Abstract

In [1] Gaunt and Stirling proposed a set of momentum and quark number sum rules, in close analogy to the well known PDF sum rules, that sensible, transverse momentum integrated, double parton distributions (DPDs) should obey. In their work they argued that these sum rules hold under LO QCD evolution, if one assumes that they are fulfilled at the starting scale. While Gaunt and Stirling used these sum rules to construct improved DPDs from single PDFs using a factorised ansatz, they quite generally place restrictions on the precise form of DPDs. Therefore we show in this work, using a Feynman diagram approach, that for the bare, i.e. unrenormalised, DPDs defined in [2] the sum rules are fulfilled to all orders in the strong coupling αs. We furthermore discuss the case of renormalised quantities. In order to make the presentation as clear as possible light-cone time ordered perturbation theory is employed and the validity of the sum rules is shown on a diagram by diagram basis. For explicitness we use a toy model of a “pion” consisting of scalar u and d quarks. The considerations for the all order proof are, however, independent of this model and also hold in full QCD for any hadron, i.e. for any valence quark content. To make things more accessible we furthermore show in an explicit $\mathcal{O}(\alpha_s)$ calculation in the scalar toy model that the sum rules are also fulfilled for UV-subtracted, i.e. renormalised quantities.

Bibliography

[1] J. R. Gaunt and W. J. Stirling, Double Parton Distributions Incorporating Perturbative QCD Evolution and Momentum and Quark Number Sum Rules, JHEP 03 (2010) 005, [0910.4347].

[2] M. Diehl, D. Ostermeier and A. Schafer, Elements of a theory for multiparton interactions in QCD, JHEP 03 (2012) 089, [1111.0910].