Recent advances in High-Energy QCD evolution equations

Javier L. Albacete

High Energy QCD: From RHIC to LHC. ECT, Trento , January 2007

INTRODUCTION

B-JIMWLK provides the best theoretical tool so far for the descriptionof high-energy scattering in QCD, but:

• It is derived at Leading Logarithmic Accuracy (s ln1/x): Fixed coupling. Need of NLO calculation.

• Valid for dense-dilute scattering: Symmetrization of the equations

• It is derived in a mean field approach: Fluctuations, Pomeron loops… not taken into account

Phenomenological consequences:

• Reduction of the energy dependence of the saturation scale

• Geometric scaling violations

• More exclusive observables…

OUTLINE

I: O(2)-corrections to B-JIMWLK from the classical equations of motion. JHEP- 0611:074, 2006, with N. Armesto and G. Milhano

- The wave function formalism- Dipole limit- Diagrammatic interpretation

II: Quark contribution to NLO-BK: some comments and numerical results.

Heribert’s talkY. Kovchegov and H. Weigert, hep-ph/0609090I. Balitsky, hep-ph/0609105

O(2)-corrections to JIMWLK from the classical equations of motion.

• B-JIMWLK can be derived using the ‘wave function formalism’ (projectile image) (Kovner, Lublinsky and Wiedemann)

• In this approach the kernel of the evolution equation depends on the classical Weizsaker-Williams (WW) fields generated by the projectile charge density.

• JIMWLK is obtained once the WW fields are taken at leading order in the projectile charge density.

• We calculate the kernel when the projectile WW-fields are taken at second order in the projectile charge density. Y

O(): JIMWLK

O(2): This work

• B-JIMWLK provides the HE evolution of a dense-dilute (target-projectile) system.

• In the Light-Cone Hamiltonian formalism and the LC gauge A+= 0

Fundamental d.o.f. and c-number (high target density)

WY

Y eikonal,independent

scattering

‘valence’ gluonsg

: Wilson lines in the adjoint representation

Averaging over target fields configurations:

The wave function formalism

Dense target

The boost opens up the phase space for new hard gluons emission

WY

YY

z

The evolution is managed boosting the projectile and leaving the target unevolved

Boosted wave function (valid only in the small projectile density limit!)

The wave function formalism

Lorentz invariance

The wave function formalism

The WW fields are determined by the classical Yang-Mills EOM

Leading order solution (independent gluon emission, ):

Expansion of the WW-fields Expansion of the evol. kernel

~O(1)

• Second order solution in the projectile density (coherent, non-linear emission):

• Left and right rotation operators:

• First correction to JIMWLK. Keeping terms up to O(g4):

Dipole limit. Corrections to BK?

Modified right and left rotation operators (ensure hermiticity of kernel):

N-expansion:

Diagrammatic interpretation

A mapping between the different terms in the kernels and diagrams can be done.(But this is a purely operational calculation!)

Diagrammatical interpretation of the kernel

LLL LRR

Final result:

The leading 1/N correction vanishes:

Using the symmetry of the wave function under the exchange of any number of dipoles (u,v) (r,p):

The corrections derived cannot be fully recasted in terms of dipoles degrees of freedom They include dipoles, quadrupoles, sextupoles, octupoles…

SUMMARY

• There are no corrections to BK coming from higher order solutions of the projectile WW classical fields in the low projectile density limit.

• The corrections to JIMWLK present a rather complicated color structure. They couple more terms to the Balitsky hierarchy

Some comments on the NLO quark contribution tosmall-x running coupling evolution

Y. Kovchegov, H. Weigert, hep-ph/0609090

I. Balitsky, hep-ph/0609105

x

y

z z1

z2

• The complete NLO quark contribution to small-x non-linear evolution has been recently calculated. It contains running coupling and non-running coupling contributions:

• LO BK equation:

x

z

y

rr1

r2

x

y

z z1

z2

x

y

z z1

z2

• K-W

• Balitsky

• The scheme dependence is due to the subtraction procedure:

• Running coupling BK:

NUMERICAL RESULTS

INITIAL CONDITION

RUNNING COUPLING

r

0.5

MV:

AN08:

Landau Pole avoided freezingThe coupling at large dipole size

AN08 init. cond

• The solutions obtained with Balitsky’s prescription propagate much slower than those obtained with KW’s.

• The solutions obtained with KW’s prescription are much closer to those obtained with the running coupling scale set at the size of the parent dipole than those obtained with Balitsky’s

MV init. cond

The asymptotic scaling function is very litle scheme-dependente

Geometric Scaling:

Old work:

LHC

KW:

Bal:

Phenom:

Fits to HERA data ??

WARNING: Just an example!!

Soyez’s talk

SATURATION SCALE

MV init cond

LHC

WARNING: Just an example!!

SATURATION SCALE

AN08 Init cond

KW:

Bal:

Phenom:

SUMMARY

(Not so) Bad News

• The new physical channels cannot be ignored: The solutions of BK equation with NLO quark contribution to the running coupling show a strong dependence on the scheme choice and on the initial condition.

Strong phenomenological consequences:

Good news:

• Two independent and coincident calculations of the complete NLO quark contribution small-x evolution are available.

• The subtraction piece is known and calculable (work in progress..).

• This would us with a good theoretical extrapolation from RHIC to the LHC