Constraining Higgs mediated dark matter interactions

The thing to know: this is the paper that closes the loophole in light Higgs-portal dark matter. Djouadi et al. had already shown the minimal portal is excluded; Greljo et al. ask whether raising the operator dimension or forbidding the two-body Higgs decay reopens it, and answer no. The key step is Section II, Eq. (6): the invisible width and the annihilation rate scale identically with the cutoff, so higher-dimensional operators make the tension worse, not better. That is a clean, correct piece of power counting, and it is the most useful result here. Section III then walks through the three-body and four-body operators, with relic-density and direct-detection numbers attached, and shows the required cutoff lands at a few hundred GeV, i.e. right where the EFT stops making sense. That is the argument, and it holds within its stated assumptions. The three models in Section IV are real work, not hand-waving: Table I's lepton-flavour bounds, the CMS triplet mass limit, the LEP mixing limit, and the global fit to the 2013 Higgs data in Appendix B are all genuine constraints, and the authors are upfront about the tuned cancellation in model (ii) and the tiny mu in model (i). Where I would push back is small. The abstract's 'only if' reads stronger than what Sections II and III deliver, since the derivation assumes order-one Wilson coefficients and no cancellations; the conclusions say 'barring fine-tuned situations', the abstract does not. That is a presentation fix, not a correctness problem. Model (i)'s flavour pattern (f_tau-tau = 1, f_ee = f_mumu = 0) is chosen to satisfy the CMS bound rather than shown to be natural, which is a fair thing for a referee to ask about but does not threaten the central message. The stress-test note is right that there is no load-bearing error here. This is for anyone working on Higgs-portal or light thermal relics, and it deserves a serious referee.

The paper's central claim is conditional: it holds under the stated assumption of O(1) Wilson coefficients and no cancellations. The authors acknowledge this explicitly in the conclusions ('barring fine-tuned situations'), and the three models are presented as realizations of the loophole. The abstract's 'only if' is slightly stronger than the derivation, but the body is careful. I do not find a load-bearing error in the power-counting argument of Sections II and III, the relic-density calculations of Appendix A, or the model constructions of Section IV. The reader's concern about model (i)'s flavor pattern is a naturalness issue, not a viability issue: the paper explicitly chooses f_ττ = 1, f_ee = f_μμ = 0 to satisfy the CMS bound mΔ > 204 GeV while obtaining the correct relic density. The reader's concern about the abstract's phrasing is a presentation issue, not a correctness issue.

The paper's main technical point—that higher-dimensional operators do not relieve the tension because B_inv and <σv> scale identically with Λ—is correct as a power-counting statement. The three models are worked through with real constraints: lepton-flavour-violating bounds in Table I, CMS triplet mass limits, LEP mixing limits, and a global fit to the 2013 LHC Higgs data in Appendix B. The reader's concern about the O(1) assumption is valid but acknowledged; the abstract's 'only if' is slightly overstated, but the body is careful. The reader's concern about model (i)'s flavor pattern is a naturalness issue, not a viability issue. I do not find a load-bearing error.

The paper makes a useful and clearly organized contribution to the Higgs portal literature. The scaling argument of Eq. (6), showing that higher-dimensional operators cannot reconcile a thermal relic with the invisible-width bound when all coefficients are O(1), is a clean way to organize the problem, and the three worked models are well motivated and demonstrate the loophole concretely. The paper is explicit about its assumptions and about the tuning required in models (i) and (ii), and it identifies falsifiable signatures (lepton-flavor-violating decays in model (i), invisible heavy-Higgs decays and mono-jet signatures in model (ii), and four-b final states in model (iii)). The relic-density and direct-detection calculations in Appendix A use standard machinery and appear correctly implemented.

The paper asks whether dark matter lighter than half the Higgs mass can still be a thermal relic while respecting the bound on invisible Higgs decays, $B(hto text{invisible})<0.19$ at 95% CL. The naive answer is no: reproducing the observed relic density needs an order-one Higgs–dark-matter coupling, while the invisible width needs a coupling below the bottom Yukawa, roughly $0.02$. The authors test whether that verdict survives leaving the minimal renormalizable portal, checking higher-dimensional operators, three-body decays $hto text{DM},text{DM},Z$ built from the Higgs vector current, four-body decays through scalar and tensor fermion currents, and a neutrino-mass-like Weinberg operator. In every case the invisible width and the annihilation rate scale with the same power of the effective cutoff, so the relic density pushes that cutoff down to a few hundred GeV, precisely where the effective description stops making sense. The conclusion is that viable light Higgs-portal dark matter requires additional particles below a few hundred GeV, and the authors exhibit three models in which the parameter controlling the relic abundance is not the one controlling the invisible width.

What carries the argument. The load-bearing device is a scaling comparison. Writing the dark-matter–Higgs interaction as an operator of dimension $4+n$ suppressed by $Lambda^n$, the authors show that the invisible branching ratio scales as $10^3 (m_h/Lambda)^{2n}$ while the annihilation cross section scales as $y_f^2 (m_h/Lambda)^{2n} (m_{text{DM}}/m_h)^k / m_h^2$, with the same power of $Lambda$ in both. Dividing the two, the relic-density requirement fixes the ratio of the two observables up to mass-suppression factors that only make the Higgs constraint stronger, so no choice of operator dimension reconciles them. The second device is model construction: in each of the three examples the coupling that enters $B(h to text{invisible})$ differs from the combination that enters $langle sigma v rangle$ — for instance $lambda_S1 sin alpha cos beta - lambda_S2 cos alpha sin beta$ versus $-lambda_S1 cos alpha cos beta - lambda_S2 sin alpha sin beta$ in the two-Higgs-doublet case, and $lambda_6 cos^2alpha + lambda_5 sin^2alpha$ versus $lambda_5 cos alpha - lambda_6 sin alpha$ in the two-singlet case — so the invisible width can be dialled below the bound while the relic abundance stays fixed.

Core claim. Within an effective field theory containing only the Standard Model and a single dark-matter field, light dark matter cannot be a thermal relic: the coupling needed for the correct relic density inevitably produces an invisible Higgs branching ratio above the experimental bound, and raising the operator dimension does not help because $B(h to text{invisible})$ and $langle sigma_{text{ann}} v rangle$ carry the same suppression in the cutoff $Lambda$. The authors show the same obstruction applies when two-body $h to text{DM},text{DM}$ is forbidden and decays proceed through the Higgs vector current (three-body, excluded by $Z$ invisible width and direct detection), through scalar or tensor fermion currents (four-body, requiring $Lambda lesssim 600$ GeV), or through the Weinberg-like leptonic operator (requiring $Lambda simeq 300$ GeV for $m_{text{DM}}=40$ GeV). The escape is a dark sector with extra light states, which the paper demonstrates with three explicit models: the Standard Model plus a Dirac dark fermion, an electroweak triplet and a singlet; a type-II two-Higgs-doublet model with a singlet scalar dark matter candidate; and the Standard Model plus dark matter and a singlet scalar lighter than the dark matter.

The premise it rests on. The exclusion rests on every dimensionless coupling being of order one and on no accidental cancellations between contributions, a premise the authors state themselves and then deliberately circumvent in their own three viable models by arranging a hierarchy or a cancellation.

If an invisible Higgs branching ratio at the few-tens-of-percent level is established, the dark matter cannot be a thermal relic unless its abundance is set by interactions that do not involve the Higgs field.

Any viable Higgs-portal thermal relic lighter than $m_h/2$ implies additional states with masses of a few hundred GeV or below, which places them within reach of the LHC rather than beyond it.

In the triplet-plus-singlet model the same couplings that fix the relic density induce charged-scalar exchange, so lepton-flavour-violating processes such as $mu^- to e^+e^-e^-$ and $mu to e gamma$ are predicted at rates the current bounds already constrain tightly.

In the two-Higgs-doublet variant the heavy CP-even scalar decays almost entirely invisibly, giving $tbar t$ plus missing energy and mono-jet signatures that can be searched for at the LHC.

In the two-singlet model the 125 GeV Higgs can decay into two light scalars at the level of about twenty percent, producing four $b$-jets that may originate from displaced vertices.

The abstract's 'only if' is more precisely 'only if, unless couplings are tuned small or two contributions cancel'; since the paper's own three examples are exactly such tunings, the result reads as a naturalness statement about the Higgs portal rather than a theorem about all possible portals.

Because the extra light states are what makes the portal work, an eventual invisible Higgs signal would point to a multi-state dark sector rather than a single dark particle, which changes what collider searches should be looking for.

If no invisible Higgs decay appears and direct detection remains null, the surviving corner is the tuned-cancellation region, where the heavy-scalar mixing angle and the dark-matter mass are the only handles — a target for precision Higgs coupling fits rather than for direct searches.

The leptophilic route ties Higgs-portal dark matter to neutrino mass generation, and the extremely small lepton-number-breaking parameter it requires, $mu lesssim 1$ eV, is itself a structural feature that future neutrino and flavour experiments could test.

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