Large-order Perturbation Theory and de Sitter/Anti de Sitter Effective Actions
Ashok Das, Gerald V. Dunne
TL;DR
The paper analyzes the large-order behavior of the weak-field expansion of the scalar effective Lagrangian in de Sitter and anti de Sitter spaces and demonstrates that perturbative information alone cannot determine the full nonperturbative physics, unlike the Euler–Heisenberg case. It shows that odd dimensions yield convergent perturbative series with no particle production, while even dimensions exhibit divergent series with distinct Borel properties for AdS and dS; notably, even-dimensional dS has a real Lagrangian despite particle production, a puzzle resolved by nonperturbative propagator structure. The authors connect the perturbative expansions to the full propagator via digamma and multiple gamma representations, explaining cancellations that align with Bogoliubov results and horizon physics. Overall, the work emphasizes that nonperturbative information embedded in propagators is essential for correctly linking perturbative series to gravitational phenomena and has implications for gravitational effective actions.
Abstract
We analyze the large-order behavior of the perturbative weak-field expansion of the effective Lagrangian density of a massive scalar in de Sitter and anti de Sitter space, and show that this perturbative information is not sufficient to describe the non-perturbative behavior of these theories, in contrast to the analogous situation for the Euler-Heisenberg effective Lagrangian density for charged scalars in constant electric and magnetic background fields. For example, in even dimensional de Sitter space there is particle production, but the effective Lagrangian density is nevertheless real, even though its weak-field expansion is a divergent non-alternating series whose formal imaginary part corresponds to the correct particle production rate. This apparent puzzle is resolved by considering the full non-perturbative structure of the relevant Feynman propagators, and cannot be resolved solely from the perturbative expansion.