Scattering Equations and S-Matrices
We introduce a new formulation for the tree-level S-matrix in theories of massless particles. Sitting at the core of this formulation are the scattering equations, which yield a map from the kinematics of a scattering process to the moduli space of punctured Riemann spheres. The formula for an amplitude is constructed by an integration of a certain rational function over this moduli space, which is localized by the scattering equations. We provide a detailed analysis of the solutions to these equations and introduce this new formulation. After presenting some illustrative examples we show how to apply this formulation to the construction of closed formulas for actual amplitudes in various theories, using only a limited set of building blocks. Examples are amplitudes in Einstein gravity, Yang-Mills, Dirac-Born-Infeld, the U(N) non-linear sigma model, and a special Galileon theory. The consistency of these formulas is checked by systematically studying locality and unitarity. In the end we discuss the implication of this formulation to the Kawai-Lewellen-Tye relations among amplitudes.