{"id":"ecc06970-86d5-4501-a270-222d802669f0","arxiv_id":"2609.09288","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"First analytical determination of the leading massive scalar triangle loop signals in cosmological collider bispectrum and trispectrum, reproduced by effective bubble diagrams.","lead":"The paper computes the leading analytical signals from a massive scalar triangle loop in inflationary bispectrum and trispectrum for the first time. These signals can be reproduced by simpler bubble diagrams with an effective pinched coupling, providing complete templates for cosmological collider searches.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"No significant objection identified; the directional cutting rule is well-justified analytically and validated numerically, and the derived signals are internally consistent and checked.","rationale":"The reader's weakest assumption—that the directional cutting rule isolates all signal-contributing contributions—is indeed the most critical point in the derivation. However, the paper provides a detailed justification in the Supplemental Material, showing that the remainder terms cannot generate the required nonanalyticities. This argument appears mathematically sound. Furthermore, the authors validate the final analytical expressions with direct numerical evaluation of the original uncut integrals, which constitutes a strong independent check. No additional internal inconsistency or missing step was identified. Therefore, the verdict should remain ACCEPT (UNCHANGED). The agreement is partial because while the reader correctly identifed the key assumption, the paper addresses it thoroughly enough that it does not raise a serious concern.","tokens_in":19518,"tokens_out":2539,"duration_ms":24354,"concrete_test":"Reproduce the numerical check for the bispectrum signal (Eq. (26)) with a independent numerical integrator for the original Schwinger-Keldysh integrals (Eq. (5)) at a specific kinematic point, e.g., nu=1.0, k3/k12=0.01, and compare the extracted oscillation frequency and amplitude to the analytical formula. Disagreement at the percent level would indicate a subtle error in the derivation or the cutting rule; full agreement (within integration error) would further confirm the robustness.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After careful analysis, I find no load-bearing flaw that would undermine the central claim. The weakest assumption identified by the reader—that the directional cutting rule isolates all signal-carrying contributions—is thoroughly addressed in the Supplemental Material. The argument proceeds by decomposing the full master integral into a cut piece and three remainder terms. For each remainder term, the authors show that step functions force the soft-energy integral to produce only integer powers of E/k1, precluding oscillatory nonanalyticities. For the soft-momentum nonanalyticity, they show that every remainder term contains a factor Delta N(s_i,s_j) which either vanishes for same-sign spectral poles (due to the sin factor) or leads to an integer sum s12 or s34, making the 1/Gamma(s12) or 1/Gamma(s34) factor in the loop integral vanish for mixed-sign poles. Both branches of the loop integral (L1 and L2) are considered; L2 is analytic in the soft momentum anyway, so no additional nonanalyticity arises. The subsequent manipulations—contour deformations, use of Barnes lemmas, reconstruction of the tree-level single-exchange seed—follow standard techniques and are validated by the direct numerical evaluation of the original SK integrals reported in the text. While one could imagine subtle issues with the analytic continuation to the boundary k/E=1, the paper handles it by first working at 0<k/E<1 and then continuing the combined signal; the remainder terms, being analytic in E, do not spoil this. The numerical agreement reinforces the analytical derivation. Therefore, the central claim stands without a pressing unaddressed vulnerability.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper computes the leading analytical cosmological collider signals from a massive scalar triangle loop in the inflationary bispectrum and trispectrum for the first time. Using a directional cutting rule and partial Mellin–Barnes representations, the authors obtain the complete four-point and three-point signals in the relevant soft limits. They show that these signals can be reproduced by the simpler bubble diagram with an effective pinched coupling, and validate their results numerically against the original Schwinger–Keldysh integrals.","tokens_in":19923,"tokens_out":2330,"duration_ms":22263,"significance":"If correct, this work fills a long-standing gap in the analytic understanding of loop-level cosmological collider signals. The directional cutting rule is justified through a careful analysis of remainder terms and provides a systematic framework for extracting signals from general loop topologies. The reduction to a bubble diagram with an effective pinched coupling is both elegant and practically useful for template construction. Numerical validation strengthens confidence in the results. The approach extends to subleading orders, arbitrary masses, spins, and interactions, making it a significant methodological advance.","major_comments":[],"minor_comments":[{"comment":"The sentence “As an independent check, we have compared the analytical trispectrum and bispectrum signals with direct numerical evaluations of the corresponding loop integrals in the soft limits considered here and found excellent agreement.” appears twice in close succession (once before the paragraph beginning “As an independent check, we numerically evaluated …” and again at the start of that same paragraph). The duplication should be removed.","section":"Main text, preceding the Discussion"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"This paper does what it says: it gives the first complete analytic expressions for the leading cosmological collider signals from a massive scalar triangle loop, in both the bispectrum and trispectrum. The directional cutting rule is the key technical advance, and the authors argue carefully that the cut piece captures all the nonanalytic signal while the remainder terms are analytic in the relevant variables. They also show that the result reduces to a bubble diagram with an effective pinched coupling, which is a nice conceptual simplification. The numerical checks against the original Schwinger–Keldysh integrals give me confidence that the algebra is correct.\n\nThe paper is well organized, the notation is consistent, and the supplemental material provides enough detail to follow the derivation. The result is significant because triangle loop signals have been a recognized gap in the template-generation pipeline, and this work fills it cleanly. For phenomenological applications, having an analytic template rather than a numerical grid is a real advantage.\n\nSoft spots: 1) The derivation relies on the hierarchy k1 ≈ k2 ≫ E ≥ k, with k1 = k2 set to k12/2 at leading order. Corrections from the difference |k1 − k2| are assumed subleading, but the paper does not quantify how large these corrections can become before the signal shape changes appreciably. 2) The cutting-rule argument, while physically well motivated, still involves a series of technical steps (contour choices, branch assignments) whose cumulative justification depends on the detailed structure of the Mellin–Barnes integrals. The authors address the weakest point (remainder terms) in the supplement, but the argument is not trivial to verify quickly. 3) The three-point signal is obtained from the same master integral but only at the boundary k/E = 1, where the two branches cancel singularities. The analytic continuation is done, but the limit is delicate; a more explicit check at finite small k/E would strengthen confidence.\n\nNone of these are fatal. The central claim holds up, and the paper is a genuine step forward.\n\nWho is this for? Anyone working on cosmological collider phenomenology or inflation correlator technology. It deserves a serious peer review. Recommendation: send to a strong journal (e.g., PRL or JHEP) with a request to verify the numerics and check the soft-momentum expansion in a few test cases.","headline":"First analytic triangle-loop signals for the cosmological collider—a clean solution to a known bottleneck, validated numerically.","tokens_in":20347,"tokens_out":885,"would_cite":true,"duration_ms":9366,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"A massive scalar triangle loop yields, for the first time, leading analytical cosmological collider signals in the bispectrum and trispectrum, reducible to a bubble diagram with one effective pinched coupling.","keywords":["cosmological collider","inflationary bispectrum","inflationary trispectrum","massive scalar triangle loop","partial Mellin-Barnes representation","directional cutting rule","pinched coupling","Schwinger-Keldysh integrals"],"falsifier":"A decisive check would be to numerically integrate the full uncut Schwinger–Keldysh triangle-loop integrals over a grid of $k_s/k_{34}$ values with $k_{34}/k_{12} \\ll 1$ and $\\tilde{\\nu} \\sim 1$, extract the oscillatory nonlocal component with frequency $2\\tilde{\\nu}$ in $\\log\\left(k_s^2/(4k_{12}k_{34})\\right)$, and compare its amplitude and phase with Eq. (23); any mismatch beyond numerical error would show that the directional cutting rule misses part of the loop signal.","tokens_in":21324,"feed_emoji":"🌌","tokens_out":14706,"duration_ms":104025,"temperature":0.7,"pith_summary":"During inflation, a heavy particle running around a loop leaves an oscillatory imprint in the correlations of density fluctuations, known as a cosmological collider signal, that encodes its mass. This paper supplies, for the first time, the leading analytical form of that signal when the loop is a triangle made of a massive scalar, for both the three-point and four-point correlators. The closed forms cover the squeezed bispectrum and the local and nonlocal branches of the trispectrum, and they are supported by direct numerical evaluation of the original loop integrals. The central result is that the entire leading triangle signal is equivalent to the corresponding bubble diagram with a single effective pinched coupling, so templates for these loop effects can be assembled from tree-level seeds rather than loop numerics.","feed_headline":"Massive triangle loop yields first analytic collider signals","feed_subtitle":"Leading triangle-loop signals collapse into a single-coupling bubble diagram, making templates fast.","key_machinery":"The engine of the computation is a directional cutting rule applied inside a partial Mellin–Barnes (PMB) representation. PMB represents massive mode functions as Mellin integrals so that time and loop-momentum integrals factor; the directional cutting rule replaces the two massive propagators ending at the low-energy vertex by their cut forms, removing time ordering across the cut and isolating the on-shell contributions that can produce collider signals. The loop-momentum factor is an isosceles massless triangle integral whose small-$k/k_{12}$ expansion separates into a soft branch and a hard branch, and only the same-sign spectral poles $s_1 = s_3 = -ic\\tilde{\\nu}/2$ with $c = \\pm 1$ contribute at leading order. After the remaining Mellin integrals are evaluated by residue sums, the leftover integrals recombine into a tree-level single-exchange seed $I_{\\tilde{\\nu}}$; pinching the hard propagator then realizes the reduction to a bubble diagram and defines the effective coupling $C_c(\\tilde{\\nu})$, which falls as $\\tilde{\\nu}^{-2}$ for large mass.","core_discovery":"Working in the Schwinger–Keldysh formalism for a massless inflaton $\\phi$ coupled to a massive scalar $\\sigma$ through $\\mathcal{L}_{\\rm int} = \\frac12 a^3 \\phi' \\sigma^2 + \\frac14 a^2 \\phi'^2 \\sigma^2$, the paper derives the leading cosmological collider signal from the one-loop triangle diagram. In the hierarchy $k_{12} \\gg k_{34} \\ge k_s$, the trispectrum signal splits into a nonlocal piece, nonanalytic in the exchange momentum $k_s$, and a local piece, nonanalytic in $k_{34}$; in the further collapsed regime $k_s \\ll k_{34}$, the local piece dominates the nonlocal piece by a factor of order $(k_{34}/k_s)^3$ and dominates the analytic background by a factor of order $k_{12}/k_{34}$. The squeezed bispectrum arises as the folded limit $k_s \\to k_{34}$, where the singularities of the two branches cancel. All three closed forms are generated by the same pinched effective vertex $O_c = a^2 C_c \\phi'^2 \\sigma^2/4$, meaning the leading triangle-loop signal is reproduced by a bubble diagram with the effective coupling $C_c$. Direct numerical evaluation of the uncut integrals for the local trispectrum and the squeezed bispectrum agrees with the analytic formulas.","pith_inferences":["Going beyond the paper: if the bubble reduction survives at subleading order, it becomes a systematic expansion in which any loop topology attached to one soft vertex is compressed into a local operator fixed by the hard subgraph.","Going beyond the paper: a direct test would be to include the first subleading corrections in $k_{34}/k_{12}$ and check whether the same pinched coupling continues to reproduce the triangle signal.","Going beyond the paper: the smooth folded limit connecting the trispectrum branches suggests that three- and four-point loop signals are continuously connected at $k=E$, a feature that may extend to other loop topologies and unify their templates.","Going beyond the paper: because the local signal dominates in collapsed configurations, searches may not need to resolve the faster nonlocal oscillation and can instead target the $(k_{34}/k_s)^3$ growth in the four-point function."],"forward_implications":["The squeezed bispectrum is an oscillation of frequency $2\\tilde{\\nu}$ in $\\log(k_3/k_{12})$, with an envelope suppressed as $e^{-2\\pi\\tilde{\\nu}}$ for large mass, giving a direct mass-reading template for data searches.","In the hierarchical collapsed regime $k_s \\ll k_{34} \\ll k_{12}$, the local trispectrum signal dominates the full triangle correlator, so this configuration is the most promising target for detecting the loop.","At leading order the triangle loop can be replaced by a bubble diagram with pinched coupling $C_c$, so template production for this topology costs no more than a single-exchange tree computation.","The same framework extends to subleading soft corrections, general internal masses, higher spins, and boost-breaking inflationary models, including cases where loop signals are expected to be the leading effect."],"supporting_citations":[{"why":"Supplies the partial Mellin-Barnes representation for inflationary correlators that factorizes time and loop-momentum integrals.","marker":"[23]"},{"why":"Supplies the phase-sensitive Mellin-Barnes treatment that makes the spectral poles of massive modes manifest.","marker":"[30]"},{"why":"Gives the closed-form tree-level single-exchange seed into which the remaining time and Mellin integrals reassemble.","marker":"[24]"},{"why":"Introduces the cutting-rule idea for isolating on-shell cosmological collider signals.","marker":"[73]"},{"why":"Provides the one-loop cutting rule, the isosceles massless-triangle expansion, and the effective pinched coupling for nonlocal signals that this work extends to local and three-point signals.","marker":"[74]"},{"why":"Establishes the nonanalyticity structure and on-shell factorization of inflation correlators used to justify extracting signals from cut diagrams.","marker":"[75]"},{"why":"Fixes the Schwinger-Keldysh diagrammatic conventions and propagators used to define the master triangle integral.","marker":"[78]"},{"why":"Supplies the triple-K representation of the massless triangle integral used for the loop-momentum factor.","marker":"[81]"},{"why":"Underlies the soft/hard asymptotic split of the loop integral into nonanalytic and analytic branches.","marker":"[82]"}],"fun_headline_variants":["Triangle loop yields first analytic cosmological collider signals","Single effective coupling captures leading triangle loop signals","Bubble diagram replaces triangle loop in leading collider signals","Analytic bispectrum and trispectrum from massive scalar triangle","First analytic signal from cosmological collider triangle loop"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The load-bearing step is the claim that cutting the two massive propagators that meet at the low-energy vertex captures every signal-carrying contribution of the triangle loop; if the uncut remainder terms ever produce a non-analytic piece in some momentum configuration, the stated formulas would be incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Triangle loop yields first analytic cosmological collider signals","Single effective coupling captures leading triangle loop signals","Bubble diagram replaces triangle loop in leading collider signals","Analytic bispectrum and trispectrum from massive scalar triangle","First analytic signal from cosmological collider triangle loop"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000511,"raw_usage":{"cost":0.000511168,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000511168,"upstream_inference_prompt_cost":0.000106568,"upstream_inference_completions_cost":0.0004046},"total_tokens":2411,"prompt_tokens":966,"completion_tokens":1445,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":256,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":1368}},"tokens_in":966,"tokens_out":1445,"cost_source":"openrouter_usage","duration_ms":14922,"image_pages":[],"temperature":1.0,"reasoning_tokens":1368,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-09-10T17:16:16.880079+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"A decisive check would be to numerically integrate the full uncut Schwinger–Keldysh triangle-loop integrals over a grid of $k_s/k_{34}$ values with $k_{34}/k_{12} \\ll 1$ and $\\tilde{\\nu} \\sim 1$, extract the oscillatory nonlocal component with frequency $2\\tilde{\\nu}$ in $\\log\\left(k_s^2/(4k_{12}k_{34})\\right)$, and compare its amplitude and phase with Eq. (23); any mismatch beyond numerical error would show that the directional cutting rule misses part of the loop signal.","supporting_citations":[{"cited_title":"Helical Inflation Correlators: Partial Mellin-Barnes and Bootstrap Equations","cited_arxiv_id":"2208.13790","evidence_quote":"Supplies the partial Mellin-Barnes representation for inflationary correlators that factorizes time and loop-momentum integrals."},{"cited_title":"Phase Information in Cosmological Collider Signals","cited_arxiv_id":"2205.01692","evidence_quote":"Supplies the phase-sensitive Mellin-Barnes treatment that makes the spectral poles of massive modes manifest."},{"cited_title":"Closed-Form Formulae for Inflation Correlators","cited_arxiv_id":"2301.07047","evidence_quote":"Gives the closed-form tree-level single-exchange seed into which the remaining time and Mellin integrals reassemble."},{"cited_title":"Cutting Rule for Cosmological Collider Signals: A Bulk Evolution Perspective","cited_arxiv_id":"2112.03448","evidence_quote":"Introduces the cutting-rule idea for isolating on-shell cosmological collider signals."},{"cited_title":"Inflation Correlators at the One-Loop Order: Nonanalyticity, Factorization, Cutting Rule, and OPE","cited_arxiv_id":"2304.13295","evidence_quote":"Provides the one-loop cutting rule, the isosceles massless-triangle expansion, and the effective pinched coupling for nonlocal signals that this work extends to local and three-point signals."},{"cited_title":"Nonanalyticity and On-Shell Factorization of Inflation Correlators at All Loop Orders","cited_arxiv_id":"2308.14802","evidence_quote":"Establishes the nonanalyticity structure and on-shell factorization of inflation correlators used to justify extracting signals from cut diagrams."},{"cited_title":"Schwinger-Keldysh Diagrammatics for Primordial Perturbations","cited_arxiv_id":"1703.10166","evidence_quote":"Fixes the Schwinger-Keldysh diagrammatic conventions and propagators used to define the master triangle integral."},{"cited_title":"Implications of conformal invariance in momentum space","cited_arxiv_id":"1304.7760","evidence_quote":"Supplies the triple-K representation of the massless triangle integral used for the loop-momentum factor."},{"cited_title":"Asymptotic expansion of Feynman integrals near threshold","cited_arxiv_id":"hep-ph/9711391","evidence_quote":"Underlies the soft/hard asymptotic split of the loop integral into nonanalytic and analytic branches."}],"review_version":1}