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Triangular invariants, three-point functions and particle stability on the de Sitter universe

Jacques Bros, Henri Epstein, Michel Gaudin, Ugo Moschella, Vincent Pasquier

TL;DR

The paper derives an exact, symmetry-rich formula for the integral of a triple product of Legendre functions on the de Sitter manifold, expressing $h_d(ppa,v,l)$ as a ratio of Gamma functions. Using geometric representations on Lobachevski space and a generalized star-triangle relation, the authors connect four interlocking structures: de Sitter two- and three-point functions, a Källén-Lehmann–type spectral representation, and a conformal limit on the hypersphere. They provide two complementary evaluations of the associated triangle invariant $J(a_1,a_2,a_3)$ and establish a duality with the coefficient $c(a_1,a_2,a_3)$, yielding explicit, analytically tractable expressions for the three-point spectral weight and the corresponding lifetime-like quantity in de Sitter space. The results imply that, unlike flat space, de Sitter decays into heavier products are permitted in the inclusive rate, with potential implications for early-universe physics and representations of noncompact groups, and they open avenues toward Yang-Baxter structures and broader QFT applications on curved backgrounds.

Abstract

We study a class of three-point functions on the de Sitter universe and on the asymptotic cone. A blending of geometrical ideas and analytic methods is used to compute some remarkable integrals, on the basis of a generalized star-triangle identity living on the cone and on the complex de Sitter manifold. We discuss an application of the general results to the study of the stability of scalar particles on the Sitter universe.

Triangular invariants, three-point functions and particle stability on the de Sitter universe

TL;DR

The paper derives an exact, symmetry-rich formula for the integral of a triple product of Legendre functions on the de Sitter manifold, expressing as a ratio of Gamma functions. Using geometric representations on Lobachevski space and a generalized star-triangle relation, the authors connect four interlocking structures: de Sitter two- and three-point functions, a Källén-Lehmann–type spectral representation, and a conformal limit on the hypersphere. They provide two complementary evaluations of the associated triangle invariant and establish a duality with the coefficient , yielding explicit, analytically tractable expressions for the three-point spectral weight and the corresponding lifetime-like quantity in de Sitter space. The results imply that, unlike flat space, de Sitter decays into heavier products are permitted in the inclusive rate, with potential implications for early-universe physics and representations of noncompact groups, and they open avenues toward Yang-Baxter structures and broader QFT applications on curved backgrounds.

Abstract

We study a class of three-point functions on the de Sitter universe and on the asymptotic cone. A blending of geometrical ideas and analytic methods is used to compute some remarkable integrals, on the basis of a generalized star-triangle identity living on the cone and on the complex de Sitter manifold. We discuss an application of the general results to the study of the stability of scalar particles on the Sitter universe.

Paper Structure

This paper contains 21 sections, 1 theorem, 146 equations.

Table of Contents

  1. Prologue
  2. Legendre functions and de Sitter Klein-Gordon fields: a short review
  3. Plane waves expansion of Legendre functions
  4. Lobatchevski space and a remarkable representation of Legendre functions
  5. Källén-Lehmann-type representation for general two-point functions
  6. Decay of de Sitter unstable particles
  7. A special class of de Sitter three-point functions.
  8. A star-triangle relation and a class of triangular invariants on the hypersphere
  9. Computing $c(a_1,a_2,a_3)$; more on the star-triangle relation
  10. Conical limit
  11. Computing $J(a_1,a_2,a_3)$: direct and indirect methods.
  12. Probabilistic interpretation of $J$
  13. Direct evaluation of $J$
  14. The $J / c-$duality and an indirect evaluation of $J(a_1,a_2,a_3)$
  15. A corollary
  16. ...and 6 more sections

Key Result

Theorem B.1

Let $f$ be a function holomorphic in the right half-plane $\{z\in {\bf C}\ :\ \mathop{\rm Re}\nolimits z >0\}$, and satisfying where $A \ge 0$ and $0 \le k < \pi$. If $f(z) = 0$ for $z=1,\ 2,\ \ldots$, then $f=0$.

Theorems & Definitions (1)

  • Theorem B.1: Carlson