Light-Front approach to $4d$ massless Higher-Spin interactions

Mattia Serrani

REVIEW 4 major objections 3 minor 300 references

Light-Front approach to $4d$ massless Higher-Spin interactions

T0 review · 4 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The thesis claims that the space of 4d massless higher-spin theories is far richer than previously assumed: infinitely many consistent local theories exist, and the full quartic constraint leaves Yang-Mills, gravity, and new quasi-chiral fa

desk verdict A careful, genuinely original solution of the light-front quartic holomorphic constraint with many new chiral higher-spin theories, but the 'all' claims need a completeness proof and the Chapter 4 determination is announced as a conjecture. read the letter →

arxiv 2607.28183 v2 pith:QIZ7H4RJ submitted 2026-07-30 hep-th

Mattia Serrani This is my paper · ORCID

classification hep-th

keywords higher-spingravitylight-frontquantizationquarticconstraintchiraltheoryself-dualYang-MillscelestialOPEassociativityspinor-helicityamplitudes

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 thesis works in the light-front formulation of 4d massless fields and asks which local, unitary higher-spin theories can actually exist. Its central claim is that the answer is much larger than previously thought: requiring the Poincaré algebra to close at quartic order leaves infinitely many consistent local higher-spin theories, some with finitely many fields and some with infinite towers. The thesis solves the holomorphic part of the quartic constraint completely, classifies all one- and two-derivative chiral theories, and shows they are subsectors of higher-spin extensions of self-dual Yang-Mills and self-dual gravity. For the full non-holomorphic constraint it recovers Yang-Mills and gravity as essentially the only lower-spin survivors, and finds new families of local quasi-chiral higher-spin theories. A sympathetic reader should care because this maps the space of possible higher-spin extensions of gravity, a candidate route to quantum gravity, much more precisely than before.

What carries the argument

The load-bearing object is the light-front Hamiltonian density h_n(P_ij, Pbar_ij, beta_k), subject to the locality rule that inverse transverse momenta 1/q and 1/qbar are forbidden while inverse beta is allowed. The quartic holomorphic constraint, obtained from the dynamical commutator [J^a-, P^-]=0, is a polynomial identity in three variables whose solution forces C_{lambda1 lambda2 omega} times C_{-omega lambda3 lambda4} to follow a factorial formula. The combinatorial engine is the 'small crystal': the six-product system (2.4.1)-(2.4.2) that lists which cubic couplings must appear together for a consistent two-derivative theory. Carrying the argument are the general even- and odd-derivati

What would settle it

Reduce the standard minimally-coupled gravitational interaction of a massless spin-3 field to light-front gauge and check whether a consistent quartic completion requires an inverse transverse momentum; if yes, the classification covers only light-front-local theories, not all local ones. Alternatively, construct a low-derivative non-chiral four-point vertex with a coloured graviton that satisfies the full quartic constraint; that would falsify the no-go for multi-graviton theories.

Watch

Extended reading notes

Core claim

The thesis establishes that the quartic light-front consistency constraint—the condition that the interacting Poincaré algebra closes at four points—does not force a unique higher-spin theory. The general solution of the holomorphic constraint has a factorial form, with products of pairs of cubic couplings fixed by a single overall constant per derivative order, and the thesis shows that the allowed sets of couplings are organized into 'small crystals': a single pair of two-derivative couplings forces six couplings to coexist, with equal products. Iterating this rule produces a complete catalogue of one- and two-derivative theories—nine families of self-dual-gravity-like theories, three infi

Load-bearing premise

The classification's 'all' is asserted inside the light-front locality class: interaction densities must be functions of P_ij, Pbar_ij and beta with no inverse transverse momenta 1/q or 1/qbar; if a manifestly local covariant theory maps to a vertex outside this class, the completeness claim does not cover it.

Editorial extensions

If this is right

If correct, every one- and two-derivative chiral higher-spin theory is a subsector of the higher-spin extension of self-dual Yang-Mills or self-dual gravity, making the known chiral theory a hub rather than an isolated example.
The crystal rules imply that turning on a single two-derivative coupling can force a whole finite or infinite spectrum; couplings cannot be switched on independently.
The non-holomorphic analysis rules out interacting multi-graviton theories at low derivatives, confirming a long-standing no-go within the light-front class, while leaving Yang-Mills and gravity as the lower-spin survivors.
All local four-point higher-spin amplitudes satisfying factorisation are determined, giving a complete on-shell output for the classified theories.
New unitary quasi-chiral families provide explicit examples of consistent higher-spin completions with both chiralities interacting, beyond the purely chiral theory.

Reading between the lines

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

Because the 'all' in the classification means all theories satisfying the light-front locality criterion, the completeness statement does not automatically cover every manifestly covariant local theory; the thesis itself notes that some light-front vertices become covariant only at the price of non-locality. A natural test is to reduce a known covariant local higher-spin interaction to light-front
The appearance of fractional helicities (2/3, 4/3, ...) among the solutions suggests the algebraic structure tolerates objects that are not standard integer-spin fields; if these are taken seriously rather than discarded, they would be a new class of 'fractional spin' representations in flat space.
The unconstrained product C_{lambda,lambda,0} C_{0,lambda',lambda'} indicates that some couplings remain free parameters even inside a classified theory; this means the classification fixes spectra and relations but not every coupling, so amplitudes may carry continuous free parameters that the factorisation analysis should expose.
The small-crystal iteration could be turned into an algorithmic classification for three- and higher-derivative vertices: if the pattern persists, the full space of local higher-spin theories would be recursively generated from finite seeds, making the quasi-chiral families the first step of an infinite ladder.

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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

The core of Chapter 2 is the real thing: the quartic holomorphic constraint is solved explicitly, and the solution is anchored by recovering SDYM, SDGR, Yang–Mills, gravity, and the multi-graviton inconsistency. The Metsaev solution is re-derived, not assumed, and the new finite- and infinite-spectrum families (seeds (2.4.13)–(2.4.51)) are new as far as I can tell. The small-crystal rules are a genuinely useful organizing principle, and the treatment of U(N), SO(N), and USp(N) adds substance.

The main soft spot is the completeness claim. The classification of all one- and two-derivative theories rests on the iterative procedure of §2.4.1: “we iterate this process up to three times and then stop.” That is a finite enumeration, not a proof that every consistent crystal is generated. Likewise, the general solution to the holomorphic constraint uses the ansatz (2.3.9), which is verified case-by-case rather than derived. These are addressable gaps, not fatal ones, but they mean the “all” and “classify all” statements in the abstract outrun what is actually shown.

The outline itself says the Chapter 4 quartic vertex set is “conjectured,” while the abstract says “determine all” amplitudes. That mismatch should be fixed in any revision. It is a minor issue but a real one, since it affects the precision of the central claim.

I could not check Chapters 3 and 4 in the available text; those parts are only visible through the outline. So I am basing my confidence on Chapter 2, which is self-contained and carefully executed. The recovered lower-spin theories and the explicit derivations give me reasonable confidence that the new solutions are correct, and the light-front locality criterion is stated up front, so the scope of the claims is at least clear.

This deserves a serious referee. The results are potentially a major organizing step for the higher-spin program, and even if the completeness proof is missing, the paper contains enough substantial, checkable mathematics to warrant careful review. I would cite it for the new families and the crystal rules. For a reading group focused on higher-spin or light-front methods, it is worth the effort; for a general hep-th audience, maybe.

Referee Report

4 major / 3 minor

Summary. The thesis develops the light-front (light-cone) approach to four-dimensional massless higher-spin interactions, focusing on closure of the Poincaré algebra at quartic order. Chapter 2 solves the holomorphic quartic constraint and claims a complete classification of one- and two-derivative chiral higher-spin theories, recovering HS-SDYM, HS-SDGR, and chiral higher-spin gravity as special cases, and finding new finite and infinite families. Chapter 3 connects the light-cone holomorphic constraint to celestial OPE associativity, Jacobi identities of the kinematical algebra, and vanishing of tree-level amplitudes. Chapter 4 analyzes the full non-holomorphic quartic constraint, recovers Yang-Mills and gravity, reproduces the multi-graviton inconsistency, and proposes new unitary and quasi-chiral higher-spin theories, with a conjectured catalogue of local quartic vertices and four-point amplitudes.

Significance. If the central claims hold, this is a substantial contribution to the classification of 4d higher-spin interactions. The manuscript is anchored by concrete external benchmarks: the constraint equations are solved rather than fitted, the Metsaev Γ-factorial solution is re-derived, and known lower-spin theories (SDYM, SDGR, Yang-Mills, gravity) are reproduced. The explicit identification of infinite families of local higher-spin theories with finite or infinite spectra, and the recovery of the multi-graviton inconsistency, are valuable results that would give new structure to the higher-spin landscape. However, the advertised completeness claims currently outrun the proofs supplied in the text.

major comments (4)

[§2.4.1, Eqs. (2.4.1)–(2.4.2)] The classification of two-derivative chiral theories is presented as complete, but the enumeration procedure states: 'we iterate this process up to three times and then stop.' No proof is given that every solution of the factorized holomorphic constraint (2.3.51) is generated by a small crystal, nor that three iterations suffice. The later assertion that all one- and two-derivative theories have all possible cubic couplings built from their spectrum is called an 'experimental fact' rather than proved. Since the Chapter 2 abstract and the thesis abstract claim a complete classification, this is a load-bearing gap: a consistent crystal requiring a fourth iteration, or a seed not reachable by the stated equivalence moves, would invalidate the 'all' claim.
[Ch. 4 outline bullet; §4.6–§4.7] The thesis abstract states that the paper 'determine[s] all local higher-spin four-point amplitudes,' but the Chapter 4 outline explicitly says: 'we conjecture the complete set of quartic vertices satisfying the light-cone consistency conditions.' A conjectured set cannot ground a determination of all amplitudes unless a separate argument shows that the amplitude catalogue is insensitive to the unproven part of the vertex set, or that the conjecture is proven. The manuscript should either provide such an argument or clearly separate the proven results from the conjectural completeness statement.
[§1.2, Eqs. (1.2.96)–(1.2.100)] The classification's universe of discourse is the space of light-front densities depending on P_ij, \bar P_ij, β_k with no inverse transverse momenta 1/q or 1/\bar q. This is a light-front notion of locality; as the manuscript itself notes via [139], light-front and covariant locality can differ. The abstract's unrestricted phrase 'all local higher-spin theories' and 'all local ... amplitudes' is therefore not supported unless the claims are explicitly scoped to this light-front locality criterion, or an argument is given that every perturbatively local covariant theory satisfies the criterion after light-cone gauge fixing.
[§2.4.1, 'only report the physical solutions'] In the same section that claims a complete classification, the text says: 'In general, the solutions may also include non-integer helicities... In the classification below, we only report the physical solutions.' This means the enumeration is implicitly restricted to integer helicities, despite the abstract's unqualified 'all.' The manuscript should state this restriction explicitly and justify it on physical grounds (e.g. unitarity, bosonic statistics) rather than leaving it as an unexplained omission from the announced classification.

minor comments (3)

[§3.6] The claimed equivalence between celestial OPE associativity, the Jacobi identity, vanishing of tree-level amplitudes, and the light-cone holomorphic constraint should be stated with explicit directionality and domain (tree-level, all helicities, with/without gauge group). Some intermediate equations in this section are not numbered, making the logical chain harder to verify.
[§2.3, Eqs. (2.3.44)–(2.3.45)] The derivation of the Metsaev solution assumes all even couplings are non-vanishing, and similarly for the odd case. This assumption should be stated before the uniqueness step, since the subsequent conclusion that no mixed even/odd solution exists depends on it.
[General] There are numerous typos and formatting issues, e.g. 'responsable', 'complexifactors', 'garzie', and an unnumbered displayed equation in §2.3.2. A careful proofread is needed before publication.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: core constraints are solved and cross-checked against known lower-spin theories; the main caveats are completeness gaps, not circular reductions.

full rationale

The central derivation chain is self-contained. The holomorphic constraint (2.3.5) is solved case-by-case, yielding the general and even-derivative solutions (2.3.50)-(2.3.51). The factorial form (2.3.9) is stated as the pattern that emerges ('As we will see, we always end up with a system of the general form...'), but the case analysis determines the polynomial functions f from the constraint itself rather than fitting parameters, and the Metsaev solution (2.3.44) is re-derived from factorization of the products C^1234_omega, not imported as an unverified input. The small-crystal rules (2.4.1)-(2.4.2) are presented as consequences of the solved two-derivative specialization of the constraint, and the classification enumerates crystals satisfying those rules; no fitted quantity is later renamed as a prediction. Known lower-spin results (SDYM, SDGR, Yang-Mills, GR) are recovered as external anchors, which is evidence that the machinery is not merely self-referential. The real weaknesses are completeness gaps, not circularity: Section 2.4.1 says 'we iterate this process up to three times and then stop' without proving that every consistent crystal is generated within three iterations, and the outline explicitly states that the Chapter 4 quartic-vertex set is 'conjecture[d]' before the abstract-style claim to 'determine all local higher-spin four-point amplitudes'. These are unsupported or overbroad claims, but they do not reduce the conclusions to the assumptions by construction. Self-citations (the chapters are stated to be identical to the author's papers [7]-[9]; the solution procedure references Appendix A of [6]) are not load-bearing: [6] is an external anchor, and the key constraints are solved in-text against known benchmarks. Score 2 reflects minor self-citation and acknowledged completeness caveats without any circular derivation step.

Assumptions & free parameters 3 free parameters · 8 assumptions · 2 invented entities

The central claims rest almost entirely on algebra: the quartic constraints are solved, not fitted, and known lower-spin theories are recovered as external anchors. The free parameters are the coupling constants themselves (constrained up to overall scales and the decoupled C_{λ,λ,0}C_{0,μ,μ} products), plus the free integer/helicity labels of the infinite families. The main structural inputs are the light-front locality criterion, the invertibility of ∂+, the quartic chiral-decoupling 'miracle' inherited from Metsaev, and the verified-but-not-derived ansatz (2.3.9). The only new entities are the quasi-chiral family solutions (announced, not fully verifiable here) and fractional-helicity fields (found, declared non-physical, excluded).

free parameters (3)

cubic coupling constants C_{λ1,λ2,λ3} = constrained by (2.3.50)-(2.3.51), (2.4.2); overall scales k± and products C_{λ,λ,0}C_{0,μ,μ} free
The couplings are the unknowns solved from the quartic constraints; relations among them are derived, but each theory retains free overall scales and the decoupled scalar-pair product.
fractional-helicity parameter in infinite crystal families = e.g. λ ≡ 2 (mod 4), λ odd, or λ ∈ Z_{≥0} in families (2.4.27)-(2.4.51)
One-parameter families of infinite theories are labelled by a free integer/helicity parameter; different values give different spectra, declared equivalent under affine reparametrization (2.4.5).
length scale ℓ_p in Metsaev couplings = C ~ (ℓ_p)^{λ123−1}/Γ(λ123)
The overall length scale of chiral higher-spin gravity vertices is a free dimensionful input (Planck-like scale), standard for the program.

assumptions (8)

domain assumption ∂+ is invertible (no zero modes p+ = 0)
The entire light-cone gauge-fixing and the 1/β kernels rely on this; paper explicitly assumes it ('we will assume the operator ∂+ to be always invertible', Ch. 1).
domain assumption Light-front locality = no inverse powers of q or q̄; inverse β allowed
Defines the space of interactions classified. The paper allows 1/β but forbids 1/q, 1/q̄; this is the notion of perturbative locality used for all 'all theories' claims (Ch. 1, 'constrained by locality').
domain assumption Holomorphic/quartic chiral-sector decoupling ('miracle')
The claim that at quartic order the chiral sector receives no contribution from H4/J4, so (2.2.10) closes on cubic data. Attributed to Metsaev [4] and used to build Chapter 2; not proven in the thesis.
ad hoc to paper Coupling symmetry C_{λ1,λ2,λ3} = (−)^{λ123} C_{λσ1,λσ2,λσ3} and cyclic symmetry (U(N) case)
Stated as harmless and without loss of generality (§2.2) to reduce sums; the classification results inherit this convention, and odd-derivative vertices with two identical helicities vanish by it.
ad hoc to paper Solution form (2.3.9): C1234ω ∝ (Λ−2)!/(2^{Λ−2}(λ12+ω−1)!(λ34−ω−1)!)
The paper states the system 'always' ends up in this form and verifies it case-by-case rather than deriving it from first principles; completeness of the subsequent classification is relative to this ansatz class.
ad hoc to paper U(N) phase θ_ω = (−)^ω
A sign/phase choice for the gauge-group commutator, equivalent to assigning Hermiticity by helicity parity; the classification of one-derivative theories depends on it.
domain assumption Kinematical constraint solution: densities h_n(P_ij, P̄_ij, β_k) with homogeneity (1.2.96)-(1.2.100)
Taken from Ponomarev-Skvortsov [6] and used without re-derivation; it fixes the allowed functional dependence of every vertex density.
standard math Combinatorial identities (2.3.17), (2.3.61): sums over ω of (Λ−2)!/(2^{Λ−3}(λ12+ω−1)!(λ34−ω−1)!) = 1
Used to close the constraint systems and to argue infinite-ladder propagation of the solution; standard binomial identities.

invented entities (2)

Fractional-helicity fields (λ = 2/3, 4/3, ...)
purpose: They are solutions of the same holomorphic constraint equations; including them would enlarge the classification beyond integer spins
Paper finds them, declares them non-physical, and omits them from the classification; no observable handle is proposed (paper speculates about 'fractional spin' in 4d).
Quasi-chiral higher-spin theories (Chapter 4 families)
purpose: New families of local theories claimed to solve the full quartic constraint, neither purely chiral nor parity-complete
Announced in abstract and Chapter 4; consistency is claimed via constraint solutions and factorization, but the full derivation was not available in the reviewed text; no observational handle.

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Cite this review

Pith. "Pith review of Light-Front approach to $4d$ massless Higher-Spin interactions." pith.science (2026). https://pith.science/paper/QIZ7H4RJ

@misc{pith2026260728183,
  author       = {Pith},
  title        = {Pith review of: Light-Front approach to $4d$ massless Higher-Spin interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QIZ7H4RJ}},
  note         = {Machine review of arXiv:2607.28183}
}

abstract

This thesis studies $4d$ massless higher-spin interactions in the Light-Front approach by analysing the closure of the Poincar\'e algebra at quartic order. We first solve the light-cone quartic holomorphic constraint in flat space and show the existence of infinitely many interacting local higher-spin theories with either a finite or infinite number of fields. We classify all one- and two-derivative theories, corresponding to higher-spin extensions of gauge and gravitational interactions. These are consistent subsectors of higher-spin extensions of self-dual Yang--Mills and gravity, themselves truncations of Chiral Higher-Spin Gravity. We then clarify the relation between the OPE associativity in celestial CFT, the vanishing of tree-level amplitudes for generic kinematics, the Jacobi identity of the associated ''gauge algebra'' (kinematical algebra), and the light-cone holomorphic constraints. Finally, we investigate the non-holomorphic quartic constraint involving both MHV and anti-MHV vertices. We recover the existence of Yang-Mills theory and gravity, and the inconsistency of interacting multi-graviton theories. We then show that once higher-derivative cubic vertices are included, nontrivial solutions to the full quartic constraint exist. We classify all unitary local higher-spin theories, identify new families of local quasi-chiral theories, and determine all local higher-spin four-point amplitudes using the spinor-helicity formalism together with locality in the form of consistent factorisation.

Figures

Figures reproduced from arXiv: 2607.28183 by the authors.

**Figure 4.1.** Generic CC¯ exchange with contact term C λ1,λ2,λ3,λ4 . When (4.1.1) is satisfied only for k = 1, 2, 3,6 additional exchange diagrams are required, as discussed in the main text. Accordingly, local quartic vertices fall into four classes: selfconsistent quartic vertices (k = 0);7 vertices requiring a single-channel exchange (k = 1); vertices requiring exchange in two channels (k = 2), as in Yang–Mills theory; and ve… view at source ↗

**Figure 4.2.** Holomorphic constraint [PITH_FULL_IMAGE:figures/full_fig_p171_4_2.png] view at source ↗

**Figure 4.3.** Non-holomorphic constraint. Starting from Eq. (4.4.8) and using the explicit form of the densities (4.3.13), we obtain the following expression: [H3, Jz− 3 ] =X λi,ω Z d 12q δ X i qi ! 9 2 h (−) ω β1(λ1 + ω − λ2) − β2(λ2 + ω − λ1) (β1 + β2) 2ω+1 β λ3 3 β λ4 4 β λ1 1 β λ2 2 × C 1234ωP¯λ12+ω−1 12 P −λ34+ω 34 i ϕ λ1 q1 ϕ λ2 q2 ϕ λ3 q3 ϕ λ4 q4 , (4.4.10) where we denote C 1234ω ≡ C λ1,λ2,ωC¯−ω,λ3,λ4 the product of the c… view at source ↗

Figures from the paper (19 more)

**Figure 4.4.** Figure 4.4: Exchange diagrams for a generic pair CC¯. Reformulation. A reformulation of the solutions is as follows. Starting from an (n, m) exchange diagram (λ1, λ2, ω, λ3, λ4) as in [PITH_FULL_IMAGE:figures/full_fig_p201_4_4.png]

**Figure 4.5.** Figure 4.5: A quartic vertex with a single exchange exists when the conditions (4.6.8) are [PITH_FULL_IMAGE:figures/full_fig_p202_4_5.png]

**Figure 4.6.** Figure 4.6: Exchange diagrams between C 2,5,1 and C¯−2,−5,−1 . This exchange diagram respects the conditions (4.6.8), with some of the fields having maximal helicity (4.6.9). In particular, we have λ1 ≤ λ2 + ω , λ2 = λ1 + ω + 2 , λ3 = λ4 + ω + 2 , λ4 ≤ λ3 + ω , (4.6.10a) λ12 + ω…

**Figure 4.7.** Figure 4.7: Relevant exchange diagrams for parity invariant pairs of cubic abelian vertices. [PITH_FULL_IMAGE:figures/full_fig_p203_4_7.png]

**Figure 4.8.** Figure 4.8: CC¯ exchange diagrams for parity invariant pairs of cubic abelian vertices. Following (4.6.8), we find the conditions λ1 ≤ λ2 + λ3 , λ2 ≤ λ1 + λ3 , λ3 ≤ λ1 + λ2 . (4.6.13) These are the usual triangular inequalities. Therefore, each triangle with integer unit length …

**Figure 4.9.** Figure 4.9: Linearised curvature terms (Rs ) 3 for higher-spins are consistent to the quartic level. The same triangular inequalities were already found in [285, 286]. The underlying idea is to construct linearised higher-spin curvatures Rs µ1ν1,...,µsνs (de Wit–Freedman curvatu…

**Figure 4.10.** Figure 4.10: Exchange diagram for a pair of cubic abelian vertices with one opposite helicity. [PITH_FULL_IMAGE:figures/full_fig_p205_4_10.png]

**Figure 4.11.** Figure 4.11: Example of abelian cubic vertices that satisfy the quartic constraint. [PITH_FULL_IMAGE:figures/full_fig_p205_4_11.png]

**Figure 4.12.** Figure 4.12: Example of non-abelian cubic vertices that satisfy the quartic constraint. [PITH_FULL_IMAGE:figures/full_fig_p206_4_12.png]

**Figure 4.13.** Figure 4.13: Generic CC¯ unitary exchange. Considering that in this case, each of the helicities can end up on the external legs, we obtain the following conditions: λ1 ≤ λ2 + λ3 , λ2 ≤ λ1 + λ3 , λ3 ≤ λ1 + λ2 . λ1 + λ2 + λ3 > 0 . (4.6.15) These conditions correspond to the trian…

**Figure 4.14.** Figure 4.14: The left diagram, by quartic consistency, implies the presence of the right one. [PITH_FULL_IMAGE:figures/full_fig_p208_4_14.png]

**Figure 4.15.** Figure 4.15: The left diagram, by quartic consistency, implies the presence of the right one. [PITH_FULL_IMAGE:figures/full_fig_p209_4_15.png]

**Figure 4.16.** Figure 4.16: The left diagram, by quartic consistency, implies the presence of the right one. [PITH_FULL_IMAGE:figures/full_fig_p210_4_16.png]

**Figure 4.17.** Figure 4.17: New exchange diagram between C λ1,λ2,n1 and C¯λ2,−λ2,−n1 . In the case that the external fields are maximal (4.6.9), we find λ1 = n1 + n2 2 , λ2 = n1 + n3 2 . (4.6.34) In any case, no higher-spin fields are allowed. Let us note that we have discarded possible soluti…

**Figure 4.18.** Figure 4.18: Exchange diagram involving a generic vertex and the anti-MHV cubic vertex of [PITH_FULL_IMAGE:figures/full_fig_p212_4_18.png]

**Figure 4.19.** Figure 4.19: The left diagram, by quartic consistency, implies the presence of the right one. [PITH_FULL_IMAGE:figures/full_fig_p213_4_19.png]

**Figure 4.20.** Figure 4.20: Holomorphic constraints between C −2,−2,2 and C λ,−λ,−2 . The holomorphic constraints fix the coefficients of the two anti-holomorphic cubic vertices C −2,−2,2 and C λ,−λ,−2 to be equal, implying C −2,−2,2 = C λ,−λ,−2 . As discussed in [7], these two anti-holomorphi…

**Figure 4.21.** Figure 4.21: (2, 2) CC¯ exchange diagram. This exchange diagram does not admit a quartic vertex that solves the quartic constraint. Interestingly, the same phenomenon also appears in the case of Yang-Mills-like interactions. In this case, we obtain the quasi-chiral HS-YM theory …

**Figure 4.22.** Figure 4.22: Generic four-point exchange diagram. Given a four-point scattering, as shown in [PITH_FULL_IMAGE:figures/full_fig_p239_4_22.png]

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Reviewed August 3, 2026 · model on record in the stance chip above.

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