Pith. sign in

REVIEW 3 major objections 6 minor 65 references

Acting on scalar-scaffolded Yang-Mills amplitudes with an (n-1)-fold differential operator built from planar variables converts the n-gluon amplitude into a single planar phi^3 diagram, and the space of mixed amplitudes with r scalars has e

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 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 15:41 UTC pith:E4Z5RPN4

load-bearing objection Solid research note with new differential-operator rules and Catalan counting; the general extraction rules are proven only for ray-like topologies and otherwise verified up to n=8, so the central claim is a well-supported conjecture rather than a theorem. the 3 major comments →

arxiv 2512.15882 v2 pith:E4Z5RPN4 submitted 2025-12-17 hep-th

On differential operators for scalar-scaffolded gluons

Jin Dong , Yong-Xiang Su , Dongyu Yang This is my paper
classification hep-th
keywords scalar-scaffolded gluonsYang-Mills amplitudesdifferential operatorsplanar phi^3 diagramsCatalan numbersmixed amplitudesuniversal expansionfactorization
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.

This paper tries to show that a Yang-Mills n-gluon tree amplitude, written in the scalar-scaffolding variables in which each physical gluon is a pair of scalars, can be reduced to a single planar phi^3 diagram by an (n-1)-fold derivative. The rules supply a fixed dictionary between the propagators of the cubic diagram and the derivative pairs, with the ray-like cases proven for all n by factorization and the remaining diagrams checked up to n=8. The same derivative logic extends to mixed amplitudes containing both scalars and gluons, and the paper argues that mixed amplitudes with r scalars span a space of dimension C_{r-2}, the (r-2)-th Catalan number, independent of the number of gluons. If correct, this yields a planar, Catalan-counting version of the universal expansion of Yang-Mills amplitudes, replacing a factorial-size summand with growth of order 5^n and exhibiting gauge invariance through nested-commutator prefactors. The load-bearing step still open is a combinatorial intersection-counting lemma used to show that all non-ray factorizations are annihilated.

Core claim

On the paper's own terms, the central claim is that the n-gluon Yang-Mills amplitude in 2n scalar variables is not merely a sum over cubic diagrams: a suitably ordered sequence of n-1 derivatives selects exactly one planar phi^3 diagram, with the amplitude's poles becoming the propagators of that diagram. The same rules, applied to a subpolygon after deleting the vertices belonging to gluons, produce mixed amplitudes in a natural planar basis; the number of linearly independent such amplitudes depends only on the number r of scalars and equals the Catalan number C_{r-2}. This is presented as a generalization of the uniqueness theorem for pure gluon amplitudes: for r=2 scalars there is one ob

What carries the argument

The carrying object is the multi-derivative operator D_{(2a',2b'),(2a_1,2b_1-1),...} = ∂^{n-1}/(∂X_{2a',2b'} ∏ ∂X_{2a_i,2b_i-1}) acting on planar variables X_{i,j}=(p_i+...+p_{j-1})^2. The central identity is the dictionary between propagators (i,j) of a cubic diagram and derivative pairs [a,b], summarized in rules (3.7)-(3.10) and anchored by the fixed derivatives [2,2n] and [1,4]. The proof mechanism is factorization: consecutive residues are evaluated recursively, while an asserted intersection-counting lemma says the only configuration with the minimal n-3 intersections between derivative lines and cut propagators is the ray-like one. For mixed amplitudes, the same dictionary is applied

Load-bearing premise

The extraction rules rest on an unproven counting lemma in Section 3.2: only the ray-like sequence of cut propagators has the minimal n-3 intersections with the derivative lines, so every other factorization is annihilated; if that count is wrong, the operators would output sums of diagrams or zero.

What would settle it

Enumerate the intersections between the derivative lines [2,2n],[1,4],[1,6],... and the cut propagators of every triangulation of the 2n-gon, and check that only the ray-like sequence attains n-3 intersections; equivalently, evaluate the corresponding operator on the scalar-scaffolded n-gluon amplitude at a non-ray factorization for n=9 and test whether the residue vanishes. A single nonzero residue for a non-ray sequence would falsify the general rule.

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

If this is right

  • If the rules hold for all n, individual planar cubic diagrams can be extracted from the full gluon amplitude by a fixed derivative, giving a direct diagram-by-diagram probe of the amplitude's pole structure.
  • The dimension C_{r-2} for mixed amplitudes means gluon insertions do not create new independent objects: any mixed amplitude with r scalars is a linear combination of amplitudes with gluons inserted into the r-scalar scalar skeletons.
  • The planar universal expansion has 2, 5, 15, 51, 188, 731 terms for n=3 through 8 and asymptotically grows like 5^n, making it substantially more compact than the factorial-counting universal expansion.
  • Each term pairs a nested-commutator prefactor with exactly one mixed amplitude in the natural basis, making gauge invariance of the internal gluons manifest and removing redundant permutations.

Where Pith is reading between the lines

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

  • The unproven intersection-counting lemma is the natural next target: a direct proof for all triangulations of the 2n-gon would upgrade the n<=8 checks to a theorem for every multiplicity.
  • The n=9 counterexamples to single-operator gauge invariance show that the paper's condition (5.18) is necessary but not sufficient; one testable extension is whether linear combinations of derivative operators can restore gauge invariance and complete the operator basis.
  • The Catalan dimension count suggests the underlying combinatorics is exactly that of triangulations of an r-gon; if so, the same counting may persist at one loop or in other colored theories with similar surfaceology descriptions, as the paper itself hints in its outlook.
  • The planar expansion, recursively expanded, could yield a reference-ordering independent presentation of kinematic numerators; the paper leaves this open, but the structure suggests a way to avoid averaging over orderings.

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 / 6 minor

Summary. The paper studies differential operators acting on scalar-scaffolded Yang-Mills amplitudes. The central claims are: (i) for any planar cubic diagram, an (n-1)-fold derivative in the 2n scalar variables converts the n-gluon amplitude into that single phi^3 diagram (§3.1); (ii) the space of mixed r-scalar/(n-r)-gluon amplitudes has dimension C_{r-2}, independent of n and of the number of gluons (§4, Appendix A.2); (iii) these mixed amplitudes can, in many cases, be obtained by the same derivative rules applied to a subpolygon, with a gauge-invariance condition controlling when this works (§5); and (iv) this yields a 'planar universal expansion' whose term count is a Catalan-weighted sum rather than the factorial-like count of the original expansion (§6). The paper gives explicit rules, many examples up to n=8, and factorization-based proofs for the two ray-like topologies at general n.

Significance. If the results are correct, they provide a significant structural simplification: tree-level gluon amplitudes would admit a planar expansion whose term count grows like a Catalan-weighted sum, and the mixed-amplitude basis would have a purely Catalan dimension. The paper is explicit and hands-on: the differential-operator rules are stated concretely, many nontrivial examples are worked out (including rank computations for the mixed amplitudes in Appendix A.1), and the ray-like cases are proved by a recursive factorization argument. The planar universal expansion with the term count (6.11) is a clear new proposal. The main value is in the confluence of the scalar-scaffolding formalism with differential operators and the resulting conjecture of a Catalan-counted basis. The paper also honestly reports a failure of the naive extension at n=9 (eq. 5.19), which is a useful diagnostic.

major comments (3)
  1. [§3.2, Step II (eqs. 3.27–3.34)] The proof that the ray-like operator annihilates all non-ray-like factorization residues relies on an unproved intersection-counting lemma: it is asserted that a derivative sequence survives a residue only if the number of intersections between the blue derivative lines and the black cut propagators is at most n−3, and that the ray-like configuration is the unique one attaining equality. This is only illustrated in Figures 10–11 and stated as 'direct to find'. Since this lemma is the sole mechanism forcing all non-ray-like residues to vanish, the general-n claim of single-diagram extraction is not established. The paper's own §5 shows that a closely related naive extension of the §3 rules fails at n=9 (eq. 5.19), so the n≤8 checks cannot safely be extrapolated. A proof or a precise combinatorial formulation of the lemma is needed.
  2. [§3.1, rules (3.7)–(3.10)] The general prescription for assigning differential operators to an arbitrary planar cubic diagram is only verified on examples up to n=8. The only general-n proof in §3.2 covers the two ray-like families (Example 3.9). For zigzag, cycle, and mixed topologies, the paper does not provide a proof that the stated operator extracts exactly the target diagram and annihilates all others. This is load-bearing for the central claim that every planar cubic diagram can be extracted for all n. The paper should either supply a general proof (perhaps extending the factorization argument) or clearly label the general rules as a conjecture supported by finite checks.
  3. [Appendix A.2] The proof of N_{n,r}=C_{r-2} is based on an explicit locality ansatz (A.3) with polynomial numerators of fixed momentum power (PMN). The soft-momentum induction has a gap at sub-sub-leading and higher orders: after defining the difference B in (A.24), the argument asserts that the Γ-type terms are ruled out by the uniqueness theorem of [54] and that the remaining terms vanish because their PMN is too low. But [54] concerns full amplitudes, not the partial kinematic factors appearing here, and the decomposition into q·p_1, q·p_n, and P_i poles is not shown to be free of overlaps. Since this appendix underpins the Catalan dimension used in §4 and §6, the proof needs to be made rigorous or the ansatz must be stated as an additional assumption.
minor comments (6)
  1. [Abstract] Typo: 'an mixed amplitude' should read 'a mixed amplitude'.
  2. [§5, Figure 15 caption] Typo: 'derivatives acing' should read 'derivatives acting'.
  3. [§3.1, Rule 4] The notion of 'closest to (1,i) or (3,i)' in Rule 4 is not defined quantitatively. For a reader trying to apply the rule to a new diagram, a definition using the cyclic order of diagonals around vertex i would help.
  4. [Appendix A.2] Repeated typo: 'anstaz' should be 'ansatz'; 'alike the case' should be 'as in the case'; 'sub s⩾3-leadingorder' should be 'subleading order for s⩾3'.
  5. [Appendix A.1] In the displayed 12×23 matrix, the tenth row is identically zero. If this corresponds to one of the listed orderings being a vanishing amplitude, that is worth a comment; otherwise it may be a typographical artifact that should be corrected.
  6. [§6] The relation to Ref. [41] is mentioned only in passing in the introduction; a sentence in §6 explaining how the present construction compares to that related work would help the reader place the novelty.

Circularity Check

0 steps flagged

No significant circularity: the differential-operator rules and Catalan-dimension theorem are derived via factorization/induction and are not equivalent to their target claims by construction.

full rationale

The paper's central derivations are not circular. The scalar-scaffolded representation is an external input from prior work ([19,20,42]); the paper does not fit its new differential operators to the amplitudes and then rename the fit as a prediction. Section 3.1 proposes a concrete map from cubic diagrams to derivative sequences, and Section 3.2 proves the ray-like cases at general n by a factorization recursion (Eqs. 3.21-3.26 and 3.28-3.33). The vanishing of non-ray-like residues in Step II rests on an intersection-counting assertion in Figures 10-11 that is stated without a general proof. This is a genuine proof gap, not a circularity: the assertion is an independent combinatorial claim whose failure would falsify the conjecture rather than expose an equivalence between input and output. Similarly, the Catalan-basis theorem in Section 4 and Appendix A.2 is proved by an induction on soft limits using the universal expansion (2.40), which was established independently in [7,8,50] and whose uniqueness argument cites the external result [54] by Rodina; the proof does not assume the dimension C_{r-2} it aims to establish. Section 5 even reports an explicit counterexample to the naive generalization at n=9 (Eq. 5.19), showing the authors distinguish verified results from conjectural extrapolation. The self-citation to [51], which includes co-author Dong, is used only to justify the derivation of the already-existing expansion (2.40); since that expansion has independent support from CHY-based works, the self-citation is not load-bearing. No step reduces by definition to its own input, so the appropriate finding is no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No empirical fit parameters appear; the gauge parameter lambda in (2.6) is a redundancy, and the operators D... are constructions, not fitted constants. No new particles or forces are introduced; scalar-scaffolded gluons are a reparametrization from prior work.

axioms (5)
  • domain assumption Scalar-scaffolded representation of YM amplitudes: gluon momenta/polarizations encoded by 2n planar variables X_{ij}, with gauge invariance (2.16)-(2.17) and factorization (2.19)-(2.22).
    Taken from refs. [19,20,42] (same group); the entire derivative construction is phrased in this representation.
  • domain assumption Universal expansion (2.40) of YM amplitudes into mixed YM+phi^3 amplitudes with BCJ prefactors.
    Theorem from [51] (includes co-author Dong); used to prove Catalan counting and to reorganize into the planar expansion.
  • domain assumption Uniqueness from gauge invariance / power-counting ansatz (A.3), including the claim that a numerator with insufficient momentum power cannot be gauge invariant.
    Relies on Rodina's uniqueness theorem [54]; used in Appendix A.2 soft-expansion induction.
  • ad hoc to paper Intersection-counting lemma: a derivative sequence vanishes on a factorization unless the number of blue/black line intersections is <= n-3, with a unique maximal configuration.
    Introduced in Section 3.2 Step II, justified by figures 10-11 rather than a proof; load-bearing for the vanishing part of the single-diagram theorem.
  • ad hoc to paper Locality ansatz for mixed amplitudes: they admit a Feynman-diagram expansion with polynomial numerators of momentum power equal to the number of gluons (Appendix A.2).
    The soft-expansion proof assumes this ansatz to conclude uniqueness at each order.

pith-pipeline@v1.3.0-alltime-deepseek · 31620 in / 12245 out tokens · 124756 ms · 2026-08-03T15:41:53.044770+00:00 · methodology

0 comments
read the original abstract

Recently, based on the curve-integral formulation for stringy Tr$\phi^3$ amplitudes, a combinatorial formulation for Yang-Mills amplitudes has been proposed which describes gluons using pairs of scalars and produces the $n$-gluon amplitude from simple kinematical shift of stringy Tr$\phi^3$ amplitudes with $2n$ scalars. It has revealed a variety of new properties and structures even for tree-level gluon amplitudes such as hidden zeros and splits, and in this note we provide another example: we study differential operators acting on Yang-Mills amplitudes with respect to $2n$-scalar kinematic variables, which convert such scalar-scaffolded gluons into scalars. In particular, we find $(n{-}1)$-fold differential operators (using $2n$-scalar variables) that turn the $n$-gluon amplitude into a single planar $\phi^3$ diagram; we then generalize such operators to those that convert $n$ gluons to mixed amplitudes with $r$ scalars and $n{-}r$ gluons (the latter can be viewed as insertions on $\phi^3$ diagrams). We also show that the number of linearly independent mixed amplitudes with $r$ scalars and $n-r$ gluons is given by the number of $\phi^3$ diagrams, the Catalan number $\mathcal{C}_{r-2}$, which can be viewed as a generalization of the ``uniqueness" theorem of gluon amplitudes (with $r=0$). Finally, our construction leads to a planar version of the universal expansion of Yang-Mills amplitudes into a sum of gauge-invariant prefactors built from nested commutators, each accompanied by an mixed amplitude in the natural basis. This formulation significantly reduces the redundancy present in the original expansion.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

65 extracted references · 53 linked inside Pith

  1. [1]

    Bern, J.J.M

    Z. Bern, J.J.M. Carrasco and H. Johansson,New Relations for Gauge-Theory Amplitudes, Phys. Rev. D78(2008) 085011 [0805.3993]

    Pith/arXiv arXiv 2008

  2. [2]

    Bern, J.J.M

    Z. Bern, J.J.M. Carrasco and H. Johansson,Perturbative Quantum Gravity as a Double Copy of Gauge Theory,Phys. Rev. Lett.105(2010) 061602 [1004.0476]

    Pith/arXiv arXiv 2010

  3. [3]

    Cachazo, S

    F. Cachazo, S. He and E.Y. Yuan,Scattering equations and Kawai-Lewellen-Tye orthogonality,Phys. Rev. D90(2014) 065001 [1306.6575]

    Pith/arXiv arXiv 2014

  4. [4]

    Cachazo, S

    F. Cachazo, S. He and E.Y. Yuan,Scattering of Massless Particles in Arbitrary Dimensions, Phys. Rev. Lett.113(2014) 171601 [1307.2199]

    Pith/arXiv arXiv 2014

  5. [5]

    Cachazo, S

    F. Cachazo, S. He and E.Y. Yuan,Scattering of Massless Particles: Scalars, Gluons and Gravitons,JHEP07(2014) 033 [1309.0885]

    Pith/arXiv arXiv 2014

  6. [6]

    Mafra, O

    C.R. Mafra, O. Schlotterer and S. Stieberger,Explicit BCJ Numerators from Pure Spinors, JHEP07(2011) 092 [1104.5224]

    Pith/arXiv arXiv 2011

  7. [7]

    Fu, Y.-J

    C.-H. Fu, Y.-J. Du, R. Huang and B. Feng,Expansion of Einstein-Yang-Mills Amplitude, JHEP09(2017) 021 [1702.08158]

    Pith/arXiv arXiv 2017

  8. [8]

    Du and F

    Y.-J. Du and F. Teng,BCJ numerators from reduced Pfaffian,JHEP04(2017) 033 [1703.05717]

    Pith/arXiv arXiv 2017

  9. [9]

    Teng and B

    F. Teng and B. Feng,Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame,JHEP 05(2017) 075 [1703.01269]

    Pith/arXiv arXiv 2017

  10. [10]

    Cheung and Z

    C. Cheung and Z. Moss,Symmetry and Unification from Soft Theorems and Unitarity,JHEP 05(2021) 161 [2012.13076]

    Pith/arXiv arXiv 2021

  11. [11]

    Edison and F

    A. Edison and F. Teng,Efficient Calculation of Crossing Symmetric BCJ Tree Numerators, JHEP12(2020) 138 [2005.03638]

    Pith/arXiv arXiv 2020

  12. [12]

    S. He, L. Hou, J. Tian and Y. Zhang,Kinematic numerators from the worldsheet: cubic trees from labelled trees,JHEP08(2021) 118 [2103.15810]

    Pith/arXiv arXiv 2021

  13. [13]

    Wei and K

    F.-S. Wei and K. Zhou,Expanding single-trace YMS amplitudes with gauge-invariant coefficients,Eur. Phys. J. C84(2024) 29 [2306.14774]

    Pith/arXiv arXiv 2024

  14. [14]

    G. Chen, L. Rodina and C. Wen,Kinematic Hopf algebra and Bern-Carrasco-Johansson numerators at finiteα’,Phys. Rev. D110(2024) L041902 [2403.04614]

    Pith/arXiv arXiv 2024

  15. [15]

    Gomez,A Novel On-Shell Recursive Relation,2507.23510

    H. Gomez,A Novel On-Shell Recursive Relation,2507.23510

    Pith/arXiv arXiv

  16. [16]

    Del Duca, L.J

    V. Del Duca, L.J. Dixon and F. Maltoni,New color decompositions for gauge amplitudes at tree and loop level,Nucl. Phys. B571(2000) 51 [hep-ph/9910563]

    Pith/arXiv arXiv 2000

  17. [17]

    Arkani-Hamed, H

    N. Arkani-Hamed, H. Frost, G. Salvatori, P.-G. Plamondon and H. Thomas,All Loop Scattering as a Counting Problem,2309.15913

    Pith/arXiv arXiv

  18. [18]

    Arkani-Hamed, H

    N. Arkani-Hamed, H. Frost, G. Salvatori, P.-G. Plamondon and H. Thomas,All Loop Scattering For All Multiplicity,2311.09284

    Pith/arXiv arXiv

  19. [19]

    Arkani-Hamed, Q

    N. Arkani-Hamed, Q. Cao, J. Dong, C. Figueiredo and S. He,Hidden zeros for particle/string amplitudes and the unity of colored scalars, pions and gluons,JHEP10 (2024) 231 [2312.16282]

    Pith/arXiv arXiv 2024

  20. [20]

    Arkani-Hamed, Q

    N. Arkani-Hamed, Q. Cao, J. Dong, C. Figueiredo and S. He,Scalar-scaffolded gluons and the combinatorial origins of Yang-Mills theory,JHEP04(2025) 078 [2401.00041]. 38

    Pith/arXiv arXiv 2025

  21. [21]

    Arkani-Hamed, Q

    N. Arkani-Hamed, Q. Cao, J. Dong, C. Figueiredo and S. He,Nonlinear Sigma model amplitudes to all loop orders are contained in the Tr(Φ3) theory,Phys. Rev. D110(2024) 065018 [2401.05483]

    Pith/arXiv arXiv 2024

  22. [22]

    Q. Cao, J. Dong, S. He and C. Shi,A universal splitting of string and particle scattering amplitudes,2403.08855

    Pith/arXiv arXiv

  23. [23]

    Bartsch, T.V

    C. Bartsch, T.V. Brown, K. Kampf, U. Oktem, S. Paranjape and J. Trnka,Hidden Amplitude Zeros From Double Copy,2403.10594

    Pith/arXiv arXiv

  24. [24]

    Y. Li, D. Roest and T. ter Veldhuis,Hidden Zeros in Scaffolded General Relativity and Exceptional Field Theories,2403.12939

    arXiv

  25. [25]

    Arkani-Hamed and C

    N. Arkani-Hamed and C. Figueiredo,All-order splits and multi-soft limits for particle and string amplitudes,2405.09608

    Pith/arXiv arXiv

  26. [26]

    Rodina,Hidden Zeros Are Equivalent to Enhanced Ultraviolet Scaling, and Lead to Unique Amplitudes in Tr(ϕ3) Theory,Phys

    L. Rodina,Hidden Zeros Are Equivalent to Enhanced Ultraviolet Scaling, and Lead to Unique Amplitudes in Tr(ϕ3) Theory,Phys. Rev. Lett.134(2025) 031601 [2406.04234]

    Pith/arXiv arXiv 2025

  27. [27]

    Q. Cao, J. Dong, S. He, C. Shi and F. Zhu,On universal splittings of tree-level particle and string scattering amplitudes,2406.03838

    Pith/arXiv arXiv

  28. [28]

    Zhou,Understanding zeros and splittings of ordered tree amplitudes via Feynman diagrams,JHEP03(2025) 154 [2411.07944]

    K. Zhou,Understanding zeros and splittings of ordered tree amplitudes via Feynman diagrams,JHEP03(2025) 154 [2411.07944]

    Pith/arXiv arXiv 2025

  29. [29]

    Y. Li, T. Wang, T. Brauner and D. Roest,Diagrammatic Derivation of Hidden Zeros and Exact Factorization of Pion Scattering Amplitudes,Phys. Rev. Lett.135(2025) 181601 [2412.14858]

    arXiv 2025

  30. [30]

    Giménez Umbert and K

    B. Giménez Umbert and K. Yeats,ϕp amplitudes from the positive tropical Grassmannian: Triangulations of extended diagrams,Phys. Rev. D110(2024) 125018 [2403.17051]

    Pith/arXiv arXiv 2024

  31. [31]

    Zhang,On the new factorizations of Yang-Mills amplitudes,JHEP02(2025) 074 [2412.15198]

    Y. Zhang,On the new factorizations of Yang-Mills amplitudes,JHEP02(2025) 074 [2412.15198]

    Pith/arXiv arXiv 2025

  32. [32]

    Zhang,New factorizations of Yang-Mills amplitudes,Phys

    Y. Zhang,New factorizations of Yang-Mills amplitudes,Phys. Rev. D111(2025) 085004 [2406.08969]

    arXiv 2025

  33. [33]

    S. De, S. Paranjape, A. Pokraka, M. Spradlin and A. Volovich,Hidden zeros of the cosmological wavefunction,JHEP07(2025) 174 [2503.23579]

    Pith/arXiv arXiv 2025

  34. [34]

    Backus and L

    J.V. Backus and L. Rodina,Emergence of Unitarity and Locality from Hidden Zeros at One-Loop Order,Phys. Rev. Lett.135(2025) 131601 [2503.03805]

    arXiv 2025

  35. [35]

    Chang, H.-Y

    Y.-C. Chang, H.-Y. Chen and Y.-t. Huang,All zeros of (super)String Theory,2506.15299

    Pith/arXiv arXiv

  36. [36]

    B. Feng, L. Zhang and K. Zhou,Hidden zeros and 2-split via BCFW recursion relation, JHEP08(2025) 205 [2504.14215]

    Pith/arXiv arXiv 2025

  37. [37]

    B. Feng, L. Zhang and K. Zhou,2-split from Feynman diagrams and Expansions, 2508.21345

    Pith/arXiv arXiv

  38. [38]

    Zhang,2-Split of Form Factors via BCFW Recursion Relation,2509.12564

    L. Zhang,2-Split of Form Factors via BCFW Recursion Relation,2509.12564

    arXiv

  39. [39]

    Zhou,Hidden zeros for higher-derivative YM and GR amplitudes at tree-level, 2510.11070

    K. Zhou,Hidden zeros for higher-derivative YM and GR amplitudes at tree-level, 2510.11070

    Pith/arXiv arXiv

  40. [40]

    Cheung, C.-H

    C. Cheung, C.-H. Shen and C. Wen,Unifying Relations for Scattering Amplitudes,JHEP02 (2018) 095 [1705.03025]. 39

    Pith/arXiv arXiv 2018

  41. [41]

    Backus and C

    J.V. Backus and C. Figueiredo,Surface gauge invariance, soft limits and the transmutation of gluons into scalars,JHEP09(2025) 069 [2505.17179]

    arXiv 2025

  42. [42]

    Arkani-Hamed, Q

    N. Arkani-Hamed, Q. Cao, J. Dong, C. Figueiredo and S. He,Surface Kinematics and the Canonical Yang-Mills All-Loop Integrand,Phys. Rev. Lett.134(2025) 171601 [2408.11891]

    Pith/arXiv arXiv 2025

  43. [43]

    Arkani-Hamed, F

    N. Arkani-Hamed, F. Cachazo and J. Kaplan,What is the Simplest Quantum Field Theory?, JHEP09(2010) 016 [0808.1446]

    Pith/arXiv arXiv 2010

  44. [44]

    Stieberger and T.R

    S. Stieberger and T.R. Taylor,Graviton as a Pair of Collinear Gauge Bosons,Phys. Lett. B 739(2014) 457 [1409.4771]

    Pith/arXiv arXiv 2014

  45. [45]

    Stieberger and T.R

    S. Stieberger and T.R. Taylor,Graviton Amplitudes from Collinear Limits of Gauge Amplitudes,Phys. Lett. B744(2015) 160 [1502.00655]

    Pith/arXiv arXiv 2015

  46. [46]

    Cachazo, S

    F. Cachazo, S. He and E.Y. Yuan,New Double Soft Emission Theorems,Phys. Rev. D92 (2015) 065030 [1503.04816]

    Pith/arXiv arXiv 2015

  47. [47]

    Arkani-Hamed, Y

    N. Arkani-Hamed, Y. Bai, S. He and G. Yan,Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet,JHEP05(2018) 096 [1711.09102]

    Pith/arXiv arXiv 2018

  48. [48]

    Cachazo, S

    F. Cachazo, S. He and E.Y. Yuan,Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM,JHEP07(2015) 149 [1412.3479]

    Pith/arXiv arXiv 2015

  49. [49]

    Cheung, G.N

    C. Cheung, G.N. Remmen, C.-H. Shen and C. Wen,Pions as Gluons in Higher Dimensions, JHEP04(2018) 129 [1709.04932]

    Pith/arXiv arXiv 2018

  50. [50]

    Lam and Y.-P

    C.S. Lam and Y.-P. Yao,Evaluation of the Cachazo-He-Yuan gauge amplitude,Phys. Rev. D 93(2016) 105008 [1602.06419]

    Pith/arXiv arXiv 2016

  51. [51]

    J. Dong, S. He and L. Hou,Universal expansions of scattering amplitudes for gravitons, gluons, and Goldstone particles,Phys. Rev. D105(2022) 105007 [2111.10525]

    Pith/arXiv arXiv 2022

  52. [52]

    Arkani-Hamed, L

    N. Arkani-Hamed, L. Rodina and J. Trnka,Locality and Unitarity of Scattering Amplitudes from Singularities and Gauge Invariance,Phys. Rev. Lett.120(2018) 231602 [1612.02797]

    Pith/arXiv arXiv 2018

  53. [53]

    Kleiss and H

    R. Kleiss and H. Kuijf,Multi - Gluon Cross-sections and Five Jet Production at Hadron Colliders,Nucl. Phys. B312(1989) 616

    1989

  54. [54]

    Rodina,Uniqueness from gauge invariance and the Adler zero,JHEP09(2019) 084 [1612.06342]

    L. Rodina,Uniqueness from gauge invariance and the Adler zero,JHEP09(2019) 084 [1612.06342]

    Pith/arXiv arXiv 2019

  55. [55]

    Bern and T

    Z. Bern and T. Dennen,A Color Dual Form for Gauge-Theory Amplitudes,Phys. Rev. Lett. 107(2011) 081601 [1103.0312]

    Pith/arXiv arXiv 2011

  56. [56]

    Q. Cao, J. Dong, S. He and F. Zhu,Superstring amplitudes meet surfaceology,2504.21676

    arXiv

  57. [57]

    Backus,New Recursions for the Canonical Scalar-Scaffolded Yang-Mills Amplitude, 2510.19901

    J.V. Backus,New Recursions for the Canonical Scalar-Scaffolded Yang-Mills Amplitude, 2510.19901

    arXiv

  58. [58]

    S. De, A. Pokraka, M. Skowronek, M. Spradlin and A. Volovich,Surfaceology for colored Yukawa theory,JHEP09(2024) 160 [2406.04411]

    Pith/arXiv arXiv 2024

  59. [59]

    Arkani-Hamed, C

    N. Arkani-Hamed, C. Figueiredo, H. Frost and G. Salvatori,Tropical Amplitudes For Colored Lagrangians,2402.06719

    Pith/arXiv arXiv

  60. [60]

    Balli, A

    F.M. Balli, A. Edison and O. Schlotterer,Pinching rules in the chiral-splitting description of one-loop string amplitudes,JHEP05(2025) 101 [2410.19641]

    Pith/arXiv arXiv 2025

  61. [61]

    Geyer, J

    Y. Geyer, J. Guo, R. Monteiro and L. Ren,Superstring amplitudes from BCJ numerators at one loop,JHEP03(2025) 017 [2410.19663]. 40

    Pith/arXiv arXiv 2025

  62. [62]

    Q. Cao, J. Dong, S. He and F. Zhu,One-loop amplitudes in gauge theories,Phys. Rev. D 111(2025) 065015 [2412.19629]

    Pith/arXiv arXiv 2025

  63. [63]

    Q. Cao, S. He, Y. Zhang and F. Zhu,Loop-Level Double Copy Relations from Forward Limits,2509.25129

    arXiv

  64. [64]

    Monteiro and L

    R. Monteiro and L. Ren,Superstring amplitudes from the field-theory limit: an n-point map at one loop,2510.02455

    arXiv

  65. [65]

    Du, C.-H

    Y.-J. Du, C.-H. Fu, Y. Wang and C. Xie,Algebraic Consistency and Explicit Construction of One-Loop BCJ Numerators of Yang-Mills and Related Theories,2511.10963. 41

    arXiv