Structure of Chern-Simons scattering amplitudes from topological equivalence theorem and double-copy

Abstract We study the mechanism of topological mass-generation for 3d Chern-Simons (CS) gauge theories, where the CS term can retain the gauge symmetry and make gauge boson topologically massive. Without CS term the 3d massless gauge boson has a single physical transverse polarization state, while adding the CS term converts it into a massive physical polarization state and conserves the total physical degrees of freedom. We formulate the mechanism of topological mass-generation at S-matrix level. For this, we propose and prove a Topological Equivalence Theorem (TET) which connects the N-point scattering amplitude of the gauge boson’s physical polarization states ( A P a $$ {A}_{\mathrm{P}}^a $$ ) to that of the transverse polarization states ( A T a $$ {A}_{\mathrm{T}}^a $$ ) under high energy expansion. We present a general 3d power counting method on the leading energy dependence of the scattering amplitudes in both topologically massive Yang-Mills (TMYM) and topologically massive gravity (TMG) theories. With these, we uncover a general energy cancellation mechanism for N -gauge boson scattering amplitudes which predicts the cancellation E 4 → E 4−N at tree level. Then, we compute the 4-gauge boson amplitudes of A P a $$ {A}_{\mathrm{P}}^a $$ -states and A T a $$ {A}_{\mathrm{T}}^a $$ -states, with which we explicitly demonstrate the TET and establish such energy cancellations for N = 4. We further extend the double-copy approach to reconstruct the massive 4-graviton amplitude of TMG from the massive 4-gauge boson amplitude of TMYM. With these, we uncover striking large energy cancellations in the 4-graviton amplitude: E 12 → E 1, and establish its correspondence to the leading energy cancellation E 4 → E 0 in the 4-gauge boson amplitude of TMYM.