Two-loop splitting in double parton distributions: the colour non-singlet case

Abstract

At small inter-parton distances, double parton distributions receive their dominant contribution from the splitting of a single parton. We compute this mechanism at next-to-leading order in perturbation theory for all colour configurations of the observed parton pair. Rapidity divergences are handled either by using spacelike Wilson lines or by applying the δ regulator. We investigate the behaviour of the two-loop contributions in different kinematic limits, and we illustrate their impact in different channels.

Double parton distributions out of bounds in colour space

“…As a consequence, distributions for different colour representations are proportional to each other, as specified in eq. ( 4.40) of [20]. This leads to the relations in (49).…”

“…The PDFs used in the splitting formula are specified in the text below (34). A weighting factor x 1 x 2 is included to keep details visible at higher momentum fractions which is natural for the perturbative splitting mechanism [20]. With this choice, the term with ζ p is zero at x 1 = x 2 .…”

“…Let us briefly comment on the factor x 1 x 2 in (42). As explained in [20,21], the definition of ζ p involves the rapidity regulator and the plus-momentum of the target proton. By contrast, x 1 x 2 ζ p refers to the regulator and the plus-momenta of the two extracted partons, and in this sense is more closely related to the scales μ 1 and μ 2 that refer to the renormalisation of the operators associated with the two partons.…”

“…It is then natural to ask whether the resulting DPDs at small y violate positivity in colour space, and if so, by how much. We address this question in the present section, using the results of the recent two-loop calculation in [20].…”

“…For small transverse distances y between the two partons, DPDs can be computed in terms of ordinary parton densities and kernels for the perturbative splitting of one parton into the two observed partons and (at higher orders) additional unobserved ones. Using the results of the recent two-loop calculation in [20], we can investigate to which extent positivity of DPDs in colour space is realised in the small y limit. This is the second goal of our work.…”