Multivariable evolution in parton showers with initial state partons

One can use more than one scale variable to define the family of surfaces in space parton splitting parameters that evolution a shower. In an earlier paper, we developed this idea for electron-positron annihilation. Here, multiple variables shower with initial state partons. Then need sophisticated analysis because distribution functions must be coordinated evolution. We make needed connections precise our work, even case just variable. develop example three variables, which leads advantages compared usual formulation only provide results Drell-Yan muon pair production.