# Probing the vacuum fluctuations in scalar ghost-free theories

## TL;DR

This paper investigates how vacuum fluctuations respond to a static potential in ghost-free scalar theories, providing exact solutions and analyzing the effects of non-locality at different scales.

## Contribution

It offers an exact solution for the Hadamard function in ghost-free scalar theories with a delta potential, revealing how non-locality influences vacuum polarization.

## Key findings

- For distances much larger than the non-locality scale, results match local theories.
- At distances smaller than the non-locality scale, differences emerge due to non-local effects.
- The vacuum polarization depends on the potential amplitude and non-locality scale.

## Abstract

We discuss the response of vacuum fluctuations to a static potential in the context of massive, ghost-free infinite-derivative scalar field theories in two dimensions. For the special case of a $\delta$-like potential, $V=\lambda \delta(x)$, the problem is exactly solvable and we calculate the corresponding Hadamard function for this quantum field. Using this exact result we determine the renormalized value of the vacuum polarization $\langle \hat{\varphi}^2(x)\rangle_\text{ren}$ as a function of the distance $x$ from the position of the potential. This expression depends on the amplitude of the potential as well as the scale of non-locality $\ell$; for distances $x\gg\ell$ the non-local and local results agree, whereas for distances $x < \ell$ there is a difference.