We present the coefficient functions for deep-inelastic scattering (DIS) via the exchange of a scalar φ directly coupling only to gluons, such as the Higgs boson in the limit of a very heavy top quark and n f effectively massless light flavours, to the third order in perturbative QCD. The two-loop results are employed to construct the next-to-next-to-leading order physical evolution kernels for the system ( F 2 , F φ ) of flavour-singlet structure functions. The practical relevance of these kernels as an alternative to MS factorization is bedevilled by artificial double logarithms at small values of the scaling variable x, where the large top-mass limit ceases to be appropriate. However, they show an only single-logarithmic enhancement at large x. Conjecturing that this feature persists to the next order also in the present singlet case, the three-loop coefficient functions facilitate exact predictions (backed up by their particular colour structure) of the double-logarithmic contributions to the fourth-order singlet splitting functions, i.e., of the terms (1−x) a ln k (1−x) with k = 4, 5, 6 and k = 3, 4, 5, respectively, for the off-diagonal and diagonal quantities to all powers a in (1−x).

We present all-order results for the highest three large-x logarithms of the splitting functions P qg and P gq and of the coefficient functions C φ,q , C 2,g and C L,g for structure functions in Higgs-and gauge-boson exchange DIS in massless perturbative QCD. The corresponding coefficients have been derived by studying the unfactorized partonic structure functions in dimensional regularization independently in terms of their iterative structure and in terms of the constraints imposed by the functional forms of the real-and virtual-emission contributions together with their Kinoshita-LeeNauenberg cancellations required by the mass-factorization theorem. The numerical resummation corrections are small for the splitting functions, but partly very large for the coefficient functions. The highest two (three for C L,g ) logarithms can be resummed in a closed form in terms of new special functions recently introduced in the context of the resummation of the leading logarithms.

We discuss the large-x behaviour of the splitting functions Pqg and Pgq and of flavour-singlet coefficient functions, such as the gluon contributions C2,g and CL,g to the structure functions F2,L, in massless perturbative QCD. These quantities are suppressed by one or two powers of (1−x) with respect to the (1−x) −1 terms which are the subject of the well-known threshold exponentiation. We show that the double-logarithmic contributions to Pqg, Pgq and CL at order α 4 s can be predicted from known third-order results and present, as a first step towards a full all-order generalization, the leading-logarithmic large-x behaviour of Pqg, Pgq and C2,g at all orders in αs.

We discuss the large-x behaviour of the splitting functions P qg and P gq and of flavour-singlet coefficient functions, such as the gluon contributions C 2,g and C L,g to the structure functions F 2,L , in massless perturbative QCD. These quantities are suppressed by one or two powers of (1−x) with respect to the (1−x) −1 terms which are the subject of the well-known threshold exponentiation. We show that the double-logarithmic contributions to P qg , P gq and C L at order α 4 s can be predicted from known third-order results.

Over the past few years considerable progress has been made on the resummation of doublelogarithmically enhanced threshold (large-x) and high-energy (small-x) higher-order contributions to the splitting functions for parton and fragmentation distributions and to the coefficient functions for inclusive deep-inelastic scattering and semi-inclusive e + e − annihilation. We present an overview of the methods which allow, in many cases, to derive the coefficients of the highest three logarithms at all orders in the strong coupling α s from next-to-next-to-leading order results in massless perturbative QCD. Some representative analytical and numerical results are shown, and the present limitations of these resummations are discussed.