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Effective potential in $SO(N)$ symmetric scalar field theories in curved spacetime

V. A. Filippov, R. M. Iakhibbaev, D. M. Tolkachev

TL;DR

The paper develops a general RG framework for leading-log quantum corrections to the effective potential in $SO(N)$-symmetric scalar theories in curved spacetime with non-minimal gravity coupling, valid in the large-$N$ limit. It derives recurrence relations for leading divergences and translates them into RG equations for the flat and curved parts of the potential, then applies them to power-like potentials ($p=4,6$) to obtain all-loop leading-log effective potentials. The authors analyze curvature effects on the potential, uncovering curvature-induced minima and a flat plateau regime, and connect the results to inflationary dynamics by transforming to the Einstein frame and computing Planck-era observables, with favorable results for $p=4$. They also point to potential applications in primordial black hole formation and outline directions for further constraints from cosmological data.

Abstract

We derive recurrence relations for leading logarithmic all-loop quantum corrections in the case of $SO(N)$ symmetric scalar theory with an arbitrary potential in curved spacetime. On this basis, a system of renormalisation group (RG) equations in the general is obtained approach for the effective potential in the large $N$ limit. As a simple illustration, we analyse the case of power-like potentials in the Jordan frame and discuss their application to inflationary cosmology.

Effective potential in $SO(N)$ symmetric scalar field theories in curved spacetime

TL;DR

The paper develops a general RG framework for leading-log quantum corrections to the effective potential in -symmetric scalar theories in curved spacetime with non-minimal gravity coupling, valid in the large- limit. It derives recurrence relations for leading divergences and translates them into RG equations for the flat and curved parts of the potential, then applies them to power-like potentials () to obtain all-loop leading-log effective potentials. The authors analyze curvature effects on the potential, uncovering curvature-induced minima and a flat plateau regime, and connect the results to inflationary dynamics by transforming to the Einstein frame and computing Planck-era observables, with favorable results for . They also point to potential applications in primordial black hole formation and outline directions for further constraints from cosmological data.

Abstract

We derive recurrence relations for leading logarithmic all-loop quantum corrections in the case of symmetric scalar theory with an arbitrary potential in curved spacetime. On this basis, a system of renormalisation group (RG) equations in the general is obtained approach for the effective potential in the large limit. As a simple illustration, we analyse the case of power-like potentials in the Jordan frame and discuss their application to inflationary cosmology.

Paper Structure

This paper contains 8 sections, 62 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Effective potential for $SO(N)$ symmetric interaction in curved spacetime
  3. Recurrence relations and RG equations in the general approach
  4. Power-like potentials
  5. Renormalisable $p=4$ case
  6. $p=6$ case
  7. Applications in inflationary cosmology
  8. Conclusion

Figures (10)

  • Figure 1: One- and two-loop diagrams giving contributions to $\textbf{V}_k$
  • Figure 2: Modified Feynman rules
  • Figure 3: One- and two-loop diagrams giving contributions to $\textbf{W}_k$
  • Figure 4: $\mathcal{R}^{\prime}$ operation for leading divergences of an $n$-loop diagram
  • Figure 5: Effective potential of the model with $p=4$ for $\xi = 0$, various numbers of fields $N$ and curvature values $R = \{R_{C1}, R_{C2}\}$. Critical curvature values $R_{C1}\simeq 50 \cdot \mu^2, R_{C2}\simeq 70 \cdot \mu^2$ are selected for the effective potential with $N=100$.
  • ...and 5 more figures