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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
ETD-HIC 3 Rainer Fries
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
ETD-HIC 4 Rainer Fries
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
ETD-HIC 5 Rainer Fries
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
ETD-HIC 6 Rainer Fries
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
ETD-HIC 7 Rainer Fries
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]
ETD-HIC 8 Rainer Fries
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]
ETD-HIC 9 Rainer Fries
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
ETD-HIC 10 Rainer Fries
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
ETD-HIC 11 Rainer Fries
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
ETD-HIC 12 Rainer Fries
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
☺
☺
ETD-HIC 13 Rainer Fries
Gluon Near Field
Before the collision: transverse fields in the nuclei E and B orthogonal
ii AxF 11
ii AxF 22
ETD-HIC 14 Rainer Fries
Gluon Near Field
Before the collision: transverse fields in the nuclei E and B orthogonal
Immediately after overlap: Strong longitudinal electric,
magnetic fields at early times0E
0B
ETD-HIC 15 Rainer Fries
Gluon Near Field
Before the collision: transverse fields in the nuclei E and B orthogonal
Immediately after overlap: Strong longitudinal electric,
magnetic fields at early times
Transverse E, B fields start to build up linearly
iE
iB
ETD-HIC 16 Rainer Fries
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?
ETD-HIC 17 Rainer Fries
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
ETD-HIC 18 Rainer Fries
Energy Momentum Tensor
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
ETD-HIC 19 Rainer Fries
Energy Momentum Tensor
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
ETD-HIC 20 Rainer Fries
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]
ETD-HIC 21 Rainer Fries
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
ETD-HIC 22 Rainer Fries
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
ETD-HIC 23 Rainer Fries
Transverse Flow @ O(1)
Free-streaming part in the MV model.
Dynamic contribution vanishes!
2
2022
221
26
ln8
1
QNNg icci
0i
ETD-HIC 24 Rainer Fries
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
ETD-HIC 25 Rainer Fries
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
ETD-HIC 26 Rainer Fries
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
ETD-HIC 27 Rainer Fries
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:
ETD-HIC 28 Rainer Fries
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
ETD-HIC 29 Rainer Fries
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]
ETD-HIC 30 Rainer Fries
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
ETD-HIC 31 Rainer Fries
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
ETD-HIC 33 Rainer Fries
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]