Matching Compact Objects to Wilson Coefficients

Symmetry determines which worldline operators are allowed, but it does not determine their coefficients. A Wilson coefficient is fixed by matching an observable computed in the full compact-object problem to the same observable computed in the long-distance effective theory. The observable, boundary conditions, field normalization, frequency regime, and subtraction scheme must all agree.

Required background. The Gibbons-Hawking-York Boundary Term supplies the gravitational variational setting, and Weak Metric Perturbations and Gauge Freedom supplies linear response and gauge translation.

Helpful background. The Einstein Equation fixes the full theory, Presymplectic Potentials and Currents clarifies charges and boundary terms, and Black-Hole Perturbation Theory supplies object-specific response calculations.

Matching as equality of observables

Suppose the effective action contains

$$S_{\mathrm{eff}} \supset \frac12 C_E(\mu) \int d\tau\,E_{\mu\nu}E^{\mu\nu}.$$

The scale $\mu$ denotes a renormalization or factorization scale, not a physical body property. To determine $C_E$, choose a long-wavelength observable sensitive to the quadrupolar electric response. Possibilities include an on-shell graviton–body scattering amplitude, a gauge-invariant asymptotic multipole induced by an applied tidal field, or a binary observable after other contributions are separated.

In a static response convention,

$$Q_{ij}(\omega=0)=-\lambda_E\,\mathcal E_{ij}.$$

The full theory computes $\lambda_E$ by solving the object’s perturbation equations with regular interior or horizon boundary conditions and a specified asymptotic normalization. The EFT computes the same relation from the local operator. Equality fixes a convention-dependent map between $C_E$ and $\lambda_E$.

The logic is

$$\mathcal O_{\mathrm{full}}(\omega,\mathbf q) = \mathcal O_{\mathrm{EFT}}(\omega,\mathbf q;C_i(\mu)) +O\!\left((\omega R)^n,(qR)^n\right).$$

Expand both sides in the same low-energy variables and solve for $C_i$. Only the complete observable is invariant. A coefficient changes when the operator basis or normalization changes, so a bare number without its action is unusable.

A quadrupolar response example

Model a dynamical quadrupole $Q_{ij}$ with

$$S_Q =\int d\tau \left[ \frac{1}{4\lambda\omega_0^2} \left( \dot Q_{ij}\dot Q^{ij} -\omega_0^2Q_{ij}Q^{ij} \right) -\frac12 Q_{ij}E^{ij} \right].$$

The equation of motion is

$$\ddot Q_{ij}+\omega_0^2Q_{ij} =-\lambda\omega_0^2 E_{ij}.$$

For harmonic time dependence $e^{-i\omega t}$,

$$Q_{ij}(\omega) =-\lambda \frac{\omega_0^2}{\omega_0^2-\omega^2} E_{ij}(\omega).$$

At $\lvert\omega\rvert\ll\omega_0$,

$$Q_{ij} =-\lambda \left[ 1+\frac{\omega^2}{\omega_0^2} +O\!\left(\frac{\omega^4}{\omega_0^4}\right) \right]E_{ij}.$$

Integrating out $Q_{ij}$ produces a tower of local operators: $E^2$, $\dot E^2$, and higher derivatives. In the normalization displayed on this page, the leading on-shell term is $+\lambda E_{ij}E^{ij}/4$. Comparing it with $+C_EE_{ij}E^{ij}/2$ therefore gives $C_E=\lambda/2$; the first derivative coefficient is correspondingly proportional to $\lambda/\omega_0^2$. Close to resonance, the derivative expansion fails and $Q_{ij}$ must remain dynamical.

The sign and pole prescription require care. A retarded response shifts $\omega\to\omega+i0$ and, with damping, develops an imaginary part. An ordinary real, single-history action gives the conservative response; an in-out functional computes time-ordered amplitudes and is not itself a retarded initial-value prescription. Absorption requires causal response data and the in-in construction.

Black holes and material bodies

For a material star, matching depends on the equation of state, internal composition, rotation, and mode spectrum. The object’s regular interior solution is joined to an exterior perturbation, and the response is read from an asymptotic ratio after the applied and induced pieces are defined.

For a black hole, the inner boundary condition is ingoing regularity at the future horizon. Static Love-number statements depend on dimension, theory, multipole, parity, spin, field variable, and subtraction convention. In four-dimensional general relativity, the standard static gravitational Love numbers of nonrotating black holes vanish in the usual asymptotic convention, while dissipative absorption remains nonzero. “Zero Love” therefore does not mean “no finite-size physics.”

The response coefficients used here are computed in Black-Hole Perturbation Theory or Relativistic Matter and Compact Objects. This volume shows how to insert them into binary dynamics; their calculation requires the perturbation problem for the individual object.

The general matching hierarchy and compact-object examples are reviewed in Porto 2016, §§6.5–8, pp. 33–87.

Scheme dependence and running

Loops in the long-distance theory can generate ultraviolet poles and logarithms. A renormalized coefficient satisfies

$$C_E^{\mathrm{bare}} =C_E(\mu)+\delta C_E(\mu), \qquad \mu\frac{dC_E}{d\mu}=\beta_E.$$

The EFT observable contains both $C_E(\mu)$ and explicit logarithms:

$$\mathcal O =C_E(\mu)\mathcal K +\beta_E\mathcal K\ln\frac{\lvert\omega\rvert}{\mu} +\cdots.$$

Scale independence requires cancellation,

$$\mu\frac{d\mathcal O}{d\mu}=0$$

through the retained order. Here the real logarithm is written for nonzero real frequency; a retarded complex logarithm additionally requires its causal branch prescription. Thus a running coefficient is not itself observable. Two subtraction schemes can quote different $C_E$ and still agree physically.

Matching should be performed at a convenient scale near the inverse body size, then evolved to the orbital scale when large logarithms would otherwise spoil perturbation theory. The original classical EFT construction and its renormalization logic are developed in Goldberger and Rothstein 2006, §§III–V.

A matching protocol

A reproducible coefficient determination records:

1. the full theory, object, dimension, background, spin, and state;
2. the worldline operator basis and field normalization;
3. the on-shell or asymptotic observable being matched;
4. frequency, momentum, and body-size expansion orders;
5. inner and outer boundary conditions;
6. regulator, subtraction scheme, and matching scale;
7. the translation from full-theory response to coefficient;
8. an invariant check in a second observable or limiting regime.

Matching a coordinate metric coefficient is generally inadequate because a field redefinition can change it. A scattering amplitude, absorption probability, induced multipole with a fixed asymptotic prescription, or binding-energy correction is more robust.

Exercises

Expand the oscillator response through $O(\omega^2)$ and identify the parametric breakdown of the local derivative expansion.

Solution

For $\lvert\omega\rvert\lt \omega_0$,

$$\frac{\omega_0^2}{\omega_0^2-\omega^2} =1+\frac{\omega^2}{\omega_0^2} +O\!\left(\frac{\omega^4}{\omega_0^4}\right).$$

The first local operator is proportional to $\lambda E^2$ and the first derivative correction to $\lambda\dot E^2/\omega_0^2$. The expansion fails when $\lvert\omega/\omega_0\rvert$ is not small, especially near the pole where the internal mode must remain explicit.

Why can two schemes have different $C_E(\mu)$ but the same tidal phase?

Solution

A scheme change shifts the finite part of $C_E$ and the corresponding explicit loop or logarithmic terms. When both are transformed consistently, their sum in the on-shell binary observable is unchanged through the calculated order. Comparing the coefficient alone omits this compensating contribution.

References

• Goldberger, Walter D., and Ira Z. Rothstein. “An Effective Field Theory of Gravity for Extended Objects.” Physical Review D 73 (2006): 104029. DOI.
• Porto, Rafael A. “The Effective Field Theorist’s Approach to Gravitational Dynamics.” Physics Reports 633 (2016): 1–104. DOI.