Axion framework with color-mediated Dirac neutrino masses

A. Batra; F.R. Joaquim; H.B. C\^amara; J.W.F. Valle; N. Nath; R. Srivastava

REVIEW 4 major objections 4 minor 116 references

Axion framework with color-mediated Dirac neutrino masses

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A single Peccei-Quinn symmetry is proposed to generate radiative Dirac neutrino masses, solve the strong CP problem, and supply the axion as dark matter.

desk verdict Table I's PQ charges forbid the charged-lepton Yukawa, the leading Dirac operator, and the Yη loop vertex, so the central model fails as written, despite a worthwhile framework direction and careful phenomenology. read the letter →

arxiv 2501.13156 v2 pith:UAUIB6H6 submitted 2025-01-22 hep-ph

A. Batra , H.B. Câmara , F.R. Joaquim , N. Nath , R. Srivastava , J.W.F. Valle This is my paper · ORCID

classification hep-ph

keywords Peccei-QuinnsymmetryKSVZaxionstrongCPproblemradiativeDiracneutrinomassvector-likequarksscalarleptoquarksdarkmatterflavor-violatingcouplings

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes that one Peccei-Quinn symmetry, of the KSVZ type in which exotic colored fermions carry the symmetry, can simultaneously solve the strong CP problem, generate radiative Dirac neutrino masses, and provide the axion as dark matter. The colored vector-like quarks and scalar leptoquarks that make the axion possible are the same particles that run in the one-loop diagram producing neutrino mass, so the framework attaches the neutrino-mass problem to the axion machinery rather than adding separate new physics. The paper explores all seven vector-like quark representations that can mix with ordinary quarks and decay into them. If the construction holds, each representation predicts a different axion-to-photon coupling, three of them predict sizable flavor-violating axion-quark couplings, and the surviving parameter space is being probed by current and next-generation axion experiments.

What carries the argument

The central object is the one-loop color-mediated neutrino mass formula, Eq. (9), together with the Peccei-Quinn charge assignments in Tables I and II. The formula sums over the five quark mass eigenstates (three standard quarks plus two vector-like quarks) and the two scalar mass eigenstates $\zeta_{1,2}$ that come from mixing the leptoquarks $\eta$ and $\chi$ through the $\kappa(\eta^\dagger\Phi)\chi$ term: $$(M_\nu)_{\$\alpha$\$\beta$} = \frac{N_c}{16\$pi^{2}$\sqrt{2}}\sum_{j,k=1}^{5}(\tilde Y_\eta)_{\$\alpha$ j}(\tilde Y_\chi)_{j\$\beta$}R_{1k}R_{2k}\frac{\tilde m_j}{m_{\zeta_k}^2 - \tilde $m_j^{2}$}\ln\left(\frac{\tilde $m_j^{2}$}{m_{\zeta_k}^2}\right).$$ This formula is what turns the axion's colored fermions into the origin of neutrino mass. The residual $\mathbb{Z}_3$ is the symmetry selector that allows only Dirac operators, while the anomaly factors $N$ and $E$ computed from the same charges feed the axion-to-photon coupling relation $g_{a\gamma\gamma} = \frac{\alpha_e}{2\pi f_a}\left(\frac{E}{N} - 1.92(4)\right)$, and the heavy-light mixing matrices $\Theta^q_X$ feed the flavor-violating axion couplings of Eq. (14).

What would settle it

Computing the Peccei-Quinn charge of the coupling term that connects the lepton doublet, the colored scalar, and the vector-like quark for each of the seven representations would settle the internal consistency: any nonzero charge would break the symmetry that the strong-CP solution, the Dirac nature of neutrinos, and the axion dark-matter picture all depend on. Experimentally, a haloscope search that covers the predicted $m_a \sim 1$–$200\,\mu$eV band without detecting axions in the predicted axion-to-photon coupling lines would rule out the framework's dark-matter claim.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a global $U(1)_{\rm PQ}$ under which the vector-like quarks $\Psi$ are chirally charged does double duty: it is the symmetry whose spontaneous breaking yields the axion, and it forbids Majorana neutrino mass terms so neutrinos are forced to be Dirac. The charge assignments leave a residual $\mathbb{Z}_3$ under which leptons and scalar leptoquarks rotate by different powers, which selects the dimension-five Dirac operator $(\bar\ell_L \tilde\Phi \nu_R)\sigma^*$ while blocking all Majorana operators. The one-loop diagram with $\Psi$, $\eta$, and $\chi$ then gives the neutrino mass matrix of Eq. (9), and the paper shows that the anomaly factors $N=1,2,3$ and the electromagnetic anomaly ratio $E/N$ vary across the seven viable vector-like quark representations, producing distinct, experimentally distinguishable axion-to-photon couplings. In three of the models, heavy-light quark mixing is large enough to induce flavor-violating axion-quark couplings, which are constrained by rare decays, meson mixing, and top decays. The axion can also account for the dark-matter abundance, with the post-inflationary case selecting $f_a$ in the range preferred by string-network simulations for the $N_{\rm DW}=1$ models.

Load-bearing premise

The load-bearing premise is that the Peccei-Quinn symmetry is an exact symmetry of every interaction term in the model, in particular that the term coupling the lepton doublet, the colored scalar, and the vector-like quark carries zero Peccei-Quinn charge for all seven representations; the paper states this condition but does not verify it case by case.

Editorial extensions

If this is right

The seven vector-like-quark representations give seven distinct $E/N$ values, so measuring $g_{a\gamma\gamma}$ at the level projected for next-generation haloscopes and helioscopes can discriminate which representation, if any, is realized.
In the post-inflationary dark-matter picture, only the isosinglet models with $N_{\rm DW}=1$ avoid the domain-wall problem; the isodoublet and isotriplet models require a bias term or a pre-inflationary breaking history.
With two vector-like quark families, the neutrino mass matrix has rank two, so the lightest neutrino is massless and no light-neutrino-exchange contribution to neutrinoless double beta decay is expected; a third family would make all three neutrinos massive.
Three models develop sizable heavy-light mixing, leading to flavor-violating axion-quark couplings; the combined bounds from rare meson decays, $D^0$–$\bar{D}^0$ mixing, and top-quark decays restrict the heavy bare mass parameter to the range $M_B\in[10^8,10^{10}]$ GeV.
Because the same PQ symmetry supplies an accidental baryon number, proton decay operators are forbidden, so the colored leptoquarks and vector-like quarks can be present at the PQ scale without destabilizing matter.

Reading between the lines

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

Beyond the paper, the residual-$\mathbb{Z}_3$ selection rule is a portable trick: any symmetry that leaves the same unbroken $\mathbb{Z}_3$ could play the role of the global PQ symmetry, which would also address the 'PQ quality' problem the authors explicitly flag.
Beyond the paper, the predicted $E/N$ ladder offers a sharp discrimination strategy: if a future axion search fixes $g_{a\gamma\gamma}$ while the dark-matter requirement fixes $f_a$, that single measurement would select one of the seven vector-like-quark representations and dictate the quantum numbers of the colored scalars for collider searches.
Beyond the paper, the models with sizable down-quark flavor-violating axion couplings are natural candidates for an axion-based explanation of the $B^+\to K^+ + E_{\rm miss}$ anomaly, an avenue the paper mentions but does not develop.
Beyond the paper, a numerical fit of Eq. (9) to current neutrino oscillation data has not been shown; doing so would test whether the minimal two-family realization can reproduce the measured splittings and mixing angles, or whether a third family is needed.

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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

Two things you should know. The paper explores a genuinely new direction: having the same colored vector-like fermions that solve the strong CP problem also mediate radiative Dirac neutrino masses at one loop, with a residual Z3 keeping neutrinos Dirac. The survey of the seven VLQ representations and their distinct E/N predictions is systematic, and the phenomenology is carefully presented. But the central charge assignments as printed do not survive contact with the paper's own Lagrangian. I checked Table I against Eqs. (2) and (6) by hand. The charged-lepton Yukawa ℓL Φ eR has PQ charge 1/6 + 0 + 1/6 = 1/3, so the PQ symmetry forbids it and the SM charged leptons are massless. The dimension-5 operator (ℓL Φ̃ νR)σ*, which the paper identifies as the leading Dirac neutrino mass term, has charge 1/6 + 4/6 − 1/2 = 1/3 as well, and it carries ω² under the residual Z3 that is supposed to allow it. The Yη ℓL η̃ ΨR vertex that drives the one-loop diagram in Fig. 1 has PQ charge 1/3 for every row in Table II. So the symmetry that is supposed to solve the strong CP problem and guarantee Diracness forbids the charged-lepton masses, the neutrino mass operator, and the loop vertex all at once. This is not a missing verification: it is a concrete internal inconsistency.

What the paper does well should be said too. The heavy-light mixing appendix is detailed and careful. The axion-to-photon coupling map in Fig. 2 and the flavor-violating axion-quark analysis of Fig. 3 are useful, with sensible treatment of the SN1987a and rare-decay constraints. The one-loop mass formula is standard, and the argument that small neutrino masses are controlled by a naturally small κ is sound. The citation pattern is appropriate; the paper builds transparently on the authors' earlier Majorana framework and on the existing Dirac-axion literature. The body is also honest about the string and domain-wall uncertainties in the post-inflationary DM story, which the abstract slightly oversells.

The likely fix is a corrected set of lepton PQ charges that make ℓL Φ eR, the Yη vertex, and the Dirac operator exactly neutral, with the Z3 structure rebuilt to match. That is a real change, since the anomaly coefficients E/N would shift. As printed, the model is not consistent. This paper is for model-builders working on axion-neutrino unification. It deserves a serious referee, but the referee's first request should be the corrected charge assignments and a term-by-term invariance check of every operator in Eqs. (1), (2), and (6). I would not cite the present version; I would watch for a corrected one.

Referee Report

4 major / 4 minor

Summary. This paper proposes a KSVZ-type axion framework in which vector-like quarks and two scalar leptoquarks generate radiative Dirac neutrino masses at one loop, while an anomalous U(1)_PQ symmetry (with the exotic fermions charged) solves the strong CP problem and provides axion dark matter. Seven representations of vector-like quarks are surveyed, leading to distinct predictions for the axion-to-photon coupling and, in three cases, sizable flavor-violating axion-quark couplings. The one-loop neutrino mass formula, the anomaly coefficients, and the heavy-light mixing appendix are standard and clearly presented. However, the charge assignments in Tables I and II do not render the Yukawa and effective operators invariant under the SM hypercharge and PQ symmetries, so the central model as written is not internally consistent.

Significance. If the symmetry assignments were corrected, the framework would be a notable unification of radiative Dirac neutrino masses with the QCD axion, with falsifiable predictions: distinct E/N ratios for each vector-like quark representation, a restricted fa range for dark matter, and flavor-violating axion couplings that can be probed in haloscopes, helioscopes, and flavor experiments. The paper's strengths are the systematic listing of the seven representations, the explicit one-loop formula in Eq. (9), the detailed heavy-light mixing appendix, and the honest acknowledgment of the PQ quality problem. Nevertheless, the hypercharge and PQ charge inconsistencies mean that the phenomenological predictions in Figs. 2 and 3 and the neutrino mass formula are not yet attached to a valid model; the present version cannot be accepted as it stands.

major comments (4)

[II B, Eq. (6), Tables I/II] The Yukawa vertices in Eq. (6) are not invariant under U(1)_Y for any of the listed representations. Taking \tilde η = iτ2 η* as the standard SU(2)-conjugate notation (the same convention used for \tilde Φ), the Yη vertex has hypercharge -1/2 - (yΨ + 1/2) + yΨ = -1, while the Yχ vertex has hypercharge yΨ + yΨ + 0 = 2yΨ, which is non-zero for every yΨ appearing in Table II. Hence the one-loop diagram in Fig. 1 is not a gauge-invariant amplitude under the stated field content. The authors must correct the hypercharges of η and χ (or the definition of \tilde η) and re-derive the allowed couplings.
[II, Table I and the first bullet] The charged-lepton Yukawa ℓL Φ eR carries PQ charge 1/6 + 0 + 1/6 = 1/3 under the charges printed in Table I, so it is forbidden by the exact PQ symmetry and the model cannot generate charged-lepton masses. The claimed leading Dirac operator (ℓL \tilde Φ νR) σ* of Eq. (2) has PQ charge 1/6 + 4/6 - 1/2 = 1/3 and is likewise forbidden. A complete, term-by-term PQ charge-conservation check should be provided for the entire Lagrangian, including the SM Yukawa sector.
[II B, Eq. (6)] The PQ charge sum of the Yη ℓL \tilde η ΨR term is 1/3 for \tilde η = iτ2 η* (independent of QPQ), so this vertex is never PQ-invariant under Table I. If instead \tilde η = η, the PQ charge is 2 QPQ - 1, which vanishes only for QPQ = 1/2, but then the Yχ ΨL χ νR and YΨ ΨL ΨR σ terms have PQ charge 2 QPQ = 1 and are forbidden. Thus, for every representation in Table II, at least one term of Eq. (6) explicitly breaks U(1)_PQ, invalidating the claimed Diracness of neutrinos and the strong-CP solution.
[II, first bullet] The residual Z3 symmetry is not realized by the stated charges. For a Z3 to survive the σ VEV, the PQ charge of σ must be a multiple of 3 in a normalization where the Z3 phase is 2π/3; with qσ = 1/2, σ transforms nontrivially under every phase, so no such discrete subgroup remains. Moreover, the statement that ℓL, eR, and νR all transform as ω under a residual Z3 is incompatible with the gauge-invariant charged-lepton Yukawa, which requires qℓ + qe = 0. The authors should specify the discrete charges explicitly and verify that they forbid every Majorana operator in Eq. (1) while allowing the Dirac operator in Eq. (2).

minor comments (4)

[III B, near Eq. (15)] The text refers to 'Ψ ∼ (3, 1, 1/6)' as one of the two models with sizable up-quark flavor-violating couplings; this should read 'Ψ ∼ (3, 2, 1/6)', the isodoublet model with sizable Θu_L in Table III.
[Eq. (6) and Table I] The symbol \tilde η is never defined; since the hypercharge and PQ invariance checks depend on whether it denotes η or iτ2 η*, please define it explicitly in the text.
[Table I caption] The PQ charges of the SM quark fields are not listed, although the text later assumes they vanish; please state this assumption in the table caption or in the main text.
[Eq. (14)] The index structure in Eq. (14) is unclear: the last equality mixes qαX and qβX without specifying the Hermitian contraction, and the relation (cqV)αβ = (cqA)αβ = cq_αβ should be stated with explicit indices and a definition of the flavor matrices involved.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Dirac neutrino loop formula, E/N anomaly ratios, and flavor-violating axion couplings are computed from the stated field content and symmetries rather than fitted to the quantities they predict. Self-citations to the prior color-mediated framework are not load-bearing for the Dirac-specific derivation.

full rationale

The paper's central derivation chain is not circular. The one-loop Dirac neutrino mass formula (Eq. 9) follows from the Yukawa interactions in Eq. (6), the scalar-potential term in Eq. (7), and the eta-chi mixing in Eq. (A3); Eq. (10) is an order-of-magnitude benchmark, not a fit of neutrino data used to define the model. The axion-to-photon coupling predictions (Eqs. 12 and 13, Fig. 2) are anomaly coefficients computed from the Table I PQ charges and are not extracted from haloscope or helioscope data. The flavor-violating axion-quark couplings in Eq. (14) are derived from the heavy-light mixing matrices computed in Appendix B, with external bounds from Refs. [98, 99] applied afterward. The statement that the PQ symmetry ensures Diracness is a model-building assignment, not a hidden prediction: the lepton PQ charges were chosen to leave a residual Z3 that forbids Majorana operators, and the testable content lies in the E/N values and flavor couplings. The self-citations to Refs. [49] and [115] supply the earlier color-mediated framework and flavor-anomaly applications, but the Dirac loop calculation and axion predictions are presented self-contained and are not reductions to those citations; hence they do not raise the circularity score. The paper also explicitly acknowledges the PQ quality problem and the unresolved topological-defect contribution to axion DM, which are limitations rather than circular steps. Separately, and as a correctness matter rather than a circularity: using the printed Table I charges, the charged-lepton Yukawa ell_L Phi e_R has PQ charge 1/6+0+1/6 = 1/3, and the claimed leading Dirac operator (ell_L Phi-tilde nu_R) sigma* has PQ charge 1/6+4/6-1/2 = 1/3; both are thus forbidden by the stated U(1)_PQ. The authors should supply a corrected charge table or an explicit term-by-term invariance check for Eqs. (2) and (6). This is a load-bearing consistency issue, but it is not an input-output circularity.

Assumptions & free parameters 7 free parameters · 6 assumptions · 4 invented entities

The central claim rests on a small number of hand-chosen input parameters: the PQ charge assignments, the number of VLQ species, Yukawa and scalar couplings, the bare heavy-light mass, and the axion scale fixed by dark matter cosmology. The model also assumes an exact enough global PQ symmetry, the standard one-loop formula, the seesaw approximation for heavy-light mixing, and unresolved numerical results for axion strings. The axion is the only invented entity with a directly testable independent handle.

free parameters (7)

Number of VLQ species NΨ = 2
Chosen as the minimal number to generate the two observed neutrino mass splittings, predicting one massless neutrino. Not fitted to data in the paper.
PQ charge assignment QPQ per VLQ representation = 0 or 1/2 depending on model
Chosen by hand in Table II to allow the required decay and mixing terms while forbidding Majorana operators. Ad hoc to the model.
Yukawa matrices Yη, Yχ, YΨ = benchmark ~10^-2 in Eq. (10)
Free couplings that set the neutrino mass scale through the loop formula; no fit to oscillation data is shown.
Scalar cubic coupling κ = 10^2 GeV benchmark in Eq. (10)
Controls η-χ mixing and the loop-induced neutrino mass; argued naturally small by t Hooft, but its value is an input.
Bare heavy-light mass MB = benchmarks 10^6 to 10^11 GeV, constrained to 10^8 to 10^10 GeV
Arbitrary bare mass controlling the ΘX mixing matrices and the flavor-violating axion couplings in Fig. 3.
Axion decay constant fa = around 5 x 10^11 GeV for theta0 of order one; post-inflationary band 5 x 10^9 to 3 x 10^11 GeV
Set by the dark matter abundance through Eq. (11) and by string simulations, not derived from the model.
Initial misalignment angle theta0 = free in pre-inflationary case; averaged to sqrt(2.152) in post-inflationary case
Enter the axion relic abundance estimate; the post-inflationary value is a statistical input from the literature.

assumptions (6)

standard math The one-loop neutrino mass formula of Eq. (9), taken from Ref. [30], applies to this color-mediated Dirac setup with the R matrix of Eq. (A3).
Used to compute Mν; assumes no other contributions and a color factor Nc=3.
domain assumption The PQ charge assignments in Table I leave a residual Z3 symmetry that forbids all Majorana operators of Eq. (1) while allowing the Dirac operator of Eq. (2).
This is the basis for Diracness and for the one-loop realization; the text states it but does not verify every operator charge sum.
domain assumption Heavy-light quark mixing is in the seesaw regime Md much less than MΨd and MΨ, so the block diagonalization in Appendix B is valid.
Used to define the ΘX matrices that control flavor-violating axion couplings.
standard math The QCD axion mass and axion-photon coupling relations from Ref. [79] are model independent at NLO.
Eqs. (4) and (12) are the basis for the ma and gaγγ predictions.
domain assumption Post-inflationary axion string simulations restrict fa to the range 5 x 10^9 to 3 x 10^11 GeV for NDW=1.
Used to identify the dark matter allowed band; the string and domain-wall contribution itself is acknowledged as unresolved.
ad hoc to paper A global U(1)PQ symmetry survives Planck-scale corrections, the so-called PQ quality problem.
The paper acknowledges the problem in Sec. II and defers a solution to future work, so the mechanism assumes acceptable PQ quality.

invented entities (4)

Right-handed neutrinos νR
purpose: Make neutrinos Dirac and enter the one-loop neutrino mass diagram.
No directly observed new interactions or mass signatures are provided beyond the model construction.
Vector-like quarks Ψ
purpose: Carry the QCD anomaly for the KSVZ axion and mediate the radiative Dirac neutrino mass.
No collider or direct search signature is quantified for the VLQ states; their scale is tied to fa.
Scalar leptoquarks η and χ
purpose: Run in the loop that generates Dirac neutrino masses and mix through the κ term.
They are constrained only indirectly through flavor and axion processes; no dedicated search prediction is given.
PQ scalar σ and its axion independent evidence
purpose: Spontaneously break U(1)PQ, solve the strong CP problem, and provide the dark matter candidate.
The axion has falsifiable couplings gaγγ and flavor-violating quark-axion couplings that haloscopes, helioscopes, and rare decay searches can probe.

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Cite this review

Pith. "Pith review of Axion framework with color-mediated Dirac neutrino masses." pith.science (2026). https://pith.science/paper/UAUIB6H6

@misc{pith2026250113156,
  author       = {Pith},
  title        = {Pith review of: Axion framework with color-mediated Dirac neutrino masses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UAUIB6H6}},
  note         = {Machine review of arXiv:2501.13156}
}

read the original abstract

We propose a KSVZ-type axion framework in which vector-like quarks (VLQ) and colored scalars generate Dirac neutrino masses radiatively. The global Peccei-Quinn symmetry (under which the exotic fermions are charged) addresses the strong CP problem and ensures the Dirac nature of neutrinos. The axion also accounts for the observed cosmological dark matter. We systematically explore all viable VLQ representations. Depending on the specific scenario, the framework predicts distinct axion-to-photon couplings, testable through haloscope and helioscope experiments, as well as potentially significant flavor-violating quark-axion interactions.

Figures

Figures reproduced from arXiv: 2501.13156 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

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Pith tools

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