Small-x%20physics%201-%20High-energy%20scattering%20in%20pQCD:%20the%20BFKL%20equation

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Small-x physics1- High-energy scattering in
pQCD the BFKL equation

• Cyrille Marquet

Columbia University
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Outstanding problems in pQCD

• What is the high-energy limit of hadronic
  scattering ?

• What is the wave function of a high-energy hadron
  ?

spin indices
momenta
wave function
color indices
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Outline of the first lecture

• Hadronic scattering in the high-energy
  limitleading logarithms and kT factorization in
  perturbative QCD
• The wave function of a high-energy hadronthe
  dipole picture and the BFKL equation
• The BFKL equation at leading orderconformal
  invariance and solutions of the equation
• BFKL at next-to-leading orderpotential problems
  and all-order resummations
• How to go beyond the BFKL approachthe ideas that
  led to the Color Glass Condensate picture

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The scattering amplitude
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High-energy scattering
light-cone variables

• before the collision

the momentum transfer is mainly transverse

• during the collision

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2 to 2 scattering at high energy
the exchanged particle has a very small
longidudinal momentum the final-state particles
are separated by a large rapidity interval
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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
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kT factorization

• from parton-parton scattering to hadron-hadron
  scattering

impact factors no Y dependence
Green function, this is what resums the powers of
aSY
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The BFKL equation

• for the unintegrated gluon distribution

real-virtual cancellation when
comes from real gluon emission
comes from virtual corrections
we will derive this equation with a wave function
calculation
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The hadronic wave function
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The wave function of a hadron

• light-cone perturbation theory (in light-cone
  gauge )

a quantum superposition of states
- 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
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The wave function of a dipole

• replace the gluon cascade by a dipole cascade

Muellers 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
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The wavefunction

• in momentum space

this selects the leading logarithm
using the wave function in the limit ,
one gets

• in mixed space

we have to compute
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Dipole cascade in position space

• the zero-th order wave function factorizes !

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The BFKL equation

• for the dipole density

no-splitting probability
splitting into a dipole of size r
using and
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Solving the leading logarithmic (LL) BFKL equation
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Conformal invariance
Lipatov (1986)

• the BFKL kernel is conformal invariant

under the conformal transformation it becomes
note the complex notation
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BFKL solutions

• a linear superposition of eigenfunctions

discrete index called conformal spin
specified by the initial condition
continuous index Mellin transformation

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BFKL at next-to-leadinglogarithmic (NLL) accuracy
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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
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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
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All-order resummations

• truncating the BFKL perturbative series generates
  singularities

Salam (1998), Ciafaloni, Colferai and Salam (1999)
has spurious singularities in Mellin (?)
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
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Salams resummation schemes
in momentum space, the poles of correspond to
the known so-called DGLAP limits k1 gtgt k2 and k1
ltlt k2 this gives information/constraints on what
add to the next-leading kernel
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Resummed NLL BFKL

• the resummed NLL-BFKL Green function

now running coupling (with symmetric scale)
effective kernel
values of 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
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Going beyond the BFKL approach
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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 shouldnt 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
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Proposals to go beyond BFKL
summing terms isnt 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
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The saturation phenomenon

• gluon recombination in the hadronic wave function

gluon density per unit area
recombination cross-section
recombinations important when
gluon kinematics
this regime is non-linear, yet weakly coupled
magnitude of Qs
x dependence
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The Color Glass Condensate
?
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
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Outline of the second lecture

• The evolution of the CGC wave functionthe JIMWLK
  equation and the Balitsky hierarchy
• A mean-field approximation the BK
  equationsolutions QCD traveling waves
• the saturation scale and geometric scaling
• Beyond the mean field approximationstochastic
  evolution and diffusive scaling
• Computing observables in the CGC
  frameworksolving evolution equation vs using
  dipole models