Scales, Operators, and Power Counting

An effective theory is predictive because it orders ignorance. Once the light fields, exact symmetries, heavy thresholds, background, and target accuracy are specified, dimensional analysis and loop counting identify which operators can affect a calculation and bound the terms that are omitted. “Modified gravity” without this hierarchy is only a list of possible terms; power counting turns it into a controlled approximation.

Required background. Einstein Equations provides the leading gravitational equations and curvature scale; Gravitational Actions and Boundary Terms provides the surface terms retained when higher-derivative operators are varied; Variational and Symplectic Geometry provides the variational language used to identify the actual dynamical variables. The relevant fields and symmetries must first be fixed as on Actions, Fields, and Symmetries.

Helpful background. Formulations of General Relativity supplies the background, constraint, and local-frame distinctions used to build invariant expansion parameters.

Canonical dimensions and operator size

In four spacetime dimensions the action is dimensionless, so a local operator $\mathcal O_i$ of mass dimension $\Delta_i$ enters as

$$\mathcal L_{\mathrm{EFT}} =\mathcal L_{\mathrm{light}} +\sum_i\frac{c_i(\mu)}{\Lambda_i^{\Delta_i-4}}\mathcal O_i.$$

Here the light fields have canonically normalized quadratic terms, $\Lambda_i$ is the threshold or suppression scale appropriate to the operator, $c_i(\mu)$ is a dimensionless Wilson coefficient at renormalization scale $\mu$, and the sum contains every operator allowed by the exact symmetries. For a process whose independent invariants are all of order $E^2$, one insertion gives the estimate

$$\epsilon_i(E) \sim c_i(E)\left(\frac{E}{\Lambda_i}\right)^{\Delta_i-4},$$

after the powers associated with external fields have been compared with the leading amplitude. This is an estimate, not an equality: helicity selection, equations of motion, a symmetry, or an accidental cancellation may make a particular observable smaller. It cannot make an operator with an uncontrolled expansion ratio safely negligible.

Several steps precede this formula.

1. Normalize every propagating field about the selected background. A large background-dependent kinetic coefficient changes the scale seen by its fluctuations.
2. Choose a nonredundant operator basis at the retained order. Integrations by parts, algebraic identities, and perturbative field redefinitions can move coefficients without changing on-shell predictions.
3. Match at each heavy threshold, then run the coefficients to a scale near the process. Logarithms of widely separated scales should not be hidden inside a nominally order-one number.
4. Estimate all independent ratios—energy, momentum transfer, curvature, field strength, background gradients, compactness, and duration—rather than forcing them into one symbol $E$.

The following page develops the basis-reduction step in detail; power counting must be reapplied after Field Redefinitions and Redundant Operators have removed operators that do not independently affect the observable.

Gravity as a derivative expansion

Let the reduced Planck mass satisfy $M_{\mathrm{Pl}}^{-2}=8\pi G$ and write a schematic parity-even metric action as

$$S_{\mathrm{grav}} =\frac{M_{\mathrm{Pl}}^2}{2}\int d^4x\sqrt{-g} \left[ R+\frac{a_2}{\Lambda^2}\mathcal R^2 +\frac{a_3}{\Lambda^4}\mathcal R^3+\cdots \right].$$

$\mathcal R^2$ and $\mathcal R^3$ denote complete bases of curvature contractions, not literal powers of one scalar. On a background whose orthonormal-frame curvature components have characteristic magnitude $\mathscr R$, successive classical terms scale relative to Einstein gravity as

$$\epsilon_2\sim a_2\frac{\mathscr R}{\Lambda^2}, \qquad \epsilon_3\sim a_3\left(\frac{\mathscr R}{\Lambda^2}\right)^2.$$

If the background also varies over a proper length $L$, operators with covariant derivatives bring independent powers of $(\Lambda L)^{-1}$. Near a black hole of areal radius $r_h$, one often has $\mathscr R\sim r_h^{-2}$, so the derivative expansion requires $\Lambda r_h\gg1$ unless a symmetry or resummation establishes a different ordering. A coordinate component that becomes large near a horizon is not by itself a cutoff violation; locally measured curvature invariants and orthonormal components are the relevant quantities.

Expanding $g_{ab}=\bar g_{ab}+2h_{ab}/M_{\mathrm{Pl}}$ canonically normalizes the graviton around weakly curved backgrounds. Einstein–Hilbert vertices carry two derivatives and powers of $M_{\mathrm{Pl}}^{-1}$. A loop built only from those vertices is suppressed relative to the corresponding tree amplitude by

$$\epsilon_{\mathrm{loop}} \sim\frac{E^2}{(4\pi M_{\mathrm{Pl}})^2},$$

up to logarithms and numerical or helicity factors. Heavy-physics corrections and graviton loops are therefore two separate expansions. A small $E/M_{\mathrm{Pl}}$ does not guarantee small $E/\Lambda$ when new physics lies well below the Planck scale. The gravitational derivative and loop counting is derived in Burgess 2004, §§3.1–3.2, pp. 23–31, equations (3.1)–(3.8).

Backgrounds require invariant expansion parameters

A homogeneous field can be large compared with $\Lambda$ while its derivatives remain small. Conversely, a field of small amplitude can vary rapidly enough to invalidate a derivative expansion. The correct parameters follow from the symmetries and operator basis. Typical examples are

$$\frac{\nabla}{\Lambda}, \qquad \frac{R_{abcd}}{\Lambda^2}, \qquad \frac{\nabla_a\phi}{\Lambda^2}, \qquad \frac{\nabla_a\nabla_b\phi}{\Lambda^3}, \qquad \frac{\omega_{\mathrm{local}}}{\Lambda}.$$

For a shift-symmetric scalar, an undifferentiated $\phi/\Lambda$ need not be an expansion parameter at all. For a generic potential it usually is. A background-dependent kinetic matrix $Z^{ab}$ can raise or lower the canonically normalized interaction scale in different directions; the eigenvalues of $Z^{ab}$ and the entire tower of higher operators must then be recounted. The strong-coupling chapter later distinguishes this fluctuation scale from the energy at which genuinely new degrees of freedom appear.

Time introduces another ratio. A correction of instantaneous size $\epsilon$ can accumulate over $N$ cycles as $N\epsilon$. Perturbation theory may need multiple-scale or resummed variables when $N\epsilon\gtrsim1$, even though every local derivative remains below $\Lambda$. Thresholds and nearly on-shell propagators likewise require a reorganized expansion rather than a blind application of canonical dimension. Burgess discusses the background and time-dependence qualifications in Burgess 2004, §3.3, pp. 31–35.

From target accuracy to a truncation

Suppose an observable is required to fractional accuracy $\delta$. A defensible truncation proceeds as follows.

Step	Calculation	Acceptance test
Domain	Bound every invariant kinematic and background ratio over the entire region of use	No retained point crosses a threshold or loses the chosen branch.
Basis	Reduce operators at a fixed order, including sources and boundary terms	No term is removed with equations of motion beyond the order at which they are valid.
Size	Estimate tree, loop, symmetry-breaking, and secular contributions	Every potentially larger-than-$\delta$ contribution is retained or bounded independently.
Remainder	Use the first omitted order and coefficient assumptions to estimate error	The error band is larger than plausible omitted terms and is propagated to the final observable.
Limits	Recover a known lower-order result and test one regime where the expansion fails	The method diagnoses, rather than hides, its boundary of validity.

“Natural” coefficients of order unity are a hypothesis about matching, not a theorem. If the conclusion relies on that hypothesis, vary the omitted coefficients over the justified range. Exact or weakly broken symmetries may protect small coefficients; unexplained cancellations generally do not. A single fitted correction with no bound on the remaining allowed operators is not an EFT prediction.

Worked normalization check

For a canonically normalized massless scalar, choose

$$\mathcal L =-\frac12(\partial\phi)^2 +\frac{c}{8\Lambda^4} \left(\partial_a\phi\,\partial^a\phi\right)^2.$$

The four-point vertex is proportional to the three pairings of external momenta. With all momenta incoming and $s+t+u=0$,

$$\mathcal A_{2\to2} =\frac{c}{4\Lambda^4}\left(s^2+t^2+u^2\right)$$

for this coefficient convention, with $s=-(p_1+p_2)^2$, $t=-(p_1+p_3)^2$, and $u=-(p_1+p_4)^2$. There are eight assignments of the identical fields to each of the three momentum pairings, canceling the factor $1/8$ in the interaction; each pairing then supplies one of $s^2/4$, $t^2/4$, or $u^2/4$. The amplitude is dimensionless, respects the shift symmetry, and scales as $c(E/\Lambda)^4$. These three checks catch a missing derivative, an inconsistent external-field normalization, or an incorrect power of the cutoff. An overall coefficient quoted from another source must still be translated to the source’s definition of $s$, external state normalization, and interaction coefficient.

Common pitfalls

A large field means EFT failure. Only symmetry-allowed dimensionless ratios decide. A shift-symmetric large field with small derivatives can be controlled; a small but rapidly varying field can be uncontrolled.

A formal extra pole is always a particle. Solving a truncated higher-derivative equation exactly can manufacture a mode at or above the cutoff. The EFT equation is solved perturbatively to its retained order unless the higher-derivative action is explicitly proposed as fundamental.

Power counting precedes basis reduction. Redundant operators can appear large while canceling from on-shell observables after a consistent field redefinition. Count the nonredundant basis, but retain all induced sources and boundary terms.

Exercises

Consider

$$S=\frac{M_{\mathrm{Pl}}^2}{2}\int d^4x\sqrt{-g} \left[R+\frac{a_2}{\Lambda^2}\mathcal R^2 +\frac{a_3}{\Lambda^4}\mathcal R^3\right]$$

on a background with $\sqrt{\mathscr R}=0.05\Lambda$ and $\Lambda=0.1M_{\mathrm{Pl}}$. Take $a_2=a_3=1$. Rank the curvature-squared, curvature-cubed, and one-loop Einstein corrections. Which terms must be kept for a target fractional accuracy $10^{-5}$ before observable-specific basis reductions?

Solution

The basic curvature ratio is

$$\epsilon=\frac{\mathscr R}{\Lambda^2}=(0.05)^2=2.5\times10^{-3}.$$

Relative to the Einstein term,

$$\epsilon_2=\epsilon=2.5\times10^{-3}, \qquad \epsilon_3=\epsilon^2=6.25\times10^{-6}.$$

Taking $E^2\sim\mathscr R$, the loop factor is

$$\epsilon_{\mathrm{loop}} \sim\frac{\mathscr R}{(4\pi M_{\mathrm{Pl}})^2} =\frac{(0.05)^2(0.1)^2}{16\pi^2} \simeq1.6\times10^{-7}.$$

The ranking is therefore curvature squared, curvature cubed, then the Einstein loop. The curvature-squared term must be retained at $10^{-5}$. The nominal curvature-cubed term lies below that target, but it sets a $6.25\times10^{-6}$ first-omitted-order estimate, so a claimed $10^{-5}$ error budget is only credible if order-one coefficient assumptions and combinatorial factors are included conservatively. The loop is smaller in this example. Finally, this ranking is not yet an observable calculation: in four-dimensional vacuum gravity some curvature-squared combinations are topological or field-redefinition redundant, while matter, boundaries, or a different dimension can restore their effect. That basis check is the next step.

References

• Burgess, C. P. (2004), “Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory,” Living Reviews in Relativity 7, article 5. DOI; Open PDF.
• Criado, Juan Carlos, and Manuel Pérez-Victoria (2019), “Field Redefinitions in Effective Theories at Higher Orders,” JHEP 03, 038. DOI; Open PDF.