  Double parton scattering in proton-proton collisions includes kinematic
regions in which two partons inside a proton originate from the perturbative
splitting of a single parton. This leads to a double counting problem between
single and double hard scattering. We present a solution to this problem, which
allows for the definition of double parton distributions as operator matrix
elements in a proton, and which can be used at higher orders in perturbation
theory. We show how the evaluation of double hard scattering in this framework
can provide a rough estimate for the size of the higher-order contributions to
single hard scattering that are affected by double counting. In a numeric
study, we identify situations in which these higher-order contributions must be
explicitly calculated and included if one wants to attain an accuracy at which
double hard scattering becomes relevant, and other situations where such
contributions may be neglected.
