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arxiv: 2607.06102 · v1 · pith:SIJAA4KP · submitted 2026-07-07 · hep-ph

Radiative Breaking of Two-Zero Neutrino Mass Minors: Revisiting the U(1)_(L_μ-L_τ) Model

Masahiro Ibe , Jun Miyamoto , Satoshi Shirai This is my paper

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classification hep-ph
keywords two-zerocorrectionsmathrmminormodelmu-lneutrinomass
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The pith

Loop corrections cut neutrino mass bound in half in U(1) flavor model

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

The paper argues that one-loop quantum corrections in the minimal U(1)_{L_mu-L_tau} neutrino mass model violate a tree-level symmetry condition called the two-zero minor structure, and that this violation is large enough to lower the model's predicted minimum total neutrino mass from roughly 0.25 eV to about 0.12 eV. At tree level, the U(1)_{L_mu-L_tau} gauge symmetry forces two specific entries of the inverse neutrino mass matrix to vanish, producing rigid correlations among neutrino masses, mixing angles, and CP phases that favor a quasi-degenerate and therefore relatively heavy neutrino spectrum. The authors compute universal one-loop threshold corrections from the type-I seesaw sector—box diagrams involving Standard Model gauge bosons and the Higgs, plus self-energy corrections—and show that logarithms of the heavy right-handed neutrino masses generate flavor-dependent contributions to the Weinberg operator that break the two-zero minor condition. When the right-handed neutrino spectrum is hierarchical (lightest mass around 10^4 GeV), these corrections produce deviation parameters of order 0.1, which is precisely the size needed to shift the allowed region in neutrino parameter space toward smaller total masses. The authors fit the one-loop-corrected model to current neutrino oscillation data and find significantly reduced chi-squared values at mass scales that were excluded at tree level, weakening the tension with cosmological upper bounds on the neutrino mass sum.

Core claim

The central mechanism is that one-loop threshold corrections to the Weinberg operator contain logarithms of the individual right-handed neutrino masses, ln(M^2 / M_hat_I^2), which do not factorize as a flavor-universal term when the right-handed neutrino spectrum is hierarchical. These non-universal logarithmic terms generate complex deviation parameters epsilon_mu_mu and epsilon_tau_tau of order 0.1 that populate the (mu,mu) and (tau,tau) entries of the inverse neutrino mass matrix—entries that vanish exactly at tree level due to the U(1)_{L_mu-L_tau} symmetry. The size of the deviation scales with the hierarchy among right-handed neutrino masses: a stronger hierarchy produces larger non-un

What carries the argument

The load-bearing object is the one-loop matching condition for the Weinberg operator coefficient (Eq. 3.32), which combines gauge-boson box diagrams, Higgs box diagrams, lepton self-energy corrections, and charged-lepton Yukawa vertex corrections. The key parameters are the deviation parameters epsilon_mu_mu and epsilon_tau_tau (Eqs. 2.19-2.20), which measure the population of the otherwise-vanishing inverse mass matrix entries. The right-handed neutrino mass hierarchy controls the size of these deviations through the non-universal logarithms.

If this is right

  • If the one-loop corrections are this large, higher-loop corrections or two-loop threshold effects could further modify the two-zero minor structure, potentially shifting the neutrino mass predictions by additional amounts.
  • The same mechanism—non-universal logarithms breaking flavor-symmetric texture conditions—should operate in other U(1)_{L_alpha - L_beta} variants and in any seesaw model where a flavor symmetry enforces texture or minor zeros at tree level.
  • Cosmological surveys that tighten the neutrino mass bound below ~0.12 eV would directly test whether the radiatively relaxed region of parameter space survives, providing a concrete observational target.
  • Neutrinoless double-beta decay experiments could probe the benchmark points directly, since the loop-corrected model predicts effective Majorana masses in the 24-94 meV range depending on the right-handed neutrino spectrum.
  • The deviation parameters epsilon_mu_mu and epsilon_tau_tau, if measured or constrained independently, could in principle be used to infer the hierarchy among right-handed neutrino masses, providing a low-energy window into high-scale seesaw parameters.

Where Pith is reading between the lines

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

  • The finding that loop corrections of order 0.1 arise from hierarchical right-handed neutrino spectra suggests that any tree-level texture-zero or minor-zero prediction in seesaw models is potentially unreliable unless the right-handed neutrino spectrum is nearly degenerate or the Yukawa couplings are very small.
  • If future cosmological data push the neutrino mass bound below ~0.10 eV, the model would face renewed tension even with loop corrections, potentially requiring additional new physics or the inclusion of the omitted U(1) gauge-boson and scalar-sector loop contributions to further relax the bound.
  • The observation that the deviation direction in parameter space depends on the complex phases of epsilon_mu_mu and epsilon_tau_tau implies that CP-violating observables at low energy could carry information about the CP structure of the right-handed neutrino sector, beyond what the tree-level model predicts.

Load-bearing premise

The paper assumes that loop corrections from the U(1)_{L_mu-L_tau} gauge boson and the symmetry-breaking scalar sector are small enough to neglect, computing only the universal Standard Model gauge and Higgs contributions. If those model-dependent corrections are comparable in size, the quantitative results—particularly the exact location of the relaxed mass bound—could shift.

What would settle it

A cosmological measurement constraining the total neutrino mass below approximately 0.12 eV (the lower bound achieved with loop corrections for the most hierarchical right-handed neutrino spectrum considered) would squeeze the viable parameter space and test whether the radiatively relaxed region survives. Additionally, if the omitted U(1) gauge-boson and scalar-sector loop corrections were shown to be of the same order as the universal corrections, the quantitative predictions would need revision.

Figures

Figures reproduced from arXiv: 2607.06102 by Jun Miyamoto, Masahiro Ibe, Satoshi Shirai.

Figure 1
Figure 1. Figure 1: Allowed region in the (r31, r21) plane obtained from the chi-square data for the neutrino mixing parameters and the Dirac CP phase in the NuFIT 6.1 analysis (IC24 with SK-atm), assuming normal neutrino mass ordering. The contours correspond to ∆χ 2 mix = 1, 4, and 9, respectively, where ∆χ 2 mix is defined in Eq. (A.2). The observed values of ∆m2 21 and ∆m2 31 are not used in drawing the contours. The red … view at source ↗
Figure 2
Figure 2. Figure 2: Allowed regions in the (r (ϵ) 31 , r (ϵ) 21 ) plane for fixed values of (ϵµµ, ϵττ ), shown in the plot. For clarity, only the contour corresponding to ∆χ 2 mix = 3 is shown. Since we use the data for the normal ordering, the consistent region corresponds to r31 > r21 > 1. The case (ϵµµ, ϵττ ) = (0, 0) corresponds to the exact two-zero minor structure and reproduces the result in [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 3
Figure 3. Figure 3: The SU(2)L box diagram contributions to the Weinberg operator for neutrino masses. There are also analogous U(1)Y box diagram contributions. In our analysis, we match the Wilson coefficient obtained at one-loop order in the UV theory onto the corresponding Wilson coefficient in the effective field theory (EFT), where the right-handed neutrinos have been integrated out. Therefore, radiative corrections that… view at source ↗
Figure 4
Figure 4. Figure 4: Higgs boson contribution to the Weinberg operator for neutrino masses. [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The self-energy contributions. Li E¯ j H H Nˆ¯ I Lk [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The correction to charged-lepton Yukawa couplings. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The effective chi-square, ∆χ 2 eff, as a function of the total neutrino mass P i mi . The colored curves show the one-loop results for different lower bounds on Mˆ 1, while the black solid curve shows the tree-level result. as a perturbativity condition. For the matching scale, we choose the geometric mean of the right￾handed neutrino masses, µM = (Mˆ 1Mˆ 2Mˆ 3) 1/3 . (4.2) The Wilson coefficient determine… view at source ↗
Figure 8
Figure 8. Figure 8: Dependence of ∆χ 2 eff on the common matching scale µM. The analysis is repeated for µM = Mˆ 1, Mˆ 2, Mˆ 3 and µM = (Mˆ 1Mˆ 2Mˆ 3) 1/3 . The colors denote the lower bound imposed on the lightest right-handed neutrino mass, as in [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
read the original abstract

The two-zero minor structure predicted by flavor symmetries is usually discussed as a tree-level relation among low-energy neutrino parameters. We point out that this relation can be significantly modified by radiative corrections even when the two-zero minor structure is enforced by an underlying symmetry at tree level. As a concrete example, we analyze the minimal $\mathrm{U}(1)_{L_\mu-L_\tau}$ model and compute the universal one-loop threshold corrections associated with the type-I seesaw sector. These corrections generate flavor-dependent contributions to the Weinberg operator and violate the exact two-zero minor conditions once the $\mathrm{U}(1)_{L_\mu-L_\tau}$ symmetry is spontaneously broken. This effect relaxes the tree-level lower bound on the total neutrino mass and thereby weakens the tension between the model and cosmological neutrino-mass constraints.

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

3 major / 6 minor

Summary. This paper studies one-loop radiative corrections to the two-zero minor structure of the neutrino mass matrix in the minimal U(1)_{L_mu-L_tau} model. At tree level, the U(1)_{L_mu-L_tau} symmetry enforces vanishing (mu,mu) and (tau,tau) entries in the inverse neutrino mass matrix, leading to strong predictions that are in tension with cosmological neutrino mass bounds. The authors compute the 'universal' one-loop threshold corrections from SM gauge and Higgs interactions with right-handed neutrinos in the electroweak symmetric phase, perform UV-EFT matching with IR divergence cancellation, and show that the resulting flavor-dependent corrections to the Weinberg operator (Eq. 3.32) break the two-zero minor structure at O(0.1) for hierarchical right-handed neutrino spectra. A numerical fit using NuFIT 6.1 data demonstrates that these corrections relax the tree-level lower bound on the total neutrino mass from approximately 0.25 eV to approximately 0.12 eV, weakening (though not fully eliminating) tension with cosmological constraints.

Significance. The paper addresses a timely and well-motivated question: whether radiative corrections can relax the well-known tension between the minimal U(1)_{L_mu-L_tau} model's two-zero minor prediction and cosmological neutrino mass bounds. The one-loop calculation in Section 3 is performed cleanly in the electroweak symmetric phase, correctly identifies that U(1)_{L_mu-L_tau}-preserving counterterms suffice for UV cancellation, and the reduced matching condition (Eq. 3.32) is cross-checked against the general results of Ref. [38] and the delta M_L of Refs. [16,17]. The benchmark points in Tables 2-3 provide concrete, falsifiable predictions for neutrinoless double-beta decay and beta-decay mass measurements. The scale-dependence check in Appendix C adds robustness to the qualitative conclusion. The main quantitative claim—that epsilon ~ O(0.1) corrections relax the mass bound—is supported by explicit numerical scans.

major comments (3)
  1. [Section 3.2] The paper's central quantitative result rests on the assumption that model-dependent radiative corrections from the U(1)_{L_mu-L_tau} gauge boson and the U(1)-breaking scalar sector are negligible compared to the 'universal' SM gauge/Higgs corrections retained in Eq. (3.32). The authors state: 'We therefore do not include them in the following analysis, assuming that the relevant couplings in these sectors are sufficiently small.' This is the load-bearing assumption of the paper. The concern is concrete: the U(1) gauge boson couples to mu and tau leptons with charge +/-1, and loop diagrams involving the U(1) gauge boson and right-handed neutrinos would generate corrections to the Weinberg operator of the same parametric form as the SM gauge box diagrams (Eqs. 3.10-3.12), but with coupling g_{L_mu-L_tau} instead of g. If g_{L_mu-L_tau} is comparable to g, these corrections would be of the
  2. [Table 2, BP4 benchmark] The benchmark points reveal that the largest deviations (epsilon_mu_mu ~ 0.15-0.19 with imaginary parts up to -0.22) occur at the lowest right-handed neutrino masses (M_1 ~ 10^4 GeV) with very small kappa_mu (~10^{-6} to 10^{-4}). At these points, the hierarchy among right-handed neutrino masses is extreme (e.g., BP4: M_ee ~ 10^{14} GeV vs. M_{e mu} ~ 10^8 GeV). The common matching scale treatment (Appendix C) introduces residual scale dependence for such hierarchical spectra, which the authors acknowledge. While Appendix C shows the qualitative conclusion is stable, the quantitative precision of the benchmark predictions (e.g., the claim that sum m_i can be as low as 0.12 eV) is limited by this unresummed large-logarithm uncertainty. The authors should more clearly quantify the theoretical uncertainty on the benchmark mass predictions, or at minimum state explicitly that the 0.12 eV
  3. [Section 4, Eq. (4.1)] The perturbativity condition tr(kappa^dagger kappa) <= pi is imposed, but several benchmark points have individual kappa values near or above 1 (e.g., BP14: kappa_e = 1.19, kappa_tau = 1.22). While tr(kappa^dagger kappa) may still satisfy the bound, the loop expansion parameter kappa^2/(16 pi^2) ~ 0.007 is still small, so this is not a problem per se. However, the paper should clarify whether the one-loop corrections remain perturbative at the benchmark points where epsilon ~ O(0.1), given that the loop corrections involve products like kappa * (ln M^2/mu^2) * kappa^dagger / (16 pi^2) and the logarithms can be large (e.g., ln(10^{14}/10^4)^2 ~ 46 for BP4). A brief comment on the convergence of the loop expansion at these points would strengthen the analysis.
minor comments (6)
  1. [Eq. (2.13)-(2.14)] The expressions for r_21 and r_31 are lengthy. A brief sentence explaining the physical content of each factor (e.g., which mixing parameters drive the constraint) would help the reader.
  2. [Figure 2] The contour labels for (epsilon_mu_mu, epsilon_tau_tau) values are somewhat crowded. Separating the panels or using a clearer color scheme would improve readability.
  3. [Section 4, last paragraph] The authors note that the scan does not quantify the volume of viable parameter space. A brief comment on whether the low-chi^2 region requires fine-tuning of the high-scale parameters (beyond the correlations already mentioned) would be useful for assessing the naturalness of the proposed solution.
  4. [Eq. (A.2)-(A.3)] The construction of the effective chi-square from marginalized NuFIT tables is clearly explained, but a one-sentence justification for why the maximum (rather than, say, a sum) is the appropriate combination would clarify the logic.
  5. [Reference [33]] The concurrent work by Ibarra and Treuer (arXiv:2604.00990) is cited in the context of model-dependent corrections. The authors should clarify whether this reference computes the U(1) gauge/scalar loop corrections that are set aside here, and if so, whether its results support or contradict the assumption that these corrections are subdominant.
  6. [Table 1] The charge assignment table is clear, but a brief note that the discrete subgroup option is mentioned to achieve the two-zero minor structure (as stated in the caption) but not elaborated in the main text would help the reader.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The main points are well-taken and we will address each in the revised manuscript. Below we respond point by point.

read point-by-point responses

  1. Referee: [Section 3.2] The paper's central quantitative result rests on the assumption that model-dependent radiative corrections from the U(1)_{L_mu-L_tau} gauge boson and the U(1)-breaking scalar sector are negligible compared to the 'universal' SM gauge/Higgs corrections retained in Eq. (3.32). [...] If g_{L_mu-L_tau} is comparable to g, these corrections would be of the same parametric size.

    Authors: The referee raises a valid and important point. We agree that the assumption stated in Section 3.2 is load-bearing and should be discussed more carefully. We will revise the manuscript to address this. Our defense of the assumption is as follows. The U(1)_{L_mu-L_tau} gauge coupling g' is a free parameter, but it is subject to experimental constraints: neutrino-electron scattering, BaBar, NA64μ, and CCSN1987A cooling constrain g' depending on the gauge boson mass. For a gauge boson in the MeV–GeV range, typical bounds are g' ≲ 10^{-3}–10^{-1}, well below the SM gauge couplings g ≈ 0.65. For heavier gauge bosons, electroweak precision tests and lepton flavor universality provide constraints. In the regime g' ≪ g, the U(1) gauge boson box diagrams are parametrically suppressed by (g'/g)² relative to the SM gauge contributions retained in Eq. (3.32). Similarly, the U(1)-breaking scalar sector contributions depend on the Yukawa couplings λ_{eμ}, λ_{eτ} and the scalar quartic couplings, which are independent free parameters that can be taken small without affecting the tree-level two-zero minor structure. That said, we acknowledge that the paper does not currently make this argument explicit. We will add a dedicated paragraph in Section 3.2 quantifying the parametric suppression and stating clearly that our results apply to the parameter region where g' and the U(1)-breaking sector couplings are sufficiently small. We will also note that for g' ~ g, the U(1) gauge boson contributions would be comparable and would need to be included, which we leave for future work as already indicated. revision: partial

  2. Referee: [Table 2, BP4 benchmark] The common matching scale treatment (Appendix C) introduces residual scale dependence for such hierarchical spectra [...] The authors should more clearly quantify the theoretical uncertainty on the benchmark mass predictions, or at minimum state explicitly that the 0.12 eV figure carries significant theoretical uncertainty.

    Authors: We agree that the quantitative precision of individual benchmark predictions is limited by the unresummed large-logarithm uncertainty, and we should state this more explicitly. From the scale-variation study in Appendix C (Fig. 8), varying μ_M among M̂_1, M̂_2, M̂_3, and the geometric mean shifts the location of the Δχ²_eff minimum for the M̂_1 = 10^4 GeV case by roughly 0.01–0.03 eV in Σ m_i. This gives an order-of-magnitude estimate of the theoretical uncertainty on the benchmark mass predictions. We will add an explicit statement in Section 4 that the 0.12 eV figure for BP4 should be understood as carrying a theoretical uncertainty of approximately ±0.03 eV from the common-scale matching prescription, and that a precision determination would require sequential decoupling of the right-handed neutrinos with RG running between thresholds. We emphasize, however, that the qualitative conclusion—that one-loop corrections relax the lower bound from ~0.25 eV to the ~0.1 eV range—is robust under the scale variations shown in Fig. 8. revision: yes

  3. Referee: [Section 4, Eq. (4.1)] The paper should clarify whether the one-loop corrections remain perturbative at the benchmark points where epsilon ~ O(0.1), given that the loop corrections involve products like kappa * (ln M^2/mu^2) * kappa^dagger / (16 pi^2) and the logarithms can be large.

    Authors: We thank the referee for raising this point. We have checked the convergence of the loop expansion at the benchmark points. The relevant expansion parameter for the logarithm-enhanced terms is (κ_i κ_j / 16π²) × ln(μ²_M / M̂²_I). For BP4, the largest individual Yukawa couplings are κ_e ≈ 0.39 and κ_τ ≈ 0.38, while the largest logarithm is ln(μ²_M / M̂²_1) ≈ 23 (using the geometric mean matching scale). The effective expansion parameter is then approximately 0.39² × 23 / (16π²) ≈ 0.022, which is comfortably small. Even for the most extreme case of taking μ_M = M̂_3 ~ 10^{14} GeV, giving ln ~ 46, the parameter becomes ~0.044, still well within the perturbative regime. For BP14, where κ_e = 1.19 and κ_τ = 1.22, the logarithms are smaller (ln ~ 2–3) due to the less hierarchical spectrum, giving an expansion parameter of order 1.2² × 3 / (16π²) ≈ 0.027. We will add a brief comment in Section 4 confirming that the loop expansion remains perturbative at all benchmark points, with the effective expansion parameter staying below ~0.05. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained with minor non-load-bearing self-citations

full rationale

The paper's central derivation chain is self-contained. The inputs are the Lagrangian (Eq. 2.1), the U(1)_{L_mu-L_tau} charge assignments (Table 1), and the resulting tree-level right-handed neutrino mass matrix (Eq. 3.1). From these, the one-loop threshold corrections (Eq. 3.32) are computed via standard Feynman diagram calculations — gauge boson box diagrams (Eqs. 3.10–3.12), Higgs box diagrams (Eq. 3.16), self-energy corrections (Eqs. 3.18–3.23), and vertex corrections (Eq. 3.24) — combined through matching (Eqs. 3.26–3.32). The deviation parameters epsilon_mu_mu and epsilon_tau_tau (Eqs. 2.19–2.20) are defined as the (mu,mu) and (tau,tau) entries of the inverse neutrino mass matrix, and their nonzero values arise from the loop calculation, not by definition. The numerical benchmark points (Table 2) are explicitly described as parameter fits to external NuFIT 6.1 data, not predictions. The self-citations to Refs. [28] and [32] (which share authors with the present paper) are used for context — the tree-level lower bound from [28] serves as a comparison baseline, and [32] is cited for noting that U(1) gauge/scalar corrections also exist but are model-dependent. Neither self-citation is load-bearing for the one-loop computation itself, which follows from the Lagrangian and standard field theory methods. The external citations to Grimus-Lavoura [16] and Aristizabal Sierra-Yaguna [17] for the delta M_L correction, and to Zhang-Zhou [38] for general matching conditions, provide independent verification of the calculation. No step in the derivation reduces to its inputs by construction.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The paper introduces no new particles, forces, or dimensions. It works within the established type-I seesaw + U(1)_{L_µ-L_τ} framework. The free parameters are the standard model parameters (right-handed neutrino masses and Yukawa couplings) fitted to data. The key ad hoc assumption is the neglect of model-dependent loop corrections from the U(1) sector.

free parameters (3)
  • M_ee, M_eµ, M_eτ, M_µτ (right-handed neutrino mass parameters) = see Table 2 benchmark points
    Four complex/real parameters of the right-handed neutrino mass matrix, fitted to neutrino oscillation data in the numerical scan.
  • κ_e, κ_µ, κ_τ (Dirac Yukawa couplings) = see Table 2 benchmark points
    Three real diagonal Dirac Yukawa couplings, fitted to data. Together with M parameters, eight real parameters total per the model.
  • M̂_1 lower bound = 10^4 to 10^14 GeV (scanned)
    Lower bound on lightest right-handed neutrino mass, treated as a scan parameter that controls the size of loop corrections.
axioms (4)
  • domain assumption U(1)_{L_µ-L_τ} charge assignments as in Table 1, with symmetry broken by single scalar field ϕ.
    Section 3.1: defines the minimal model structure. Standard assumption for this model class.
  • ad hoc to paper Model-dependent radiative corrections from U(1) gauge boson and U(1)-breaking scalar sector are negligible.
    Section 3.2: 'We therefore do not include them in the following analysis, assuming that the relevant couplings in these sectors are sufficiently small.' This is a load-bearing simplifying assumption.
  • domain assumption Common matching scale µ_M = (M̂_1 M̂_2 M̂_3)^{1/3} is adequate for the qualitative conclusion.
    Section 4 and Appendix C: the authors acknowledge residual scale dependence for hierarchical spectra but argue the qualitative conclusion is stable. Sequential decoupling is deferred to future work.
  • domain assumption NuFIT 6.1 marginalized chi-square tables provide a conservative approximation to the full 6D likelihood.
    Appendix A: the effective chi-square (Eq. A.3) uses max of available 1D/2D/3D tables, which may underestimate the full chi-square. The authors state this makes constraints conservative.

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