Effective Hamiltonian for High energy QCD

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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
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Beyond the B-JIMWLK equation
Problems with the B-JIMWLK equation found
(IancuMueller,
MuellerShoshi, IancuTriantafyllopoulos)
A number of attempts to modify/improve the
B-JIMWLK equation
Connection to statistical physics problem
(MunierPeschanski,
Iancu,MuellerMunier)
Dipole model in large Nc
(IancuTriantafyllopoulos, Mueller,ShoshiWong,
LevinLublinsky, Levin)
Projectile wavefunction approach
(KovnerLublinsky)
Combination of the CGC formalism and the
effective action approach
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B-JIMWLK Whats missing?
The B-JIMWLK equation assumes
A dilute projectile scatters off a dense target.

Gluon recombination
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Effective action approach
Lipatov, VerlindeVerlinde, Balitsky


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From the effective action to the
Hamiltonian(quantization)
A rule of thumb
Kovner Lublinsky
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Construction of the effective action
The total gauge field
Coulomb gauge light-cone gauge
classical semi-hard soft
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High density regime JIMWLK Hamiltonian
Start with
in the background Coulomb gauge,
( )
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Low density regime BREM Hamiltonian
Start with
in the background light cone gauge,
dual to JIMWLK
c.f. Kovner Lublinsky
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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
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Two-dimensional effective theory of Wilson lines
is invariant under
, i.e., is self-dual.

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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.

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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.