MSc defence by Alexander Boccaletti

Multi-loop Feynman integrals in maximally symmetric gauge theory

Scattering amplitudes are indispensable objects for studying the interaction of subatomic particles in different particle physics theories. They are a key ingredient for calculating the cross section, which is the main physical observable measured in collider experiments at the LHC. These scattering amplitudes also reflect the symmetries of the theory they are calculated in and therefore, provide a ground of study for yet undiscovered symmetries and mathematical structures. The recently found dual superconformal symmetry in planar N = 4 sYM  combines with the long known ordinary (spacetime related) superconformal symmetry enjoyed by the same theory, to cre- ate an infinite dimensional symmetry algebra acting on scattering amplitudes called Yangian symmetry. The action of this algebra determines that planar N = 4 sYM has an integrability structure; the integrability of the theory gives additional constraints on the structure of scattering amplitudes.

The thesis mainly focuses on the integrals contributing to these scattering amplitudes, rather than on the amplitudes themselves. More precisely, in this thesis we compute the symbol of the 2-loop 10-point double-pentagon integral Idp contribut- ing to 2-loop MHV scattering amplitude in planar N = 4 supersymmetric Yang-Mills (sYM) theory in d = 4 spacetime dimensions, whose alphabet evaluates to both rational and algebraic letters. The relevant mathematical objects and structures for studying such integrals are reviewed beforehand. The calculation is performed using the duality of a certain class of Feynman-integrals with the vacuum expectation value (VEV) of null polygonal Wilson loops. However, some of these integrations over the edges require the rationalization of the square roots contained in the integrand. In total, we obtain a resulting symbol alphabet with 122 rational letters and 54 algebraic letters.