Initial fields and instabilities in the classical model of the heavy-ion collision
22
July 2007 at ETD-HIC Initial fields and instabilities in the classical model of the heavy-ion collision Kenji Fukushima (RIKEN BNL Research Center)
description
Initial fields and instabilities in the classical model of the heavy-ion collision. Kenji Fukushima (RIKEN BNL Research Center). Outline. Assume the McLerran-Venugopalan model as a classical model of the heavy-ion collision. - PowerPoint PPT Presentation
Transcript of Initial fields and instabilities in the classical model of the heavy-ion collision
July 2007 at ETD-HIC
Initial fields and instabilitiesin the classical model
of the heavy-ion collision
Kenji Fukushima(RIKEN BNL Research Center)
July 2007 at ETD-HIC
Outline
Assume the McLerran-Venugopalan model as a classical model of the heavy-ion collision.
Calculate the initial energy density at early times analytically in the MV model.
Perturb the CGC background with rapidity dependent fluctuations.
July 2007 at ETD-HIC
What and why the MV model?
Preparation (two kinetic variables)□ Transverse Momentum Q2
Transverse resolution (size inverse) of partons□ Bjorken's x
Longitudinal fraction of parton momentum
~)( 02
1 zPPPHigh Energy Hadron Target
1Q
Pxp1x
)0,,( 0 qqq
July 2007 at ETD-HIC
Going to smaller x with fixed Q2
Gluon increases with a fixed transverse area
Graphically
small-x Dense Gluon Matter
July 2007 at ETD-HIC
Going to larger Q2 with fixed x
Gluon slowly increases with a decreasing areaGraphically, in the same way,
When does the distribution come to overlap?large Q Dilute Gluon Matter
1~ )1(
),(222Asc
sRQN
xQxg
Gluons with ktQs(x)are saturated.
July 2007 at ETD-HIC
Dense-Dense Scattering
Scattering amplitude in the Eikonal approx.
}{ }{}{ }{
][W][W~p pt t
VWS pxtx
Dense Target Dense Projectile
xx
July 2007 at ETD-HIC
Classical approximation
Stationary-point approximation
},{
4
sourceYM},{
}{ }{}{ }{
)(exp][][W][W~
],,,[exp][][W][W
][W][W~
pt
pt
p pt t
aapa
atYMpxtx
ptpxtx
pxtx
AAxdiiSDA
VWiSiSDA
VWS
Stationary-point approx. is made at ap
at
ΑAaAS
YM
July 2007 at ETD-HIC
Equations of Motion
Coordinates
Equations to be solved (in A=0 gauge)
jj
jijii
ii
FDE
FDFDE
AEAE
1
1
1 ,
)]/()ln[( rapidity eproper tim
21
22
ztztzt
July 2007 at ETD-HIC
Boundary Conditions
Two-source problemxx
)()2(xi)()1(
xi
?
],[
0
0
)0( conelight On the
)2()1(
)2()1(
ii
iii
igE
E
A
A
i
July 2007 at ETD-HIC
McLerran-Venugopalan Model
Gaussian weight
Once A is known, observables like the field energies are calculable in unit of .
Two steps: solve A[t,p] and take
2
23
2|)(|exp][W
xx
xxd
x is related to Qs(x)
larger = larger = dense gluons = larger Qs
pt
BE
,
22
pt ,
July 2007 at ETD-HIC
Parameter Choice
Model parameter for RHIC
Classical description works till ~1/Qs~ 0.1fm□ Conventional choice is g2= 2GeV and s = 1/
by Krasnitz, Nara, Venugopalan, Lappi, Romatschke, Kapusta, Fries ...
GeV 2-1~)/( 6/100 AxxQQs Q0, x0, from DIS
210~~ spx tWe focus on "soft" physics pt ~1GeV. c.f. "hard" physics = pQCD
July 2007 at ETD-HIC
We are set . . . in principle
We have formulated the model.
We have fixed the model parameters.
In principle, we are set . . .
July 2007 at ETD-HIC
Only Longitudinal
Initial fields at = 0
Longitudinal fields□ Order 0
Transverse fields are vanishing
0 as )/ln(21
32)1()(
)()(
)(2
22
)0(42
22
)0(42
2
aacL
NNEg
gBg
g cc
a
Lk
kdx/
/
1)2(
2)2(1222
2
222 ~)(~~Roughly
Only TransverseOnly Transverse
L scale infra.
a scale ultra.
July 2007 at ETD-HIC
What happens if expanded in
Transverse fields become non-vanishing□ Order 2
Similar in Longitudinal fields2
32
2)2(42
22
)2(42
2
1
)(64
)1()()(
)()(
a
ONNEg
gBg
g ccii
Expansion not in / but in /a !
c.f. Fries-Kapusta-Li ('06)
July 2007 at ETD-HIC
Log-Ansatz
Naive expansion reads
Log-ansatz
2
22
42
2
32)1(
)( aNN
gg cc
0 as ln)(16
3
)()()(ln
41
43
21QCD42
22
2
2222
222
agg
gagLgc
July 2007 at ETD-HIC
Results
Ansatz works
K.F.('07)c.f. T.Lappi ('06)
~ 0.1fm
~ 130GeV/fm350GeV/fm3 (L1/QCD)
c.f. ~1fm ~ 5.1GeV/fm3
July 2007 at ETD-HIC
Instability ?
So far, we have established the CGC solution which is boost invariant with no dependence.QUESTION:What happens if small fluctuations depending on exist in the initial state ?ANSWER:(Some of) depending modes exponentially grow as a function of .
Romatschke-Venugopalan ('05)
July 2007 at ETD-HIC
Linearized Equations of Motion
Expand in Ai
Solutions□ > 0
□ < 0
cij
acbabji
abkkij
abij
bj
abij
ai
ai
FfgDDDDG
AGAA
2)()(1
121
2 4 6 8 10
-500
500
1000
1500
2 4 6 8 10
-400
-200
200
400
)( iI
)||( iJ
(for indep. )
July 2007 at ETD-HIC
Initial CGC Fields
Instability "Tendency"□ The background is frozen at =0 for simplicity
and consider the time evolution of fluctuations.
01 G Oscillatory modes dominant around the mean fields.
))(( 11 GG disperses including <0
Ensemble average over CGC contains contributions with <0
July 2007 at ETD-HIC
Initial Fluctuations
In Fourier space (Ai)2
~ 1/(Ei)2
~
22
22
)(~)(
)(~)(ii
ii
EE
AB
Same order of magnitude
Law Gauss ,0 EAKF-Gelis-McLerran ('06)
July 2007 at ETD-HIC
After ensemble average
)||( iJ
Initially oscillatory
is predominant.
Afterwards, exponentiallygrowing modes with <0whose weight was smallinitially as become dominant.
)2/()( 22
~ e
K.F.('07)
July 2007 at ETD-HIC
Summary
Classical description works in the heavy-ion collision till ~1/Qs (~ 0.1fm at RHIC).
Initial fields are calculable analytically with the log-ansatz.Unstable modes grow up.□ CGC fields evolve as times goes. neglected□ Real instability should be weaker? or stronger??□ Dynamical problem is very interesting!
(under investigation)
