On the next-to-next-to-leading order evolution of flavour-singlet fragmentation functions

Abstract

We present the third-order contributions to the quark-gluon and gluon-quark timelike splitting functions for the evolution of fragmentation functions in perturbative QCD. These quantities have been derived by studying physical evolution kernels for photon-and Higgs-exchange structure functions in deep-inelastic scattering and their counterparts in semi-inclusive annihilation, together with constraints from the momentum sum rule and the supersymmetric limit. For this purpose we have also calculated the second-order coefficient functions for one-hadron inclusive Higgs decay in the heavy-top limit. A numerically tolerable uncertainty remains for the quark-gluon splitting function, which does not affect the endpoint logarithms for small and large momentum fractions. We briefly discuss these limits and illustrate the numerical impact of the third-order corrections. Compact and accurate parametrizations are provided for all third-order timelike splitting functions.

Timelike structure functions and hadron multiplicities

“…In the following after reviewing the history which lead to the latest important improvements in "timelike" QCD of Refs. [12,9] and sketching our formalism, we present, as anticipated, the result of a slightly different global fit than in [8] where also the strong coupling contstant has been added among the free parameters confirming the good quality of the fit.…”

“…[33,34] and finally in Ref. [12]. Here we mention only that the analytic continuation x → 1/x becomes subdle (see Ref.…”

“…[25]) for the logarithmic terms singular in x → 1 for which 5) and that additional constaints (see Ref. [12] and referencies therein) from the momentum sum rules, the supersymmetric limit (C A = C F = n f ) and from the relations found in Ref. [22] are needed to close the problem in a satisfactory way.…”

“…[12,9] represent the beginning of the happy end of a quite long story: In the 1972 it was realized in Ref. [13] that at the lowest order the "timelike" splitting functions (occuring in Semi-Inclusive electron positron Annihilation (SIA)) are the same as their "spacelike" counterparts (occuring in Deep Inelastic Scattering (DIS)).…”

Heavy quark fragmenting jet functions

“…The non-diagonal entries of the singlet matrix, P (2) gq and P (2) qg , were recently determined in ref. [39]. Notice that the splitting functions P (0) gg and P (0) gq have a singularity at z → 0, associated with soft gluon emissions.…”