Small-x physics

1- High-energy scattering in pQCD: the BFKL equation

Cyrille Marquet

Columbia University

wavefunction

Outstanding problems in pQCD• What is the high-energy limit of hadronic scattering ?

gggggqqqqqqgqqq .........hadron color indices

spin indices momenta

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

t

s for , what is the behavior ofthe scattering amplitude ?

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

The scattering amplitude

High-energy scattering

• before the collision

light-cone variables

+ z

the momentum transfer is mainly transverse• during the collision

final-state particles

pseudo rapidity

rapidity

• after the collision

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:

• consider 2 to 2 scattering with (Regge limit):

initial momenta and

final state parametrized by and

using the perturbative expansion is not the right approach

• next-order diagram:

the new final-state gluon yields the factor

this contribution goes as and

is as large as the zeroth order in

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

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

only gluons contribute in the LLA, and the coupling doesn’t run

• the leading-logarithmic approximation

n-th order

kT factorization

impact factorsno Y dependence

Green function, this is what

resums the powers of αSY

• from parton-parton scattering to hadron-hadron scattering

kT factorization is also proved at

NLL but there are many complicationsFadin et al. (2005-2006)

• 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

The BFKL equation• for the unintegrated gluon distribution

we will derive this equation with a wave function calculation

comes from realgluon emission

comes from virtualcorrections

real-virtualcancellation

when

• the solution of this linear equation

initial condition obtainedfrom the impact factor

the saddle point at givesthe high-energy behavior

The hadronic wave function

The wave function of a hadron• light-cone perturbation theory (in light-cone gauge )

the wave functions can be computed followingFeynman rules a bit different from those of standard

covariant perturbation theoryBrodsky and Lepage

- the partons in the wave function are on-shell

- their virtuality is reflected by the non-conservation of momentum in the x- direction

a quantum superposition of states

- the unintegrated gluon distribution is given by

• 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 evolutionof 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

using the wave function in the limit , one gets

this selects theleading logarithm

• in mixed space

we have to compute

Dipole cascade in position space• the zero-th order wave function factorizes !

interpretation:two-dipole

wave functionoriginal dipolewave function

= xamplitude probabilityfor dipole splitting

dipolesplitting probability

• the evolution of the hadron wave function is that of a dipole cascade

the evolution the density of dipoles in the hadron wave function

is the same as the evolution of the unintegrated gluon distribution

using and :

The BFKL equation• for the dipole density no-splitting probability

splitting into adipole of size r

or equivalently

• back to momentum spacethe BFKL equation for the unintegrated gluon distribution can be recovered using

Solving the leading logarithmic (LL) BFKL equation

Conformal invarianceLipatov (1986)

under the conformal transformationit becomes

note the complex notation:

• eigenfunctions: labeled by one discrete index n and one continuous ν

• eigenvalues:

• the BFKL kernel is conformal invariant:

BFKL solutions

discrete indexcalled conformal spin

specified by theinitial conditioncontinuous index

~ Mellin transformation

• a linear superposition of eigenfunctions

after Fourier transforming to momentum space, one recovers thesolution given earlier for the unintegrated gluon distribution

at high energies, one can neglect

non-zero conformal spins

• the saddle point at high energies is

BFKL at next-to-leadinglogarithmic (NLL) accuracy

The NLL-BFKL Green functionit 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 impact factors are knownthe first complete NLL-BFKL calculation was for

Bartels et al.

Bartels, Colferai and Vacca (2002)

the NLO impact factors are very difficult to compute

Ivanov and Papa (2006)

it took about 10 years to compute the photon impact factors

the impact factors are known but after 5 years there is still no numerical result

for deep inelastic scattering

for jet production in hadron-hadron collisions

for vector meson production

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)

has spurious singularities in Mellin (γ) space, they lead to unphysicalresults, this is an artefact of the truncation of the perturbative series

to produce meaningful NLL-BFKL results, one has to add the higher ordercorrections which are responsible for the canceling the singularities

NLL

Ciafaloni, Colferai, Salam, Stasto, Altarelli, Ball, Forte, Brodsky, Lipatov, Fadin, … (1999-now)

• different resummation schemes

there are different proposal to add the relevant higher-order corrections

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 schemesin 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

NLL 3SregularizationStrategy:

implicit equationeff

there is some arbitrary:different schemes S3, S4, …

in practice, each value k1k2

leads to a different effective kernel

the S3 kernel (extended to p ≠ 0) :

with

expanding in powers of , one recovers

Resummed NLL BFKL

now running coupling(with symmetric scale)

effective kernel

• the resummed NLL-BFKL Green function

the power-law growth of

scattering amplitudes with

energy is

slowed down compared

to the LLA result

values of at the saddle point

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

this so-called infrared diffusioninvalidates the perturbative treatment

this leads to unitarity violations, for instance for the total cross-section,the Froissart bound cannot be verified at high energies

• the growth of scattering amplitudes with energy

• the growth of gluon density with increasing rapidity

what did we do wrong ? use a perturbative treatment when we shouldn’t have

even if this initial condition is a fully perturbativewave function (no gluons with small )

the BFKL evolution populatesthe non perturbative region

in this approach, hadronic scattering is described by the exchange ofquasi-particles called Reggeized gluons (or Reggeons)

summing terms isn’t enough, high-density effects are missingto deal with this many body problem, one needs effective degrees of freedom

Proposals to go beyond BFKL

• the modified leading logarithmic approximation (MLLA)

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 BFKL growth is due to the approximation thatgluons in the wave function evolve independently

• the color glass condensate (CGC)

the CGC sums both and

in this approach, the hadronic wave function is described by classical fields

when the gluon density is large enough, gluon recombination becomes importantthe idea of the CGC is to take into account this effect via strong classical fields

The saturation phenomenon

gluon kinematics

recombination cross-section

gluon density per unit area

the saturation regime: for with

• gluon recombination in the hadronic wave function

recombinations important when

McLerran and Venugopalan (1994)

• 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

magnitude of Qs x dependencethis regime is non-linear,yet weakly coupled

lifetime of the

fluctuations ~

The Color Glass Condensategggggqqqqqqgqqq .........hadron

separation between high-x partons ≡ static sources

and low-x partons ≡ dynamical fieldseffective wave function

for the dressed hadron

CGC][hadron xD

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

short-lived fluctuations

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 wavesthe 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