Effective Hamiltonian Effective Hamiltonian for High energy QCDfor High energy QCD

Yoshitaka Hatta (RIKEN BNL)

in collaboration with E. Iancu, L. McLerran, A. Stasto, D. Triantafyllopoulos

Ref. hep-ph/0504182

Beyond the Beyond the B-JIMWLKB-JIMWLK equation 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)

Projectile wavefunction approach (Kovner&Lublinsky)

Dipole model in large Nc (Iancu&Triantafyllopoulos, Mueller,Shoshi&Wong, Levin&Lublinsky, Levin)

Combination of the CGC formalism and the effective action approach

B-JIMWLK: What’s missing?B-JIMWLK: What’s missing?

Gluon recombination

The B-JIMWLK equation assumes…

A dilute projectile scatters off a dense target.…

Gluon splitting

…

…

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

A A

ia

A A A A

…

)exp(P AdxigV

AVVAS ],[JIMWLK

A A A A

ia

A A

AWWAS ],[BREM

)exp(P AdxigW

…

EffectiveEffective action approach action approachLipatov, Verlinde&Verlinde, Balitsky…

From the effective action to the HamiltonianFrom the effective action to the Hamiltonian(“quantization”)(“quantization”)

],[JIMWLK VVH AWWAH ],[BREM

)exp(P

dxgW

A

iiA

2A rule of thumb

Kovner & Lublinsky

Full Hamiltonian containing BOTH gluon saturation and fluctuation must be self-dual.

],[],[ fullfull VWHWVH

Construction of the effective actionConstruction of the effective action

AaA

by

jijab

ax

i

xy

FDyxGFDS ))(,()(2

1eff

The total gauge field

classical semi-hard soft

b

AaAiSDaAAiS ];,,[exp];,[exp sourceYMeff

Coulomb gauge

light-cone gauge iA

A

x xAA

x x

A

High density regime: JIMWLK Hamiltonian

by

jijab

ax

i

xy

FDAyxGFDS )2)(;,()2(2

1eff

in the background Coulomb gauge,

x

y

Start with

)()()()()(1

ln);,(,, TxyTyxTT xVxyxVyxyx

bAyxiG

AF ii

A( )

iA

Low density regime: BREM HamiltonianLow density regime: BREM Hamiltonian

by

jijab

ax

i

xy

FDAyxGFDS )2)(;,()2(2

1eff

Start with

x x

A

in the background light cone gauge,

x

y

)()()()()(1

ln);,(,, TxyTyxTT xWxyxWyxyx

bAyxiG

ii AF

dual to JIMWLK

c.f. Kovner & Lublinsky

The general case: full effective actionThe general case: full effective action

by

jiijab

ax

i

xy

FDAAyxGFDS )2)(,;,()2(2

1eff

Start again with

in the background light cone gauge, VVg

iAF iii

x x

iA A

)()()()()(1

ln);,(,, TxyTyxTT xWxyxWyxyx

bAyxiG

zero curvature in the ( ) plane

0F

xx ,

(Same as the BREM case)

xx

W

W

V

V

Two-dimensional effective theory of Wilson lines

is invariant under , i.e., is self-dual.

effS WV effS

ConclusionsConclusions

• 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.

The commutation relation

implies that the charges are noncommutative

Hilbert space of the BREM Hamiltonian

The observable is a product of the charges with specified ordering.The BREM Hamiltonian, together with the commutation relation, give unambiguous evolution equations.

],[][ BREM OHO