On the Infrared and Ultraviolet Behavior of Scattering Amplitudes and Wavefunctions

Abstract

We extend the on-shell program on three fronts: finite Λ, finite ε, and finite α’ where these are the cosmological constant, the dimensional regularization parameter ε = (4-D)/2, and a scale for putative ultraviolet completions of tree-level amplitudes with graviton exchange (string scale) respectively. In the first case we study conformally coupled scalars in the associated de Sitter background, whose wavefunction contributions are transcendental functions of uniform weight and are closely related to the density perturbations of direct interest for studying inflation. We derive a recursive rule for the symbol of such functions and study the implications for singularities, differential equations, and identify the symbol letters with graph associahedra. Most of this work appeared in [1]. In the second case we study the structure of UV and IR divergences for scalar Feynman integrals in the context of Feynman polytopes and tropical fans, providing a unified description for both and a prescription to compute the leading divergence. In the IR case we apply this to the study of infinite classes of diagrams appearing in the expansion of the four-point amplitude in N = 4 SYM. This is based on work with Nima Arkani-Hamed and Sebastian Mizera [2]. In the case of UV divergences, we introduce u-variables which realize an associated binary geometry and facilitate a novel subtraction scheme in dimensional regularization to compute the full Laurent expansion of the associated Feynman integral. On the third front, we consider unitarity constraints on a class of tree-level UV completions of amplitudes with graviton exchange. In particular, we study completions of Higgs scattering and gauge boson scattering in the Standard Model and gauge boson scattering in diverse dimensions. We find surprising constraints connecting with intuitions from the swampland. This is based on work with Brad Bachu [3].