Extending Weinberg's EFT: effective scalar-tensor theories up to sixth order

TL;DR

This work extends Weinberg's scalar–tensor EFT to include six-derivative terms, constructing a complete on-shell operator basis with $5$ even-parity and $3$ odd-parity structures, each weighted by functions of the scalar field. The authors develop a systematic procedure to remove redundancies via field redefinitions, IBP, and EOMs, yielding a minimal, complete set at order $6$ in derivatives. They corroborate the counting with on-shell scattering amplitudes in four dimensions, showing agreement between amplitude-based and Lagrangian approaches and highlighting the parity correspondences. The resulting framework enables robust exploration of quantum/stringy corrections, parity-violating gravitational phenomena, and strong-curvature effects in cosmology, black holes, and gravitational waves, and points to future extensions to higher-derivative orders and matter couplings.

Abstract

We present a systematic construction of the six-derivative effective scalar-tensor theories, extending the four-derivative framework previously developed by Steven Weinberg. The on-shell effective field theory comprises five parity-even and three parity-odd independent six-derivative scalar-tensor interactions, representing all inequivalent deformations consistent with general covariance. We further confirm this operator counting through an independent analysis using the scattering amplitude formalism in four-dimensional flat spacetime. The six-derivative Lagrangian constructed here provides the next-to-leading-order extension of scalar-tensor gravity, furnishing a robust framework for exploring quantum or stringy corrections, parity-violating interactions, and strong-curvature effects in cosmology, black hole physics and gravitational wave observations.

Figures (1)

• Figure 1: The diagram illustrates how a typical set of terms $\mathcal{S}_r^j$ transforms under the application of Bianchi identities, dimensional-dependent identities, Ricci identities, integration by parts, and the leading-order equations of motion (\ref{['eom']}). The use of symmetries, Bianchi identities, and dimensional-dependent identities does not generate terms belonging to other groups. In contrast, Ricci identities, integration by parts, and the leading-order equations of motion (\ref{['eom']}) may produce terms outside the original set, as indicated by the arrows in the diagram.