Abstract
We give a description of double parton scattering with measured transverse momenta in the final state, extending the formalism for factorisation and resummation developed by Collins, Soper and Sterman for the production of colourless particles. After a detailed analysis of their colour structure, we derive and solve evolution equations in rapidity and renormalisation scale for the relevant soft factors and double parton distributions. We show how in the perturbative regime, transverse momentum dependent double parton distributions can be expressed in terms of simpler nonperturbative quantities and compute several of the corresponding perturbative kernels at one-loop accuracy. We then show how the coherent sum of single and double parton scattering can be simplified for perturbatively large transverse momenta, and we discuss to which order resummation can be performed with presently available results. As an auxiliary result, we derive a simple form for the square root factor in the Collins construction of transverse momentum dependent parton distributions.
Highlights
Proton-proton collisions at high energies are sensitive to regions of phase space where partons have small momentum fractions
We show how the theory simplifies for intermediate transverse momenta Λ ≪ qT ≪ Q, where transverse-momentum dependent double parton distributions (DPDs) can be matched on transverse-momentum integrated distributions, with the transverse-momentum dependence being computed in perturbation theory
We explicitly show how the soft factors relevant for the cross section can be entirely absorbed into DTMDs or DPDFs, and we derive the resulting evolution equations in ζ, as well as the ones in μ
Summary
Proton-proton collisions at high energies are sensitive to regions of phase space where partons have small momentum fractions. The most frequent and best studied case of such multiple hard interactions is double parton scattering (DPS) This mechanism can be especially prominent in cross sections depending on transverse momenta in the final state. Correlations in spin and in colour have been classified systematically [27, 28, 32] and will play an important role in the present work Their size is poorly known, but can be limited by positivity bounds [33, 34], which have similar theoretical status as positivity constraints on single parton distribution functions (PDFs). Our definition generalises the combination of collinear and soft factors in [65] to double parton distributions, and it provides an alternative form of this construction for single parton TMDs. The colour structure of DPS is significantly more complicated than the one of SPS, and we show in section 4 how this structure can be handled in a general and efficient way. A variety of technical details and results are given in the appendices
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