The overarching topic of this thesis is double parton scattering (DPS), which describes the situation when two individual hard scattering reactions occur in a single hadron- hadron collision. In some regions of phase space DPS may give sizeable contributions to the production of multi-particle final states and thus be an important background to single parton scattering (SPS). Not only this, but DPS is also an interesting phenomena in its own right, as it gives insight into the correlations of partons inside of hadrons. Therefore a theoretical description of such processes from first principles is required. Such a prescription is obtained in the form of a factorisation theorem akin to the one known from SPS, with a central building block being the double parton distributions (DPDs). However, these DPDs are presently basically unknown as experimental data is still lacking.
One of the few general theoretical constraints for DPDs are the number and momentum sum rules proposed by Gaunt and Stirling. In chapter 3 of this thesis a proof is presented that the DPD sum rules are valid at all orders in the strong coupling for renormalised distributions. As by-products of this proof the all order form of the inhomogeneous evolution equation for momentum space DPDs can be derived and it can be shown how the inhomogeneous term in this equation is related to the contribution of a short distance 1 -> 2 splitting to the DPDs. It can furthermore be shown that the 1 -> 2 evolution kernels in the inhomogeneous term fulfil number and momentum sum rules closely resembling the ones for DPDs.
In chapter 4 the sum rules considered in chapter 3 are used to construct improved position space DPD models. To this end it is first shown how position space DPDs can be matched onto the momentum space DPDs for which the sum rules have been shown to be valid. Following this an initial DPD model consisting of an intrinsic part and a contribution from the perturbative 1 -> 2 splitting is iteratively refined in order to obtain the best possible agreement with the DPD sum rules. In particular, this highlighted that a good agreement with the momentum and equal flavour number sum rules is not possible without taking into account the 1 -> 2 splitting contribution. Finally the dependence of the agreement with the sum rules on the renormalisation scale and the cut-off scale introduced by the matching onto momentum space DPDs is investigated.
As the 1 -> 2 splitting contribution has been shown to be quite important for DPS it is extensively studied in chapter 5. In particular, the next-to-leading order (NLO) expres- sion for this splitting is the only missing quantity required for NLO DPS calculations.
Therefore this splitting is calculated at NLO in perturbation theory for unpolarised colour singlet DPDs in all partonic channels using state of the art techniques. From this momentum and position space 1 -> 2 splitting kernels as well as the 1 -> 2 evolution kernels needed in the inhomogeneous evolution equation of momentum space DPDs are extracted at NLO. As a cross-check for the correctness of the results the agreement of the 1 -> 2 evolution kernels with the sum rules derived in chapter 3 is explicitly verified. Finally various kinematic limits of the momentum space splitting kernels and the evolution kernels are discussed
