AMPLITUHEDRA FOR PHI^3 THEORY AT TREE AND LOOP LEVEL

In this thesis we explore a novel connection between scattering amplitudes and positive geometries, which are semi-algebraic varieties iteratively defined by the property of possessing a boundary structure which reduces into lower dimensional version of themselves. The relevance of positive geometries in physics was first discovered in the context of scattering amplitudes in N = 4 SYM and led to the definition of the Amplituhedron. An analogue structure was very recently found to tie tree level scattering amplitudes in the bi-adjoint scalar theory to the Stasheff polytope and the moduli space of Riemann surfaces of genus zero. Here we further extend this framework and show how the 1-loop integrand in bi-adjoint theory, or more generally in a planar scalar cubic theory, is connected with moduli spaces of more general Riemann surfaces. We propose hyperbolic geometry to be a natural language to study the positive geometries living in various moduli spaces, then we illustrate convex realizations of polytopes which are combinatorially equivalent to them, but live directly in the kinematical space of amplitudes and integrands. Finally, we show how to exploit these constructions to provide novel and efficient recursive formulae for both tree level amplitudes and 1-loop integrands.