1. Kenji Fukushima (RIKEN BNL Research Center)
Initial fields and instabilities in the classical model of the heavy-ion collision
Kenji Fukushima (RIKEN BNL Research Center)
July 2007 at ETD-HIC

2. 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

3. 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
High Energy Hadron Target
July 2007 at ETD-HIC

4. 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

5. Going to larger Q2 with fixed x
Gluon slowly increases with a decreasing area
Graphically, in the same way,
When does the distribution come to overlap?
large Q  Dilute Gluon Matter
Gluons with kt << Qs(x) are saturated.
July 2007 at ETD-HIC

6. Dense-Dense Scattering
Scattering amplitude in the Eikonal approx.
Dense Target
Dense Projectile
July 2007 at ETD-HIC

7. Classical approximation
Stationary-point approximation
Stationary-point approx. is made at
July 2007 at ETD-HIC

8. Equations of Motion Coordinates
Equations to be solved (in At =0 gauge) h t
July 2007 at ETD-HIC

9. Boundary Conditions
Two-source problem ?
July 2007 at ETD-HIC

10. McLerran-Venugopalan Model
Gaussian weight
Once A is known, observables like the field energies are calculable in unit of m.
Two steps: solve A[rt,rp] and take
mx is related to Qs(x)
larger m = larger r = dense gluons = larger Qs
July 2007 at ETD-HIC

11. Parameter Choice Model parameter for RHIC
Classical description works till t ~1/Qs~ 0.1fm
Conventional choice is g2m = 2GeV and as = 1/p by Krasnitz, Nara, Venugopalan, Lappi, Romatschke, Kapusta, Fries ...
We focus on "soft" physics pt ~1GeV.
c.f. "hard" physics = pQCD
Q0, x0, l from DIS
July 2007 at ETD-HIC

12. 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

13. Initial fields at t = 0 Longitudinal fields
Order t 0
Transverse fields are vanishing
Only Longitudinal
Only Transverse
Only Transverse
July 2007 at ETD-HIC

14. What happens if expanded in t
Transverse fields become non-vanishing
Order t 2
Similar in Longitudinal fields
Expansion not in t /m but in t /a !
c.f. Fries-Kapusta-Li ('06)
July 2007 at ETD-HIC

15. Log-Ansatz
Naive expansion reads
Log-ansatz
July 2007 at ETD-HIC

16. Results Ansatz works e ~ 130GeV/fm350GeV/fm3 (L1/LQCD) t ~ 0.1fm
c.f. t ~1fm e ~ 5.1GeV/fm3
K.F.('07)
c.f. T.Lappi ('06)
July 2007 at ETD-HIC

17. Instability ?
So far, we have established the CGC solution which is boost invariant with no h dependence.
QUESTION: What happens if small fluctuations depending on h exist in the initial state ?
ANSWER: (Some of) h depending modes exponentially grow as a function of t.
Romatschke-Venugopalan ('05)
July 2007 at ETD-HIC

18. Linearized Equations of Motion
Expand in dAi
Solutions
l > 0
l < 0 l
(for t indep. l)
July 2007 at ETD-HIC

19. Initial CGC Fields Instability "Tendency"
The background is frozen at t =0 for simplicity and consider the time evolution of fluctuations.
Oscillatory modes dominant around the mean fields.
l disperses including l <0
Ensemble average over CGC contains contributions with l <0
July 2007 at ETD-HIC

20. Initial Fluctuations In Fourier space (dAi)2 ~ 1/n (dEi)2 ~ n
KF-Gelis-McLerran ('06)
(dAi)2
~ 1/n
(dEi)2
~ n
Same order of magnitude
July 2007 at ETD-HIC

21. After ensemble average
Initially oscillatory is predominant.
Afterwards, exponentially growing modes with l <0 whose weight was small initially as
become dominant.
K.F.('07)
July 2007 at ETD-HIC

22. Summary
Classical description works in the heavy-ion collision till t ~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)
July 2007 at ETD-HIC