Probing the vacuum fluctuations in scalar ghost-free theories

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.