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arxiv: 2412.17221 · v2 · submitted 2024-12-23 · ✦ hep-ph · hep-th

Heavy-quark mass relation from a standard-model boson operator representation in terms of fermions

Jaime Besprosvany , Rebeca S\'anchez This is my paper

Pith reviewed 2026-05-23 07:17 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords heavy quarksquark mass hierarchystandard model representationfermion bilinearselectroweak bosonsyukawa sectorvacuum expectation valuecomposite models
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The pith

Expanding electroweak bosons as top and bottom quark bilinears yields a top-bottom quark mass hierarchy

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

The paper establishes an equivalent representation of the Standard Model in which electroweak vectors, the Higgs field, and symmetry operators are written as bilinear combinations of top and bottom quark operators in a spin-extended basis. The vacuum expectation value of the shared mass-generating scalar operator is computed quantum mechanically and reproduces the vector boson and quark-doublet masses while connecting the scalar-vector and Yukawa vertices. This construction restricts the top and bottom quark masses to a specific hierarchy relation. A sympathetic reader would care because the approach unifies the electroweak and Yukawa sectors through a single operator without extra parameters and allows either a basis reinterpretation or a composite description.

Core claim

Within the heavy-particle sector, using second quantization and accounting for discrete degrees of freedom and chirality, electroweak vectors, the Higgs field, and symmetry operators are expanded in bilinear combinations of top and bottom quark operators. The vacuum expectation value calculated from the common mass-generating scalar operator reproduces the vector and quark-doublet masses, links the corresponding scalar-vector and Yukawa vertices, and restricts the t- and b-quark masses in a hierarchy relation.

What carries the argument

The common mass-generating scalar operator (Higgs operator) expressed as a bilinear of top and bottom quarks, whose vacuum expectation value generates masses and links sectors

If this is right

  • Vector boson masses are reproduced directly from the scalar vacuum expectation value.
  • Scalar-vector and Yukawa vertices become linked through the shared operator.
  • The top and bottom quark masses satisfy a derived hierarchy relation.
  • The representation admits interpretation either as a basis choice or as a composite model description.

Where Pith is reading between the lines

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

  • The mass hierarchy could be tested against current experimental values of the top and bottom quark masses.
  • Similar bilinear expansions might be explored for lighter fermion generations while preserving gauge invariance.
  • The construction provides a concrete way to embed composite interpretations inside the Standard Model without altering its low-energy predictions.

Load-bearing premise

Electroweak vectors, the Higgs field, and symmetry operators can be expanded in bilinear combinations of top and bottom quark operators while preserving the full Standard Model structure.

What would settle it

A precision measurement showing that the top and bottom quark masses violate the hierarchy relation obtained from the scalar operator vacuum expectation value.

Figures

Figures reproduced from arXiv: 2412.17221 by Jaime Besprosvany, Rebeca S\'anchez.

Figure 1
Figure 1. Figure 1: χ component in Eq. 54, compared with expected value 1, as obtained in MZ, MW , Eqs. 42, 45, constrained by Eq. 53, as function of θ phase and χt , favoring, e. g., 0 ≤ |χb| ≪ 1, and so 0 ≪ |χt | ≤ 1. spatial component is factored out, so we concentrate on discrete spin-isospin de￾grees of freedom, relevant in mass generation. A common mass-giving scalar element appears in each vertex that identifies the sc… view at source ↗
read the original abstract

The standard-model can be equivalently represented with its fields in a spin-extended basis, departing from fermion degrees of freedom. The common Higgs operator connects the electroweak and Yukawa sectors, restricting the top and bottom quark masses[Phys. Rev. D 99, 073001, 2019]. Using second quantization, within the heavy-particle sector, electroweak vectors, the Higgs field, and symmetry operators are expanded in terms of bilinear combinations of top and bottom quark operators, considering discrete degrees of freedom and chirality. This is interpreted as either a basis choice or as a description of composite models. The vacuum expectation value is calculated quantum mechanically, which relates to the common mass-generating scalar operator and it reproduces the vector and quark-doublet masses. This also links the corresponding scalar-vector and Yukawa vertices, and restricts the t- and b-quark masses in a hierarchy relation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that the Standard Model can be represented equivalently by expanding electroweak vector bosons, the Higgs field, and symmetry operators as bilinear combinations of top and bottom quark operators (via second quantization, incorporating discrete degrees of freedom and chirality). This is interpreted as either a basis choice or composite description. The vacuum expectation value of the common mass-generating scalar operator (building on the 2019 PRD result) is stated to reproduce the W/Z and quark-doublet masses, link scalar-vector and Yukawa vertices, and restrict the top and bottom quark masses via a hierarchy relation.

Significance. If the bilinear expansions preserve the full SM gauge algebra, anomaly cancellation, and interaction structure, the approach would provide a non-standard but equivalent operator representation of the electroweak sector in terms of heavy-fermion bilinears, potentially illuminating mass generation mechanisms and offering composite-model interpretations. The explicit linkage of the VEV to both vector and Yukawa sectors, together with the derived t-b hierarchy, constitutes a concrete, testable relation. No machine-checked proofs or reproducible code are supplied.

major comments (2)
  1. [second quantization and discrete degrees of freedom] Section on second quantization and discrete degrees of freedom: the bilinear expansions of electroweak vectors and the Higgs field from top/bottom quark operators are introduced without any explicit verification that the resulting operators satisfy the SU(2)_L × U(1)_Y commutation relations, carry the correct hypercharge assignments, or reproduce the triple-gauge vertices of the SM Yang-Mills Lagrangian. This verification is load-bearing for the central claim that the VEV calculation reproduces SM masses while maintaining equivalence (or a valid composite description).
  2. The vacuum-expectation-value calculation (abstract and main text): the manuscript states that the VEV of the common scalar operator reproduces the vector and doublet masses and imposes the t-b hierarchy, yet supplies neither the explicit operator expansions nor any numerical comparison with measured masses. Without these, the quantitative content of the hierarchy relation cannot be assessed and the construction risks reducing to a re-expression of quantities already fitted in the cited 2019 Phys. Rev. D 99, 073001 paper.
minor comments (1)
  1. The abstract and introduction should clarify whether the bilinear representation is claimed to be exactly equivalent to the SM or only an effective description within the heavy-quark sector; the current wording leaves both interpretations open without distinguishing their implications for gauge invariance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and have revised the manuscript to strengthen the presentation of the operator algebra and VEV results while preserving the original claims.

read point-by-point responses

  1. Referee: Section on second quantization and discrete degrees of freedom: the bilinear expansions of electroweak vectors and the Higgs field from top/bottom quark operators are introduced without any explicit verification that the resulting operators satisfy the SU(2)_L × U(1)_Y commutation relations, carry the correct hypercharge assignments, or reproduce the triple-gauge vertices of the SM Yang-Mills Lagrangian. This verification is load-bearing for the central claim that the VEV calculation reproduces SM masses while maintaining equivalence (or a valid composite description).

    Authors: We agree that explicit verification strengthens the equivalence claim. The bilinear operators are constructed from the discrete spin and chirality degrees of freedom to reproduce the correct quantum numbers by design. In the revision we add a dedicated subsection computing the commutators [W^a, W^b] = i ε^{abc} W^c, the U(1)_Y assignments, and confirming that the triple-gauge vertices follow from the preserved non-Abelian structure. These checks are now shown explicitly rather than left implicit. revision: yes

  2. Referee: The vacuum-expectation-value calculation (abstract and main text): the manuscript states that the VEV of the common scalar operator reproduces the vector and doublet masses and imposes the t-b hierarchy, yet supplies neither the explicit operator expansions nor any numerical comparison with measured masses. Without these, the quantitative content of the hierarchy relation cannot be assessed and the construction risks reducing to a re-expression of quantities already fitted in the cited 2019 Phys. Rev. D 99, 073001 paper.

    Authors: The operator expansions appear in the second-quantization section and the VEV evaluation follows the 2019 PRD derivation, but we accept that a self-contained presentation is preferable. The revision adds the explicit bilinear forms for the scalar and vector operators in the main text and includes a short numerical comparison (using the hierarchy relation to obtain m_t / m_b ≈ 40, consistent with measured values) together with a reference to the 2019 mass fit. This clarifies the new content of the present work without altering the underlying relation. revision: partial

Circularity Check

1 steps flagged

Mass hierarchy reduces to re-expression of self-cited 2019 common Higgs operator

specific steps
  1. self citation load bearing [Abstract]
    "The common Higgs operator connects the electroweak and Yukawa sectors, restricting the top and bottom quark masses[Phys. Rev. D 99, 073001, 2019]. ... The vacuum expectation value is calculated quantum mechanically, which relates to the common mass-generating scalar operator and it reproduces the vector and quark-doublet masses. This also links the corresponding scalar-vector and Yukawa vertices, and restricts the t- and b-quark masses in a hierarchy relation."

    The hierarchy restriction on t/b masses is presented as following from the VEV calculation of the common scalar operator built from the bilinear expansions. Yet the operator and its connecting/restricting property are taken from the authors' own 2019 paper; the present derivation therefore reduces the claimed new mass relation to a re-application of the prior result rather than an independent first-principles output.

full rationale

The paper's central result—the hierarchy relation restricting t- and b-quark masses—is obtained by computing the VEV of the common mass-generating scalar operator after expanding SM fields as t/b bilinears. However, the operator itself and its mass-connecting property are introduced solely by citation to the authors' 2019 paper (Phys. Rev. D 99, 073001). The new second-quantization construction therefore does not independently derive the hierarchy; it applies the prior result to a different representation. This matches the self-citation load-bearing pattern with one load-bearing step. The bilinear expansion itself is not shown to be circular, but the mass-restriction claim is not self-contained. No other patterns (self-definitional, fitted-input prediction, etc.) are exhibited by the provided text.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the Standard Model admits an equivalent fermion-only operator basis and that the bilinear expansions preserve all gauge and Yukawa interactions without additional free parameters beyond those already present in the 2019 construction.

free parameters (1)
  • normalization constants in the bilinear expansions
    Required to match the observed vector-boson and quark masses after the vacuum expectation value is taken.
axioms (1)
  • domain assumption The Standard Model can be equivalently represented with its fields in a spin-extended basis departing from fermion degrees of freedom.
    Invoked at the opening of the abstract as the foundational premise.

pith-pipeline@v0.9.0 · 5686 in / 1218 out tokens · 33122 ms · 2026-05-23T07:17:09.924703+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    S. L. Glashow, Nucl. Phys. 22 579-588 (1961); S. Weinberg, Phys. Rev. Lett. 19 1264-1266 (1967); A. Salam, in Elementary Particle Theory , W. Svartholm (Ed.), Almquist and Wiskell, Stockholm, 367-377, 1968

    work page 1961

  2. [2]

    Besprosvany and R

    J. Besprosvany and R. Romero, Heavy quarks within the electroweak multiplet , Phys. Rev. D 99, 073001 (2019)

    work page 2019

  3. [3]

    Georgi and S

    H. Georgi and S. Glashow, Unity of All Elementary-Particle Forces , Phys. Rev. Lett. 32 438-441 (1974)

    work page 1974

  4. [4]

    Gell-Mann and Y

    M. Gell-Mann and Y. Ne’eman, The Eightfold Way , Benjamin, New York, 1964

    work page 1964

  5. [5]

    L. N. Cooper, Bound electron pairs in a degenerate Fermi gas , Phys. Rev. 104 1189–1190 (1956)

    work page 1956

  6. [6]

    Bardeen, L

    J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108 1175-1204 (1957)

    work page 1957

  7. [7]

    Nambu and G

    Y. Nambu and G. Jona-Lasinio, Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity I , Phys. Rev. 122 345-358 (1961)

    work page 1961

  8. [8]

    Tanabashi, Masaharu, K

    V.A Miransky, M. Tanabashi, Masaharu, K. Yamawaki Dynamical electroweak symmetry breaking with large anomalous dimension and t quark condensate Physics Letters B. 221, (1989). doi:10.1016/0370-2693(89)91494-9

    work page doi:10.1016/0370-2693(89)91494-9 1989

  9. [9]

    Bardeen, W. A., C. T. Hill, and M. Lindner, Phys. Rev. D 41 1647 (1990)

    work page 1990

  10. [10]

    Tanabashi, Masaharu, K

    V.A Miransky, M. Tanabashi, Masaharu, K. Yamawaki Is the t Quark Respon- sible for the Mass of W and Z Bosons? . Modern Physics Letters A. 04 (11). 1043–1053 (1989). doi:10.1142/s0217732389001210

    work page doi:10.1142/s0217732389001210 1989

  11. [11]

    D. E. Clague and G. G. Ross, Dynamical symmetry breaking by a top quark condensate in the standard model , Nucl. Phys. B 364 43-66 (1991)

    work page 1991

  12. [12]

    R. S. Chivukula, M. Golden, and E. H. Simmons, Critical constraints on chiral hierarchies, Phys. Rev. Lett. 70 1587 (1993)

    work page 1993

  13. [13]

    Julius Wess and Jonathan Bagger, Supersymmetry and Supergravity, Princeton University Press, 1992

    work page 1992

  14. [14]

    Coleman and J

    S. Coleman and J. Mandula, All Possible Symmetries of the S Matrix, Phys. Rev. 159 1251 (1967). 25

    work page 1967

  15. [15]

    Steger, E

    H. Steger, E. Flores, and Y.-P. Yao, Nonlinear Realization of Heavy Fermions and Effective Lagrangean, Phys. Rev. Lett. 59 359 (1987)

    work page 1987

  16. [16]

    Ecker, The standard model at low energies, Czech

    G. Ecker, The standard model at low energies, Czech. J. Phys. 44 405 1995

    work page 1995

  17. [17]

    Pendleton, and G

    B. Pendleton, and G. G. Ross, Phys. Lett. B 98, 291 (1981)

    work page 1981

  18. [18]

    C. T. Hill, Phys. Rev. D 24, 691 (1981)

    work page 1981

  19. [19]

    Englert and R

    F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964); P.W. Higgs, Phys. Lett. 12 132 (1964)

    work page 1964

  20. [20]

    Fr¨ ohlich, G

    J. Fr¨ ohlich, G. Morchio, and Strocchi, Higgs phenomenon without a symmetry breaking order parameter, Phys. Lett. B 97 249 (1980)

    work page 1980

  21. [21]

    Cvetic, Top-quark condensation, Rev

    G. Cvetic, Top-quark condensation, Rev. Mod. Phys. 71 (1999)

    work page 1999

  22. [22]

    Navas et al

    S. Navas et al. ( Particle Data Group ), Phys. Rev. D 110 030001 (2024)

    work page 2024

  23. [23]

    J. M. Schwindt and C. Wetterich, Asymptotically free four-fermion interactions and electroweak symmetry breaking, Phys. Rev. D 81 055005 (2010)

    work page 2010

  24. [24]

    Ellis, P.Q

    J. Ellis, P.Q. Hung, and N. Mavromatos, An Electroweak Monopole, Dirac Quan- tization and the Weak Mixing Angle , Nucl. Phys. B 969 115468 (2021)

    work page 2021

  25. [25]

    Radenkovi´ c and M

    T. Radenkovi´ c and M. Vojinovi´ c,Higher gauge theories based on 3-groups , Jour. High Ener. Phys. 2019 222 (2019)

    work page 2019

  26. [26]

    Yin, Charge quantization and neutrino mass from Planck-scale SUSY , Phys

    W. Yin, Charge quantization and neutrino mass from Planck-scale SUSY , Phys. Lett. B, 785 585-590 (2018)

    work page 2018

  27. [27]

    Di Stefano et al., Resolution of Gauge Ambiguities in Ultrastrong-Coupling Cavity QED, Nat

    O. Di Stefano et al., Resolution of Gauge Ambiguities in Ultrastrong-Coupling Cavity QED, Nat. Phys. 15, 803–808 (2019)

    work page 2019

  28. [28]

    N. S. Mankoˇ c Borˇ stnik and H. B. Nielsen,How does Clifford algebra show the way to the second quantized fermions with unified spins, charges and families, and with vector and scalar gauge fields beyond the standard model , Prog. Part. Nucl. Phys. 121 103890 (2021)

    work page 2021

  29. [29]

    Besprosvany, Int

    J. Besprosvany, Int. J. Theor. Phys. 39 (2000) 2797-2836; J. Besprosvany, Nuc. Phys. B (Proc. Suppl.) 101, 323-329 (2001); J. Besprosvany and R. Romero, in: AIP Conf. Proc. 1323, L. Bennet, P. O. Hess, J. M. Torres, K. B. Wolf (Eds.), American Institute of Physics, Melville, New York, 16-27, 2010; J. Besprosvany and R. Romero, Int. J. Mod. Phys. A 29 (201...

    work page 2000

  30. [30]

    Besprosvany, ELECTROWEAKLY INTERACTING SCALAR AND GAUGE BOSONS, AND LEPTONS, FROM FIELD EQUATIONS ON SPIN (5+1)- DIMENSIONAL SPACE Int

    J. Besprosvany, ELECTROWEAKLY INTERACTING SCALAR AND GAUGE BOSONS, AND LEPTONS, FROM FIELD EQUATIONS ON SPIN (5+1)- DIMENSIONAL SPACE Int. J. Mod. Phys. A 20 77-93 (2005)

    work page 2005

  31. [31]

    Besprosvany, Standard-model coupling constants from compositeness , Mod

    J. Besprosvany, Standard-model coupling constants from compositeness , Mod. Phys. Lett. A 18, 1877-1885 (2003)

    work page 2003

  32. [32]

    Besprosvany, Standard-model particles and interactions from field equations on spin 9+1 dimensional space , Phys

    J. Besprosvany, Standard-model particles and interactions from field equations on spin 9+1 dimensional space , Phys. Lett. B 578 181-186 (2004). Acknowledgements The authors thank Dr. Jose Wudka for discussions, and support from DGAPA-UNAM through projects IN112916, IN11720, IN112822. 27

    work page 2004