

53

162

OV iii

2sinh2coshcoshsinh2sinh2cosh

coshsinhsinhcosh

2sinh2coshsinhcosh2sinh2cosh

f

CBBAWVCB

WVWV

CBWVCBBA

Tii

iiii

ii


The Plasma Phase

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

For central collisions (use radial symmetry):

Non-vanishing stress tensor: Stress 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

V

pA

VpApe

162

8

24

3

0

2

0

2

0

rV

C

A

ii

22 VpAV

Cr

rz

Small times:


Application to the MV Model

Apply to the MV case At early times C = 0

Radial flow velocity at early times Assuming p = 1/3 e Independent of cutoff

tanhv

v

22

z

rr

r

rr

pA

V

pA

VpApe

22

21

22

21

2

3v

r

r

0rz


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 Rough estimate:

aavayy 121coshcosh 00*

m

fa 0

[Kapusta, Mishustin][Mishustin 2006]

20 GeV/fm 9f


Summary

Recursive solution for Yang Mills equations (boost-invariant case)

Strong initial longitudinal gluon fields

Negative longitudinal pressure baryon stopping

Transverse energy flow of energy starts at = 0

Use full energy momentum tensor to match to hydrodynamics

Constraining hydro initial conditions


Backup


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]


Energy Matching

Total energy content (soft plus pQCD) RHIC energy.

early time evolution of high energy nuclear collisions

Documents

initial values

energy momentum conservation

energy momentum tensor

initial interaction

total kinetic energy

forward light coneinfinite

forward direction

large nucleus