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arxiv: 2605.15270 · v1 · pith:UXM6FI6Fnew · submitted 2026-05-14 · ✦ hep-ph

Light states in real 3HDMs with spontaneous CP violation and softly broken symmetries

Jos\'e M. Camacho , Carlos Mir\'o , Miguel Nebot , Daniel Queiroz , Tom\'as Tobarra This is my paper

Pith reviewed 2026-05-19 15:29 UTC · model grok-4.3

classification ✦ hep-ph
keywords 3HDMthree Higgs doublet modelspontaneous CP violationscalar massesperturbativitydiscrete symmetrysoftly broken symmetries
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The pith

In three-Higgs-doublet models with spontaneous CP violation, new scalar masses stay near the electroweak scale once quartic couplings obey perturbativity.

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

The paper shows that in three-Higgs-doublet models with a CP-invariant potential and a CP-violating vacuum, perturbativity requirements on the quartic couplings prevent additional scalar states from becoming much heavier than the electroweak scale. This bound holds even when the quadratic mass terms are allowed to take arbitrarily large values. The analysis imposes an extra discrete symmetry on the quartic sector to obtain analytic and numerical results, then extracts phenomenological features that follow from the scalar spectrum alone. A reader cares because the result implies that any such extended Higgs sector cannot hide its new particles at high mass scales; instead the extra states remain potentially accessible to collider searches and precision measurements.

Core claim

In the minimal three-Higgs-doublet model with spontaneous CP violation, the requirement that quartic couplings remain perturbative forces the masses of the new scalar particles to lie close to the electroweak scale, regardless of how large the free quadratic terms are made. The discrete symmetry that shapes the quartic part of the potential produces explicit mass relations and spectra that are examined both analytically and numerically.

What carries the argument

The quartic sector of the three-Higgs-doublet potential, constrained by a discrete symmetry, together with the CP-violating vacuum expectation values, which together generate the scalar mass matrix whose eigenvalues are bounded by perturbativity.

If this is right

  • Additional scalars remain light enough to contribute to electroweak precision observables.
  • The scalar spectrum exhibits specific mass patterns fixed by the discrete symmetry.
  • Phenomenological predictions for Higgs decays and couplings follow directly from the scalar-sector properties.
  • The model cannot decouple the extra states into a heavy, invisible sector.

Where Pith is reading between the lines

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

  • Similar light-mass bounds may appear in 3HDMs without the discrete symmetry once the full set of quartic couplings is required to stay perturbative.
  • The result could restrict the parameter space available for generating sufficient CP violation for electroweak baryogenesis within these models.
  • Experimental searches for light additional scalars gain motivation independent of specific Yukawa textures.

Load-bearing premise

The quartic part of the Higgs potential must be shaped by a discrete symmetry.

What would settle it

An explicit parameter choice in which all quartic couplings stay below the perturbativity threshold yet at least one new scalar mass exceeds several TeV.

Figures

Figures reproduced from arXiv: 2605.15270 by Carlos Mir\'o, Daniel Queiroz, Jos\'e M. Camacho, Miguel Nebot, Tom\'as Tobarra.

Figure 1
Figure 1. Figure 1: FIG. 1: Masses of the new scalars [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: H [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Interplay among couplings, [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Interplay among couplings, [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Couplings [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Couplings [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Maximal decay widths of H [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
read the original abstract

Scalar sectors with several Higgs doublets, a CP invariant potential, and a CP violating vacuum, possess a mass spectrum in which, unexpectedly, new scalars cannot have masses much larger than the electroweak scale once perturbativity requirements are imposed on the quartic couplings of the Higgs potential and despite the presence of free quadratic (mass) terms that can be arbitrarily large. The minimal model in which this behavior is manifest involves 3 Higgs doublets. We analyze and illustrate in detail that kind of scenario including an additional simplifying assumption: the quartic part of the potential is shaped by some discrete symmetry. Besides analytic results, a numerical analysis is presented together with some phenomenological consequences derived only from the properties of the scalar sector alone.

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

1 major / 3 minor

Summary. The manuscript examines real three-Higgs-doublet models (3HDMs) with a CP-conserving scalar potential but a spontaneously CP-violating vacuum. It argues that perturbativity constraints on the quartic couplings impose an upper bound on the masses of additional scalar states near the electroweak scale, even when the quadratic mass parameters are allowed to take arbitrarily large values. The analysis incorporates an additional discrete symmetry that shapes the quartic sector, presents both analytic derivations and numerical scans, and extracts phenomenological consequences from the scalar sector properties alone.

Significance. If the central mass bound holds under the stated assumptions, the result offers a mechanism for keeping extra scalars light in multi-Higgs models without fine-tuning the quadratic terms, which has direct implications for collider phenomenology and model building. The combination of spontaneous CP violation with softly broken symmetries and the provision of both analytic results and numerical evidence constitute clear strengths. The focus on falsifiable mass bounds derived solely from the scalar potential adds to the paper's value for the field.

major comments (1)
  1. [Abstract and §1] Abstract and §1: The lightness result is stated for CP-invariant potentials with CP-violating vacua in multi-Higgs models, yet the detailed analytic and numerical work is performed only after imposing an additional discrete symmetry on the quartic couplings. While the abstract notes this as a simplifying assumption, the manuscript should explicitly address whether the same upper bound on scalar masses survives in the general real 3HDM without this symmetry, where the larger number of independent quartic parameters could relax the perturbativity constraints while still satisfying bounded-from-below and unitarity conditions.
minor comments (3)
  1. [§3.2, Eq. (12)] §3.2, Eq. (12): the definition of the perturbativity criterion on the quartic couplings should include an explicit numerical threshold (e.g., |λ| < 4π or similar) and a brief justification for the choice, as this directly affects the reported mass upper bound.
  2. [Table 2] Table 2: the caption should specify the range of quadratic mass parameters scanned and whether any points violate tree-level unitarity, to allow readers to assess the robustness of the numerical results.
  3. [§4] §4: a short discussion of how the softly broken symmetry affects the CP-violating phases in the vacuum would improve clarity, even if the main focus remains on the mass spectrum.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation for minor revision. We address the major comment below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: The lightness result is stated for CP-invariant potentials with CP-violating vacua in multi-Higgs models, yet the detailed analytic and numerical work is performed only after imposing an additional discrete symmetry on the quartic couplings. While the abstract notes this as a simplifying assumption, the manuscript should explicitly address whether the same upper bound on scalar masses survives in the general real 3HDM without this symmetry, where the larger number of independent quartic parameters could relax the perturbativity constraints while still satisfying bounded-from-below and unitarity conditions.

    Authors: We agree that the role of the discrete symmetry should be addressed more explicitly. This symmetry is imposed on the quartic sector to reduce the number of independent parameters, enabling analytic derivations of the mass bounds and efficient numerical scans while preserving the spontaneous CP violation and the soft breaking of the symmetry by quadratic terms. In the general real 3HDM without the symmetry, additional quartic parameters exist. However, the unitarity constraints on 2-to-2 scalar scattering and the bounded-from-below conditions become correspondingly more restrictive, which does not relax but rather reinforces the upper bound on the masses of the additional scalars to maintain perturbativity up to high scales. We will add a clarifying paragraph at the end of §1 and a brief remark in the conclusions explaining that the lightness result is illustrated in the symmetric case but is expected to hold qualitatively in the general case, with a full exploration of the parameter space without the symmetry left for future work. This is a minor addition that clarifies the scope. revision: yes

Circularity Check

0 steps flagged

No circularity: mass bound follows from perturbativity on quartics independent of quadratic terms and explicit symmetry assumption

full rationale

The paper states that the lightness of new scalars follows from imposing perturbativity on the quartic couplings of a CP-invariant potential with a CP-violating vacuum, even when quadratic mass terms are free and arbitrarily large. The discrete symmetry is explicitly introduced as an 'additional simplifying assumption' for the numerical analysis rather than a hidden premise that forces the bound. No equations reduce a prediction to a fitted input by construction, no self-citation chain is load-bearing for the central claim, and the derivation remains self-contained once the stated assumptions (CP invariance, vacuum structure, and optional symmetry) are granted. This is the normal non-circular outcome for a paper whose result is an upper bound derived from unitarity/perturbativity constraints.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the perturbativity requirement applied to quartic couplings and on the discrete symmetry that shapes those couplings; quadratic mass terms are treated as free but are over-constrained by the former.

free parameters (1)
  • quadratic mass parameters
    Allowed to be arbitrarily large yet ultimately bounded indirectly by quartic perturbativity.
axioms (2)
  • domain assumption Potential is CP invariant while vacuum spontaneously breaks CP
    Standard setup for spontaneous CP violation in multi-Higgs models.
  • ad hoc to paper Quartic part of the potential is shaped by a discrete symmetry
    Explicitly introduced as an additional simplifying assumption.

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Works this paper leans on

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