The Scattering Equations and Their Off-Shell Extension

摘要

1. Introduction There are two beautiful theories, both of which can be formulated on gauge principles. One is general relativity,~1 which has passed all the classical tests, but has resisted consistency with quantum corrections. The other is non-Abelian~2 (Yang-Mills) theory and its extensions, which are the backbones of electroweak and strong interactions. As we also know, while the gauge principle~3 is elegant, calculations based on perturbation expansion in the past were cumbersome. The situation has been fundamentally changed in the last 15 years or so, because analyticity has been better understood and used, which results in recurrence relations,~4 gauge freedom has been better deployed in the language of spinors~5 and taken advantage of, and most surprisingly, many hidden symmetries in the amplitudes have been unveiled.~(6-7) The last discovery leads to a much better understanding (and more manageable) of the connection between gauge and graviton amplitudes.~8 We shall review some of these progresses, which give a historical perspective of the scattering equation formulation,~9 which seems to give the possibility of a new interaction principle, perhaps beyond local field theories. In the original formulation, CHY (Cachazo-He-Yuan)~9 is for on-shell tree amplitudes. We~(10) shall show that by a simple modification, which is mostly dictated by M?bius covariance and off-shell energy-momentum conservation, CHY is also valid for tree Green's functions. This establishes CHY as a viable unitary theory.