Effective action for high energy QCD Y.H., Iancu, McLerran, Stasto, Triantafyllopoulos : NPA 764 (2006) Y.H. : NPA 768 (2006); NPA 781 (2007) Yoshitaka Hatta (Saclay)

Regge limit of QCD What is the asymptotic behavior of the S-matrix in hadronic collision?

The BFKL Pomeron … Rapidity ordering

Ways to go — beyond BFKL LLA BFKL predicts indefinite growth of the gluon number. Next to leading-log approximation (NLLA) Many reggeized gluon exchange à la BKP Nonlinear equations from gluon saturation Must be tamed (How?)

The BFKL Pomeron … free propagator Rapidity ordering

Evolution in the presence of saturation dressed propagator Hadrons are characterized by color fields of order ~ Need to sum all orders in, or, or both. Calculating the effective action is a systematic, and convenient way to do this.

Outline Introduction Eikonal approximation in QED/GR/QCD Source terms in QCD JIMWLK/BREM/diamond effective actions

The eikonal approximation Scattering amplitude in the impact parameter space Multiple scattering exponentiates

QED eikonal S-matrix boost Aberbanel & Itzykson, `69

Scattering at Planckian energies Newtonian gravity Boost ~ `t Hooft, `87 null geodesics Gravitational shock wave Aichelburg & Sexl, `71

effective action Verlinde & Verlinde `93 cross singularity of Wilson lines Korchemsky `94 AdS/CFT Janik & Peschanski `99 Rho, Sin & Zahed `99 stochastic vacuum model Shoshi, Steffen Dosch & Pirner `02 complete two-loop pQCD Balitsky & Babansky `03 semiclassical QCD eikonal scattering Nachtmann `91 Many approaches

Path integral of spin operators “…path integrals suffer most grievously from a serious defect. They do not permit a discussion of spin operators…in a simple and lucid way. Nevertheless, spin is a vital part of real quantum mechanical systems.” R. Feynman Spin-magnetic interaction (1+1)-D Wess-Zumino term a.k.a., Polyakov’s spin factor, Berry’s phase, Kirillov form, symplectic form…

“dilute” “dense” Source terms in QCD: Double functional integral representation of the S-matrix Note : For a color singlet projectile, complete Wilson loops by attaching transverse Wilson lines at infinity. Y.H., NPA781 Saddle point equation

Use the residual gauge freedom to relegate interactions to the initial or final time. Kovchegov & Mueller `98 Iancu, Leonidov & McLerran `00 Belitsky, Ji & Yuan `02 time-like transverse Y.H., NPA781 Source terms in the light-cone gauge

Scattering in the light-cone gauge Mueller, in Cargese lectures

An example of unitary S-matrix: A lesson from two-dimensional integrable models Sum of QED-type diagrams SU(N) ``Thirring model” with the coupling elastic intermediate states only

The ‘ ’ approximation Lipatov, `88 Keep all powers of Exact S-matrix of 2D SU(N) models Berg, Karowski, Weisz, Kurak, `78

High energy effective action Effective action : Sum of all vertices in a given rapidity interval computed in the leading log approximation

Total gauge field Construction of the effective action hard semihard soft

~~ The case of asymmetric scattering dilutedense

The JIMWLK effective action where,

The eikonal phase allows one to replace Evolution equation for the S-matrix From the effective action to the evolution kernel Virtual part—functional derivative of the real part Weigert, `00 Recovered from the ordering of

The Bremsstrahlung Hamiltonian The eikonal phase allows one to replace Kovner & Lublinsky `05 “dual” to JIMWLK Return to Note that the color charges depend on time.

Beyond the BK-JIMWLK : Pomeron loops Gluon merging merging + splitting = Pomeron loops Saturation effect in the right mover, (fluctuation effect in the left mover) missing in BK-JIMWLK Saturation effect in the left mover, included in BK-JIMWLK Gluon splitting

The role of the gluon splitting : Gluon number fluctuation boost Salam, `96 Start with a single color dipole

Strong current induces strong semihard field Effective action with two strong sources ~ ~ ~ Effective action = sum of all tree diagrams with cubic and quartic couplings ! ….

The diamond effective action Symmetric in V and W. The ``leading” term that contains the JIMWLK action. =

High energy evolution with Pomeron loops Necessarily a two—source problem Need to construct symmetric effective actions Little is known beyond the ``leading order ” Balitsky computed Approach with simpler models E.g., theory coupled with a strong current Gelis & Venugopalan ‘ 06 Conclusions