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Multi-particle production in
QCD at high energies
Raju Venugopalan
Brookhaven National Laboratory
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)
Nucleus-Nucleus Collisions…leading order graphs
All such diagrams
of Order O(1/g)
Inclusive multiplicity even to leading order
requires 2 -> n Feynman amplitudes
- completely non-perturbative problem!
F. Gelis, RV
hep-ph/0601209
Yang-Mills Equations for two nuclei
Kovner,McLerran,Weigert
Initial conditions
from matching
eqns. of motion
on light cone
Longitudinal E and B fields created right
after the collision - non-zero
Chern-Simons charge generated
Kharzeev,Krasnitz,RV; Lappi, McLerran
Lattice Formulation
 Hamiltonian in
Krasnitz, RV
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
Gluon Multiplicity
with
# dists. are infrared finite
PRL 87, 192302 (2001)
Dispersion
relation:
Just as for a
Debye screening
mass
Classical field
Classical field
/ Particle
Particle
Melting CGC to QGP
L. McLerran, T. Ludlam,
Physics Today
Glasma…
The “bottom up” scenario
Baier, 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 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
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 transverse
configurations:
Generate Gaussian random
function 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
LSZ
Probability of producing n particles in theory with sources:
Generating Function of moments:
Action of
D[j_+,j_-] generates
all the connected
Green’s functions
of the Schwinger-Keldysh
formalism
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 )
+
Similar to Schwinger mechanism in QED
Remarkably, both terms can be computed by solving
the small fluctuations equations of motion with
retarded boundary conditions!
Gelis & RV
In QCD, for example,
2
+
Summary and Outlook
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
 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)