# Two-loop splitting in double parton distributions

### Submission summary

 As Contributors: Markus Diehl · Jonathan Gaunt Arxiv Link: https://arxiv.org/abs/1902.08019v2 (pdf) Date accepted: 2019-07-24 Date submitted: 2019-07-19 02:00 Submitted by: Diehl, Markus Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Phenomenology Approach: Theoretical

### Abstract

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.

### Ontology / Topics

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Published as SciPost Phys. 7, 017 (2019)

We thank the referees for their careful reading of our manuscript and for the constructive remarks. Let us reply to the reports of Referees 1 and 3 in turn.

# Referee 1

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Referee 1
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Indeed, the paper gives a fair amount of technical detail and is not short. However, we believe that this is justified: we give as much detail as we think is necessary for a QCD practitioner to understand what we compute and how we do it, in the spirit of our work being reproducible. The formulae regarding dimensional regularisation and the running coupling are given to set up our notation and to make the paper self consistent (rather than forcing a reader to look elsewhere for definitions that are needed to read the present manuscript). The same holds for convolutions, which are in a large part non-standard because they involve functions of two momentum fractions.

Concerning a lack of conceptual discussion mentioned by the referee, we think that we present the relevant methods as they are used for our specific calculation. A more general discussion, e.g. of the theory of double parton scattering would go beyond the scope of our work and can be found elsewhere in the literature. We have augmented the references given in the introduction, such that an interested reader can more easily locate the relevant papers.

The introductory paragraph we added at the beginning of section 2 is meant to guide the reader by giving an overview of what is to come in that section.

As to the physics impact of our calculation, we have added a section 5.4, where we present a numerical example for NLO effects in double parton distributions. An analysis at the level of physical cross sections will involve a significant number of further steps. Performing and presenting such calculations is a project in itself, much beyond the scope of the present work.

To provide a better motivation for our work, we have extended the introduction, giving more specific reasons to study DPS and to analyse it at NLO accuracy. For a more general or more extended discussion of DPS, our paper would not be the most suitable place. We instead point the reader to the existing literature, especially to the monograph [39], which also covers phenomenological and experimental aspects.

We would like to point out that, by the nature of the presented material, the present work will be most interesting and accessible to theorists working on DPS - and possibly to practitioners of higher-order computations - rather than to experimentalists. In this respect, we agree with the assessment in the Report of Referee 2.

# Referee 3

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Referee 3
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In the last paragraph of the Conclusion, we now give a bit more detail about where we stand and what are the next steps for studying parton splitting in DPS. Some particularly interesting DPS processes are identified in the introduction.

We hope that our revised manuscript addresses the referee reports in an adequate manner and send our best regards,

The authors

### List of changes

1. In the introduction (p.3) we have added some general physics motivation to study double parton scattering (DPS), along with references to experimental results (thus providing a connection to phenomenology). For a broader introduction to DPS, we refer to a recent monograph (Ref [39]).

2. Also in the introduction (p.4) we give more arguments for NLO computations of DPS in general, and the perturbative splitting mechanism in particular.

3. For the general orientation of the reader, we have added a paragraph at the beginning of section 2 (p.5), and a sentence at the end of the first paragraph of section 4 (p.21).

4. In a new section 5.4, we give a numerical illustration of the difference between a double parton distribution computed at leading and at next-to-leading order. While this is for a particular parton combination in particular kinematics, it shows that the difference between the two perturbative orders can be numerically important. We are aware that this is not equivalent to a study of DPS cross sections at NLO, mentioned by Referee 1. Such a study will require significant additional work - well beyond the computation of the NLO kernels - and we think it is justified to leave this to a future project.

5. In the last paragraph of the Conclusion, we give a slightly more detailed outlook on future work.