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In this review, we describe the status of transverse momentum dependence (TMD) in double parton scattering (DPS). The different regions of TMD DPS are discussed, and expressions are given for the DPS cross section contributions that make use of as much perturbative information as possible. The regions are then combined with each other as well as single parton scattering to obtain a complete expression for the cross section. Particular emphasis is put on the differences and similarities to transverse momentum dependence in single parton scattering. We further discuss the status of the factorisation proof for double colour singlet production in DPS, which is now on a similar footing to the proofs for TMD factorisation in single Drell-Yan, discuss parton correlations, and give an outlook on possible research on DPS in the near future.

Double parton scattering (DPS) is the process in which one has two hard scatterings, producing two sets of particles that we can label as ‘

While TMD factorisation in SPS has been rigorously proven for colour singlet production (see, e.g., [

In the production of two colour singlets, such as two vector bosons, the TMD SPS factorisation theorem can be applied as usual to study the region where the sum of the two transverse momenta is small. If, however, the transverse momenta of both bosons are measured to be much smaller than the hard scale, standard TMD factorisation alone is no longer sufficient. For these observables, DPS contributes at the same power as SPS, and no leading-power factorisation theorem can be derived without simultaneously taking care of SPS and DPS, including their interference. An overview of the different factorisation theorems in hadron collisions and the treatment of the initial state in different regions of the sum and difference of the transverse momenta of the two colour singlets is shown in Figure

Transverse momentum regions and descriptions of the initial state in the leading power factorisation theorems. The power counting behaviour of the differential cross section in each region is indicated on the figure.

An important issue to address when describing processes which can receive significant contributions from DPS is the consistent combination of SPS and DPS, avoiding double counting. First approaches to this problem are described in [

A description of the transverse momentum dependent DPS cross section was pursued at the leading logarithmic (LL) level in [

We note in passing that double parton distributions depending on transverse momentum arguments, sometimes referred to as ‘unintegrated’ double parton distributions (UDPDFs), appear in approaches designed to describe the DPS cross section at small

The TMD distributions in DPS (DTMDs) depend on the longitudinal momentum fractions

In the rest of this review, we will take a closer look at the current status of TMD and spin physics in DPS. In Section

The TMD DPS cross section formula can be written in terms of two hard coefficients and two DTMDs as [

Although the structure of this formula is very similar to the TMD factorisation for SPS, there are several interesting differences as we will see as we have a closer look at the different ingredients.

In TMD measurements in SPS, it is well known that the collinear and soft momentum regions individually contain rapidity divergences. To tame these divergences and obtain one function describing each of the hadrons, the soft function is split up and combined with the collinear regions. There are several techniques for this procedure depending on the choice of regulator used for the rapidity divergences, but in essence it boils down to separating soft radiation on each side of a rapidity parameter [

In processes producing colourless particles, one needs the soft factor

The subtracted DTMDs, i.e., with the rapidity divergences subtracted through the combination with the soft factor as illustrated in Figure

Rapidity subtractions and definition of right and left moving DTMDs.

It can be useful to compare this result to the definition of the subtracted (single parton scattering) TMD; see, e.g., chapter 13 in [

The DTMD evolves two renormalisation scales, related to the two partons and one rapidity scale [

Similar to the small impact parameter expansion possible in TMD SPS, the small

Regions of

region | approximations |
---|---|

DPS, large | |

DPS, small | |

SPS | |

The large-

When

The splitting contribution

The term

To avoid large logarithms in the matching coefficients, one should perform this matching at the scale

The full DPS cross section can be obtained from the combination of the large- and small-

In the region

The DPS cross section discussed in the last section can be combined with SPS in a consistent manner following [

A discussion of the perturbative order at which the ingredients in the SPS as well as DPS cross sections and logarithmic accuracy are achievable is given in section 6.6 of [

Essentially all the steps towards a proof of factorisation for TMD DPS producing two uncoloured systems have now been completed [

In derivations of factorisation theorems the most difficult momentum region to treat is the Glauber region; this region is characterised by the momentum of the particle being mainly transverse (technically, a Glauber momentum

A further important step of the proof is to show that the central soft gluon attachments into the collinear factors can ultimately be disentangled into attachments into the soft Wilson lines given in (

To achieve a description of the TMD cross section in which the soft factor is absorbed into the separate TMDs, it is necessary that the soft factor has the property described in (

At this point the factorisation status for TMD DPS is essentially at the same level as that for TMD SPS. One remaining technical issue relates to Wilson line self-interactions; see section 9 of [

One of the most exciting aspects of DPS is the access it provides to the correlations between two partons bound inside the proton. The study of these correlations actually dates back all the way to the 80s [

The description of spin correlations in DPS has many parallels and similarities to the spin correlations in TMD physics. The main difference is actually in the physical interpretation, where the correlations between the hadron spin and the spin of a parton is replaced by parton-parton spin correlations. The correlations between the transverse momenta and spin of a parton and/or proton in SPS TMDs are in DTMDs replaced by correlations involving (one or two) parton spins, their two transverse momenta, and the transverse distance between them. The physical difference means that intuition on the size of the correlations built up from SPS can not be applied, but the similarities in the calculations means that large parts of calculations for TMDs can be recycled for DTMDs or DPDFs.

Colour correlations in DPS do not have a simple analogy in QCD TMD studies; however, certain similarities may be found in recent work on resumming electroweak (EW) logarithms (i.e., logarithms of the hard scale

The recent progress in DPS in general and for DTMD factorisation in particular has made the different ingredients necessary for a lot of interesting phenomenology available. What has never been done is taking the theoretical framework, producing all the ingredients, and applying them to a particular process such as same-sign W-boson production or double Drell-Yan. This, however, is a rather daunting task if one aims to directly treat all possible correlations, all momentum regions, interferences, etc. A more approachable way might be to start the development towards this goal, step by step. It would, for example, be very interesting to see the results of the RGE and rapidity evolution on the DPDs, in particular for the different colour channels. While it is known that the colour correlations decrease with evolution scale, this has never been investigated taking the full evolution equations into account, and simplified studies only explicitly treated the quark nonsinglet DPDF [

The authors declare that they have no conflicts of interest.

Tomas Kasemets acknowledges support from the Alexander von Humboldt Foundation.

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