Momentum sum rule and factorization of double parton distributions

Abstract

We show that the momentum sum rule is a necessary condition for factorization of double parton distributions into a product of two single parton distributions for small values of the parton momentum fractions $x$ and large enough values of the evolution scale ${Q}^{2}$. This is a somewhat surprising result since the momentum sum rule involves integration over all values of the momentum fraction. In essence, the momentum sum rule provides a proper relation between the double and single parton distributions, which is necessary for the small $x$ factorization at large ${Q}^{2}$.