Protons and neutrons (alias, the nucleons) constitute the building blocks of matter, accounting for almost all the mass of our world. Even if we are still far from understanding their physical inner structure, many efforts have been made to deepen our knowledge about them.

Over the past few years, thanks to a fruitful synergy of theoretical and experimental progress, we have started opening the study of new multi-dimensional images of the structure of proton, investigating the behavior of its fundamental constituents, the quarks and gluons.

When we look into nucleons with extremely high resolution, we are in the regime of perturbative QCD (in other words, a regime where we can really work out mathematical calculations) and quarks and gluons appear almost free. With the due caveats, we can compare the situation to observing water at extreme magnifications, and seeing quasi-free water molecules. As we reduce the magnification, we realize that the molecules clump together in heavier, composite droplets. Eventually, at low magnification they form a single object, like the proton.

Pursuing the analogy, when we are looking at a proton at rest (not smashed inside a collider, for example) it is as if we were unable to describe water starting from the dynamics of molecules. This is because confinement, the reason for quarks and gluons being inescapably bound inside a proton, is left without any rigorous mathematical justification. Confinement is the most crucial characteristic of the theory and represents one of the hardest physics problems of today.

What we can do is to just describe this jam of quarks and gluons giving rise to a proton through mathematical objects specifically introduced in order to “parametrize” our ignorance about its structure: these are what are called parton distribution functions (PDFs), which shape the probability of finding quarks and gluons within a proton.

The knowledge of the multi-dimensional structure of protons allows the analysis of properties otherwise inaccessible. The situation may be compared to diagnostic studies: electrocardiography, for example, gives us mono-dimensional information about the hearth activity. It is of fundamental importance, but it does not give detailed information about the multidimensional inner structure. Instead, more important for this purpose are multi-dimensional tomographies of heart activity (MRI, CT and others). The enormous advantages of medical diagnostic imaging literally revolutionized medicine and surgery. In a similar way, the latest “multi-dimensional” pictures of the nucleon obtained with QCD phenomenology can improve the current status of hadronic physics and aim at better understanding particle physics in general.

Although one-dimensional (collinear) parton distribution functions are extremely useful for studying any process involving hadrons (including the proton-proton collisions taking place at the LHC), from the point of view of nucleon tomography they are rather limited, because they describe the distribution of partons in a single dimension.

More informative distributions are the so-called transverse-momentum-dependent distributions (TMDs). They represent pictures of three-dimensional probabilities in momentum space. The distributions change depending on the energy scale at which they are probed (in a way that is calculable using evolution equations from perturbative QCD) and on the value of the longitudinal fractional momentum.

There are many nontrivial questions concerning TMDs that do not have an answer yet, like their most truthful mathematical representation. At present, we know that experimental data form proton-proton and electron-proton collisions point towards Gaussian shapes (if the spin of quarks is neglected), but other forms could do the job as well.

An important question concerns the flavor dependence of transverse-momentum-dependent distributions: are up quarks moving in the nucleon with greater velocity than the down ones, or vice versa? What about sea quarks? Are they faster than the other ones? Part of my research activity is devoted to the investigation of this topic, which could be quite relevant both from the theoretical and experimental point of view. After lot of struggling with data analysis, we now know that sea quarks are likely to be faster than up quarks, which are then faster than down ones.

The statistical analysis of the huge amount of data collected at hadron colliders like the Tevatron and the LHC strongly relies on the detailed knowledge of parton distribution functions, both in 1D and 3D (the TMDs!). Before now, data analysis has been carried out assuming that quarks have all the same velocity, but we now know that this is not the case! This means that it will be important to refine the knowledge of quark (and gluons too, in the future) velocities, in order to improve the accuracy and reliability of data analysis. That’s what a PhD student can do during his/her amazing time in scientific research!