Gauge theory amplitudes using on-shell methods and positive geometry [HBNI Th265]

Abstract

In this thesis, we investigate the amplitudes of gauge theories in two different contexts using the on-shell methods and positive geometry. We study the amplitudes and so-called on-shell functions for the Coulomb branch of maximal supersymmetric N = 4 super Yang-Mills. On-shell functions are obtained by arbitrary gluings of three-point ampli- tudes using on-shell internal legs. The simplest such on-shell diagram construction is a BCFW-bridge: joining a transverse line between two external legs [1]. Momentum conservation at each vertex forces the intermediate bridge momentum to be precisely the BCFW deforming momentum. We study such a construction for the massive theories, focusing on the Coulomb branch of N = 4 SYM. We find the on-shell function realization of BCFW shifts involving massive particles. For the arbitrary mass configurations, we find new mass-deforming BCFW shifts, understood best using the variables suitable for three-body special kinematics. Using these BCFW bridges, we calculate bigger on-shell functions, like the maximal cut of loop diagrams. The equivalence of the triple cut, bridge as an on-shell function, and the BCFW deformation leads to the equivalence between the maximal cut of the box diagram and the BCFW computation for a tree-level amplitude. In the second half of the thesis, we study the amplitudes of the pure Yang-Mills theory from the perspective of positive geometry. We use the so-called Corolla differentials. introduced by D. Kreimer, M. Sars, and W. D. van Suijlekom in [2], to ‘spin-up’ the canonical form of the associahedron (and its loop variants) into a color-ordered gluon amplitude (loop integrand). The program of positive geometry implies that the amplitudes are equivalent to the differential forms. We argue that sensible amplitudes can be interpreted as appropriate scalars obtained by the contraction of these differential forms with appropriate functionals, the multi-vector fields (MVFs). For instance, an appropriate MVF constructed out of the color factors leads to the color-dressed scalar amplitude, and an MVF constructed out of the Corolla di↵erentials leads to the color-ordered gluon am- plitudes. Recently, the colored 3 scalar amplitudes (at arbitrary loops) have been written as integrals over global Schwinger parameters, and are termed curve integral formula [3]. We use the Corolla differentials in the Schwinger parameters to write down the analogous gluon curve integral formula.