Classical soft theorems meet the post-Newtonian expansion

Riccardo Sturani; Samim Akhtar

REVIEW

Classical soft theorems meet the post-Newtonian expansion

Samim Akhtar, Riccardo Sturani

General Relativity and Quantum Cosmology · Submitted 24 Sep 2026

Pith's verdict: The n=1 tail-of-memory is a clean, checkable NRGR result, but the all-order tower and its exponential resummation rest on a 1/n! that the paper's own n=2 display contradicts.

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arXiv ID2609.29895 (version 2)

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CitationRiccardo Sturani, Samim Akhtar. "Classical soft theorems meet the post-Newtonian expansion." arXiv:2609.29895 (2026). https://pith.science/paper/27NHPABU

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Editor's note

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The n=1 tail-of-memory computation in Section 3.2 is the real content: reducing the two-loop amplitude to the on-shell memory form factor times the master integral I1(ω), with the 1/ε_IR pole and the physical log ω in (3.21), is carefully documented and independently checkable. There is also something genuinely new here. The paper makes the bound-binary hereditary ladder explicit in NRGR as a realization of the Sahoo–Sen expansion, which was not written out before: [68] conjectures the tower, [75] does it in PM, but the NRGR route for bound systems is new. The authors are honest about scope. They state that bound inspirals lack Coulomb-type logarithms, they say the spin-memory sector is not expected to match PN–MPM, and footnote 5 concedes the exact coefficient was verified only through n=1.

The second-pass stress-test concern lands. The explicit n=2 result, eq. (4.8), reads A_ttom ~ (iGMω)² A_mem [L(ω)]² with coefficient 1, and the general statement (4.20) likewise carries coefficient 1. But eq. (4.18) inserts a 1/n!, and the resummation (5.3) is iA = e^{iGMωL}/ω · iσ*_ij Q̃_ij, which exists only if the coefficient is 1/n!. If the coefficient is really 1, the series sums to 1/(1−iGMωL) and the claimed all-order Sahoo–Sen correspondence collapses. If the coefficient is really 1/n!, then the displayed n=2 computation is missing a factor. The wider resummation literature expects the exponential, which makes me suspect the §4 display is the slip, but the paper cannot have both, and footnote 5 tells us the n≥2 numbers were never checked. The factorization itself is also asserted by iteration rather than derived: the exact denominators (4.15) are coupled through R_a = Σ_{b≥a} r_b, and the clean product of I1's only emerges after a nested pinching approximation whose corrections are never shown to be harmless for the leading logⁿ coefficient.

Minor items: the UV beta-function running is sketched, not computed, and the manuscript contains extensive corrupted equation passages that make several steps unverifiable as presented. A referee will need the clean source.

Who should read this: anyone working on the IR structure of compact-binary waveforms. The n=1 tail-of-memory result is worth having regardless of the tower; the all-order claim is currently a conjecture wearing a derivation's hat. Send it to referees. The asks are concrete and finite: settle the 1/n! versus coefficient-1 mismatch at n=2, prove or explicitly flag the factorization, and restore the equations.

Referee report

Significance

The n=1 tail-of-memory computation in Section 3.2 is a genuine strength: the two-loop amplitude is reduced without approximation to an off-shell memory form factor acting on a tail kernel, and the IR pole plus physical log ω are isolated in the master integral I_1(ω) (eqs. 3.13–3.21, C.1–C.8). The Appendix D polarization-sector analysis, showing that only the all-tensor {σ²} channel produces the IR pinch, is also careful and useful. The paper is honest about what it has verified (footnote 5: coefficient checked only through n=1). If the all-order tower (1.2) and its exponential resummation (5.3) were established, this would be a valuable bridge between PN hereditary effects and classical logarithmic soft theorems. However, the novelty of the all-order claim is substantially narrower than presented: the leading-log power counting is fixed by loop counting (each static insertion contributes one factor I_1(ω) ∼ (1/ω)log ω, as the paper itself states in Section 2.1 and eq. (2.16)), so the nontrivial content is precisely the numerical coefficient — the part unverified beyond n=1. Since the displayed n=2 result and the asserted 1/n!

Major objections (3)

[§4.1–§4.2, §5.1 (eqs. 4.8, 4.18, 4.19, 5.2, 5.3)] The paper is internally inconsistent on the n=2 coefficient, and this inconsistency invalidates the exponential resummation. The explicitly computed tail-of-tail-of-memory result, eq. (4.8), is displayed as A^TT_ttom|_IR ∼ (iGMω)² A^TT_mem [L(ω)]², i.e. with coefficient 1, and eq. (4.19) likewise carries no 1/n!. Yet the asserted n-insertion factorization, eq. (4.18), and the resummation input, eq. (5.2), carry an explicit 1/n!. For n=2 these cannot both be correct. If the true coefficient is 1, the resummed series is Σ_{n≥0}(iGMωL)^n = 1/(1−iGMωL), a geometric series, not the exponential e^{iGMωL} claimed in eq. (5.3), and the all-order correspondence to the Sahoo–Sen hierarchy fails. If the true coefficient is 1/2, then the displayed n=2 computation (4.8) and the general formula (4.19) are wrong. Either way, the central claims of the paper — the universal soft-dressed exponential (5.3) and the 'first PN/NRGR derivation of the all-order tower' (b55) — are not supported by the manuscript's own results.
[§4.2, eqs. 4.15–4.18; footnote 5] The generalization to n insertions is asserted, not derived. The exact denominators in eq. (4.15) are coupled: −2k·R_a + R_a² with R_a = Σ_{b≥a} r_b, so each denominator depends on all later r_b. The product-of-I_1(ω) structure in eq. (4.18) emerges only after the nested approximation [−2k·(r_a + r_{a+1} + ⋯) + (r_a + r_{a+1} + ⋯)²] ≈ (−2k·r_a + r_a²), pinching r_n → 0 first and iterating. The paper states only ('Iterating the previous construction... one can find', just above eq. 4.18) that this holds, and footnote 5 concedes the numerical coefficient was verified only through n=1. There is no estimate or proof that the cross terms and the O(r_1,...,r_n) numerator terms in (4.17) are subleading in the leading-soft sense for all n. Since the product structure and the 1/n! combinatorics are the entire basis for the exponentiation in §5.1, the derivation needs either a complete n=2 evaluation (including the coupled-denominator corrections) or a general argument establishing which terms contribute to the (log ω)^n coefficient at fixed net power ω^{n−1}.
[§2.1, eq. (2.16); §5.2; b55] The claimed novelty should be recalibrated. The leading-log tower (GM)^n ω^{n−1}(log ω)^n is already exhibited in the paper's own Section 2.1 (eq. 2.16 and the discussion below it) as the soft counting of the PN-MPM hierarchy, and it follows almost by construction from loop counting: each static mass insertion contributes one factor I_1(ω) ∼ (1/ω)log ω. The nontrivial content of the all-order claim in eq. (1.2) and b55 is therefore the overall coefficient and the 1/n! structure — precisely the part that footnote 5 says has not been verified beyond n=1 and that is inconsistent with eq. (4.8) as discussed above. The authors should either provide the complete n=2 (and n=3) coefficient from an explicit evaluation, or restrict the claim to the power-counting statement, which is not a 'first derivation' of the tower.

Minor objections (5)

[Eqs. (2.42), (3.17), (4.6), (4.18)] The manuscript text as received contains long runs of corrupted glyph sequences inserted inside equations (e.g. '⌟⟨⟨⟪rl⟫mo⟨⌟...' in eqs. (2.42), (3.17), (4.6), and (4.18)); these need to be cleaned and the surrounding equations checked, as they currently interrupt the derivations.
[References] References [68] and [93] are the same item (A. Sen, 'Gravitational wave tails from soft theorem: a short review', [2408.08851]) listed twice; one should be removed and the in-text citations reconciled.
[Eq. (2.12)] The displayed radiative-quadrupole formula (2.12) omits the 4PN tail-of-memory term even though the surrounding text discusses it; the authors note this ('not included in the compact formula') but a reader tracking the hereditary hierarchy in one formula would benefit from the term being shown separately or explicitly referenced.
[§5.3, eq. (5.8)] The renormalization of the bare multipole operator in eq. (5.8) and the stated beta-function structure are asserted rather than derived; since the UV sector is declared subleading relative to the IR tower this is acceptable, but a sentence pointing to the derivation in [24] at the point of eq. (5.8) would clarify that this is a summary rather than a new result.
[Appendix C, eq. (C.1)] The integral I_1(ω) is called a 'mixed massive-massless bubble' although the second propagator is static (zero frequency); the terminology is confusing and should be aligned with the static-propagator language used in the main text.

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The paper

The paper claims

The paper claims that the post-Newtonian hereditary ladder for bound binaries — memory at 2.5PN, tail-of-memory at 4PN, and the n-tail terms at (2.5+1.5n)PN — is the frequency-space realization of the Sahoo–Sen logarithmic soft graviton…

The reading

The paper tries to show that the hereditary corrections to gravitational waveforms from bound compact binaries — the displacement memory at 2.5PN, the tail-of-memory at 4PN, and their higher generalizations — are nothing other than the bound-binary realization of the universal logarithmic soft graviton theorem. Working in the NRGR effective field theory, it computes the corresponding one- and multi-loop emission amplitudes and finds that the n-th member of the ladder, the (n)-tail-of-memory at $(2.5+1.5n)$PN, scales as $(GM)^n E_{\rm rad}\,\omega^{n-1}(\ln\omega)^n$. The whole tower is claimed to resum into a single exponential soft-dressing factor, matching the Sahoo–Sen tower order by order and giving the bound-binary hereditary hierarchy a closed, universal form. If correct, this supplies the first post-Newtonian derivation of the all-order leading-logarithm tower previously built on the post-Minkowskian side, and it identifies where bound and scattering cases split: the Coulomb-type logarithms are absent for bound binaries, so the backscattering sector alone realizes the hierarchy.

What carries the argument

The engine of the ladder is the master integral $I_1(\omega)=\int_q [(k-q)^2(q^2-(\omega+i0^+)^2)]^{-1}$, the one-loop integral over one retarded and one static propagator. Each additional mass insertion $M$ adds one factor of this integral through a $\sigma\sigma\phi$ vertex, so $n$ insertions yield $[I_1(\omega)]^n$ and hence $\omega^{n-1}(\ln\omega)^n$, with the $1/n!$ factors supplied by the attachment combinatorics. The multiplicand is the STF memory-source tensor $\tilde Q_{ij}(\omega)=G\int(d\omega'/2\pi)(\omega-\omega')^3\omega'^3\tilde M_{\langle ai}(\omega-\omega')\tilde M_{j\rangle a}(\omega')$, the same tensor at every order, so the entire tower multiplies one universal object. Appendix D isolates the all-tensor $\{\sigma^2\}$ polarization channel as the sole source of the IR poles and the leading soft logarithms, with the other channels contributing only UV poles and finite constants.

Extended reading notes

Core claim

Within the NRGR effective field theory, each scattering of the emitted radiation off the static background mass $M$ is encoded by one static propagator that evaluates to the master integral $I_1(\omega)\sim-(1/8\pi\omega)[1/\epsilon_{\rm IR}-\tfrac12\ln(\omega^2/\tilde\mu^2)+i\pi/2\,\mathrm{sgn}(\omega)]$, converting the leading $1/\omega$ memory pole into one power of the hereditary logarithm. The amplitude with $n$ mass insertions then factorizes into $(1/n!)[I_1(\omega)]^n$ times the same on-shell memory form factor, giving the universal soft behavior $A_n|_{\omega\to0}\sim(GM)^n E_{\rm rad}\,\omega^{n-1}(\ln\omega)^n$ that matches the Sahoo–Sen prediction at each subleading order. Summing the ladder produces the soft-dressed factor $iA = e^{iGM\omega L(\omega)}/\omega\cdot i\sigma^*_{ij}\tilde Q_{ij}(\omega)$, the claimed infrared exponentiation of the radiation effective theory. The all-tensor polarization channel is the unique source of the IR poles and soft logarithms, while UV logarithms first appear at $n\geq2$ and remain subleading relative to the IR tower.

Load-bearing premise

The paper assumes in section 4.2, equation (4.18), that the $n$-tail amplitude factorizes cleanly as $(1/n!)$ times a product of $n$ copies of the master integral $I_1(\omega)$, with the $1/n!$ combinatorics correct and with no leading-soft contamination from the $O(r_1,\ldots,r_n)$ numerator terms that are dropped, even though the authors state the numerical coefficient was verified only up to the $n=1$ tail-of-memory case.

Editorial extensions

If this is right

The whole post-Newtonian chain from memory through tail-of-memory to the n-tail terms reproduces the universal soft expansion coefficient by coefficient, tying bound-binary hereditary radiation directly to the Sahoo–Sen logarithmic soft theorem.
The tower sums to $e^{iGM\omega L(\omega)}/\omega$, so all leading infrared logarithms of bound-orbital emission are captured by a single exponential soft-dressing factor multiplying a universal memory-source tensor.
For bound binaries the Coulomb-type logarithms of hard-particle trajectories are absent, so the backscattering (tail) sector alone generates the hierarchy, in contrast to the hyperbolic scattering case where both contributions enter.
The IR logarithms enter the waveform phase and cancel from the flux, whereas the UV logarithms that first appear at $n\geq2$ carry a multipole-dependent renormalization-group running that is subleading to the IR tower.
The dependence on the regulator scale $\tilde\mu$ in the log corresponds to an unobservable overall time shift $t\to 2GE\,\ln\lambda$, so the dressed phase is universal across multipoles.

Reading between the lines

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

If the exponential structure holds, the cleanest quantitative test is the ratio of successive ladder terms, predicted to be exactly $(GM)\omega/(n+1)$; confirming this order by order against direct PN-MPM or numerical-relativity waveforms would settle the factorization claim.
The soft-dressing factor is independent of spin at leading order, which suggests the spin-quadrupole tail and the spin-memory ladder should obey the same exponential with only their $S_n$ coefficients modified; this is a checkable extension the paper does not carry out.
Comparing the bound-binary tower with the post-Minkowskian tower, which contains both Coulomb and backscattering logs, the bound-case result predicts the two differ precisely by the Coulomb term — an interpolation testable in the small-velocity limit of hyperbolic scattering.
The UV beta-function running of the multipole operators at $n\geq2$ could be matched against known 4PN and 5.5PN energy-flux results as an independent check of the subtraction procedure.

Original abstract

Classical soft theorems govern the non-analytic structure of electromagnetic and gravitational radiation in the low-frequency regime, with the gravitational memory effect providing the canonical realization of the classical soft graviton theorem. While the soft behavior of radiation from hyperbolic scattering is well understood, its realization in gravitationally bound systems remains less explored. In this work, we study the soft structure of gravitational waveforms generated by compact binaries, focusing on the scattering of emitted radiation off the static background curvature within the multipolar post-Newtonian framework. We isolate the infrared (IR) logarithms associated with hereditary effects, including the tail-of-memory, and their higher generalizations with n-tail insertions. The resulting amplitudes exhibit a Laurent expansion in frequency space consistent with the Sahoo-Sen logarithmic soft theorems, thereby establishing a direct correspondence between gravitational memory effects, their higher-order tail corrections, and the soft expansion. We also comment on ultraviolet (UV) logarithms arising from short-distance sensitivity of the multipole expansion and their associated renormalization-group running.

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-25: 1 request, 5,079 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.21 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.59 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Classical soft theorems govern the non-analytic structure of electromagnetic and gravitational radiation in the low-frequency regime, with the gravitational memory effect providing the canonical realization of the classical soft graviton theorem. While the soft behavior of radiation from hyperbolic scattering is well understood, its realization in gravitationally bound systems remains less explored. In this work, we study the soft structure of gravitational waveforms generated by compact binaries, focusing on the scattering of emitted radiation off the static background curvature within the multipolar post-Newtonian framework. We isolate the infrared (IR) logarithms associated with hereditary effects, including the tail-of-memory, and their higher generalizations with n-tail insertions. The resulting amplitudes exhibit a Laurent expansion in frequency space consistent with the Sahoo-Sen logarithmic soft theorems, thereby establishing a direct correspondence between gravitational memory effects, their higher-order tail corrections, and the soft expansion. We also comment on ultraviolet (UV) logarithms arising from short-distance sensitivity of the multipole expansion and their associated renormalization-group running.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Classical soft theorems govern the non-analytic structure of electromagnetic and gravitational radiation in the low-frequency regime, with the gravitational memory effect providing the canonical realization of the classical soft graviton theorem.	background 1.00	0.04	0.03	asserted 1.00
s1	While the soft behavior of radiation from hyperbolic scattering is well understood, its realization in gravitationally bound systems remains less explored.	gap 0.99	0.04	0.02	asserted 0.97
s2	In this work, we study the soft structure of gravitational waveforms generated by compact binaries, focusing on the scattering of emitted radiation off the static background curvature within the multipolar post-Newtonian framework.	contribution 0.97	0.90	0.04	asserted 0.93
s3	We isolate the infrared (IR) logarithms associated with hereditary effects, including the tail-of-memory, and their higher generalizations with n-tail insertions.	contribution 0.59	0.93	0.06	asserted 0.97
s4	The resulting amplitudes exhibit a Laurent expansion in frequency space consistent with the Sahoo-Sen logarithmic soft theorems, thereby establishing a direct correspondence between gravitational memory effects, their higher-order tail corrections, and the soft expansion.	result 0.77	0.91	0.86	asserted 0.93
s5	We also comment on ultraviolet (UV) logarithms arising from short-distance sensitivity of the multipole expansion and their associated renormalization-group running.	contribution 0.70	0.67	0.03	asserted 0.61

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
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levels run

P abstract

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Memory diagram. One-loop process with two retarded propagators. No static propagator — the TT radiative amplitude is finite, with no divergence. The 1/ω soft pole is the Weinberg soft graviton factor. Here Nij ab,cd denotes the complete contraction of the two source-vertex tensor structures, the radiative propagator numerators, and the gauge-fixed σσσ vertex. Since the interaction vertex carries two derivatives, the loop integral in (3.2) reduces to the following structures: B0 ≡ ∫ q 1 D1D2 , Bi ≡ ∫ q qi D1D2 , Bij ≡ ∫ q qiqj D1D2 . (3.3) The scalar master integral B0 is c

**Figure 2.** Tail-of-memory diagram. Two-loop process with three retarded propagators (wavy lines) and one static propagator (dashed). through the σσϕ vertex V2. We denote the complete TT contraction of V1 with σ fields propagators sourced by I ab , I cd, with V1 ab cd mn(ω1,q;ω2,p−q). Its explicit KK rule follows from the cubic bulk action [29, 98]. The vertex V2 connects one internal radiative mode, the static mode, and the external σ ∗ ij . It involves one time derivative for each σ (see e.g. eq. (13) of [98]), and its contraction with a σ propagator Pσσ gives PσklσabV (3) 2 abij = −2i

**Figure 3.** Tail-of-tail-of-memory diagram. Three-loop process with four retarded propagators (wavy lines) and two static propagators (dashed). The finite memory subgraph connects the two source quadrupoles. The tail-of-tail-of-memory is a three-loop process obtained by adding a second mass insertion to the tail-of-memory diagram. The fields sourced by the quadrupoles Iab(ω −ω ′ ) and Icd(ω ′ ) meet at the same σσσ vertex V (3) 1 , while the two subsequent σσϕ vertices generate nested tail kernels. – 22 –

**Figure 4.** General n-tail-of-memory diagram. The process has n + 2 retarded propagators and n static propagators (dashed). The finite memory subgraph connects the two source quadrupoles. Keeping the finite memory subgraph off-shell, the exact n-tail-of-memory amplitude can be organized as iA TT tnom = (i2 Mω2 Λ2 ) n σ ∗ ijΠijkl(nˆ)∫ p1,...,pn Hmem kl (ω,p1) n ∏ a=1 [p 2 a − (ω + i0) 2 ] n−1 ∏ a=1 (pa+1 − pa) 2 (k − pn) 2 . (4.12) This is the direct n-tail extension of (4.1): the two source propagators and the complete V1 contraction remain inside Hmem kl . To expose

Figures reproduced from arXiv:2609.29895 by the authors.

References

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Exact two-loop gravitational tail integrals and their ultraviolet-infrared structure
gr-qc 2026-10 conditional novelty 7.0 of 10

The entire integer numerator domain of the coupled two-loop gravitational tail-of-tail integral family is solved in closed form in generic dimension, with a complete UV/IR pole map near d=3.
Gravitational multipoles, antipodal matching relations, and the logarithmic soft graviton theorem
hep-th 2026-09 conditional novelty 7.0 of 10

A position-space derivation of the classical logarithmic soft graviton theorem for massive-particle scattering, built on a new tower of antipodal Weyl-scalar matchings in harmonic gauge.
All Order Gravitational Wave Tails from Classical Soft Theorem for Subleading Post-Newtonian Expansion
hep-th 2026-10 conditional novelty 6.0 of 10

The coefficients of the entire infinite tower of gravitational wave tail terms are derived at first subleading post-Newtonian order and shown to be universal functions of asymptotic scattering data.

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