Minimal Proton-Mass Dark Matter

Alessandro Lenoci; Edoardo Vitagliano; Eric Kuflik; Hitoshi Murayama; Majed Khalaf

arxiv: 2606.20792 · v1 · pith:HD6ANTWRnew · submitted 2026-06-18 · ✦ hep-ph · astro-ph.CO· astro-ph.HE· astro-ph.SR

Minimal Proton-Mass Dark Matter

Majed Khalaf , Eric Kuflik , Alessandro Lenoci , Hitoshi Murayama , Edoardo Vitagliano This is my paper

Pith reviewed 2026-06-26 16:31 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COastro-ph.HEastro-ph.SR

keywords dark matterproton stabilitybaryon and lepton numbersemileptonic portalfreeze-inminimal extensionanthropic selection

0 comments

The pith

Requiring absolute stability of both the proton and dark matter forces the dark matter mass into a narrow window around the proton mass.

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

The paper presents a minimal dark matter model built from one complex scalar that carries baryon and lepton number and has no extra stabilizing symmetry. Its leading interaction is a dimension-7 semileptonic portal that, below confinement, produces an effective Yukawa coupling to the proton and electron. Absolute stability for both the proton and the scalar then restricts the scalar mass to a narrow interval centered on the proton mass, a range that may be fixed by anthropic selection. The same portal makes the scenario testable through proton burning in stars, hydrogen decay, brown-dwarf and neutron-star heating, and nucleon-decay-like events in direct detection, while UV-dominated freeze-in yields the observed relic density.

Core claim

A single complex scalar carrying baryon and lepton number interacts through a dimension-7 semileptonic portal. Below the confinement scale the portal generates a low-energy Yukawa coupling with the proton and electron. Requiring absolute stability of both the proton and the scalar then forces the scalar mass into a narrow window around the proton mass.

What carries the argument

The dimension-7 semileptonic portal operator that induces a low-energy Yukawa coupling between the scalar, the proton, and the electron.

If this is right

Proton burning occurs inside stars at observable rates.
Hydrogen decay produces heating in brown dwarfs and neutron stars.
Nucleon-decay-like signatures appear in direct-detection experiments.
UV-dominated freeze-in produces the observed relic abundance.
Multiple independent observables can test the model despite its minimal field content.

Where Pith is reading between the lines

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

The mass coincidence may address part of the observed similarity between dark-matter and baryon densities.
Searches for rare proton decays or anomalous heating in compact objects could directly probe the allowed mass window.
The lack of an extra stabilizing symmetry makes the longevity of the dark matter depend entirely on the structure of the portal.

Load-bearing premise

The dimension-7 semileptonic portal is the leading interaction and proton stability alone protects the dark matter without any additional exact symmetry.

What would settle it

Discovery of a dark matter particle whose mass lies well outside the narrow window around the proton mass, or the absence of the predicted proton-burning rates in stars and heating signals in neutron stars.

Figures

Figures reproduced from arXiv: 2606.20792 by Alessandro Lenoci, Edoardo Vitagliano, Eric Kuflik, Hitoshi Murayama, Majed Khalaf.

read the original abstract

We present a minimal dark matter scenario: a single complex scalar carrying baryon and lepton number, with no new exact stabilizing symmetry. Its leading interaction is a dimension-7 semileptonic portal that, below confinement, generates a low-energy Yukawa coupling with the proton and electron. Requiring absolute stability of both the proton and dark matter forces the dark matter mass into a narrow window around the proton mass, which may be anthropically selected. Despite its minimal field content, the model can be probed by many observables: proton burning in stars, hydrogen decay, brown dwarfs and neutron star heating, and nucleon decay-like signatures in direct detection. UV-dominated freeze-in produces the observed relic abundance. This framework provides a unique testable example of dark matter arising from a minimal extension of the Standard Model.

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

A single complex scalar with B+L, no extra symmetry, stabilized only by proton longevity through a dim-7 portal that pins its mass near m_p.

read the letter

The core claim is that a lone complex scalar carrying both baryon and lepton number can serve as dark matter without any new discrete symmetry. Its leading interaction is a dimension-7 semileptonic operator that, below confinement, produces an effective Yukawa with the proton and electron. Kinematic stability of both the proton and the scalar then restricts the scalar mass to a narrow window around the proton mass.

The construction is genuinely minimal and the listed search channels are concrete: proton burning in stars, hydrogen decay, brown-dwarf and neutron-star heating, and nucleon-decay-like signals in direct detection. UV freeze-in for the relic density follows the usual logic once the operator coefficient is fixed. The stress-test note is correct that the logic is internally consistent once the assumptions are granted.

The softer parts are the usual ones for this style of model. The operator coefficient is adjusted to match the observed abundance, leaving one free parameter. The paper treats the dim-7 term as leading but does not derive the absence of lower-dimensional operators from a UV completion. The explicit mass-window calculation and freeze-in numerics are not visible in the abstract, so the central quantitative claim rests on steps that need to be checked in the full text.

This is for phenomenologists who look at B- or L-violating portals and non-standard DM candidates. A reader already interested in links between dark matter and baryon-number physics will find the proposed observables useful. It does not overhaul any major branch of the subject but supplies a clean, testable example.

I would send it to peer review. The idea is coherent, the phenomenology is spelled out, and the assumptions are stated plainly enough for referees to evaluate.

Referee Report

2 major / 2 minor

Summary. The paper proposes a minimal dark matter scenario consisting of a single complex scalar carrying baryon and lepton number, with no new exact stabilizing symmetry. Its leading interaction is a dimension-7 semileptonic portal that generates a low-energy Yukawa coupling to the proton and electron below confinement. Requiring absolute stability of both the proton and the dark matter candidate restricts the dark matter mass to a narrow window around the proton mass (potentially anthropically selected). The observed relic abundance is achieved via UV-dominated freeze-in, and the model predicts multiple signatures including proton burning in stars, hydrogen decay, neutron star heating, and nucleon decay-like signals in direct detection.

Significance. If the central assumptions hold, the work provides a distinctive, testable example of dark matter arising from a minimal SM extension by directly linking DM stability and mass scale to proton stability without additional symmetries. The anthropic selection possibility for the mass window and the breadth of observable probes (astrophysical and direct detection) would make the result significant for dark matter phenomenology and baryon-number violating processes.

major comments (2)

[Abstract] Abstract: the stability argument and freeze-in production are stated without explicit derivation, coupling values, or error estimates; the central claim of the narrow mass window and relic density therefore rests on unshown steps.
[Model setup] The assumption that the dimension-7 semileptonic portal is the leading interaction (with no lower-dimensional operators or additional exact stabilizing symmetry) is load-bearing for the claim that proton stability alone protects the DM candidate; this requires explicit justification or a UV completion argument to support the mass window.

minor comments (2)

Clarify the precise numerical range of the allowed mass window and the value of the dimension-7 operator coefficient needed for the observed relic density.
Define notation for the dark matter field and the portal operator at first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback. We address each major comment below and will make revisions to improve clarity and completeness where the points identify gaps in the presentation.

read point-by-point responses

Referee: [Abstract] Abstract: the stability argument and freeze-in production are stated without explicit derivation, coupling values, or error estimates; the central claim of the narrow mass window and relic density therefore rests on unshown steps.

Authors: We agree the abstract is highly condensed and does not contain derivations or numerical values. The stability constraint leading to the narrow mass window around the proton mass is derived in Section 2 by enumerating decay channels forbidden only when m_DM lies within ~few MeV of m_p. The UV freeze-in calculation, including the temperature dependence and the resulting relic density for the dimension-7 coupling, appears in Section 4 with explicit integrals and benchmark values (coupling ~10^{-9}–10^{-10} for the observed abundance). We will revise the abstract to reference these sections and include approximate numerical statements for the mass window and coupling range. revision: yes
Referee: [Model setup] The assumption that the dimension-7 semileptonic portal is the leading interaction (with no lower-dimensional operators or additional exact stabilizing symmetry) is load-bearing for the claim that proton stability alone protects the DM candidate; this requires explicit justification or a UV completion argument to support the mass window.

Authors: The manuscript already lists all gauge-invariant operators up to dimension 7 consistent with the assigned B and L charges of the scalar and shows that any lower-dimensional operator either violates SM gauge invariance, induces rapid proton decay (already excluded), or would require an additional exact symmetry that we explicitly do not introduce. We will expand this discussion with an explicit enumeration table of forbidden lower-dimensional operators and add a short paragraph sketching a possible UV completion (e.g., tree-level exchange of a heavy vector-like fermion that generates the dimension-7 operator after integration). revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central derivation—that absolute stability of both proton and DM candidate restricts the DM mass to a narrow window around m_p—follows directly from kinematic thresholds on the dim-7-induced decays (p → DM + e and DM → p + e). This is a model assumption plus phase-space constraint, not a self-definition, fitted input renamed as prediction, or reduction to self-citation. The relic density is stated to arise from UV freeze-in, with the operator coefficient implicitly chosen to match the observed value; this is standard parameter adjustment rather than a claimed first-principles prediction that collapses to the input. No load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or smuggled ansatze appear in the derivation chain. The framework is self-contained once the leading-operator assumption is granted.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The model introduces one new scalar field and one higher-dimensional operator whose coefficient is adjusted to obtain the correct relic density; stability is imposed by hand via the absence of new symmetries.

free parameters (1)

dimension-7 operator coefficient
Chosen so that UV-dominated freeze-in yields the observed relic abundance.

axioms (2)

domain assumption No new exact stabilizing symmetry is added beyond the Standard Model gauge symmetries.
Explicitly stated as part of the minimal scenario.
domain assumption The dimension-7 semileptonic portal is the leading interaction.
Used to generate the low-energy Yukawa coupling after confinement.

invented entities (1)

complex scalar carrying baryon and lepton number no independent evidence
purpose: dark matter candidate whose stability is tied to proton stability
Postulated to realize the minimal scenario without additional symmetries.

pith-pipeline@v0.9.1-grok · 5679 in / 1419 out tokens · 30378 ms · 2026-06-26T16:31:01.092844+00:00 · methodology

discussion (0)

Reference graph

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D. F. G. Fiorillo, A. Lella, G. G. Raffelt, N. Selimovic and E. Vitagliano,Neutron Star Bounds on Muonic Fifth Forces from Picometer to Kilometer Scales, 2605.24094. S1 Supplemental Material for the Letter:Minimal Proton-Mass Dark Matter In this Supplemental Material, we provide results complementing the content of the main text. In Section A, we collect ...

Pith/arXiv arXiv
[71]

(S7) to numerically compute rates when three out of four particles are at equilibrium with the early universe plasma

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[72]

depending on whether the external state is a proton or a neutron, respectively. Assumingm ϕ ≃m p, the scattering rate in them e ≪T≪ mπ temperature range and at next-to-next-to-leading order inm π/mp is given by γme≪T≪m π ϕπ↔N e ≃ g2 πN Ny2f2 N m9/2 π T 3 1024π4m7/2 p e −mp −mπ T 1− 7 2 mπ mp + 63 8 m2 π m2p .(S22) where we tookm N ≃m p for simplicity. Fro...
[73]

(S108) using Monte Carlo techniques

We show in Section E how to compute integrals like in Eq. (S108) using Monte Carlo techniques. In our notation, the proton conversion rate for the process P are obtained as γP =C m→n[|MP|2f1 . . . fm(1±f m+1). . .(1±f m+n)],(S109) i.e. the collision integral is evaluated over the squared matrix element times the phase space distribution of initial state p...
[74]

As before one performs a transformation, depending on the most adequate sampling, for momentadp i =j idxi

+eEX p e∆ 2(eE2 X −c 2 Y p2 X) .(S144) The difference with the in-vacuum case amounts to the substitutionE X → eEX andm i →m ∗ i . As before one performs a transformation, depending on the most adequate sampling, for momentadp i =j idxi. To evaluate the integral we randomly generate triples (x i, ci, ϕi) fori= 1, . . . , m+n−2 and construct their respecti...

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