Duality Symmetry in High Energy Scattering

Alex Prygarin

University of Hamburg

Wed, Mar. 10th 2010, 14:15

Salle Claude Itzykson, Bât. 774, Orme des Merisiers

The hadron scattering amplitude at high energies $\sqrt{s}$ in the Leading Logarithmic Approximation (LLA) is obtained by summing contributions of the order $(g^2 \log s)^n$, where each power of the (square of the ) coupling constant is accompanied by the same power of the logarithm of the energy. In this limit only gluons contribute, and the QCD amplitude is the same as one calculated in $\mathcal{N}=4$ super Yang-Mills theory. In the LLA the t-channel gluons reggeize an the BFKL Pomeron is a composite state of two reggeized gluons in the color singlet state, which is described by the Schrodinger-like BFKL equation for the evolution in the rapidity (imaginary time) with the Hamiltonian dependent on only transverse momenta. Lipatov found that beyond the conformal symmetry, in the multicolor limit the BFKL Hamiltonian possesses also the {\it duality symmetry}, which leaves the form of the BFKL Hamiltonian invariant if one replaces the transverse momentum by its conjugate coordinate. The duality symmetry is related to the integrability of the BFKL and the BKP equations. The high energy evolution in the large $N_c$ can be alternatively described by a system of color(less) dipoles instead of gluons. We show that the duality symmetry can be viewed as a symmetry between the reggeized gluon formulation and the dipole picture. We extend the analysis to the non-linear case of the BK equation and give the duality symmetry a physical interpretation as a symmetry under rotation of the BFKL Kernel in the transverse space from the t-channel (reggeized gluons) to the s-channel (color dipoles) and back. In this picture, the upper momentum of the reggeized gluon is dual to the initial dipole size, while the transferred momentum is dual to the difference in the sizes of a non-diagonal dipole, in contrast to the conventional wisdom, which relates the transferred momentum to the impact parameter.

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