Structure of Chern-Simons Graviton Scattering Amplitudes from Topological Graviton Equivalence Theorem and Double Copy

Abstract

Gravitons naturally acquire topological masses in the 3d topologically massive gravity (TMG) theory that includes the gravitational Chern-Simons term. We present a Weyl-transformed TMG (WTMG) formulation by introducing an unphysical dilaton field through the Weyl transformation. We perform the BRST quantization of the WTMG, which reduces to the conventional TMG in the unitary gauge. We demonstrate that this WTMG theory conserves the physical degrees of freedom (DoF) in the massless limit, under which the physical massive graviton becomes an unphysical massless graviton and its physical DoF is converted to the massless dilaton. With these, we newly establish a Topological Graviton Equivalence Theorem (TGRET), which connects each scattering amplitude of physical gravitons to the corresponding dilaton scattering amplitude in the high energy limit. The TGRET provides a general mechanism to guarantee all the large energy cancellations in any massive graviton scattering amplitudes. Applying the TGRET and using the generalized gravitational power counting rule, we prove that the $N$-point massive graviton amplitudes ($N\!\!\geqq\!4$) have striking energy cancellations by powers proportional to $\frac{5}{2}N$ ($\frac{7}{2}N$) in the Landau (unitary) gauge. For four graviton scattering amplitudes, this explains their large energy cancellations of $E^{11}\to E^1$ (Landau gauge) and $E^{12}\to E^1$ (unitary gauge). We compute the four-point graviton (dilaton) amplitudes and explicitly demonstrate the TGRET and these large energy cancellations. With the extended massive double-copy approach, we systematically construct the three- and four-point graviton (dilaton) scattering amplitudes in the WTMG theory from the corresponding gauge boson (adjoint scalar) amplitudes in the topologically massive Yang-Mills theory.