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

TL;DR

This work develops a covariant formulation of 3d topologically massive gravity by embedding a dilaton via a conformal transformation and applying BRST quantization, ensuring a consistent massless limit where the massive graviton’s DoF converts to a physical dilaton. It then establishes the Topological Graviton Equivalence Theorem (TGRET), linking high-energy massive-graviton amplitudes to dilaton amplitudes, and uses a generalized gravitational power-counting scheme to explain large energy cancellations that scale as $\frac{5}{2}N$ (Landau) or $\frac{7}{2}N$ (unitary) for $N$-point amplitudes. The paper provides explicit 3- and 4-point calculations verifying TGRET and demonstrates that the extended massive double-copy construction maps graviton and dilaton amplitudes in TMG from gauge-theory amplitudes in the corresponding 3d topologically massive Yang–Mills theory, including nontrivial checks against gauge choices and massless limits. It also clarifies the relation to 3d massless theories (GRD3) in the limit $m\to0$, showing a consistent gravity–gauge duality structure in 3d. Overall, TGRET supplies a robust mechanism for large-energy cancellations in massive 3d gravity without Higgs-like fields and anchors a 3d gravity–gauge double-copy correspondence.

Abstract

Gravitons naturally acquire topological masses in the 3d topologically massive gravity (TMG) theory that includes the gravitational Chern-Simons term. We present a covariant formulation of the TMG theory by introducing an unphysical dilaton field through the conformal transformation. We conduct the BRST quantization of the covariant TMG theory, which reduces to the conventional TMG in the unitary gauge. We demonstrate that this covariant TMG 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 ($N\geqslant 4$) have striking energy cancellations by powers proportional to $\frac{5}{2}N$ ($\frac{7}{2}N$) in the Landau (unitary) gauge. This explains the large energy cancellations of $E^{11}\to E^1$ (Landau gauge) and $E^{12}\to E^1$ (unitary gauge) for the four graviton amplitudes. 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 graviton (dilaton) scattering amplitudesin the TMG theory from the corresponding gauge boson (adjoint scalar) amplitudes in the topologically massive Yang-Mills theory.

Figures (11)

• Figure 1: Schematic outline and summary of the present analyses: 1). the topological graviton equivalence theorem (TGRET) in the covariant TMG theory, in comparison with the topological gauge equivalence theorem (TGAET) in the TMYM theory; 2). the massive double-copy constructions of both the graviton amplitudes and dilaton amplitudes for the covariant TMG theory from the gauge boson amplitudes and adjoint scalar amplitudes in the TMYM-Scalar theory (including adjoint scalars). Further explanations are given in the text.
• Figure 2: Three-point gauge boson scattering in the TMYM theory (left diagram) and the three-point graviton scattering in the TMG theory (right diagram). The amplitudes of these two scattering processes are connected by the double copy.
• Figure 3: Feynman diagrams for the triple graviton scattering (left diagram) and for the $\phi\space\phi\space h_{\rm{P}}^{}$ scattering (right diagram), which are used for verifying the TGRET \ref{['eq:TGRET']}.
• Figure 4: Relevant Feynman diagrams of the three-point graviton (dilaton) scattering for verifying the Slavnov-Taylor-type identities \ref{['eq:hhF2-TGETID']} and \ref{['eq:hhF3-TGETID']}.
• Figure 5: Four-point massive graviton scattering (left diagram) versus the corresponding four dilaton scattering (right diagram), and the demonstration of the topological gravitational equivalence theorem (TGRET) in the high energy limit. The $4h_{\rm{P}}^{}$ diagram represents the dressed four-point scattering amplitudes, which have absorbed the contributions from the $4\space h_{\rm{P}}^{}$ contact diagram, whereas the leading four-dilaton amplitude is given by the $4\space\phi$ diagram (right diagram) via graviton exchanges alone.