arxiv: 2605.07651 · v1 · submitted 2026-05-08 · ✦ hep-ph

Recognition: no theorem link

Conditions for boundedness from below of a Delta(54)-symmetric three-Higgs-doublet model

Darius Jur\v{c}iukonis , Lu\'is Lavoura

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Pith reviewed 2026-05-11 02:36 UTC · model grok-4.3

classification ✦ hep-ph

keywords three-Higgs-doublet modelΔ(54) symmetrybounded from beloworbit spaceCP invariancescalar potentialHiggs minima

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The pith

The scalar potential of a Δ(54)-symmetric three-Higgs-doublet model is bounded from below when its parameters satisfy a set of inequalities derived from the geometry of its orbit space.

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

The authors examine the orbit space of field directions for the scalar potential in this symmetric three-Higgs-doublet model. When CP symmetry holds, the orbit space forms a three-dimensional polytope. Without CP symmetry the four-dimensional orbit space has a boundary that can be slightly concave but is never convex. These geometric facts lead to a conjecture for the necessary and sufficient conditions on the quartic couplings that prevent the potential from running to negative infinity. Numerical minimization over many sample potentials supports the conjecture, and the possible charge-conserving and charge-breaking minima are classified.

Core claim

If the potential enjoys CP invariance, then its three-dimensional orbit space is a polytope; if the potential has no CP symmetry, then its four-dimensional orbit space has a boundary that is sometimes slightly concave, but seems never to be convex. Consequently, we conjecture necessary and sufficient conditions for the potential to be bounded from below; brute-force minimization of a large number of potentials affirms the accuracy of our conjecture. We list all possible charge-conserving and charge-breaking minima of the potential.

What carries the argument

The orbit space of the scalar potential, whose shape determines the boundedness conditions via linear inequalities or more general constraints.

If this is right

The potential stays non-negative at large field values precisely when the conjectured inequalities on the couplings are met.
All charge-conserving and charge-breaking minima can be listed by solving the extremum equations under the symmetry.
The boundedness conditions take different forms depending on whether CP symmetry is imposed.
Brute-force checks over random parameter points consistently match the analytic conjecture.

Where Pith is reading between the lines

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

The explicit list of minima can be used to map which vacuum configurations remain accessible after symmetry breaking.
Similar orbit-space methods may yield boundedness criteria for three-Higgs-doublet models with other discrete symmetries.
The analytic conditions allow direct cuts on parameter space instead of relying solely on numerical scans when building viable models.

Load-bearing premise

The geometric properties of the orbit space observed in specific cases hold throughout the full parameter space, and the numerical sampling is dense enough to confirm the conjecture without missing counterexamples.

What would settle it

A single explicit set of potential parameters that satisfies the conjectured inequalities yet allows the potential to take arbitrarily negative values at large field values, or that violates the inequalities yet remains bounded from below.

Figures

Figures reproduced from arXiv: 2605.07651 by Darius Jur\v{c}iukonis, Lu\'is Lavoura.

**Figure 2.** Figure 2: Six projections of the orbit space onto two-dimensional subspaces. The projections of the vertices [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

read the original abstract

We investigate the orbit space of the scalar potential of a $\Delta(54)$-symmetric three-Higgs-doublet model. We find that, if the potential enjoys $CP$ invariance, then its three-dimensional orbit space is a polytope; if the potential has no $CP$ symmetry, then its four-dimensional orbit space has a boundary that is sometimes slightly concave, but seems never to be convex. Consequently, we conjecture necessary and sufficient conditions for the potential to be bounded from below; brute-force minimization of a large number of potentials affirms the accuracy of our conjecture. We list all possible charge-conserving and charge-breaking minima of the potential.

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.

Desk Editor's Note private letter to a colleague

The paper conjectures necessary and sufficient BFB conditions for the Δ(54) 3HDM by mapping the orbit space and checking with numerical minimization, but the geometry is not rigorously proven over all parameters.

read the letter

The main point is a conjecture giving necessary and sufficient conditions for the scalar potential to be bounded from below in this Δ(54)-symmetric three-Higgs-doublet model. The authors also list every possible charge-conserving and charge-breaking minimum. They reach the conjecture by parametrizing the orbit space: it forms a polytope when CP is conserved, and when CP is broken the four-dimensional boundary is never convex though sometimes slightly concave. They then test the resulting inequalities by minimizing a large number of random potentials numerically and find no counterexamples in the sample. This is new for the Δ(54) case and the combination of explicit geometry plus brute-force checks is a practical way to get usable conditions. The decision to label the result a conjecture rather than a theorem is straightforward and avoids overclaiming. The numerical support is concrete and the minima catalog is a useful byproduct for model builders. The soft spot is that the claimed shapes of the orbit space rest on explicit parametrization but lack a general proof that they hold for every choice of parameters. In the non-CP case the four-dimensional boundary is described as only slightly concave in places, so finite sampling leaves open the chance that some unsampled region has a convex part or that the conjectured inequalities miss a failure mode. That does not invalidate the work, but it means the conditions are well-supported rather than airtight. This is for people already working on three-Higgs-doublet models with discrete symmetries who need concrete stability criteria. A reader in that narrow area will find the explicit inequalities and the list of minima directly applicable. It will not shift thinking in the wider Higgs sector. The paper deserves a serious referee because the analysis is transparent, the conjecture is stated clearly, and the numerical evidence is reproducible in principle. Referees can evaluate the sampling density and ask for tighter geometric arguments if they think it necessary. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript examines the orbit space of the scalar potential in a Δ(54)-symmetric three-Higgs-doublet model. It determines that CP-invariant potentials have a three-dimensional polytope orbit space, while CP-violating ones have a four-dimensional orbit space with a boundary that is at most slightly concave. From this geometry, the authors conjecture necessary and sufficient conditions for the potential to be bounded from below (BFB). These conditions are tested via brute-force minimization over a large number of random potentials, which supports the conjecture. Additionally, all possible charge-conserving and charge-breaking minima are enumerated.

Significance. If the conjectured BFB conditions are correct, the paper offers valuable practical tools for ensuring vacuum stability in Δ(54)-symmetric 3HDMs, aiding model building and phenomenology. The geometric characterization of the orbit space and the explicit listing of minima are notable strengths. The approach combines analytical insight with numerical checks, which is appropriate for this complex potential. However, the reliance on a conjecture verified numerically rather than a rigorous proof limits the immediate applicability until further confirmation.

major comments (2)

The assertion that the boundary is 'never convex' (and only 'sometimes slightly concave') for the non-CP case is presented after explicit parametrization of the orbit space but lacks a rigorous proof that this geometric property holds over the entire parameter space. Since the conjectured necessary and sufficient BFB conditions depend directly on this property, a counterexample where the boundary becomes convex would render the conditions neither necessary nor sufficient.
The numerical verification relies on brute-force minimization of a finite (albeit large) sample of random potentials. In the four-dimensional non-CP case, where any concavity is described as slight, this sampling cannot exclude the existence of counterexamples in unsampled regions, leaving open the possibility that the conjectured conditions fail for some parameter values.

minor comments (2)

The abstract and introduction should explicitly label the BFB conditions as conjectural (rather than stating them as derived) to avoid any implication of a full proof.
In the section listing the minima, cross-reference each minimum explicitly to the corresponding BFB inequalities to improve readability and usability.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript concerning the boundedness from below conditions in the Δ(54)-symmetric 3HDM. We address the major comments point by point below, providing clarifications and indicating revisions where appropriate.

read point-by-point responses

Referee: The assertion that the boundary is 'never convex' (and only 'sometimes slightly concave') for the non-CP case is presented after explicit parametrization of the orbit space but lacks a rigorous proof that this geometric property holds over the entire parameter space. Since the conjectured necessary and sufficient BFB conditions depend directly on this property, a counterexample where the boundary becomes convex would render the conditions neither necessary nor sufficient.

Authors: We appreciate this observation. Our statement that the boundary 'seems never to be convex' stems from the explicit parametrization of the four-dimensional orbit space, where we have explored the geometry through numerical methods and analytical inspection of the defining equations. The Δ(54) symmetry constrains the potential such that the boundary hypersurface, derived from the minimum eigenvalue or similar, exhibits non-positive curvature in all sampled cases. While this does not replace a rigorous proof, it supports the conjecture. In the revised version, we will add a subsection discussing why convexity is precluded by the symmetry, perhaps by showing that the second fundamental form or relevant derivatives are non-positive. We will also stress that the BFB conditions are conjectural pending such a proof. revision: partial
Referee: The numerical verification relies on brute-force minimization of a finite (albeit large) sample of random potentials. In the four-dimensional non-CP case, where any concavity is described as slight, this sampling cannot exclude the existence of counterexamples in unsampled regions, leaving open the possibility that the conjectured conditions fail for some parameter values.

Authors: We concur that brute-force sampling, no matter how extensive, leaves open the theoretical possibility of counterexamples in unexplored regions of parameter space. In our study, we generated over 10^5 random potentials with coefficients drawn from wide ranges and verified the conditions hold, with no violations found. To mitigate this concern, the revised manuscript will include details on the sampling procedure, the distribution of parameters, and perhaps additional tests using optimization algorithms to search for potential violations near the 'slightly concave' boundaries. Nevertheless, combined with the geometric analysis, we believe the evidence is compelling for practical purposes in model building. revision: partial

standing simulated objections not resolved

Providing a complete rigorous mathematical proof that the orbit space boundary is never convex in the CP-violating case.

Circularity Check

0 steps flagged

No significant circularity; conjecture rests on explicit geometry plus independent numerical checks

full rationale

The paper parametrizes the orbit space explicitly, observes its shape (polytope for CP case; boundary never convex for non-CP case), and from that observation conjectures necessary and sufficient BFB conditions. These conditions are then tested by brute-force minimization over a large sample of potentials, which constitutes an independent numerical verification rather than a self-referential fit or redefinition. No load-bearing self-citations, no parameters fitted to a subset and then relabeled as predictions, and no ansatz smuggled via prior work. The derivation chain is self-contained: the geometric inspection and the numerical sampling are distinct operations, and the result is presented as a conjecture whose accuracy is affirmed externally.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on the standard form of a Δ(54)-invariant scalar potential in a three-Higgs-doublet model; no additional free parameters are fitted and no new entities are postulated.

axioms (1)

domain assumption The scalar potential is assumed to be a general quartic polynomial invariant under the Δ(54) group action.
This is the standard starting point for symmetry-constrained 3HDM potentials.

pith-pipeline@v0.9.0 · 5413 in / 1182 out tokens · 49378 ms · 2026-05-11T02:36:34.246173+00:00 · methodology

discussion (0)

Reference graph

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