Large N expansion for scattering amplitudes in QCD

Quantum Chromodynamics (QCD) is the quantum theory describing strong interactions in the framework of the Standard Model (SM). It is a non-abelian gauge theory based upon the symmetry group SU(N), with N = 3, characterized by a single coupling constant gS running with respect to the energy scale. In the perturbative regime and high energy scales, scattering amplitudes are expanded perturbatively in powers of gS. However, at low energy scales, where perturbation theory breaks down, it is useful to find another expansion parameter. A possibility suggested by ’t Hooft that turned out to be very powerful consisted in considering the large N limit of the theory, expanding with respect to the inverse of the number of colors. The main goal of this thesis is to explore the efficacy of the large N expansion even in the perturbative regime, in conjunction with the usual expansion in gS, in order to simplify the computation of gluon scattering amplitudes with many external particles. Using modern techniques, like different color decompositions and recursive approaches, we compute and classify first the complexity then the accuracy of the expansion up to 10 external gluons at tree-level, and up to 8 at loop-level. Numerical results for tree-level amplitudes show how the expansion at lowest order in the number of colors becomes less and less accurate increasing the number of external gluons, while at the next order it is extremely accurate. In addition, the classification of color factors up to 10 gluons allows us to make guesses about their structure and their maximum values for each order for an higher number of external gluons. Finally, we prove that the number of lowest order color factors for another color decomposition known as color-flow is given by a suitable binomial coefficient growing very slowly for an increasing the number of external gluons. The results we obtained pave the way to a more efficient approach to the computation of scattering amplitudes in QCD.