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REVIEW 3 major objections 5 minor 112 references

This paper claims that the vanishing of the tree-level Higgs–dark-scalar portal in a dark-matter model follows from five-dimensional locality, with the portal regenerated radiatively by bulk sterile neutrinos, and that this geometry shifts

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2026-08-03 00:30 UTC pith:XE6XVIDD

load-bearing objection A clean new geometric origin for the zero tree-level portal, but the quoted mass windows rest on an uncomputed KK-tower contribution that the paper's own diagnostics suggest is not negligible. the 3 major comments →

arxiv 2607.28754 v1 pith:XE6XVIDD submitted 2026-07-30 hep-ph hep-th

A geometric origin for the radiative neutrino portal to secluded dark matter

Mattia Di Mauro This is my paper
classification hep-ph hep-th
keywords secluded dark matterradiative neutrino portalseesaw mechanismfive-dimensional sequesteringbulk sterile neutrinosHiggs-singlet mixingdirect detectionBig Bang nucleosynthesis
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to give a theoretical origin for the radiative neutrino portal to secluded dark matter: why the tree-level Higgs–dark-scalar coupling is exactly zero at the ultraviolet scale. It proposes a five-dimensional construction in which the Standard Model lives on one brane and the dark sector on another, so that a local contact operator is geometrically forbidden. The only messengers are bulk sterile neutrinos, whose loop exchange regenerates a small portal that inherits the seesaw suppression of neutrino masses. The central consequence is that the portal strength, and hence direct-detection and collider signals, is tied to the heavy‑neutrino mass scale, while the need for the hidden mediator to decay before Big Bang nucleosynthesis bounds it from below. A fair reader would care because the model explains why laboratory signals can be tiny without appealing to an ad hoc small coupling.

Core claim

The core claim is that the boundary condition κ(Λ_UV)=0 for the Higgs–dark‑scalar portal is not imposed by hand but follows from five‑dimensional locality: the Higgs and the hidden scalar are localized on different branes, so the delta‑function overlap δ(y)δ(y−L) vanishes for separated branes. The same heavy‑neutrino sector that generates light neutrino masses through the seesaw mechanism then induces the portal at one loop, with a threshold contribution κ_loop = −Σ_I y_N,I² (Y_ν†Y_ν)_II/(4π²). In the aligned seesaw limit this is proportional to the light‑neutrino masses, making the mixing angle scale as M_N². The resulting phenomenology confines the heavy‑neutrino scale to roughly 10 TeV <

What carries the argument

The central object is the five‑dimensional sequestered construction with a compact S¹/Z₂ interval: visible fields at y=0, hidden fields at y=L, and bulk sterile Dirac fermions with a bulk mass M₅. The machinery is the combination of (i) the locality argument that forbids a tree‑level portal because the two boundary delta functions have disjoint support, and (ii) the one‑loop brane‑to‑brane box diagram in which the heavy‑neutrino propagators are replaced by bulk propagators. The key identity is the profile factor F₀(x)=(2x/(1−e^(−2x)))³ e^(−2x), with x=|M₅|L, which multiplies the flat‑profile loop result and encodes the exponential suppression as the bulk mass grows relative to the compactifi

Load-bearing premise

The quantitative results rely on the assumption that the contribution of the massive Kaluza–Klein tower to the radiative portal is much smaller than the light‑state contribution, an assumption the paper notes is not verified and whose violation could change the normalization and the inferred mass bounds.

What would settle it

Compute the complete, regulated KK spectral sum for the one‑loop portal with a consistent cutoff prescription and compare |Δκ_KK| to |κ_loop⁽⁰⁾| for benchmark values such as |M₅|/M_KK = 1 and 2; if the ratio is of order one or larger, the predicted 10 TeV–100 PeV window is not robust. Alternatively, measure the spin‑independent dark‑matter cross section at the level predicted for M_N ≈ 10 PeV; its absence or presence would directly confirm or exclude the central mass‑scaling relation.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the construction is correct, direct‑detection cross sections and Higgs‑mixing observables in secluded dark‑matter models are naturally suppressed below current and near‑future experimental reach for heavy‑neutrino masses up to tens of PeV.
  • The same portal also sets the lifetime of the hidden scalar mediator; the BBN requirement that the mediator decay within about one second forces the heavy‑neutrino scale to lie above roughly 10 TeV (or 100 TeV with stronger sequestering).
  • The heavy‑neutrino mass window 10 TeV–10 PeV (or 100 TeV–100 PeV with |M₅|=2M_KK) becomes the key phenomenological target, with the lower bound driven by BBN and the upper bound by direct detection.
  • Laboratory signals from active–sterile mixing remain tiny because the seesaw predicts θ∼√(m_ν/Mₙ) ≈ 10⁻⁷, so prompt collider signatures are negligible in the canonical aligned regime.
  • The model predicts a tower of sterile‑neutrino KK states with masses m²_n = M₅² + n²M_KK²; if the compactification scale is multi‑TeV, these states are heavy and decouple from low‑energy processes except through threshold corrections.

Where Pith is reading between the lines

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

  • A testable extension would be to compute the full regulated KK spectral sum and check whether the massive‑level contributions, which the paper estimates can be comparable to the zero‑mode term, preserve the sign and the M_N⁴ scaling of the mixing angle; if they partially cancel, the allowed mass window could widen or narrow.
  • Because the loop‑induced portal is proportional to the light‑neutrino masses in the aligned limit, a measurement of the Higgs‑singlet mixing angle at the level of 10⁻¹⁰–10⁻⁹ would indirectly probe the absolute neutrino mass scale, offering a rare high‑energy handle on the neutrino sector.
  • The paper leaves the thermal history of the hidden sector only partially specified; a dedicated coupled Boltzmann analysis, especially in the regime |M₅| ≳ M_KK where light‑state thermal contact is exponentially suppressed, would either validate the relic‑density estimate or shift the preferred couplings and mass range.
  • If direct‑detection experiments reach the neutrino‑fog floor, the predicted σ_SI ∝ (M_N/m_Hp)⁴ sin²α could be used to infer the heavy‑neutrino mass even without observing collider signatures, effectively using dark‑matter scattering as a low‑energy probe of the seesaw scale.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a five-dimensional sequestered realization of the radiative neutrino portal to secluded dark matter. The SM is localized at y=0 and the dark sector at y=L of an S^1/Z_2 interval, with sterile neutrinos propagating in the bulk. Five-dimensional locality forbids a tree-level Higgs–hidden-scalar contact term, giving κ(Λ_UV)=0; the portal is regenerated by one-loop brane-to-brane sterile-neutrino exchange. The paper derives the 4D one-loop matching, the 5D KK decomposition, the brane-to-brane propagator, and a profile factor F_0(M_5,L) for the light zero-mode contribution. The phenomenological analysis combines direct-detection and BBN constraints on the induced Higgs–singlet mixing, yielding mass windows for the heavy-neutrino scale: roughly 10 TeV ≲ M_N ≲ 10 PeV in the 4D model and 100 TeV ≲ M_0 ≲ 100 PeV for |M_5|=2 M_KK in the 5D model.

Significance. The proposed mechanism is attractive: if the KK-tower contribution can be controlled, the construction gives a genuine geometric origin for the otherwise ad hoc boundary condition κ(Λ_UV)=0, and it ties the smallness of direct-detection signals to the seesaw scale. The paper's technical core is largely transparent and correct as far as it goes: the 4D one-loop matching in App. B is standard; the brane-to-brane kernel in Sec. III F and App. D is derived with a closed form; the finite-truncation diagnostics in App. G are honest and instructive; and the benchmark arithmetic is internally consistent. The central quantitative claims, however, are not yet established because the massive KK contribution Δκ_KK is never computed, and the 5D phenomenological curves use a thermal-relic estimate that the paper itself marks as invalid in the strong-localization regime.

major comments (3)
  1. [Sec. III F, Eq. (153); App. G, Eqs. (G19)–(G23)] The load-bearing assumption for all quantitative results—Eqs. (72), (163) and Figs. 5–6—is |Δκ_KK| ≪ |κ_loop^(0)|, but Δκ_KK is never computed. The text itself states that its quantitative effect 'cannot be determined without the complete regulated spectral sum' and that the zero-state curves 'should be regarded as illustrative.' This is not merely a cosmetic caveat: App. G's own diagnostics give R_pure_1 ≃ 0.54 and R_pure_2 ≃ 2.20 at |M_5|/M_KK = 1, and 2.31 and 36.1 at |M_5|/M_KK = 2. These numbers omit phases and mixed terms, but they directly contradict the assumption that individual massive levels are parametrically negligible. Since the direct-detection bound scales as M_N^4 and the BBN lower bound is also set by the same portal, an O(1) or larger KK correction shifts both sides of the quoted windows. The paper should either compute the regulated spectral sum (or provide a rigorous
  2. [Sec. III G 2; Figs. 5–6] The 5D curves in Figs. 5 and 6 use y_p ≃ 0.4 from the standard secluded thermal-relic estimate, Eq. (59). But for |M_5|/M_KK = 1 and 2, the paper's own criterion gives K_0 ≃ 45 e^{-2π|M_5|/M_KK} ≃ 0.084 and 1.6×10^{-4}, respectively. The text states that in this regime light-state thermal contact is 'strongly suppressed' and that the standard thermal-relic estimate 'must be replaced by a coupled Boltzmann analysis.' Using the standard estimate for exactly these curves is therefore internally inconsistent. The resulting mass window in Eq. (163) cannot be considered a firm prediction until the thermal history is either computed or the curves are clearly flagged as indicative under an additional unverified assumption.
  3. [Sec. III F and Sec. IV] The abstract and conclusions present the heavy-neutrino mass windows as established results, while Sec. III F explicitly warns that the KK threshold can modify the normalization and the inferred bounds. This mismatch is not just presentational: the headline quantitative payload of the paper—the shift from Eq. (72) to Eq. (163)—is precisely what depends on the uncomputed Δκ_KK. The paper would be substantially strengthened if the conclusions were rephrased in terms of what is actually shown: a viable geometric mechanism whose quantitative realization requires a controlled KK summation.
minor comments (5)
  1. [Sec. II A] Typo: 'mass matrixes' should be 'mass matrices'.
  2. [Fig. 5 caption] Typo: 'heavy neutrini mass' should be 'heavy neutrino mass'.
  3. [Sec. I] Repetition: the text says the resonant mechanism 'relies on a special relation between the DM and mediator masses, which relies on a special mass relation that is not generic'—the clause is duplicated and should be streamlined.
  4. [Acknowledgments] The personal note addressed to the referee is unconventional and inappropriate for a serious journal; it should be removed.
  5. [Sec. III G 1] The notation M_ref^N and M_0 is used interchangeably in places; since Eq. (157) relates the two, the figure axes and the quoted bound in Eq. (163) should make clear which quantity is plotted and which is constrained.

Circularity Check

0 steps flagged

No circularity: the radiative portal is computed from seesaw inputs and checked against external constraints; the uncomputed KK contribution is a robustness gap, not a definitional loop.

full rationale

The central derivation is self-contained rather than circular. The loop-induced portal is obtained by an explicit one-loop matching calculation (Sec. II C and Appendix B), giving κ_loop = −Σ_I y_N,I² (Y_ν†Y_ν)_II/(4π²) from the same Yukawa couplings that enter the type-I seesaw. The Casas-Ibarra parametrization is used only to express (Y_ν†Y_ν)_II in terms of physical light-neutrino masses and heavy-neutrino masses; no fitted parameter is later renamed as a prediction. The 5D boundary condition κ(Λ_UV)=0 follows from the assumed brane-separated geometry, Eq. (94): ∫ dy δ(y)δ(y−L)=0 for L≠0. This is a model premise, not an output derived from the portal it is used to explain. The phenomenological predictions (σ_SI, τ_Hp, mass windows) are then compared with external LZ/XENONnT and BBN constraints, providing independent falsifiability. The paper does rely on the author's earlier Ref. [28] for the 4D framework and thermal-history details, but the one-loop portal is re-derived here and the 5D sequestering argument is new; no load-bearing claim rests solely on an unverified self-citation. The manuscript itself flags a genuine limitation: Sec. III F states that the full KK threshold 'can change their normalization and may also modify the inferred numerical bounds' and that its 'quantitative effect cannot be determined without the complete regulated spectral sum,' while Appendix G shows R_pure_1 ≃ 0.54 and R_pure_2 ≃ 2.20. This is a completeness and robustness concern about the single-state approximation, not circularity: the zero-mode result is not defined in terms of the uncomputed KK sum, and no equation forces the quoted bounds to equal their own inputs by construction.

Axiom & Free-Parameter Ledger

8 free parameters · 10 axioms · 5 invented entities

The observable consequences of the model are engineered to sit below present and near-future sensitivity; the quantitative predictions are derived from inputs (m_nu, y_N, M_N, |M5|/M_KK) rather than fitted, but those inputs are benchmarks chosen by hand, and the two most load-bearing procedural assumptions (KK truncation, thermalization of the hidden sector) are acknowledged in the text as unverified.

free parameters (8)
  • m_Hp (hidden-scalar mass) = 10 GeV benchmark
    Chosen by hand as low-energy input. The paper acknowledges delta m^2_Hp ~ y_N^2 M_N^2/(16 pi^2) destabilizes a light singlet (Sec. II G), so 10 GeV requires tuning.
  • y_N (hidden Yukawa) = O(1), taken = 1 in benchmarks
    Hidden-sector Yukawa controlling the portal magnitude (Eqs. (35)-(36)); chosen, not derived.
  • m_nu (light-neutrino mass) = 0.05 eV (aligned seesaw benchmark)
    Representative light-neutrino mass setting the portal scale via Eq. (35); input from oscillation data, value chosen for illustration.
  • |M5|/M_KK (localization parameter) = 0, 1, 2 in figures
    Bulk-mass-to-KK-scale ratio controlling the sequestering factor F_0 (Eq. (152)); scanned by hand to illustrate suppression.
  • Lambda_UV = 10^10 GeV in Fig. 2
    Scale where kappa(Lambda_UV)=0 is imposed; UV-to-threshold running is neglected in benchmark estimates (Sec. II D), so results carry implicit Lambda_UV dependence.
  • y_p (scalar-DM Yukawa) = 0.4 from Eq. (59)
    Fixed by the approximate secluded freeze-out estimate using x'_f, zeta, g_* assumptions; treated as an input thereafter.
  • M_bare (bare Majorana mass of sterile neutrinos) = 0 (baseline)
    Set to zero so M_N = v_r Y_N/sqrt(2); symmetry-protected in the Z4 realization, otherwise an assumed UV boundary condition (Sec. II A).
  • U(1)_X spectator sector (m_Z', g_X, kinetic mixing epsilon_X) = negligible / prompt
    Stueckelberg mass, zero kinetic mixing, and prompt Z' decays are assumed so the gauge sector drops out of all observables (Sec. II A).
axioms (10)
  • domain assumption Type-I seesaw relation m_nu ~ -m_D M_N^{-1} m_D^T
    Eq. (14); the entire link between neutrino masses and the portal rests on the seesaw hierarchy.
  • domain assumption Casas-Ibarra parametrization with aligned limit O = 1_3x3
    Eqs. (24)-(26); non-aligned O could change flavor factors in the portal, but benchmarks use the aligned limit.
  • domain assumption 5D locality plus regulated delta(y)delta(y-L) = 0
    Eqs. (91)-(95); the geometric heart of the paper assumes a standard local 5D EFT with no fundamental nonlocal brane operators.
  • domain assumption Secluded thermal history: heavy-neutrino bridge equilibrates sectors; freeze-out in hidden bath; p-wave chi chi -> H_p H_p
    Sec. II E; the paper's own K_0 criterion suggests this fails at |M5| >= M_KK (Sec. III G 2).
  • ad hoc to paper Z4 charge assignment (R -> -R, N_R -> i N_R, L -> i L, ...)
    App. A; keeps kappa_eff ~ kappa, forbids R^3, R Phi^dagger Phi, N^c N, and makes M_bare = 0 natural. A model-building choice, not derived.
  • domain assumption |Delta kappa_KK| << |kappa_loop^(0)| (single-state truncation of the portal)
    Sec. III F; acknowledged in the text as unverified; App. G's diagnostics suggest individual KK levels can be comparable or larger.
  • domain assumption Reheating hierarchy M_0 <~ T_RH << m_1 (KK tower not thermally populated)
    Eq. (173); needed for the thermal history to remain the standard secluded one.
  • domain assumption Flat, stabilized 5D background; radion neglected
    Sec. III G 3-4; Goldberger-Wise-type stabilization is invoked but not specified.
  • domain assumption Electroweak and singlet naturalness not imposed; m_Hp treated as a low-energy input
    Sec. II G; the largest-M_N region requires cancellations in delta mu_H^2 and delta m_Hp^2.
  • domain assumption Perturbativity of all couplings up to the cutoff
    Sec. II A; stated as a requirement for the EFT treatment.
invented entities (5)
  • Hidden scalar R (real singlet mediator) no independent evidence
    purpose: Dark mediator; carrier of the radiative portal and of chi chi -> H_p H_p annihilation
    All SM couplings are suppressed by sin alpha <~ 10^-5 in the benchmarks; the only 'signal' is further null results.
  • Dirac dark-matter fermion chi with vector-like U(1)_X charge no independent evidence
    purpose: Thermal secluded DM
    Designed to be invisible; p-wave suppression of chi chi -> H_p H_p removes present-day indirect signals.
  • U(1)_X gauge boson Z' (Stueckelberg) no independent evidence
    purpose: Spectator enforcing chi stability
    Explicitly assumed to decay promptly or never be populated, so it provides no falsifiable handle.
  • Fifth dimension S^1/Z_2 with branes at y=0 and y=L no independent evidence
    purpose: Geometric sequestering; origin of kappa(Lambda_UV)=0
    The KK tower is the only potential probe, but the model keeps it heavy (multi-TeV+) and weakly mixed, explicitly avoiding constraints.
  • Bulk sterile neutrinos N_I (5D Dirac fermions, right-handed zero mode) no independent evidence
    purpose: Seesaw messengers and portal generators
    Active-sterile mixing theta ~ 7x10^-8 (m_nu/0.05 eV)^{1/2} (10 TeV/M_N)^{1/2} (Eq. (66)) puts them out of reach of laboratory searches; no new positive signature.

pith-pipeline@v1.3.0-alltime-deepseek · 57413 in / 25336 out tokens · 254140 ms · 2026-08-03T00:30:03.482037+00:00 · methodology

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read the original abstract

We study a secluded dark-matter scenario in which the smallness of laboratory signals is linked to the origin of neutrino masses. In the four-dimensional theory, the dark sector contains a hidden scalar, a fermionic dark-matter particle, and heavy Majorana neutrinos. The same heavy-neutrino sector that generates light neutrino masses via the seesaw mechanism also induces the Higgs-dark-scalar portal at one loop. This portal is tied to the small observed neutrino masses, naturally leading to very small Higgs-singlet mixing and suppressed signals in direct-detection and collider experiments. We then provide a geometric origin for the boundary condition of the tree level portal coupling being zero $\kappa(\Lambda_{\rm UV})=0$ via a five-dimensional sequestered setup. In this construction, the Standard Model fields and the hidden sector are localized on different branes, with sterile neutrinos propagating in the bulk. Five-dimensional locality forbids a fundamental local tree-level Higgs-dark-scalar contact interaction, while heavy-neutrino loops induce a small residual portal coupling. The main phenomenological parameter controlling the viability of the model is the heavy-neutrino mass scale: direct detection bounds it from above, while Big Bang nucleosynthesis bounds it from below through the requirement that the hidden scalar decays sufficiently early. Depending on the amount of five-dimensional sequestering, the viable region can span heavy-neutrino masses from the multi-TeV scale to the PeV scale, or even higher.

Figures

Figures reproduced from arXiv: 2607.28754 by Mattia Di Mauro.

Figure 1
Figure 1. Figure 1: FIG. 1. One-loop box diagram generating the radiative [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Running and threshold matching of the radiatively [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic representation of the five-dimensional sequestered setup. The visible sector is localized on the brane at [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Geometrical profile factor [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Spin-independent direct-detection cross section as a [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Combined direct-detection and BBN constraints in [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. One-loop box diagram generating the radiative [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗

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Reference graph

Works this paper leans on

112 extracted references · 58 linked inside Pith

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    In particular, a visible-boundary vertex probes the field aty= 0, whereas a hidden-boundary vertex probes it aty=L

    Right-handed second-order Green kernel To compute amplitudes involving interactions localized on different boundaries, we need the propagator of the bulk fermion between two points in the fifth dimension y. In particular, a visible-boundary vertex probes the field aty= 0, whereas a hidden-boundary vertex probes it aty=L. The quantity relevant for communic...

  2. [2]

    The spectral representation de- rived from the KK decomposition in Appendix C is GR(pE;y, y′) = f (0) R (y)f (0) R (y′) p2 E + ∞X n=1 f (n) R (y)f (n) R (y′) p2 E +m 2n

    Spectral representation and exact orbifold kernel The right-handed Green kernel can be represented ei- ther as a sum over the free KK eigenstates or, equiva- lently, in a closed form obtained by solving the differ- ential equation directly. The spectral representation de- rived from the KK decomposition in Appendix C is GR(pE;y, y′) = f (0) R (y)f (0) R (...

  3. [3]

    This quantity compares the separation between the two boundaries with the characteristic Euclidean propagation length 1/ω

    Momentum regimes and brane-to-brane suppression The behavior of the exact cross-boundary kernel de- pends on the dimensionless quantityωL= p p2 E +M 2 5 L. This quantity compares the separation between the two boundaries with the characteristic Euclidean propagation length 1/ω. Two distinct regimes can be identified. For ωL≪1,(D23) 39 the propagation leng...

  4. [4]

    Indeed, for every mode with n≥1, 1 p2 E +m 2n ≃ 1 m2n , p 2 E ≪m 2 n,(D32) so that the corresponding contribution remains finite as p2 E →0

    Low-energy zero-mode propagation For Euclidean momenta well below the first KK mass, p2 E ≪m 2 1,(D31) the excited KK contributions are suppressed relative to the zero-mode contribution. Indeed, for every mode with n≥1, 1 p2 E +m 2n ≃ 1 m2n , p 2 E ≪m 2 n,(D32) so that the corresponding contribution remains finite as p2 E →0. By contrast, the free zero-mo...

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    This limit provides a simple analytical expression that can be evaluated directly and compared with the corresponding four-dimensional the- ory

    Perturbative single-state limit Although the one-generation result derived above still contains the complete infinite tower of physical Majo- rana states, it is useful to identify a controlled limit in which the amplitude can be approximated by the contri- bution of a single light state. This limit provides a simple analytical expression that can be evalu...

  6. [6]

    Brane interactions and dimensions of the couplings It is useful to distinguish the five-dimensional brane coefficients bY5,ν and bY5,N from the dimensionless four- dimensional Yukawa couplings obtained after compacti- fication. With the orbifold parity assignment used in this work, the relevant brane interactions involving the fields NI are Svis ⊃ − Z d4x...

  7. [7]

    Consequently, in addition to the ordinary contrac- tion⟨N N ⟩, one must also include the anomalous con- tractions involvingNandN c

    Majorana mass, physical eigenstates, and physical profiles Once the Majorana interaction is included, sterile- fermion number is not conserved and the physical mass eigenstates are Majorana fermions, for which the parti- cle and antiparticle are not independent degrees of free- dom. Consequently, in addition to the ordinary contrac- tion⟨N N ⟩, one must a...

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    (E13) The first maps sterile-flavor space into lepton-flavor space, while the second maps lepton-flavor space back into sterile-flavor space

    Boundary F eynman rules and loop topology Since the visible-boundary interaction Lagrangian is written as Lvis ⊃ −δ(y) bY5,ν αI Lα eΦN IR + h.c.(E10) it is convenient to define the transition operators ΓLN eΦ αI = bY5,ν αI PR,(E11) ΓN L eΦ† Iα = bY † 5,ν Iα PL.(E12) In terms of these operators, the visible-boundary in- teraction can be written in the comp...

  9. [9]

    (E23) Matching this one-loop vertex to Eq

    General one-loop amplitude The complete mixed-representation one-loop ampli- tude, with all external fields stripped off, is Γ(1) RRΦΦ† (0) =−N comb Z d4pE (2π)4 Trspin,flav,NG h ΓRN NSN (pE;L, L)ΓRN NSN (pE;L,0)Γ N L eΦ† SL(pE)ΓLN eΦ SN (pE; 0, L) i . (E23) Matching this one-loop vertex to Eq. (E21) gives κ5,loop =− 1 2 Γ(1) RRΦΦ† (0).(E24) for two exter...

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    One five-dimensional sterile generation To obtain an expression that is sufficiently simple for a transparent analytical and numerical evaluation, we first restrict the discussion to a single five-dimensional sterile generation. This assumption removes the additional fla- vor structure associated with several bulk fermions and isolates the effects of the ...

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    (E37), the lightest physical state is dominated by the free zero mode, so that FR,0(y)≃f (0) R (y),(E45) where the normalized right-handed zero-mode profile is given in Eq

    Matching to the flat-profile four-dimensional reference result In the weak-mixing approximation of Eq. (E37), the lightest physical state is dominated by the free zero mode, so that FR,0(y)≃f (0) R (y),(E45) where the normalized right-handed zero-mode profile is given in Eq. (C41). At the matching scaleµ=M 0, the single-state contri- bution to the portal ...

  12. [12]

    Beyond the single-state approximation The single-state formulas are reliable only when M0 ≪m 1, ϵ n ≪1.(E65) If the boundary-induced mixing is not perturbative, the physical profilesF R,Ia must be retained and can differ substantially from the free profiles. The complete coeffi- cient may be organized as κ5,loop(µ) =κ (0) 5,loop(µ) + ∆κKK(µ),(E66) Here ∆κ...

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    (C2) and (C73)

    Dimensional reduction of the action The five-dimensional action is the sum of the bulk sterile-fermion action and the visible- and hidden- boundary actions, S5 =S bulk +S vis +S hid.(F1) The explicit form of these terms was given in Eqs. (C2) and (C73). The visible and hidden fields are already four- dimensional because they are confined to the boundaries...

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    Their effects are encoded in operators suppressed by in- verse powers of the KK scale and in finite threshold cor- rections to the parameters of the four-dimensional theory

    Low-energy four-dimensional limit A finite four-dimensional effective theory is obtained when the characteristic energy of the process is much smaller than the first KK mass: E≪m 1, m 1 = r M 2 5 + π2 L2 .(F6) The KK-dominated states can then be integrated out. Their effects are encoded in operators suppressed by in- verse powers of the KK scale and in fi...

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    Relation between the five- and four-dimensional masses Before hidden symmetry breaking, the five- dimensional theory contains a massless right-handed chiral zero mode and a tower of free Dirac states with masses m2 I,n =M 2 5,I + n2π2 L2 , n≥1,(F11) as derived in Eq. (C54). The bulk parametersM 5,I there- fore determine the localization of the zero modes ...

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    Its boundary values satisfy f (0) R (0) f (0) R (L) =e −µL.(F16) Equation (F8) then shows that the effective visible Yukawa coupling is exponentially suppressed

    Sequestering and the effective four-dimensional couplings For the sequestered branch relevant to the present model, M5 =−µ, µ >0,(F15) the light sterile state is localized toward the hidden boundary aty=L. Its boundary values satisfy f (0) R (0) f (0) R (L) =e −µL.(F16) Equation (F8) then shows that the effective visible Yukawa coupling is exponentially s...

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    Reduction of the radiative portal In the five-dimensional theory, a local tree-level op- eratorR 2Φ†Φ is absent becauseRand Φ are localized on different boundaries. The corresponding ultraviolet boundary condition is therefore κtree(ΛUV) = 0.(F18) After compactification, the physical Majorana states cou- ple to both boundaries throughλ ν andg N . Their ex...

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    Pure first-level estimate A simple analytic diagnostic is obtained by neglecting the boundary-induced mixing in the massive sector and retaining only the pure contribution of one free KK level. For then-th level, the field-dependent Weyl mass matrix may be written as Mn(R, ϕ) =   0λ nϕ0 λnϕ gnnR mn 0m n 0   ,(G13) whereϕdenotes a fixed neutral compo...

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    Five-dimensional action and induced metrics The gravitational action on the interval is Sgrav = M 3 ∗ 2 Z M5 d5X p |g5| R5 +S GHY,(H3) where d5X≡d 4xdy, g 5 ≡detg AB,(H4) andS GHY denotes the Gibbons–Hawking–York boundary term required for a well-defined metric variation on a manifold with boundaries [84, 85]. Explicitly, SGHY =M 3 ∗ X i=vis,hid ϵi Z y=yi...

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    Five-dimensional Einstein equations The variation of the gravitational action with respect to the inverse metric gives δSgrav = M 3 ∗ 2 Z d5X p |g5|G 5,AB δg AB,(H24) after including the boundary term in Eq. (H5). The five- dimensional Einstein tensor is G5,AB =R 5,AB − 1 2 gABR5.(H25) The variation of the matter actions is δSmatter =− 1 2 Z d5X p |g5| T ...

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    Kaluza–Klein decomposition of the graviton The universal coupling of the graviton zero mode to both branes can be seen directly by expanding the five- dimensional metric around the flat background: gAB(x, y) =ηAB + 2 M 3/2 ∗ hAB(x, y).(H44) The normalization in Eq. (H44) is chosen so that the five- dimensional graviton field has a canonical kinetic term. ...

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    At linear order, δγi,µν = 2 M 3/2 ∗ hµν(x, yi).(H58) Substituting this relation into Eq

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    Newtonian limit To obtain the Newtonian limit, we write the weak-field four-dimensional metric as ds2 4 = (1 + 2ΦN ) dt2 −(1−2Ψ N )δ ijdxidxj.(H71) For nonrelativistic matter with negligible anisotropic stress, T00 ≃ρ,|T 0i| ≪T00,|T ij| ≪T00,(H72) and ΦN = ΨN .(H73) At linear order, the 00 component of the Einstein tensor is G(4) 00 ≃2∇ 2ΦN .(H74) Equatio...

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    Cosmological equations The same conclusion follows in a homogeneous and isotropic cosmological background. Once the compactifi- cation radius is stabilized, the ordinary four-dimensional Friedmann equations are recovered at energies below the gravitational KK and radion scales [74, 86]. The four- dimensional metric is ds2 4 = dt2 −a 2(t)δijdxidxj.(H99) Th...

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