Two-loop splitting in double parton distributions

Double parton distributions (DPDs) receive a short-distance contribution from a single parton splitting to yield the two observed partons. We investigate this mechanism at next-to-leading order (NLO) in perturbation theory. Technically, we compute the two-loop matching of both the position and momentum space DPDs onto ordinary PDFs. This also yields the 1 ->2 splitting functions appearing in the evolution of momentum-space DPDs at NLO. We give results for the unpolarised, colour-singlet DPDs in all partonic channels. These quantities are required for calculations of double parton scattering at full NLO. We discuss various kinematic limits of our results, and we verify that the 1 ->2 splitting functions are consistent with the number and momentum sum rules for DPDs.

In double parton scattering (DPS), two partons from a proton could have arisen as a result of one parton perturbatively splitting into (at least) two: 1 → 2 mechanism This will be dominant contribution at small (perturbative) transverse separation between partons, y All-order form of double parton distribution F at small y is: PDF Perturbative kernel Overall 1/ 2 dependence This Τ 1 2 dependence causes a power divergence when naïve formulation of DPS cross section is used: ∫ 2 . Related to leaking of DPS into SPS region.
Use double parton distributions (DPDs) in y space, insert cut-off into y integration: 1 2 Cuts off integral for ≲ 1/ , regulates power divergence Use subtraction term in sum of SPS and DPS to avoid double counting: DPS cross section with both DPDs replaced by fixed order splitting expression dependence cancelled order by order must reduce to perturbative expression at small y. When modelling we used a sum of two terms: [JHEP 1706[JHEP (2017  One can also consider Δ-space DPDs, where all divergences regularised using dimreg + , and compute matching onto PDFs: Evolution of Δ-space DPDs involves an inhomogeneous 1 → 2 splitting term: We compute also and at NLO. For our purpose: s are needed to link our y-space DPDs to Δ-space DPDs, latter of which satisfy momentum and number sum rules at Δ = 0. Allows us to check to what extent our models for satisfy the sum rules, and construct improved models. (see talk by Peter tomorrow) In this talk, I'll focus on computation of matching coefficients and splitting functions for colour-singlet, unpolarised DPDs, for all parton channels. These will be made available shortly in [arXiv:1812.xxxxx].
Integration-by-parts reduction to master integrals (LiteRed) Construct differential equations in 1 and solve (Fuchsia) Results for bare graphs! Computation of 3 → 0 limit of master integrals using method of regions (boundary conditions) • Full computation of bare graphs done using light-cone and covariant Feynman gauge ✓ • Master integrals satisfy differential equation in 2 ✓ • Master integrals all checked numerically at 10 random points using FIESTA ✓ • Individual graphs have poles in up to −3 , as well as rapidity divergences.
−3 pole + rapidity divergences cancel after summing over graphs, −2 pole is as predicted by renormalisation group equation ✓ • Splitting functions 1 2 , 0 (1) satisfy constraints related to number and momentum sum rules: Interesting processes/regions for studying DPS typically involve small values (higher density of partons→greater chance of DPS, plus smaller such that power suppression is reduced).
→ Interesting to study matching coefficients and splitting functions in limits of small . For example, small 1 , 2 limit of , . We find that this is not the casein fact we observe that above ( • NLO matching of DPDs onto PDFs, and NLO 1 → 2 splitting functions, computed in unpolarised colour-singlet case ✓ • Corresponding matching coefficients to come for polarised + colournonsinglet channels.
• Then numerics! • Look at effect of NLO corrections on DPD y-profiles, parton luminosities, cross sections, etc. • Investigate perturbative convergence of DPS cross sections • Look for observables where we might be able to detect differences between LO and NLO predictions.