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This paper establishes an infinite tower of two-sided positivity bounds that every effective field theory of heavy scalar fields on de Sitter must satisfy if it is to come from a unitary, causal UV completion.

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2026-08-03 14:19 UTC pith:SMPYQM3Y

load-bearing objection First analytic two-sided positivity bounds for in-in dS EFTs, but the pivotal Eq. (30) bridging moments to EFT coefficients is asserted without derivation. the 2 major comments →

arxiv 2512.20706 v2 pith:SMPYQM3Y submitted 2025-12-23 hep-th gr-qc

Propagator positivity bounds for cosmological correlators

Mang Hei Gordon Lee , Scott Melville This is my paper
classification hep-th gr-qc MSC 81T1083F05 PACS 04.62.-v11.10.-z98.80.Cq
keywords positivity boundsde Sittereffective field theoryKällén-Lehmann representationin-in propagatorcosmological colliderprimordial non-Gaussianityself-energy
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.

Using unitarity and causality alone, this paper establishes an infinite tower of two-sided positivity bounds that every effective field theory of heavy scalar fields on de Sitter must satisfy — at every order in the EFT expansion — if it is to come from a consistent UV completion. The bounds apply to the in-in (Bunch-Davies) two-point function used for cosmological correlators, so they single out EFT correlators that can never emerge from a unitary, causal inflationary model. A sympathetic reader cares because this is the first analytic bound of this kind on de Sitter EFT coefficients, and because it ties the damping of cosmological-collider oscillations directly to the shape of the non-oscillating EFT background. The concrete payoff is a checkable restriction on primordial non-Gaussianity, e.g. a lower bound on the EFT background in terms of the measured decay rate of collider oscillations.

Core claim

The paper's main result is a tower of two-sided positivity bounds, displayed in (24): for the EFT combinations g_n defined by the subtraction formula (30), one must have g_2 > 0, 0 < g_4/g_2 < g_3/g_2 < 1 and (g_3/g_2)^2 < g_4/g_2, at every order in the EFT expansion. These g_n are the positive moments of the spectral density ρ of the de Sitter self-energy after subtracting the dissipative (decay) part, which is why unitarity (ρ > 0) implies the inequalities. Because on de Sitter there is no mass gap — a light particle can decay into heavier particles — the usual Minkowski positivity of the coefficient c_2 is replaced by c_2 > ∫_{−Λ}^{Λ} dμ/(π μ) Im Σ(μ)/μ^4, with the H→0 limit recovering c_

What carries the argument

The central object is the Källén-Lehmann representation of the in-in (Bunch-Davies) self-energy, Σ(ν) = ∫ dμ'^2/π ρ(μ')/(μ'^2 − ν^2 − iε), with positive spectral density ρ = Im Σ > 0 enforcing unitarity and the prescribed analytic structure enforcing causality. Because the EFT must capture both the conservative and dissipative parts of the propagation, the paper doubles the fields (ϕ±) and writes the EFT self-energy as Σ_EFT(ν) = Σ_n (c_n + i γ_n tanh(πν)) ν^{2n}. The load-bearing identity is the subtraction formula g_n(Λ) = Λ^{2n} c_n − Σ_j 2Λ^{2j+1} γ'_j/(1 + 2j − 2n), which maps the physical EFT coefficients onto the positive spectral moments (28). It is the absence of a mass gap on de Si

Load-bearing premise

The subtraction formula (30), which converts the EFT coefficients c_n and γ_n into the positive moments g_n, is asserted without derivation; if this subtraction is incorrect or needs additional counterterms, the bounds (24) do not apply to the physical EFT coefficients.

What would settle it

Start from a unitary, causal UV completion (e.g. the ϕχ^2 bubble in d=3) and add a local counterterm that shifts c_2 by an arbitrary real constant while leaving the spectral density ρ unchanged. If for some counterterm value the combinations g_4/g_2 − (g_3/g_2)^2 or 1 − g_3/g_2 computed via (30) become negative, the bounds are violated by a healthy theory and (30) must be incomplete; alternatively, if every such shift keeps the combinations in the allowed leaf, the bounds survive this test.

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

If this is right

  • EFT correlators that violate the inequalities (24) — for instance with (g_3/g_2)^2 > g_4/g_2 — cannot be produced by any unitary, causal UV completion, so they are ruled out as descriptions of an inflationary phase.
  • In the flat-space limit H→0 the bounds reduce to the familiar Minkowski positivity c_n > 0; on de Sitter the sign of c_2 is instead controlled by the decay width via c_2 > ∫ dμ/(π μ) Im Σ(μ)/μ^4.
  • In the squeezed bispectrum, the damping rate γ̃ of the oscillatory cosmological-collider signal is set by Im Σ, and the bounds translate into concrete restrictions on the non-oscillating EFT background — e.g. c_2 Λ^3 > −4γ̃/(3π) at leading order — so a measured decay rate forces a minimum size of background corrections.
  • The in-in EFT has a radius of convergence set by the smallest mass difference (|ν|^2 = |μ1−μ2|^2 + d^2/4), not the lightest mass, so truncation errors at intermediate ν can be O(1) unless enough orders are kept — an explicit warning for EFT truncation in cosmology.
  • Positivity of Σ(iα) for imaginary ν — an extra condition beyond the g_n bounds — implies the EFT background is always made larger by heavy physics in the squeezed limit, a distinct, testable prediction.

Where Pith is reading between the lines

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

  • If the subtraction formula (30) fails for some UV completion with additional counterterms, the bounds would not apply to c_n and γ_n as stated; a complete proof (or a counterexample) of (30) would settle this and is a natural next step the paper does not take.
  • The leaf-shaped allowed region for {g_3/g_2, g_4/g_2} is the seed of a de Sitter 'EFThedron'; adding further physical requirements (spin dominance, fixed spatial dimension d=3) is likely to shrink the allowed region and produce sharper bounds.
  • Because the bounds tie the oscillatory and non-oscillatory parts of the bispectrum, they provide a direct falsifiable link between future measurements of primordial non-Gaussianity and the unitarity/causality of the underlying inflation model — a logical template that could extend to boost-breaking correlators if the spectral representation can be generalised.
  • The numerical verification of (30) in tree, bubble and sunset examples is suggestive but not exhaustive; testing the bounds in a UV completion with a known local counterterm that changes c_2 without changing ρ would stress the subtraction scheme.

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

2 major / 5 minor

Summary. The paper constructs an in-in effective field theory for a light scalar field propagating on a fixed de Sitter background, with quadratic Wilson coefficients c_n (conservative) and γ_n (dissipative). Using a Källén-Lehmann representation for the self-energy with a positive spectral density, it defines cut-off moments g_n(Λ) and asserts a formula (Eq. (30)) relating these moments to the EFT coefficients. From positivity of the spectral density it derives two-sided moment inequalities (Eq. (24)) and states that any unitary/causal UV completion must produce EFT coefficients satisfying them. It then checks the bounds in explicit UV completions: tree-level mass mixing, a one-loop bubble, and a two-loop sunset, all using spectral densities from prior work. Finally it discusses implications for the cosmological collider signal, claiming that the damping rate of oscillations is tied to the non-oscillating EFT background.

Significance. If the central relation (30) is established, the paper would provide a genuinely new class of analytic positivity constraints on de Sitter EFTs, going beyond Minkowski bounds by explicitly incorporating the particle-production (dissipative) part of the self-energy. The claimed two-sided bounds constrain combinations of EFT coefficients that have not previously been bounded, and the connection to cosmological collider phenomenology is potentially impactful. The paper has real strengths: it uses a physically motivated Källén-Lehmann representation with positive spectral density, considers nontrivial UV completions with explicit spectral densities, and includes numerical verification of the inequalities as well as a careful discussion of when the EFT expansion breaks down. However, the bridge between the positive moments and the EFT coefficients—Eq. (30)—is asserted without derivation, and this is the load-bearing step for the main claim. The significance is therefore conditional on that gap being filled.

major comments (2)
  1. [§IV, Eq. (30)] Equation (30) is the only connection between the positive moments g_n(Λ) defined in (28) and the EFT coefficients c_n, γ'_n appearing in (15)/(19), yet it is introduced with the phrase 'we therefore have' and no derivation. Substituting the EFT spectral density (29) into the sub-Λ part of the moment integral yields integrals of the form ∫_0^{Λ^2} dµ'^2 ρ(µ')/µ'^{2n+2}, which are IR-divergent whenever 2j−2n+1≤0. Equation (30) is precisely what one obtains by discarding the divergent lower-limit contributions, i.e. by an analytic/minimal subtraction, but the paper does not state this prescription or specify the renormalization scheme in which the c_n of the action (15) are defined. Different IR regulators or finite counterterm choices shift c_n by scheme-dependent constants, and the inequalities (24) do not automatically survive such shifts. The UV-completion checks in §V are suggestive bu
  2. [§V.B] The EFT expansion (29) converges only for |ν| below the lowest singularity (e.g. for the bubble, |ν| ≲ 2μ in the d=2 example). For cut-offs Λ above that scale, the infinite sum in (30) diverges and cannot be used to compute g_n from the EFT coefficients. The text acknowledges this in footnote 76, but the abstract and §IV state that the bounds hold 'at every order in the EFT expansion' without this caveat. The distinction between bounds on the UV-defined g_n and bounds on the low-energy EFT coefficients must be made explicit in the main text; as written, a reader could apply (30) outside its domain of validity.
minor comments (5)
  1. [Eq. (1)] The illustrative bound in Eq. (1), c_2 > ∫_{-Λ}^{Λ} dµ/(πµ) ImΣ(µ)/µ^4, appears inconsistent with the bound derived later in §VI, c_2Λ^3 > −2γ'_0/3 + ... . The integral as written is IR-divergent and does not reduce to the stated combination. Please correct or clarify the intended expression.
  2. [Eqs. (28)–(30)] The notation for the spectral density is inconsistent: (28) integrates over dµ'^2 with ρ(µ'), while (29) defines ρ(ν) as a function of ν. The change of variables from µ' to ν should be stated explicitly, since the powers in (30) depend on this convention.
  3. [Figure 1] The caption says the bounds 'forbid EFT coefficients in the grey region', but the axes are ratios of g_n, which are nonlinear combinations of c_n and γ'_n via (30). Please clarify that the forbidden region is in the space of these g-combinations, not the bare Wilson coefficients.
  4. [Footnote 76] Footnote 76 states that Fig. 2 plots all values of Λ even when the EFT series (30) does not converge. It would be helpful to explain in the main text how g_n is obtained in that regime (presumably from the UV integral (28) directly), and to mark the region where the EFT calculation of (30) is valid.
  5. [General] There are several typos and minor grammatical issues: 'Kallen-Lehamnn' (should be Källén–Lehmann), 'analytiticity', 'phenoemonlogical', 'renormalisaiton'. These should be corrected in a final version.

Circularity Check

0 steps flagged

No significant circularity: the central bounds follow from an external Källén–Lehmann spectral representation; the self-citations are peripheral and Eq. (30), while terse, is an algebraic moment identity rather than a fitted input.

full rationale

The central derivation is not circular. The de Sitter Källén–Lehmann representation (25)-(26) is imported from prior work with no author overlap ([30,40,44,67-72]), and unitarity is converted into positivity Im Sigma > 0 inside this paper via (11)-(13). The EFT coefficients c_n, gamma_n are fixed by matching Sigma_EFT in (19) to the UV self-energy, not by fitting to any quantity that the bounds are then said to predict. The positive moments g_n in (28) are defined directly from the positive spectral density rho; Eq. (30) is the algebraic expression of those moments once the EFT expansion (29) is substituted. Even though (30) is asserted in a single sentence and its low-nu subtraction is formally divergent without an explicitly specified renormalization scheme, that is a derivation/scheme gap (a correctness risk), not a circular step: the bounds (24) are not equivalent to assuming (30), and the numerical UV completions in Sec. V are used as consistency checks, not to set constants. Self-citations appear but are peripheral: [41] (co-authored by Melville) supplies mode functions/vertex constants used in the phenomenological appendix, and [63] is cited for the elementary moment inequality g_{n+1}<g_n; neither supports the main positivity claim, which would stand if those citations were removed. Thus no prediction is a renamed fit and no load-bearing premise reduces to author self-citation. The score of 2 reflects one minor non-load-bearing self-citation in the appendix, not circularity of the central derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The bounds rest on the standard dS spectral representation with positive measure, plus the paper-specific in-in EFT construction and an unproved subtraction formula (30). No free parameters are fitted to data; the EFT coefficients are the constrained outputs, and the cut-off Λ is an input scale.

axioms (6)
  • domain assumption dS Källén-Lehmann representation (25)-(26) with positive spectral density
    Imported from prior work [30,40,44,47,71,72]; positivity of ρ follows from inserting a complete set of states (unitarity). The derivation assumes this holds for the in-in self-energy.
  • domain assumption Unitarity implies Im Σ(ν) > 0 for ν > 0
    Eq (13) identifies ρ = Im Σ > 0; standard QFT on dS, used throughout §IV.
  • domain assumption Causality implies Σ(ν) is analytic in the lower half ν-plane
    Invoked to justify the spectral representation and contour arguments; cited to [43].
  • domain assumption In-in EFT action (15) with doubled fields and matching condition κ_n = γ_n (18)
    This is the paper's EFT construction; it is a modeling choice, not derived from a more fundamental principle, but it is internally consistent.
  • ad hoc to paper Subtraction formula (30): g_n = Λ^{2n} c_n − Σ_j 2Λ^{2j+1} γ'_j/(1+2j−2n)
    The mapping from g_n to physical EFT coefficients is stated without derivation; it is the paper's central technical assumption and the main derivation gap.
  • domain assumption Principal-series assumption (real μ, m^2 ≥ d^2/4)
    Only heavy principal-series fields are integrated out; complementary-series fields are left explicit. This scopes the bounds, and the paper notes the extension would require deforming the contour in (25).

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0 comments
read the original abstract

Using unitarity and causality, we derive an infinite tower of two-sided positivity bounds on the effective field theory coefficients which describe the propagation of heavy fields on de Sitter spacetime. We design this EFT to describe propagators with the in-in boundary conditions that are relevant for cosmological correlators. Our positivity bounds therefore identify EFT correlators that can never emerge from a consistent underlying model of inflation. This implies non-trivial constraints on primordial non-Gaussianity; for instance the cosmological collider oscillations in the squeezed bispectrum from the exchange of a heavy scalar are tied to the shape of its EFT background.

Figures

Figures reproduced from arXiv: 2512.20706 by Mang Hei Gordon Lee, Scott Melville.

Figure 1
Figure 1. Figure 1: For instance, the first EFT interaction beyond mass and kinetic counterterms, schemati￾cally c2∂ 4ϕ 2 , must obey c2 > 0 on Minkowski. On de Sitter, we show that c2 is no longer always pos￾itive and we establish a new bound: c2 > Z Λ −Λ dµ πµ Im Σ(µ) µ4 (1) where Λ is the EFT cut-off and Im Σ is the de￾cay width of ϕ due to the spacetime background. In the Minkowski limit H → 0, this decay width vanishes e… view at source ↗
Figure 3
Figure 3. Figure 3: For ν < 1 they decrease uniformly as ex￾pected from Minkowski, so the series can be safely truncated at any order. But for 1 < ν < 4µ 2 + 1, the dominant contribution is from a finite order in the EFT series, and truncating before this order will lead to O(1) errors in the EFT approximation of ρ. For ν > 4µ 2 + 1, the series coefficients grow more and more negative and unitarity is violated if the EFT seri… view at source ↗
Figure 2
Figure 2. Figure 2: The region of EFT parameter space traced [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The relative size of the first 25 terms appear [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

discussion (0)

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Reference graph

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