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

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