Rapidity Divergences and Deep Inelastic Scattering in the Endpoint Region

Abstract: The deep inelastic scattering cross section in the endpoint region, $x \sim 1$, has been subjected to extensive analysis. We revisit this process, and show that in the endpoint individual factors in the factorized hadronic tensor have rapidity divergences. We regulate these divergences using a recently introduced rapidity regulator, and find that each factor requires a different scale to minimize large rapidity logarithms. Interestingly, in this case running in rapidity is non-perturbative. In addition, we give an operator definition of the parton distribution function in the endpoint region, and remark on the issues that should be considered in constructing this function.