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Seven scattering matrices capture every tree-level unitarity constraint on models with extra doublets, neutral singlets, and charged singlets.

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

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2026-08-04 12:43 UTC pith:XF3LF5WZ

load-bearing objection Useful, mostly correct systematization of unitarity bounds for singlet-doublet scalar sectors; the main claim holds, with a concrete conjugation typo that needs fixing.

arxiv 2510.02434 v3 pith:XF3LF5WZ submitted 2025-10-02 hep-ph

Perturbative unitarity for models with singlet and doublet scalars

Carolina T. Lopes , Andr\'e Milagre , Jo\~ao P. Silva This is my paper
classification hep-ph
keywords perturbative unitarityscalar extensionsquartic couplingspartial-wave analysismulti-Higgs-doublet modelsBounDSscattering matricesdark matter models
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 aims to give a complete, minimal description of perturbative unitarity bounds on the quartic scalar couplings of any model with arbitrary numbers of SU(2) doublets, neutral singlets, and charged singlets—precisely the scalar content common in dark-matter extensions. It claims that after classifying two-particle states by electric charge, hypercharge, and total isospin, and discarding redundant channels, all independent tree-level 2-to-2 unitarity constraints reduce to the eigenvalues of just seven scattering matrices, with the modulus of each eigenvalue bounded by 8π. If correct, this gives model-builders a universal checklist: build those seven matrices, diagonalize, and the quartic sector is fully constrained regardless of vacuum structure or mass mixings. The paper also supplies a Mathematica notebook, BounDS, that constructs the potential and matrices for a user-specified scalar content and symmetries, and it verifies the method against a series of known models.

Core claim

For a renormalizable SU(2)×U(1) theory with n_D doublets, n_c charged singlets, and n_n neutral singlets, the authors derive the most general quartic scalar potential, with hermiticity and index-symmetry relations, and show that in the high-energy limit where propagators vanish, the tree-level 2→2 amplitude is the negative fourth derivative of that potential. Grouping two-particle states by the conserved quantum numbers |Q, Y, T⟩ and removing states whose amplitudes are identical by symmetry leaves exactly the seven independent scattering matrices of Table I. The eigenvalues of these matrices give the complete set of zeroth-partial-wave unitarity bounds, |Λ| ≤ 8π, for any flavour or generali

What carries the argument

The central object is the set of seven coupled-channel scattering matrices constructed in the high-energy limit from second derivatives of the quartic scalar potential; the amplitude for A B → C D is the negative fourth derivative of V4. The classification of two-particle states by total electric charge Q, hypercharge Y, and total isospin T—together with the removal of redundant channels via Clebsch-Gordan symmetries—is what reduces the problem to a minimal basis. The unitarity condition is the eigenvalue bound |Λ| ≤ 8π on each matrix. BounDS is the companion automation that assembles these matrices from the field content and imposed symmetries.

Load-bearing premise

The derivation rests on the high-energy limit in which all propagator contributions vanish, so only quartic scalar self-interactions grow with energy; if gauge-boson or fermion channels, or non-quartic terms, also saturate unitarity at the relevant scale, the seven-matrix bound is not the complete constraint.

What would settle it

Take a specific model, e.g., n_D=2, n_n=1, n_c=1, and compute the full tree-level zeroth-partial-wave eigenvalues of all 2-to-2 channels, including those with longitudinal gauge bosons via the equivalence theorem; if any eigenvalue exceeds 8π while all seven Table I eigenvalues remain below 8π, the completeness claim is falsified. Alternatively, numerically scan random quartic couplings and check whether the largest eigenvalue of the full state space ever exceeds the largest eigenvalue of the seven-matrix set, which would disprove the claimed minimality.

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

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If this is right

  • For any model in this class, checking the seven matrices is sufficient to impose all tree-level partial-wave unitarity constraints on quartic couplings, independent of the vacuum or the quadratic/cubic terms.
  • The bounds are directly applicable to dark-matter-motivated scalar extensions, many of which contain exactly doublets, neutral singlets, and charged singlets.
  • The |Q,Y,T⟩ labeling, with Q shown to be redundant in the minimal basis, provides a template for simplifying scattering-matrix construction in other multi-scalar theories.
  • BounDS lets users specify field content and symmetries, discrete or continuous, abelian or non-abelian, and outputs the potential, the seven matrices, and their eigenvalues where closed forms exist.
  • Converting these quartic-coupling bounds into mass or mixing-angle bounds requires additional case-by-case work, defining the full potential, vacuum, and mass diagonalization, which the paper explicitly leaves out.

Where Pith is reading between the lines

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

  • Because the bounds are formulated purely in terms of quartic couplings, they are invariant under field redefinitions and do not depend on how symmetry breaking is realized; this makes them robust probes but also means they cannot alone bound masses without extra assumptions.
  • The same reduction technique could plausibly be extended to include fermionic or gauge-boson initial states in the coupled-channel analysis, which would test whether the scalar-only matrices are truly the complete unitarity conditions away from the strict high-energy limit.
  • The automated block-diagonalization by additional quantum numbers suggests a general strategy: for any finite symmetry group, one could systematically decompose the full scattering matrix into irreps, reducing large numerical scans to small analytic blocks.
  • One testable extension is to apply BounDS to a model with a non-abelian continuous flavour symmetry, which the paper states is supported, and compare the resulting bounds with independent calculations, checking both the code and the claimed generality.

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

0 major / 5 minor

Summary. The paper derives tree-level partial-wave unitarity bounds on the quartic couplings of a general renormalizable scalar sector containing n_D SU(2) doublets with Y=1/2, n_n real neutral singlets with Y=0, and n_c singly charged singlet scalars with Y=1. The authors write the most general quartic potential, impose hermiticity and index-symmetry relations, classify two-particle scattering states by |Q,Y,T>, reduce to a minimal set of seven coupled-channel matrices (Table I), and impose the standard |Λ| ≤ 8π condition. Explicit amplitude formulas, several worked examples (SM, singlet extensions, 2HDM, 2HDM with singlets, Z3 3HDM), and the Mathematica notebook BounDS are provided. The paper claims that this constitutes a complete and minimal description of the unitarity constraints for this model class.

Significance. If the completeness claim holds, this is a valuable reference result: it unifies and extends previously scattered unitarity analyses into one framework and provides a publicly available automated tool. The derivation is self-contained, and the literature comparisons in Section V are used as consistency checks rather than as inputs. The Appendix A redundancy argument is standard and, on my reading, internally sound; the concern that a T=2 doublet-doublet channel is missing does not land, because two doublets combine only to T=0 or T=1, and the external T=0 singlets do not mix with T=1 doublet-doublet states. The main caveat is that several displayed matrices contain local errors (listed below) that should be corrected before the paper is used as a formula source. These errors do not, in my assessment, invalidate the central classification or the general method.

minor comments (5)
  1. [Eq. (118)] The off-diagonal entries of M_{|1,1,0>} are not hermitian as printed: both entries are written with κ*_{12,11} but with opposite signs. Eq. (32) gives the (1,2) element as -2√2 i κ_{12,11}, so the (2,1) element should be +2√2 i κ*_{12,11} (or the overall sign convention reversed consistently). The eigenvalue formula in Eq. (122), which depends only on |κ_{12,11}|^2, is consistent with a hermitian matrix and confirms the intended form. This is a typographical error, but it should be fixed because Eq. (118) is one of the paper's explicit scattering matrices.
  2. [Eq. (119)] The state label |1,1,2> is impossible: two SU(2) doublets combine only into T=0 or T=1, and charged/neutral singlets have T=0. The displayed 3×3 matrix is exactly the |2,1,1> matrix for a general 2HDM (compare Eq. (71)); the label should be |2,1,1>, not |1,1,2>. This is presumably a typo in the state label, but it is confusing and should be corrected.
  3. [Eqs. (51), (79), (105)] The |0,0,0> matrices are not symmetric as printed, which is inconsistent with hermiticity of the scattering matrix. For example, Eq. (51) has M_{12}=2√2 γ_{11,11} and M_{21}=2 γ_{11,11}; from Eq. (40) with N_{11}=1/√2, both entries should be 2γ_{11,11}. This is also required by the eigenvalue formula Eq. (52), which uses 4γ^2. Similar √2/N-factor asymmetries appear in Eqs. (79) and (105). The authors should check all displayed matrices against Eqs. (28)–(43), especially the N factors for identical neutral-singlet pairs.
  4. [Appendix A] The notation Φ_i Φ*_j is used for both the T=0 combination in Eq. (A8) and the T=1 combination in Eq. (A9). This makes Eq. (A11) ambiguous: the equality with M[ϕ+_a ϕ0*_b → ...] can only hold for the T=1 combination. Please use distinct symbols (e.g. (ΦΦ*)_0 and (ΦΦ*)_1) to avoid confusion.
  5. [General presentation] There are several minor language/typo issues: “dully diagonalized” in Section VI should be “fully diagonalized”; “respectivelly” in Appendix A; “one one wishes” in Section VI. These do not affect the physics but should be cleaned up.

Circularity Check

0 steps flagged

Derivation is self-contained; self-citations are only consistency checks, so no circularity.

full rationale

The paper's central derivation starts from the quartic potential (Eq. 4) and constructs 2-to-2 amplitudes via Eq. (20), then imposes partial-wave unitarity (Eq. 24) on the eigenvalues of the coupled-channel matrices. No parameter is fitted and no predicted quantity is equal to an input by construction. The minimal-basis claim in Table I is supported by the explicit algebraic redundancy relations in Appendix A (Eqs. A1-A11), not by a self-cited uniqueness theorem. The cited works with overlapping authors (Refs. [19], [28], [30]) are used as comparisons, dictionaries, or technical alternatives, and are not load-bearing inputs to the derivation. The high-energy/propagator-vanishing approximation is a stated physical assumption, not a circular step. The skeptical concern about possible missing channels for n_D>2 is a potential completeness or correctness question in Appendix A, not circularity, since completeness is not defined in terms of the cited results.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No numbers are fitted; all constraints are analytic inequalities on symbolic quartic couplings. The derivation rests on standard partial-wave unitarity, the high-energy/equivalence-theorem limit, and the assumed renormalizable potential form.

axioms (4)
  • standard math Tree-level partial-wave unitarity bound |Re a0| ≤ 1/2 (equivalently |Λ| ≤ 8π) applies to the eigenvalues of each coupled-channel scattering matrix.
    Section III A, Eq. (23); the standard S-matrix unitarity condition used throughout.
  • domain assumption In the high-energy limit only quartic scalar interactions contribute to 2-to-2 amplitudes; s-, t-, and u-channel propagator diagrams vanish.
    Section II B: 'We perform all calculations in the high-energy limit where all contributions from propagators vanish'; and Section III A.
  • domain assumption Longitudinal gauge-boson scattering can be replaced by Goldstone/scalar scattering via the Equivalence Theorem.
    Invoked in the Introduction via Ref. [14]; justifies ignoring explicit gauge-boson external states.
  • domain assumption The potential in Eq. (4) with hermiticity relations (5)-(18) is the most general renormalizable quartic potential for the specified field content.
    Section II B; the entire matrix construction follows from this parametrization.

pith-pipeline@v1.3.0-alltime-deepseek · 23074 in / 23833 out tokens · 238817 ms · 2026-08-04T12:43:15.738646+00:00 · methodology

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Cite this review

Pith. "Pith review of Perturbative unitarity for models with singlet and doublet scalars." pith.science (2026). https://pith.science/paper/XF3LF5WZ

@misc{pith2026251002434,
  author       = {Pith},
  title        = {Pith review of: Perturbative unitarity for models with singlet and doublet scalars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XF3LF5WZ}},
  note         = {Machine review of arXiv:2510.02434}
}
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read the original abstract

We provide a complete description of perturbative unitarity bounds on the gauge-scalar sectors of models with extra $SU(2)$ doublet, neutral singlet, and charged singlet scalars. Such additions are very frequent in models beyond the Standard Model, and, in particular, they are almost universal in models explaining the dark matter problem. We propose a specific classification and minimal set of scattering matrices containing all the relevant information. We also developed a Mathematica implementation of our results, BounDS, and we use it to fully study a number of simple cases, comparing with the literature, when available. The Mathematica notebook BounDS is provided via a public GitHub repository.

discussion (0)

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Reference graph

Works this paper leans on

46 extracted references · 22 linked inside Pith

  1. [1]

    These matrices have rank-1 and take the following form: M|1,0,1⟩ =M |2,1,1⟩ = 2λ11,11,(45) M|0,0,0⟩ = 6λ11,11.(46) 8

    Scattering Matrices For this minimal scalar content, there are only three non-zero scattering matrices. These matrices have rank-1 and take the following form: M|1,0,1⟩ =M |2,1,1⟩ = 2λ11,11,(45) M|0,0,0⟩ = 6λ11,11.(46) 8

  2. [2]

    Unitarity Bounds Applying the partial wave unitarity condition of Eq. (24) to the eigenvalues of these scattering matrices, we find |2λ11,11| ≤8π,|6λ 11,11| ≤8π=⇒λ 11,11 ≤ 4π 3 .(47) In order to facilitate the comparison of our results with those of Ref. [19], it is useful to provide a shortdictionary of notations. Table III lists the correspondence betwe...

  3. [3]

    Scattering Matrices The set of non-zero scattering matrices for this model are given by: M|1,0,1⟩ =M |2,1,1⟩ = 2λ11,11,(49) M|1, 1 2 , 1 2 ⟩ = 2γ11,11,(50) M|0,0,0⟩ = " 6λ11,11 2 √ 2γ11,11 2γ11,11 12β11,11 # .(51)

  4. [4]

    Unitarity Bounds We apply the partial wave unitarity condition of Eq. (24) to each of the eigenvalues of the zero partial wave amplitude matrix and find 6β11,11 + 3λ11,11 ± q 9 (λ11,11 −2β 11,11) 2 + 4γ2 11,11 ≤8π,(52) |2λ11,11| ≤8π,(53) |2γ11,11| ≤8π.(54) C. TheZ 2-Symmetric 2HDM We now consider a 2HDM model with a discreteZ 2 symmetry that acts on the t...

  5. [5]

    Scattering Matrices The set of non-zero scattering matrices for this model reads M|2,1,1⟩ =   λ1 λ5 0 λ5 λ2 0 0 0λ 3 +λ 4   ,(59) M|1,0,1⟩ =   λ1 λ4 0 0 λ4 λ2 0 0 0 0λ 3 λ5 0 0λ 5 λ3   ,(60) M|0,0,0⟩ =   3λ1 2λ3 +λ 4 0 0 2λ3 +λ 4 3λ2 0 0 0 0λ 3 + 2λ4 3λ5 0 0 3λ 5 λ3 + 2λ4   ,(61) M|1,1,0⟩ =λ 3 −λ 4.(62)

  6. [6]

    Eigenvalues By imposing the unitarity condition of Eq. (24) to each of the eigenvalues of the zero partial wave amplitude matrix we derive |λ3 ±λ 4| ≤8π,(63) 10 |λ3 ±λ 5| ≤8π,(64) |λ3 + 2λ4 ±3λ 5| ≤8π,(65) 1 2 λ1 +λ 2 ± q (λ1 −λ 2)2 + 4λ2 4 ≤8π,(66) 1 2 λ1 +λ 2 ± q (λ1 −λ 2)2 + 4λ2 5 ≤8π,(67) 1 2 3λ1 + 3λ2 ± q 9 (λ1 −λ 2)2 + 4 (2λ3 +λ 4)2 ≤8π.(68) We find...

  7. [7]

    If we setλ 6 =λ 7 = 0 (along with their complex conjugates), we re-obtain, as expected, the results presented in Section V C 1

    Scattering Matrices The set of non-zero scattering matrices for this model reads M|2,1,1⟩ =   λ1 √ 2λ ∗ 6 λ∗ 5√ 2λ 6 λ3 +λ 4 √ 2λ ∗ 7 λ5 √ 2λ 7 λ2   ,(71) 11 M|1,0,1⟩ =   λ1 λ6 λ∗ 6 λ4 λ∗ 6 λ3 λ∗ 5 λ∗ 7 λ6 λ5 λ3 λ7 λ4 λ7 λ∗ 7 λ2   ,(72) M|0,0,0⟩ =   3λ1 3λ∗ 6 3λ6 2λ3 +λ 4 3λ6 λ3 + 2λ4 3λ5 3λ7 3λ∗ 6 3λ∗ 5 λ3 + 2λ4 3λ∗ 7 2λ3 +λ 4 3λ∗ 7 ...

  8. [8]

    However, some can be computed analytically, and unitarity bounds are imposed accordingly, |λ3 −λ 4| ≤8π.(75) E

    Unitarity Bounds The eigenvalues of the scattering matrices in this case are, in general, too complex to write down in closed form. However, some can be computed analytically, and unitarity bounds are imposed accordingly, |λ3 −λ 4| ≤8π.(75) E. 1 Scalar Doublet and 2 Neutral Scalar Singlets In this case, the most general quartic part of the scalar potentia...

  9. [9]

    Scattering Matrices The set of non-zero scattering matrices for this model reads M|1,0,1⟩ =M |2,1,1⟩ = 2λ11,11,(77) M|1, 1 2 , 1 2 ⟩ = " 2γ11,11 2γ11,12 2γ11,12 2γ11,22 # ,(78) M|0,0,0⟩ =   6λ 11,11 2 √ 2γ 11,11 2 √ 2γ 11,12 2 √ 2γ 11,22 2γ 11,11 12β 11,11 12 √ 2β 11,12 12β 11,22 2 √ 2γ 11,12 12 √ 2β 11,12 24β 11,22 12 √ 2β 12,22 2γ 11,22 12β 11,22 1...

  10. [10]

    Unitarity Bounds Again, certain eigenvalues of the scattering matrices are too complicated to be expressed in closed form. However, others can be written analytically, and, for those, the corresponding unitarity bounds are then applied, as: γ11,11 +γ 11,22 ± q 4γ2 11,12 + (γ11,11 −γ 11,22) 2 ≤8π,(80) |2λ11,11| ≤8π,(81) (82) In order to compare our results...

  11. [11]

    Scattering Matrices The set of non-zero scattering matrices for this model reads M|1, 1 2 , 1 2 ⟩ = " 2γ11,11 0 0 2γ 22,11 # ,(86) M|2,1,1⟩ =   2λ11,11 2λ∗ 12,12 0 2λ12,12 2λ22,22 0 0 0 2λ 11,22 + 2λ12,21   ,(87) M|1,0,1⟩ =   2λ11,11 2λ12,21 0 0 2λ12,21 2λ22,22 0 0 0 0 2λ 11,22 2λ∗ 12,12 0 0 2λ 12,12 2λ11,22   ,(88) M|0,0,0⟩ =   6λ1...

  12. [12]

    Unitarity Bounds Some eigenvalues are too complicated to be written in closed form. Therefore, we show only the results of imposing partial-wave unitarity on the remaining eigenvalues of the zero partial-wave amplitude matrix, |2γ11,11| ≤8π,(91) |2γ22,11| ≤8π,(92) 2|λ11,22 ±λ 12,21| ≤8π,(93) 2|λ 11,22 ± |λ12,12|| ≤8π,(94) λ11,11 +λ 22,22 ± q (λ11,11 −λ 22...

  13. [13]

    ) refers to a block diagonal matrix, whose entries are the matrices A, B,

    Scattering Matrices The set of non-zero scattering matrices for this model reads M|1, 1 2 , 1 2 ⟩ =   2γ11,11 2γ11,12 0 0 2γ11,12 2γ11,22 0 0 0 0 2γ 22,11 2γ22,12 0 0 2γ 22,12 2γ22,22   ,(100) M|2,1,1⟩ =   2λ11,11 2λ∗ 12,12 0 2λ12,12 2λ22,22 0 0 0 2λ 11,22 + 2λ12,21   ,(101) M|1,0,1⟩ =   2λ11,11 2λ12,21 0 0 2λ12,21 2λ22,22 0 0 0 0 2λ ...

  14. [14]

    Unitarity Bounds As several eigenvalues are too involved to express analytically, we show only the results of imposing partial-wave unitarity on the remaining eigenvalues of thes-wave amplitude matrix, that can be computed explicitly, 2|λ 11,22 ±λ 12,21| ≤8π,(107) 2|λ 11,22 ± |λ12,12|| ≤8π,(108) 2|λ 11,22 + 2λ12,21 ±3|λ 12,12|| ≤8π,(109) λ11,11 +λ 22,22 ±...

  15. [15]

    Scattering Matrices The set of non-zero scattering matrices are: M|2,2,0⟩ = 2α11,11,(115) M|2, 3 2 , 1 2 ⟩ = " δ11,11 δ∗ 12,11 δ12,11 δ22,11 # ,(116) M|1, 1 2 , 1 2 ⟩ =   δ11,11 δ12,11 0−2iκ ∗ 12,11 δ∗ 12,11 δ22,11 2iκ∗ 12,11 0 0−2iκ 12,11 2γ11,11 2γ∗ 12,11 2iκ12,11 0 2γ 12,11 2γ22,11   ,(117) M|1,1,0⟩ = " 2λ11,22 −2λ 12,21 2i √ 2κ∗ 12,11 −2i √ ...

  16. [16]

    Unitarity Bounds Since certain eigenvalues are too complex to evaluate analytically, unitarity constraints are shown only for the remaining eigenvalues of the zero partial-wave amplitude matrix. These read ζ11,11 +λ 11,22 −λ 12,21 ± q (−ζ11,11 −λ 11,22 +λ 12,21)2 −4 (ζ 11,11λ11,22 −ζ 11,11λ12,21 −2|κ 12,11|2) ≤8π,(122) 1 2 δ11,11 +δ 22,11 ± q (δ11,11 −δ 2...

  17. [17]

    2λ11,33 + 2λ13,31 2 √ 2λ∗ 12,32 2 √ 2λ12,32 2λ22,22 # ,(128) B=

    Scattering Matrices The scattering matrices are all block diagonal. We find M|2,1,1⟩ = blkdiag(A, B, C),(127) where A= " 2λ11,33 + 2λ13,31 2 √ 2λ∗ 12,32 2 √ 2λ12,32 2λ22,22 # ,(128) B= " 2λ11,11 2 √ 2λ∗ 12,13 2 √ 2λ12,13 2λ22,33 + 2λ23,32 # ,(129) C= " 2λ33,33 2 √ 2λ13,23 2 √ 2λ∗ 13,23 2λ11,22 + 2λ12,21 # ,(130) M|1,0,1⟩ = blkdiag(D, E, F),(131) 18 where ...

  18. [18]

    Φ1Φ∗ 1 Φ1Φ∗ 1 # =U

    Unitarity Bounds We show the results of partial wave unitarity bounds to the eigenvalues that can be computed analytically, 2|λ11,22 −λ 12,21| ≤8π,(140) 2|λ11,33 −λ 13,31| ≤8π,(141) 2|λ22,33 −λ 23,32| ≤8π,(142) λ11,11 +λ 22,33 +λ 23,32 ± q (λ11,11 +λ 22,33 +λ 23,32)2 −4 (λ 11,11λ22,33 +λ 11,11λ23,32 −2|λ 12,13|2) ≤8π,(143) λ11,33 +λ 13,31 +λ 22,22 ± q (λ1...

    arXiv 2024

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    1 |1,0,1⟩ ϕ+ 1 ϕ0∗ 1 1 |0,0,1⟩ Φ1Φ∗ 1 1 |0,0,0⟩ Φ1Φ∗ 1 1 |0,1,1⟩ ϕ0 1ϕ0 1 1 TABLE XI: Basis of two-particle states labelled by|Q, Y, T⟩. As a result, the scattering matrix for the states|0,0⟩, gets diagonalized as (M|0,0⟩)new =U(M |0,0⟩)oldU † = " 6λ11,11 0 0 2λ 11,11 # .(B4) Similarly, for the|1,1⟩states, only the symmetric combination survives ϕ+ (1ϕ0

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    2λ11,11 2λ12,12 2λ12,12 2λ22,22 # ,(D2) M|2,1,1,−1⟩ = 2 (λ 11,22 +λ 12,21),(D3) M|1,1,0,−1⟩ = 2 (λ 11,22 −λ 12,21),(D4) M|1,0,1,+1⟩ =

    = 1√ 2 h 1 1 i " ϕ+ 1 ϕ0 1 ϕ0 1ϕ+ 1 # ,(B5) meaning (M|1,1⟩)new = 2λ11,11.(B6) Collecting all cases, the scattering eigenvalues are M|0,0,0⟩ = 6λ11,11,(B7) M|0,0,1⟩ =M |1,1,0⟩ =M |2,1⟩ =M |1,0⟩ =M |0,1⟩ = 2λ11,11.(B8) The eigenvalues, and therefore the unitarity bounds, follow directly. This demonstrates the advantage of organizing states in the|Q, Y, T⟩b...

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