Structure of Chern-Simons scattering amplitudes from topological equivalence theorem and double-copy
Yan-Feng Hang (Tsung-Dao Lee Institute & School of Physics and Astronomy, Key Laboratory for Particle Astrophysics and Cosmology (MOE), Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University, Shanghai, 200240, China); Hong-Jian He (Tsung-Dao Lee Institute & School of Physics and Astronomy, Key Laboratory for Particle Astrophysics and Cosmology (MOE), Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University, Shanghai, 200240, China, Institute of Modern Physics and Department of Physics, Tsinghua University, Beijing, 100084, China, Center for High Energy Physics, Peking University, Beijing, 100871, China); Cong Shen (Tsung-Dao Lee Institute & School of Physics and Astronomy, Key Laboratory for Particle Astrophysics and Cosmology (MOE), Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University, Shanghai, 200240, China)
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 ( $$ {A}_{\mathrm{P}}^a $$ ) to that of the transverse polarization states ( $$ {A}_{\mathrm{T}}^a $$ ) 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 $$ {A}_{\mathrm{P}}^a $$ -states and $$ {A}_{\mathrm{T}}^a $$ -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.
