On sum rules for double and triple parton distribution functions and PYTHIA's model of multiple parton interactions

Multi-parton distributions in a proton, the nonperturbative quantities needed to make predictions for multiple scattering rates, are poorly constrained from theory and data and must be modelled. All Monte Carlo event generators that simulate multiple parton interactions (e.g. PYTHIA) contain such a model of multi-parton PDFs. One important theoretical constraint for the case of double parton distributions is provided by the socalled number and momentum sum rules. In this paper we investigate to what extent the double parton distribution functions used in the PYTHIA event generator obey these sum rules. We also derive the number and momentum sum rules for the triple parton distribution functions and discuss how one can use the PYTHIA code to construct triple parton distribution functions which approximately satisfy these sum rules.