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

Technical Summary: Extending Weinberg's EFT to Sixth-Order Scalar-Tensor Theories

Problem Statement
The paper addresses the systematic construction of the most general effective field theory (EFT) for scalar-tensor gravity, extending the framework originally established by Steven Weinberg up to fourth-order derivatives. While Horndeski and beyond-Horndeski theories have dominated recent literature due to their manifestly second-order equations of motion, the EFT perspective suggests that higher-derivative operators naturally arise when integrating out heavy degrees of freedom. The challenge lies in identifying a minimal, non-redundant basis of operators at higher orders (specifically sixth order in derivatives) that respects general covariance, accounting for the vast number of potential terms and the various redundancies introduced by integration by parts (IBP), geometric identities, and field redefinitions.

Methodology
The authors employ a two-pronged approach to construct and verify the operator basis:

1. Covariant Lagrangian Construction (On-Shell EFT):

   • Algorithm: The authors generate all possible six-derivative operators consistent with diffeomorphism invariance. These are classified by the number of Riemann tensors ($r$) and "dressed scalars" (scalar fields with at least one derivative, $j$).
   • Redundancy Elimination: A systematic flow is applied to eliminate redundant terms using:
     • Symmetries of the Riemann and Levi-Civita tensors.
     • Bianchi identities and Dimensional Dependent Identities (DDIs).
     • Integration by parts (IBP).
     • On-Shell Reduction: Crucially, the authors substitute the leading-order equations of motion (Einstein and scalar field equations) to remove terms proportional to $\Box \phi$ and the Ricci tensor/scalar. This transitions the theory from an off-shell to an on-shell EFT, where operators proportional to the equations of motion are treated as redundant via field redefinitions.
   • Parity Separation: The analysis is split into parity-even and parity-odd sectors to manage the complexity of the operator space.

2. Scattering Amplitude Verification:

   • To independently verify the operator counting, the authors utilize on-shell scattering amplitude techniques in four-dimensional flat spacetime.
   • They analyze contact diagrams (local interactions without propagator poles) for processes involving gravitons and scalars up to six-point interactions.
   • Using the spinor-helicity formalism, they construct amplitudes that respect little-group scaling and Lorentz covariance. The number of independent amplitude structures at a given derivative order is matched against the number of independent EFT operators.

Key Contributions and Results

• Operator Counting: The paper establishes that the minimal on-shell basis for six-derivative scalar-tensor theories in four dimensions consists of eight independent operators:

  • Five Parity-Even Operators:
    1. $R_{\alpha\beta\gamma\delta}R^{\alpha\beta}_{\phantom{\alpha\beta}\zeta\eta}R^{\gamma\delta\zeta\eta}$ (Cubic curvature)
    2. $R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}\nabla_\zeta\phi\nabla^\zeta\phi$ (Curvature-squared coupled to scalar kinetic term)
    3. $R_{\alpha\beta\gamma\delta}\nabla^\alpha\phi\nabla^\gamma\phi\nabla^\delta\nabla^\beta\phi$ (Mixed curvature-scalar derivative)
    4. $(\nabla_\alpha\phi\nabla^\alpha\phi)^3$ (Pure scalar kinetic term)
    5. $\nabla_\alpha\phi\nabla^\alpha\phi\nabla^\gamma\nabla^\beta\phi\nabla_\gamma\nabla_\beta\phi$ (Pure scalar derivative)
  • Three Parity-Odd Operators:
    1. $\epsilon_{\zeta\eta\theta\kappa}R_{\alpha\beta\gamma\delta}R^{\alpha\beta\zeta\eta}R^{\gamma\delta\theta\kappa}$ (Cubic curvature with Levi-Civita)
    2. $\epsilon_{\gamma\delta\zeta\eta}R^{\alpha\beta}_{\phantom{\alpha\beta}\gamma\delta}R_{\alpha\beta}^{\phantom{\alpha\beta}\zeta\eta}\nabla_\theta\phi\nabla^\theta\phi$ (Curvature-squared coupled to scalar kinetic term)
    3. $\epsilon_{\gamma\delta\zeta\eta}R^{\alpha\beta}_{\phantom{\alpha\beta}\gamma\delta}\nabla_\alpha\phi\nabla^\zeta\phi\nabla^\eta\nabla_\beta\phi$ (Mixed curvature-scalar derivative)

• Explicit Lagrangian: The authors provide the explicit form of the sixth-order Lagrangian ($S_6[g, \phi]$) with arbitrary scalar-dependent coupling functions ($e_n(\phi)$ and $o_n(\phi)$), scaling as $1/\Lambda^2$ where $\Lambda$ is the cutoff scale.

• Verification: The scattering amplitude analysis confirms the operator count derived from the covariant Lagrangian method. The amplitude analysis reveals that mixed-helicity amplitudes vanish at this order, leading to a one-to-one correspondence between even- and odd-parity operators when restricting to terms containing at least one Riemann tensor.

Significance and Claims
The paper claims to provide the next-to-leading-order (NLO) extension of scalar-tensor gravity, furnishing a robust framework for exploring:

• Quantum and Stringy Corrections: Six-derivative operators represent the next-to-next-to-leading order in derivative expansions, capturing cubic curvature terms and mixed scalar-curvature structures expected in low-energy string expansions and UV completions.
• Parity-Violating Phenomena: The inclusion of parity-odd operators allows for the study of effects such as gravitational wave birefringence (chiral modes) and cosmic birefringence, which are relevant for cosmological observations and gravitational wave astronomy.
• Strong-Field Regimes: The framework is applicable to black hole physics and cosmology, particularly in scenarios where fourth-order corrections might be suppressed (e.g., due to approximate shift symmetry or flat potentials), allowing sixth-order terms to dominate the dynamics.

The authors emphasize that while the specific choice of basis is not unique (due to field redefinitions and IBP), the number of independent operators is fixed. They note that their results agree with previous shift-symmetric studies (specifically Ref. [21]) in the appropriate limit but extend beyond that by including general scalar-dependent couplings and both parity sectors. The work is presented as a foundational step for future phenomenological studies and for exploring matter couplings in open EFTs, though the current analysis is restricted to the vacuum sector away from external sources.