On the Drell-Levy-Yan relation to O (αs2)

We study the validity of a relation by Drell, Levy and Yan (DLY) connecting the deep inelastic structure (DIS) functions and the single-particle fragmentation functions in e(+)e(-) annihilation which are defined in the space-like (q(2) < 0) and time-like (q(2) > 0) regions respectively. Here q denotes the momentum of the virtual photon exchanged in the deep inelastic scattering process or the annihilation process. An extension of the DLY-relation, which originally was only derived in the scaling parton model, to all orders in QCD leads to a connection between the two evolution kernels determining the q(2)-dependence of the DIS structure functions and the fragmentation functions, respectively. In relation to this we derive the transformation relations between the space- and time like splitting functions up to next-to-leading order (NLO) and the coefficient functions up to NNLO both for unpolarized and polarized scattering. It is shown that the evolution kernels describing the combined singlet evolution for the structure functions F-2(x, Q(2)), F-L(x, Q(2)) where Q(2) = \q(2)\ or F-2(x, Q(2)), partial derivative F-2 (x, Q(2))/partial derivative ln(Q(2)) and the corresponding fragmentation functions satisfy the DLY relation up to next-to-leading order. We also comment on a relation proposed by Gribov and Lipatov. (C) 2000 Elsevier Science B.V. All rights reserved.