Topological Equivalence Theorem and Double-Copy for Chern-Simons Scattering Amplitudes

We study the mechanism of topological mass-generation for 3d Chern-Simons gauge theories and propose a brand-new Topological Equivalence Theorem to connect scattering amplitudes of the physical gauge boson states to that of the transverse states under high energy expansion. We prove a general energy cancellation mechanism for $N$-point physical gauge boson amplitudes, which predicts large cancellations of $E^{4-L}\to E^{(4-L)- N}$ at any $L$-loop level ($L\geqslant 0$). We extend the double-copy approach to construct massive graviton amplitudes and study their structures. We newly uncover a series of strikingly large energy cancellations $E^{12}\to E^1$ of the tree-level four-graviton scattering amplitude under high energy expansion and establish a new correspondence between the two energy cancellations in the topologically massive Yang-Mills gauge theory and the topologically massive gravity theory. We further study the scattering amplitudes of Chern-Simons gauge bosons and gravitons in the nonrelativistic limit.