On the Drell-Levy-Yan Relation to $O(α_s^2)$

TL;DR

This work investigates the Drell–Levy–Yan crossing between spacelike DIS structure functions and timelike fragmentation functions within perturbative QCD. By formulating scheme-invariant evolution kernels and deriving transformation rules for spacelike and timelike splitting and coefficient functions up to NNLO, the authors show that physical kernels for key observable pairs are invariant under DLY up to NLO, while individual coefficient and splitting functions generally violate DLY due to scheme dependence. The analysis clarifies how Gribov–Lipatov relations behave across orders, finding LO validity but NLO violation for many quantities, and outlines the requirements (3-loop inputs) to extend DLY invariance tests to NNLO. The results provide a rigorous framework for cross-channel consistency checks and sharpen understanding of how perturbative QCD encodes spacelike-timelike relations in a scheme-invariant manner. Overall, the paper advances the theoretical foundation for connecting DIS and e+e− fragmentation through perturbative evolution, with implications for precision QCD phenomenology.

Abstract

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 spacelike (q^2<0) and timelike (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 $F_2(x,Q^2), \partial F_2(x,Q^2)/\partial \ln(Q^2)$ 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.