Higher-order threshold resummation for semi-inclusive e+e− annihilation

The complete soft-enhanced and virtual-gluon contributions are derived for the quark coefficient functions in semi-inclusive e+e− annihilation to the third order in massless perturbative QCD. These terms enable us to extend the soft-gluon resummation for the fragmentation functions by two orders to the next-to-next-to-next-to-leading logarithmic (N3LL) accuracy. The resummation exponent is found to be the same as for the structure functions in inclusive deep-inelastic scattering. This finding, together with known results on the higher-order quark form factor, facilitates the determination of all soft and virtual contributions of the fourth-order difference of the coefficient functions for these two processes. Unlike the previous (N2LL) order in the exponentiation, the numerical effect of the N3LL contributions turns out to be negligible at LEP energies.