Generalized threshold resummation for semi-inclusive e+e- annihilation

Abstract

Recently methods have been developed to extend the resummation of large-x double logarithms in inclusive deep-inelastic scattering (DIS) to terms not addressed by the soft-gluon exponentiation.Here we briefly outline our approach based on fixed-order results, the general large-x structure in dimensional regularization and the all-order factorization of mass singularities, which is directly applicable also to semi-inclusive e + e − annihilation (SIA). We then present some main results for the corresponding timelike splitting functions and transverse and longitudinal fragmentation functions. The close relation between DIS and SIA facilitates the determination of additional third-order results for the latter function which is fully known only at the next-to-leading order. Therefore all above quantities can be resummed at next-to-next-to-leading logarithmic accuracy.

Generalized threshold resummation in inclusive DIS and semi-inclusive electron-positron annihilation

“…Considerable progress has been made in the past seven years on the resummation of large-x (or, in Mellin space, large-N) threshold logarithms [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51] beyond those addressed by the soft-gluon exponentiation (SGE) [25][26][27][28][29]. This holds for sub-leading contributions, in terms of powers of (1−x) or 1/N for x → 1 or N → ∞, to quantities to which the SGE is applicable for the leading terms, as well as for which the SGE is not applicable at all.…”

“…This holds for sub-leading contributions, in terms of powers of (1−x) or 1/N for x → 1 or N → ∞, to quantities to which the SGE is applicable for the leading terms, as well as for which the SGE is not applicable at all. So far most of the explicit large-x results for higher-order splitting functions and coefficient functions have been obtained by studying physical evolution kernels [36][37][38][39][40][48][49][50] and the structure of unfactorized cross sections in dimensional regularization [32][33][34][35]51] (see Refs. [53,54] for an analogous small-x resummation in SIA).…”

“…We now address the final-state ('time-like') off-diagonal splitting functions, which have not been presented at NNLL accuracy before, for the NLL expressions see Ref. [33]. Since these quantities are closely related to their initial-state ('space-like') counterparts, the formulae for the two cases are very similar.…”

Leading large- x logarithms of the quark–gluon contributions to inclusive Higgs-boson and lepton-pair production

“…The presence of 2n − 1 terms in the sums (25) represents a crucial difference to W (n) H,gg in the N 0 soft-gluon limit, where only the n even values of ℓ occur [13], and inclusive DIS and semiinclusive e + e − annihilation (SIA), where the corresponding sums run from ℓ = 1 to ℓ = n [26,28]. In those cases, an N n LO calculation leads to a N n LL resummation with a large number of relations to spare.…”

“…Finally this table can be used to find and verify the all-order resummation formula for the quark-gluon coefficient functions, (28) which involves the same ingredients as its counterpart for DIS [24] but is slightly more complicated. The corresponding coefficient function for the Drell-Yan process can be obtained from (28) by C F → T f in the numerator of the prefactor and C A ↔ C F everywhere else, including the argument of the function B 0 . Expansion of Eq.…”