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

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 High Energy Hadron Target 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 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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