  In the phenomenology of strong interactions most physical states acquire a
substantial width, and thus can only be defined in a model-independent way by
pole positions and residues of the $S$-matrix. This information is incorporated
in the K\"all\'en-Lehmann representation, whose spectral function characterizes
the shape of the resonance and can be constrained by the dominant decay
channels. Here, we argue that similar effects become important whenever
beyond-the-Standard-Model particles possess a sizable decay width - as possible
for instance in cases with a large branching fraction to a dark sector or
strongly coupled scenarios - and show how their widths can be incorporated in
the calculation of loop observables. As an application, we consider the
anomalous magnetic moment of the muon, including both the direct effect of new
physics and the possible indirect impact of a broad light $Z'$ on
$e^+e^-\to\text{hadrons}$ cross sections. Throughout, we provide results for a
general spectral function and its reconstruction from the one-loop imaginary
part, where the latter captures the leading two-loop effects.
