UC Davis

A geometric formulation of scattering amplitudes in perturbative quantum field theory has surfaced in recent years where, in place of the summations of all off-shell Feynman diagrams for an amplitude, a single geometric object in the kinematic space of on-shell scattering data completely dictates the final answer. By far the most sophisticated example of this new formulation is the \emph{amplituhedron} geometry for the planar $\mathcal N=4$ super Yang-Mills theory, whose boundary structure encodes all singularities of any amplitude at any loop-order.

While an amplituhedron describes an amplitude (prior to loop integration) as a differential form with logarithmic singularities on its boundary, a \emph{dual amplituhedron} whose \emph{bulk volume} directly reproduces the same amplitude is much desired. In chapter two of this thesis, we provide nontrivial evidence for the dual amplituhedron from the perspective of \emph{local triangulation}. We classify all the elementary geometric spaces relevant for one-loop Feynman integrands which allow for further identification of geometrical spaces associated with the chiral pentagon-integrands in the local expansion of one-loop MHV amplitudes. Using projective duality maps on certain two-dimensional boundaries where the geometry reduce to polygons, we show that the real purpose of life of chiral pentagons is to triangulate the putative dual amplituhedron.

The discovery of amplituhedron originated from a reformulation of amplitudes at the level of the \emph{integrand} --the unintegrated sum of Feynman diagrams, in terms of \emph{positive Grassmannians} and \emph{on-shell diagrams}. Extending the geometric formulation to more general theories after the example of $\mathcal N=4$ sYM requires both a wealth of theoretical data from explicit computations of loop integrands and further understanding of the connection between on-shell diagrams and Grassmannian geometry without planarity.

In chapter three, we discuss one-loop \emph{integrand} construction for planar Yang-Mills amplitudes with $1\leq \mathcal N < 4$ supersymmetry based on a \emph{generalized unitarity} method. We trace the source of ambiguity in defining ``the loop integrand'' for non-maximally supersymmetric amplitudes to the ill-defined non-singlet massless “bubble-cuts”, and pinpoint two prescriptions for \emph{defining} the loop integrand, paving the way for future exploration of amplituhedron-like geometric objects for less supersymmetric Yang-Mills theories.

In chapter four, we go beyond the planar sector of $\mathcal N=4$ sYM, exploring the geometry of non-planar on-shell diagrams --- essential building blocks of any non-planar amplitudes --- that arise in the context of BCFW recursion relations of tree-amplitudes using non-adjacent shifts of momenta. We show that individual terms in the non-adjacent BCFW expansions are associated with a particular class of non-planar positive geometries which generalize the positroid cells of the positive Grassmannian into the Grassmannian.