ar X iv : h ep - p h / 99 05 51 7 v 1 2 7 M ay 1 99 9 1 Initial conditions and evolution of off - diagonal distributions

We briefly discuss the problem of specifying initial conditions for evolution of off-diagonal (skewed) parton distributions. We present numerical results to show that evolution rapidly washes out differences of input. Off-diagonal (or skewed) parton distributions provide important information about the nonper-turbartive structure of the nucleon [ 1]. They can in principle be measured in such processes as deeply virtual Compton scattering, diffractive vector meson production or diffractive high-p T jet production. Just as for the ordinary (diagonal) parton distributions, the QCD evolution equations [ 2] play an important role in the determination of skewed parton distributions. As usual input distributions are required. However, this is more complicated than in the diagonal case since the skewed parton distributions depend on additional variable – the asymmetry parameter ξ ∼ p − p ′. We use the symmetric formulation of Ji [ 1] in which the skewed parton distributions are given by functions H(x, ξ) with support −1 ≤ (x, ξ) ≤ 1, see [ 3] for more details. Here we discuss various ways to specify the initial conditions and show how the differences in the input distributions disappear on evolution. In particular we illustrate the conclusion of [ 3] that the skewed distributions H(x, ξ), at small x and ξ, are fixed by the conventional diagonal partons. The distributions H(x, ξ) have to fulfil several conditions with respect to the variable ξ. First, the time-reversal invariance and hermiticity impose the condition [ 1] H(x, ξ) = H(x, −ξ). (1) Thus H(x, ξ) is an even function of ξ. The second condition states that in the limit ξ = 0 we recover the ordinary diagonal parton distributions H(x, 0) = H diag (x) .