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About electroweak domain walls in Majoron models

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Electroweak instantons generate a potential for the Majoron only when B+L is explicitly broken, and even then the induced mass is so small that the proposed domain walls cannot harm cosmology.

desk verdict The rotation-away argument is right and the 2019 PRL claim does not survive; but the quantitative mass estimate feeding the cosmology is built on an operator the appendix shows is Majoron-blind in the minimal model, so the parameter-space plots are not yet trustworthy. read the letter →

arxiv 2506.02910 v4 pith:CCMKIWU5 submitted 2025-06-03 hep-ph astro-ph.COhep-th

Maximilian Berbig This is my paper
classification hep-phastro-ph.COhep-th
keywords MajoronelectroweakinstantonsdomainwallsB+Lviolationleptonnumberbiastermultralightdarkmatterdynamicalenergy
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 re-examines a 2019 claim that electroweak instantons break global lepton number down to a discrete Z3 and thereby force Majoron models to face a cosmological domain-wall problem. It argues that the effect can be completely rotated away: as long as baryon number plus lepton number (B+L) is not explicitly broken, the Majoron's coupling to the electroweak topological term is unphysical and no potential is generated. When explicit B+L breaking is present, the induced Majoron mass is exponentially small and the resulting domain walls carry far too little energy to disturb cosmology. A sympathetic reader should care because, if the argument holds, a whole class of model-building constraints evaporates and the weak anomaly becomes a possible asset rather than a liability.

What carries the argument

The load-bearing object is the coupling of the Majoron to the electroweak topological term, $(g_W^2/32\pi^2)(j/v_L) W\widetilde W$, together with the $U(1)_{B+L}\otimes \mathrm{SU}(2)_W^2$ anomaly identity that lets field redefinitions shuffle the Majoron between operators. When no explicit $B+L$ violation exists, the Majoron is aligned with the anomaly-free combination $U(1)_{B-L}$ and the coupling is pure gauge; the mass estimate then relies on a dimension-six $B+L$-violating operator, a single-instanton 't Hooft vertex with three insertions of that operator, and the instanton-size integral of Ref.~[77], which is UV-dominated and carries the exponential factor $\exp(-8\pi^2/g_W^2(M_{\rm UV}))$. The numerical smallness can be softened by adding scalar $\mathrm{SU}(2)_W$ multiplets that flatten the gauge-coupling running, encoded in Eqs. (15)--(16). The appendix's chiral vector-like doublets of Eq. (18) are the mechanism that actually makes the $B+L$ operator pick up the Majoron phase in the minimal fermion content.

What would settle it

A first-principles instanton calculation, or a lattice computation, of the effective potential in a model containing Eq. (5) plus the vector-like doublets of Eq. (18) would settle the claim: if the induced Majoron mass turns out to be independent of the $B+L$-violating coefficient $c_L$, or nonzero when $B+L$ is not explicitly broken, the central argument fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that electroweak $\mathrm{SU}(2)_W$ instantons do not generate a potential for the Majoron unless $B+L$ is explicitly broken by additional interactions. The reason is a field-redefinition identity: rephasing lepton and quark doublets shifts the electroweak vacuum angle, so a Majoron inserted into $W\widetilde W$ can be moved into other operators; without explicit $B+L$ violation every such operator is invariant along an anomaly-free $U(1)_{B-L}$ direction, and the Majoron mass vanishes. With a dimension-six $B+L$-violating operator of the form $LQQQ/M_{\rm UV}^2$, the paper estimates the induced mass using a one-instanton 't Hooft-vertex calculation and obtains a value suppressed by $\exp(-8\pi^2/g_W^2)$, of order $10^{-29}\,$eV for benchmark parameters. The associated domain-wall energy-density fractions are at most about $10^{-16}$ today, far below observational bounds, and thermal sphalerons do not change the picture because they act as friction rather than generating a potential. The paper then shows that the tiny electroweak mass can serve as a bias term that collapses walls generated by a larger lepton-number-breaking source, producing ultralight Majoron dark matter, or, if it is the leading contribution, can implement an electroweak Majoron as dynamical dark energy.

Load-bearing premise

Everything quantitative rests on the assumption that an explicit baryon-plus-lepton-number-violating interaction really does pick up the Majoron field after fermion redefinitions; the appendix shows this requires adding new fermions that are not vector-like under baryon-minus-lepton number, because the minimal Standard-Model operator accidentally conserves that combination.

Editorial extensions

If this is right

  • In generic Majoron models without explicit $B+L$ breaking, electroweak instantons generate no Majoron potential, so the domain walls proposed in Ref. [61] do not form.
  • When $B+L$ is broken, the instanton-induced mass is exponentially small; the present-day energy fraction of the resulting walls is of order $10^{-16}$ or less, so no cosmological catastrophe follows.
  • The tiny electroweak mass can act as a bias term that collapses a wall network generated by a larger lepton-number-breaking source, and the decay can produce the observed ultralight Majoron dark-matter relic density.
  • If the electroweak instanton is the leading source of the Majoron mass, the Majoron can serve as thawing-quintessence dark energy, although the preferred parameter region conflicts with the weak gravity and swampland distance conjectures.
  • Thermal sphalerons do not generate the Majoron potential; their effect is friction, so wall formation and evolution should be described by the instanton-induced potential rather than by the sphaleron mass scale.

Reading between the lines

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

  • The rotation argument is more general than the Majoron: any pseudo-Goldstone boson of a global symmetry whose only anomaly is electroweak will be blind to instantons unless an explicit $B+L$-breaking operator survives field redefinitions, so the same test should be applied to other lepton-number-like symmetries.
  • The appendix's accidental $B-L$ conservation suggests that many would-be $B+L$-violating operators built from Standard-Model fermions cannot generate a Majoron mass at all; model builders should classify operators by their $B-L$ charge before estimating instanton effects.
  • The reversal of the axion bias-term logic, using the smallest explicit-breaking contribution to collapse walls produced by a larger one, may extend to other pseudo-Goldstone dark-matter and dark-energy scenarios where the hierarchy of explicit breakings is usually assumed to work in the opposite direction.
  • A concrete next step is to compute Eq. (19) in an explicit ultraviolet completion with the vector-like doublets, since the NDA estimate's cancellation for $N_S=1$ suggests an exact result worth checking.
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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

3 major / 4 minor

Summary. The paper revisits the claim of Ref. [61] that electroweak SU(2)_W instantons generate a potential for the Majoron and thereby lead to cosmological domain walls in Majoron models. The authors argue, using the anomaly structure and chiral field redefinitions, that in the absence of explicit B+L breaking the Majoron can be rotated away from the electroweak topological term, so no instanton-induced potential or domain-wall network arises. They then estimate the instanton-induced Majoron mass from a B+L-violating operator using the NDA one-instanton method of Ref. [77], and study the cosmological consequences: the resulting domain walls are harmless, the tiny mass can act as a bias term to collapse domain walls from a larger lepton-number-breaking source (potentially producing ultralight Majoron dark matter), or the Majoron can serve as a dynamical-dark-energy candidate. An appendix, however, concedes that the operator used in the main text conserves B−L and does not couple to the Majoron in the minimal SM fermion content; the repair requires vector-like fermions chiral under B−L, which modify the mass estimate.

Significance. If the technical issues are resolved, the paper would provide a useful and largely correct correction to a recent PRL claim. The central symmetry argument is standard and applied cleanly: the absence of a Majoron potential without B+L breaking follows from the anomaly-free nature of U(1)_{B−L} and the ability to remove the Majoron from the topological term by field redefinitions. The paper is also commendably explicit about its assumptions, including the axion-quality-like problem for the bias potential and the strong UV sensitivity of the one-instanton estimate. Its main quantitative applications, however, currently rest on a mass formula whose operator is admitted in the appendix not to couple to the Majoron in the minimal model; this makes the cosmological parameter windows and dark-energy discussion conditional on the corrected estimate. The paper does not contain machine-checked proofs or code, but its symmetry argument is checkable analytically and the appendix already flags the main subtlety.

major comments (3)
  1. [The Argument, Eq. (6); Appendix] The derivation of m_j^2 in Eq. (6) assumes that the B+L-violating operator in Eq. (5) acquires a Majoron phase under the field redefinitions described in the text. The appendix shows that this is false for the minimal SM fermion content: the operator conserves B−L, while the Majoron couples through α_B − α_L, and Eq. (5) transforms with α_B + α_L and therefore remains j-independent. Thus Eq. (6) is not the correlator that generates the Majoron mass in the minimal model. The repair in Eq. (18) introduces new fermionic zero modes and changes the estimate to Eq. (19). Because Eqs. (7)–(14), the enhancement in Eq. (15), and Figs. 2–3 all use Eq. (6) (plus scalar-only beta-function enhancement), the paper's quantitative cosmology is computed with a mass formula whose operator does not couple to the Majoron in the stated minimal setup. The appendix's assertion that the conclusions are unchanged needs to be substantiated by redoing the relevant figures and constraints with Eq. (19), including the relation M_Ψ = Y_Ψ v_L/√2.
  2. [Appendix, Eq. (19)] The text claims that for N_S = 1 the suppression from the fermionic zero modes is exactly canceled by the enhancement from the changed running. However, substituting N_S = 1 into Eq. (19) gives m_j^2 → m_j^2 × (M_Ψ/M_UV) × (M_UV/M_Ψ)^{5/3} = m_j^2 (M_UV/M_Ψ)^{2/3}, which is an enhancement for M_UV > M_Ψ, not an exact cancellation. Please clarify the definition of N_S or correct Eq. (19) and the surrounding discussion. This point matters because the appendix uses this cancellation to argue that the main-text estimates are unaffected.
  3. [Abstract and Introduction] The statement that the Majoron can only couple to the electroweak topological term if B+L is explicitly broken is, as the appendix shows, only a necessary condition. The explicit operator must also break B−L, or the theory must contain B−L-chiral fermions (as in Eq. (18)), for the Majoron to acquire a coupling. The paper should state this qualification in the main text and abstract rather than relegating it to the appendix, since the illustrative operator in Eq. (5) does not work in the minimal SM content.
minor comments (4)
  1. [Dark Energy section] The text contains an artifact, 'https://www.overleaf.com/learn', immediately before 'm_j ≃ H_0'; this should be deleted.
  2. [Fig. 1 caption] The phrase 'closed up by three insertions' is awkward; consider 'contracted' or 'closed off' instead.
  3. [Eq. (8)] The definition of |˜c_L| is hasty; write explicitly |c_L| [cos(θ_EW + 3δ_L)]^{1/3} to avoid ambiguity with powers of the cosine.
  4. [Eq. (3) and surrounding text] The text presents a single-generation toy model in Eq. (3) but uses N_g = 3 in the anomaly argument in Eq. (2); clarify that the single-generation choice is for illustration only and that the anomaly count is taken from the full three-generation Standard Model.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central symmetry argument is derived from anomaly structure and field redefinitions, and the sole self-citation is contextual and not load-bearing.

full rationale

The central claim, that electroweak instantons do not generate a Majoron potential unless B+L is explicitly broken, is established in the text by an explicit field-redefinition and anomaly argument centered on Eq. (2), not by assuming the conclusion. The rotation of the Majoron into the topological term and back is shown step by step for the lepton and quark doublets, and the conclusion is benchmarked against prior independent work cited in Refs. [57, 68-73]. The mass estimate in Eq. (6) is taken from the independent instanton-NDA framework of Ref. [77]; no parameter is fitted to the quantity being predicted, so there is no fitted-input-renamed-as-prediction. The only self-citation, Ref. [159], appears in a list of prior thawing-quintessence attempts and carries no load-bearing weight in the derivation. The appendix's concession that the operator in Eq. (5) 'does not actually pick up a coupling to the Majoron' in the minimal field content is an internal-consistency correction to the numerical bridge: it changes the estimate via Eq. (19) and affects the applicability of Figs. 2-3, but it is not a circular reduction, because the mass formula is not assumed in order to derive itself. It is derived from Ref. [77] and then amended when the operator is shown to be B-L conserving. The DESI dark-energy section similarly compares model parameter space with the preferred ranges from Ref. [138] rather than fitting an input and calling it a prediction. I therefore find no step in which an output is equivalent by construction to an input; the score reflects only the presence of a single non-load-bearing self-citation and the internal inconsistency flagged in the appendix, which is a correctness concern rather than circularity.

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The central symmetry claim is parameter-free and relies only on anomaly structure and field redefinitions. The new mass estimate introduces EFT parameters that are chosen by hand, and the appendix adds a model-building ingredient that was not emphasized in the main text. No entity is invented to explain an observation; the extra fermions are a consistency requirement.

free parameters (6)
  • c_L = O(1), assumed real in benchmarks
    Coefficient of the B+L violating operator in Eq. (5); the mass estimate scales as |c_L|^{3/2} and the plots set |c_L| cos(theta_EW + 3 delta_L)^{1/3} = 1.
  • M_UV = 10^16 GeV in benchmarks
    Cut-off of the EFT in Eq. (5); enters with power (M_UV)^4 in Eq. (6) and is chosen near the GUT scale.
  • v_L = 10^13 GeV for bias benchmarks; O(10^18 GeV) for dark energy
    Lepton number breaking scale; the mass scales as 1/v_L and is selected by astrophysical bounds and dark matter or dark energy requirements.
  • M_j (dominant Majoron mass from other breaking) = 10^-18 or 10^-17 eV benchmarks
    In the bias scenario, the leading source of lepton number breaking via Eq. (11) is an input and is scanned over in Fig. 2.
  • Additional scalar representations (d, M, N_S) = d=4, M=5 TeV; d=5, M=20 TeV; d=3, M=10 TeV
    They change the SU(2) beta function and enhance the instanton mass via Eq. (15); benchmarks are chosen to reach the desired m_j.
  • n (dimension of operator in Eq. 11) = n=17 for v_L=10^13 GeV, M_j=10^-18 eV
    Controls the leading explicit breaking in the bias scenario; chosen to match the desired mass scale.
assumptions (5)
  • domain assumption The U(1)_{B+L} x SU(2)_W^2 anomaly is the only non-perturbative source of Majoron potential at the electroweak scale.
    Invoked in 'The Argument' and used to justify the form of the instanton-induced potential proportional to cos(3 theta_j).
  • standard math Constrained instanton (one-instanton) approximation with IR cutoff 1/(g_W v_H) and UV cutoff 1/M_UV.
    Used via Ref. [77] to derive Eq. (6); it is an approximation, not an exact QFT result.
  • domain assumption The Standard Model fermion content with N_g=3 and no new fermionic zero modes unless explicitly added.
    The t'Hooft vertex in Fig. 1 assumes three generations; the appendix shows extra chiral fermions are needed for the Majoron coupling to survive.
  • ad hoc to paper No other sources of explicit lepton number breaking contribute in the pure instanton scenario.
    Stated in the text: 'Throughout this paper we do not consider other sources of explicit symmetry breaking for lepton number.' This isolates the instanton contribution.
  • ad hoc to paper A mechanism exists to protect the quality of the bias potential, suppressing all lower-dimensional operators in Eq. (11).
    Stated in the Bias term section: 'we have to assume a mechanism to protect the required quality of the potential in Eq. (11) by preventing the appearance of all lower dimensional operators with n < 17'.
invented entities (1)
  • Vector-like doublet fermions Psi_L,R with chiral B-L charges
    purpose: Make the B+L violating operator in Eq. (5) pick up a Majoron coupling under field redefinitions; see appendix and Eq. (18).
    No direct observational handle is provided; their mass M_Psi and Yukawa Y_Psi are model inputs, and they alter the instanton estimate by Eq. (19).

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Cite this review

Pith. "Pith review of About electroweak domain walls in Majoron models." pith.science (2026). https://pith.science/paper/CCMKIWU5

@misc{pith2026250602910,
  author       = {Pith},
  title        = {Pith review of: About electroweak domain walls in Majoron models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CCMKIWU5}},
  note         = {Machine review of arXiv:2506.02910}
}
abstract

Some time ago it was claimed in "Spontaneous Breaking of Lepton Number and Cosmological Domain Wall Problem" (Phys. Rev. Lett. 122, 151301 (2019)) that non perturbative instantons of the weak interaction $\text{SU}(2)_\text{W}$ lead to the formation of domain walls in Majoron models owing to the anomaly of the spontaneously broken global lepton number $L$ symmetry $\text{U}(1)_L$ with respect to $\text{SU}(2)_\text{W}$. We point out that it has long been known, that this effect can be completely rotated away unless there is a source of explicit $B+L$ breaking present, where $B$ denotes baryon number. We further estimate the tiny instanton induced Majoron mass from $B+L$ breaking and analyze the cosmological impact of such domain walls including possible finite temperature effects. In general this scenario does not lead to a cosmological catastrophe and we demonstrate that the tiny instanton induced mass can act as a bias term to collapse walls induced by a larger source of lepton number breaking. Alternatively this electroweak Majoron could act as dynamical dark energy.

Figures

Figures reproduced from arXiv: 2506.02910 by the authors.

Figure 1
Figure 1. FIG. 1. Diagrammatic representation of the single instanton [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Parameter space for the production of ultra-light Majoron dark matter from domain wall decay. Here we depict a [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Parameter space that reproduces the preferred range of Majoron masses and decay constants [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.