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)