Worldline Effective Field Theory

• Worldline EFT is a framework that replaces compact objects with worldlines and localized operators to encode multipole structures, finite-size effects, and dissipation.
• It integrates out short-distance degrees of freedom to yield effective long-distance descriptions for gravitational dynamics, scattering, and radiation.
• The formalism enables matching of Wilson coefficients, renormalization, and extension to include environmental effects in modified gravity or gauge theory.

Worldline effective field theory replaces a compact object or heavy source by a worldline endowed with localized operators that encode multipole structure, finite-size response, and dissipation, while short-distance degrees of freedom are integrated out to produce a long-distance description of dynamics, scattering, and radiation. In gravitational applications this framework is used for black holes, spinning compact objects, post-Newtonian and post-Minkowskian two-body dynamics, and tidal response; closely related constructions also describe extended charged sources as Wilson lines traced by the worldline of a finite charge distribution (Wong et al., 2019, Ben-Shahar, 2023, Bini et al., 28 Apr 2025, Plestid, 2024).

1. Core worldline construction

A standard starting point is a bulk action plus one or more point-particle worldline actions. For a nonspinning black hole interacting with a light scalar field, the bulk theory is written as

$$S_{\rm bulk}[g,\phi] =\int d^4x\,\sqrt{-g}\Bigl[\,2\bar m^2R-\tfrac12(\nabla\phi)^2-\tfrac12\,\mu^2\phi^2\Bigr],$$

and the black hole of mass $M$ is replaced, in the far zone, by a worldline $z^\mu(\tau)$ with velocity $u^\mu=dz^\mu/d\tau$, $u^2=-1$, together with a tower of localized composite operators $q^L(\tau)$ that capture finite-size response (Wong et al., 2019). The corresponding point-particle action is

$$S_{\rm pp} =-\!M\!\int d\tau +\sum_{\ell=0}^\infty\!\frac{1}{\ell!}\int d\tau\;q^{L}(\tau)\,\nabla_{L}\phi\bigl(z(\tau)\bigr) +\dots\,,$$

with $L=(i_1\cdots i_\ell)$ an STF multi-index in the local rest frame.

In the tidal sector, a static and spherically symmetric black hole in GR is described by a worldline with internal degrees of freedom $X$ and couplings to gravito-electric and gravito-magnetic tidal tensors,

$$S_{\rm WL} = S_{\rm pp}[x^\mu(\tau),h_{\mu\nu}] + S_{X}[X] - \sum_{\ell\ge2}\!\int d\tau\, \Bigl[ Q^{L}_{\rm E}(X)\,{\cal E}_{L}(\tau) + Q^{L}_{\rm B}(X)\,{\cal B}_{L}(\tau) \Bigr],$$

where $M$0 and $M$1 are STF tidal tensors built from the Weyl tensor (Kobayashi et al., 16 Nov 2025).

For spinning compact objects the worldline variables are enlarged to include the position $M$2, conjugate momentum $M$3, antisymmetric spin tensor $M$4, and body-fixed Lorentz frame $M$5. The gauge-fixed action enforcing the covariant spin-supplementary condition $M$6 contains a minimal sector plus nonminimal curvature couplings through $M$7 (Ben-Shahar, 2023). In the notation of that construction, the nonminimal terms include the electric and magnetic parts of the curvature, $M$8 and $M$9, and higher-derivative operators with Wilson coefficients such as $z^\mu(\tau)$0, $z^\mu(\tau)$1, and $z^\mu(\tau)$2.

A general worldline EFT allows all scalar operators consistent with diffeomorphism invariance. In modified-gravity applications, the point-particle action may contain couplings

$z^\mu(\tau)$3

although in vacuum or pure black-hole backgrounds $z^\mu(\tau)$4 and $z^\mu(\tau)$5 can be removed by field redefinitions, and for nonspinning black holes in GR $z^\mu(\tau)$6 (Kulkarni et al., 2024).

2. Integrating out internal modes and causal response

A defining step is the integration over localized worldline degrees of freedom. For black holes with induced scalar charge, the worldline composite operators $z^\mu(\tau)$7 are integrated out using the in-in, or Schwinger-Keldysh, formalism, in which every field is doubled and the effective worldline influence functional is

$z^\mu(\tau)$8

If the worldline theory is Gaussian, the effective action becomes quadratic in the external fields, with kernels $z^\mu(\tau)$9 in CTP/Keldysh space (Wong et al., 2019).

In the classical limit, the retarded response $u^\mu=dz^\mu/d\tau$0 and the commutator $u^\mu=dz^\mu/d\tau$1 survive, whereas the Hadamard kernel $u^\mu=dz^\mu/d\tau$2 does not enter the classical equations of motion. The retarded correlator admits a low-frequency expansion

$u^\mu=dz^\mu/d\tau$3

where the real even coefficients are reactive Wilson coefficients and the imaginary odd coefficients encode dissipation (Wong et al., 2019). This separation between even-in-$u^\mu=dz^\mu/d\tau$4 conservative response and odd-in-$u^\mu=dz^\mu/d\tau$5 dissipative response reappears in the tidal framework, where

$u^\mu=dz^\mu/d\tau$6

defines conservative $u^\mu=dz^\mu/d\tau$7-coefficients and dissipative $u^\mu=dz^\mu/d\tau$8-coefficients (Kobayashi et al., 16 Nov 2025).

For a black hole in a long-wavelength scalar background $u^\mu=dz^\mu/d\tau$9, the induced scalar charge follows from the same response kernel. Writing $u^2=-1$0, one finds

$u^2=-1$1

and matching to the full solution fixes $u^2=-1$2, so that

$u^2=-1$3

with $u^2=-1$4 the horizon area (Wong et al., 2019). In that construction the induced scalar charge is inextricably linked to accretion of the background environment, because both effects stem from the same parent term in the effective action. A plausible implication is that dissipation and long-range response are not separate sectors of the EFT, but two projections of the same retarded correlator.

3. Matching, Wilson coefficients, and renormalization

Worldline EFT is predictive only after matching to a full theory or to exact solutions. In the scalar black-hole problem, matching to the Kerr-background solution fixes the dissipative Wilson coefficient $u^2=-1$5 and thereby the induced charge (Wong et al., 2019). For spinning compact objects, the tree-level Compton amplitude and the one-loop $u^2=-1$6 scattering amplitude are matched to analytic solutions of the Teukolsky equation, fixing the Kerr values

$u^2=-1$7

together with

$u^2=-1$8

and vanishing $u^2=-1$9 analogues, while the single SSC-breaking coefficient is set to $q^L(\tau)$0 (Ben-Shahar, 2023). With these values, the same-helicity Compton amplitude exponentiates perfectly and the one-loop amplitude matches the Kerr-Kerr scattering angle through fourth order in both spins.

For a Kerr-Newman black hole, matching proceeds through one-point functions rather than two-point or scattering observables. Comparing the EFT long-distance fields to the ACMC multipole expansion gives

$q^L(\tau)$1

corresponding respectively to a vanishing electric dipole, magnetic dipole equal to the charge, electric quadrupole equal to the charge, and unit mass-quadrupole coefficient (Zheng, 2 Jan 2026).

In nonspinning black-hole sectors of GR, a recurrent result is the vanishing of static tidal Wilson coefficients. For the modified-gravity WEFT treatment, one has $q^L(\tau)$2 for nonspinning black holes in GR (Kulkarni et al., 2024). In the 5PM scattering framework, all Love-number-type coefficients $q^L(\tau)$3, $q^L(\tau)$4, and higher tidal operators vanish for classical black holes in four dimensions, so that through 5PM one need only keep the leading mass-monopole coupling (Bini et al., 28 Apr 2025). A common misconception is that this eliminates black-hole response altogether. The tidal-response analysis of non-rotating black holes instead shows that the renormalized tidal response function is subject to inevitable ambiguities associated with the choice of renormalization scheme and with the initial condition of the renormalization flow equation, and once these ambiguities are fixed one obtains scheme-dependent dynamical tidal Love numbers (Kobayashi et al., 16 Nov 2025).

The renormalization structure is explicit in the MST-to-worldline matching. The bare response is read off from

$q^L(\tau)$5

and dimensional regularization introduces a pole in $q^L(\tau)$6 that must be removed by a counterterm (Kobayashi et al., 16 Nov 2025). The renormalized response obeys

$q^L(\tau)$7

so only the logarithmic running and the dissipative part are universal; the finite conservative part can be shifted by scheme choices without affecting observables (Kobayashi et al., 16 Nov 2025).

4. Conservative dynamics, radiation, and environmental effects

In binary dynamics, one expands the metric around flat space, fixes gauge, and integrates out near-zone potential modes perturbatively. In a Kaluza-Klein parametrization,

$q^L(\tau)$8

with $q^L(\tau)$9, the fields scale as $$S_{\rm pp} =-\!M\!\int d\tau +\sum_{\ell=0}^\infty\!\frac{1}{\ell!}\int d\tau\;q^{L}(\tau)\,\nabla_{L}\phi\bigl(z(\tau)\bigr) +\dots\,,$$0, $$S_{\rm pp} =-\!M\!\int d\tau +\sum_{\ell=0}^\infty\!\frac{1}{\ell!}\int d\tau\;q^{L}(\tau)\,\nabla_{L}\phi\bigl(z(\tau)\bigr) +\dots\,,$$1, and $$S_{\rm pp} =-\!M\!\int d\tau +\sum_{\ell=0}^\infty\!\frac{1}{\ell!}\int d\tau\;q^{L}(\tau)\,\nabla_{L}\phi\bigl(z(\tau)\bigr) +\dots\,,$$2, and a single $$S_{\rm pp} =-\!M\!\int d\tau +\sum_{\ell=0}^\infty\!\frac{1}{\ell!}\int d\tau\;q^{L}(\tau)\,\nabla_{L}\phi\bigl(z(\tau)\bigr) +\dots\,,$$3-exchange reproduces the usual $$S_{\rm pp} =-\!M\!\int d\tau +\sum_{\ell=0}^\infty\!\frac{1}{\ell!}\int d\tau\;q^{L}(\tau)\,\nabla_{L}\phi\bigl(z(\tau)\bigr) +\dots\,,$$4 Newtonian potential (Kulkarni et al., 2024). This is the basic near-zone matching step of the WEFT.

For black-hole binaries embedded in a fuzzy-dark-matter halo, each worldline carries an induced scalar charge $$S_{\rm pp} =-\!M\!\int d\tau +\sum_{\ell=0}^\infty\!\frac{1}{\ell!}\int d\tau\;q^{L}(\tau)\,\nabla_{L}\phi\bigl(z(\tau)\bigr) +\dots\,,$$5 and a slowly varying mass $$S_{\rm pp} =-\!M\!\int d\tau +\sum_{\ell=0}^\infty\!\frac{1}{\ell!}\int d\tau\;q^{L}(\tau)\,\nabla_{L}\phi\bigl(z(\tau)\bigr) +\dots\,,$$6. Integrating out potential modes in the PN regime gives the instantaneous scalar potential

$$S_{\rm pp} =-\!M\!\int d\tau +\sum_{\ell=0}^\infty\!\frac{1}{\ell!}\int d\tau\;q^{L}(\tau)\,\nabla_{L}\phi\bigl(z(\tau)\bigr) +\dots\,,$$7

which yields a Newtonian-order attractive fifth force (Wong et al., 2019). The same EFT gives a fifth-force equation of motion,

$$S_{\rm pp} =-\!M\!\int d\tau +\sum_{\ell=0}^\infty\!\frac{1}{\ell!}\int d\tau\;q^{L}(\tau)\,\nabla_{L}\phi\bigl(z(\tau)\bigr) +\dots\,,$$8

with $$S_{\rm pp} =-\!M\!\int d\tau +\sum_{\ell=0}^\infty\!\frac{1}{\ell!}\int d\tau\;q^{L}(\tau)\,\nabla_{L}\phi\bigl(z(\tau)\bigr) +\dots\,,$$9 describing mass growth by accretion (Wong et al., 2019). The leading scalar-dipole radiation arises at 1.5PN, while accretion and dynamical friction produce drag forces scaling as $L=(i_1\cdots i_\ell)$0PN and $L=(i_1\cdots i_\ell)$1PN but are suppressed by $L=(i_1\cdots i_\ell)$2 (Wong et al., 2019).

In a dark sector containing a Proca field and an axion-like scalar, the worldline action includes masses $L=(i_1\cdots i_\ell)$3, electric charges $L=(i_1\cdots i_\ell)$4, and dark-photon charges $L=(i_1\cdots i_\ell)$5. The conservative dynamics is computed up to 1PN order for gravitational, electromagnetic, and Proca fields and up to 2PN order for the scalar field; the effect of the axion-electromagnetic coupling $L=(i_1\cdots i_\ell)$6 arises in the conservative dynamics at 2.5PN order, while the kinetic-mixing constant $L=(i_1\cdots i_\ell)$7 enters at 1PN order (Bhattacharyya et al., 2023). In the radiative sector, multipole couplings of the remaining long-wavelength fields yield scalar, electromagnetic, Proca, and gravitational power formulas. The $L=(i_1\cdots i_\ell)$8 contribution to scalar radiation appears at $L=(i_1\cdots i_\ell)$9, to gravitational radiation at $X$0, and these radiative corrections vanish for any orbit confined to a plane because of the existence of a binormal like term in the effective radiative action, while they are non-zero for orbits that lie in three dimensions (Bhattacharyya et al., 2023).

Modified-gravity worldline EFTs deform the conservative potential in a calculable way. For an $X$1 correction, the propagator changes from $X$2 to

$X$3

leading to the Yukawa-like potential

$X$4

whereas a cubic $X$5 invariant first contributes at two-loop order in the near zone and produces a short-range $X$6 correction to the potential (Kulkarni et al., 2024). The formalism therefore supports both environmental effects and deformations of GR within the same operator-based framework.

5. Worldline QFT, scattering amplitudes, and PM observables

A major development is the recasting of spinning-object EFTs into a worldline quantum field theory expansion. One expands around straight-line backgrounds,

$X$7

introduces worldline fluctuation fields $X$8, and evaluates scattering observables from the path integral with de Donder-gauge graviton propagators and worldline propagators for $X$9, $$S_{\rm WL} = S_{\rm pp}[x^\mu(\tau),h_{\mu\nu}] + S_{X}[X] - \sum_{\ell\ge2}\!\int d\tau\, \Bigl[ Q^{L}_{\rm E}(X)\,{\cal E}_{L}(\tau) + Q^{L}_{\rm B}(X)\,{\cal B}_{L}(\tau) \Bigr],$$0, and spin fluctuations (Ben-Shahar, 2023). This produces compact Feynman rules for one-graviton, two-graviton, and mixed fluctuation-graviton vertices, including nonminimal $$S_{\rm WL} = S_{\rm pp}[x^\mu(\tau),h_{\mu\nu}] + S_{X}[X] - \sum_{\ell\ge2}\!\int d\tau\, \Bigl[ Q^{L}_{\rm E}(X)\,{\cal E}_{L}(\tau) + Q^{L}_{\rm B}(X)\,{\cal B}_{L}(\tau) \Bigr],$$1, $$S_{\rm WL} = S_{\rm pp}[x^\mu(\tau),h_{\mu\nu}] + S_{X}[X] - \sum_{\ell\ge2}\!\int d\tau\, \Bigl[ Q^{L}_{\rm E}(X)\,{\cal E}_{L}(\tau) + Q^{L}_{\rm B}(X)\,{\cal B}_{L}(\tau) \Bigr],$$2, and derivative insertions.

At tree level, the Compton amplitude through fourth order in spin is obtained from three worldline diagrams, and in the same-helicity sector the result takes the form

$$S_{\rm WL} = S_{\rm pp}[x^\mu(\tau),h_{\mu\nu}] + S_{X}[X] - \sum_{\ell\ge2}\!\int d\tau\, \Bigl[ Q^{L}_{\rm E}(X)\,{\cal E}_{L}(\tau) + Q^{L}_{\rm B}(X)\,{\cal B}_{L}(\tau) \Bigr],$$3

so the characteristic exponential in $$S_{\rm WL} = S_{\rm pp}[x^\mu(\tau),h_{\mu\nu}] + S_{X}[X] - \sum_{\ell\ge2}\!\int d\tau\, \Bigl[ Q^{L}_{\rm E}(X)\,{\cal E}_{L}(\tau) + Q^{L}_{\rm B}(X)\,{\cal B}_{L}(\tau) \Bigr],$$4 emerges up to quartic order (Ben-Shahar, 2023). At one loop, the classical $$S_{\rm WL} = S_{\rm pp}[x^\mu(\tau),h_{\mu\nu}] + S_{X}[X] - \sum_{\ell\ge2}\!\int d\tau\, \Bigl[ Q^{L}_{\rm E}(X)\,{\cal E}_{L}(\tau) + Q^{L}_{\rm B}(X)\,{\cal B}_{L}(\tau) \Bigr],$$5 amplitude through $$S_{\rm WL} = S_{\rm pp}[x^\mu(\tau),h_{\mu\nu}] + S_{X}[X] - \sum_{\ell\ge2}\!\int d\tau\, \Bigl[ Q^{L}_{\rm E}(X)\,{\cal E}_{L}(\tau) + Q^{L}_{\rm B}(X)\,{\cal B}_{L}(\tau) \Bigr],$$6 is recovered by combining two copies of the worldline action and reducing the loop integrals to scalar master integrals in a transverse tensor basis (Ben-Shahar, 2023).

In post-Minkowskian scattering, worldline EFT—sometimes called NRGR in this context—organizes conservative and radiative corrections to two-body scattering up to 5PM. The action is split into Einstein-Hilbert, gauge-fixing, and two point-particle worldline sectors, and for spinless black holes through 5PM only the mass-monopole coupling is needed because all tidal coefficients vanish (Bini et al., 28 Apr 2025). Conservative and radiation-reaction corrections to the impulse are computed from Feynman diagrams with potential and radiation modes. The resulting $$S_{\rm WL} = S_{\rm pp}[x^\mu(\tau),h_{\mu\nu}] + S_{X}[X] - \sum_{\ell\ge2}\!\int d\tau\, \Bigl[ Q^{L}_{\rm E}(X)\,{\cal E}_{L}(\tau) + Q^{L}_{\rm B}(X)\,{\cal B}_{L}(\tau) \Bigr],$$7PM–$$S_{\rm WL} = S_{\rm pp}[x^\mu(\tau),h_{\mu\nu}] + S_{X}[X] - \sum_{\ell\ge2}\!\int d\tau\, \Bigl[ Q^{L}_{\rm E}(X)\,{\cal E}_{L}(\tau) + Q^{L}_{\rm B}(X)\,{\cal B}_{L}(\tau) \Bigr],$$8PM conservative terms agree term-by-term with Tutti Frutti once the gauge-invariant phases are identified, and the $$S_{\rm WL} = S_{\rm pp}[x^\mu(\tau),h_{\mu\nu}] + S_{X}[X] - \sum_{\ell\ge2}\!\int d\tau\, \Bigl[ Q^{L}_{\rm E}(X)\,{\cal E}_{L}(\tau) + Q^{L}_{\rm B}(X)\,{\cal B}_{L}(\tau) \Bigr],$$9PN tail-of-tail nonlocality is reproduced by in-out radiation-zone diagrams (Bini et al., 28 Apr 2025).

Worldline EFT also treats mixed gravitoelectromagnetic scattering. For a spinning, charged source matched to Kerr-Newman, the tree-level graviton-photon conversion amplitude is computed to $M$00 and is manifestly gauge invariant (Zheng, 2 Jan 2026). For an unpolarized incoming photon and unobserved outgoing graviton polarization, the differential conversion cross section is

$M$01

At $M$02 the amplitude factorizes into the photon-Compton amplitude times a universal kinematic conversion factor, but once classical spin is included the worldline EFT exhibits new non-factorizing structures at $M$03 and $M$04 (Zheng, 2 Jan 2026).

6. Extensions beyond minimal gravity and related worldline theories

The formalism is not restricted to vacuum GR. For generic non-rotating compact objects in GR, an exterior Regge-Wheeler solution

$M$05

defines a transfer function $M$06, and the bare response becomes

$M$07

Observables can then be computed directly from the bare response without fixing a renormalization scheme (Kobayashi et al., 16 Nov 2025). For black holes in theories beyond GR or with environmental fields, the master equation is deformed and the worldline EFT must be extended by adding higher-curvature terms in $M$08 or additional worldline multipoles, after which the same matching and renormalization procedures go through (Kobayashi et al., 16 Nov 2025).

A conceptually parallel construction appears in gauge theory as the EFT of extended Wilson lines. In the heavy-mass limit $M$09 with inverse size $M$10 held fixed, a heavy spatially extended charge distribution is represented by a worldline current

$M$11

or equivalently by a worldline Lagrangian with multipole corrections (Plestid, 2024). Expanding the source density in moments produces an infinite tower of operators involving derivatives of $M$12, and resumming them yields an extended Wilson line $M$13 whose expansion reproduces form-factor-dressed Feynman rules (Plestid, 2024). In loop diagrams where a soft photon attaches to the heavy line, the propagator eikonalizes and the soft Coulomb region factorizes; the same leading-power factorization applies with electromagnetic or weak charged-current insertions (Plestid, 2024).

This suggests a broad unifying view of worldline EFT. Whether the object is a black hole, a spinning compact body, a binary in dark matter, or an extended heavy charge, the central move is the same: replace short-distance structure by a worldline operator algebra, integrate out the appropriate bulk or internal modes, and determine the Wilson coefficients by matching to exact solutions, perturbation theory, or measured long-distance observables. The specific content of the operator basis, the causal prescription, and the renormalization scheme then determine which conservative, dissipative, tidal, or radiative effects are manifest at a given order.