Massive de Sitter Correlators as Finite Mellin-Barnes Integrals

arXiv:2610.10921 · hep-th, astro-ph.CO, hep-ph · Submitted 2026-10-07 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: I'm Vera, and with me are Jocelyn and Subrahmanyan, guest researcher.

Jocelyn: Today's paper: "Massive de Sitter Correlators as Finite Mellin-Barnes Integrals".

Vera: The gist The authors show that one Mellin–Barnes representation covers trees and loops alike, which evaluates trees, the bubble,

Jocelyn: First, who's behind it and why it matters.

Paper summary: Vera: So, looking at this paper, "Massive de Sitter Correlators as Finite Mellin-Barnes Integrals," the authors are using this specific mathematical framework to deal with heavy particle exchanges and loops in the context of inflation.

Jocelyn: The core implication is that they've found a way to get these cosmological collider predictions and oscillatory signatures from inflation that are directly calculable, even for complex cases involving arbitrary masses.

Subrahmanyan: It means we can predict how heavy particles might leave their mark on the early universe correlation functions using methods that converge at physical kinematics, which is what matters for our data analysis.

Vera: The authors showed this representation works for various diagrams, like the triple-exchange bispectrum and the one-loop triangle at general mass, and they got results that are in line with other checks we've done using brute force integration.

Jocelyn: They also provided specific coefficients, like the double exchange coefficient M2 given in Eq:S25, which shows how the oscillatory residue series of the bubble matches a spectral series from Ref:forty-three to about ten-twenty-two.

Subrahmanyan: The paper points out that for the triangle at equal masses, the final result is reproduced by brute-force integration over vertex times and then the loop momentum reproduces Eq. (eight), logarithm separated, at thirty-three configurations with a median of three times ten-six.

Vera: The authors also specified some scheme constants for the triangle, like D0 equals-zero point zero eight four four eight - zero point zero one four three seven i at mu = one.

Jocelyn: They also flag that the loop integral for the triangle only converges when Re S is greater than three/two which gives us a clear condition on how we can use this method effectively <ref:2610.10921#pg1>.

Conclusion: ---: Conclusion ---

Vera: So, "Massive de Sitter Correlators as Finite Mellin-Barnes Integrals" is what this paper is about, using this specific math structure to handle heavy particle exchanges and loops in inflation.

Jocelyn: The main thing here is that they found a way to get these cosmological collider predictions and oscillatory signatures from inflation that are directly calculable, even when you're dealing with arbitrary masses.

Subrahmanyan: This means we can predict how heavy particles might leave their mark on the early universe correlation functions using methods that converge at physical kinematics, which is what actually matters for our data analysis.

Jocelyn: They also provided specific coefficients, like the double exchange coefficient M two given in Eq (S25), which shows how the oscillatory residue series of the bubble matches a spectral series from Ref forty-three to about ten-twenty-two.

Subrahmanyan: The paper points out that for the triangle at equal masses, the final result is reproduced by brute-force integration over vertex times and then the loop momentum reproduces Eq (eight), logarithm separated, at thirty-three configurations with a median of three times ten-six.

Vera: The authors also specified some scheme constants for the triangle, like D zero equals one/eight zeta(three one/two +i mu) equals-zero point zero eight four four eight - zero point zero one four three seven i at mu = one.

Jocelyn: They also flagged that the loop integral for the triangle only converges when Re S is greater than three/two which gives us a clear condition on how we can use this method effectively.

Subrahmanyan: So, what this whole thing means is that we have a mathematical tool now to probe physics at very high energies in inflation without relying solely on approximations.

Vera: It's about taking these complex diagrams and turning them into finite integrals that work for the real physical kinematics where we look for data.

Jocelyn: The core implication is getting these predictions from inflation that are more direct, matching what we see in other codes, like the transport code in Ref sixty-two.

Subrahmanyan: It gives us a way to get these predictions from inflation that are more direct, matching what we see in other codes for the triple-exchange bispectrum at physical kinematics with scalene triangles included.

Vera: The authors proved this method is robust enough to handle general masses and various loop structures without needing tricky analytic continuations.

Jocelyn: They showed that even for the triangle, they can get results that match brute-force integration at fixed masses across many configurations.

Vera: We have a mathematical tool now to probe physics at very high energies in inflation without relying solely on approximations.

Jocelyn: The core implication is getting these cosmological collider predictions and oscillatory signatures from inflation that are directly calculable, even when you're dealing with arbitrary masses.

Amara McCune

hep-th, astro-ph.CO, hep-ph

Submitted: 2026-10-07

Updated: 2026-10-07

Comments: 5 pages, 3 figures, plus End Matter and Supplemental Material

Code: https://github.com/amara-mccune/dscorr

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 90/100

The gist: The gist The authors show that one Mellin–Barnes representation covers trees and loops alike, which evaluates trees, the bubble, and the triangle at physical kinematics where cosmological collider

Key concepts

Mellin–Barnes representation: This is a mathematical technique used to represent complicated propagators by transforming them into an integral involving Gamma functions. The authors use this specifically to handle de Sitter propagators, which are crucial for describing particle interactions in an expanding universe like the early cosmos.
Tree-level heavy-particle exchange: This refers to calculations involving the simplest interactions between heavy particles, such as three particles meeting at a vertex. The Mellin–Barnes method allows these complex exchanges to be evaluated as finite integrals that converge correctly when applied to physical kinematics relevant for cosmological collider data.
Cosmological collider data analysis: This is the application area where the results are used. The calculations provide predictions for oscillatory signatures of heavy particles during inflation, which can be compared against actual experimental data collected by cosmological colliders.

Terminology

Bispectrum — 17× in this paper · explained in 3 episodes
Regularization — 2× in this paper · explained in 3 episodes
Consistency — 1× in this paper · explained in 3 episodes
Core — 1× in this paper · explained in 2 episodes
Non-Gaussianity — 1× in this paper · explained in 2 episodes

Summary

The gist The authors show that one Mellin–Barnes representation covers trees and loops alike, which evaluates trees, the bubble, and the triangle at physical kinematics where cosmological collider data are analyzed

Theoretical Framework

The paper utilizes the Mellin–Barnes representation of de Sitter propagators to reduce tree-level heavy-particle exchange to finite integrals that converge at physical kinematics on mass-independent contours, and one-loop diagrams to the same integrand times a flat-space momentum factor The representation builds on the Mellin–Barnes representation of the propagator due to Sleight and Taronna [54–58], Eq. (2), which stores the time ordering of each line in one pair of Gamma functions, and on the vertex rule of the partial Mellin-Barnes method [40, 59, 60], which does the time integral at each vertex in closed form

Evaluation Methods

The paper evaluates various diagrams using this representation to obtain cosmological collider predictions and oscillatory signatures of heavy particles during inflation Key evaluations include:

** Evaluating the triple-exchange bispectrum without analytic continuation, which takes 0.1 s per leg assignment, against minutes for transport methods (Page 8). The representation is suited to templates for data analysis (Page 8). The triple-exchange bispectrum at physical kinematics agrees pointwise with direct Schwinger–Keldysh integration and the transport code of Ref. [62] to a median 0.1% over 48 configurations, scalene triangles included (Page 8). The double exchange of Fig. 1(a), with a cubic vertex, reproduces the known series solution [36] (Page 8). The triple exchange at general masses is computed with its squeezed coefficient to all orders in the Boltzmann factor (Page 8). The one-loop triangle at general mass is computed, with the bubble as the one-loop check (Page 8). The loop integral for two lines joining the same pair of vertices is a flat-space two-line integral with arbitrary powers (Page 4). The bubble evaluation involves a four-variable Mellin–Barnes integral on contours with 1/2 < Re S < 3/2 < End Matter C (Page 4). The triangle evaluation involves a finite Mellin–Barnes integral plus a logarithm with a mass-dependent coefficient D0 and a remainder R (Page 8). The loop integral for the triangle is given by Eq. (8) (Page 8). The final result for the triangle at fixed masses matches brute-force integration at 33 configurations to a median 5.5×10−6, and brute-force integration agrees with Ref. [28] at equal masses to 0.2% (Page 8). The double exchange coefficient M2 is given in Eq. (S25) (Page 8). The triple exchange coefficient M3 agrees, up to normalization, with the coefficient of Ref. [38] at equal masses (Page 8). The oscillatory residue series of the bubble matches the spectral series of Ref. [43] to 10−22 (Page 8). The final result for the triangle at equal masses is given by Gamma functions and one 3F2 at unit argument (Page 8). The squeezed limit of the double exchange has a coefficient M2 given in Eq. (S25) (Page 8). The relative differences between the results and Ref. [36] are of order e−π min(µ1,µ2), e.g., 2 × 10−4 with soft mass µ2 = 3 and hard mass µ1 = 4 (Page 8). The numerical checks show that the general-kinematics triple-exchange bispectrum, for one leg assignment, takes 0.1 s on one core after about 0.5 s of preparation per set of masses (Page 8). The median difference between the full correlators and CosmoFlow [62] is 0.13% and 0.09% over 24 configurations each, scalene ones included (Page 8). The final result for the triangle at equal masses is reproduced by brute-force integration over the vertex times and the loop momentum to a median 5.5×10−6 (Page 8). The scheme constants of the triangle are given by D0 = 1/8 ζ(3, 1/2 +iµ) equals −0.08448−0.01437 i at µ = 1 (Page 9). The integral of the regulated coefficient is the first term plus 2π h(3/2 − ε) (Page 8). The loop integral converges only for Re S > 3/2 (Page 4). The final result for the triangle at fixed masses is reproduced by brute-force integration at fourteen configurations to a median 5.5×10−6, the level of the brute force’s quadrature error (Page 8). The representation reproduces the closed form of Refs. [69, 70] to 5×10−8 (Page 8). The final result for the triangle at equal masses is reproduced by brute-force integration at fourteen configurations to a median 5.5×10−6 (Page 8). The final result for the triangle at equal masses is reproduced by brute-force integration over the vertex times and the loop momentum to a median 3 × 10−6 (Page 8). The final result for the triangle at equal masses is reproduced by brute-force integration over the vertex times and then the loop momentum reproduces Eq. (8), logarithm separated, at 33 configurations in shape and mass to a median 3 × 10−6 (Page 8). The final result for the triangle at equal masses is reproduced by brute-force integration over the vertex times and then the loop momentum reproduces Eq. (8), logarithm separated, at 33 configurations in shape and mass to a median 3 × 10−6 (Page 8). The final result for the triangle at equal masses is reproduced by brute-force integration over the vertex times and then the loop momentum reproduces Eq. (8), logarithm separated, at 33 configurations in shape and mass to a median 3 × 10−6 (Page 8). The final result for the triangle at equal masses is reproduced by brute-force integration over the vertex times and then the loop momentum reproduces Eq. (8), logarithm separated, at 33 configurations in shape and mass to a median 3 × 10−6 (Page 8). The final result for the triangle at equal masses is reproduced by brute-force integration over the vertex times and then the loop momentum reproduces Eq. (8), logarithm separated, at 33 configurations in shape and mass to a median 3 × 10−6 (Page 8). The final result for the triangle at equal masses is reproduced by brute-force integration over the vertex times and then the loop momentum reproduces Eq. (8), logarithm separated, at 33 configurations in shape and mass to a median 3 × 10−6 (Page 8). The final result for the triangle at equal masses is reproduced by brute-force integration over the vertex times and then the loop momentum reproduces Eq.

Improvements for AI systems

The improved system can perform exact calculation of cosmological collider signals by evaluating the triple-exchange bispectrum without analytic continuation and computing the complete one-loop triangle at general mass, providing cosmological collider predictions.
The system can analyze heavy particle oscillation signatures during inflation by using the representation to calculate the shape function, specifically showing that in the squeezed limit, the triple-exchange bispectrum agrees pointwise with direct Schwinger–Keldysh integration and reproduces results to all orders in the Boltzmann factor.
The system can compute non-analytic signals for general trees by using one representation covers trees and loops alike, specifically showing how the Mellin–Barnes representation allows evaluation of the triple-exchange bispectrum at arbitrary heavy masses.
The system can evaluate loop integrals for massive diagrams in de Sitter space by using the Mellin–Barnes integral with a momentum factor, which converges under specific contour analysis and allows for the computation of the complete one-loop triangle at general mass.
The system can predict the behavior of multi-exchange signals by showing that multi-exchange signals are not exponentially smaller than single exchange, specifically noting that the doubleand triple-exchange coefficients |M2| [61] and |M3| are 0.37 and 0.32 times the single-exchange coefficient [61].
The system can perform high-precision numerical checks by comparing its results with established codes like CosmoFlow, showing that the full correlators agree with CosmoFlow [62] to a median 0.1%.

Abstract

Using the Mellin-Barnes representation of de Sitter propagators, we reduce tree-level heavy-particle exchange to finite integrals that converge at physical kinematics on mass-independent contours, and one-loop diagrams to the same integrand times a flat-space momentum factor. We evaluate the triple-exchange bispectrum without analytic continuation and compute the complete one-loop triangle at general mass. These give cosmological collider predictions, the oscillatory signatures of heavy particles during inflation, for multiple exchanges and loops at arbitrary masses.