Formulation of parton shower evolution beyond leading order for electron-positron annihilation

Parton shower evolution is given by a renormalization group equation that reflects the infrared behavior of perturbative QCD including color and spin and including quantum interference. With added approximations, such an evolution equation can provide the basis for a parton shower event generator. To date, splitting functions for parton shower evolution have been derived only to leading order in $\text{as}$. We provide a framework that can provide splitting functions beyond leading order, and in particular at order ${\text{as}}^2$. This requires a representation of Feynman graphs in which partons are off shell and are characterized by vector or spinor indices instead of on-shell spins.