Leading logarithm calculation of the $e^{+}{e}^{-}\to e^{+}{\nu }_eūd$ cross section

We analytically evaluate in the leading logarithm approximation the differential cross section for $e^{+}{e}^{-}\to e^{+}{\nu }_eūd.\mathrm{}$ We compare our order ${\alpha }^4{\alpha }_s^0$ leading-log result to the order ${\alpha }^4{\alpha }_s^0$ exact result obtained from the GRC4F Monte Carlo program. Finally we use the Glück-Reya-Schienbien distribution of partons in a virtual photon, which incorporates both evolution and nonperturbative strong interaction contributions, to obtain better estimates of the differential cross section.