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 ),

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We prove that the validity of the recently proposed dressed, asymptotic Bethe ansatz for the planar AdS/CFT system is indeed limited at weak coupling by operator wrapping effects. This is done by comparing the Bethe ansatz predictions for the four-loop anomalous dimension of finite-spin twist-two operators to BFKL constraints from high-energy scattering amplitudes in N = 4 gauge theory. We find disagreement, which means that the ansatz breaks down for length-two operators at four-loop order. Our method supplies precision tools for multiple all-loop tests of the veracity of any yet-to-be constructed set of exact spectral equations. Finally we present a conjecture for the exact four-loop anomalous dimension of the family of twist-two operators, which includes the Konishi field.

We present the results of two-loop calculations of the anomalous dimension matrix for the Wilson twist-2 operators in the N=4 Supersymmetric Yang-Mills theory for polarized and unpolarized cases. This matrix can be transformed to a triangle form by the same similarity transformation as in the leading order. The eigenvalues of the anomalous dimension matrix are expressed in terms of an universal function with its argument shifted by integer numbers. In the conclusion we discuss relations between the weak and strong coupling regimes in the framework of the AdS/CFT correspondence.Comment: LaTeX file, 10 pages, no figure

A new method of massive Feynman integrals calculation which is based on the rule of integration by parts is presented. This rule is expanded to the massive case. By applying this rule, we obtain a differential equation with respect to the mass for the initial diagram. The right-hand side of the equation contains simpler diagrams (i.e., containing only loops, not chains). This can be done by applying the procedure consecutively. These loops can be calculated either by the standard Feynman-parameter procedure or by a procedure which decreases the number of loops step-by-step. We demonstrate the capacities of this method for various complicated diagrams and make an attempt to analyze other possible massive Feynman diagrams calculations.

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We investigate the Eden-Staudacher equation for the anomalous dimension of the twist-2 operators at the large spin s in the N = 4 super-symmetric gauge theory. This equation is reduced to a set of linear algebraic equations with the kernel calculated analytically. We prove that in perturbation theory the anomalous dimension is a sum of products of the Euler functions ζ(k) having the property of the maximal transcendentality with the coefficients being integer numbers. The radius of convergency of the perturbation theory is found. It is shown, that at g = ∞ the kernel has an essential singularity. The analytic properties of the solution of the Eden-Staudacher equation are investigated. In particular for the case of the strong coupling constant the solution has an essential singularity on the second sheet of the variable j appearing in its Laplace transformation. Similar results are derived also for the Beisert-Eden-Staudacher equation which includes the contribution from the phase related to the crossing symmetry of the underlying S-matrix. We show, that its singular solution at large coupling constants reproduces the anomalous dimension predicted from the string side of the AdS/CFT correspondence.

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