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In the latest years the theoretical and phenomenological advances in the factorization of several collider processes using the transverse momentum dependent distributions (TMD) have greatly increased. I attempt here a short resume of the newest developments discussing also the most recent perturbative QCD calculations. The work is not strictly directed to experts in the field and it wants to offer an overview of the tools and concepts which are behind the TMD factorization and evolution. I consider both theoretical and phenomenological aspects, some of which have still to be fully explored. It is expected that actual colliders and the Electron Ion Collider (EIC) will provide important information in this respect.

The knowledge of the structure of hadrons is a leitmotiv for the study of quantum chromodynamics (QCD) for decades. Apart from the notions of quarks and gluons (we call them generically “partons” in the following), the natural question is how the momenta of these particles are distributed inside the hadrons and how the spin of hadrons is generated. Phenomenologically it is possible to access at this problem only in some particular kinematical conditions, as provided, for instance, in experiments like (semi-inclusive) deep inelastic scattering, vector and scalar boson production,

The ideal description of the process in (

The study of factorization [

The scale

In the rest of this review I will try to give an idea on how all these problems can be consistently treated, which can be useful also to explore new and more efficient solutions. Several parts of this review use material that can be originally found in [

The factorization of the cross sections into TMD matrix elements has been provided by several authors and it has been object of many discussions [

Diagrams of regions for TMD factorization (original figure in [

Because the soft radiation is not finally measured, its interactions should be included (and resumed) in the collinear parts, which become sensitive to a rapidity scale which acts in a way similar to the usual factorization scale. It is possible to define the soft radiation through a “soft factor”; that is, by an operator matrix element,

The direct calculation of the soft factor is all but trivial and the way the calculation is performed can influence directly the final formal definition of the transverse momentum dependent distribution used by different authors. In fact a simple perturbative calculation shows that in the soft factor there are divergences which cannot be regularized dimensionally (say, they are not explicitly ultraviolet (UV) or infrared (IR)) which occur when the integration momenta are big and aligned on the light-cone directions. The divergences that arise in this configuration of momenta are generically called rapidity divergences and regulated by a rapidity regulator. One can understand the necessity of a specific regulator observing that the light-like Wilson lines are invariant under the coordinate rescaling in their own light-like directions. This invariance leads to an ambiguity in the definition of rapidity divergences. Indeed, the boost of the collinear components of momenta

The factorization theorem to all orders in perturbation theory relies on the peculiar property of soft function of being at most linear in the logarithms generated by the rapidity divergences. Then it comes natural to factorize it in two pieces [

Another fundamental ingredient in the formulation of the factorization theorem is represented by the definition of the TMD operators that are involved. We use here the notation of [

Some comments finally are necessary for the zero-bin overlap problem. In principle an overlap factor affects the rapidity renormalization factor as

An important aspect of factorization is finally represented by the cancellation of unphysical modes, the Glauber gluons. A check of this cancellation has been provided in [

The practical implementation of the TMD for data analysis benefits from asymptotic limits of the distribution. These limits allow defining the TMDs at different scale and constraining the nonperturbative behavior of the TMDs. Commonly one starts with the large transverse momentum limit of the TMD. In this case one can refactorize the TMDs in terms of Wilson coefficient and collinear parton distribution functions (PDF), following the usual rules for operator product expansion (OPE). At operator level one finds

The calculation of the Wilson coefficients in (

Summary of available perturbative calculations of quark TMD distributions and their leading matching at small-b.

Name | Function | Leading matching function | Twist of leading matching | Maximum known order of coef. function | Ref. | Mix with gluon |
---|---|---|---|---|---|---|

unpolarized | | | tw-2 | NNLO ( | [ | yes |

Sivers | | | tw-3 | NLO ( | [ | yes |

| ||||||

helicity | | | tw-2 | NLO ( | [ | yes |

worm-gear T | | | tw-2/3 | LO ( | [ | yes |

transversity | | | tw-2 | NNLO( | [ | no |

Boer-Mulders | | | tw-3 | LO ( | [ | no |

worm-gear L | | | tw-2/3 | LO ( | [ | no |

pretzelosity | | – | tw-4 | – | – | – |

The evolution of the TMDs in the factorization,

Notation for TMD anomalous dimensions used in the literature.

rapidity evolution scale | TMD anomalous dimension | cusp anomalous dimension | vector form factor anomalous dimensions | rapidity anomalous dimension | |
---|---|---|---|---|---|

[ | | | | | |

[ | | | | | |

[ | – | – | | | |

[ | | | | – | |

Quark and gluon rapidity anomalous dimensions are related up to three loops by the Casimir scaling (see [

The consistency of the differential equations ((

As an example of application of the TMD formalism I review the study of unpolarized TMD parton distribution functions in Drell-Yan and Z-boson production following [

Namely, I consider the process

In (

Evaluating the lepton tensor and combining together all factors one obtains the cross section for the unpolarized Drell-Yan process at leading order of TMD factorization, in the form [

The evolution of the TMDs play a special role in (

The double-evolution equation of the TMDs can be formulated as in [

The evolution field presented in the previous section is conservative only when the full perturbative expansion of the evolution equations is known. In practice only a few terms of the evolution are calculated, so that it is important to understand in which sense the evolution field remains conservative. Using the Helmholtz decomposition, the evolution field is split into two parts

The immediate consequence of the fact that the evolution field

The path independence of the evolution is crucial for the implementation of the perturbative formalism, as its absence can derive into uninterpretable extractions of TMDs or big theoretical errors. The path independence can be achieved observing that

In the literature one can find a typical way to implement the evolution that one can call the improved

Paths for the improved

The presence of the intermediate scale

Because we have a double scale running of the TMD one can find lines of null evolution in the

The

I provide here some description of the

The last choice give us much more freedom to model the nonperturbative part of the TMD and the definition of the initial scale is unique and nonperturbatively defined. The choice of

A plot for the

The nonperturbative part of the evolution kernel can be modeled in different ways. For instance, one can introduce a simple ansatz like a the modification

The expression for the cross section with the optimal TMD definition is particularly compact and reads

The derivation of the saddle point using formula (

Let me conclude this section discussing a comparison of the optimal TMD construction with a more traditional implementation on data following the recent fits in [

Comparison of error bands obtained by the scale-variations for cross sections at NNLO in traditional (upper figure) and optimal (lower figure) TMD implementation. Here, the kinematics bin-integration, etc., is for the Z-boson production measure at ATLAS at 8 TeV [

The formulation of factorization theorems in terms of TMDs is a first fundamental step for the study of the structure of hadrons and the origin of spin. The use of the effective field theory appears essential to correctly order the QCD contributions. Properties of TMDs like evolution and their asymptotic limit at large values of transverse momentum can be systematically calculated starting from the definition of correct operators and the evaluation of the interesting matrix elements. A key point for the renormalization of TMDs is represented by the so-called soft matrix element which is common in the definition of all spin dependent leading twist TMD.

Still, all this is just a starting point for the study of TMDs. In fact a correct implementation of evolution requires a control of all renormalization scales that appear in the factorization theorem. I have described here some of these possibility putting the accent on some recent interesting developments which, at least theoretically, allow a better control of the resumed QCD series. The understanding of factorization allows also precisely defining the range of ideal experimental conditions where this formalism can be applied. A full analysis of present data using all the theoretical information collected so far is still missing and it will certainly be an object of research in the forthcoming years. The formalism described in this work is the one developed for unpolarized distributions. However the evolution factors are universal; that is, they are the same for polarized and unpolarized leading twist TMDs and they are valid in Drell-Yan, SIDIS experiments, and

The authors declare that they have no conflicts of interest.

I would like to thank Alexey Vladimirov and Daniel Gutierrez Reyes for discussing this paper. Ignazio Scimemi is supported by the Spanish MECD grant FPA2016-75654-C2-2-P.

_{T}, transverse parton distributions and the collinear anomaly

_{T}: Infrared safety from the collinear anomaly

_{T}

_{T}and transverse-momentum distributions on-the-light-cone

_{T}spectra of Drell-Yan and Z-boson production

_{T}-space