Small-x physics
1- High-energy scattering in
pQCD: the BFKL equation
Cyrille Marquet
Columbia University
Outstanding problems in pQCD
• What is the high-energy limit of hadronic scattering ?
for
, what is the behavior of
the scattering amplitude
?
s
t
• What is the wave function of a high-energy hadron ?
spin indices
wave
function
hadron  qqq  qqqg  ...  qqq......ggggg
momenta
color indices
Outline of the first lecture
• Hadronic scattering in the high-energy limit
leading logarithms and kT factorization in perturbative QCD
• The wave function of a high-energy hadron
the dipole picture and the BFKL equation
• The BFKL equation at leading order
conformal invariance and solutions of the equation
• BFKL at next-to-leading order
potential problems and all-order resummations
• How to go beyond the BFKL approach
the ideas that led to the Color Glass Condensate picture
The scattering amplitude
High-energy scattering
light-cone variables
• before the collision
+z
• during the collision
• after the collision
final-state particles
rapidity
pseudo rapidity
the momentum transfer is mainly transverse
2 to 2 scattering at high energy
• consider 2 to 2 scattering with
(Regge limit):
initial momenta
and
final state parametrized by and
the exchanged particle has a very small longidudinal momentum:
the final-state particles are separated by a large rapidity interval:
• next-order diagram:
the new final-state gluon yields the factor
this contribution goes as
is as large as the zeroth order in
 using the perturbative expansion
and
is not the right approach
Summing large logarithms
• the relevant perturbative expansion in the high-energy limit:
: leading-logarithmic approximation (LLA), sums
: next-to-leading logarithmic approximation (NLLA), sums
.
.
.
Balitsky, Fadin, Kuraev, Lipatov
• the leading-logarithmic approximation
only gluons contribute in the LLA, and the coupling doesn’t run
n-th order
this is schematic, but the actual summation of the leading
logs by BFKL confirms this power-law growth with energy
in practice, NLL corrections are large
kT factorization
• from parton-parton scattering to hadron-hadron scattering
impact factors
Green function, this is what
no Y dependence resums the powers of αSY
• the unintegrated gluon density
and
obey the same
BFKL equation, with different initial condition
is related to the hadron’s wave
function which we will study in the following
kT factorization is also proved at
NLL but there are many complications
Fadin et al. (2005-2006)
The BFKL equation
• for the unintegrated gluon distribution
comes from real
gluon emission
real-virtual
cancellation
when
comes from virtual
corrections
we will derive this equation with a wave function calculation
• the solution of this linear equation
the saddle point at
the high-energy behavior
gives
initial condition obtained
from the impact factor
The hadronic wave function
The wave function of a hadron
• light-cone perturbation theory (in light-cone gauge
)
a quantum superposition of states
the wave functions can be computed following
Feynman rules a bit different from those of standard
covariant perturbation theory Brodsky and Lepage
- the unintegrated gluon distribution is given by
- the partons in the wave function are on-shell
- their virtuality is reflected by the non-conservation
of momentum in the x- direction
• the simplest light-cone wave function
~ energy denominator like in quantum mechanics
The wave function of a dipole
• replace the gluon cascade by a dipole cascade
Mueller’s idea to compute the evolution
of the unintegrated gluon distribution
 simple derivation of the BFKL equation
this dipole picture of a hadron is used a lot in small-x physics
• the key to the simplicity of this approach : the
mixed space
the wave function depends only on the dipole size r because of momentum conservation
The
wavefunction
• in momentum space
this selects the
leading logarithm
using the
• in mixed space
we have to compute
wave function in the limit
, one gets
Dipole cascade in position space
• the zero-th order wave function factorizes !
interpretation:
two-dipole
wave function
=
amplitude probability
for dipole splitting
x
original dipole
wave function
• the evolution of the hadron wave function is that of a dipole cascade
dipole
splitting probability
the evolution the density of dipoles in the hadron wave function
is the same as the evolution of the unintegrated gluon distribution
The BFKL equation
• for the dipole density
no-splitting probability
splitting into a
dipole of size r
using
and
:
or equivalently
• back to momentum space
the BFKL equation for the unintegrated gluon distribution can be recovered using
Solving the leading logarithmic
(LL) BFKL equation
Conformal invariance
• the BFKL kernel is conformal invariant:
Lipatov (1986)
under the conformal transformation
it becomes
note the complex notation:
• eigenfunctions:
• eigenvalues:
labeled by one discrete index n and one continuous ν
BFKL solutions
• a linear superposition of eigenfunctions
discrete index
called conformal spin
continuous index
~ Mellin transformation
specified by the
initial condition
• the saddle point at high energies is
at high energies, one can neglect
non-zero conformal spins
after Fourier transforming to momentum space, one recovers the
solution given earlier for the unintegrated gluon distribution
BFKL at next-to-leading
logarithmic (NLL) accuracy
The NLL-BFKL Green function
it took about 10 years to compute the NLL Green function
Fadin and Lipatov (1998), Ciafaloni (1998)
up to running coupling effects, the eigenfunctions are unchanged
the eigenvalues are
with
On the NLO impact factors
the NLO impact factors are very difficult to compute
for deep inelastic scattering
it took about 10 years to compute the photon impact factors
Bartels et al.
for jet production in hadron-hadron collisions
the impact factors are known but after 5 years there is still no numerical result
Bartels, Colferai and Vacca (2002)
for vector meson production
the impact factors are known
the first complete NLL-BFKL calculation was for
Ivanov and Papa (2006)
but the results are very unstable when varying the renormalization scheme
 impossible to make reliable predictions
All-order resummations
• truncating the BFKL perturbative series generates singularities
Salam (1998), Ciafaloni, Colferai and Salam (1999)
 NLL
has spurious singularities in Mellin (γ) space, they lead to unphysical
results, this is an artefact of the truncation of the perturbative series
to produce meaningful NLL-BFKL results, one has to add the higher order
corrections which are responsible for the canceling the singularities
• different resummation schemes
there are different proposal to add the relevant higher-order corrections
Ciafaloni, Colferai, Salam, Stasto, Altarelli, Ball, Forte, Brodsky, Lipatov, Fadin, … (1999-now)
there are equivalent at NLL accuracy and produce similar numerical results
Salam’s schemes are the only ones used so far for
phenomenological studies because they are easy to implement
Salam’s resummation schemes
in momentum space, the poles of
correspond
to the known so-called DGLAP limits k1 >> k2 and k1 << k2
this gives information/constraints on what add to the next-leading kernel
Strategy:
 NLL regularization
S3
there is some arbitrary:
different schemes S3, S4, …
the S3 kernel (extended to p ≠ 0) :
with
expanding in powers of
, one recovers
implicit equation
 eff
in practice, each value k1k2
leads to a different effective kernel
Resummed NLL BFKL
• the resummed NLL-BFKL Green function
now running coupling
(with symmetric scale)
values of
effective kernel
at the saddle point
the power-law growth of
scattering amplitudes with
energy
is
slowed down compared
to the LLA result
the growth with rapidity of the gluon density
in the hadronic wave function is also slower
Going beyond the BFKL
approach
The problem with BFKL
• the growth of scattering amplitudes with energy
this leads to unitarity violations, for instance for the total cross-section,
the Froissart bound
cannot be verified at high energies
what did we do wrong ? use a perturbative treatment when we shouldn’t have
• the growth of gluon density with increasing rapidity
even if this initial condition is a fully perturbative
wave function (no gluons with small )
the BFKL evolution populates
the non perturbative region
this so-called infrared diffusion
invalidates the perturbative treatment
Proposals to go beyond BFKL
summing
terms isn’t enough, high-density effects are missing
to deal with this many body problem, one needs effective degrees of freedom
• the modified leading logarithmic approximation (MLLA)
in this approach, hadronic scattering is described by the exchange of
quasi-particles called Reggeized gluons (or Reggeons) Bartels, Ewertz, Lipatov, Vacca
the BFKL approximation corresponds to the exchange of two Reggeons
(a Pomeron), the idea of the MLLA is to include multiple exchanges
• the color glass condensate (CGC)
in this approach, the hadronic wave function is described by classical fields
the BFKL growth is due to the approximation that
gluons in the wave function evolve independently
when the gluon density is large enough, gluon recombination becomes important
the idea of the CGC is to take into account this effect via strong classical fields
the CGC sums both
and
The saturation phenomenon
• gluon recombination in the hadronic wave function
gluon density per unit area
recombination cross-section
gluon kinematics
the saturation regime: for
this regime is non-linear,
yet weakly coupled
recombinations important when
with
magnitude of Qs
x dependence
• an effective theory to describe the saturation regime of QCD
the numerous small-x gluons can be described by large color fields
which can be treated as classical fields
higher-x gluons act as static color sources for these fields
McLerran and Venugopalan (1994)
The Color Glass Condensate
hadron  qqq  qqqg  ...  qqq......ggggg

hadron   D  x [  ]   CGC
short-lived fluctuations
lifetime of the
fluctuations ~
separation between high-x partons ≡ static sources
and low-x partons ≡ dynamical fields
effective wave function
for the dressed hadron
when computing the unintegrated gluon distribution
we recover the BFKL equation in the low-density regime
what I will cover: how the wave function
evolves with x
how do we “measure” it with well-understood probes
what I will not cover: how this formalism is applied to heavy ion collisions
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