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 ($$ {A}_{\mathrm{P}}^a $$ A P a ) to that of the transverse polarization states ($$ {A}_{\mathrm{T}}^a $$ A 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 E4→ E4−N at tree level. Then, we compute the 4-gauge boson amplitudes of $$ {A}_{\mathrm{P}}^a $$ A P a -states and $$ {A}_{\mathrm{T}}^a $$ A 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: E12→ E1, and establish its correspondence to the leading energy cancellation E4→ E0in the 4-gauge boson amplitude of TMYM.

Scattering amplitudes of Kaluza-Klein strings and extended massive double-copy

“…The Cachazo-He-Yuan (CHY) formalism [22][23][24][25][26] shows that the KLT kernel can be interpreted as the inverse amplitudes of bi-adjoint scalars, and this can be generalized to double-copy relations for other field theories [25,26]. So far substantial efforts have been made to formulate and test the double-copy constructions between the massless gauge theories and massless GR [20], and some recent works attempted to extend the doublecopy method to the 4d massive Yang-Mills (YM) theory versus Fierz-Pauli-like massive gravity [27][28][29][30], to the KK-inspired effective gauge theory with extra global U(1) [31], to compactified 5d KK gauge/gravity theories [32], and to the 3d Chern-Simons theories with or without supersymmetry [33][34][35][36][37][38][39][40].…”

“…Such geometric "Higgs" mechanisms can be quantitatively formulated at the scattering S-matrix level by the KK gauge boson equivalence theorem (KK GAET) [41,44,45] and KK gravitational ET (KK GRET) [32], which generally ensure much better high energy behaviors of the KK scattering amplitudes than that of other ill-defined massive theories (with explicitly broken gauge/gravity symmetries) and thus hold real promise for successful double-copy construction. The 3d CS gauge/gravity theories [46,47] naturally realize topological massgeneration for the gauge bosons (gravitons) in a gauge-invariant (diffeomorphism-invariant) way, which can also ensure good high energy behaviors of the scattering amplitudes [40] and realize successful double-copy constructions [33][34][35][36][37][38][39][40].…”

The SAGEX Review on Scattering Amplitudes, Chapter 14: Classical Gravity from Scattering Amplitudes

“…[209,210] discussed the application of the double copy to anyons and to topological quantisation conditions. Another interesting example of the double copy relating three-dimensional theories is that of topologically massive gravity and topologically massive Yang-Mills theory [211][212][213][214][215]. In this context, a natural analogue of the Weyl double copy has recently been presented: the Cotton double copy, based on the Cotton tensor [216,217].…”

Off-Shell Color-Kinematics Duality for Chern-Simons

“…Much more recently, there has been renewed interest in the double copy applied to 3D massive theories. A new massive double copy has been applied to topologically massive 3D Yang-Mills theory [186][187][188][189][190][191][192], which includes a Chern-Simons term in the Lagrangian. In the current paper, we will not directly make used of these quite different constructions, but instead follow the standard formulation [1].…”