The Gravitational EFT Action

The gravitational EFT action is the most general local action built from the light fields and their symmetries, ordered by derivatives and field powers. General covariance fixes the operator classes but not their Wilson coefficients. Newton’s constant and the cosmological constant are leading measured parameters; higher-curvature coefficients summarize short-distance matching and run under low-energy loops.

Required background. Flat-Spacetime Quantum Field Theory supplies operator expansions and Wilson coefficients. Gravitational Actions and Boundary Terms supplies the variational principle.

Helpful background. Einstein Equations supplies the leading equation, Quantum Information supplies unitarity language, and Quantum Gravity supplies the scale hierarchy.

Local covariant operator basis

In four dimensions, with the metric and declared light matter fields retained, the bulk density begins

$$\mathcal L_{\mathrm{EFT}}=\sqrt{-g}\left[ \frac{R-2\Lambda_{\mathrm{cc}}}{16\pi G} +c_1R^2+c_2R_{\mu\nu}R^{\mu\nu} +c_3R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} +\frac{1}{\Lambda^2}\sum_i d_i\mathcal O_i^{(6)}+\cdots \right].$$

Here $\Lambda_{\mathrm{cc}}$ is the cosmological constant with mass dimension two; $\Lambda$ is a heavy suppression scale with mass dimension one. The notation $\mathcal O_i^{(6)}$ denotes six-derivative scalars such as curvature-cubic terms or derivatives acting on curvature. Matter adds covariant operators involving its fields, masses, and field strengths. Discrete symmetries must be stated: parity-odd densities are allowed unless excluded.

Dimensional analysis checks the hierarchy. In natural units $[R]=2$, $[G]=-2$, $[c_i]=0$, and a six-derivative coefficient has dimension $-2$. At a characteristic momentum $E$, an $R^2$ insertion is suppressed relative to Einstein exchange by a dimensionless combination of order $Gc_iE^2$, while a six-derivative insertion carries an additional $E^2/\Lambda^2$ relative to an appropriate four-derivative term.

Redundancy and boundary qualifications

An operator list is not yet a minimal observable basis. In four dimensions,

$$E_4=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} -4R_{\mu\nu}R^{\mu\nu}+R^2$$

is the Euler density. On a closed manifold its bulk contribution is topological; with a boundary, the corresponding boundary completion matters. Local field redefinitions can also move operators proportional to the leading Einstein equation. For pure on-shell graviton scattering about flat space, Ricci-squared terms at the relevant order can therefore be redundant, while matter, off-shell correlators, nonzero cosmological constant, or finite boundaries can make the same-looking terms contribute.

The correct basis is observable and regime dependent but predictions are not: two bases related by a nonsingular local field redefinition give the same on-shell amplitudes when coefficients and external operators are transformed consistently. Dropping an operator solely because it vanishes on one background is unsafe if fluctuations or matter make it nonzero.

Higher-derivative actions require boundary terms compatible with the chosen variational data. For asymptotically flat scattering those terms usually organize surface variations rather than new bulk amplitudes. For finite regions or thermodynamic ensembles they can contribute directly and must be included.

Wilson coefficients and matching

Write a renormalized coefficient as $c_i(\mu)$. Loops of light gravitons and matter generate scale dependence; matching to a heavy theory at scale $M$ fixes a boundary value. A low-energy amplitude then combines

$$c_i(\mu)\,P_i(p)+b_iP_i(p)\log\!\left(\frac{p^2-i0}{\mu^2}\right),$$

so the running of $c_i$ cancels the explicit $\mu$ dependence. Here $P_i$ is the local momentum polynomial and $b_i$ the logarithm’s coefficient. At the displayed order, treating $b_i$ as constant gives $\mu\,dc_i/d\mu=2b_i$; $b_i$ itself is not the beta function. Its value follows from the light spectrum and the lower-order couplings, while the finite part of $c_i$ is matching data. At higher orders, loops with previously matched higher-derivative vertices can also contribute to nonanalytic terms.

Matching compares the same infrared observable in the full and effective theories, expanded at $p^2/M^2\ll1$. The difference is analytic and determines local coefficients. Matching one coefficient is not evidence for every feature of the proposed ultraviolet theory, and a coefficient can receive contributions from several heavy sectors.

Donoghue formulates quantum GR precisely in this Wilsonian, symmetry-based sense and emphasizes that finitely many parameters enter at any fixed low-energy order Donoghue 2023, §§2–5.

Truncation and independent checks

Suppose the target accuracy is relative order $E^2/\Lambda^2$. Include every operator that contributes at that order, every loop of lower-order operators at the same order, and the counterterms required to renormalize them. The first omitted correction is then the larger of the next derivative order, next loop order, or an independently small curvature or mass ratio.

Two limits check the construction. As $E/\Lambda\to0$, higher-derivative tree contributions vanish and the Einstein action is recovered. If a nominally heavy field becomes light, $E/M$ is no longer small; its nonanalytic threshold cannot be represented by the local series, so it must be restored as an explicit degree of freedom.

Do not resum a finite truncation into an exact propagator near $E\sim\Lambda$. Extra roots then depend on operators omitted at the same order and cannot be assigned reliable particle status. Perturbative insertion of higher derivatives is the controlled EFT procedure.

Exercises

Exercise. In four dimensions, compare the size of an $R^2$ term with coefficient $c$ to the Einstein term on a smooth background with curvature scale $R\sim E^2$. Derive the dimensionless expansion ratio and state when the truncation is controlled.

Solution

The Einstein density scales as $E^2/G$, while $cR^2$ scales as $cE^4$. Their ratio is

$$\epsilon_{R^2}\sim cGE^2.$$

The correction is perturbative when $\lvert c\rvert GE^2\ll1$ and every other retained ratio is also small. If matching gives $c\sim1/(GM^2)$, the ratio becomes $E^2/M^2$, as expected for a heavy threshold. A large $c$ can lower the actual EFT control scale below the Planck scale, so $GE^2\ll1$ alone is not sufficient.

References

• Donoghue, John F. “Quantum General Relativity and Effective Field Theory.” 2023. arXiv:2211.09902.