Minimal Decaying Dark Matter and the LHC

Worth your time if you work on light-ish decaying DM or on displaced-vertex signatures, less so if you want a collider reach study — this isn't one, and the authors say as much in §6 (they call it a qualitative analysis, no boost factors, no detector simulation). What's genuinely new is the FIMP/SuperWIMP branch: two new states, a Majorana singlet and one colored or electroweak scalar, no stabilizing symmetry, and the relic density built as freeze-in plus SuperWIMP decay of the scalar. The WIMP branch they explicitly hand off to Garny et al. (ref. 48), which is the right call and they state it plainly in §4.2. The useful technical bridge is Eq. (3.5): the DM lifetime is tied directly to the scalar decay length, so an ID signal and a displaced vertex are two faces of the same two couplings. That relation is the paper's best idea and it's cleanly derived. Appendix A is real work — they solve the coupled Boltzmann system numerically and show 2→2 scattering contributes at most a percent (Eqs. A.14–A.16), which justifies the additive decomposition. So the reader's worry about decoupling the scalar equation is, I think, defused by the paper's own appendix and by the gauge-equilibrium argument in §4. I'd not raise it as the weak point. The weak point is the one the stress test flags, and it lands. §3 says the hadronic antiproton limits span 10²⁶ to 10²⁸ s depending on diffusion and background, then adopts a flat 10²⁷ s 'for each single channel' and applies it to q qbar ν, u dbar dbar, u dbar l and the leptonic channels alike. Since l_Σ,SM ∝ τ_ψ (Eq. 6.3) and the branching ratio scales as τ_ψ⁻¹ (Eq. 6.4), a factor-ten shift in the applicable limit moves the decay length by an order of magnitude — exactly the gap between 'displaced vertex' and 'detector stable' in the §6 classification and in Figs. 9–12. The figures look quantitative but inherit that spread without propagating it. That's a fixable weakness, not a fatal one: a band rather than a line in those figures would do it. Minor: the proton-decay bound |λ'λ''| < 10⁻²⁷ for the Σd case is waved away by assuming a coupling hierarchy, which is fine but thin. Overall this is careful, self-aware phenomenology with a modest but real result. Send it out.

The paper's central quantitative bridge is Eq. (3.1) combined with a single reference lifetime τ=10^27 s, which then fixes λ′ via Eq. (6.2) and hence the scalar decay length in Eq. (6.3) and the displaced-vertex/track classification of §6. But §3 itself states that the hadronic antiproton limits 'range between 10^26 s and 10^28 s' depending on diffusion parameters and background treatment, and that limits are stronger for heavier quarks. The paper then adopts 'a reference limit on the DM lifetime of 10^27 s for each single channel' and applies it uniformly to q qbar ν, u dbar dbar, u dbar l, and the leptonic l lbar ν channels. Since l_Σ,SM ∝ τ_ψ (Eq. 6.3) and BR ratios scale as τ_ψ^{-1} (Eq. 6.4), a factor-10 shift in the applicable limit moves the predicted decay length by an order of magnitude, which is precisely the difference between 'displaced vertex' and 'detector stable' in the §6 classification. The reader's weakest_assumption (scalar not in thermal equilibrium / additive FIMP+SuperWIMP decomposition) is a real modeling choice, but the paper addresses it directly: §4 argues gauge interactions keep Σf in equilibrium until freeze-out, Appendix A solves the coupled Boltzmann system numerically and reports that 2→2 scattering contributes at most percent level (Eqs. A.14–A.16), and Fig. 4 shows the FIMP and SuperWIMP regimes separating cleanly. That is a supported approximation, not the weakest link. The weakest link is that the entire LHC-reach discussion inherits a benchmark lifetime that the paper's own cited source says varies by two orders of magnitude across channels, and the paper does not propagate that spread into the displaced-vertex boundaries of Figs. 9–12.

The paper's core claim is that relic density plus an indirect-detection-reachable lifetime selects two regions (FIMP/SuperWIMP and WIMP) with distinguishable LHC signatures. The reader's weakest_assumption targets the decoupling of the scalar and DM Boltzmann equations in §4. That is a legitimate modeling question, but the paper confronts it head-on: §4 states the scalar 'feels gauge interactions ... which guarantee efficient annihilation and scattering processes keeping it in thermal equilibrium until it freezes-out,' Appendix A sets up the coupled system (Eqs. A.1–A.2), solves it numerically, and shows the scattering term is at most percent level (Eqs. A.14–A.16, I≈0.46). Fig. 4 demonstrates the FIMP and SuperWIMP contributions separating as λ varies. So the reader's chosen weak point is, in my read, adequately defended; I only partially agree with the reader's identification. The genuinely load-bearing soft spot is elsewhere: the paper's own §3 acknowledges that the cited antiproton limits span 10^26–10^28 s across channels and diffusion assumptions, yet adopts a single 10^27 s benchmark 'for each single channel' and carries it into Eq. (6.2), Eq. (6.3), and the displaced-vertex classification of §6. Because l_Σ,SM ∝ τ_ψ and the BR ratio scales as τ_ψ^{-1}, a one-decade shift in the applicable limit moves the predicted decay length by a decade, which is exactly the scale separating 'displaced vertex' from 'detector stable' in Figs. 9–12. This does not overturn the existence of the two regions, but it does mean the §6 reach boundaries are channel-dependent in a way the figures do not show. The reader's other stated concerns (qualitative LHC reach without a detector study, WIMP branch overlapping ref. 48) are fair and already reflected in the CONDITIONAL verdict. I would keep the verdict at CONDITIONAL and not move it: the central claim survives, but the collider-reach numbers should be presented with the channel-dependent lifetime spread made explicit. The paper is candid about its approximations and the numerical cross-check in Appendix A is real support, so this is a tightening request, not a soundness failure.

The paper is a useful, self-contained template for decaying DM with renormalizable couplings, complementary to gravitino and R-parity-violating scenarios. The observation that the FIMP coupling fixed by the relic density, combined with a lifetime near the indirect-detection limit, lands the scalar decay length in the displaced-vertex window is genuinely interesting and worth developing. The numerical cross-check in Appendix A, showing that 2→2 scattering contributes at most percent level (Eqs. A.14–A.16), is a strength, and the parameter-space scan is reproducible from the quoted formulas. The main caveat is that the LHC-reach classification inherits a single lifetime benchmark whose channel-to-channel variation the paper itself quotes.

This paper asks whether a dark matter candidate that is permitted to decay — with no symmetry imposed to keep it stable — can simultaneously have the observed cosmic abundance and a lifetime short enough for indirect-detection telescopes to register. The model adds only two particles to the Standard Model: a Majorana fermion as the dark matter and one scalar that is either colored or electroweakly charged, coupled to the dark matter and to ordinary fermions through two Yukawa-like couplings. Because those same two couplings control the dark matter's three-body decay rate, the scalar's decay length at a collider, and the dark matter's production in the early universe, the authors argue that cosmology and the LHC are measuring the same numbers from opposite ends. Working through the relic-density bookkeeping, they find two viable regimes — a weakly coupled freeze-in/SuperWIMP regime and an order-one-coupling thermal WIMP regime — and map which of the two survives existing indirect-detection, direct-detection, and collider bounds.

What carries the argument. The load-bearing device is the three-body decay rate $Gamma_{rm DM} = |lambda_{psi f}|^2 |lambda'|^2 m_psi^5 / (128 (2pi)^3 m_{Sigma_f}^4)$, which contains both couplings and both masses and therefore links three otherwise separate observables: the dark matter lifetime seen by telescopes, the production and decay of the scalar at a collider, and — through the branching ratio of the scalar into dark matter versus Standard Model fermions — the early-universe yield. The authors turn this into a chain: fix $lambda$ from the freeze-in relic density, fix $lambda'$ from the assumed lifetime, and read off the scalar's decay length $c tau$. Which regime applies follows from which term of the relic density dominates, the freeze-in term scaling as $lambda^2 x$ with $x = m_psi/m_{Sigma_f}$ and the SuperWIMP term scaling as $x$ times the branching ratio into dark matter times the scalar's would-be thermal relic density.

Core claim. The central claim is that the joint requirement of (i) the measured dark matter density $Omega_{rm CDM} h^2 simeq 0.11$ and (ii) a decay lifetime near the current indirect-detection sensitivity of about $10^{27}$ s selects two disconnected patches of the parameter space in the dark matter mass, the scalar mass, and the two couplings, and that these patches are distinguishable at the LHC. In the first patch the dark matter coupling is tiny, $lambda lesssim 10^{-7}$, so the dark matter never thermalises; its abundance is the sum of a freeze-in term produced while the charged scalar is still in the thermal bath and a SuperWIMP term from the scalar's later out-of-equilibrium decay. Fixing $lambda$ from the freeze-in contribution and then imposing $tau_psi sim 10^{27}$ s forces the scalar-to-Standard-Model coupling $lambda'$ to a value whose corresponding scalar decay length is tens of metres for a TeV-scale scalar and a light dark matter particle — inside the detector, so the signature is a displaced vertex or a long-lived charged track rather than the usual missing-energy event. In the second patch $lambda sim 1$ and the dark matter is an ordinary thermal WIMP; the same lifetime then requires $lambda' sim 10^{-22}$, which makes the decay invisible, and the authors show this branch is largely excluded by direct detection and monojet searches unless the coupling is preferentially to third-generation quarks.

The premise it rests on. The whole map rests on the charged scalar staying in thermal equilibrium through its own gauge interactions until it freezes out, so that its evolution decouples from the dark matter's and the relic density can be written as a simple sum of a freeze-in piece and a SuperWIMP piece; the authors check this numerically in the appendix, but every coupling value and region boundary they quote inherits it.

In the freeze-in/SuperWIMP branch the LHC signature is a displaced vertex or a long-lived charged particle from the scalar, not missing energy, so searches built around a stable invisible particle are poorly matched to it.

A collider measurement of the scalar's mass and gauge quantum numbers fixes its thermal relic density, which by itself tests whether the SuperWIMP contribution is needed to reach the observed abundance.

If the scalar is found and its decay length measured, combining that with an indirect-detection lifetime determines both couplings and closes the model at the LHC.

The WIMP branch with couplings to light quarks is essentially ruled out by direct detection and monojet searches, leaving only third-generation couplings alive.

Leptonic realizations survive existing bounds far more easily than hadronic ones, so a null result in colored channels would not kill the scenario.

The multi-TeV SuperWIMP corner the authors mention in passing is the natural target for future antiproton and gamma-ray observatories, and it is the one corner where the scalar is likely beyond LHC reach while the decay signal is still real — so the two detection strategies are complementary rather than redundant there.

Because no symmetry stabilizes the dark matter, a positive indirect-detection signal would be a measurement of an operator coefficient rather than the discovery of a new conservation law; that changes what would count as confirming the picture.

For the scalar with the quantum numbers of a right-handed down squark, proton-decay limits on the product of the two Standard-Model couplings force a strong flavour hierarchy in any viable freeze-in realization — a structural requirement rather than a free choice, and one that collider flavour signals could check.

The same two-coupling structure would arise in R-parity-violating supersymmetry with one light sfermion and a light neutralino, so a displaced-vertex signal alone would not separate the two; a lifetime measurement from indirect detection would be needed to break the degeneracy.

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