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

We present the calculation of the order $\alpha_s^2$ 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 ($d\sigma_L(x,Q^2)/dx$) and transverse cross sections ($d\sigma_T(x,Q^2)/dx$) 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 $D_p^H(x,M^2)$ ($p=q,g$; $H=\pi^\pm, K^\pm, P, \bar{P}$). Our analysis reveals that the order \alpha_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 $\sigma_L$ and $\sigma_T$ in an independent way. The sum $\sigma_{tot} = \sigma_L + \sigma_T$, corrected up to order \alphastwo, agrees with the well known result in the literature providing us with an independent check an our calculations.