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

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