Gravity as Effective Field Theory

Gravity is predictive as a quantum effective field theory at sufficiently low energies. Power counting separates calculable long-distance effects from local coefficients that encode shorter-distance physics, without assuming ultraviolet completeness.

Enter this chapter

Begin with Applying EFT Power Counting to Gravity to identify the gravitational degrees of freedom and expansion parameters.

Helpful background. Effective Field Theory of Gravity: Architecture and Power Counting and Local Field Redefinitions and the Equivalence Theorem develop the definitions used by the chapter’s opening pages.

Topics

1. Applying EFT Power Counting to Gravity — Assigns momentum, curvature, loop, and species order.
2. Curvature Operator Bases and Field Redefinitions — Reduces local terms without discarding physical data.
3. Background-Field Quantization, Gauge Fixing, and Ghosts — Constructs the metric and ghost Hessians.
4. Local Supersymmetry and Supergravity as Low-Energy EFT — Adds the gravitino and local-supersymmetry constraints.
5. One-Loop Graviton EFT — Isolates graviton/ghost divergences and nonanalytic amplitudes.
6. Matter Contributions to Gravitational Matching — Matches matter loops without double counting.
7. Heavy Thresholds, Species, and the Gravitational Cutoff — Makes active-field and multiplicity estimates scale dependent.
8. Matching Amplitudes to Background Observables — Translates common invariant data between asymptotic and curved settings.
9. Graviton and Matter Nonlocal Form Factors — Keeps sector coefficients and causal prescriptions separate.
10. Long-Distance Quantum Corrections — Extracts infrared tails from nonanalytic momentum dependence.
11. Relational and Gauge-Invariant Gravitational Observables — Attaches predictions to physical quantities.
12. Vacuum Energy and the Cosmological Constant — Separates threshold sensitivity from the measured parameter.
13. Higher-Derivative Poles, Ghost Diagnostics, and Order Reduction — Classifies poles relative to the cutoff.
14. Validity, Unitarity, and Breakdown — Combines the independent stopping conditions.

How the ideas fit together

Local curvature operators absorb ultraviolet sensitivity order by order. Nonanalytic momentum dependence generated by light propagation has a different origin and produces long-distance corrections that cannot be removed by changing local coefficients Donoghue 1994, §§ II–IV.

The claimed correction must still be attached to an observable with specified gauge and field-redefinition treatment. Matter loops, graviton loops, and a semiclassical expectation value represent different approximations. Extra poles obtained by resumming a finite higher-derivative truncation also need not represent physical particles within the EFT’s domain.

The comparison below separates the assumptions and tests needed for these different conclusions.

Domain and failure conditions

Calculation or observable	Dynamical fields and internal loops	Local basis and field redefinitions	Gauge-invariant datum	Threshold or nonanalytic input	Expansion and decisive failure
Tree graviton–matter amplitude	External matter/gravitons; no loop	EH plus required contact operators; EOM-redundant terms movable	On-shell amplitude	Heavy physics in Wilson coefficients	$p/\Lambda_{\mathrm{EFT}}$; partial waves or omitted operators become large
Matter effective action	Internal matter only; metric prescribed	Curvature basis with matter-threshold matching	Renormalized background response	Mass-dependent form factors and cuts	Curvature/derivative expansion fails near threshold or strong background
Graviton one-loop amplitude	Internal graviton and ghost lines, plus declared mixed diagrams	Four-derivative counterterm closure	On-shell nonanalytic amplitude	Massless cuts and $\log(-p^2)$	$p/(4\pi M_{\mathrm{Pl}})$ or species-enhanced loop reaches unity
Heavy-threshold matching	Heavy fields integrated out once	Local Wilson coefficients below the mass	Equality of low-energy amplitudes/responses	Smooth decoupling around $p\sim m$	Explicit heavy loop plus matched coefficient double counts
Weak-background response	Fields declared in the 1PI or in-in action	Basis translated before variation	Smeared curvature or relational response	Retarded continuation of nonlocal kernels	No asymptotic S-matrix assumed; instability or curvature expansion fails
Long-distance tail	Massless loop sectors resolved	Analytic terms remain coefficient/scheme dependent	Scattering angle, time delay, or other invariant response	Nonanalytic $\lvert p\rvert$ and $\log(-p^2)$ pieces	Extrapolation to $r\sim\Lambda_{\mathrm{EFT}}^{-1}$ is forbidden
Relational observable	Gauge-fixed fields internally, clocks/dressings externally	Invariant under local EOM redefinitions to retained order	Event-defined curvature, asymptotic charge, or dressed amplitude	Matching residual tested across gauges/bases	Bare coordinate component or incomplete clock system
Vacuum energy	All thresholds contributing to the zero-derivative term	Renormalized $\Lambda$ is an independent measured coupling	Curvature response after matching	Contributions scale as heavy masses to the fourth power	Regulator power divergence mistaken for an observable, or naturalness claimed solved
Higher-derivative propagator	EH pole plus perturbative insertions below cutoff	EOM-redundant pieces removed only within observable	Pole/residue of a controlled low-energy amplitude	Extra exact pole compared with $\Lambda_{\mathrm{EFT}}$	Above-cutoff root treated as an EFT particle or order reduction used across a genuine low pole

Where to go next

Continue to Quantum Fields in Cosmology to derive canonically normalized modes on an expanding background.

References

• Donoghue, J. F. “General Relativity as an Effective Field Theory: The Leading Quantum Corrections.” Physical Review D 50, 3874–3888 (1994). doi:10.1103/PhysRevD.50.3874. Open PDF