$O(α_s^2)$ Contributions to the longitudinal fragmentation function in $e^+\,e^-$ annihilation
P. J. Rijken, W. L. van Neerven
TL;DR
This paper computes the O(alpha_s^2) corrections to the longitudinal fragmentation function F_L(x,Q^2) in e+e- annihilation through the timelike coefficient functions C_L,i(z,Q^2/mu^2) for i = q,g, and presents the results in the MSbar and annihilation schemes. It provides explicit expressions for the non-singlet, singlet, and gluon contributions and discusses the scheme transformations. Numerical evaluation with alpha_s(M_Z) = 0.126 and n_f = 5 yields large corrections to F_L, ranging roughly from 45% to 67% across 0.01 < x < 0.9, and gives sigma_L/sigma_tot ≈ 0.054 at Q^2 = M_Z^2, in good agreement with OPAL's 0.057 ± 0.005. The work also addresses small-x logarithms, gluon dominance at low x, and the role of heavy quark and higher-twist effects for a complete phenomenology.
Abstract
We present the order $α_s^2$ contributions to the coefficient functions corresponding to the longitudinal fragmentation function $F_L(x,Q^2)$. A comparison with the leading order $α_s$ 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 $σ_L/σ_{\rm tot}$ amounts to 0.054. This number is very close to the most recent value obtained by the OPAL collaboration which obtained $0.057\pm 0.005$.