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We find one-loop correction to the integral kernel of the BFKL equation for the total cross section of the high energy scattering in QCD and calculate the next-to-leading contribution to anomalous dimensions of twist-2 operators near j = 1.The BFKL equation is very important for the theory of the Regge processes at high energies √ s in the perturbative QCD [1]. In particular, it can be used together with the DGLAP evolution equation [2] for the description of structure functions for the deep inelastic ep scattering at small values of the Bjorken variable x = −q 2 /(2pq), where p and q are the momenta of the proton and the virtul photon correspondingly. But up to recent years the integral kernel for the BFKL equation was known only in the leading logarithmic approximation (LLA), which did not allow one to find its region of applicapability, including the scale in transverse momenta fixing the argument of the QCD coupling constant α(ck 2 ⊥ ) and the longitudinal scale √ s 0 for the minimal initial energy. In this paper we calculate the QCD radiative corrections to this kernel.In LLA the gluon is reggeized and the Pomeron is a compound state of two reggeized gluons. One can neglect multi-gluon components of the Pomeron wave function also in the next-to-leading logarithmic approximation (NLLA) and express the total cross-section σ(s) for the high energy scattering of colourless particles A, B in terms of their impact factors Φ i ( − → q i ) and the t-channel partial wave G ω ( − → q , − → q ′ ) for the reggeized gluon scattering at t = 0:Here − → q and − → q ′ are transverse momenta of gluons with the virtualities − − → q 2 ≡ −q 2 and − − → q ′ 2 ≡ −q ′2 correspondingly, s = 2p A p B is the squared invariant mass of the colliding particles with momenta p A and p B . Note, that the dependence of the Regge factor from 1

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We derive the DGLAP and BFKL evolution equations in the N = 4 supersymmetric gauge theory in the next-to-leading approximation. The eigenvalue of the BFKL kernel in this model turns out to be an analytic function of the conformal spin |n|. Its analytic continuation to negative |n| in the leading logarithmic approximation allows us to obtain residues of anomalous dimensions γ of twist-2 operators in the non-physical points j = 0, −1, ... from the BFKL equation in an agreement with their direct calculation from the DGLAP equation. Moreover, in the multi-color limit of the N = 4 model the BFKL and DGLAP dynamics in the leading logarithmic approximation is integrable for an arbitrary number of particles. In the next-to-leading approximation the holomorphic separability of the Pomeron hamiltonian is violated, but the corresponding Bethe-Salpeter kernel has the property of a hermitian separability. The main singularities of anomalous dimensions γ at j = −r obtained from the BFKL and DGLAP equations in the next-to-leading approximation coincide but our accuracy is not enough to verify an agreement for residues of subleading poles.PACS: 12.38.BxIt is well known, that equation (2) is simplified after its Mellin transformation to the Lorentz spin j representation: d d ln Q 2 f a (j, Q 2 ) = b γ ab (j)f b (j, Q 2 ),

We review the parton model and the Regge approach to the QCD description of the deep-inelastic ep scattering at the small Bjorken variable x and demonstrate their relation with the DGLAP and BFKL evolution equations. It is shown, that in the leading logarithmic approximation the gluon is reggeized and the pomeron is a compound state of two reggeized gluons. The conformal invariance of the BFKL pomeron in the impact parameter space is used to investigate the scattering amplitudes at high energies and fixed momentum transfers. The remarkable properties of the Schrödinger equation for compound states of an arbitrary number of reggeized gluons in the multi-colour QCD are reviewed. The gauge-invariant effective action describing the gluon-Reggeon interactions is constructed. The known next-to-leading corrections to the QCD pomeron are discussed.