{"id":"35f61bf6-233a-482b-8644-7a7e96692d7a","arxiv_id":"2411.12807","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A CHY-level derivation showing that the scaffolding residue of 2n-scalar Yang-Mills-Scalar amplitudes reproduces n-gluon amplitudes, with the pfaffian of A turning into the pfaffian of Psi.","lead":"This paper derives gluon scattering amplitudes from scattering amplitudes of twice as many scalar particles, using the CHY worldsheet formalism. The same derivation converts scalar amplitudes in an Einstein-Maxwell-Scalar theory into graviton amplitudes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central identity Pf'(A)=Pf'(Ψ) relies on an unproved invariance of the reduced Pfaffian under the paper's row/column congruence; the property is true for a suitable deletion choice but is asserted without proof in §2.1.1 and §4.1.","rationale":"The reader's weakest assumption correctly isolates the invariance of the reduced Pfaffian under the row/column congruence as the load-bearing step in the central derivation. My independent analysis shows the step is true for the specific S when the deleted pair is chosen appropriately, so the paper's result is very likely correct; however, the manuscript asserts rather than proves the property, and the general statement is false for arbitrary determinant-1 congruences. This justifies keeping the CONDITIONAL verdict pending a written proof or a numerical/analytic verification. I also note a likely typo in eq. (3.4), where det X should contain (∏ x_i)^2, not (∑ x_i)^2, based on squaring eq. (2.10); this does not change the final result but should be corrected alongside the Pfaffian lemma.","tokens_in":19137,"tokens_out":31825,"duration_ms":298093,"concrete_test":"Supply an independent proof of Pf'(S A S^T) = Pf'(A) for the paper's S by choosing the reduced-Pfaffian deletion to be rows/columns (1,2) and noting the remaining (2n−2)×(2n−2) submatrices are related by a determinant-1 congruence; then verify the equality numerically for a 6×6 A with generic massless momenta, generic u's, and τ = 1, using the same deletion. If the equality fails, eq. (2.21) is unjustified; if it holds, the gap is fillable. As a secondary check, confirm that eq. (2.23)'s off-diagonal W entry should read ε_a·k_b/u_ab rather than k_a·ε_b/u_ab to agree with eq. (2.33).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equations (2.21) and (2.35) require Pf' A|_{x°} = Pf' Ψ. The paper obtains A_final = S A S^T via col i → col i + col i' and row i → row i + row i', then asserts Pf'(A_final) = Pf'(A) because the matrices are related by row/column operations. This is not a general property of reduced Pfaffians: for a fixed deletion pair (i,j), Pf' transforms nontrivially under a general determinant-1 congruence unless that congruence restricts to a determinant-1 transformation of the remaining submatrix. For the particular S used here, the equality does hold when the deleted rows/columns are chosen among the unprimed indices, because the induced congruence on the remaining submatrix has determinant 1; however, the paper neither states this restriction nor proves it, and it silently relies on the standard but nontrivial independence of Pf' from the deletion pair. Both Section 2.1.1 (after eq. 2.29) and Section 4.1 (after eq. 4.18) assert the step without proof. If the missing lemma failed, eq. (2.35) and the EMS statement eq. (3.6) would not be established. The lemma is true and can be proved in a few lines, so this is a rigor gap rather than a demonstrated error, but it is load-bearing because it is the step that identifies the simple scalar integrand with the gluon Pfaffian.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a CHY-formalism derivation of the 'scaffolding residue' of 2n-scalar Yang-Mills-Scalar (YMS) amplitudes: in the limit s_{ii'} -> 0 for all i, the most singular term is shown to equal the n-gluon Yang-Mills amplitude multiplied by the product of 1/s_{ii'} (eq. 2.35). The central technical step is the identification Pf' A|_{x_i=x_i^o} = Pf' Psi (eq. 2.21), where the simple momentum matrix A is transformed by row/column operations into the rich Psi matrix of the gluon CHY integrand, with the derived map k_i = k_i + k_i' and epsilon_i = k_i'. The same computation is repeated for Einstein-Maxwell-Scalar (EMS) scalars, yielding the n-graviton amplitude (eq. 3.6). The paper also treats partial (non-maximal) multi-collinear limits, obtaining amplitudes with m gluons and n-m scalars (eq. 4.22), and discusses color-dressed amplitudes, deriving the appearance of structure constants in the pole expansion (eq. 4.25).","tokens_in":19415,"tokens_out":14908,"duration_ms":133375,"significance":"If the derivation is made fully rigorous, the paper provides a self-contained CHY-level proof of the scaffolding construction of gluon amplitudes from scalar data, explicitly showing how the complicated Psi matrix arises from the simple A matrix. The map epsilon_i = k_i' is derived rather than assumed, and the derivation contains no fitting parameters, which are notable strengths. The extension to gravity and to partial multi-collinear limits, as well as the treatment of color-dressed amplitudes, broadens the scope. The main weakness is a missing linear-algebra lemma about the behavior of reduced Pfaffians under the congruence A -> S A S^T; this property is true for the specific S used here but is asserted without proof in two places. The physical results are consistent with expectations from ref. [1], so the novelty is moderate, but the CHY derivation is a useful contribution.","major_comments":[{"comment":"The equality Pf'(A_final) = Pf'(A) is asserted without proof. The paper justifies it by noting that A_final is obtained from A by row and column operations, but the reduced Pfaffian is not invariant under an arbitrary determinant-1 congruence for a fixed deletion pair: the induced congruence on the deleted (2n-2)x(2n-2) submatrix need not have determinant 1. For the specific matrix S here, the equality does hold when the deleted rows and columns are chosen among the unprimed indices, because the operations then act trivially on the deleted rows/columns and with determinant 1 on the remaining submatrix; the claim then follows from the standard independence of Pf' from the deletion pair. This argument is absent, and the step is load-bearing for eqs. (2.21) and (2.35). Please add a short lemma with proof.","section":"Section 2.1.1, after eq. (2.29)"},{"comment":"The analogous statement Pf'(Psi~) = Pf'(A) for the partial multi-collinear limit is also asserted without proof. Here the row and column operations are followed by block swaps, so the determinant of the induced congruence on the reduced submatrix is not immediate. Since this equality underlies eq. (4.22), a proof of the congruence property, or at least a precise statement of the deletion choice that makes it valid, is needed here as well.","section":"Section 4.1, after eq. (4.18)"}],"minor_comments":[{"comment":"The determinant of X is stated as 1/(tau^{2n} (sum x_i)^2), but from (Pf X)^2 = det X and eq. (2.10) it should be 1/(tau^{2n} (prod x_i)^2). The subsequent integration over x_i implicitly uses the product form, so this appears to be a typo rather than a substantive error.","section":"Eq. (3.4)"},{"comment":"The notation k_j is used ambiguously: in the sum (s_{i'j}+s_{i'j'})/u_{ij} it denotes the combined gluon momentum k_j+k_j', while elsewhere k_j denotes the unprimed scalar momentum. Please clarify to avoid confusion.","section":"Eqs. (2.17)-(2.18)"},{"comment":"The overall sign of the leading term of Pf X is not tracked; the combination of signs from the Parke-Taylor factor and the Pfaffian is left implicit. Since the final amplitude is compared with the standard CHY formula, please state the sign convention used for the leading term.","section":"Section 2.1.1, around eqs. (2.8)-(2.10)"},{"comment":"The statement that M = O(tau^{-1}) is imprecise, as the submatrix of M with both indices unprimed is O(1). What is needed is only that the leading singular part of Pf(M) is 1/(tau^m prod x_i). Please rephrase.","section":"Section 4.1, after eq. (4.11)"}],"recommendation":"major_revision","confidential_remarks":"This is a well-written PSI winter school project note. The central gap is the missing proof of the reduced-Pfaffian congruence property, which is a few lines and does not appear to be a genuine error. Once that lemma is added, the paper should be suitable for publication as a technical note. The originality is moderate, as the physical scaffolding result was introduced in ref. [1], but the CHY derivation and the extensions to gravity, partial multi-collinear limits, and color-dressed amplitudes are new and appropriate for JHEP."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good news first: this is a clean, mostly explicit CHY computation that does what it says. The scaffolding construction itself is from Arkani-Hamed et al., but this paper gives the first detailed CHY-level derivation of the residue, shows how Pf' Psi emerges from Pf' A via row/column operations and the xi equations, and extends the story to multi-collinear limits and color-dressed amplitudes. The map epsilon_i = k_i' is derived, not assumed, and the emergence of the C_ii diagonal entries from the scattering equations is a genuinely nice observation. For the EMS case, the same computation correctly yields graviton amplitudes from scalar amplitudes. The paper is also honest about its provenance as a PSI winter project, and the writing is clear.\n\nThe soft spot is the one the stress-test flagged, and it is load-bearing. In Section 2.1.1, after eq. (2.29), and again in Section 4.1, after eq. (4.18), the paper asserts that Pf'(A_final) = Pf'(A) because the matrices are related by row and column operations. That is not generically true for reduced Pfaffians. For a fixed deletion pair, a determinant-1 congruence on the full matrix does not by itself preserve the reduced Pfaffian; one needs the induced congruence on the deleted submatrix to have determinant 1. For the specific operations used here, the equality does hold when the deletion is chosen among the unprimed indices, and the stress-test note says the lemma can be proved in a few lines. But the paper does not state or prove it. Since this equality is exactly what identifies the scalar integrand with the gluon Pfaffian, a referee should ask for that proof before the central identity (2.35) is taken as established.\n\nThere are minor issues too: the 'trivially generalises' claim for non-canonically-ordered partial amplitudes in Section 4.2 is too quick, and there are no numerical checks. None of this makes the central claim look false; it is a rigor gap rather than a demonstrated error. The paper is worth a serious referee, and the missing Pfaffian lemma is easily fixed.\n\nWho is this for? People who work on CHY formulations, combinatorial origins of amplitudes, or the scaffolding program. A reader in that area gets value from the explicit derivation and the multi-collinear extensions. I would not cite it in my own work next year, but I would bring it to a reading group focused on CHY.","headline":"A useful, explicit CHY computation of the scaffolding residue with one load-bearing unproved Pfaffian step that should be fixed before publication.","tokens_in":19969,"tokens_out":3032,"would_cite":false,"duration_ms":24508,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Scaffolding scalar amplitudes yields pure gluon amplitudes via the CHY formalism.","keywords":["scaffolding residue","Yang-Mills-Scalar amplitudes","CHY formalism","scattering equations","Pfaffian","multi-collinear limits","Einstein-Maxwell-Scalar","color-dressed amplitudes"],"falsifier":"Take a small antisymmetric matrix $A$, say $6\\times 6$ or $8\\times 8$ with generic entries, build $S$ from the operations column $i \\to$ column $i$ + column $i'$ and row $i \\to$ row $i$ + row $i'$, and compare $\\mathrm{Pf}'$ of $A$ with $\\mathrm{Pf}'$ of $S A S^T$; any mismatch disproves the asserted invariance and thus eq. (2.21). Alternatively, evaluate both sides of eq. (2.35) numerically at a random kinematic point for $n=4$ in the $s_{ii'} \\to 0$ limit and check equality including the product of poles.","tokens_in":18867,"feed_emoji":"⚛️","tokens_out":12459,"duration_ms":107912,"temperature":0.7,"pith_summary":"Within the CHY (Cachazo-He-Yuan) formulation of tree-level amplitudes, the paper shows that the scaffolding limit of a $2n$-scalar Yang-Mills-Scalar amplitude — sending each paired invariant $s_{ii'}$ to zero — reproduces the $n$-gluon Yang-Mills amplitude multiplied by a product of $1/s_{ii'}$ factors. The result is consequential because it exhibits a pure gluon amplitude, polarization vectors and all, as the most singular residue of a much simpler scalar integrand built from the matrix $A$. The same computation turns $2n$-scalar Einstein-Maxwell-Scalar amplitudes into $n$-graviton amplitudes, and partial multi-collinear limits yield mixed gluon-scalar amplitudes as well as the expected poles in color-dressed amplitudes.","feed_headline":"2n scalars become n gluons in the scaffolding limit","feed_subtitle":"The most singular collinear residue of 2n-scalar amplitudes is exactly the n-gluon amplitude, polarizations included.","key_machinery":"The machinery is the scaffolding residue computed inside the CHY integral: parameterize the limit by $s_{ii'} = \\tau \\hat{s}_{ii'}$ and $u_{i'} = u_i + \\tau x_i$, so that the Parke-Taylor factor, the Pfaffian of $X$, and the integration measure each supply powers of $1/\\tau$; the primed scattering equations then fix the $x_i$ and produce the diagonal entries of $\\Psi$. The load-bearing linear-algebra step is the congruence $A_{\\mathrm{final}} = S A S^{T}$ (row $i \\to$ row $i$ + row $i'$, column $i \\to$ column $i$ + column $i'$) with $\\det S = 1$, under which the reduced Pfaffian (the Pfaffian with two rows and columns removed, as used in CHY formulas) is asserted to be invariant, so that $\\mathrm{Pf}' A$ evaluated at the solutions $x_i^\\circ$ equals $\\mathrm{Pf}' \\Psi$. The explicit map $k_i := k_i + k_{i'}$, $\\varepsilon_i := k_{i'}$ geometrizes gauge invariance: shifting the pair $(k_i, k_{i'})$ parallel to $k_i + k_{i'}$ leaves the amplitude unchanged.","core_discovery":"The central claim is eq. (2.35): in the limit $s_{ii'} \\to 0$, the YMS scalar amplitude $\\mathcal{A}^{\\mathrm{YMS}}(1,1',\\ldots,n,n')$ equals $\\left(\\prod_i 1/s_{ii'}\\right)\\mathcal{A}^{\\mathrm{YM}}(1,\\ldots,n)$ plus subleading terms, with the reduced Pfaffian of the simple momentum matrix $A$ evaluated on the scattering-equation solutions equal to the reduced Pfaffian of the polarization-rich matrix $\\Psi$ (eq. 2.21). The proof passes through a row-and-column operation on $A$ that turns it into $\\Psi$ once the identifications $k_i := k_i + k_{i'}$ and $\\varepsilon_i := k_{i'}$ are made, and once the primed scattering equations supply the diagonal entries of $\\Psi$. The analogous statement for the EMS theory is eq. (3.6), producing $n$-graviton amplitudes. The paper also extends the argument to partial scaffolding (some pairs collinear, others untouched), obtaining mixed gluon-scalar amplitudes, and to color-dressed amplitudes, where the collinear poles come with the structure constants $f_{jj'j^\\star}$.","pith_inferences":["We would expect the asserted $\\mathrm{Pf}'$ congruence invariance to be checkable by direct numerical evaluation on random small antisymmetric matrices; a counterexample there would break the derivation before any physics enters.","The same row-and-column mechanism may generate the richer CHY matrix $\\Pi$ of Einstein-Yang-Mills amplitudes from a simpler scalar integrand, a step the paper mentions only as future work.","The geometric picture of gauge invariance as vertex shifts along $k_i + k_{i'}$ suggests a wider family of 'scaffolded' theories in which any polygon vertex carries a pair of scalar momenta, possibly extending the construction beyond tree level.","Because the residue formula (2.35) is exact at leading order in $\\tau$, an explicit $n=4$ evaluation on both sides would provide a sharp end-to-end test of the sign and normalization of the product of poles."],"forward_implications":["Any $n$-gluon CHY amplitude can be recovered as the most singular term of a $2n$-scalar YMS amplitude, so gluon polarization data need not be inserted by hand.","The same identity upgrades to gravity: the scaffolding residue of $2n$ EMS scalars is the pure $n$-graviton amplitude (eq. 3.6).","Partial scaffolding produces mixed amplitudes with $m$ gluons and $n-m$ scalars, taking the expected Yang-Mills-scalar form (eq. 4.22).","In color-dressed amplitudes the collinear poles appear multiplied by the structure constants $f_{jj'j^\\star}$, so color and kinematics factor in the multi-collinear limit (eq. 4.25).","The diagonal entries $C_{ii}$ of $\\Psi$ emerge from the primed scattering equations (eq. 2.34), resolving a previously opaque part of the CHY matrix from scalar kinematics alone."],"supporting_citations":[{"why":"introduced the scaffolding residue and the combinatorial picture of gluons from scalar data; the paper recomputes that residue in CHY without assuming its map.","marker":"[1]"},{"why":"gave the Parke-Taylor factor used throughout as one of the CHY integrands.","marker":"[2]"},{"why":"supplies the color decomposition used in Section 4.2 for color-dressed amplitudes.","marker":"[21]"},{"why":"established the CHY scattering-equation formula for Yang-Mills tree amplitudes, the target identity the scaffolding residue reproduces.","marker":"[32]"},{"why":"provided CHY formulations for scalars, gluons, and gravitons used as the starting and ending points of the derivation.","marker":"[34]"},{"why":"showed how compactification builds YMS and related CHY integrands from Yang-Mills, yielding the Pf X Pf' A integrand.","marker":"[35]"},{"why":"proved the CHY Yang-Mills formula in arbitrary dimension, the standard result the scaffolding derivation must match.","marker":"[36]"}],"fun_headline_variants":["Scaffolding limit turns 2n scalars into n gluons","Collinear scaffolding residue yields gluon amplitudes","From 2n-scalar amplitudes to n-gluon via scaffolding","Scaffolding: scalar pairs collapse to single gluons","Gluon amplitudes from scalar scaffolding residue"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the unproved linear-algebra premise that the reduced Pfaffian ($\\mathrm{Pf}'$) is invariant under the row-and-column congruence $A_{\\mathrm{final}} = S A S^T$ with $\\det S = 1$; if that invariance fails, the identification $\\mathrm{Pf}' A = \\mathrm{Pf}' \\Psi$, and with it the central scaffolding identity, does not follow. This is a separate mathematical premise from the collinear limit itself.","fun_headline_variants_meta":{"raw":{"variants":["Scaffolding limit turns 2n scalars into n gluons","Collinear scaffolding residue yields gluon amplitudes","From 2n-scalar amplitudes to n-gluon via scaffolding","Scaffolding: scalar pairs collapse to single gluons","Gluon amplitudes from scalar scaffolding residue"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000839,"raw_usage":{"total_tokens":3660,"prompt_tokens":949,"completion_tokens":2711,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":565,"completion_tokens_details":{"reasoning_tokens":2629}},"tokens_in":565,"tokens_out":2711,"duration_ms":19305,"temperature":1.0,"reasoning_tokens":2629,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:12:50.358169+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small antisymmetric matrix $A$, say $6\\times 6$ or $8\\times 8$ with generic entries, build $S$ from the operations column $i \\to$ column $i$ + column $i'$ and row $i \\to$ row $i$ + row $i'$, and compare $\\mathrm{Pf}'$ of $A$ with $\\mathrm{Pf}'$ of $S A S^T$; any mismatch disproves the asserted invariance and thus eq. (2.21). Alternatively, evaluate both sides of eq. (2.35) numerically at a random kinematic point for $n=4$ in the $s_{ii'} \\to 0$ limit and check equality including the product of poles.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gave the Parke-Taylor factor used throughout as one of the CHY integrands."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the color decomposition used in Section 4.2 for color-dressed amplitudes."},{"cited_title":"Proof of the Formula of Cachazo, He and Yuan for Yang-Mills Tree Amplitudes in Arbitrary Dimension","cited_arxiv_id":"1311.5200","evidence_quote":"proved the CHY Yang-Mills formula in arbitrary dimension, the standard result the scaffolding derivation must match."}],"review_version":1}