It is generally accepted that there are four fundamental forces in nature: the electro-magnetic force, the weak force, the strong force and the gravitational force. In this thesis we will focus on the strong force, which is described by a gauge theory that is known as Quantum Chromo-Dynamics (QCD).

Considering nucleons, of which protons and neutrons are the simplest exam- ples, we know that they are composed of quarks and gluons whose interactions are described by QCD. On the other hand, the fact that quarks and gluons are confined in nucleons is still not very well understood. We know that at low-energy the strong force becomes, well... strong (i.e. the coupling constant, which describes how strongly particles interact with each other, becomes large ～ 1) but this alone is not enough to explain confinement. For the moment the question of how and if confinement can be derived from theory is still open, so we will not discuss this in any more detail in this text.

Focusing on low-energy phenomena, the fact that the coupling constant becomes large is conceptually easy to understand but, from a calculation point of view, making predictions in this regime is far from trivial. The main reason for this is due to the fact that when the coupling constant gets large we can no longer use standard perturbative methods to do calculations. In other words, we have entered the non-perturbative regime of the theory (QCD). This means that if we want to "calculate" or predict low-energy or large-distance phenomena, starting from the QCD Lagrangian without inserting anything by hand, we need to have access to non-perturbative calculation methods. There are several ongoing attempts to construct such methods of which we mention: Lattice QCD [1], Borel Resummation [2], AdS/CFT [3] and BRST inspired techniques [4], none of which are completely satisfactory for the moment.

In this thesis we will discuss another approach, which makes use of a special kind of loop space, where with loop space we refer to the space constructed of all possible loops in the (space-time) manifold under consideration or put in more mathematical terms: the space of all (not necessarily continuous) maps from the circle to the manifold. The "special kind" will refer to the fact that we consider a Generalized loop space, for which a detailed construction will be given. The interesting part about this approach is that it might give access to some restricted parts of the non-perturbative sector and has an extra motivation coming from another question in nuclear physics: How does the proton (nucleon) inherits it’s spin from the constituent particles [5]? Naively one could answer this question by stating that a proton is a composition of three quarks, two up and one down, where each of these quarks has spin 1/2, such that one could conclude that the spins of two quarks cancel and the remaining one gives the proton the known spin 1/2. Problem solved..., but not quite. From scattering experiments in the second half of the 20th century, and from renormalization theory, we know that the (collinear) distribution functions of quarks in a proton depend on the energy scale at which we do the experiments and on the "longitudinal fraction of momentum" the quarks are carrying with respect to the proton. As a result we conclude that the naive picture from above is too simple and fails to reproduce the results of the high-energy experiments beyond the collinear approximation. Now, also taking gluon contributions to the spin into account, it is far from trivial how the proton gets to have a fixed spin 1/2 value. The situation even gets worse when we realize that the quarks and gluons are moving around in the proton (nucleon), such that there is possibly also an angular momentum contribution. So how can we try to solve this problem?

In order to investigate what is happening inside the proton we turn to scat- tering experiments, which in the last century have given us the insights we have today in particle physics. Dedicated scattering experiments to unravel the proton’s structure, performed in the second half of the last century, mainly measured the longitudinal Parton momentum Distribution Functions giv- ing us in some sense only one-dimensional information. These distributions describe, in some reference frame, the probability to find a parton (quarks, anti-quarks and gluons) inside the proton (or nucleon) with a certain momen- tum fraction at some energy scale. Using a Fourier transform, this momentum information is then transformed in position information.

This information is clearly inadequate to describe the full internal structure of the proton, for instance if we are interested in the angular momenta of the parton’s contribution to the protons spin we obviously will need something more than just the information provided by the longitudinal distribution functions. Several generalizations of the longitudinal Distribution Functions have been proposed. We mention the Generalized Parton Distributions that give us access to the three dimensional structure of the nucleon, and the Transverse Momentum Distributions describing the probability to find a parton in a nucleon with a certain momentum fraction and a certain transverse momentum with respect to the nucleon direction of motion. Many others exist (see for instance [6]), but in this text we will restrict ourselves to these Transverse Momentum Distributions and in the last chapter we will focus on the half-Fourier transform of such a distribution, a Transverse Distance Dependent (TDD).

However, we point out that there is still an ongoing discussion about the correct definition for these transverse distributions and that in this text we will use one specific definition [7].

The problem with these distribution functions, both the longitudinal and the transverse, is that they cannot be calculated from theory because they originate in the non-perturbative sector of the theory, which is problematic as we discussed above. All we can do at this time is to measure them over as large as possible ranges of their parameter space. Having this information we can then investigate if we can find evolution equations with respect to the different parameters hopefully providing us with some more insight about the underlying physics. In other words, starting from a certain value of one of the parameters, one can go over to another value for this parameter by means of the evolution equations. We mention as examples of such evolution equations the famous Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations with respect the mass parameter μ [8–11] and the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation for evolution with respect to the momentum fraction x [12, 13]. Both equations, unfortunately, do not cover the entire ranges possible for their evolution parameters so there is still room for improvement. For the transverse-dependent case we mention the Collins-Soper equations [14, 15], which again have only a restricted application range with respect to their parameter space.

The quest of this Dissertation is to look for a consistent field-theoretically motivated and mathematically correctly formulated set of evolution equations with respect to all the scales (ultra-violet, rapidity, infra-red, if needed) for the Transverse Momentum Distributions (TMDs). As we will discuss, using the standard quantum field-theoretical renormalization techniques for these TMDs do not always work. In the hope of dealing with some of the renormalization issues we turned our attention to the Generalized Loop Space formalism [16, 17]. In this formalism we only considered the subset of Wilson loops, where we then applied a kind of geometrical renormalization to arrive at an evolution equation. These equations should in principle be testable at the Thomas Jefferson National Accelerator Facility (Newport New, VA), planned Electron-Ion Collider, LHC (CERN) and other machines.