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Initial fields and instabilities in the classical model of the heavy-ion collision

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.

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Initial fields and instabilities in the classical model of the heavy-ion collision

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  1. Initial fields and instabilitiesin the classical modelof 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/pby 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 t2 • 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) c.f. t ~1fme ~ 5.1GeV/fm3 t ~ 0.1fm 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 simplicityand 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 KF-Gelis-McLerran ('06) (dEi)2 ~ n Same order of magnitude July 2007 at ETD-HIC

  21. After ensemble average Initially oscillatory is predominant. Afterwards, exponentiallygrowing modes with l <0whose weight was smallinitially 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

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