1. Multi-particle production in QCD at high energies Raju Venugopalan Brookhaven National Laboratory

2. Outline of Lectures Lecture I: EFT approach to high energy QCD-The Color Glass Condensate; multi-particle production in the CGC Lecture II: Hadronic scattering in the CGC-multiple scattering & quantum evolution effects in limiting fragmentation & quark pair production Lecture III: Plasma instabilities & thermalization in the CGC; computing particle production in Heavy Ion collisions to next-to-leading order (NLO)

3. All such diagrams of Order O(1/g) Nucleus-Nucleus Collisions…leading order graphs Inclusive multiplicity even to leading order requires 2 -> n Feynman amplitudes - completely non-perturbative problem! F. Gelis, RV hep-ph/0601209

4. Initial conditions from matching eqns. of motion on light cone Yang-Mills Equations for two nuclei Kovner,McLerran,Weigert

5. Longitudinal E and B fields created right after the collision - non-zero Chern-Simons charge generated Kharzeev,Krasnitz,RV; Lappi, McLerran

6.  Hamiltonian in gauge; per unit rapidity, For ``perfect’’ pancake nuclei, boost invariant configurations  Solve 2+1- D Hamilton’s equations in real time for space-time evolution of glue in Heavy Ion collisions Lattice Formulation Krasnitz, RV

7. Gluon Multiplicity with # dists. are infrared finite PRL 87, 192302 (2001)

8. Dispersion relation: Just as for a Debye screening mass

9. Classical field / Particle Particle

10. Melting CGC to QGP L. McLerran, T. Ludlam, Physics Today Glasma…

11. The “bottom up” scenario Baier, Mueller, Schiff, Son Scale for scattering of produced gluons (for t > 1/Q_s) set by Multiple collisions:

12. Occupation # Radiation of soft gluons important for Thermalization for: and

13. A flaw in the BMSS ointment - Weibel instabilities… Arnold,Lenaghan,Moore,Yaffe; Rebhan, Romatschke, Strickland; Mrowczynski Anisotropic momentum distributions of hard modes cause -exponential growth of soft field modes Changes sign for anisotropic distributions

14. Effective potential interpretation: -ve eigenvalue => potential unbounded from below Large magnetic fields can cause O(1) change in hard particle trajectories on short time scales - - possible mechanism for isotropization of hard modes

15. THE UNSTABLE GLASMA

16. Instabilities from violations of boost invariance ? Boost invariance is never realized: a) Nuclei always have a finite width at finite energies b) Small x quantum fluctuations cause violations of boost invariance that are of order unity over Possible solution: Perform 3+1-D numerical simulations of Yang-Mills equations for Glasma exploding into the vacuum Romatsche + RV

17. Construct model of initial conditions with fluctuations: i) ii) Method: Generate random transverse configurations: Generate Gaussian random function in \eta This construction explicitly satisfies Gauss’ Law

18. Compute components of the Energy-Momentum Tensor

19. Violations of boost invariance (3+1 -D YM dynamics) - leads to a Weibel instability Romatschke, RV PRL 96 (2006) 062302 For an expanding system,

20. ~ 2 * prediction from HTL kinetic theory

21. Instability saturates at late times-possible Non-Abelian saturation of modes ?

22. Distribution of unstable modes also similar to kinetic theory Romatschke, Strickland

23. Very rapid growth in max. frequency when modes of transverse magnetic field become large - “bending” effect ?

24. Accompanied by growth in longitudinal pressure…

25. And decrease in transverse pressure… Right trends observed but too little too late…

26. How do we systematically compute multi-particle production beyond leading order ? Problem can be formulated as a quantum field theory with strong time dependent external sources

27. Power counting in the theory: Order of a generic diagram is given by with n_E = # of external legs, n_L the # of loops and n_J the # of sources.  Order of a diagram given by # of loops and external legs

28. In standard field theory, For theory with time dependent sources, Generating functional of Green’s functions with sources

30. Probability of producing n particles in theory with sources: LSZ Generating Function of moments:

31. Action of D[j_+,j_-] generates all the connected Green’s functions of the Schwinger-Keldysh formalism

32. Inclusive average multiplicity: [ ]

33. I) Leading order: O (1 / g^2) Obtained by solving classical equations - result known! Krasnitz, RV; Krasnitz, Nara, RV; Lappi

34. II) Next-to-leading order: O ( g^0 ) + Remarkably, both terms can be computed by solving the small fluctuations equations of motion with retarded boundary conditions! Gelis & RV Similar to Schwinger mechanism in QED

35. In QCD, for example, 2 +

36. We now have an algorithm (with entirely retarded b.c.) to systematically compute particle production In AA collisions to NLO - particularly relevant at the LHC Pieces of this algorithm exist:  Pair production computation of Gelis, Lappi and Kajantie very similar  Likewise, the 3+1-D computation of Romatschke and RV Summary and Outlook

37.  Result will include  All LO and NLO small x evolution effects  NLO contributions to particle production  Very relevant for studies of energy loss, thermalization at the LHC  Conceptually issues at a very deep level - diffraction for instance - will challenge our understanding of the separation between “evolution” and “production” - factorization in QCD (Gelis & RV, in progress)