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Classical Black Hole Scattering to Celestial Amplitudes in Effective Theories of Gravity

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Pith's one-line read This thesis computes classical black-hole scattering observables in four effective theories of gravity beyond General Relativity, using worldline and amplitude methods to produce analytic results for potentials, impulses, waveforms…

desk verdict A technically careful PhD thesis that compiles five published papers; the computations check out, but the headline dCS spin-eikonal result is blocked by an unresolved IR issue the thesis itself admits. read the letter →

arxiv 2608.00784 v1 pith:245RDZUB submitted 2026-08-01 hep-th gr-qchep-ph

Saptaswa Ghosh This is my paper · ORCID
classification hep-thgr-qchep-ph
keywords blackholescatteringeffectivefieldtheoryworldlinequantumpost-MinkowskianexpansioneikonalphasecelestialamplitudesdynamicalChern-Simonsgravity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This thesis argues that modern quantum-field-theoretic methods—worldline effective field theory, worldline quantum field theory, and on-shell amplitude expansions—can be turned into a reliable analytic toolkit for classical two-body gravity beyond General Relativity. It demonstrates this through explicit computations: conservative potentials and scalar radiation for binaries in axion and dark-photon environments, 2PM impulses and waveforms in scalar-tensor gravity, a 3PM spin-dependent eikonal phase in dynamical Chern-Simons gravity, an infrared-subtracted potential and scattering angle in Einstein-Maxwell-dilaton theory, and phase-dressed celestial eikonal amplitudes in quadratic gravity. If these calculations are right, they supply analytic benchmarks for gravitational-wave searches and for flat-space celestial holography in modified gravity.

What carries the argument

The load-bearing machinery is the worldline approach to classical scattering, in which compact objects are point particles and the gravitational and matter fields are integrated out: in the inspiral regime via non-relativistic general relativity's split into potential and radiation modes, and in the scattering regime via worldline quantum field theory, where worldline fluctuations are themselves quantized. The central generating object is the eikonal phase, understood as the classical limit of the logarithm of the S-matrix and computed with Magnus/Murua causal weights that cancel spurious infrared poles; impulse, spin kick, and scattering angle follow from Poisson-bracket expansions built on it. Multi-loop master integrals are evaluated through integration-by-parts reduction adapted to exponential Fourier kernels and through canonical dlog-form differential equations, and celestial amplitudes are obtained by Mellin transforming the eikonal-resummed amplitude.

What would settle it

Perform the 3PM dynamical Chern-Simons computation including the radiation-reaction or dissipative sector and check whether the 1/$epsilon^{2}$ infrared poles of the eikonal phase cancel. If no consistent subtraction scheme exists, the spin-dependent scattering angle quoted in Chapter 4 cannot be regarded as a well-defined classical observable.

Watch

Extended reading notes

Core claim

The central claim is a set of concrete classical observables computed to stated orders: the 1PN–2.5PN conservative potentials and scalar radiation up to N(4)LO for binaries in an axion and dark-photon environment; the 2PM impulse and waveforms in scalar-tensor theory with smooth massive limits; the linear-in-spin eikonal phase at 3PM in dynamical Chern-Simons gravity; the infrared-subtracted conservative potential and scattering angle in Einstein-Maxwell-dilaton theory; and the phase-dressed celestial eikonal amplitude in quadratic gravity, where, unlike in Einstein gravity, the u- and s-channel contributions do not cancel. The thesis presents these as faithful analytic results that can serve as benchmarks, and it reports a notable structural finding: the axion-photon coupling contributes to radiation only for orbits that are genuinely three-dimensional, vanishing for planar orbits in the non-spinning case.

Load-bearing premise

The 3PM dynamical Chern-Simons eikonal phase rests on an unproven assumption that its infrared divergences can be cancelled by some prescription; the thesis states that the conservative sector alone cannot remove them and leaves the systematic treatment to future work.

Editorial extensions

If this is right

  • The 1PN–2.5PN potentials and scalar radiation up to N(4)LO give concrete templates for scalar and vector dark-matter effects on binary inspiral.
  • The 2PM scalar-tensor impulse and waveform have smooth massless scalar limits, so they can serve as benchmark data for gravitational-wave waveform models beyond general relativity.
  • The 3PM linear-in-spin eikonal phase in dynamical Chern-Simons gravity yields a spin-dependent scattering angle that, once the infrared issue is resolved, predicts parity-violating deviations in high-energy black-hole scattering.
  • The Einstein-Maxwell-dilaton conservative potential and scattering angle reduce smoothly to the general-relativity and charge-free limits, providing amplitude-based benchmarks for charged compact-object dynamics.
  • In quadratic gravity the u- and s-channel contributions to the phase-dressed celestial eikonal do not cancel, changing the conformal data and OPE of the would-be celestial CFT relative to Einstein gravity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The vanishing of the axion-photon radiation for planar orbits suggests that the most promising gravitational-wave probes of axionic dark matter are eccentric or inclined binaries, or spinning binaries with spins precessing out of the orbital plane; the thesis notes the spinning case but does not frame it as a search strategy.
  • If the infrared cancellation in dynamical Chern-Simons gravity requires the dissipative sector, then the conservative eikonal phase is not a standalone observable, and the 3PM spin-dependent scattering angle would need to be redefined in a way that parallels the Magnus prescription already needed in general relativity.
  • The exponential-Fourier integration-by-parts method used for waveform master integrals could be applied to other observables with oscillatory factors, such as memory effects or radiation reaction, where standard rational-function IBP packages fail.
  • The non-vanishing u/s-channel contributions in quadratic gravity suggest a richer celestial holographic dictionary for higher-derivative gravity; a direct test would be to compute the shadow OPE coefficients from the momentum-space Born amplitude rather than from the eikonal resummation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This thesis develops quantum-field-theoretic tools for classical two-body gravitational dynamics beyond General Relativity and applies them to five settings: PN conservative and radiative dynamics of binaries in axion-like-particle and dark-photon environments (Chapter 2); impulse and waveforms in scalar-tensor gravity from worldline QFT (Chapter 3); a linear-in-spin eikonal phase at 3PM in dynamical Chern-Simons gravity (Chapter 4); IR-subtracted conservative potential and scattering angle in Einstein-Maxwell-dilaton theory (Chapter 5); and phase-dressed celestial eikonal amplitudes in quadratic gravity (Chapter 6). The exposition is unusually complete: Feynman rules, PN/PM power counting, master integral evaluations, and appendices are provided, and several internal consistency checks are reported, including smooth massless limits (e.g., Eqs. (2.51) and (2.170)) and the LO gravitational quadrupole formula reproducing the standard 32G/5 mu^2 r^4 Omega^6 result. The manuscript is presented as a PhD thesis and is largely based on five published co-authored papers.

Significance. If the results are correct, the thesis supplies analytic benchmarks for beyond-GR binary dynamics: the first N(4)LO scalar radiation in the axion/dark-photon model, 2PM scalar-tensor observables, a 3PM linear-in-spin dCS eikonal phase, an IR-finite EMD potential and scattering angle, and a non-vanishing s/u-channel contribution to the celestial eikonal in quadratic gravity. Credit is due for the unusually detailed derivations, the explicit master-integral technology, and the multiple internal consistency checks. The central caveat is that the dCS eikonal phase, the main result of Chapter 4, is not presented as a well-defined observable because its IR cancellation is explicitly deferred; until that is supplied or the corresponding claims are qualified, the benchmark status of the Chapter 4 observables is not established.

major comments (2)
  1. [Ch. 4, Secs. 4.1 and 4.5; Ch. 1, p. 11 (after Fig. 1.3)] The thesis's own statements block the central claim of Chapter 4. It states that "in our computation of the eikonal phase in dCS theory, we find that the IR divergences cannot be removed by considering only the conservative sector" and that a systematic treatment is "left as an important direction for future work." Since the eikonal phase is the generating quantity for the spin-dependent impulse and scattering angle, the 3PM linear-in-spin phase derived in Sec. 4.4 is scheme-dependent as presented; the GR cancellation of the leading 1/epsilon^2 pole via Magnus/Murua weights (Eqs. (1.33)-(1.35)) has no demonstrated analogue here. The chapter should either supply a concrete IR-cancellation prescription and show scheme independence, or explicitly present the eikonal phase and its derived observables as provisional and remove the corresponding benchmark claims.
  2. [Ch. 2, Secs. 2.4.6 and 2.5.2] The claimed N(4)LO precision of the scalar radiation result (Eq. (2.114)) is not yet established because the source multipoles in Eq. (2.110) are evaluated on the uncorrected Keplerian/circular orbit. Sec. 2.4.6 states that corrections to the orbit from the higher-order conservative potential are neglected, but no power-counting argument is given to show that these orbit corrections enter only beyond N(4)LO. The authors should either include the orbit corrections at the claimed orders or demonstrate by power counting that they are subleading; otherwise the N(4)LO label is unsupported.
minor comments (4)
  1. [Ch. 1, Sec. 1.4] The general Feynman-integral definition is numbered (2.138) inside the Introduction, and related display equations in the same section are also numbered in the 2.x series; the equation numbering should be made consistent with the chapter structure.
  2. [Ch. 2, Eq. (2.164)] The displayed gravitational radiation power formula is extremely long and difficult to verify as typeset; it should be split into a compact leading term plus a table of coefficient functions, with the bracketing carefully checked.
  3. [Ch. 2, Sec. 2.5.2] The quantity T(\dot{\bar\phi}) is introduced in Eq. (2.99) and then re-expressed in Eq. (2.101), but the relationship between the two forms is not explicitly stated; adding one line of explanation would improve readability.
  4. [Front matter / acknowledgments] The front matter contains non-standard material (an AI-generated image caption and an epigraph) that is unrelated to the scientific content; for archival purposes this should be removed or clearly separated from the technical presentation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivations are self-contained; the dCS IR caveat is a completeness limitation, not a circular reduction.

full rationale

The manuscript is a compilation of published computations, but the derivations are carried out inside the thesis rather than imported as black boxes. Each observable (potential, radiation, impulse, waveform, eikonal phase) is obtained from stated Lagrangians and WEFT/WQFT/Feynman rules with explicit integrals and master-integral reductions. There is no parameter fit to a subset of data that is later renamed a prediction. The only self-referential elements are bibliographic (the list of co-authored papers) and do not carry the derivations. The dCS eikonal IR caveat is explicitly acknowledged: 'we do not attempt such an analysis, and instead leave the systematic treatment of this issue as an important direction for future work.' That is a limitation on the well-definedness of the observable, not a circular reduction: the eikonal phase is not defined in terms of the scattering angle it is used to compute. Similarly, Chapter 2 explicitly says radiative power is evaluated on the leading-order Keplerian orbit; that is a stated approximation, not an input-output equivalence. No equation in the provided text sets a predicted quantity equal by construction to an input. Therefore there is no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The ledger is short on fitted parameters but long on domain assumptions. The computations assume the standard WQFT/PM-EFT picture: point particles, tree-level worldline diagrams, soft-messenger classical limits, and eikonal exponentiation with Magnus-determined causality prescriptions. The most fragile entries are (i) the point-particle idealization in modified-gravity theories where black holes can carry scalar hair, and (ii) the unproven IR-cancellation prescription in dCS (Ch. 1, p. 11). Additionally, the radiative analysis in Ch. 2 evaluates power on the unperturbed Keplerian orbit, and the celestial analysis assumes convergence of Mellin/dispersion integrals. No invented entities and no fitted numbers were found; the couplings and masses are all inputs from the cited models. This makes the derivations parameter-free in the fitting sense, with the risk concentrated in the modeling assumptions rather than in tuned constants.

assumptions (6)
  • domain assumption The two-body system is modelled by structureless point particles; finite-size (tidal) couplings C_R and C_V in the worldline action are set to zero.
    Ch. 2, Eq. (2.10) and Sec. 2.3 state 'we will only consider non-spinning binaries, hence we can ignore the finite size effects.' Standard in GR for Schwarzschild black holes, but nontrivial in the modified-gravity chapters because black holes in scalar-tensor and dCS theories can carry scalar hair and tidal responses.
  • domain assumption The classical limit is extracted from tree-level worldline diagrams with soft messenger loops; purely quantum loops are dropped.
    Ch. 1 Sec. 1.2 and Ch. 2 Secs. 2.2-2.3; the selection rules (no pure messenger loops, no intersecting matter lines) are stated and attributed to [27]. This is the defining approximation of WQFT/PM-EFT.
  • domain assumption The WQFT partition function equals e to the i chi, and the Magnus/Murua weighting fixes the causal i-epsilon prescription so that spurious 1/epsilon squared poles cancel.
    Ch. 1 Sec. 1.2, Eqs. (1.22)-(1.35), built on [33]. The thesis notes this prescription does not remove all IR divergences in dCS, leaving that cancellation unproven.
  • domain assumption Radiative-sector results are evaluated on the zeroth-order Keplerian circular orbit; corrections to the orbit from the 1PN-2PN potentials are neglected.
    Stated at the end of Sec. 2.4.6: the paper works 'with the leading order solution, i.e. Keplerian orbit solution (in fact, the circular orbit).' The induced error at the claimed N(4)LO and 2.5PN orders is not estimated.
  • standard math Dimensional regularization, IBP identities (including the exponential-shift modification of Eqs. (1.45)-(1.47)), and differential-equation/canonical-basis methods are valid for the master integrals used.
    Ch. 1 Sec. 1.4; the exponential IBP trick is attributed to [53]. This is the technical backbone of all chapters.
  • domain assumption In the celestial analysis, eikonal resummation before Mellin transformation renders the celestial amplitude well-defined or definable by analytic continuation, and the dispersion relations over the phase-dressed amplitude converge.
    Ch. 6 Secs. 6.2.3-6.2.4; convergence of the integrals is assumed, with the thesis citing the GR eikonal literature for the pattern.

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Pith. "Pith review of Classical Black Hole Scattering to Celestial Amplitudes in Effective Theories of Gravity." pith.science (2026). https://pith.science/paper/245RDZUB

@misc{pith2026260800784,
  author       = {Pith},
  title        = {Pith review of: Classical Black Hole Scattering to Celestial Amplitudes in Effective Theories of Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/245RDZUB}},
  note         = {Machine review of arXiv:2608.00784}
}
read the original abstract

This thesis investigates quantum field theoretic approaches to classical gravity beyond General Relativity. Using worldline effective field theory, worldline quantum field theory, and on shell scattering amplitudes, it investigates compact-object dynamics in dark-sector models, scalar tensor gravity, dynamical Chern Simons gravity, and Einstein Maxwell dilaton theory. The thesis derives a range of conservative and radiative observables, including impulses, waveforms, eikonal phases, two-body potentials, and scattering angles, while incorporating effects associated with spin, parity violation, infrared structure, and multi-loop integrals. It also constructs celestial eikonal amplitudes and analyzes their conformal data in gravitational theories beyond General Relativity. Collectively, these results broaden the application of modern quantum field theoretic techniques to gravitational dynamics and contribute to a more detailed understanding of gravity beyond the Einsteinian framework. (Detailed abstract in the thesis.)

Figures

Figures reproduced from arXiv: 2608.00784 by the authors.

Figure 1.1
Figure 1.1. Schematic bubble notation for nested Poisson brackets. Popping a bubble produces [PITH_FULL_IMAGE:figures/full_fig_p026_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Schematic graphical content of Murua’s recursion. The displayed rooted tree has two [PITH_FULL_IMAGE:figures/full_fig_p028_1_2.png] view at source ↗
Figure 1.3
Figure 1.3. Causality-prescription diagrams for the IR-sensitive master integral. The [PITH_FULL_IMAGE:figures/full_fig_p028_1_3.png] view at source ↗
Figures from the paper (27 more)
Figure 1.4
Figure 1.4. Figure 1.4: Celestial description of a four-dimensional scattering process. A momentum-space [PITH_FULL_IMAGE:figures/full_fig_p031_1_4.png]
Figure 1.5
Figure 1.5. Figure 1.5: Geometry and contour of nested integrals [PITH_FULL_IMAGE:figures/full_fig_p035_1_5.png]
Figure 2.1
Figure 2.1. Figure 2.1: Tree level diagram (left) vs. loop level (one loop) (right) diagram in WEFT [PITH_FULL_IMAGE:figures/full_fig_p046_2_1.png]
Figure 2.2
Figure 2.2. Figure 2.2: Different scales in the binary inspiral problem. We reproduce the figure from [ [PITH_FULL_IMAGE:figures/full_fig_p047_2_2.png]
Figure 2.3
Figure 2.3. Figure 2.3: Diagrams contributing to the gravitational bound sector at 1PN. [PITH_FULL_IMAGE:figures/full_fig_p055_2_3.png]
Figure 2.4
Figure 2.4. Figure 2.4: Diagrams contributing to the electromagnetic bound sector up to 1PN. [PITH_FULL_IMAGE:figures/full_fig_p056_2_4.png]
Figure 2.5
Figure 2.5. Figure 2.5: Diagrams contributing to the Proca bound sector up to 1PN. [PITH_FULL_IMAGE:figures/full_fig_p057_2_5.png]
Figure 2.6
Figure 2.6. Figure 2.6: Scalar-Electromagnetic interaction diagrams which contribute to the bound sector. [PITH_FULL_IMAGE:figures/full_fig_p057_2_6.png]
Figure 2.7
Figure 2.7. Figure 2.7: 1PN diagrams for the scalar sector with 3 point vertex coming from scalar-graviton [PITH_FULL_IMAGE:figures/full_fig_p060_2_7.png]
Figure 2.8
Figure 2.8. Figure 2.8: 1PN scalar diagrams that contribute to the bound sector. [PITH_FULL_IMAGE:figures/full_fig_p061_2_8.png]
Figure 2.9
Figure 2.9. Figure 2.9: Scalar diagrams coming from the 3-point vertices that contribute at 2 PN order in [PITH_FULL_IMAGE:figures/full_fig_p062_2_9.png]
Figure 2.10
Figure 2.10. Figure 2.10: Diagrams contributing to scalar bound sector at 2 PN order coming from worldline [PITH_FULL_IMAGE:figures/full_fig_p065_2_10.png]
Figure 2.11
Figure 2.11. Figure 2.11: Diagrams contributing to the bound sector at 1PN order due to the Proca [PITH_FULL_IMAGE:figures/full_fig_p066_2_11.png]
Figure 2.12
Figure 2.12. Figure 2.12: Self-energy diagram contributing to the effective action for the source term [PITH_FULL_IMAGE:figures/full_fig_p071_2_12.png]
Figure 2.13
Figure 2.13. Figure 2.13: Radiative diagrams for the scalar field that contribute up to [PITH_FULL_IMAGE:figures/full_fig_p073_2_13.png]
Figure 2.14
Figure 2.14. Figure 2.14: Radiative diagrams for electromagnetic field up to [PITH_FULL_IMAGE:figures/full_fig_p078_2_14.png]
Figure 2.15
Figure 2.15. Figure 2.15: LO radiation coming from the pure gravitational sector. [PITH_FULL_IMAGE:figures/full_fig_p083_2_15.png]
Figure 2.16
Figure 2.16. Figure 2.16: Radiative diagrams for gravitational fields. [PITH_FULL_IMAGE:figures/full_fig_p085_2_16.png]
Figure 2.17
Figure 2.17. Figure 2.17: α(m;r) vs mr plot Another useful relevant integral for (2.56) has the following form, β(m;r) = Z k e ik·⃗r k 2 + m2 Z k1 1 k 2 1 [(k1 − k) 2 + m2] = Z k e ik·⃗r k 2 + m2    1 8|k| [1 − 2 π arctan(m/|k|)] k 2 > m2 1 4π|k| arctan(|k|/m) k 2 < m2 , = Z k e ik·r (k 2…
Figure 2.18
Figure 2.18. Figure 2.18: β(m;r) vs mr plot 83 [PITH_FULL_IMAGE:figures/full_fig_p100_2_18.png]
Figure 2.19
Figure 2.19. Figure 2.19: Correspondence between PN diagrams (left) and one-loop scalar Feynman diagram [PITH_FULL_IMAGE:figures/full_fig_p101_2_19.png]
Figure 3.1
Figure 3.1. Figure 3.1: Sketch of the celestial sphere under consideration. [PITH_FULL_IMAGE:figures/full_fig_p177_3_1.png]
Figure 3.2
Figure 3.2. Figure 3.2: Figure showing dependency of the integral [PITH_FULL_IMAGE:figures/full_fig_p182_3_2.png]
Figure 5.1
Figure 5.1. Figure 5.1: Diagrammatic representation of Lippmann-Schwinger equation [PITH_FULL_IMAGE:figures/full_fig_p267_5_1.png]
Figure 6.1
Figure 6.1. Figure 6.1: Figure describing the s, t,u-channel diagram respectively, relevant for our computation quadratic effective field theories (EFTs) of gravity. It is well understood that quantum corrections to Einstein gravity generically induce higher-derivative terms, including thos…
Figure 6.2
Figure 6.2. Figure 6.2: Figure depicting the chosen contour for dispersion relation in the complex- [PITH_FULL_IMAGE:figures/full_fig_p292_6_2.png]
Figure 6.3
Figure 6.3. Figure 6.3: Plot depicting the matching for extraction of OPE coefficient using BC expansion [PITH_FULL_IMAGE:figures/full_fig_p304_6_3.png]

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