Worldsheet formulations of quantum field theories have had wide ranging impact on the study of scattering amplitudes. However, the mathematical framework becomes very challenging on the higher-genus worldsheets required to describe loop effects. I will describe how in such worldsheet models based on the scattering equations, formulae on higher-genus surfaces can be transformed to remarkably simple expressions on the Riemann sphere. My talk will focus on both supersymmetric and non-supersymmetric Yang-Mills theory and gravity, and discuss a proposal for the all-loop integrand.
