Spin Dependent Drell Yan in QCD to $O(\alpha_s^2)$ (I). (The Non-Singlet Sector)

A study of the order $\alpha_s^2$ corrections to the Drell Yan (non-singlet) cross section for incoming states of arbitrary longitudinal helicities is presented. The study is performed in the case of the transverse momentum distributions, $(q_T)$, of the lepton pair and extends the calculation of Ellis Martinelli and Petronzio (EMP) to the polarized case. We use the $\bar{MS}$ scheme and the t'Hooft-Veltman regularization for the helicity projectors. From our results one can obtain the bulk of the totally inclusive NNLO cross section for the production of a Drell Yan pair in the non-singlet sector by a simple integration over the heavy photon momentum. We show that in the $\bar{MS}$ helicity is not conserved, unless a finite renormalization is done to reenter into a physical scheme ($\bar{MS}_p$). This aspect of the calculation follows the same trend as in polarized production of single and double photon to $O(\alpha_s^2)$. In the unpolarized limit we reproduce all the results of EMP.