  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.
