Initial Energy Density, Momentum and Flow in Heavy Ion Collisions Rainer Fries Texas A&M University...
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Transcript of Initial Energy Density, Momentum and Flow in Heavy Ion Collisions Rainer Fries Texas A&M University...
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
LHC: Last Call 3 Rainer Fries
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
LHC: Last Call 4 Rainer Fries
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
LHC: Last Call 5 Rainer Fries
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
LHC: Last Call 6 Rainer Fries
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
LHC: Last Call 7 Rainer Fries
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]
LHC: Last Call 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]
LHC: Last Call 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
LHC: Last Call 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
LHC: Last Call 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
2,
2
2,
00 LO
10LO
kJAA
kJ
k
AA
ii kk
kk
ii AA 21 ,
In transverse Fourier space
LHC: Last Call 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
☺
☺
LHC: Last Call 13 Rainer Fries
Gluon Near Field
Before the collision: transverse fields in the nuclei E and B orthogonal
ii AxF 11
ii AxF 22
LHC: Last Call 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
LHC: Last Call 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
LHC: Last Call 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]; ...
Caveat: there might be structure on top (corrections from non-boost invariance, fluctuations)
LHC: Last Call 17 Rainer Fries
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
LHC: Last Call 18 Rainer Fries
Energy Momentum Tensor
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
LHC: Last Call 19 Rainer Fries
Energy Momentum Tensor
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
LHC: Last Call 20 Rainer Fries
Energy Momentum Tensor
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
LHC: Last Call 21 Rainer Fries
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]
LHC: Last Call 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
LHC: Last Call 23 Rainer Fries
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
LHC: Last Call 24 Rainer Fries
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
LHC: Last Call 25 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]
LHC: Last Call 26 Rainer Fries
Estimating Energy Density
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
LHC: Last Call 27 Rainer Fries
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
LHC: Last Call 28 Rainer Fries
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
LHC: Last Call 29 Rainer Fries
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
LHC: Last Call 30 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
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
LHC: Last Call 31 Rainer Fries
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
LHC: Last Call 32 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]
LHC: Last Call 33 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 We conclude:
aavayy 121coshcosh 00*
m
fa 0
[Kapusta, Mishustin]
20 GeV/fm 9f
LHC: Last Call 34 Rainer Fries
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
LHC: Last Call 35 Rainer Fries
Backup
LHC: Last Call 36 Rainer Fries
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
LHC: Last Call 37 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
LHC: Last Call 38 Rainer Fries
Estimating Energy Density
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
LHC: Last Call 39 Rainer Fries
Energy Matching
Total energy content (soft plus pQCD) RHIC energy.