gluon fields at early times and initial conditions for hydrodynamics rainer fries university of...
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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
Gluon Fields at Early Times 3 Rainer Fries
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
Gluon Fields at Early Times 4 Rainer Fries
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)
Gluon Fields at Early Times 5 Rainer Fries
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
Gluon Fields at Early Times 6 Rainer Fries
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
Gluon Fields at Early Times 7 Rainer Fries
Outline
Minijets
Color ChargesJ
Class. GluonField F
FieldTensor Tf
Plasma
Tensor Tpl
Hydro
Gluon Fields at Early Times 8 Rainer Fries
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
Gluon Fields at Early Times 9 Rainer Fries
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
Gluon Fields at Early Times 10 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
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
Gluon Fields at Early Times 11 Rainer Fries
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
Gluon Fields at Early Times 12 Rainer Fries
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 Fields at Early Times 13 Rainer Fries
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 Fields at Early Times 14 Rainer Fries
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
Gluon Fields at Early Times 15 Rainer Fries
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
Gluon Fields at Early Times 16 Rainer Fries
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
Gluon Fields at Early Times 17 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
Gluon Fields at Early Times 18 Rainer Fries
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
Gluon Fields at Early Times 19 Rainer Fries
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
Gluon Fields at Early Times 20 Rainer Fries
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
Gluon Fields at Early Times 21 Rainer Fries
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 )
Gluon Fields at Early Times 22 Rainer Fries
Going into the Forward Light Cone
Compare to the full result
Numerical result by McLerran & Lappi
GeV
/fm
3
Preliminary
O(2 )
Gluon Fields at Early Times 23 Rainer Fries
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
Gluon Fields at Early Times 24 Rainer Fries
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
Gluon Fields at Early Times 25 Rainer Fries
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*
Gluon Fields at Early Times 26 Rainer Fries
Initial Conditions for the QGP Flow starts to build up linearly with time: System starts to flow before thermalization.
210 ~~ iii TB f
Preliminary
Gluon Fields at Early Times 27 Rainer Fries
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
Gluon Fields at Early Times 28 Rainer Fries
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.
Gluon Fields at Early Times 30 Rainer Fries
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
Gluon Fields at Early Times 31 Rainer Fries
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