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Effective potentials for de Sitter and anti de Sitter quantum fields

Alfio Bonanno, Sergio Luigi Cacciatori, Ugo Moschella

Abstract

We derive a systematic treatment of one-loop effective potentials for interacting scalar fields in curved spacetimes, providing a general formula valid in arbitrary geometries and explicit results for de Sitter and anti-de Sitter backgrounds. We then compute the effective potential for a scalar $O(N)$ theory on a de Sitter space in any integer dimension. In $d=3$ and dimensional regularization, we extend the calculation up to two loops and compute the $β$-function and the anomalous mass dimension. They coincide exactly with flat-space results, despite dramatic curvature modifications to physical masses/couplings. The flat limit $R\to\infty$ recovers Coleman-Weinberg, confirming consistency. Working in $d=3$ dimensions, we repeat the calculation for $AdS_3$ by using point-splitting regularization, obtaining analogous results for the $β$-function and anomalous mass dimension.

Effective potentials for de Sitter and anti de Sitter quantum fields

Abstract

We derive a systematic treatment of one-loop effective potentials for interacting scalar fields in curved spacetimes, providing a general formula valid in arbitrary geometries and explicit results for de Sitter and anti-de Sitter backgrounds. We then compute the effective potential for a scalar theory on a de Sitter space in any integer dimension. In and dimensional regularization, we extend the calculation up to two loops and compute the -function and the anomalous mass dimension. They coincide exactly with flat-space results, despite dramatic curvature modifications to physical masses/couplings. The flat limit recovers Coleman-Weinberg, confirming consistency. Working in dimensions, we repeat the calculation for by using point-splitting regularization, obtaining analogous results for the -function and anomalous mass dimension.
Paper Structure (15 sections, 129 equations, 2 figures)

This paper contains 15 sections, 129 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The 1-loop effective potential in curved spacetime: a general formula
  3. The $O(N)$ model on the Euclidean de Sitter sphere
  4. The 1-loop correction
  5. Even dimensions
  6. Odd dimensions
  7. Two-loops in $d=3$
  8. Renormalization
  9. Flat limit
  10. $\beta$-function and anomalous mass dimension
  11. Anti de Sitter effective potentials
  12. The one loop case
  13. Two-loop at $d=3$
  14. $\beta$-function and anomalous mass dimension
  15. Conclusions

Figures (2)

  • Figure 1: 1-loop effective potential in the de Sitter $d=3$ case. The bare coupling constant is $c_R=0.1$. The masses from left to right: principal series $m_R=10$, limit case $m_R=1$, complementary series $m_R=0.5$ and $m_R=0.001$.
  • Figure 2: 1-loop effective potential in the de Sitter $d=3$ case. The bare coupling constant is $c_R=4$. The masses from left to right: principal series $m_R=10$, limit case $m_R=1$, complementary series $m_R=0.5$ and $m_R=0.001$. One observes the curious behavior of the complementary series. Further analysis is needed with the renormalization group.