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Effective Hamiltonian for High energy QCD

Effective Hamiltonian for High energy QCD. Yoshitaka Hatta (RIKE N BNL). in collaboration with E. Iancu, L. McLerran, A. Stasto, D. Triantafyllopoulos. Ref. hep-ph/0504182. Beyond the B-JIMWLK equation.

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Effective Hamiltonian for High energy QCD

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  1. Effective Hamiltonian for High energy QCD Yoshitaka Hatta (RIKEN BNL) in collaboration with E. Iancu, L. McLerran, A. Stasto, D. Triantafyllopoulos Ref. hep-ph/0504182

  2. Beyond the B-JIMWLK equation Problems with the B-JIMWLK equation found (Iancu&Mueller, Mueller&Shoshi, Iancu&Triantafyllopoulos) A number of attempts to modify/improve the B-JIMWLK equation Connection to statistical physics problem (Munier&Peschanski, Iancu,Mueller&Munier) Dipole model in large Nc (Iancu&Triantafyllopoulos, Mueller,Shoshi&Wong, Levin&Lublinsky, Levin) Projectile wavefunction approach (Kovner&Lublinsky) Combination of the CGC formalism and the effective action approach

  3. When the density is low, a new type of diagrams (gluon splitting) becomes important. … An effect which “slows down” saturation … Can one derive the evolution equation which describes gluon splitting, or more generally, both the gluon recombination and splitting? Pomeron loops Gluon splitting B-JIMWLK: What’s missing? The B-JIMWLK equation assumes… A dilute projectile scatters off a dense target. … Gluon recombination

  4. Effective action approach Lipatov, Verlinde&Verlinde, Balitsky… … …

  5. Full Hamiltonian containing BOTH gluon saturation and fluctuation must be self-dual. From the effective action to the Hamiltonian(“quantization”) A rule of thumb Kovner & Lublinsky

  6. Construction of the effective action The total gauge field Coulombgauge light-cone gauge classical semi-hard soft

  7. High density regime: JIMWLK Hamiltonian Start with in the background Coulomb gauge, ( )

  8. Low density regime: BREM Hamiltonian Start with in the background light cone gauge, dual to JIMWLK c.f.Kovner & Lublinsky

  9. The general case: full effective action Start again with in the background light cone gauge, (Same as the BREM case) zero curvature in the ( ) plane

  10. Two-dimensional effective theory of Wilson lines is invariant under , i.e., is self-dual.

  11. Conclusions • The effective action approach is a powerful method to construct an evolution Hamiltonian at high-energy. • The full effective action is self-dual, and reduces to JIMWLK and BREM in appropriate limits. • Quantization of the full effective action.

  12. Hilbert space of the BREM Hamiltonian The commutation relation implies that the charges are noncommutative The observable is a product of the charges with specified ordering. The BREM Hamiltonian, together with the commutation relation, give unambiguous evolution equations.

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