Covariant Power Counting

Covariant power counting compares terms before their full tensor algebra is evaluated. It tracks derivatives, loops, Wilson coefficients, background curvature, and external normalization in dimensionless ratios. The key comparison is not “tree versus loop” by itself: a higher-curvature tree and an Einstein loop can contribute at the same momentum order, so a consistent prediction must retain both.

Required background. Flat-Spacetime Quantum Field Theory supplies loop and operator counting. Gravitational Actions and Boundary Terms supplies the covariant action.

Helpful background. Einstein Equations supplies the leading background, Quantum Information supplies unitarity checks, and Quantum Gravity supplies invariant scale criteria.

Canonical normalization and scales

Declare the reduced Planck mass

$$\bar M_{\mathrm{Pl}}^2=\frac{1}{8\pi G}=\frac{4}{\kappa^2}.$$

With $g_{\mu\nu}=\eta_{\mu\nu}+2h_{\mu\nu}/\bar M_{\mathrm{Pl}}$, an Einstein $n$-graviton vertex scales as

$$V_n^{(2)}\sim \kappa^{n-2}E^2 =\left(\frac{2}{\bar M_{\mathrm{Pl}}}\right)^{n-2}E^2.$$

A local operator with $d$ derivatives and coefficient written schematically as $c_d\Lambda^{4-d}$ contributes additional powers of $E/\Lambda$, together with the same field-normalization factors. Exact coefficients depend on the chosen operator basis; the energy exponent follows from derivative counting.

The independent small quantities may include

$$\epsilon_G=GE^2, \qquad \epsilon_\Lambda=\frac{E^2}{\Lambda^2}, \qquad \epsilon_R=\frac{\lvert R\rvert}{\Lambda^2},$$

plus masses, impact parameters, occupation numbers, and state fluctuations. The Ricci scalar here is only a representative curvature scale: Ricci-flat backgrounds require tidal or other Riemann data. A single “energy” is insufficient when $s$, $t$, curvature, and thresholds are parametrically different.

Loop and derivative formula

For a connected graph with $L$ loops and vertices of derivative order $d_v$, topological identities give the generic momentum degree

$$D=2+2L+\sum_v(d_v-2).$$

Each Einstein loop adds the relative factor

$$\epsilon_{\mathrm{loop}}\sim \frac{\kappa^2E^2}{16\pi^2} =\frac{E^2}{4\pi^2\bar M_{\mathrm{Pl}}^2} \sim GE^2,$$

where the final relation suppresses convention-dependent numerical factors. Each insertion of a four-derivative operator raises $D$ by two. Donoghue derives this low-energy organization for graviton and matter amplitudes Donoghue 2023, §§3–5.

For example, let the action contain $cR^2$. Relative to an Einstein tree contribution at scale $E$,

$$\frac{\mathcal A_{R^2}^{\mathrm{tree}}}{\mathcal A_{\mathrm{EH}}^{\mathrm{tree}}} \sim c\kappa^2E^2,$$

while

$$\frac{\mathcal A_{\mathrm{EH}}^{\mathrm{1\ loop}}}{\mathcal A_{\mathrm{EH}}^{\mathrm{tree}}} \sim\frac{\kappa^2E^2}{16\pi^2}.$$

They are the same derivative order when the operator contributes to the observable. This is generic power counting, not a claim that $R^2$ changes the four-dimensional pure-graviton on-shell amplitude: there the curvature-squared bulk basis is redundant up to the Euler term at this order. With matter or in a nonredundant sector, the relevant local coefficients absorb the corresponding loop divergence and scale dependence.

A mixed graph provides a sharper check. Take one loop and one four-derivative $R^2$ insertion. The general index gives

$$D=2+2(1)+(4-2)=6.$$

Relative to a leading Einstein tree amplitude, its generic size is

$$\frac{\mathcal A_{R^2}^{\mathrm{1\ loop}}} {\mathcal A_{\mathrm{EH}}^{\mathrm{tree}}} \sim \left(c\kappa^2E^2\right) \left(\frac{\kappa^2E^2}{16\pi^2}\right).$$

It is therefore a six-derivative contribution and renormalizes operators at that order. Counting the loop or the insertion alone would place it incorrectly. The conclusion survives changes in graph topology because $L=I-V+1$ absorbs any replacement of a higher-point vertex by lower-point vertices and propagators.

Smooth backgrounds and external legs

On a smooth background, derivatives acting on fluctuations scale with their wavelength while background curvatures count as two derivatives. For a curvature-cubic term, write schematically $\mathcal R=\mathcal R_{\mathrm{bg}}+\delta\mathcal R$ and $\delta\mathcal R\sim\kappa E^2h$. Its density contains terms of the form

$$\frac{1}{\Lambda^2} \left[\mathcal R_{\mathrm{bg}}^3 +3\mathcal R_{\mathrm{bg}}^2\delta\mathcal R +3\mathcal R_{\mathrm{bg}}(\delta\mathcal R)^2 +(\delta\mathcal R)^3\right],$$

where tensor contractions and further nonlinear fluctuation terms depend on the operator. Every displayed term has mass dimension four. For example, the quadratic term scales as $\kappa^2\mathcal R_{\mathrm{bg}}E^4h^2/\Lambda^2$; relative to the Einstein kinetic term $E^2h^2$, its ratio is $\kappa^2\mathcal R_{\mathrm{bg}}E^2/\Lambda^2$. Comparing terms requires the same number of external fields and their canonical normalization.

External canonical normalization matters. Counting an $n$-point Green function of $g_{\mu\nu}$ differs by powers of $\bar M_{\mathrm{Pl}}$ from an amplitude of canonical $h_{\mu\nu}$. A reported scaling must say whether the momentum-conserving delta function and external wave functions have been removed. Ratios to a leading observable are often safer because those conventions cancel.

Background equations can reduce an operator basis but should be used only at the declared order. Replacing $R_{\mu\nu}$ by a matter stress tensor with the leading Einstein equation moves contributions between pure-curvature and matter operators; it does not eliminate their physical content.

For a weak encounter with impact parameter $b$ and source mass $M$, the Fourier wave number resolving that distance is of order $1/b$, while background tidal curvature scales as $GM/b^3$. Long-distance quantum corrections can therefore sample $G/b^2$, while a curvature-to-cutoff comparison samples $GM/(\Lambda^2b^3)$. The Fourier wave number is not generally the total classical impulse, which can involve many coherent exchanges. The classical weak-deflection ratio $GM/b$ must also be controlled. Large center-of-mass energy alone consequently does not diagnose large local curvature at large impact parameter.

A consistent truncation recipe

First choose the observable and desired relative accuracy. Second list every light field and invariant scale. Third enumerate all local operators and loops through that order. Fourth renormalize in a declared scheme and match the necessary coefficients. Fifth vary gauge, scale, and operator basis, and verify that the complete on-shell result is unchanged through the retained order.

The first omitted error is the largest among the next derivative insertion, loop, curvature ratio, threshold expansion, and secular enhancement. A small $GE^2$ does not protect an observable if $E$ approaches a lighter threshold $\Lambda$, if $c_d$ is anomalously large, or if forward poles amplify a different scale.

Two limits provide independent checks. Sending $E/\Lambda\to0$ removes higher-derivative insertions. Sending $G\to0$ at fixed nongravitational data removes graviton loops and Einstein interactions. At $E\sim\bar M_{\mathrm{Pl}}$ or $R\sim\Lambda^2$, the ordered expansion fails and no finite truncation supplies a reliable ultraviolet propagator.

Exercises

Exercise. Compare a tree insertion of $cR^2$ with a one-loop Einstein correction at energy $E$. Compute their ratio and state what must be retained if $c$ is of natural one-loop size.

Solution

Both are relative order $\kappa^2E^2$ up to their loop and Wilson-coefficient factors. Their ratio is

$$\frac{\mathcal A_{R^2}^{\mathrm{tree}}} {\mathcal A_{\mathrm{EH}}^{\mathrm{1\ loop}}} \sim16\pi^2c.$$

If $c\sim1/(16\pi^2)$, the two allowed contributions have the same generic size and must both be considered. This assumes the chosen operator has a nonzero matrix element: a redundant $R^2$ term in pure four-dimensional on-shell gravity is an exception, not a counterexample to the counting. In a nonredundant sector, omitting the required local term leaves uncanceled scale dependence. A large matching coefficient can lower the practical cutoff.

References

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