Higher order QCD corrections to the transverse and longitudinal fragmentation functions in electron-positron annihilation

TL;DR

This paper advances perturbative QCD predictions for fragmentation in $e^+e^-$ annihilation by computing order $α_s^2$ corrections to the fragmentation coefficient functions for the longitudinal and transverse structure functions, and by validating the results against the known total cross section. It develops a comprehensive framework using dimensional regularization to separate non-singlet, singlet, and purely singlet contributions, and to perform renormalization and mass-factorization in both the ${\overline{MS}}$-scheme and the annihilation (A) scheme. The numerical results show that the ${α_s^2}$ corrections to $σ_L$ are substantial while those to $σ_T$ are modest, and that $F_L$ is particularly sensitive to the gluon fragmentation density, guiding refined extractions of $D_g^H$. The study achieves an improved understanding of scale dependence and enables a full NLO treatment of $F_L$, while indicating that NNLO effects for $F_T$ are likely small due to missing three-loop timelike splitting functions. By comparing with LEP-era data using various fragmentation-density parametrizations, the work highlights the importance of higher-order corrections for precise fragmentation modeling and motivates future refinements that include heavy-quark masses and potential higher-twist contributions.

Abstract

We present the calculation of the order $α_s^2$ corrections to the coefficient functions contributing to the longitudinal ($F_L(x,Q^2)$) and transverse fragmentation functions ($F_T(x,Q^2)$) measured in electron-positron annihilation. The effect of these higher order QCD corrections on the behaviour of the fragmentation functions and the corresponding longitudinal ($dσ_L(x,Q^2)/dx$) and transverse cross sections ($dσ_T(x,Q^2)/dx$) are studied. In particular we investigate the dependence of the above quantities on the mass factorization scale ($M$) and the various parameterizations chosen for the parton fragmentation densities $D_p^H(x,M^2)$ ($p=q,g$; $H=π^\pm, K^\pm, P, \bar{P}$). Our analysis reveals that the order α_s^2 contributions to $F_L(x,Q^2)$ are large whereas these contributions to $F_T(x,Q^2)$ are small. From the above fragmentation functions one can also compute the integrated cross sections $σ_L$ and $σ_T$ in an independent way. The sum $σ_{tot} = σ_L + σ_T$, corrected up to order \alphastwo, agrees with the well known result in the literature providing us with an independent check an our calculations.

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