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REVIEW 3 major objections 5 minor 23 references

A nonlocal Stueckelberg portal to the dark sector can leave the dark photon mass unconstrained for nonlocality scales above about 1 TeV while still allowing a thermal dark-matter relic.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-13 18:02 UTC pith:YD5W4QTY

load-bearing objection Clean meson-level extension of nonlocal Stueckelberg portal; the unconstrained-m_A' claim for Λ_NL ≳ 1 TeV is real for their form factor but rests on an unvaried IR power. the 3 major comments →

arxiv 2603.25712 v2 pith:YD5W4QTY submitted 2026-03-26 hep-ph

Nonlocal Portal to the Dark Sector

Sergey Kovalenko , Sergey Kuleshov , Valery E. Lyubovitskij , Alexey S. Zhevlakov This is my paper
classification hep-ph
keywords dark photonStueckelberg portalnonlocalitymeson decaysinvisible decaysdark sectorform factor
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 proposes that the only link between the Standard Model and a dark sector is a massive dark photon that couples nonlocally to ordinary quarks and leptons. In the strict local limit the two sectors fully decouple. The nonlocality is encoded in a momentum-dependent form factor that vanishes both at vanishing momentum transfer and at infinite nonlocality scale. Matching the nonlocal quark currents onto an effective theory of light mesons, the authors compute widths for invisible vector-meson decays and semi-invisible pseudoscalar decays. Because the form factor is strongly suppressed when the dark-photon mass is far below the nonlocality scale, existing meson, collider and cosmological limits lose sensitivity once that scale exceeds roughly 1 TeV, leaving the dark-photon mass free. The same infrared suppression weakens direct-detection scattering while leaving thermal annihilation intact, so an observable relic density remains compatible with null results.

Core claim

A nonlocal realization of the Stueckelberg portal, implemented by the entire form factor ê(k^{2})=(k^{2}/Λ_NL^{2})^{2} exp(k^{2}/Λ_NL^{2}), makes the Standard Model and dark sector decouple in the local limit and renders existing meson-decay, collider and cosmological bounds insensitive to the dark-photon mass once Λ_NL ≳ 1 TeV, while still permitting a thermal dark-fermion relic.

What carries the argument

The nonlocal Stueckelberg form factor ê(k^{2}) that multiplies every dark-photon–fermion vertex; it converts constant portal couplings into a momentum-dependent suppression that vanishes both as k^{2} o0 and as Λ_NL o∞, and is matched onto meson effective Lagrangians via vector-meson dominance and Wess–Zumino–Witten terms.

Load-bearing premise

The concrete entire-function shape chosen for the form factor is not derived from any ultraviolet completion; a different infrared power would redraw the exclusion contours and could restore bounds below 1 TeV.

What would settle it

A positive signal for on-shell dark-photon production in rare meson decays (for example NA64 π^{0},η oγ+invisible) at a mass well below 1 TeV with a coupling strength that contradicts the predicted form-factor suppression would falsify the claimed decoupling.

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

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 nonlocal realization of the Stueckelberg portal connecting the Standard Model to a U(1)_D Dark Sector. A massive Dark Photon A' couples to SM quarks and leptons through flavor-dependent vector and axial couplings that are promoted to nonlocal operators via an entire form factor ε̂(k²)=(k²/Λ_NL²)² exp(k²/Λ_NL²). After matching the quark-level interactions onto a chiral-plus-resonance effective Lagrangian (including WZW terms), the authors derive widths for A'→ℓℓ̄, P→γA', V→χχ̄ and P→χχ̄γ. They recast existing collider, cosmological and meson-decay bounds into the (m_A',Λ_NL) plane and conclude that for Λ_NL≳1 TeV the Dark-Photon mass is left unconstrained, while the same infrared suppression can reconcile a thermal relic density with null direct-detection results.

Significance. If the nonlocal portal is realized as claimed, the work supplies a concrete, meson-level map between Stueckelberg couplings and laboratory observables (invisible and semi-invisible light-meson decays) that is useful for fixed-target and missing-energy experiments. The effective Lagrangian matching via vector-meson dominance and WZW terms is standard and transparent, and the decay-width formulae (Eqs. 18, 23–25) are ready for recasting. The idea that an entire form factor can simultaneously suppress t-channel direct detection while leaving s-channel annihilation viable is of genuine phenomenological interest and extends earlier nonlocal-kinetic-mixing proposals. The result is, however, tied to a specific infrared power of the form factor that is not derived from a UV completion; the significance of the unconstrained-m_A' claim therefore remains conditional on that choice.

major comments (3)
  1. [Sec. V, Fig. 6, Eq. (7)] Sec. V and Fig. 6: the central claim that m_A' is unconstrained for Λ_NL≳1 TeV is obtained by recasting local limits via ε̂(m_A'²)g/e<ε_exp(m_A'), using the single form factor of Eq. (7) whose infrared factor (k²/Λ_NL²)² supplies the decisive suppression. The paper never varies the infrared power (e.g. n=1 or n=4), never compares pure exponential entire functions, and never shows that the unconstrained region survives under such variations. Because entire-function requirements only constrain analyticity and growth, different infrared powers remain allowed and would quantitatively alter (or remove) the claimed decoupling. A robustness scan, or an explicit UV-motivated derivation of the power, is needed before the claim can be regarded as model-independent within the nonlocal framework.
  2. [Sec. V] Sec. V: the statement that meson-decay bounds also leave m_A' unconstrained for Λ_NL>1 TeV is asserted qualitatively (“we found that…”) without presenting the corresponding exclusion contours or the numerical values of the recast limits from BABAR, NA64 or projected NA64 pion-beam sensitivities. Given that the meson widths (Eqs. 23–25) are the paper’s main calculational contribution, the recasting should be shown explicitly (analogous to Fig. 6) so that the reader can verify the Λ_NL≳1 TeV threshold.
  3. [Sec. II, Eq. (7)] Sec. II, Eq. (7): the form factor is written with exp(+k²/Λ_NL²). The text states that ultraviolet and infrared convergence hold in Euclidean space, but for the time-like kinematics that dominate the phenomenology (on-shell A' production, k²=m_A'²>0) the same expression grows with m_A'. The relation between the Minkowski matrix elements used in Secs. IV–V and the Euclidean entire function should be stated clearly, and any unitarity or growth constraints that restrict the allowed range of m_A'/Λ_NL should be discussed.
minor comments (5)
  1. [Sec. IV.D, Eq. (25)] Eq. (25): the phase-space factor contains (q²-m_η'²) while the process is written for a general pseudoscalar P; this should be (q²-m_P²) (or the formula should be restricted to η').
  2. [Sec. III, Eq. (14)] Eq. (14): the list of couplings repeats g_ρ^{0}ηγ twice with different numerical values (1.55 and 2.73 GeV^{-1}); the second entry is presumably g_ρ^{0}η'γ.
  3. [Fig. 1] Fig. 1: the three analytic forms shown in the legend are hard to parse from the text rendering; the caption should state the three functions explicitly and clarify which one is used for the exclusion plot.
  4. Throughout: several typographical slips (“strightforwardly”, “digram”, “V ector”, inconsistent spacing in “m_A'”) should be corrected in a revision.
  5. [Sec. II] Sec. II: the statement that kinetic mixing regenerated by loops is absorbed into g^{v,a} and fixed at a scale μ_0 is left without a quantitative estimate of the residual running; a short remark on the size of the regenerated mixing between μ_0 and the GeV scale would help the reader.

Circularity Check

1 steps flagged

No load-bearing circularity: the unconstrained-m_A' claim is a direct consequence of an explicit form-factor ansatz applied to external bounds, not a self-derived prediction.

specific steps
  1. self citation load bearing [Sec. II, paragraph introducing Stueckelberg portal; Refs. [5]]
    "the last dimension-4 operator, with g^{v,a}_{ij} flavor non-diagonal couplings, originates from a dimension-5 operator arising in the Stueckelberg realization of U(1)_D symmetry group. These flavor non-diagonal SM−DS portal was recently found in Ref. [5]."

    The local Stueckelberg portal that is later made nonlocal is justified solely by the authors’ own prior papers. The present work treats those couplings as free inputs rather than re-deriving them, so the self-citation is background, not a closed loop that forces the nonlocal phenomenology. Minor and non-load-bearing for the central claim.

full rationale

The paper proposes a nonlocal Stueckelberg portal by promoting free flavor couplings g^{v,a}_{ij} to a momentum-dependent entire form factor (Eqs. 6–7), matches those operators onto a standard meson EFT with WZW and VMD terms (Sec. III), computes decay widths, and recasts published local Dark-Photon limits via ê(m_A'²) g/e < ε_exp(m_A'). The IR suppression that opens the unconstrained region for Λ_NL ≳ 1 TeV (Fig. 6, Sec. V) is therefore an immediate algebraic consequence of the chosen power (k²/Λ_NL²)², not a quantity fitted to the same data or forced by a uniqueness theorem. Self-citations to the authors’ earlier local Stueckelberg-portal papers supply the starting local operators, which remain free parameters; they do not close a definitional loop with the present nonlocal results. No step equates a claimed prediction with its own input by construction. The form-factor choice is an ansatz (and is acknowledged as UV-challenging), which is a model-building assumption rather than circular reasoning.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 1 invented entities

The central claim rests on three free parameters (Λ_NL, g_A′, m_A′), one ad-hoc entire-function form factor, the assumption that the portal is the unique SM–DS communication, and the standard chiral+resonance matching. No new particle beyond the already-postulated Dark Photon and dark fermion χ is introduced; the nonlocality itself is the novel entity.

free parameters (3)
  • Λ_NL (nonlocality scale)
    Controls the strength of the form factor; chosen by hand and scanned; no first-principles value is derived.
  • g_A′ (universal Dark-Photon–quark coupling)
    Overall strength of the Stueckelberg portal; free parameter of the effective theory.
  • m_A′ (Dark-Photon mass)
    Free mass parameter of the Stueckelberg vector; scanned against experimental bounds.
axioms (4)
  • ad hoc to paper The SM–DS portal is realized solely by a nonlocal Stueckelberg operator that vanishes as Λ_NL → ∞.
    Stated in Sec. II; no derivation from a UV theory is given.
  • domain assumption The nonlocal form factor must be an entire function of k² (Efimov-type condition).
    Invoked via Refs. [8–11] to guarantee UV/IR convergence; standard in nonlocal QFT literature.
  • domain assumption Vector-meson dominance and WZW anomaly matching correctly embed the quark-level A′ currents into the light-meson Lagrangian.
    Standard low-energy QCD assumption used in Sec. III.
  • domain assumption Kinetic mixing regenerated by loops can be absorbed into a redefinition of the flavor-diagonal couplings at a chosen scale μ₀.
    Sec. II; conventional field-redefinition argument.
invented entities (1)
  • Nonlocal Stueckelberg portal form factor ε̂(k²) no independent evidence
    purpose: Provides the unique SM–DS communication that decouples in the local limit and suppresses low-momentum processes.
    Introduced by hand in Eqs. (6)–(7); no independent experimental handle outside the paper’s own recasting of existing bounds.

pith-pipeline@v1.1.0-grok45 · 14284 in / 3103 out tokens · 36708 ms · 2026-07-13T18:02:48.134330+00:00 · methodology

0 comments
read the original abstract

We propose a nonlocal realization of the Stueckelberg portal between the Standard Model and Dark Sector, which decouple in the local limit. This implies that the mediator, $U(1)_{D}$ Dark Photon $A'$ with a Stueckelberg mass, interacts nonlocally with the Standard Model quarks and leptons. We study phenomenological implications of this scenario for the meson decays into semi-invisible and invisible channels. We discuss the experimental limitations on the model parameters, including the nonlocality scale.

discussion (0)

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Reference graph

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