  Lattice gauge theories are lattice approximations of the Yang-Mills theory in
physics. The abelian lattice Higgs model is one of the simplest examples of a
lattice gauge theory interacting with an external field. In a previous
paper~\cite{flv2021}, we calculated the leading order term of the expected
value of Wilson loop observables in the low-temperature regime of the abelian
lattice Higgs model on $ \mathbb{Z}^4 ,$ with structure group $G = \mathbb{Z}_n
$ for some $ n \geq 2. $ In the absence of a Higgs field, these are important
observables since they exhibit a phase transition which can be interpreted as
distinguishing between regions with and without quark confinement. However, in
the presence of a Higgs field, this is no longer the case, and a more relevant
family of observables are so-called open Wilson lines. In this paper, we extend
and refine the ideas introduced in~\cite{flv2021} to calculate the leading
order term of the expected value of the more general Wilson line observables.
Using our main result, we then calculate the leading order term of several
natural ratios of expected values and confirm the behavior predicted by
physicists.
