Minimal Proton-Mass Dark Matter
Minimal Proton-Mass Dark Matter
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.
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The paper is largely internally consistent. The DM mass window (Eq. 4) is used consistently throughout all phenomenological sections. The Yukawa coupling y is connected to the UV scale Λ via lattice matrix elements (y ≃ 0.0144 GeV^3/Λ^3) and this relation is used consistently in both the freeze-in and constraint analyses. The parameter δ = Δ/me is defined once and used uniformly across stellar, brown dwarf, and hydrogen-decay sections. One inconsistency: the main text states that the freeze-in coupling required for the observed abundance is y ∼ 1.2×10^{-21} (Treh/1 TeV)^{5/2} (Eq. 7), but the SupM derives the same result (Eq. S71) without the factor of (Treh/1 TeV)^{5/2} dependence on the same numerical prefactor (1.2×10^{-21}) — note this is consistent; both give the same expression. The freeze-out discussion states the scenario is excluded, but the specific threshold y > 5×10^{-9} (hadronic) and y > 3×10^{-8} (electromagnetic) for freeze-out viability are not explicitly derived in the main text; they are stated as results. This is a minor gap but does not create a contradiction. The anthropic discussion is logically coherent: the tuning is framed as a selection effect operating when both DM and proton stability conditions are required. No self-contradictions were found in the central logical chain.
The mathematical framework is sound in its core structure, but several key derivations are compressed and rely on results stated without full derivation in the main text. The central result — the UV freeze-in coupling (Eq. 7) — is presented as 'The total rate and the resulting relic abundance are calculated in the SupM.' The SupM derivation (Eqs. S37–S71) shows the matrix elements and the rate γ_ϕ ∝ T^{10}/Λ^6, and then states the result with a numerical factor 1.4×10^{23} GeV × T_{reh}^5/Λ^6. The step from the rate to the Boltzmann equation integration is not fully algebraically expanded; the dependence on g_{s}, g_{ ho} is not explicitly tracked. The conversion from Λ to y uses the lattice estimate y ≃ 0.0144 GeV^3/Λ^3, which is an external input and not derived here. The neutron star constraints (strongest bounds in the paper) rely on a numerical integration over the APR equation of state density profile; the main text shows only the order-of-magnitude estimate and states that the 'careful numerical evaluation' matches it. The SupM provides the framework (Monte Carlo method in Section E, rates in Section D) but the final integration is presented as a result without showing numerical convergence or the full integral. The brown dwarf, stellar, and hydrogen-decay bounds are all based on order-of-magnitude estimates with clear physical reasoning, which is acceptable for constraints. The invisible neutron decay rate (Eq. 21) contains a loop factor that is not derived from the UV operator; it is stated as an estimate. Overall, the mathematical structure is physically motivated and the scaling laws are correct dimensionally, but the central quantitative results depend on compressed derivations and external numerical inputs that are not fully validated in the text.
This is highly falsifiable phenomenological model-building. The core claim produces a tightly delimited mass window around the proton mass, quantitative relations between the effective Yukawa coupling y and multiple observables, and several distinct experimental/astrophysical channels that can independently exclude the model. The work does not merely say 'there may be effects'; it gives concrete scaling laws and benchmark bounds for hydrogen decay, stellar proton depletion, brown-dwarf heating, neutron-star stability/heating, invisible neutron decay, and dark-matter-induced nucleon destruction in large detectors. A particularly strong feature is that the same minimal interaction controls both cosmological production and late-time signatures, so discovery or exclusion is not arbitrarily decoupled from the rest of the framework.
The paper could be even stronger if it explicitly stated falsification criteria in one dedicated paragraph, e.g. 'if no signal is seen in X and Y down to coupling Z, the entire freeze-in realization with reheating above QCDPT is excluded for such-and-such parameter region.' Still, in practice the exclusion logic is very clear from the plots and bounds. Importantly, several tests are current rather than purely futuristic, so this is not a numerically precise but operationally empty theory. Because there are multiple quantitative, near-term or existing probes, the highest score is justified.
The paper is generally well organized at the section level: model setup, low-energy interaction, stability window, relic abundance, then a sequence of phenomenological constraints. A graduate-level reader in particle/astroparticle physics can follow the high-level story, and the central conceptual points are communicated effectively. The abstract and introduction are strong, and the figure usefully summarizes the parameter-space logic.
However, clarity is not exceptional. The manuscript is dense, often moves quickly from qualitative setup to formulae without enough verbal signposting, and relies heavily on the supplement for derivational context. Some observational constraints are presented as order-of-magnitude estimates in the main text, with details deferred, which is acceptable for a letter but makes the immediate chain of reasoning harder to digest. There is also symbol overload/reuse (notably delta/Delta and various mu notations), and the sheer number of channels and rates can make the phenomenology read as a catalog rather than a guided narrative. Because the red-flag instruction caps clarity at 3 when symbol/term redefinition is detected, and because the prose does require re-reading in several places, a 3 is the appropriate score.
The main novelty is the synthesis: a single complex scalar carrying baryon and lepton number, with no extra exact stabilizing symmetry, whose absolute stability is enforced kinematically by being nearly degenerate with the proton. That is a distinctive conceptual move. The model ties together proton stability, dark-matter stability, near-threshold kinematics, UV freeze-in, and a broad suite of astrophysical/laboratory signals in a compact framework. The anthropic discussion is secondary, but the central mechanism—dark matter stabilized by the same mass-window logic that preserves baryons rather than by an imposed parity—is a genuinely fresh framing with clear phenomenological consequences.
That said, many ingredients are individually familiar: higher-dimensional portals, baryon/lepton-charged dark sectors, hydrogen-decay constraints, and neutron-star arguments all exist in prior literature, some of which the authors cite appropriately. So the work is not introducing an entirely new mathematical structure or radically new dynamical principle. Its originality lies more in a clean, minimal recombination of known ingredients that generates an unusually sharp and testable parameter space. That supports a 4 rather than a 5.
The paper is substantially complete relative to its own goals. It specifies the field content, quantum numbers, leading operator, low-energy reduction, mass window, production mechanism, and a broad set of observational consequences. It also provides a UV completion example and a large supplement containing amplitudes, rates, freeze-in/freeze-out details, neutron-star modeling, and phase-space methods. This makes the overall program followable and fairly self-contained.
The work is especially strong in covering edge regimes relevant to its thesis: it discusses the full proton/DM stability window, distinguishes production before vs. after the QCD phase transition, states the EFT validity condition Treh ≲ Λ, and notes when neutron-star heating arguments apply only in the perturbative-structure regime. It also explicitly states that additional lower-dimension portals may exist but are not the focus, and it separates the dimension-7 phenomenology from those effects.
The main reason this is not a 5 is that some support is uneven. Several phenomenological bounds in the main text are order-of-magnitude estimates whose assumptions are only partly exposed there, and some boundary-condition discussions remain incomplete. Examples include the reliance on approximate stellar-core numbers without a full stellar-evolution treatment in the main paper, the simplified treatment of hydrogen diffuse-background and 21 cm recasts, and the neutron-star conclusions' sensitivity to equation of state, in-medium pion physics, self-interactions, and possible condensate behavior. The authors do acknowledge some of these limitations, but not always with full quantitative scope. There are also minor presentational gaps: some notation is dense or introduced in compressed form, and some claims defer crucial details to the supplement rather than summarizing them clearly in the main text. Still, the central argument is developed enough to support the stated claims.
This paper presents a minimal dark matter model — a single complex scalar φ carrying baryon and lepton number B=L=−1, interacting via a dimension-7 semileptonic portal — that generates a rich and testable phenomenological program despite containing only one new field. The central mechanism is conceptually elegant: absolute stability of both dark matter and the proton is enforced kinematically by placing the dark matter mass inside the narrow window mp−me ≤ mφ ≤ mp+me (Eq. 4). This double-stability requirement, derived from forbidding φ→p̄e+ and p→φ*e+, is a genuinely fresh organizational principle: dark matter stability emerges from proton stability rather than from a new symmetry. The UV freeze-in production via the dimension-7 contact operator yields a relic abundance scaling as Y∝Treh^5/Λ^6 (Eq. 7), consistent with the operator's dimensionality, and the low-energy Yukawa y controls a broad suite of astrophysical and laboratory signatures. Novelty specialists scored this 4/5, reflecting a synthesis that is meaningfully original — not merely recombining known ingredients — especially the direct-detection signature of dark-matter-induced nucleon destruction with invariant mass near 2mp and a harder pion spectrum, distinguishable from standard nucleon-decay searches. Falsifiability is rated 5/5 by both science specialists: the model makes multiple quantitative, channel-specific predictions already constrained by or testable with current instruments (Super-Kamiokande, SNO+, Borexino, JUNO, neutron-star observations), and a single portal governs both cosmological production and late-time signatures so that exclusion in one regime has direct implications for others.
The completeness specialists (3/3 specialists, 4/5) recognize that the Supplemental Material is unusually thorough: it supplies full matrix elements for all relevant binary and multi-body scattering channels (Sections A–B), detailed Boltzmann equation integration for both pre- and post-QCD phase transition freeze-in, in-medium neutron-star modeling using the APR equation of state with effective masses, self-energies, chemical potentials, Pauli blocking, and a general Monte Carlo phase-space method (Sections D–E). The neutron-star constraints in particular are derived with notable rigor, with radial density profile integration over two stellar mass profiles (2M⊙ and 1.5M⊙), explicit treatment of processes (S81a–f), and numerical verification that the order-of-magnitude estimates in the main text (Eqs. 14–15) match the careful SupM computations to within expected factors.
However, the mathematical validity specialists (3/5, high confidence, zero spread) and internal consistency specialists (4/5, moderate confidence, spread 2) identified several specific load-bearing points where the derivational chain is compressed or unverified. The most critical is the hadronic matching coefficient y≃0.0144 GeV³/Λ³ (text following Eq. 3), which maps the quark-level dimension-7 operator to the hadronic Yukawa: the operator basis, chiral projection conventions, RG running from 2 GeV, and hadronization details are not derived in the paper but are asserted via citation of Ref. [9] (Aoki et al., lattice proton-decay matrix elements). Since all phenomenological bounds and the relic-abundance curves in Fig. 1 scale as y²∝Λ⁻⁶, errors in this matching would shift the entire exclusion map. A second flagged point is the UV freeze-in abundance Eq. (7): the SupM derives the total rate γ∝7776T^{10}/((2π)^7 Λ^6) (Eq. S70), but the step from this rate through the Boltzmann equation integration to the numerical coefficient 1.4×10^{23} GeV and the T_reh^5 scaling is not algebraically expanded step-by-step; the dependence on g_{s}(T) and g_{ρ}(T) is not tracked explicitly. The proton-decay width formula in footnote/Ref. [10] has a notation issue: the Källén function arguments appear to be written as λ(mp², mφ, me) with unsquared masses rather than λ(mp², mφ², me²), which is a dimensional inconsistency in the formula as printed, though the underlying kinematic logic of Eq. (4) is unaffected. The invisible neutron decay estimate Eq. (21) includes a W-boson loop factor without derivation, and the direct-detection cross section Eq. (20) uses the full dark-matter density without distinguishing the φ vs φ components in a symmetric freeze-in scenario where only one component destroys ordinary nucleons through the stated channel. The process label H→φ+γ in the hydrogen decay section should read H→φ+γ for charge consistency with Eq. (9).
One internal narrative tension deserves explicit attention. The text following Eq. (6), in the context of the UV completion, states that baryon and lepton numbers are conserved and do not induce 'dangerous processes such as proton decay.' However, the paper simultaneously writes an explicit proton-decay width for p→e+φ* (footnote near Eq. 4), applies Super-Kamiokande and SNO+ proton-decay experimental limits as exclusions in Fig. 1, and uses those limits to define the lower boundary of the viable parameter space. The resolution is that in the UV completion the mediators S and S' carry definite baryon/lepton number and the effective operator conserves B+L, so no B-violating decay (in the GUT/Baryon-number-violating sense) is induced; but the process p→φ*+e+ is nonetheless a nucleon-destruction process constrained by the same proton-decay search experiments. This is physically coherent once the two meanings of 'proton decay' are distinguished, but the paper's phrasing oscillates between them and a reader following the logic sequentially will encounter an apparent contradiction. A single clarifying sentence would resolve this. The clarity specialists were split (scores 3 and 5, spread 2), reflecting that the high-level structure is excellent but the local exposition requires re-reading in several places, particularly around notation reuse for δ/Δ across different sections and the dense compressed presentation of astrophysical rate estimates in the main text.
Overall this is a scientifically strong, phenomologically rich, and highly falsifiable paper that makes a genuine conceptual contribution. The specific mathematical risk flags — the unverified hadronic matching at Eq. (3), the incompletely expanded Boltzmann integration underlying Eq. (7), the notation issue in the proton-decay width formula, the compressed NS heating numerical derivation behind Eq. (15), the loop-level invisible neutron decay estimate Eq. (21), and the direct-detection density counting in Eq. (20) — should be addressed or explicitly acknowledged as externally verified inputs to make the quantitative conclusions fully reproducible. None of these issues overturn the central stability-window mechanism or the qualitative phenomenological program, but they are the primary barriers to a higher mathematical-validity score.
This work departs from mainstream consensus physics in the following ways. These are not penalties - they are informational flags that highlight where the author proposes alternative interpretations of physical phenomena. The scores above evaluate rigor, not orthodoxy.
- ◈The paper introduces a dark matter candidate with baryon number B=−1 and lepton number L=−1, coupling through a dimension-7 semileptonic portal: this departs from the standard assumption that dark matter is a gauge-singlet with no Standard Model baryon or lepton charge.
- ◈The model achieves dark matter stability without any new exact stabilizing symmetry (no new Z2, U(1), or discrete parity), relying instead on kinematic mass-window enforcement: this departs from the standard approach of introducing an ad hoc stabilizing symmetry.
- ◈The paper invokes an anthropic selection argument to explain the ~2×10^{-3} mass tuning required to place mφ within the window mp−me to mp+me, proposing that observers exist only in regions of parameter space where both matter and dark matter survive: this is a non-standard explanatory framework not part of the mainstream dark matter model-building program.
- ◈The invisible neutron decay channel n→φ*+ν̄e (Eq. 21) represents a non-standard neutron decay mode not present in the Standard Model, constrained by but distinct from ordinary neutron lifetime measurements.
- ◈The proton-burning process p+e−→γ+φ* in stellar interiors (Eq. 9) converts hydrogen into dark matter, providing a non-standard energy-loss and hydrogen-depletion mechanism in main-sequence stars beyond any Standard Model process.
1 model failed to respondReduced Panel (8/9)
anthropic/claude-opus-4-7(math)
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Key Equations (3)
Leading dimension-7 semileptonic portal in the UV that couples the scalar dark matter φ to first-generation right-handed quarks and charged lepton; Λ is the cutoff/mediator scale and C = i γ^2 γ^0.
Low-energy effective Yukawa interaction generated after confinement: φ couples to a proton and a right-handed electron with dimensionless coupling y.
Stability window enforced by requiring both φ and the proton to be kinematically stable against decays φ\to\bar p + e^+ and p\to \phi^* + e^+; forces m_φ to lie within roughly ±m_e of the proton mass.
Other Equations (4)
Estimate (from lattice proton matrix elements) relating the low-energy Yukawa y to the UV scale Λ (evaluated at 2 GeV).
Approximate UV-dominated freeze-in relation: the produced relic abundance (normalized to the observed value) as a function of the Yukawa coupling y and reheating temperature T_reh, valid when production before QCD transition dominates.
Proton decay rate into φ^* + e^+ (used to compare to experimental nucleon-decay limits); λ is the Källén function for phase space.
Estimated direct-detection event rate for DM-induced nucleon destruction in a water Cherenkov detector of mass M (fiducial Super-Kamiokande reference) as a function of coupling y.
Testable Predictions (5)
Hydrogen decay H→φ + γ produces monoenergetic photons and is constrained by diffuse-photon backgrounds and 21 cm observations; existing diffuse-background limits imply hydrogen lifetimes τ_H ≳ O(10^{19} s) depending on photon energy, and Borexino-type searches set τ_H ≳ O(10^{28} yr) for accessible energies.
Falsifiable if: Detection of the predicted monoenergetic photons at the energies set by m_φ (or a sky-averaged 21 cm anomaly) at rates inconsistent with the model's allowed parameter space would support the scenario; non-detection improving these lifetime limits below the model prediction for a given y and m_φ would exclude that parameter point.
Direct-detection experiments can see nucleon-destruction events (φ + p/n → e^+ + π + ...) with an event rate R ≳ 1/yr in a 22.5 kton detector for y ≳ 7×10^{-13}; events have distinctive kinematics (invariant mass near ∼2 m_p and a harder pion spectrum) requiring dedicated recasts of nucleon-decay searches.
Falsifiable if: A dedicated search in Super-K/Hyper-K (or equivalent large detectors) finding no nucleon-destruction events at rates above the predicted R for the corresponding y would exclude those coupling values; observation of such events with the predicted kinematics would support the model.
Absolute stability of both φ and the proton requires m_φ to lie within the narrow interval m_p ± m_e (i.e. |m_φ - m_p| ≲ m_e).
Falsifiable if: Detection of stable dark-matter particles with mass outside the interval m_p ± m_e while simultaneously observing a stable proton (or conversely observed proton decay consistent with p→φ^* + e^+ while φ remains cosmologically stable).
UV-dominated freeze-in can produce the observed relic abundance for parametrically tiny Yukawa couplings; for T_reh ~ 1 TeV the required coupling is y ~ 1.2×10^{-21}.
Falsifiable if: Robust cosmological/thermal-history measurements or alternative production-channel bounds that rule out freeze-in with y ≲ O(10^{-21}) for plausible reheating temperatures, or a confirmed relic abundance inconsistent with the predicted yield for any allowed T_reh and Λ in this model.
Neutron-star structure and cooling place strong upper limits on the coupling: requiring that neutron conversion never removes more than 10% of neutrons over a 2 M_☉ NS lifetime gives y ≲ 1.2×10^{-19}, while heating constraints from the cold NS PSR J2144–3933 imply y ≲ 5.2×10^{-23}. Future observations of colder NSs (surface T ≃ 1000 K) could probe down to y ∼ 4×10^{-26}.
Falsifiable if: Observation of old, heavy or very cold neutron stars with properties incompatible with these upper bounds (e.g. measured NS temperatures or mass/radius combinations implying stronger limits) would rule out the model parameter region above the stated y; conversely, detecting effects (excess heating or structural softening) consistent with conversions at y above these bounds would support the model.
Tags & Keywords
Keywords: dimension-7 semileptonic portal, baryon-number carrying scalar, UV freeze-in, proton stability, neutron-star heating, hydrogen decay, dark-matter induced nucleon destruction, leptoquark UV completion
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