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REVIEW 2 major objections 4 minor 58 references

Dirac neutrinos in a $SU(2)$ left-right symmetric model

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proposes a left-right symmetric model with three scalar bidoublets in which neutrinos remain purely Dirac and neutrino masses arise from a tiny vacuum expectation value rather than tiny Yukawa couplings, removing a fine-tuning.

desk verdict A solid two-bidoublet Dirac-neutrino LR model undercuts its own fine-tuning-free pitch for three bidoublets. read the letter →

arxiv 1908.02828 v2 pith:F43AMNEF submitted 2019-08-07 hep-ph hep-ex

Henry Diaz Chavez , V. Pleitez , O. P. Ravinez This is my paper · ORCID
classification hep-phhep-ex PACS 12.60.Fr12.15.-y14.60.Pq
keywords left-rightsymmetryDiracneutrinosscalarbidoubletsneutrinomasshierarchyYukawafine-tuningvacuumalignmentinertdoubletright-handedgaugebosons
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

Working in a left-right symmetric gauge theory with only scalar bidoublets and two scalar doublets, the paper tries to show that neutrinos can be purely Dirac particles at all orders in perturbation theory. With two bidoublets, the model reproduces the observed tiny neutrino masses only if the neutrino Yukawa couplings are about ten orders of magnitude smaller than the charged-lepton Yukawas, which is a fine-tuning. The central claim is that a third bidoublet removes that fine-tuning: the neutrino mass scale is then set by a very small vacuum expectation value, $k_\nu$, while the charged-lepton and quark masses come from GeV-scale VEVs, so all Yukawa couplings can be of order one. If this vacuum alignment exists, Dirac neutrinos become a natural option in a well-motivated extension of the Standard Model without scalar triplets, extra fermions, or lepton-number violation.

What carries the argument

The load-bearing object is the scalar sector: $n$ bidoublets plus two doublets, with discrete symmetries assigning one bidoublet to leptons, another to quarks, and protecting one doublet as inert. The argument works through the vacuum expectation value hierarchy: in the two-bidoublet case a single 2 GeV VEV sets both neutrino and charged-lepton masses, so the ratio of neutrino to charged-lepton masses must be absorbed into a $10^{-11}$ Yukawa suppression; in the three-bidoublet case a tiny neutrino VEV $k_\nu$ carries the neutrino mass and a GeV-scale $k'_l$ carries charged-lepton masses, converting the suppression from Yukawa couplings into a VEV hierarchy. The paper illustrates how such a hierarchy can arise using a non-Hermitian quadratic term in a two-doublet example, but states that the complete three-bidoublet potential needs separate study.

What would settle it

Minimize the full three-bidoublet, two-doublet scalar potential and scan its parameter space: if every point with $k_\nu$ near $10^{-11}$ GeV and the charged-lepton and quark VEVs near GeV requires cancellations among the $\mu^2$ parameters at the level of one part in $10^{10}$ or worse, the central no-fine-tuning claim is refuted.

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Extended reading notes

Core claim

The paper constructs an $SU(2)_L \times SU(2)_R \times U(1)_{B-L}$ model, with parity identified with left-right exchange, in which the scalar sector contains bidoublets plus two doublets and no triplets. The parity and discrete symmetries ensure lepton number is conserved, so neutrinos acquire only Dirac masses; with two bidoublets the neutrino mass matrix is $M_\nu = G k_1/\sqrt{2}$ with $k_1$ around 2 GeV, forcing the neutrino Yukawa matrix entries to be of order $10^{-11}$, exactly the kind of tuning the construction aims to avoid. The paper's central discovery is that a third bidoublet, together with a discrete $D$ symmetry forbidding the tilde-conjugate Yukawa couplings, lets the neutrino mass come from a separate tiny VEV $k_\nu$ while the charged leptons get their mass from a GeV-scale VEV $k'_l$; then the neutrino and charged-lepton Yukawa matrices can both have entries of order one. The authors are explicit that this relies on a hierarchical vacuum alignment whose full scalar-potential analysis is not carried out in the paper.

Load-bearing premise

The full scalar potential with three bidoublets and two doublets can produce the hierarchy $k_\nu$ of order $10^{-11}$ GeV and GeV-scale $k'_l$, $k_q$ without fine-tuning among the quadratic mass parameters.

Editorial extensions

If this is right

  • Neutrinos stay strictly Dirac at every order, so there is no neutrinoless double beta decay and no lepton-number violation.
  • A third bidoublet lets the neutrino, charged-lepton, and quark Yukawa matrices all be order one, eliminating the $10^{-11}$ hierarchy that the two-bidoublet version shares with the Standard Model.
  • The right-handed $W_2$ and $Z_2$ bosons must be heavier than about 7.2 TeV and 9.3 TeV respectively, with $W_L$-$W_R$ mixing below $10^{-4}$, so the model is testable at high-energy colliders.
  • The doublet $\chi_L$ can be an inert scalar whose stability is protected by left-right symmetry, providing a dark-matter candidate.
  • Because neutrinos are Dirac, the same-sign dilepton and trilepton signals expected from Majorana seesaw processes are absent or suppressed by the tiny neutrino masses, giving a collider distinction between the two neutrino natures.

Reading between the lines

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

  • The obvious next step the paper does not take is to minimize the full three-bidoublet, two-doublet scalar potential; if $k_\nu$ near $10^{-11}$ GeV can only be obtained through cancellations among the quadratic parameters, the fine-tuning has moved from the Yukawa sector to the scalar potential.
  • The VEV-hierarchy trick is generic: the same transfer from a small Yukawa to a small VEV could be applied to other Dirac-neutrino models, where a dedicated scalar analysis would decide whether the trick is natural.
  • The model's relative phase between left- and right-handed charged currents gives a quark electric dipole moment close to the current neutron EDM bound, so improved EDM measurements could constrain or falsify the phase structure.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper constructs a left-right symmetric model based on SU(2)_L x SU(2)_R x U(1)_{B-L} with a scalar sector of two doublets and either two or three bidoublets, with the goal of keeping neutrinos purely Dirac to all orders in perturbation theory. The two-bidoublet version is analyzed in detail: the scalar potential and minimization conditions are given, the charged and neutral gauge-boson mass matrices are solved, and the neutrino and charged-lepton Yukawa matrices are explicitly constructed from the measured masses and PMNS matrix. From the exact MW/MZ ratio the authors derive vR > 24 TeV and consequently MW2 > 7.2 TeV and MZ2 > 9.28 TeV. The central, headline claim is that adding a third bidoublet, with a hierarchy of VEVs k_nu, k'_nu, k_l << k'_l << k_q, k'_q, removes the fine-tuning in the neutrino Yukawa couplings, allowing O(1) couplings. The three-bidoublet mechanism is described qualitatively in Sec. VII.B; the full scalar potential for that case is not written down or minimized.

Significance. If the central claim could be established, this would be a useful contribution: the model keeps neutrinos Dirac to all orders, provides an explicit two-bidoublet benchmark with Yukawa matrices built from data, and gives a clean vR bound from the measured MW/MZ ratio with concrete W2 and Z2 mass and width predictions. The construction of the Yukawa couplings in Eqs. (37)-(38) is explicit and is not a disguised prediction, since it uses measured masses and mixing as input. The gauge-boson analysis in Sec. IV is internally consistent and appears sound. However, the advertised novelty, namely that the three-bidoublet version avoids the two-bidoublet fine-tuning, is not demonstrated in the manuscript; the relevant scalar potential and vacuum analysis are explicitly left for future work. The significance of the paper therefore depends on a missing load-bearing analysis.

major comments (2)
  1. [Abstract and Sec. VII.B] The abstract's claim that adding a third bidoublet removes the fine-tuning of the neutrino masses is not supported by the analysis in Sec. VII.B. The section defines the three bidoublets and the D symmetry in Eq. (64), states the desired hierarchy k_nu, k'_nu, k_l << k'_l << k_q, k'_q, and then concedes that the full scalar potential with three bidoublets and two doublets 'needs a separately study.' No potential, no minimization conditions, and no existence proof for such a vacuum are given. Since this is the main novel result advertised in the abstract, this missing analysis is load-bearing rather than a presentation issue.
  2. [Sec. VII.B, Eqs. (66)-(68)] The illustrative two-doublet mechanism does not remove the fine-tuning; it relocates it to the scalar potential. For O(1) neutrino and tau Yukawa couplings, Eq. (66) requires k_nu/k'_l ~ m_nu/m_tau ~ 10^-10. Equation (68) gives u ~ -mu^2_12 v / [mu^2_2 + (lambda_3 + lambda_4) v^2], so the hierarchy is obtained only if |mu^2_12|/mu^2_2 ~ 10^-10. The D symmetry in Eq. (64) does not protect this ratio, because Phi_nu and Phi_l carry the same charge under D and the quadratic term mu^2_{nu l} Tr(Phi_nu^dagger Phi_l) is invariant. The paper identifies no symmetry that makes the 10^-10 scalar mass-parameter ratio natural. Unless a full three-bidoublet potential with a protected small parameter is provided, the central claim in the abstract is not established.
minor comments (4)
  1. [Eq. (59)] The entry s^2_23 = 0.0.512 contains a typographical double decimal and should presumably read 0.512.
  2. [Sec. VII.B and Sec. X] The phrases 'needs a separately study' and 'these is what we have done' should be corrected in a revised version.
  3. [Sec. IV and Sec. IX] The lower bound vR > 24 TeV in Sec. IV uses gL = gR at all scales, but Sec. IX argues that gL and gR differ below the parity-breaking scale; the authors should state explicitly whether the bound is robust to this running effect, or restrict the claim to the scale where parity is exact.
  4. [Sec. VIII] The estimate for the electron EDM and the phase sin phi_l < 10^-2 would benefit from a clearer statement of which experimental bound is being used and whether the WL-WR mixing contribution is included at the same order as the quoted limit.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the Yukawa couplings are reconstructed from measured masses, and the three-bidoublet claim is an acknowledged unproven scalar-potential hierarchy, not a self-referential reduction.

full rationale

No circular step meets the required standard. In Sec. V, Eqs. (37)-(38), the lepton Yukawa matrices are explicitly constructed from the measured charged-lepton and neutrino masses and the PMNS matrix, and the numerical entries in Eqs. (60)-(63) are this reconstruction, not a prediction; therefore no fitted input is renamed as a prediction. The Dirac character of neutrinos is a consequence of the field content without scalar triplets and of lepton-number conservation, not an input. The lower bound vR > 24 TeV comes from the measured MW/MZ ratio and is a constraint, not a prediction. The one place where the abstract's main claim could have become circular is Sec. VII.B: the required VEV hierarchy k_nu, k'_nu, k_l << k'_l << k_q, k'_q is asserted only as 'it may be possible', and the paper explicitly concedes that 'the full scalar potential with three bidoublets and two doublets is rather complicated and needs a separately study.' No minimization of that potential is performed, so the absence of fine-tuning is not demonstrated. That is a derivation gap or an overstated conclusion, not a circular reduction. The small-VEV illustration is attributed to Ma [45], an independent external result, and the only self-citations are background or motivation (e.g., refs. [2,3,30,50]); none carries the argument. Accordingly, the manuscript does not exhibit self-definitional, fitted-input-as-prediction, or self-citation load-bearing circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 2 invented entities

The model's freedom is concentrated in the scalar sector: dozens of VEVs and couplings are introduced and mostly unconstrained. The fine-tuning claim for the three-bidoublet case rests on an assumed VEV hierarchy with no derived potential, which is the main unredeemed assumption.

free parameters (4)
  • VEV k1 (lepton bidoublet) = It is set to 2 GeV to fit the charged lepton masses.
    Chosen so the F Yukawa matrix reproduces charged lepton masses; not derived from the scalar potential.
  • VEVs k2, k'_1, k'_2, vL, vR = They are imposed as hierarchical: k2 ~ 174 GeV, vL = 0 (inert), vR > 24 TeV.
    The VEV hierarchy is imposed at the outset; vR is bounded by MW/MZ but not dynamically determined.
  • Scalar potential couplings (mu^2_ii, lambda_ii, etc.) = They are left unconstrained in the paper.
    Many quartic and quadratic couplings are written down but never fit; the naturalness of the VEV hierarchy depends on them.
  • Neutrino Yukawa matrix G = Entries are roughly 0.1 to 2 times 10^-11 in the two-bidoublet case.
    Constructed from input neutrino masses and PMNS angles (Eqs. (59)-(63)); not a prediction.
assumptions (4)
  • domain assumption The gauge group is SU(2)_L x SU(2)_R x U(1)_{B-L} with parity P forcing gL = gR at every scale.
    Sec. II states this equality is an approximation, since renormalization-group running would split the couplings.
  • domain assumption Only renormalizable operators are kept, with no scalar triplets, so no lepton-number-violating terms exist.
    This is the defining choice that keeps neutrinos Dirac; it is a strong assumption about the UV completion, noted in Sec. V and elsewhere.
  • ad hoc to paper Discrete symmetries Z2 x Z2', Z5, and D are imposed to shape the scalar potential and Yukawa couplings.
    These symmetries are introduced in Secs. II, III, and VII.B to achieve the desired couplings; they are not derived from the gauge structure.
  • ad hoc to paper For three bidoublets, a vacuum hierarchy k_nu << k_l can be realized by a soft term as in E. Ma's two-doublet example.
    Sec. VII.B invokes E. Ma's example (Eqs. (67)-(68)) instead of analyzing the actual three-bidoublet potential, so the hierarchy is assumed for the fine-tuning claim.
invented entities (2)
  • Second and third scalar bidoublets (Phi_2 and Phi_nu, Phi_l)
    purpose: Generate quark and lepton masses with separate VEVs so neutrinos can get O(1) Yukawa couplings without a tiny charged-lepton VEV.
    New scalars are added beyond the minimal LR model; no mass spectrum, decay modes, or collider signatures are computed.
  • Inert doublet chi_L
    purpose: Dark matter candidate, protected from fermion couplings by the left-right symmetry when its VEV is zero.
    Sec. III proposes it as a dark matter candidate based on its inert character, but no relic density or direct detection analysis is given.

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Cite this review

Pith. "Pith review of Dirac neutrinos in a $SU(2)$ left-right symmetric model." pith.science (2026). https://pith.science/paper/F43AMNEF

@misc{pith2026190802828,
  author       = {Pith},
  title        = {Pith review of: Dirac neutrinos in a $SU(2)$ left-right symmetric model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F43AMNEF}},
  note         = {Machine review of arXiv:1908.02828}
}
read the original abstract

In a left-right symmetric model, with the scalar sector consisting of several bi-doublets and two doublets, neutrinos remain as Dirac fermions in all order in perturbation theory. Although with only two bi-doublets the neutrino masses need still a fine tuning, this is not the case when a third bi-doublet is added. One of the scalar doublet may be of the inert type since it is protected by the left-right symmetry.

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Reviewed August 14, 2026 · model on record in the stance chip above.