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On the Infrared Sensitivity of the Longitudinal Cross Section in e+ e- Annihilation


We have calculated the contributions proportional to $\beta_0^n\alpha_s^{n+1}$ to the longitudinal fragmentation function in $e^+e^-$ annihilation to all orders of perturbation theory. We use this result to estimate higher-order perturbative corrections and nonperturbative corrections to the longitudinal cross section $\sigma_L$ and discuss the prospects of determining $\alpha_s$ from $\sigma_L$. The structure of infrared renormalons in the perturbative expansion suggests that the longitudinal cross section for hadron production with fixed momentum fraction $x$ receives nonperturbative contributions of order $1/(x^2Q^2)$, whereas the total cross section has a larger, $1/Q$ correction. This correction arises from very large longitudinal distances and is related to the behaviour of the Borel integral for the cross section with fixed $x$ at large values of the Borel parameter.

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