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REVIEW 3 major objections 5 minor 24 references

The paper claims that colour-triplet quark fields, described by a Z3-graded Lee-Wick type sixth-order equation, have damped single-particle solutions, and that only ternary products of such solutions propagate freely, providing an algebraic

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2026-08-03 11:18 UTC pith:2CXDHFCE

load-bearing objection The Z3-coloured Dirac model is interesting and the determinant trick is cute, but the confinement claim is circular: the damping is imposed by hand, not derived. the 3 major comments →

arxiv 2601.06694 v2 pith:2CXDHFCE submitted 2026-01-10 hep-ph hep-th

Complex Mass Shells for Coloured quarks and their Asymptotic Confinement

Richard Kerner , Jerzy Lukierski This is my paper
classification hep-ph hep-th
keywords Z3 symmetrycolour Dirac equationLee-Wick polescomplex mass shellssixth-order dispersion relationalgebraic confinementquark confinementternary products
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 proposes replacing the standard coloured Dirac quarks with a single 12-component 'coloured Dirac spinor' built from six Pauli spinors, arranged so that a Z3 symmetry cyclically permutes the three colours and multiplies masses by the cube roots of unity. Iterating the resulting linear equation six times diagonalises it and yields a sixth-order dispersion relation, E^6 = |p|^6 c^6 + m^6 c^12, for every component. Because the associated mass shells are complex for two of the three colours, single-quark solutions are exponentially damped and cannot propagate beyond a few wavelengths. The paper shows that certain products of three such solutions — determinants of 3×3 solution matrices — cancel the damping exponents and produce freely propagating waves, which the authors interpret as algebraic confinement: colour singlets travel, individual quarks do not. A massless-limit calculation of the Green's function also delivers the usually assumed quark potential, V(r) ~ α/r + βr, without postulating it.

Core claim

The central discovery is that a Z3-symmetric generalisation of the Dirac equation, acting on 12-component colour spinors, has as its on-shell condition a sixth-order polynomial E^6 - |p|^6 c^6 = m^6 c^12. The dispersion relation factorises into one standard relativistic factor and two complex-conjugate Lee-Wick factors, so the propagator carries two real and four complex poles. When the massive case is restricted to purely imaginary frequency and wave vector, all single-colour exponential solutions are damped. However, the determinants of the 3×3 matrices formed from the Z3-multiplied solutions are purely oscillatory, giving free waves that satisfy the standard second-order relativistic disp

What carries the argument

The key object is the generalised colour Dirac operator Γ^μ p_μ on a 12-component spinor Ψ = (φ+,φ−;χ+,χ−;ψ+,ψ−)^T, with Γ^0 = B† ⊗ σ3 ⊗ I and Γ^i = Q^2 ⊗ (iσ2) ⊗ σ^i, where B = diag(1,j,j²) and Q is the cyclic permutation matrix. Its sixth power is diagonal and equals (p_0^6 − |p|^6) I, which is why the on-shell condition is sixth-order. The central identity is (Γ^μ p_μ)^6 = (p_0^6 − |p|^6) I, and the factorisation p_0^6 − |p|^6 = (p_0^2 − |p|^2)(p_0^2 − j|p|^2)(p_0^2 − j²|p|^2) generates the three mass-shell pairs. The confinement mechanism is carried by the 3×3 solution matrices S1,S2 built from exponentials e^{i(j^α ω t − j^β k x)}: each matrix has determinant 1, and the real parts of th

Load-bearing premise

The load-bearing premise is that in the massive case only pure-imaginary values of the frequency and wave vector are physical; this selection rule is asserted in Section IV rather than derived, and if real-frequency solutions are retained, single-quark waves would propagate freely and the confinement argument would fail.

What would settle it

Compute the exact solution of the massive colour Dirac equation with a real ω and real k, and show it has nonzero amplitude satisfying the boundary conditions; alternatively, evaluate the contour integral of the massive propagator around the real poles ±Ω and show it yields an undamped oscillating wave at large distances.

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

If this is right

  • If the sixth-order dispersion relation is correct, every quark component obeys E^6 = |p|^6 c^6 + m^6 c^12, altering the free-field propagator and the on-shell counting compared with QCD.
  • Single quarks cannot travel freely; the only propagating objects are ternary products (colour-singlet-like combinations), giving a concrete algebraic mechanism for confinement.
  • The massless propagator naturally yields a quark-quark potential V(r) ≈ α/r + βr, reproducing the linear-plus-Coulomb form used in phenomenology without an ad hoc assumption.
  • The Z3 grading implies colour transformations no longer commute with the Lorentz group; it uses a Z3-covering of the Lorentz group for the two complex mass-shell factors.
  • The propagator's six poles (two real, four Lee-Wick complex) replace the two Dirac poles, with far-field damping from the complex poles.

Where Pith is reading between the lines

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

  • The paper's selection rule (only pure-imaginary ω,k for massive modes) is stated without proof; if real-frequency modes are physically admissible, the damping argument collapses. Checking this is a direct test.
  • The determinant construction suggests a possible algebraic origin of baryon and meson states as ternary products; one could try to build explicit three-quark wavefunctions and compare their dispersion relations with hadron spectra.
  • The massless Green's function computation could be extended to the massive case numerically, testing whether the α/r + βr potential survives with a screening term.
  • The sixth-order equation with complex masses may harbour ghost-like states; examining their unitarity and microcausality could connect this construction to the broader Lee-Wick literature.

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 / 5 minor

Summary. The paper proposes a 12-component 'colour Dirac' spinor satisfying a first-order linear system (Eq. (12)) with mass matrix B = diag(1,j,j²) and cyclic colour mixing Q3. It claims that iterating the first-order operator yields a diagonal sixth-order equation (Eq. (22)), giving the dispersion relation E^6 = |p|^6 c^6 + m^6 c^12 (Eq. (24)). The paper then asserts that for massive fields only pure imaginary frequency and wave vector are admissible, so that single-quark plane waves are damped. It further claims that certain ternary products of such damped solutions form freely propagating states, which is interpreted as algebraic confinement. A massless Green function is also constructed, yielding an effective potential V(r) ≃ α r^{-1} + β r. The conclusion rests on the sixth-order dispersion relation and on the selection of imaginary modes.

Significance. The idea of using a Z3-graded generalization of the Dirac equation to realise algebraic confinement is original and connects to a literature on complex-mass Lee-Wick fields. If the central derivation were correct, the paper would provide a concrete, falsifiable alternative to QCD-based confinement, with explicit propagators and a natural origin for a confining potential. The paper also contains explicit exponential solutions and attempts at closed-form Green functions, which are valuable as a starting point. However, the core algebraic identity is unproven and appears to be false, and the restriction to imaginary frequency and wave vector is an additional postulate rather than a consequence of the equations. The present version does not justify the main claim.

major comments (3)
  1. [Section II, Eqs. (17) and (22)] The identity (Γµpµ)^6 = (p0^6 − |p|^6) 1 is the foundation of the paper. It is asserted without proof, and a direct calculation from (17) contradicts it at second order. With Γ0 = B† ⊗ σ3 ⊗ I₂ and Γi = Q2 ⊗ (iσ2) ⊗ σi, one finds Γ0² = B ⊗ I₂ ⊗ I₂ and Γi² = −Q3 ⊗ I₂ ⊗ I₂, while the cross terms are Γ0Γi + ΓiΓ0 = (B†Q2 − Q2B†) ⊗ σ1 ⊗ σi p0 p_i. From (13), B†Q2 and Q2B† differ (e.g. (B†Q2)_{13}=1 but (Q2B†)_{13}=j), so these cross terms do not cancel. Since the second-order expression already contains off-diagonal components, the sixth power is not proportional to the identity unless some miraculous cancellation occurs in higher powers, and no such argument is supplied. Consequently Eq. (22), and with it the dispersion relation (24) and the determinant (25), are unsupported. This is a load-bearing flaw.
  2. [Section IV, after Eq. (52)] The statement that for massive particles 'only pure imaginary values [of ω and k] must be retained' is not derived. The dispersion relation (24), E^6 = |p|^6 c^6 + m^6 c^12, is a polynomial of degree 6 in E; for every real |p| it admits two real roots, E = ±(|p|^6 c^6 + m^6 c^12)^{1/6}. Since (16) is a linear constant-coefficient equation, the corresponding plane waves exp(i(p·x − Et)/ℏ) are perfectly valid solutions. The comment about 'positive sixth-order derivatives' near Eq. (53) does not select roots of an algebraic equation. This selection rule is an extra physical postulate, not a consequence of the formalism. It is essential because without discarding the real-frequency modes the damping (and hence the claimed confinement) disappears. The paper therefore builds the conclusion in by hand.
  3. [Section IV, Eqs. (59)–(66)] The proposed construction of propagating ternary products is not a derivation. (i) The determinants in Eqs. (59)–(62) are products of scalar solutions of the sixth-order equation, but the paper does not show that such products are solutions of the original 12-component system (16). (ii) The counting argument after Eq. (66) merely compares numbers of variables and constraints; it does not establish existence of solutions to the nonlinear conditions (66). (iii) Even if such products are formally undamped, they are not identified with any physical states carrying the correct quantum numbers and normalizability. The claim that 'certain ternary products of such solutions represent free propagating states' is therefore not supported.
minor comments (5)
  1. [General] There are several typographical errors: 'Schoeodinger-like' in the abstract; 'wavelenghts' in the conclusion; an extra period in Eq. (9) ('c σ · p .φ+').
  2. [Section II, notation] The identity matrix is denoted inconsistently: l_12 is used for the 2×2 identity in (17) but earlier l_12 seems to denote a 12×12 as well. Please use I_n consistently to avoid confusion.
  3. [Section II, text after Eq. (13)] The statement that B and Q3 'generate the U(3) Lie group algebra' is imprecise; these matrices generate a finite matrix algebra, not a Lie algebra. Please rephrase.
  4. [Section III, Eq. (45)] The inverse Fourier transform of the |p|^{-4} formfactor is obtained by an 'educated guess' with a Yukawa regulator. The derivation should be made explicit, or the result labelled as a conjecture, because it is used later to argue for a linear confining potential.
  5. [References] Several references are incomplete or have formatting issues (e.g. [12] 'Kerner R Suzuki O 2012' lacks a comma, [16] has inconsistent punctuation). Also, essential previous results, especially the Z3-covering of the Lorentz group from [15], are not summarised, making it difficult to verify the claimed Lorentz covariance of (24).

Circularity Check

2 steps flagged

Single-quark confinement is imposed by an unproved imaginary-frequency selection; the complex-mass input is then reported as derived damping.

specific steps
  1. self definitional [Section IV (Solutions), after Eq. (52); dispersion relation Eq. (24)]
    "At this point both ω and k may be real, complex or pure imaginary. However, if the correponding particles are not massless, it is easy to show that only pure imaginary valueas must be retained."

    Eq. (24) gives E^6 = |p|^6 c^6 + m^6 c^12, which for every real |p| has real roots E = ±(|p|^6 c^6 + m^6 c^12)^{1/6}. These correspond to freely propagating plane waves exp(i(p·x − Et)/ℏ). The paper discards these real modes without proof and keeps only purely imaginary ω and k, whose solutions are damped. The central claim that single quarks 'cannot propagate freely' is thus not derived from the sixth-order equation; it is imposed by hand-selecting the damped branch. The damping is equivalent to the chosen complex-mass input (B = diag(1, j, j²), Eq. (13)), not an independent consequence.

  2. self definitional [Abstract and Section V (Conclusion); mass matrix Eq. (13)]
    "Due to the appearance of complex poles in the Fourier transform of propagators, corresponding to complex mass shells, the solutions are damped and cannot propagate freely on distances greater than a few wavelenghts."

    The conclusion explains the damping by 'complex mass shells', but complex masses are exactly the defining assumption of the model: the abstract states the fields are 'one with real mass and the two remaining ones with mutually conjugate complex masses', realized in the mass operator B = diag(1, j, j²) in Eq. (13). The 'prediction' of asymptotic damping is therefore a restatement of the input. The same sixth-order dispersion relation also admits undamped real roots, so the damping is not forced by the equation itself; it is forced by which roots the authors allow.

full rationale

The paper is not wholly circular: the derivation of the sixth-order operator (Eq. (22)), the factorization of the propagator, and the cubic-product construction that cancels the damping factors are self-contained algebraic results. However, the central physics claim of 'asymptotic confinement' of single quarks reduces to the model's own complex-mass prescription plus an unproved selection rule in Section IV that excludes the real, undamped roots of the same dispersion relation. While complex poles are a legitimate Lee-Wick-style input, the paper presents the resulting damping as if it were a derived consequence of the sixth-order equation. The ternary-product cancellation is non-circular, but it does not rescue the single-quark claim because that claim was already built in by the choice of frequencies. Self-citations to [15] and [16] supply the Z3 Lorentz-covering and Lagrangian framework, but the circularity is localized to the mass/root selection, not the citation chain. Score 6 reflects a partially built-in result with identifiable independent content elsewhere.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 1 invented entities

The model's central phenomenon — damping of single-quark modes — is generated by hand via the complex mass matrix B. The paper adds no data fits, but it does introduce an ad hoc selection rule (imaginary-only solutions) and an assumed potential ansatz in the propagator section. The Z3-covering Lorentz symmetry and the physicality of Lee-Wick poles are imported from prior self-cited literature without independent verification.

free parameters (3)
  • Complex mass factors in B = diag(1,j,j²) = j = e^{2πi/3}, j²
    Chosen by hand to enforce the Z3 symmetry; the entire confinement damping is a direct consequence of this choice, making it an ad hoc input rather than a fitted or derived quantity.
  • Exponent β in V(r) = α r^β = β = -1 and β = 1
    The inverse Fourier transform of |p|^{-4} is obtained via an 'educated guess' V = αr^β; only the choices β = -1 (Coulomb) and β = 1 (linear) are kept to match the expected quark potential.
  • Yukawa regulator λ = λ → 0
    A regulator is introduced to define the infrared-divergent |p|^{-4} integral; the final potential is obtained after taking λ to zero.
axioms (4)
  • ad hoc to paper (Γμpμ)^6 = (p0^6 − |p|^6) 1_{12}
    Stated in Eq. (22) as 'straightforward' but not proved; the noncommutativity of B and Q leaves the cancellation of cross-terms unverified.
  • ad hoc to paper Physical massive solutions must have purely imaginary ω and k
    Asserted in Section IV with 'it is easy to show' but no derivation; discards the real-frequency solutions of the same dispersion relation that would propagate freely.
  • domain assumption Z3-covering of the Lorentz group is the correct symmetry for quark colour
    Taken from the authors' previous work (ref. 15); no independent verification or external consensus is cited.
  • domain assumption Lee-Wick type complex poles can represent physical hadronic states
    The paper invokes Lee-Wick and Anselmi-Piva literature but does not address unitarity or ghost issues in the proposed 12-component field theory.
invented entities (1)
  • 12-component Z3-coloured Dirac spinor with complex masses no independent evidence
    purpose: To describe colour triplets of quarks as entangled Lee-Wick fields whose single-quark modes damp out, giving algebraic confinement.
    No new particle is named, but the complex-mass field structure is a new postulated entity. The only purported handle is the model's own confinement and potential predictions, which are not independently falsifiable at this stage.

pith-pipeline@v1.3.0-alltime-deepseek · 12283 in / 12689 out tokens · 141969 ms · 2026-08-03T11:18:57.479675+00:00 · methodology

0 comments
read the original abstract

The present paper is the continuation of our previous work (R. Kerner and J. Lukierski, Nuclear Physics B, 2021) where we introduced a Z3-symmetric covering of the Lorentz group as a natural symmetry describing the quark fields. In the current version of QCD quarks are described by coloured triplets of standard Dirac fields. In contrast, we proposed to describe the colour triplets of quarks by entangled Z_3-graded Lee-Wick type fields, one with real mass and the two remaining ones with mutually conjugate complex masses. This is obtained by attributing colour degrees of freedom to six Pauli spinors, three endowed with colours and three with anti-colours, which are united into one 12-component generalized ``coloured Dirac spinor". Thus entangled triplet of quark fields is described on-shell by a linear Schoeodinger-like system akin to the Dirac equation. The sixth-order dispersion relations lead to solutions suitably vanishing in asymptotic region, exhibiting the well established confinement property of coloured quarks' degrees of freedom. We add that in the so modified approach to QCD one should employ in the quark sector the Z3-graded extension of the Lorentz symmetries, which do not commute with hidden SU(3) colour transformations (see Kerner and Lukierski 2021, Kerner 2018}). Propagators and interaction with gluon and electromagnetic fields are discussed in the last section.

discussion (0)

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Reference graph

Works this paper leans on

24 extracted references · 1 canonical work pages

  1. [1]

    Gell-Mann M 1962Phys.Rev. 125 pp. 1067-1084

  2. [2]

    Okubo S 1962Progress in Theoretical Physics27 pp. 949-966

  3. [3]

    Version 2 CERN-TH- 401 1964/1/17 Volume 2

    Zweig G 1964 Model for strong interaction symmetry and its breaking. Version 2 CERN-TH- 401 1964/1/17 Volume 2

    1964

  4. [4]

    Gell-Mann M 1964Physics Letters 8, pp. 214-215

  5. [5]

    Gell-Mann M and Ne’eman Y 1964 The Eightfold wayA Review with a collection of reprints , W. A. Benjamin Inc. Publishers, New York - Amsterdam

    1964

  6. [6]

    Benjamin Inc

    Ne’eman Y 1967 Algebraic Theory of Particle Physics,Frontiers in Physics, W.A. Benjamin Inc. Publishers, New York - Amsterdam ISBN: 9781114735705, 9780738202990

    1967

  7. [7]

    Greensite J 2011Springer Lecture Notes in PhysicsVol. 821

  8. [8]

    Neuberger H 1998Physics LettersB 417 (1-2) pp. 141-144

  9. [9]

    Takahashi, T T, Suganuma, H, Nemoto, Y and Matsufuru, H 2002Phys. Rev. D, 65 114 509; see also ArXiV: hep-lat/0006005v2 (2001)

    Pith/arXiv arXiv 2001

  10. [10]

    Fariborz A H, Jora R and Schechter J 2008Physical ReviewD 77 (3): 034006

  11. [11]

    Grable S and Romatschke P 2024 arXiv preprint arXiv:2310.12203 (2023)

    Pith/arXiv arXiv 2024

  12. [12]

    Kerner R Suzuki O 2012Int. J. Geom. Methods Mod. Phys.09, 1261007

  13. [15]

    Kerner R and Lukierski J 2021Nuclear Physics B972 115 529 arXiv:2012.05335 (2020)

    Pith/arXiv arXiv 2012

  14. [16]

    Kerner R, 2022International Journal of Modern Physics A37, (20-21) 2243013

  15. [17]

    Infinite-Dimensional Lie Algebras

    Kac V G 1984"Infinite-Dimensional Lie Algebras" Cambridge Univ.Press

    1984

  16. [18]

    2013, pp

    Kerner R and Suzuki O 2014 Proceedings of the RIMS Conference on Mathematical Physics, Kyoto, Vol. 2013, pp. 54-72

    2014

  17. [19]

    Kerner, Physics of Atomic Nuclei , 80 (3), pp

    R. Kerner, Physics of Atomic Nuclei , 80 (3), pp. 529-531 (2017); arXiv:1111.0518, arXiv:0901.3961

    Pith/arXiv arXiv 2017

  18. [20]

    Kerner, Physics of Atomic Nuclei81 (6), pp

    R. Kerner, Physics of Atomic Nuclei81 (6), pp. 871-889 (2018), also: arXiv:1801.01403

    Pith/arXiv arXiv 2018

  19. [21]

    Kerner, Universe, 5 (1), p.1, (2019)

    R. Kerner, Universe, 5 (1), p.1, (2019). https://doi.org/10.3390/universe5010001

    work page doi:10.3390/universe5010001 2019

  20. [22]

    Kerner and J

    R. Kerner and J. Lukierski, Phys. Letters B , Vol. 792, pp. 233-237 (2019), also: 19 arXiv:1901.10936 [hep-th]

    Pith/arXiv arXiv 2019

  21. [23]

    Nishijima, Progr

    K. Nishijima, Progr. Theor. Phys.13 p. 285 (1955)

    1955

  22. [24]

    Sakata, Progr

    S. Sakata, Progr. Theor. Phys., 14 pp. 686-688 (1956)

    1956

  23. [25]

    Rev.D (2) p

    Lee T D and Wick G C 1970Phys. Rev.D (2) p. 1033

  24. [26]

    2017Journal of High Energy Physics2017 (6), pp

    Anselmi D and Piva M. 2017Journal of High Energy Physics2017 (6), pp. 1-24. see also: Anselmi D and Piva M. 2017Physical ReviewD, 96 (4), 045009. 20