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$O(α_s^2)$ Contributions to the longitudinal fragmentation function in $e^+\,e^-$ annihilation

P. J. Rijken·W. L. van Neerven
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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 $0.057\pm 0.005$.

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P. J. Rijken, W. L. van Neerven. 1996-04-30. $O(α_s^2)$ Contributions to the longitudinal fragmentation function in $e^+\,e^-$ annihilation. https://doi.org/10.1016/0370-2693(96)00898-2

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