Institut fur Theoretische Teilchenphysik, Universitat Karsruhe,

D-76128 Karsruhe, GERMANY;

E-mail: kotikov@particle.uni-karlsruhe.de

and

Bogoliubov Lab. of Theoretical Physics,

Joint Institute for Nuclear Research,

141980 Dubna, Moscow Region, RUSSIA

E-mail: kotikov@thsun1.jinr.ru

**Equations of evolution in supersymmetric gauge theories**

**Abstract:**

The Dokshitzer-Gribov-Lipatov-Altareli-Parisi (DGLAP) and Balitsky-Fadin-Kuraev-Lipatov
(BFKL) evolution equations in the *N* = 4 supersymmetric gauge theory
in the next-to-leading approximation are derived. 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 to obtain residues of anomalous
dimensions g 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 g 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.