Asymptotic nonlocality

TL;DR

This paper develops a theory of scalar fields that smoothly transitions from Lee-Wick models with multiple poles to a nonlocal infinite-derivative theory as the number of poles approaches infinity, offering insights into divergence regulation.

Contribution

It introduces an auxiliary-field formulation that unifies Lee-Wick and nonlocal theories, providing a new framework for addressing quadratic divergences and the hierarchy problem.

Findings

The large-N limit yields a nonlocal infinite-derivative theory.

The effective scale for divergence regulation can be lower than the lightest resonance.

The construction offers potential solutions to the hierarchy problem.

Abstract

We construct a theory of real scalar fields that interpolates between two different theories: a Lee-Wick theory with $N$ propagator poles, including $N-1$ Lee-Wick partners, and a nonlocal infinite-derivative theory with kinetic terms modified by an entire function of derivatives with only one propagator pole. Since the latter description arises when taking the $N\rightarrow\infty$ limit, we refer to the theory as "asymptotically nonlocal." Introducing an auxiliary-field formulation of the theory allows one to recover either the higher-derivative form (for any $N$) or the Lee-Wick form of the Lagrangian, depending on which auxiliary fields are integrated out. The effective scale that regulates quadratic divergences in the large-$N$ theory is the would-be nonlocal scale, which can be hierarchically lower than the mass of the lightest Lee-Wick resonance. We comment on the possible utility of this construction in addressing the hierarchy problem.