$O({\alpha }_s^2)$ Contributions to the longitudinal fragmentation function in $e^{+}{e}^{-}$ annihilation

Abstract

We present the order ${\alpha }_s^2$ contributions to the coefficient functions corresponding to the longitudinal fragmentation function $F_L(x,Q^2)$. A comparison with the leading order ${\alpha }_s$ result for $F_L(x,Q^2)$ shows that the corrections are large and vary from 44\% to 67\% in the region $0.01<x<0.9$ at $Q^2=M_Z^2$. Our calculations also reveal that the ratio of the longitudinal and total cross section ${\sigma }_L/{\sigma }_{\mathrm{tot}}$ amounts to 0.054. This number is very close to the most recent value obtained by the OPAL collaboration which obtained $0.057\pm 0.005$.

Comments: 11 pages, uses 'mcite', and 'amsfonts', dvi-file available at http://rulgm4.leidenuniv.nl/preprints/longfrag.dvi