Width effects of broad new resonances in loop observables and application to $(g-2)_\mu$

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.