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Finite parts of inflationary loops II: A streamlined UV in-in algorithm and distinguishable signatures

Guillermo Ballesteros, Jesús Gambín Egea, Flavio Riccardi

TL;DR

The paper tackles the long-standing difficulty of computing ultraviolet contributions in in-in cosmological loops by presenting a streamlined dimensional-regularization approach that cleanly separates UV from IR parts and decouples time and momentum integrals in the UV. It demonstrates how high-momentum expansions and the iϵ prescription enable simplifications that reduce the problem to manageable momentum integrals and a single remaining time integral, while also revealing structural obstacles to renormalization in multi-vertex loops. Through a detailed application to the one-loop scalar bispectrum in an EFT of inflation with a constant M3^4, it shows that certain loop effects are intrinsically distinguishable from tree-level counterterms, though others can be absorbed into counterterm freedom. The work thus provides a practical, symmetry-respecting framework for inflationary loop computations and clarifies when loop corrections yield genuine physical information versus scheme-dependent, counterterm-induced effects.

Abstract

We introduce a streamlined method for evaluating in-in loop integrals using dimensional regularization for diagrams with an arbitrary number of external legs and vertices, which complements earlier work and facilitates the extraction of the ultraviolet contributions. The method leads us to identify an apparent difficulty to renormalize with Hamiltonian counterterms within the in-in formalism. We also discuss the importance of the finite parts of loop corrections that can be distinguished from their associated counterterm contributions. As an application, we examine the one-loop primordial bispectrum in the context of the effective field theory of inflation, considering a specific set of interactions, and identifying a contribution distinguishable from its tree-level counterpart.

Finite parts of inflationary loops II: A streamlined UV in-in algorithm and distinguishable signatures

TL;DR

The paper tackles the long-standing difficulty of computing ultraviolet contributions in in-in cosmological loops by presenting a streamlined dimensional-regularization approach that cleanly separates UV from IR parts and decouples time and momentum integrals in the UV. It demonstrates how high-momentum expansions and the iϵ prescription enable simplifications that reduce the problem to manageable momentum integrals and a single remaining time integral, while also revealing structural obstacles to renormalization in multi-vertex loops. Through a detailed application to the one-loop scalar bispectrum in an EFT of inflation with a constant M3^4, it shows that certain loop effects are intrinsically distinguishable from tree-level counterterms, though others can be absorbed into counterterm freedom. The work thus provides a practical, symmetry-respecting framework for inflationary loop computations and clarifies when loop corrections yield genuine physical information versus scheme-dependent, counterterm-induced effects.

Abstract

We introduce a streamlined method for evaluating in-in loop integrals using dimensional regularization for diagrams with an arbitrary number of external legs and vertices, which complements earlier work and facilitates the extraction of the ultraviolet contributions. The method leads us to identify an apparent difficulty to renormalize with Hamiltonian counterterms within the in-in formalism. We also discuss the importance of the finite parts of loop corrections that can be distinguished from their associated counterterm contributions. As an application, we examine the one-loop primordial bispectrum in the context of the effective field theory of inflation, considering a specific set of interactions, and identifying a contribution distinguishable from its tree-level counterpart.
Paper Structure (22 sections, 139 equations, 1 figure)

This paper contains 22 sections, 139 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Simplifying the calculation of UV contributions to in-in correlators
  3. Distinguishable loop effects
  4. Renormalization of in-in correlators with several vertices
  5. Summary and structure of the paper
  6. Dimensional regularization: computing the UV contribution
  7. The high-momentum limit of the loops
  8. One-loop diagrams with a single vertex insertion
  9. One-loop diagrams with two vertices
  10. One-loop diagrams with more than two vertices
  11. On the structure of tree- and loop-level contributions
  12. $P(\phi,X)$ models in the effective theory of inflation
  13. Background renormalization
  14. $M_3^4\neq 0$ in the decoupling limit
  15. One-loop bispectrum for $M_3^4 \neq 0$
  16. ...and 7 more sections

Figures (1)

  • Figure 1: Integration domains (shaded regions) for the integrals Eq. \ref{['int1']} (left) and Eq. \ref{['int2']} (right). The regions around which the integrals are expressed as an expansion in time derivatives, Eq. \ref{['noremantint']}$(\tau"=\tau'=\tau)$ and Eq. \ref{['remantint']}$(\tau"=\tau')$ respectively, are indicated encircled.