Radiative corrections to the ratio ofZ- andW-boson production

We calculate the order-${\alpha }_s$ ${\mathrm{}}^2$ corrections to Z-boson production in pp¯ collisions for which there are no corresponding corrections to W-boson production. These corrections affect the ratio of Z- and W-boson production, a quantity which is useful for counting the number of species of neutrinos. We find that these corrections are much smaller than the uncertainty in the ratio stemming from the quark and gluon distribution functions and the value of ${\Lambda }_{\mathrm{QCD}}$.