We have calculated the complete matrix of three-loop helicity-difference (`polarized) splitting functions Delta P_ik^(2), i,k = q,g, in massless perturbative QCD. In this note we briefly discuss some properties of the polarized splitting functions an
d our non-standard determination of the hitherto missing lower-row quantities Delta P_gq^(2) and Delta P_gg^(2). The resulting next-to-next-to-leading order (NNLO) corrections to the evolution of polarized parton distributions are illustrated and found to be small even at rather large values of the strong coupling constant alpha_s.
Over the past few years considerable progress has been made on the resummation of double-logarithmically enhanced threshold (large-x) and high-energy (small-x) higher-order contributions to the splitting functions for parton and fragmentation distrib
utions 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 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.
We study the splitting functions for the evolution of fragmentation distributions and the coefficient functions for single-hadron production in semi-inclusive electron-positron annihilation in massless perturbative QCD for small values of the momentu
m fraction and scaling variable x, where their fixed-order approximations are completely destabilized by huge double logarithms of the form alpha_s^n 1/x ln^(2n-a) x. Complete analytic all-order expressions in Mellin-N space are presented for the resummation of these terms at the next-to-next-to-leading logarithmic accuracy. The poles for the first moments, related to the evolution of hadron multiplicities, and the small-x instabilities of the next-to-leading order splitting and coefficient functions are removed by this resummation, which leads to an oscillatory small-x behaviour and functions that can be used at N=1 and down to extremely small values of x. First steps are presented towards extending these results to the higher accuracy required for an all-x combination with the state-of-the-art next-to-next-to-leading order large-x results.
We have derived the coefficients of the highest three 1/x-enhanced small-x logarithms of all timelike splitting functions and the coefficient functions for the transverse fragmentation function in one-particle inclusive e^+e^- annihilation at (in pri
nciple) all orders in massless perturbative QCD. For the longitudinal fragmentation function we present the respective two highest contributions. These results have been obtained from KLN-related decompositions of the unfactorized fragmentation functions in dimensional regularization and their structure imposed by the mass-factorization theorem. The resummation is found to completely remove the huge small-x spikes present in the fixed-order results for all quantities above, allowing for stable results down to very small values of the momentum fraction and scaling variable x. Our calculations can be extended to (at least) the corresponding as^n ln^(2n-l) x contributions to the above quantities and their counterparts in deep-inelastic scattering.
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 a
re suppressed by one or two powers of 1-x with respect to the 1/(1-x) 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 alpha_s^4 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 P_qg, P_gq and C_2,g at all orders in alpha_s.
We analyze the iterative structure of unfactorized partonic structure functions in the large-x limit, and derive all-order expressions for the leading-logarithmic off-diagonal splitting functions P_gq and P_qg and the corresponding coefficient functi
ons C_phi,q and C_2,g in Higgs- and gauge-boson exchange deep-inelastic scattering. The splitting functions are given in terms of a new function not encountered in perturbative QCD so far, and vanish maximally in the supersymmetric limit C_A - C_F to 0. The coefficient functions do not vanish in this limit, and are given by simple expressions in terms of the above new function and the well-known leading-logarithmic threshold exponential. Our results also apply to the evolution of fragmentation functions and semi-inclusive e^+ e^- annihilation.
We report on the first calculation of the structure function g_1 in polarised deep-inelastic scattering to the third order in massless perturbative QCD. The calculation follows the dispersive approach already used for the corresponding unpolarised ca
ses of F_2,L, but additionally involves higher tensor integrals and the Dirac matrix gamma_5 in D unequal 4 dimensions. Our results confirm all known two-loop expressions including the coefficient functions of Zijlstra and van Neerven not independently verified before. At three loops we extract the helicity-difference next-to-next-to-leading order (NNLO) quark-quark and gluon-quark splitting functions Delta P_qq and Delta P_qg. The results exhibit interesting features concerning sum rules and the momentum-fraction limits x to 1 and x to 0.
