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The paper proves that one-loop Einstein-Yang-Mills integrands with a gluon loop expand into ordinary one-loop Yang-Mills integrands, with coefficients identical to those in the Yang-Mills-scalar expansion.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-03 17:43 UTC pith:BBPDZPAA

load-bearing objection Plausible one-loop EYM-to-YM expansion with the same coefficients as YMS, but the proof leans on an unproven companion-paper identity; worth refereeing if that dependency is resolved. the 3 major comments →

arxiv 2512.08354 v3 pith:BBPDZPAA submitted 2025-12-09 hep-th

Expansion formula of one-loop Einstein-Yang-Mills integrand

Yi-Jian Du , Chongsi Xie This is my paper
classification hep-th
keywords scattering amplitudesEinstein-Yang-Mills integrandsYang-Mills integrandsYang-Mills-scalar theoryBCJ numeratorscolour-kinematic dualityforward limitone-loop amplitudes
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

One-loop Einstein-Yang-Mills integrands with a gluon running in the loop can be rewritten as ordinary one-loop Yang-Mills integrands with quadratic propagators. The paper proves the expansion formula (3.3): the EYM integrand equals a sum over shuffles of scalars and gravitons of coefficients C times YM integrands, with exactly the same coefficients that appear in the Yang-Mills-scalar expansion. The proof first writes the EYM integrand as a forward limit of tree-level YM amplitudes, expands those into bi-adjoint-scalar amplitudes, then uses consistency conditions on the coefficients and a partial-fraction identity to convert cyclic sums of linear-propagator diagrams into quadratic-propagator diagrams. A sympathetic reader should care because this gives the one-loop EYM and YMS theories a shared kinematic structure, which is the necessary raw material for constructing BCJ numerators in both theories.

Core claim

On its own terms, the paper's central result is the identity I^EYM(l;1,...,r||H) = sum over sigma in {2,...,r} shuffle perms H of C(l;1,sigma) I^YM(l;1,sigma). The left side is the colour-stripped single-trace EYM integrand with a gluon loop, expressed in the forward limit; the right side is a combination of conventional one-loop YM integrands with quadratic propagators. The coefficients C are the same as in the YMS expansion formula. The equality is an integrand-level statement up to loop-momentum shifts, which the paper explicitly notes. The derivation reorganizes the forward-limit sum, quotients by cyclic divisions, and applies the consistency conditions to factor out coefficients, with t

What carries the argument

The central engine is a shuffle expansion of one-loop integrands plus a diagrammatic partial-fraction identity. The shuffle A shuffle B interleaves two ordered sets while preserving relative order within each; it controls which gravitons sit between the scalars and labels the sum in the expansion formula. The consistency conditions say the kinematic coefficients are unchanged when the loop momentum is shifted so that the scalar 1 moves through the permutation; this is what makes the coefficients pull out of the cyclic sums. The actual conversion from forward-limit linear propagators to conventional quadratic propagators is done by identity (B.3): a cyclic sum of tree-level Feynman diagrams a

Load-bearing premise

The proof imports a partial-fraction identity (B.3) — a cyclic sum of forward-limit Feynman diagrams with linear loop propagators equals a single one-loop diagram with quadratic propagators — stated but not derived here, and it also inherits a cited cancellation of tadpole diagrams; if either fails for some diagram topology, the claimed collapse does not follow from this argument.

What would settle it

Evaluate both sides of eq. (3.3) for a generic two-scalar, three-graviton configuration at a fixed loop momentum, expanding in the same bi-adjoint-scalar basis; any nonzero remainder that cannot be absorbed by a loop-momentum shift would refute the expansion. The most targeted check is identity (B.3) itself: test whether the cyclic sum of forward-limit linear-propagator diagrams equals a single quadratic-propagator diagram for a topology in which two subcurrents share momentum support.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For any one-loop EYM integrand with a gluon loop, the expansion reduces the problem of constructing BCJ numerators to the same coefficient data already used for YMS integrands.
  • The explicit coefficient formulas supplied for up to three gravitons give concrete, usable EYM expansion formulas for those cases.
  • Combining the formula with the known GR-to-EYM relations yields an expansion of one-loop gravity integrands in terms of one-loop YM integrands.
  • Because the paper establishes that any YMS expansion formula with consistent coefficients implies an EYM formula, future YMS results automatically transfer to EYM.
  • The proof isolates the technical content: the quadratic-propagator conversion and the coefficient consistency conditions. Checking those for new topologies is all that is needed to extend the expansion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the imported partial-fraction identity could be established by a different method or for a wider class of diagram topologies, the expansion formula would likely survive even where the forward-limit reorganization becomes cumbersome.
  • The coincidence of coefficients between EYM and YMS hints at a universal numerator module determined by the shuffle of scalars and gravitons, independent of which particle runs in the loop; testing this with other loop-particle content would clarify how far the double-copy structure reaches.
  • A natural next step, not taken in the paper, is an explicit four-graviton check: the consistency conditions and the partial-fraction identity should produce coefficient formulas algorithmically, and numerical integrand comparison would test the imported identity directly.
  • The paper's equality is stated only up to loop-momentum shifts; an editorial reading is that a canonical routing prescription would be needed before the coefficients can be promoted to fully gauge-invariant local numerators.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper claims a proof of an expansion formula for one-loop gluon-loop Einstein-Yang-Mills (EYM) integrands: under the assumption that the kinematic coefficients in two expansion steps satisfy the one-loop consistency conditions (2.5)-(2.6), the EYM integrand can be written as a sum of conventional one-loop Yang-Mills integrands with quadratic propagators, with coefficients identical to those in the YMS expansion (2.8). The proof uses the forward-limit representation of the EYM integrand (3.1)-(3.2), decomposes tree-level BS amplitudes into Feynman diagrams, and then reorganizes the sums using the consistency conditions and a graphical partial-fraction identity (B.3) to convert linear propagators into quadratic ones. A two-gluon/two-graviton example is worked out in detail, and the general proof is outlined in §4.2. The paper is explicit that the existence of consistent coefficients follows from [21] and that the identity (B.3) and tadpole cancellations are taken from [22,23].

Significance. If the result holds, it provides a one-loop analogue of the tree-level expansion/BCJ-numerator construction for EYM, connecting EYM integrands to YM and YMS integrands and offering a foundation for constructing one-loop BCJ numerators in EYM and related theories. The worked example is internally consistent, and the paper is honest about its dependencies on companion works. However, the central technical lemma (B.3) is not derived in this manuscript, making the main theorem conditional on an external result. The paper does not provide machine-checked proofs or parameter-free derivations, but it does give a concrete algorithmic proof outline and an explicit example, which are strengths.

major comments (3)
  1. [Appendix B, identity (B.3)] The identity (B.3) is the central mechanism that converts sums of forward-limit linear-propagator diagrams into a single quadratic-propagator one-loop diagram. It is used explicitly in (4.5) and again in (4.22) for the general proof. However, it is only stated with the phrase 'Partial fraction identity induces the following identity' and no derivation is provided. As written, the proof of (3.3) is conditional on this unproven graphical lemma. Please include a self-contained proof of (B.3), or at least a complete algebraic derivation for general I, and make explicit the partial-fraction steps. Without this, the central claim is not established within the manuscript.
  2. [§4.1, eqs. (4.7)-(4.12)] The cancellation of tadpole diagrams is essential to the reorganization leading to (4.15). The manuscript states 'As demonstrated in [23], the sum of all such tadpole diagrams that are related via cyclic permutations of gluons vanishes' and then proceeds to ignore them. Since these terms are boundary contributions in the sum over Feynman diagrams, if they do not cancel the expansion formula (3.3) would receive extra contributions. Please include the tadpole cancellation argument explicitly, or formulate it as a lemma with a proof, so that the step from (4.8) to (4.12) is verifiable.
  3. [§4.2, eqs. (4.20)-(4.22)] The passage from the cyclic sum in (4.20) to the single quadratic-propagator term in (4.22) is compressed. It is asserted that 'According to the identity (B.3), the sum of diagrams in (4.20) produces a quadratic-propagator BS Feynman diagram.' For the general case, the matching of the subcurrents in (4.20) to the cyclic divisions required by (B.3) is not demonstrated. Please spell out the diagrammatic correspondence and define the objects in (4.22) explicitly, so that the general proof can be followed without re-deriving the combinatorics.
minor comments (5)
  1. [§2, eq. (2.2)] The notation 'η μν Δ μν → D−2' is ambiguous. It should be stated that η is the flat metric and Δ^{μν} is the polarization sum; the contraction η_{μν} Δ^{μν} = D−2.
  2. [§4.1, eq. (4.6)] The second definition in (4.6) appears to have a typo: it should be ẽN(ℓ;p,1,2,q,−ℓ) rather than ẽN(ℓ;p,1,2,q,−).
  3. [§4.1, eqs. (4.12)-(4.13)] The notation for the BS integrand and for the T-functions is inconsistent: I^BS(1, p, q,2|ℓ; 1,2, q, p) mixes the loop-momentum position, and T(l−k_A +k_P ; ...) uses a lowercase 'l'. Use ℓ uniformly and adopt a consistent placement for the loop-momentum argument.
  4. [References] The proof depends on several companion preprints: [20], [21], and [23] are listed as unpublished. If the final version appears, these should be updated to published versions or their key statements (existence of consistent coefficients, identity (B.3), tadpole cancellation) should be included as appendices or lemmas.
  5. [Appendix B, eq. (B.3)] The diagrammatic identity (B.3) is stated without a figure in the text. In a published version, include an explicit diagram or an algebraic expression for both sides of (B.3) to make the content of the identity unambiguous.

Circularity Check

0 steps flagged

No circularity: Eq. (3.3) is derived from the forward limit and consistency conditions; the load-bearing partial-fraction identity (B.3) is an external lemma, not an assumed form of the conclusion.

full rationale

The central formula (3.3) is not taken as an input. It is obtained in Sec. 4 by reorganizing the forward-limit expression (3.2) and applying the consistency conditions (2.5)-(2.6). The coefficients C are defined independently as tree-level YMS/EYM expansion coefficients satisfying consistency (eqs. (2.6), (2.9)), and the proof shows how they factor out of the diagram sum. The key conversion from linear-propagator forward-limit diagrams to quadratic-propagator one-loop integrands is identity (B.3), stated in Appendix B and cited to the authors' prior works [22,23]. This is a load-bearing external lemma whose proof is not reproduced here; the tadpole cancellation is cited to [23], and existence of consistent coefficients to [21]. These are parameter-free lemmas whose assumptions do not include the target expansion (3.3), so they are independent support rather than a circular re-use of the conclusion. The paper is explicit about these dependencies (Sec. 5, App. B). No fitted quantity is relabeled as a prediction, and no self-citation is used to forbid alternatives. Verification of (B.3) is a correctness check, not a circularity check.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper introduces no free parameters, no fitted constants, and no new entities. Its burden is entirely in assumptions: the forward-limit representation of EYM integrands, the consistency-condition framework, the imported graphical rule (B.3), shift-invariance of loop integrals, and momentum conservation. Every one of these is either standard in the field or explicitly imported from same-group companion works.

axioms (5)
  • domain assumption Forward-limit expression of the one-loop EYM integrand (eq. (3.1)-(3.2)), including the tree-level EYM expansion into tree-level YM amplitudes with YMS coefficients and the subsequent YM→BS expansion.
    The paper's starting point; relies on the validity of the forward limit for EYM loop integrands from worldsheet/ambitwistor approaches [13-19] and on the tree-level EYM=YMS expansion [3], none of which is re-derived here.
  • domain assumption Existence of kinematic coefficient sets C and Ñ satisfying the consistency conditions (2.5)-(2.6); the theorem (3.3) is explicitly conditional on them.
    Section 2 defines the conditions and Section 5 states existence was proven in [21] (same first author). If no such coefficients exist for a given process, (3.3) has no content.
  • domain assumption Identity (B.3): cyclic sums of forward-limit linear-propagator Feynman diagrams equal a single quadratic-propagator one-loop diagram (refined graphic rule from [22, 23]).
    Stated in Appendix B as 'induced by' a partial fraction identity, with no derivation in this paper; used to complete the collapse of diagram sums at (4.5) and (4.22).
  • standard math Loop-momentum redefinition invariance of the integrals (dimensional regularization shift invariance).
    Stated in Section 2 and used throughout §4.1 to identify T(ℓ) with T'(ℓ) (eqs. (4.11)-(4.12)) and to define the loop momentum as the one to the left of particle 1.
  • standard math On-shell momentum conservation of the external legs, k_1 + ... + k_n = 0.
    Used implicitly; e.g., the matching of momentum shifts in (4.4) requires k_1 + k_p + k_q + k_2 = 0.

pith-pipeline@v1.3.0-alltime-deepseek · 13405 in / 26114 out tokens · 236362 ms · 2026-08-03T17:43:53.237139+00:00 · methodology

0 comments
read the original abstract

Building upon the algebraic consistency construction of one-loop Bern-Carrasco-Johansson (BCJ) numerators for Yang-Mills (YM) and Yang-Mills-scalar (YMS) theories, we explore the expansion formula of one-loop Einstein-Yang-Mills (EYM) integrands (with a gluon loop) in terms of conventional one-loop YM integrands with quadratic propagators. We first express the EYM integrand by tree-level amplitudes according to the forward limit approach. Employing a two-step expansion strategy, the gluon-loop EYM integrand is decomposed into tree-level YM amplitudes under the forward limit, which are subsequently expanded into tree-level bi-adjoint scalar (BS) ones. We then prove that when the kinematic coefficients in both expansion steps satisfy the one-loop consistency conditions, the EYM integrand is finally expanded as a combination of YM integrands with quadratic propagators. The coefficients in this expansion formula coincide exactly with those in the expansion formula for YMS integrands. This correspondence highlights a shared kinematic structure, providing the proper foundation for constructing BCJ numerators in both YMS and EYM theories at one loop.

discussion (0)

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Forward citations

Cited by 1 Pith paper

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  1. Off-shell recursion for all-loop planar integrands in Yang-Mills theory

    hep-th 2026-04 unverdicted novelty 5.0

    Yang-Mills planar loop integrands admit an off-shell recursion that organizes the pure-gluon sector into matrix form and incorporates ghost contributions, yielding a concrete two-loop strategy.

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