First correction to JIMWLK evolution from the classical EOMs N. Armesto 19th International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions (QM2006) Shanghai, November 15th 2006 Néstor Armesto Departamento de Física de Partículas and IGFAE, Universidade de Santiago de Compostela with Javier L. Albacete (Ohio State) and José Guilherme Milhano (IST Lisbon) Preprint hep-ph/ (to appear in JHEP) 1

Contents: N. Armesto First correction to JIMWLK evolution from the classical EOMs 1. Introduction. 2. The wave function formalism and JIMWLK. 3. First correction from the EOMs. 4. Relevance for phenomenology? 5. Summary. 2

1. Introduction: hdQCD Iancu, Venugopalan in QGP3, hep-ph/ N. Armesto First correction to JIMWLK evolution from the classical EOMs 3 ● QCD at high densities (energies or nuclear sizes): domain of large gluon densities, recombination at work --> saturation (CGC). ● Basic interest: unitarity in QCD and its phenomenological consequences, link with successful pre-QCD ideas (RFT, Glauber theory). ● Linear evolution equations: DGLAP or BFKL, modified to non-linear equations: JIMWLK, BK. Pioneering ideas in GLR, MQ:

1. Introduction: the MV setup N. Armesto First correction to JIMWLK evolution from the classical EOMs 4 ● McLerran-Venugopalan: valence quarks as sources for the classical dynamics of slow partons. For high enough energies (and/or sizes), BFKL evolution drives the system dense at any large scale: perturbative methods applicable. ● Effective FT: independence on the fast-slow separation scale gives a renormalization group equation (JIMWLK, Balitsky, Kovner-Lublinsky) ● Mean-field approximation: BK equation, well studied and understood.

2. The WF formalism: setup N. Armesto First correction to JIMWLK evolution from the classical EOMs 5 ● Evolution in the wave function of the projectile (Kovner, Wiedemann, '01; Kovner, Lublinsky, '05) : superposition of fast gluons in the + direction ● S ab : single gluon scattering matrix, eikonal. ● Scattering matrix of the projectile: average over target configurations ● Evolution: small shift in rapidity, ● b i : WW fields of the projectile, solutions of the Yang-Mills EOMs. (S ab =  ab for initial, low density)

2. WF -> JIMWLK: N. Armesto First correction to JIMWLK evolution from the classical EOMs 6 ● Evolution for the weight functional: dipole model ● EOMs (MV, Kovchegov '96, '97) : A + =0, ● To 1 st order in g  : JIMWLK

3. 1st corrections from EOMs: N. Armesto First correction to JIMWLK evolution from the classical EOMs 7 ● Limitations in JIMWLK: S's as c-numbers: high density target, and lowest order in g  : dilute projectile --> asymmetric configuration (Kovner '05; Triantafyllopoulos '05, Soyez '06). ● Both problems linked, exact solution unknown: dense-dilute duality, 2-->1vertex, sFKPP equation... ● This work: higher order in g  in the EOMs 

3. 1st corr.: dipole model N. Armesto First correction to JIMWLK evolution from the classical EOMs 8 ● Projectile made of dipoles: LLL LLR ● Classifying in terms of the number of dipoles: : no leading 1/N correction to JIMWLK -> BK. ● Symmetry in the dipoles of the projectile allows further simplifications: complicated expressions in terms of higher poles, no projectile recombination.

4. Relevance for phenomenology: scaling? N. Armesto First correction to JIMWLK evolution from the classical EOMs 9 ● Numerical studies needed to assess the practical relevance of these corrections for e.g. the LHC: experimental consequences? Analogy with reaction- diffusion processes: sFKPP. ● Geometric scaling is a most compelling evidence of saturation (Stasto et al '00; Rummukainen et al '03; NA et al ‘04; Gelis et al '06). It appears naturally in the JIMWLK context, understood both analytically and numerically from the BK equation. ● Correlations and pomeron loops (Mueller-Shoshi-Iancu-Munier '04) neglected in JIMWLK/BK, relevant for dilute-dilute, break geometric scaling -> diffusive scaling (sFKPP) (Hatta et al '06). LHC??

4. Relevance for phenomenology: correlations? N. Armesto First correction to JIMWLK evolution from the classical EOMs 10 ● Analogy of QCD evolution to a reaction- diffusion process: old RFT is directed percolation, similar to sFKPP (Iancu et al ’06; Bondarenko et al ’06). ● All this suggests searching the effects of fluctuations or PhT on correlations. ● Rapidity correlations may help: try to identify differences between CGC predictions (no phase transition without loops) (Kovchegov-Levin-McLerran- NA-Pajares '01, '06) and those including a phase transition e.g. phenomenological models like percolation, or realizations of dense-dense systems in hdQCD (Braun '00; ‘05). B F y2y2 y1y1

5. Summary: N. Armesto First correction to JIMWLK evolution from the classical EOMs 11 ● JIMWLK evolution is suitable for dilute-dense scattering: corrections needed for less asymmetric situations. ● New ingredients proposed to mend this situation: correlations in the source, pomeron loops in the evolution. ● We have analyzed the corrections to JIMWLK of order g  coming from higher orders in the solution of the classical equations of motion for the slow glue: * They give no correction to BK. * They cannot be expressed in terms of dipoles but give contributions to higher correlators (in Balitsky hierarchy). ● Studies on the numerical/phenomenological relevance of these new ingredients under development: correlations may offer an experimental testing ground at the LHC.

Backup I: complete expressions N. Armesto First correction to JIMWLK evolution from the classical EOMs 12

Backup II: y-correlations in the CGC N. Armesto First correction to JIMWLK evolution from the classical EOMs 13 NA, McLerran, Pajares ’06 to appear in NPA Uncorrelated diagram: Correlated diagram: ● For gluons the long range rapidity correlation increases with energy, centrality and decreases with |y 1 -y 2 |. Baryons -> smaller correlation. 1/g g BF y2y2 y1y1