Narrow Results

We analyze the structure of the eigenvalue of the color-singlet Balitsky–Fadin–Kuraev–Lipatov (BFKL) equation in N=4 SYM in terms of the meromorphic functions obtained by the analytic continuation of harmonic sums from positive even integer values of the argument to the complex plane. The meromorphic functions we discuss have pole singularities at negative integers and take finite values at all other points. We derive the reflection identities for harmonic sums at weight five decomposing a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or non-negative values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible set of bilinear reflection identities at weight five which presents the main result of the paper. We show how the reflection identities can be used to restore the functional form of the next-to-leading eigenvalue of the color-singlet BFKL equation in N=4SYM, i.e. we argue that it is possible to restore the full functional form on the entire complex plane provided one has information how the function looks like on just two lines on the complex plane. Finally we discuss how nonlinear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums.

A phenomenological study of the two-photon reaction within the discrete BFKL approach is performed. We estimate the BFKL amplitude in terms of a discrete set of eigenfunctions and use the HERA $F_2^p$ data to constrain the free parameters of the model. As a consequence, we derive a parameter free prediction of the energy dependence of the total $\gamma \gamma$ cross-section and the Bjorken-$x$ dependence of the photon structure function. We demonstrate that the LEP data is quite well described by the discrete BFKL approach and that experimental data from the future $e^{+}{e}^{-}$ colliders can be useful to constrain the description of the QCD dynamics at high energies.

The solution of the next-to-leading order BFKL equation is obtained by constructing the eigenfunctions as a perturbative expansion of the conforma leading order conformal eigenfunctions.

We remind the Gribov approach to the hadron high energy scattering. It was based on the effective field theory for the Pomeron interactions. In QCD and in gravity the gluons and gravitons are reggeized and therefore at high energies one should reformulate these theories in terms of the reggeons and their interactions. We review the basic ideas of the BFKL approach in QCD and in supersymmetric models and generalize them in the framework of the gauge-invariant effective theory for the reggeized gluon interactions. The similar generally covariant action for the reggeized graviton interactions is formulated in terms of the effective currents satisfying a non-linear evolution equation.