Multi-particle production in QCD at

high energiesRaju Venugopalan

Brookhaven National Laboratory

Outline of LecturesLecture 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)

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, RVhep-ph/0601209

Initial conditionsfrom matchingeqns. of motion on light cone

Yang-Mills Equations for two nuclei

Kovner,McLerran,Weigert

Longitudinal E and B fields created right after the collision - non-zero Chern-Simons charge generated

Kharzeev,Krasnitz,RV; Lappi, McLerran

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 FormulationKrasnitz, RV

Gluon Multiplicity

with

# dists. are infrared finite

PRL 87, 192302 (2001)

Dispersionrelation:

Just as for a Debye screening mass

Classical field Classical field / Particle

Particle

Melting CGC to QGPL. McLerran, T. Ludlam,Physics Today

Glasma…

The “bottom up” scenarioBaier, Mueller, Schiff, Son

Scale for scattering of produced gluons (for t > 1/Q_s) set by

Multiple collisions:

Occupation #

Radiation of soft gluons important for

Thermalization for:

and

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

Effective potential interpretation:

-ve eigenvalue => potential unboundedfrom below

Large magnetic fields can cause O(1) change in hard particle trajectories on short time scales -

- possible mechanism for isotropization of hard modes

THE UNSTABLE GLASMA

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

Construct model of initial conditions with fluctuations:

i)

ii) Method:Generate random transverseconfigurations:

Generate Gaussian randomfunction in \eta

This construction explicitly satisfies Gauss’ Law

Compute components of the Energy-Momentum Tensor

Violations of boost invariance (3+1 -D YM dynamics)- leads to a Weibel instability Romatschke, RV

PRL 96 (2006) 062302

For an expanding system,

~ 2 * prediction from HTL kinetic theory

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

Distribution of unstable modes also similar to kinetic theory

Romatschke, Strickland

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

Accompanied by growth in longitudinal pressure…

And decrease in transverse pressure…

Right trends observed but too little too late…

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

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

In standard field theory,

For theory with time dependent sources,

Generating functional of Green’s functions with sources

Probability of producing n particles in theory with sources:

LSZ

Generating Function of moments:

Action of

D[j_+,j_-] generates all the connectedGreen’s functions of the Schwinger-Keldyshformalism

Inclusive average multiplicity:

[ ]

I) Leading order: O (1 / g^2)

Obtained by solving classical equations - result known!

Krasnitz, RV; Krasnitz, Nara, RV;Lappi

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

In QCD, for example,

2

+

We now have an algorithm (with entirely retarded b.c.) to systematically compute particle productionIn 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

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)