arxiv: 2604.07434 · v1 · submitted 2026-04-08 · ✦ hep-ph · astro-ph.CO· hep-th

Recognition: no theorem link

Scalars at the Cosmological Collider: Full Shapes of Tree Diagrams and Bispectrum Searches using Planck Data

Soubhik Kumar , Qianshu Lu , Zhong-Zhi Xianyu , Yisong Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:43 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COhep-th

keywords cosmological colliderprimordial bispectrumnon-Gaussianitymassive scalarsinflationPlanck datachemical potential

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The pith

Massive scalars during inflation generate explicit oscillatory bispectrum shapes from three tree diagrams that Planck data can test, showing a 1.5 sigma hint when a chemical potential enables heavier particles.

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

The paper derives unified shape functions for the bispectrum contributions from single-exchange, double-exchange, and triple-exchange diagrams involving massive scalar particles that decay into inflaton quanta. These functions capture the non-analytic oscillatory non-Gaussianity and remain valid for every combination of momenta in the kinematic space. Searches using these templates on Planck observations find no significant signal in the minimal setup. When the model is extended with a scalar chemical potential that allows production of particles with masses much larger than the Hubble scale, a global 1.5 sigma preference for non-zero non-Gaussianity appears for parameters where the chemical potential minus the mass is roughly three times the Hubble rate. This supplies a concrete way to look for new heavy particles by examining the statistics of primordial density fluctuations.

Core claim

We provide a unified evaluation of all three diagrams and derive the explicit shape functions for the bispectrum, valid across the entire kinematic space. We perform a search for these three processes with the Planck data, finding no evidence for NG. We also consider simple extensions of the minimal scenario that can counter the exponential suppression of the non-analytic signature, and produce on-shell particles with masses M ≫ H. In particular, we focus on the scalar chemical potential mechanism and extend our previous search to a wider range of chemical potential (ω) and M, finding global 1.5σ evidence for non-zero NG for the parameter space ω - M ≃ 3H.

What carries the argument

The explicit bispectrum shape functions obtained from a unified calculation of the single, double, and triple tree-level exchange diagrams of massive scalars.

If this is right

The derived shapes supply ready-to-use templates that improve the sensitivity of bispectrum searches in current and upcoming CMB data sets.
Current observations already constrain the allowed range of masses and couplings for these massive scalars in the minimal scenario.
Chemical potential effects lift the exponential suppression, permitting tests of particles with masses several times the Hubble scale during inflation.
Absence of a strong signal in the minimal model places upper limits on the interaction strengths that produce such non-Gaussianity.

Where Pith is reading between the lines

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

The same shape functions could be applied to other correlation functions or to data from next-generation surveys to test whether the mild hint strengthens.
Similar unified calculations might be performed for particles with spin or for higher-order diagrams to enlarge the set of searchable signatures.
The parameter region around ω - M ≃ 3H could be prioritized in theoretical model building that connects inflationary dynamics to high-energy particle spectra.

Load-bearing premise

The non-analytic oscillatory signatures arise specifically from the decay of massive scalar particles into inflaton quanta, and the chemical-potential extension accurately captures the production mechanism for M ≫ H without additional confounding effects.

What would settle it

A higher-precision bispectrum measurement from a future CMB experiment that shows no oscillatory feature at the amplitude and frequency predicted for ω - M ≃ 3H would rule out the interpretation of the 1.5 sigma hint.

Figures

Figures reproduced from arXiv: 2604.07434 by Qianshu Lu, Soubhik Kumar, Yisong Zhang, Zhong-Zhi Xianyu.

**Figure 2.** Figure 2: FIG. 2. An illustration of a CMF bulk-to-boundary propa1 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. The shape function of the single exchange model, [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. The shape function of double exchange diagrams [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. The shapes of triple exchange diagrams for selected [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. The shape function in the scalar chemical potential [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. CMB-BEST result for single exchange model. In the [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. The [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. CMB-BEST result of double exchange model. We [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13. Comparison of shapes of triple exchange diagrams [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗

**Figure 15.** Figure 15: FIG. 15. The best-fit values of [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗

**Figure 14.** Figure 14: FIG. 14. In the top panel, we show the local SNR for the [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗

**Figure 17.** Figure 17: FIG. 17. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p017_17.png] view at source ↗

**Figure 18.** Figure 18: FIG. 18. Comparison of CMF and bootstrap result for the [PITH_FULL_IMAGE:figures/full_fig_p019_18.png] view at source ↗

**Figure 20.** Figure 20: FIG. 20. Comparison of CMF and bootstrap result for the [PITH_FULL_IMAGE:figures/full_fig_p020_20.png] view at source ↗

**Figure 19.** Figure 19: FIG. 19. The shapes of SE diagrams in isosceles configura [PITH_FULL_IMAGE:figures/full_fig_p020_19.png] view at source ↗

**Figure 21.** Figure 21: FIG. 21. The shapes of double exchange diagrams for [PITH_FULL_IMAGE:figures/full_fig_p021_21.png] view at source ↗

read the original abstract

The Cosmological Collider (CC) provides a unique opportunity to probe the particle spectrum and fundamental interactions at extremely high energies. Massive particles, via their decay into inflaton quanta, can induce a non-analytic, oscillatory, primordial non-Gaussianity (NG), including the bispectrum. At tree level, three classes of such processes contribute to the bispectrum: 'single exchange', 'double exchange', and 'triple exchange', depending on the number of massive particle propagators. We provide a unified evaluation of all three diagrams and derive the explicit shape functions for the bispectrum, valid across the entire kinematic space. We perform a search for these three processes with the Planck data, finding no evidence for NG. We also consider simple extensions of the minimal scenario that can counter the exponential suppression of the non-analytic signature, and produce on-shell particles with masses $M\gg H$, the Hubble scale during inflation. In particular, we focus on the 'scalar chemical potential' mechanism and extend our previous search to a wider range of chemical potential ($\omega$) and $M$, finding global 1.5$\sigma$ evidence for non-zero NG for the parameter space $\omega - M \simeq 3H$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They supply explicit full-kinematic bispectrum shapes for the three tree-level exchange diagrams and report a marginal 1.5 sigma hint only after adding a chemical-potential extension.

read the letter

The paper works out the complete shape functions for single-exchange, double-exchange, and triple-exchange diagrams in the cosmological collider and gives them in a form that covers the whole kinematic range. It then runs a search on Planck data, finding nothing in the minimal case and a 1.5 sigma global signal once a scalar chemical potential is turned on to ease the Boltzmann suppression for heavy masses. That is the core contribution: the unified expressions and the extended parameter scan are new relative to earlier partial treatments. The calculations follow standard de Sitter Feynman rules and the search uses public maps, so the setup is straightforward and the null result in the basic model is stated plainly. The explicit shapes should be directly usable by anyone building templates for future analyses. The 1.5 sigma hint is the obvious soft spot. It is reported at the right level of caution and does not get oversold, but it remains consistent with a fluctuation; any reader will want to see how the significance moves under reasonable changes in priors or sky cuts. The chemical-potential mechanism itself is introduced cleanly to lift the exponential suppression, though it adds two free parameters whose physical origin could be spelled out more. This is for people who need ready-to-use bispectrum templates for collider-type signals or who are planning next-generation NG searches. The derivations look solid enough on the evidence given that the paper deserves a full referee rather than a desk rejection. I would send it out for review; the shapes are a practical addition even if the data result stays marginal.

Referee Report

0 major / 2 minor

Summary. The paper claims to provide a unified evaluation of the three tree-level diagrams (single-exchange, double-exchange, and triple-exchange) for the bispectrum induced by massive scalar particles in the cosmological collider. It derives explicit shape functions valid across the full kinematic space and performs a search for these signals in Planck CMB data, reporting no detection in the minimal model but a global 1.5σ hint for non-zero non-Gaussianity when extending to a scalar chemical potential for parameters satisfying ω - M ≃ 3H.

Significance. The explicit derivation of complete shape functions for all three diagrams supplies ready-to-use templates that advance cosmological collider phenomenology and enable more systematic data analyses in future CMB and LSS surveys. The Planck search, while returning only marginal evidence, concretely demonstrates the application of these templates and quantifies the reach of current data, particularly highlighting the role of the chemical-potential extension in mitigating Boltzmann suppression for M ≫ H.

minor comments (2)

The abstract states that the shapes are 'valid across the entire kinematic space' but does not specify the precise boundaries or any numerical checks performed at the edges of phase space; adding a brief statement or reference to a validation plot would strengthen clarity.
In the data-search section, the treatment of the look-elsewhere effect for the 'global' 1.5σ significance should be stated explicitly, including the number of independent parameter points scanned in the (ω, M) plane.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation for minor revision. The referee's description accurately captures the unified tree-level bispectrum shapes for single-, double-, and triple-exchange diagrams as well as the Planck search results, including the marginal hint in the chemical-potential extension. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No circularity: derivations from standard dS QFT and independent Planck search

full rationale

The paper derives explicit bispectrum shape functions for single-, double-, and triple-exchange tree diagrams via unified evaluation of Feynman rules in de Sitter space, valid over the full kinematic range. These steps rely on standard perturbative QFT techniques rather than any self-referential definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The Planck data search is performed on public maps and returns a statistically independent result (no detection in the minimal case, 1.5σ global in the chemical-potential extension). Although the text references extending a prior search, this does not reduce the new shape derivations or the reported evidence to an internal loop. The overall chain is self-contained against external benchmarks and standard methods.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard quantum field theory in de Sitter space during inflation, the assumption that tree-level diagrams dominate the bispectrum, and the validity of the chemical-potential mechanism for producing on-shell heavy scalars.

free parameters (1)

chemical potential ω and mass M
The 1.5 sigma hint is reported for the specific combination ω - M ≃ 3H, indicating these parameters are scanned or fitted against the data.

axioms (1)

domain assumption Quantum field theory calculations in de Sitter space during inflation are valid and tree-level diagrams capture the leading non-analytic signatures.
The entire cosmological collider framework invoked in the abstract relies on this standard assumption of inflationary cosmology.

pith-pipeline@v0.9.0 · 5530 in / 1409 out tokens · 140820 ms · 2026-05-10T17:43:49.477687+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

84 extracted references · 74 canonical work pages · 2 internal anchors

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background

Cosmological Bootstrap The central idea behind the Bootstrap approach is to derive a set of differential equations that the cosmological correlators satisfy. These equations can then be solved, subject to boundary conditions, to obtain the general kinematic form of the correlators. To illustrate the basic idea, consider the SE diagram with some interactio...
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background

Coupled Mode Function Besides the bootstrap method which provides analyt- ical results for certain diagrams, we will introduce the Coupled Mode Function (CMF) method as an efficient numerical approach for evaluating cosmological correla- tors. This method deals with the quadratic mixing of fields non-perturbatively, resulting in a set of coupled mode func...
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Single Exchange Firstly we consider a model for the SE diagram, where thenon-Gaussianityissourcedbythescale-invariantcou- plings Lint ⊃ ˜λ2Λa3φ′σ+ ˜λ(1) 3 2Λ a2φ′2σ,(31) in which we have introduced a cutoff scaleΛto make the coupling constants ˜λdimensionless. In Fig. 3, we show the shape functions in isosceles con- figurationS(k, k, k 3)for various masse...
[4]

Double Exchange Now we consider a model for the DE process. The couplings are given as Lint ⊃ ˜λ2Λa3φ′σ+ ˜λ(2) 3 a3φ′σ2.(36) Following the same procedure as in the SE model and using (26), one can construct the shape functions of DE diagrams in the(k1, k2, k3)space. The shapes in isosceles configurationsS(k 1, k1, k3)of selected masses are shown in Fig. 4...
[5]

Triple Exchange We consider the following model for TE, Lint ⊃ ˜λ2Λa3φ′σ+ 1 6 ˜λ(3) 3 Λa4σ3.(40) 0.001 0.005 0.01 0.05 0.1 0.5 1 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.001 0.005 0.01 0.05 0.1 0.5 1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 FIG. 4. The shape function of double exchange diagrams including all permutations (solid blue) and after normalizing to unity at equila...
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Chemical Potential For the scenarios described above, the non-analytic signal mediated by the heavy scalar become exponen- tially suppressed in theM≫Hregime, and instead the NG is dominated by ‘EFT’ contributions peaking in the equilateral configurations. Therefore, to explore the non-analytic signatures ofM≫Hparticles, we consider mechanisms where the ex...

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The result deviates from the equilateral shape only when˜ν≲1

The result becomes constant at large mass˜ν≳3and the value is almost identical to that of the standard equi- lateral shape, givingfNL = 18±42. The result deviates from the equilateral shape only when˜ν≲1. The most significant result appears at˜ν= 1.4, withfNL = 20±39 and a local significance of 0.52σ.7 1 2 3 4 5 -100 -50 0 50 100 0 1 2 3 4 5 0.0 0.2 0.4 0...
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complete

The peak in thef NL constraint provides a hint that the CMB data prefers an oscillatory bispectrum more than a smooth (equilateral) one, although it does not necessarily mean that the TE model correctly describes the underlying physics. We will expand more on this point in the discussion of the chemical potential model in the following subsection. B. Chem...
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To investigate the appearance of numerical instability, we calculate the shapeS(k 1, k2, k1)produced by the leftmost diagram in Fig

Single Exchange For SE, the folded limit of the bootstrap formulae appears when one of the external linesk 1 andk 2 is squeezed, andu≡2k 3/(k1 +k2 +k3)→1. To investigate the appearance of numerical instability, we calculate the shapeS(k 1, k2, k1)produced by the leftmost diagram in Fig. 1 for˜ν= 2(as a representative value) as a function ofk 2/k1 with bot...
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Double Exchange Unlike the case of SE diagrams, the bootstrap formulae for the DE diagrams suffer a lot more from the numerical instability. As in the previous case, we evaluate the shape S(k1, k2, k1)produced by the middle diagram in Fig. 20 for˜ν= 2with both methods, and the result is shown in Fig. 20. Since the bootstrap formulae of the DE dia- grams i...
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