Positive and Negative Ladders in Loop Space

Ross Glew; Tomasz Lukowski

REVIEW 3 major objections 6 minor 2 cited by

Positive and Negative Ladders in Loop Space

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper derives a closed all-loop formula for ladder geometries in loop space for MHV amplitudes, expressing every canonical form as sums over quadruple cuts.

desk verdict First all-loop, all-n ladder formula for the MHV momentum amplituhedron is a real step forward, but the load-bearing split identity (5.5) is asserted rather than proved, so the main result stands on an unverified leg. read the letter →

arxiv 2411.14989 v1 pith:HEUAJGZO submitted 2024-11-22 hep-th

Ross Glew , Tomasz Lukowski This is my paper · ORCID

classification hep-th

keywords scatteringamplitudesN=4superYang-Millsmomentumamplituhedronnegativegeometriesladdersinloopspacecanonicalformschiralpentagonsquadruplecuts

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to establish a closed all-loop formula for the canonical forms of ladder geometries in loop space for MHV amplitudes in planar $\mathcal{N}=4$ super-Yang-Mills. Ladders are the simplest graphs obtained by relaxing or flipping the mutual positivity conditions among loop momenta, and their canonical forms are differential forms that organize the MHV integrand into sums over quadruple cuts. The main result, equation (5.14), expresses the all-negative ladder at any loop order as a double sum over quadruple-cut points, with each term factorized into products of one-loop fiber forms, chiral-box factors, and generalized pentagon factors. Positive and mixed ladders are obtained by flipping edge colors and replacing the summation range $V^{-}$ by $V^{+}$. If correct, the formula reduces a recursive geometry problem to a finite cut-sum and supplies the integrand-level input needed for computing finite Wilson-loop objects with a Lagrangian insertion at arbitrary multiplicity.

What carries the argument

The machinery is the fibration-of-fibrations recursion. Decompose the last loop's region into one-loop chambers, factor the fiber form into a chamber form times a sum of chiral boxes, then repeat the same step on the remaining ladder. The new ingredient is the splitting of a chiral box by an extra quadruple-cut constraint, equation (5.5), which is organized into five cases according to whether two chords cross, kiss, or coincide; this produces the generalized pentagon factors that make the recursion close. The graphical notation represents each loop as a vertex, positivity and negativity as green and red edges, frozen quadruple cuts as square vertices, and chambers by sign patterns of distances to quadruple cuts; one-loop chambers are counted by Eulerian numbers.

What would settle it

Evaluate both sides of equation (5.5) at a generic five-point kinematic configuration for each one-loop chamber and each pair of quadruple cuts $\ell^*_{ij},\ell^*_{kl}$; any mismatch in the differential forms would break the recursion and invalidate the final formula (5.14).

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Extended reading notes

Core claim

The central claim is that the fibration-of-fibrations structure found at two loops extends to every ladder: for any $n$, the canonical form of the $L+1$-loop all-negative ladder is $$\Omega_{\text{ladder}} = \sum_{\ell^*_{kl}\in $V^{{-}}$}\sum_{\ell^*_{ij}\in $V^{{-}}$}\Omega_{y_1,\ell^*_{kl}}\wedge\Omega_{\ell^*_{kl},y_1,y_2,y_L,\ell^*_{ij}}\wedge\Omega_{\ell^*_{ij},y_L,y_{L+1}},$$ where each factor is a one-loop fiber form, a chiral box, or a generalized pentagon produced by the recursion, and the sums run over quadruple-cut points on the one-loop fiber $\Delta(x)$. Flipping any negative edge to a positive edge changes the summation range from $V^{-}$ to $V^{+}$, so the same formula covers positive, negative, and mixed ladders. The paper checks the four-point limit against known results and verifies that the formula is symmetric under reversing the order of the loop variables.

Load-bearing premise

The entire all-loop ladder formula rests on the unproved identity (5.5): that a constrained region of the negative two-loop fiber decomposes term by term into the known one-loop chiral-box integrands, with no remainder.

Editorial extensions

If this is right

For any number of particles and any loop order, ladder contributions to the MHV integrand are expressed entirely as sums over maximal cuts, without needing explicit forms for the one-loop chambers.
The same formula handles positive, negative, and mixed ladders by a uniform rule: replace $V^{-}$ with $V^{+}$ where edges are positive.
At four points, where the one-loop fiber is a single chamber, the formula reduces to the known four-point ladder results and to products of chiral pentagons.
The alternative expansion (5.17), obtained by recursing from both ends, is stated to be the key ingredient for integrating over all but one loop and obtaining Wilson loops with a Lagrangian insertion at any multiplicity.

Reading between the lines

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

If identity (5.5) survives close scrutiny, the same splitting-by-extra-cut idea may apply to higher-valency vertices in loop-space trees, making the fibration picture viable beyond ladder topologies once explicit chamber forms are known.
A promising indirect test is to integrate the all-loop formula over $L-1$ loop variables at $n>5$: the result should develop non-trivial cross-ratio dependence, and matching it with Wilson-loop data would confirm the recursion without checking (5.5) term by term.
The five-case classification of split chiral boxes suggests that maximal-cut terms for arbitrary graphs in loop space could be indexed by the crossing, kissing, and coincidence relations among chords, giving a purely combinatorial handle on the integrand.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The headline is that this paper gives the first all-loop, all-multiplicity formula for ladder canonical forms in the MHV momentum amplituhedron. That is a genuine advance. The two-loop fibration and four/five-point negative ladders already existed, but the extension to arbitrary n and L, with the split identities (5.7)–(5.11) and generalized pentagon factors, is new. The graphical notation is helpful and the presentation is clear. The n=4 limit reproduces known results and the reflection symmetry check gives some confidence.

The soft spot is exactly where the stress-test puts it: equation (5.5) is asserted, not derived. The paper says "remarkably, we find that it can indeed be constructed" and then lists the split formulae. Everything after that—the recursion (5.12) and the main formula (5.14)—rides on that decomposition. The checks don't actually test it: at n=4 there is only one chamber so the identity trivializes, and the reflection symmetry check tests the final sum, not the term-wise validity of (5.5). Missing too is a direct comparison with the five-point two-loop negative ladder results from [14,16] that the paper cites as motivation. That comparison would be the obvious place to see the split identity in action.

Let me be clear: I don't think this is fatal. The formula has a lot of structural plausibility, and the authors are transparent about the assumption. But the central claim is not yet verified beyond the two-loop and four-point cases. A referee should ask for a proof of (5.5), or at minimum a numerical/analytical check at n=5 for L=2.

Who gets value: anyone working on positive geometries, the momentum amplituhedron, or N=4 integrands at loop level. It deserves a serious referee; the result is important enough to warrant the effort, and the missing proof is a fixable gap rather than a wrong idea.

My recommendation: send it to review, with the clear instruction that the split identity needs to be established or independently checked before publication.

Referee Report

3 major / 6 minor

Summary. The paper studies canonical forms of ladder-type geometries in loop space for MHV_n amplitudes in planar N=4 SYM, working in dual-momentum space. It extends the graphical notation of negative geometries by adding frozen square vertices for quadruple-cut points and by using the one-loop chamber decomposition of [12]. The main result, Eq. (5.14), expresses the (L+1)-loop all-negative ladder as a double sum over quadruple cuts, with each term factorized into a one-loop form, chiral-box factors, and generalized chiral-pentagon factors; positive ladders are obtained by the stated red-to-green flip of edges and corresponding changes of summation ranges. The derivation proceeds through a recursion, Eq. (5.12), which relies on a new split identity, Eq. (5.5), for the negative two-loop fiber with an added constraint. The paper checks the four-point limit against [14] and reports a reflection-symmetry consistency check of the final formula.

Significance. If established, Eq. (5.14) is a compact and surprisingly simple all-loop, all-multiplicity formula for a whole family of loop-space geometries, and it would significantly extend the fibration-of-fibrations picture beyond two loops. The paper is genuinely useful: the recursion is explicit, parameter-free, and the final formula has a clear combinatorial structure reminiscent of the chiral-pentagon expansion. The main strength is that the derivations are concrete rather than schematic. However, the central all-loop claim is not yet fully supported, because the key algebraic identity (5.5) is asserted rather than proved, and no independent check at n=5 is provided. The paper therefore reports a very plausible and interesting result whose correctness is contingent on an unverified identity.

major comments (3)

[Sec. 5, Eq. (5.5)] Equation (5.5) is the sole new algebraic input on which the recursion (5.12) and the final formula (5.14) rest. The manuscript itself states, “While it is not obvious that such an expansion exists, remarkably, we find that it can indeed be constructed,” but no derivation is supplied. The five displayed cases (5.7)–(5.11) are the content of the claimed split, not a proof that the split is exact for every chamber and every pair of chords. The checks reported in the paper do not cover this step: at four points there is only one chamber and (5.15) trivializes the decomposition, while the reflection-symmetry check tests only the symmetry of the final summed expression. Since every loop order beyond L=2 and every multiplicity beyond n=4 inherits (5.5), the authors should provide either a direct proof of (5.5) or a detailed independent verification, e.g., by symbolic comparison of both sides on representative chambers at n=5 and n=6.
[Sec. 5, Eq. (5.14)] The paper motivates the calculation by citing recent five-point negative-ladder results in [14,16], yet it never compares Eq. (5.14) with those known results. The five-point case is the first nontrivial one, with eleven one-loop chambers and three cyclic chamber classes as in Eq. (4.18); it is also the case where the chord-pair cases (5.7)–(5.11) become genuinely distinct. An explicit check of (5.14) against the five-point two-loop (or higher-loop) negative ladder would provide independent support for the split identity (5.5) and for the chamber-dependence of the final formula. Its absence is a significant gap, especially because the paper states the n=4 limit is drastically simplified.
[Sec. 5, “Arbitrary ladder” paragraph after Eq. (5.14)] The extension from the all-negative ladder to an arbitrary ladder is stated in one sentence: “The formula for an arbitrary ladder can be obtained by simultaneously flipping red edges to green edges, and the corresponding sum range from V^- to V^+.” This is plausible, but it is not demonstrated. The derivation of (5.14) uses the specific form of the negative two-loop fiber decomposition (4.30) and the split (5.5); the analogous statements for positive fibers involve a different set of quadruple cuts and a different one-loop geometric factor (4.27) versus (4.32). The authors should spell out that every step of the recursion goes through with the sign flip, or explicitly prove the positive-ladder case. As written, the abstract’s claim covering both positive and negative ladders is stronger than what is explicitly established.

minor comments (6)

[Sec. 4, Eq. (4.31)] In Eq. (4.31), the summation range is displayed as $\ell^*_{ij}\in V^+_\alpha$, but consistency with Eq. (4.30) and with the surrounding text describing the negative two-loop ladder suggests it should be $V^-_\alpha$. Please check and correct.
[Sec. 5, text after Eq. (5.3)] The text refers to the additional constraint $(y_2-\ell^*_{ij})^2<0$, whereas the diagrams in Eqs. (5.3)–(5.5) use the variables $y_{L-1}$ and $y_L$. Please make the notation consistent.
[Sec. 5, reflection-symmetry sentence] The sentence “We have checked the formula (5.14) is symmetric under exchanging $y_a\leftrightarrow y_{y_{L-a+2}}$” contains an apparent typo; it should read $y_a\leftrightarrow y_{L-a+2}$.
[Sec. 6, Conclusion] The conclusion states that “the canonical forms for all one-loop chambers, currently unknown, will be required,” but Section 3 says explicit chamber forms are not needed and does not state they are unknown. Please clarify what is known from [12] and what remains unknown.
[Sec. 5, Eqs. (5.13) and (5.18)] The definitions of the multi-loop factors in (5.13) and (5.18) use nested sums and ellipses; please specify the precise index ranges, the order of summation, and the empty-sum/empty-product conventions so that (5.14) is unambiguous.
[Title page] The author name “Tomasz /suppress Lukowski” appears to contain a LaTeX command that was not removed; please correct the metadata.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the all-loop ladder formula is a conditional derivation from prior fibration/chamber results and an explicitly stated new algebraic identity; the unproved status of (5.5) is a soundness gap, not a circular reduction.

full rationale

The derivation chain is: (i) use the one-loop chamber decomposition and the two-loop fibration/chiral-box results of [12], e.g. (4.24) and (4.30); (ii) assume the term-wise factorised chamber expansion (5.3) and the new constrained-fiber expansion (5.5); (iii) iterate the recursion (5.12) to obtain the main formula (5.14). Each step is an equality between an independently defined geometric canonical form and an explicit sum over chiral-box-type integrands. No parameter is fitted, and no target quantity is inserted into its own definition. The self-citations to [12] are load-bearing, but they are prior parameter-free derivations with stated assumptions, so under the rubric they count as independent support rather than as circularity. The genuine weakness is that (5.5) is asserted without proof: the paper says 'while it is not obvious that such an expansion exists, remarkably, we find that it can indeed be constructed', and the reported checks do not isolate this identity. However, an unproved or under-verified algebraic identity is a correctness/soundness concern, not a circularity. The four-point simplification (5.15) and the reflection-symmetry check of (5.14) are consistency checks, not inputs that make the conclusion equivalent to the premises. Therefore no significant circularity is found.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

No free parameters appear: the formula is a combinatorial and algebraic statement, not a fit. The central claim relies on prior chamber and fibration results from [10,12], mostly from the same research group, and on a new asserted split identity (5.5) that is not proved. The invented entities are notation and integral building blocks, without independent evidence.

assumptions (4)

domain assumption The MHV momentum amplituhedron and its one-loop fiber and chamber decomposition are as defined in references [8,10,12].
The paper inherits the sign-flip definition, equations (3.3)-(3.5), and the chamber count from [10,12] without re-deriving them.
domain assumption The two-loop fibration formula, equations (4.23)-(4.25), of reference [12] holds termwise.
Used as the base of the recursion and for the graphical identities in Section 4; not re-proved here.
ad hoc to paper The new split identities (5.5)-(5.11) are correct algebraic decompositions of the constrained negative two-loop fibers.
Asserted with 'we find it can indeed be constructed'; no proof or independent verification is provided, and they are load-bearing for the main formula.
standard math Canonical forms and wedge products of forms satisfy the usual positive-geometry pullback rules and factorize under split sign patterns.
Background formalism of positive geometries from reference [1]; no controversial step.

invented entities (2)

Frozen square vertices labelled by subsets of quadruple-cut points l*_ij
purpose: Bookkeeping device to encode signs of distances (y - l*_ij)^2 inside one-loop chambers
New graphical convention only; no new dynamical object and no external falsifiable prediction.
Generalized chiral pentagon factors with two quadruple-cut arguments, as in equations (5.7)-(5.11) and (5.13)
purpose: Building blocks for the recursion that express constrained two-loop fibers
Introduced in this paper as simple generalizations of chiral pentagons; their only support is the asserted identities.

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Cite this review

Pith. "Pith review of Positive and Negative Ladders in Loop Space." pith.science (2026). https://pith.science/paper/HEUAJGZO

@misc{pith2026241114989,
  author       = {Pith},
  title        = {Pith review of: Positive and Negative Ladders in Loop Space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HEUAJGZO}},
  note         = {Machine review of arXiv:2411.14989}
}

abstract

Motivated by a new term-wise factorised formula for the two-loop MHV integrand for scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills (SYM), together with recent results for the five-point negative ladders in loop space, we present the canonical forms for general ladders in loop space for an arbitrary number of particles to all loops. We make use of the graphical notation introduced in the negative geometries literature, where each loop momentum is represented as a vertex, and mutual positivity (resp. negativity) conditions as a positive (resp. negative) edge. In this paper we extend this notation to include the notion of chambers of the one-loop momentum amplituhedron. Equipped with this new graphical notation, we find the canonical form of the $L$-loop (negative/positive) ladders for all MHV$_n$ amplitudes. Our final formula is remarkably simple and reminiscent of the chiral pentagon expansion of the one and two loop momentum amplituhedron. It expresses ladder contributions as sums over maximal cuts, with each term appearing in the sum factorising into products of either chiral pentagons or their simple generalisations.

Discussion (0). Continue with ORCID to comment.

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Reference graph

Works this paper leans on

21 extracted references · 21 linked inside Pith · cited by 2 Pith papers

[12]

The Two-loop MHV Momentum Amplituhedron from Fibrations of Fibrations

L. Ferro, R. Glew, T. Lukowski and J. Stalknecht, “The Two-loop MHV Momentum Amplituhedron from Fibrations of Fibrations” , arxiv:2407.12906

arXiv
[14]

Nonperturbative negative geometries: amplitudes at strong coupling and the amplituhedron

N. Arkani-Hamed, J. Henn and J. Trnka, “Nonperturbative negative geometries: amplitudes at strong coupling and the amplituhedron” , JHEP 2203, 108 (2022) , arxiv:2112.06956

arXiv 2022
[1]

Positive Geometries and Canonical Forms

N. Arkani-Hamed, Y. Bai and T. Lam, “Positive Geometries and Canonical Forms” , JHEP 1711, 039 (2017) , arxiv:1703.04541

arXiv 2017
[2]

The Amplituhedron

N. Arkani-Hamed and J. Trnka, “The Amplituhedron”, JHEP 1410, 030 (2014) , arxiv:1312.2007

arXiv 2014
[3]

All-Loop Four-Point Aharony-Bergman-Jaﬀeris-Maldacena Amplitudes from Dime nsional Reduction of the Amplituhedron

S. He, C.-K. Kuo, Z. Li and Y.-Q. Zhang, “All-Loop Four-Point Aharony-Bergman-Jaﬀeris-Maldacena Amplitudes from Dime nsional Reduction of the Amplituhedron”, Phys. Rev. Lett. 129, 221604 (2022) , arxiv:2204.08297

arXiv 2022
[4]

The ABJM Amplituhedron

S. He, Y.-t. Huang and C.-K. Kuo, “The ABJM Amplituhedron”, JHEP 2309, 165 (2023) , arxiv:2306.00951, [Erratum: JHEP 04, 064 (2024)]

arXiv 2023
[5]

Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet

N. Arkani-Hamed, Y. Bai, S. He and G. Yan, “Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet” , JHEP 1805, 096 (2018) , arxiv:1711.09102. – 19 –

arXiv 2018
[6]

The Correlahedron

B. Eden, P. Heslop and L. Mason, “The Correlahedron”, JHEP 1709, 156 (2017) , arxiv:1701.00453

arXiv 2017

Show all 21 references

[7]

All-loop geometry for four-point correlation functions

S. He, Y.-t. Huang and C.-K. Kuo, “All-loop geometry for four-point correlation functions” , Phys. Rev. D 110, L081701 (2024) , arxiv:2405.20292

2024 arXiv
[8]

The Momentum Amplituhedron

D. Damgaard, L. Ferro, T. Lukowski and M. Parisi, “The Momentum Amplituhedron”, JHEP 1908, 042 (2019) , arxiv:1905.04216

2019 arXiv
[9]

The Loop Momentum Amplituhedron

L. Ferro and T. Lukowski, “The Loop Momentum Amplituhedron” , JHEP 2305, 183 (2023) , arxiv:2210.01127

2023 arXiv
[10]

Prescriptive unitarity from positive geometries

L. Ferro, R. Glew, T. Lukowski and J. Stalknecht, “Prescriptive unitarity from positive geometries”, JHEP 2403, 001 (2024) , arxiv:2308.02438

2024 arXiv
[11]

Momentum Amplituhedron for N=6 Chern-Simons-Matter Theory: Scattering Amplitudes from Conﬁgurations of Point s in Minkowski Space

T. Lukowski and J. Stalknecht, “Momentum Amplituhedron for N=6 Chern-Simons-Matter Theory: Scattering Amplitudes from Conﬁgurations of Point s in Minkowski Space” , Phys. Rev. Lett. 131, 161601 (2023) , arxiv:2306.07312

2023 arXiv
[13]

The m=2 amplituhedron and the hypersimplex: signs, clusters, tilings, Eulerian numbers

M. Parisi, M. Sherman-Bennett and L. K. Williams, “The m=2 amplituhedron and the hypersimplex: signs, clusters, tilings, Eulerian numbers .”, Commun. Am. Math. Soc. 3, 329 (2023) , arxiv:2104.08254

2023 arXiv
[15]

Loops of loops expansion in the amplituhedron

T. V. Brown, U. Oktem, S. Paranjape and J. Trnka, “Loops of loops expansion in the amplituhedron”, JHEP 2407, 025 (2024) , arxiv:2312.17736

2024 arXiv
[16]

Positivity properties of ﬁve-point two-loop Wilson loops with Lagrangian insertion

D. Chicherin, J. Henn, J. Trnka and S.-Q. Zhang, “Positivity properties of ﬁve-point two-loop Wilson loops with Lagrangian insertion” , arxiv:2410.11456

arXiv
[17]

The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM

N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, “The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM” , JHEP 1101, 041 (2011) , arxiv:1008.2958

2011 arXiv
[18]

Local Integrals for Planar Scattering Amplitudes

N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo and J. Trnka, “Local Integrals for Planar Scattering Amplitudes”, JHEP 1206, 125 (2012) , arxiv:1012.6032

2012 arXiv
[19]

Integrated negative geometries in ABJM

J. M. Henn, M. Lagares and S.-Q. Zhang, “Integrated negative geometries in ABJM” , JHEP 2305, 112 (2023) , arxiv:2303.02996

2023 arXiv
[20]

Higher-loop integrated negative geometries in ABJM

M. Lagares and S.-Q. Zhang, “Higher-loop integrated negative geometries in ABJM” , JHEP 2405, 142 (2024) , arxiv:2402.17432

2024 arXiv
[21]

Integrating the full four-loop negative geometries and al l-loop ladder-type negative geometries in ABJM theory

Z. Li, “Integrating the full four-loop negative geometries and al l-loop ladder-type negative geometries in ABJM theory” , JHEP 2410, 124 (2024) , arxiv:2402.17023. – 20 –

2024 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [12]

The Two-loop MHV Momentum Amplituhedron from Fibrations of Fibrations

L. Ferro, R. Glew, T. Lukowski and J. Stalknecht, “The Two-loop MHV Momentum Amplituhedron from Fibrations of Fibrations” , arxiv:2407.12906

arXiv

[2] [14]

Nonperturbative negative geometries: amplitudes at strong coupling and the amplituhedron

N. Arkani-Hamed, J. Henn and J. Trnka, “Nonperturbative negative geometries: amplitudes at strong coupling and the amplituhedron” , JHEP 2203, 108 (2022) , arxiv:2112.06956

arXiv 2022

[3] [1]

Positive Geometries and Canonical Forms

N. Arkani-Hamed, Y. Bai and T. Lam, “Positive Geometries and Canonical Forms” , JHEP 1711, 039 (2017) , arxiv:1703.04541

arXiv 2017

[4] [2]

The Amplituhedron

N. Arkani-Hamed and J. Trnka, “The Amplituhedron”, JHEP 1410, 030 (2014) , arxiv:1312.2007

arXiv 2014

[5] [3]

All-Loop Four-Point Aharony-Bergman-Jaﬀeris-Maldacena Amplitudes from Dime nsional Reduction of the Amplituhedron

S. He, C.-K. Kuo, Z. Li and Y.-Q. Zhang, “All-Loop Four-Point Aharony-Bergman-Jaﬀeris-Maldacena Amplitudes from Dime nsional Reduction of the Amplituhedron”, Phys. Rev. Lett. 129, 221604 (2022) , arxiv:2204.08297

arXiv 2022

[6] [4]

The ABJM Amplituhedron

S. He, Y.-t. Huang and C.-K. Kuo, “The ABJM Amplituhedron”, JHEP 2309, 165 (2023) , arxiv:2306.00951, [Erratum: JHEP 04, 064 (2024)]

arXiv 2023

[7] [5]

Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet

N. Arkani-Hamed, Y. Bai, S. He and G. Yan, “Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet” , JHEP 1805, 096 (2018) , arxiv:1711.09102. – 19 –

arXiv 2018

[8] [6]

The Correlahedron

B. Eden, P. Heslop and L. Mason, “The Correlahedron”, JHEP 1709, 156 (2017) , arxiv:1701.00453

arXiv 2017