1. Small-x physics 2- The saturation regime of QCD and the Color Glass Condensate
Cyrille Marquet
Columbia University

2. The saturation regime of QCD
the BFKL approximation
summing terms leads to a growth of the gluon density
in the hadronic wave function but high-density effects are missing
when the gluon density is large enough, gluon recombination is important
gluon recombination in the hadronic wave function
gluon density per unit area
recombination cross-section
gluon kinematics
recombinations important when
the saturation regime: for with
magnitude of Qs
x dependence
this regime is non-linear,
yet weakly coupled
the idea of the CGC is to take into account this effect via strong classical fields

3. The Color Glass Condensate
effective description of a high-energy hadron/nucleus
the CGC is moving in the direction and the gauge is
the current is
solving Yang-Mills equations gives 
short-lived fluctuations
separation between the long-lived high-x
partons and the short-lived low-x gluons
CGC wave function
in practice we deal with CGC averages such as
we are interested in the evolution of with x, which sums both
the BFKL equation is recovered in the low-density regime

4. Outline of the second lecture
The evolution of the CGC wave function the JIMWLK equation and the Balitsky hierarchy
A mean-field approximation: the BK equation solutions: QCD traveling waves
the saturation scale and geometric scaling
Beyond the mean field approximation stochastic evolution and diffusive scaling
Computing observables in the CGC framework solving evolution equation vs using dipole models

5. The evolution of the CGC wave function: the B-JIMWLK equations

6. The JIMWLK equation a functional equation for the x evolution of :
Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner
a functional equation for the x evolution of :
with
the JIMWLK equation gives the evolution of the wave function for small enough x
the equivalent Balitsky equations are obtained by considering
the scattering of simple test projectiles (dipoles) off the CGC
the Wilson lines sum powers of
adjoint representation

7. Scattering off the CGC eigenstates ? eigenvalues
their interaction should conserve their spin, polarisation, color, momentum …
in the high-energy limit, the eigenstates are simple when the partons
transverse momenta are Fourier transformed to transverse coordinates
in the large-Nc limit, further simplification: the eigenstates are only made of dipoles
x, y : transverse coordinates
other quantum numbers aren’t explicitely written, the
eigenvalues depend only on the transverse coordinates
eigenvalues
scattering amplitude off the (dense) target CGC
the interaction with the CGC conserves transverse position

8. Dipoles as test projectiles
dipoles are ideal projectiles to probe small distances
the dipole:
u : quark space transverse coordinate
v : antiquark space transverse coordinate
scattering of the quark:
dependence kept implicit in the following
JIMWLK equation → evolution equation for the dipole correlators

9. The Balitsky hierarchy
equations for dipoles scattering of the CGC
Balitsky (1996)
the BFKL kernel
an hierarchy of equation involving correlators with more and more dipoles
BFKL
saturation
general structure:
we will now derive the first equation of the hierarchy
solving the B-JIMWLK equations gives , …
which can then be used to compute observables
in the large Nc limit, the hierarchy is restricted to dipoles

10. Recall the dipole wave function
the valence component (using the mixed space):
such that only perturbative
sizes r << 1/ QCD are included
x : quark space transverse coordinate
y : antiquark space transverse coordinate
other degrees of freedom are not explicitely written
from QCD at order gS
color
index
the dipole rapidity:
the dipole dressed with one gluon

11. Elastic scattering of the dipole
the dipole scattering is described by Wilson lines
scattering of the quark
scattering of the antiquark
for the qqg component:
let’s compute the elastic scattering amplitude Ael(Y ) in the
dipole-CGC collision where is the total rapidity
Y = YCGC + Yd
in the frame in which and

12. Frame independence of Ael(Y )
in the frame in which and
using the following identity to get rid of the adjoint Wilson line
= 1/2 -1/(2Nc)
one obtains
our two expressions for Ael(Y )
should be identical, this
requirement in the limit
Y - YCGC  dY  0 gives
the first Balitsky equation
frame independence:

13. A mean field approximation: the Balitsky-Kovchegov equation

14. The BK equation obtain by neglecting correlations
the BK equation is a closed equation for obtained by assuming
solutions: qualitative behavior
r = dipole size
let’s consider impact-parameter independent solutions
at small Y, is small, and the quadratic term can be neglected, the equation
reduces then to the linear BFKL equation and rises exponentially with Y
as gets close to 1 (the stable fixed point of the equation), the non-linear term becomes important, and , saturates at 1
with increasing Y, the unitarization scale get bigger

15. Coordinate vs momentum space
let’s go to momentum space 
due to conformal invariance, the linear part of the equation is the same for and
coordinate space
momentum space
dipole scattering amplitude
unintegrated gluon distribution
linear BFKL equation
linear BFKL equation
genuine saturation:
no real saturation:
both equation are in the same universality class as the F-KPP equation

16. The F-KPP equation same features as the BK equation
Fisher, Kolmogorov, Petrovsky, Piscounov
time derivative
space derivative
linear term meaning
exponential growth
non linear term
meaning saturation
dictionary F-KPP → BK
when expanding to second order,
the equations are the same (in momentum space)
in spite of small difference, same universality class
the precise forms of the space derivatives and of the non-linear term doesn’t matter
these equations belong to the same universality class
F-KPP and BK asympotic (in t or Y) solutions are the same: traveling waves

17. Traveling wave solutions
what is a traveling wave
position:
the speed of the wave is determined
only by the linear term of the equation
saturation
region
wave traveling at speed v,
independently of the initial condition
provided it is steep enough
there the initial condition
has not been erased yet
BK solutions: same rapidity evolution
quantitative features can be derived
the formation of the traveling wave

18. Recall the BFKL solutions
a superposition of waves with speeds
initial condition
in Mellin space
the minimal speed
is obtained for and its value is
this will be the speed of the
asymptotic traveling wave
the saturation exponent
for the traveling wave to form, the initial condition
must feature

19. QCD traveling waves the initial condition is steep enough
Munier and Peschanski (2004)
in QCD
asymptotic solutions of BK
same features for except
in the saturation region Y0
Y >Y0
the saturation scale 
with
then
called geometric scaling

20. Numerical solutions numerical simulations confirm the results
for the scattering amplitude
for the saturation scale
sub-asymptotic corrections (≡ geometric scaling violations) are also known:

21. Running coupling corrections
running coupling corrections to the BK equation
taken into account by the substitution
Kovchegov
Weigert
Balitsky
consequences
similar to those first obtained by the simpler substitution
running coupling corrections slow down the increase of Qs with energy
also confirmed by numerical simulations, however
this asymptotic regime is reached for larger rapidities

22. Beyond the mean field approximation

23. Recall the Balitsky hierarchy
equations for dipole correlators
it describes the so-called dense regime where
the BK equation is a
good approximation
general structure
at large Nc
the equation for is fine, its dilute limit is the BFKL equation, however the other equations
are incomplete, they do not agree with Mueller’s dipole model in the limit
incorrect results for have consequences for
still there is something missing
describing properly the dilute regime is also important
the B-JIMWLK equations do not describe it properly

24. A new hierarchy of equations
the hierarchy must be completed
Iancu and Triantafyllopoulos (2005)
fluctuations (included to recover Mueller’s
dipole model in the dilute limit) important when
BFKL
saturation
important when
this QCD evolution is equivalent to a stochastic process
reformulation into a Langevin problem
average over the
realisations of the stochastic process
the correlators can all be obtained from a single
stochastic quantity which obeys a Langevin equation
CGC wave function

25. The stochastic F-KPP equation
the QCD Langevin equation
deterministic part of the equation: BK !
noise
high-energy QCD evolution = stochastic process in the universality
class of reaction-diffusion processes, of the sF-KPP equation
Iancu, Mueller and Munier (2005)
the reduction to one dimension introduces the noise strength parameter 
noise
r = dipole size
the stochastic F-KPP equation

26. A stochastic saturation scale
the stochastic evolution
the Langevin evolution generates, at each rapidity,
a stochastic ensemble of solutions for
the average speed of the waves:
the dispersion coefficient: D
analytic expressions for v and D are known only for very small
equivalent to a stochastic saturation scale Qs
(for )
Qs is a stochastic variable distributed
according to a Gaussian probability law:
corrections to the Gaussian law for improbable fluctuations
also known, picture confirmed by numerical simulations
C.M., Soyez and Xiao
average saturation scale:
the probability distribution of Qs

27. A new scaling law the average dipole scattering amplitude
: the diffusion is negligible and
one recovers geometric scaling
: the diffusion is important and
new regime: diffusive scaling
very high energies

28. The diffusive scaling regime
important properties
in the diffusive scaling regime:
the amplitudes are dominated by events that
feature the hardest fluctuation of
in average the scattering is weak, yet saturation is the relevant physics
running coupling corrections
Dumitru et al.
push the diffusive scaling regime up to asymptotically high energies
conclusion at the moment
using the mean field approximation is good enough for phenomenology

29. Computing observables in the CGC framework

30. The CGC averages the Wilson line correlators
in the CGC framework, any cross-section is determined by colorless
combinations of Wilson lines, averaged over the CGC wave function
the energy evolution of cross-sections is encoded in the evolution of
this wave function is mainly non-perturbative, but its evolution is known
 Balitsky/BK equation for the correlators
this is the most common average, (for instance it determines deep inelastic scattering)
its Fourier transform is the unintegrated gluon distribution, it obeys the BK equation
the 2-point function or dipole amplitude
for 2-particle correlations
more complicated correlators
for instance

31. The strategy what should be done in principle what is done in practice
- express the cross-sections in terms of correlators of Wilson lines
the same correlators enter in the formulation of very different processes
the more exclusive the final state is, the more complicated the correlators are
- use B-JIMWLK or BK equation to compute the correlators
use appropriate initial conditions
include non-perturbative effects such as the impact parameter dependence
running coupling effects should also be included
many processes have been studied in the CGC framework using dipole models
what is done in practice
solving evolution equations with running coupling is not the favored approach
instead one uses parametrizations with features inspired from theoretical results
it makes it easier to deal with impact parameter dependence of the dipole amplitude
famous dipole models: GBW, IIM, IPsat

32. Outline of the third lecture
The hadron wave function summary of what we have learned
The saturation models from GBW to the latest ones
Deep inelastic scattering (DIS) the cleanest way to probe the CGC/saturation allows to fix the model parameters
Diffractive DIS and other DIS processes these observables are predicted
Forward particle production in pA collisions and the success of the CGC picture at RHIC