Loop correction and resummation of vertex functions for a self interacting scalar field in the de Sitter spacetime

TL;DR

This paper studies the quantum corrections to vertex functions of a self-interacting scalar field in de Sitter space, focusing on resummation of secular logarithms and their impact on late-time behavior.

Contribution

It introduces a resummation method for local secular logarithms in vertex functions using the dynamically generated mass in de Sitter space, extending understanding beyond flat spacetime.

Findings

Local secular logarithms grow unboundedly over time

Resummation indicates vertex functions are suppressed compared to tree level

Results differ from Minkowski spacetime and align with stochastic formalism

Abstract

We consider a massless and minimally coupled self interacting quantum scalar field theory in the inflationary de Sitter background of dimension four. The self interaction potential is taken to be either quartic, $\lambda \phi^4/4!$, or quartic plus cubic, $\lambda \phi^4/4!+\beta \phi^3/3!$ ($\lambda \,{\ensuremath >}\,0$). We compute the four and three point vertex functions up to two loop. The purely local or partly local part of these renormalised loop corrected vertex functions grow unboundedly after sufficient number of de Sitter $e$-foldings, due to the appearances of secular logarithms. We focus on the purely local part of the vertex functions and attempt a resummation of them in terms of the dynamically generated mass of the scalar field at late times. Such local logarithms have sub-leading powers compared to the non-local leading ones which can be resummed via the stochastic formalism. The variation of these vertex functions are investigated with respect to the tree level couplings numerically. Since neither the secular effect, nor the dynamical generation of field mass is possible in the Minkowski spacetime, the above phenomenon has no flat spacetime analogue. We have also compared our result with the ones that could be found via the recently proposed renormalisation group techniques. All these results suggest that at late times the value of the non-perturbative vertex function should be less than the tree level coupling.