Next-to-Next-to-Leading Order Evolution of Non-Singlet Fragmentation Functions

Abstract

We have investigated the next-to-next-to-leading order (NNLO) corrections to inclusive hadron production in e^+e^- annihilation and the related parton fragmentation distributions, the `time-like' counterparts of the `space-like' deep-inelastic structure functions and parton densities. We have re-derived the corresponding second-order coefficient functions in massless perturbative QCD, which so far had been calculated only by one group. Moreover we present, for the first time, the third-order splitting functions governing the NNLO evolution of flavour non-singlet fragmentation distributions. These results have been obtained by two independent methods relating time-like quantities to calculations performed in deep-inelastic scattering. We briefly illustrate the numerical size of the NNLO corrections, and make a prediction for the difference of the yet unknown time-like and space-like splitting functions at the fourth order in the strong coupling constant.

Figures (2)

• Figure 1: Comparison of the coefficient function $c^{}_{T,\rm ns}$ for the (time-like) process $e^+e^- \rightarrow\, h + X$ with its counterpart $c^{}_{1,\rm ns}$ in (space-like) deep-inelastic scattering for $Q^2 \simeq M_Z^2\,$. Left plot: Mellin moments, right plot: convolutions (\ref{['eq:Fah']}) with a schematic input shape denoted by $f$.
• Figure 2: The differences $\delta\, P^{\,\rm ns} = P^{\,\rm ns}_{\sigma=1} - P^{\,\rm ns}_{\sigma=-1}$ between the time-like ($\sigma=1$) and space-like ($\sigma=-1$) non-singlet splitting functions at a 'low' scale characterized by the (order-independent) value $\alpha_{\rm s} = 0.2$ of the strong coupling constant. Left: moment-space comparison with the higher-order corrections in the space-like case. Right: convolutions with a schematic input shape.