Solution of the off-forward leading logarithmic evolution equation based on the Gegenbauer moments inversion

Abstract

Using the conformal invariance the leading-log evolution of the off-forward structure function is reduced to the forward evolution described by the conventional Dokshitzer-Gribov-Lipatov-Altarelli-Parisi ͑DGLAP͒ equation. The method relies on the fact that the anomalous dimensions of the Gegenbauer moments of the off-forward distribution are independent of the asymmetry, or skewedness, parameter and equal to the DGLAP ones. The integral kernels relating the forward and off-forward functions to the same Mellin and Gegenbauer moments are presented for an arbitrary asymmetry value. ͓S0556-2821͑99͒07321-X͔ PACS number͑s͒: 11.15.BtThe logarithmic evolution of the off-forward structure function is described by the generalization of the DokshitzerGribov-Lipatov-Altarelli-Parisi ͑DGLAP͒ equation for the parton distribution. However the explicit dependence on the longitudinal momentum transfer makes the splitting functions much more complicated since they include different pieces in different kinematics regions ͓1͔.On the other hand, it is well known that there is a set of twist-two operators whose leading-log evolution is exactly diagonal due to conformal symmetry remaining to be valid at the one-loop level ͑see e.g., ͓2,3͔͒. 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 nonsinglet offforward 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. This property allows, in principle, to reduce the nondiagonal leading-log evolution to the diagonal one: if one could replace in some way the powers of x in the solution of the conventional DG-LAP equation by the Gegenbauer polynomials the result returns a solution of the off-forward equation. However the Gegenbauer polynomials do not form a complete set on the full interval, where the structure function is defined, therefore this substitution cannot be done straightforward. It is the aim of the present paper to construct explicitly the transformation relating the off-forward and forward evolution.

Reconstruction of non-forward evolution kernels

“…Several methods have been offered so far to solve the off-forward evolution equation: numerical integration [8], expansion of OFPD w.r.t. an appropriate basis of polynomials [9,10], mapping to the forward case 2 [12,13] and solution in the configuration space [14,15]. The last three methods are based on the well-known fact that operators with definite conformal spin do not mix in the one-loop approximation.…”

Scale dependence of QCD string operators

“…However, it is clear that before a non-perturbative information can be reliably extracted from experimental data, all perturbative aspects, such as QCD evolution, have to be understood. So far, the main effort has been devoted to studies of evolution equations for skewed parton distributions in the momentum representation [1,2,5,12,13,14,15,16,17,18]. This is certainly the most natural choice in the forward, deep inelastic scattering limit, but, as observed recently [19], as soon as there is a longitudinal momentum transfer between initial and final hadron states, the corresponding amplitude can be as conveniently represented in terms of momentum-as coordinate-space skewed parton distributions.…”

Dual parametrization of generalized parton distributions in two equivalent representations

“…4 The integral kernel K(x, η|y) contains further support restrictions and its explicit expression was found by means of the dispersive approach [30,31]. In loose words, it consists in presenting the conformal PW expansion (2.11) as a discontinuity of a certain formal series of generalized functions.…”

“…Another possibility of summing up the conformal PW expansion (2.11) is to employ a map of a GPD to the forward-like function F(y, η, t), 3 by requiring that the conformal moments (2.6) of the GPD are obtained from the Mellin transform in the auxiliary variable y [30],…”

“…• The approach [29][30][31] based on the Shuvaev-Noritzsch transform. In this approach one is dealing with the so-called forward-like functions of an auxiliary momentumfraction-type variable.…”