Solution of the Off-forward Leading Logarithmic Evolution Equation Based on the Gegenbauer Moments Inversion

Using the conformal invariance the leading-log evolution of the off-forward structure function is reduced to the forward evolution described by the conventional DGLAP equation. The method relies on the fact that the anomalous dimensions of the Gegenbauer moments of the off-forward distribution are independent on the asymmetry, or skewedness, parameter and equal to the DGLAP ones. The integral kernels relating the forward and off-forward functions with the same Mellin and Gegenbauer moments are presented for arbitrary asymmetry value. 1. The logarithmic evolution of the off-forward structure function is described by the generalization of DGLAP equation for the parton distribution. However the explicit dependence on the longitudinal momentum transfer makes the splitting functions to be much more complicated since they include the pieces different in different kinematics regions [1]. From the other hand it is well known that there is the set of twist-two operators, which leading-log evolution is exactly diagonal due to conformal symmetry remaining to be valid at the one loop level. The off-forward matrix elements of these operators are the Gegenbauer polynomials, they turn into the simple powers of Bjorken x in the forward kinematics. The Gegenbauer moments of the non-singlet off-forward function have a simple multiplicative evolution like the forward ones. There is a mixture of quark and gluon channels for the singlet function but only between the moments of the same order, the anomalous dimensions being equal for the forward and off-forward cases.