Renormalization, Running, and Classical Logarithms

Classical two-body calculations develop ultraviolet divergences whenever the long-distance theory probes a compact object more finely than its worldline description can resolve. Renormalization does not make the underlying body quantum: it absorbs short-distance sensitivity into coefficients of allowed worldline operators. The subtraction scale and operator coefficients are scheme dependent, while a correctly matched binding energy, scattering angle, flux, or waveform is not.

Required background. The Gibbons-Hawking-York Boundary Term supplies the gravitational action and boundary variation, and Weak Metric Perturbations and Gauge Freedom supplies perturbative field and gauge redefinitions.

Helpful background. The Einstein Equation fixes the bulk equations, Presymplectic Potentials and Currents clarifies invariant charges, and Black-Hole Perturbation Theory supplies full-object response data for matching.

Classical ultraviolet sensitivity

Replacing an extended body by a delta-function source makes near-worldline fields singular. A near-zone diagram or direct field integral can then contain a short-distance contribution of the schematic form

$$I(\epsilon) =\mu^{2\epsilon} \int\frac{d^{3-2\epsilon}\mathbf k}{(2\pi)^{3-2\epsilon}} \frac{\mathcal N(\mathbf k,\mathbf p)}{(\mathbf k^2)^n},$$

where dimensional regularization continues the spatial dimension to $3-2\epsilon$. When the degree of divergence is logarithmic,

$$I(\epsilon) =\frac{A}{\epsilon} +2A\ln\frac{\mu}{Q} +B_{\mathrm{scheme}} +O(\epsilon).$$

Here $Q$ is a physical orbital momentum or inverse distance. The factor two comes from expanding $(\mu/Q)^{2\epsilon}$ multiplying the simple pole. The pole must be absorbed by an allowed local counterterm, a redundant equations-of-motion operator, or a contribution from an overlapping region; it does not always signal a new independent finite-size coefficient. Symmetry fixes the allowed structures, and matching fixes the physical finite coefficients.

Power divergences often vanish in dimensional regularization. That convenient fact is not a claim that the corresponding short-distance physics is absent. It means that those analytic contributions have been assigned to local coefficients in a particular scheme. A hard cutoff displays them explicitly, and the matched observable agrees after the coefficients are translated.

The separation of object, orbital, and radiation scales and the associated classical renormalization are developed in Goldberger and Rothstein 2006, §§II–V.

Counterterms and running coefficients

Let an allowed worldline operator $\mathcal O$ have a bare coefficient $C_{\mathrm B}$. In a minimal-subtraction-like scheme write

$$C_{\mathrm B} =\mu^{q\epsilon} \left[ C(\mu)+\frac{a_1(C)}{\epsilon} +\frac{a_2(C)}{\epsilon^2}+\cdots \right],$$

where $q$ compensates the continued dimension. The bare action cannot depend on the arbitrary scale $\mu$, so

$$0=\mu\frac{dC_{\mathrm B}}{d\mu} \quad\Longrightarrow\quad \mu\frac{dC}{d\mu}=\beta_C(C).$$

The beta function is classical when the divergent diagrams arise entirely from classical potential or radiation fields. Loops here count nonlinear field iterations and separated momentum regions; a loop symbol does not by itself imply an $\hbar$ correction.

At leading order with constant $\beta_C=\gamma$,

$$C(\mu)=C(\mu_0)+\gamma\ln\frac{\mu}{\mu_0}.$$

The natural matching scale $\mu_0$ is of order the inverse object size, where the compact-object calculation supplies $C(\mu_0)$. Renormalization-group evolution then transports the coefficient toward an orbital or radiative scale. This can resum powers of a large scale logarithm even when the nonlogarithmic terms at the same perturbative order are not known.

Changing the operator basis by equations of motion or field redefinitions changes individual beta functions. Therefore a quoted running coefficient must include the action, basis, regulator, and subtraction prescription.

Cancellation in an observable

Suppose a dimensionless observable through one logarithmic order is

$$\mathcal A(Q) =K\,C(\mu) -K\gamma\ln\frac{\mu}{Q} +\mathcal A_{\mathrm{finite}}(Q).$$

Using $\mu\,dC/d\mu=\gamma$ gives

$$\mu\frac{d\mathcal A}{d\mu} =K\gamma-K\gamma +O(\text{higher order}) =0.$$

The coefficient and explicit orbital logarithm are separately scale dependent, but their sum is not. This cancellation is an essential independent check. If residual $\ln\mu$ remains at the retained order, then a counterterm, a momentum region, a matching contribution, or a running term is missing.

A finite scheme change illustrates the same point. Let

$$C'(\mu)=C(\mu)+\Delta, \qquad \mathcal A'_{\mathrm{finite}} =\mathcal A_{\mathrm{finite}}-K\Delta.$$

Then $\mathcal A'=\mathcal A$. Neither $C$ nor the local finite remainder is an observable. This is why comparing Hamiltonian coefficients before a canonical transformation, or comparing worldline coefficients in different bases, can produce a false disagreement.

The EFT hierarchy and the translation between running quantities and observables are reviewed in Levi 2020, §§2.2–5.1, pp. 13–54.

Near-zone and radiation-zone logarithms

Two-body gravity contains more than one kind of logarithm. A near-zone ultraviolet pole is local from the orbital viewpoint and renormalizes a worldline or composite-system operator. A radiation-zone infrared or ultraviolet pole may reflect overlap with the near zone or the short-time limit of a nonlocal tail interaction. The labels “UV” and “IR” therefore refer to a momentum region, not automatically to body-scale microphysics.

For a hereditary tail, the time-domain action contains a nonlocal kernel schematically of the form

$$S_{\mathrm{tail}} \sim G^2M \int dt\, I_{ij}^{(3)}(t) \operatorname{Pf}_{T} \int_{-\infty}^{\infty} \frac{dt'}{|t-t'|}\, I_{ij}^{(3)}(t').$$

Here $M=m_1+m_2$ is the binary mass monopole at the accuracy displayed. The finite-part scale $T$, equivalently an inverse subtraction scale $\mu\sim T^{-1}$ up to a scheme-dependent constant, cancels against a local near-zone contribution. The factor $1/|t-t'|$ is required both by dimensions and by the 4PN tail kernel. Fourier transformation converts its finite part into a logarithm of a dimensionless frequency ratio together with the causal or time-symmetric prescription appropriate to the observable. Dropping either region destroys the cancellation.

One must also keep conservative and dissipative pieces distinct. A time-symmetric tail contributes to conservative dynamics; a retarded tail also contains radiation reaction. The same logarithmic function can appear with different imaginary prescriptions, so copying a coefficient between in-out and in-in calculations is unsafe. The potential/radiation split, tails, and renormalization-group flow are treated in Porto 2016, §§7.1–7.7, pp. 40–68.

A reliable renormalization check

For each classical logarithm:

1. identify the momentum region and the scale that bounds it;
2. name the local operator or nonlocal kernel that absorbs the pole;
3. state the regulator, subtraction scheme, and operator basis;
4. derive the beta function from the pole coefficient;
5. match the finite coefficient at a declared physical scale;
6. differentiate the final observable with respect to $\ln\mu$;
7. compare a gauge-invariant limit, such as the test-mass limit or scattering angle.

Renormalization is justified only while the effective hierarchy remains separated. If the orbital frequency approaches an internal-mode frequency, or the separation approaches the body scale, adding more local counterterms does not repair the expansion; the missing degree of freedom or a different description must be introduced.

Exercises

Solve the leading renormalization-group equation and show explicitly that the observable $\mathcal A(Q)$ is independent of the intermediate scale.

Solution

Integrating $dC/d\ln\mu=\gamma$ from $\mu_0$ to $\mu$ gives

$$C(\mu)=C(\mu_0)+\gamma\ln\frac{\mu}{\mu_0}.$$

Substitution yields

$$\mathcal A(Q) =K C(\mu_0) +K\gamma\ln\frac{Q}{\mu_0} +\mathcal A_{\mathrm{finite}}(Q),$$

which contains no intermediate $\mu$. The reference $\mu_0$ is still a freely chosen matching scale, not a new observable: changing it requires evolving $C(\mu_0)$ as well. If matching expresses the coefficient through a physical body scale, the combined answer contains a logarithm of physical scale ratios. Differentiation with respect to $\ln\mu$ gives zero through the calculated order.

Explain why setting every power divergence to zero in dimensional regularization cannot prove that a compact object has no finite-size response.

Solution

Dimensional regularization assigns scaleless power-divergent integrals to zero, but finite-size information resides in matched local coefficients. A different regulator shifts analytic short-distance terms between diagrams and those coefficients. Only a matched observable can establish that a response coefficient vanishes in a specified theory and convention.

References

• Goldberger, Walter D., and Ira Z. Rothstein. “An Effective Field Theory of Gravity for Extended Objects.” Physical Review D 73 (2006): 104029. DOI.
• Levi, Michèle. “Effective Field Theories of Post-Newtonian Gravity: A Comprehensive Review.” Reports on Progress in Physics 83 (2020): 075901. DOI.
• Porto, Rafael A. “The Effective Field Theorist’s Approach to Gravitational Dynamics.” Physics Reports 633 (2016): 1–104. DOI.