Effective action for $φ^4$-Yukawa theory via 2PI formalism in the inflationary de Sitter spacetime

TL;DR

This work develops the local part of the two-loop 2PI effective action for a $\phi^4$ theory Yukawa-coupled to fermions in inflationary de Sitter space, using Hartree resummation to capture nonperturbative effects and secular logarithms. The authors derive renormalized 1PI effective actions and effective potentials, revealing nonperturbative counterterms and a dynamically generated scalar mass that stabilizes the potential. A key finding is the necessity of a nonperturbative cancellation condition, implemented via a parameter choice $k=-1$, which yields a finite local action and imposes a high-field stability bound $\lambda \gtrsim 16 g^2$. They show qualitative differences with standard 1PI perturbation theory and highlight how curved spacetime can qualitatively alter vacuum structure and secular behavior, with potential implications for stochastic inflation and reheating. The work also sets the stage for incorporating non-local self-energy effects through the full Schwinger-Dyson framework in this curved background.

Abstract

We consider a scalar field theory with quartic self interaction, Yukawa coupled to fermions in the inflationary de Sitter spacetime background. The scalar has a classical background plus quantum fluctuations, whereas the fermions are taken to be quantum. We derive for this system the effective action and the effective potential via the two particle irreducible (2PI) formalism. This formalism provides an opportunity to find out resummed or non-perturbative expressions for some series of diagrams. We have considered the two loop vacuum graphs and have computed the local part of the effective action. The various resummed counterterms corresponding to self energies, vertex functions and the tadpole have been explicitly found out. The variation of the renormalised effective potential for massless fields has been investigated numerically. We show that for the potential to be bounded from below, we must have $λ\gtrsim 16 g^2$, where $λ$ and $g$ are respectively the quartic and Yukawa couplings. We emphasise the qualitative differences of this non-perturbative calculation with that of the standard 1PI perturbative ones in de Sitter. The qualitative differences of our result with that of the flat spacetime has also been pointed out.

Figures (10)

• Figure 1: (Left) The two loop (lowest order) 2PI vacuum diagram for the quartic self interaction. (Right) The two loop (lowest order) 2PI diagram for the Yukawa interaction. Solid line stands for scalar, whereas a dashed line stands for fermion. The propagators are exact here.
• Figure 2: Variation of the effective potential \ref{['59eff23']}, with respect to the background field $v$, for negative rest mass squared $m_0^2 \sim -0.063H^2$. The bar over the quantities denote scaling with respect to appropriate power of the de Sitter Hubble rate, $H$. The blue, red and black curves respectively correspond to $\lambda$ values $0.10$, $0.11$ and $0.15$ respectively. This non-trivial feature was reported first in Arai:2012shArai:2013jnaLopezNacir:2013alw.
• Figure 3: Variation of the effective potential \ref{['59eff23']}, with respect to the background field $v$. The The left and right set of curves correspond to $m^2_0=0$ and $m_0^2 \sim 0.01H^2$ respectively. The blue, red and black curves respectively correspond to $\lambda$ values $0.10$, $0.11$ and $0.15$.
• Figure 4: Three loop (next to the leading order) 2PI vacuum diagram for $\phi^4$ self interaction. The other three loop diagram is a connected three-bubble, which is not 2PI. Although we have not considered this graph for non-perturbative computations in this paper, we have computed its renormalised expression for a massless minimal scalar in \ref{['A']}, with the tree level propagtor, \ref{['props2']}. See main text for discussion.
• Figure 5: One loop self energy diagrams for the Yukawa interaction. Solid and dashed lines respectively correspond to scalar and fermion propagators. The propagators are exact here.