1. Color dipoles from Bremsstrahlung in high energy evolution Yoshitaka Hatta (RIKEN BNL) with E. Iancu, L. McLerran, A. Stasto, Nucl. Phys. A762 (2005) 272 DIS06, Tsukuba

2. Outline  BK-JIMWLK and beyond  Effective action approach  Bremsstrahlung Hamiltonian  Generalized dipole model  Evolution equations in the dilute regime  Summary

3. Color Glass Condensate formalism McLerran & Venugopalan ‘93 A high energy hadron (nucleus) is replaced with classical color charges distributed according to a weight function. Hadron-CGC scattering amplitudes are first calculated with fixed background charges, then averaged over. satisfies the JIMWLK equation small-x evolution equation for : rapidity

4. B-JIMWLK equation The dipole—CGC scattering amplitude An infinite hierarchy of coupled nonlinear equations Realizes the black disc limit at a fixed impact parameter

5. Beyond BK-JIMWLK equation Balitsky-Kovchegov-JIMWLK equation: A sort of mean field approximation, does not take into account gluon number fluctuations developed in the dilute regime. E. Iancu, plenary talk G. Soyez, previous talk gluon recombination gluon splitting (Bremsstrahlung)

6. Effective action approach The total gauge field hard semihard soft Functionally integrate out intermediate rapidity gluons in the presence of two background fields (target & projectile). Y.H., Iancu, McLerran, Stasto, Triantafyllopoulos Nucl.Phys.A764

7. JIMWLK and cousins Gluon splittingPomeron loop Gluon recombination Wilson line Effective action Hamiltonian

8. The Bremsstrahlung Hamiltonian acts on non-commutative color charges Kovner & Lublinsky ‘05

9. Color dipole model Mueller, ‘94 dipole = heavy quark—antiquark pair (“onium”) large boost splitting probability dipole kernel

10. Onium as a color glass Iancu & Mueller, ‘03 dipole “creation operator” Each dipole emits two gluons Master equation for the probability distribution loss gain

11. Dressed dipole creation operator Generalized dipole model Weight function: Each dipole emits arbitrary number of gluons consistent with the master equation satisfied by Evolution equation

12. Evolution equation for the dipole densities Dipole number operator Dipole number density Dipole pair density ~ fluctuation term “seed” of

13. Application to scattering problem two gluon exchange with two dipoles four gluon exchange with a single dipole eventually grows like Pomeron squared, dominant at high energy See, also, Marquet et al., ‘05. Two dipoles scatter off an onium

14. Summary  We have worked out the dipole sector of the BREM Hamiltonian (dual to JIMWLK).  We constructed the weight function and dipole densities, and derived their evolution equation by acting with the Hamiltonian.  Bremsstrahlung effects beyond two gluon exchange per dipole are subleading at high energy.