Conformal Gravity from Gauge Theory

TL;DR

The paper develops a novel dimension-six gauge theory whose color-kinematics duality enables a double-copy construction of conformal gravity amplitudes. By matching the CK-dual gauge-theory amplitudes to gravitational amplitudes, the authors reproduce the Berkovits-Witten conformal gravity results and systematically generate non-minimal conformal supergravities via supersymmetric extensions and deformations. Dimensional analysis shows the left gauge theory must be marginal in six dimensions, and the double copy yields four-derivative (Weyl^2) gravity and its extensions, with a rich web of interpolations to Einstein gravity through mass and scalar deformations. The work provides explicit amplitudes up to eight points and a concrete framework for exploring UV properties and loop extensions of conformal gravity.

Abstract

We use the duality between color and kinematics to obtain scattering amplitudes in non-minimal conformal N=0,1,2,4 (super)gravity theories. Generic tree amplitudes can be constructed from a double copy between (super-)Yang-Mills theory and a new gauge theory built entirely out of dimension-six operators. The latter theory is marginal in six dimensions and contains modes with a wrong-sign propagator, echoing the behavior of conformal gravity. The dimension-six Lagrangian is uniquely determined by demanding that its scattering amplitudes obey the color-kinematics duality. The conformal gravity amplitudes obtained from the double copy are compared with the Berkovits-Witten twistor string and shown to agree up to at least eight points in the MHV sector. Our construction can be generalized in a number of ways. Adding scalars to the dimension-six theory gives Maxwell-Weyl gravity, and further adding phi^3 self-interactions among these scalars gives Yang-Mills-Weyl gravity. The latter is identified with Witten's twistor string for maximal N=4 supersymmetry. Deforming the dimension-six theory by adding a Yang-Mills term, m^2 F^2, gives a gauge theory that interpolates between marginal D=6 and D=4 theories. The corresponding double copy gives an interpolation between conformal gravity and Einstein gravity.

Figures (3)

• Figure 1: (a): The relation (\ref{['1stID']}) that enforces that $(T_R^a)^{\alpha \beta}$ is a covariant tensor under infinitesimal gauge-group rotations. (b): The Jacobi identity (enforces that $f^{abc}$ is a covariant tensor under infinitesimal gauge-group rotations). (c): A nice but redundant relation; it follows directly from the identity in figure \ref{['GeneralizedJacobis']}(a).
• Figure 2: (a): The reduction relation (\ref{['Id1a']}) for contracting two $(C^{\alpha})^{ab}$ tensors. (b): The relation (\ref{['2ndID']}) that enforces that $(C^{\alpha})^{ab}$ is a covariant tensor under infinitesimal gauge-group rotations.
• Figure 3: (a): The relation (\ref{['3rdID']}) that enforces that $d^{\alpha \beta \gamma}$ is a covariant tensor under infinitesimal gauge-group rotations. (b): The reduction relation (\ref{['Id1b']}) for $d^{\alpha\beta\gamma}$ contracted with $(C^{\gamma})^{ab}$; it can be more compactly written in terms of anti-commutators: $d^{\alpha\beta\gamma}(C^{\gamma})^{ab} = \{iT_R^{a},iT_R^{b}\}^{\alpha\beta} + \{C^{\alpha},C^{\beta}\}^{ab}$.