The interplay between SM precision, BSM physics and measurements of $\alpha_{\rm{had}}$ in $\mu$-$e$ scattering

Muon electron scattering experiments such as the proposed MUonE experiment, offer an opportunity for an improved measurement of the Leading Order hadronic running of $\alpha$, denoted $\Delta \alpha_{\rm{had}}$. Such a measurement could be utilized to reduce the theoretical uncertainty on the prediction of the anomalous magnetic moment of the muon, $g-2$. Currently there is a discrepancy between theory and data for this observable which could potentially be explained by Beyond the Standard Model (BSM) physics. Here we investigate the possible impact of missing Standard Model (SM) higher order corrections and BSM physics on the proposed measurement of $\Delta \alpha_{\rm{had}}$. In principle either could be indirectly fitted into $\Delta \alpha_{\rm{had}}$, causing inconsistencies if used in a $g-2$ application. The literature suggests a target of 10 ppm on the cross section for the theoretical accuracy. We assess the validity of this target in detail using a variety of methods, finding that a 1 ppm target is a more conservative estimate to ensure missing higher orders do not dominate the theoretical uncertainty. For the potential BSM contributions we study various models which contribute first at tree- and loop-level. Of particular interest is the impact from dark photon models, which can potentially affect the measurement of $\Delta \alpha_{\rm{had}}$ at the desired accuracy. At loop-level there exists in general a kinematic suppression adequate to reduce the BSM contributions to a level which can be neglected for the extraction of $\Delta \alpha_{\rm{had}}$.