Model-independent extrapolation of MUonE data with Pad\'e and D-Log approximants

The MUonE experiment is designed to extract the hadronic contribution to the electromagnetic coupling in the space-like region, $\Delta \alpha_{\rm had}(t)$, from elastic $e\mu$ scattering. The leading order hadronic vacuum polarization contribution to the muon $g-2$, $a_\mu^{\mathrm{HVP, \,LO}}$, can then be obtained from a weighted integral over $\Delta \alpha_{\rm had}(t)$. This, however, requires knowledge of $\Delta \alpha_{\rm had}(t)$ in the whole domain of integration, which cannot be achieved by experiment. In this work, we propose to use Pad\'e and D-Log Pad\'e approximants as a systematic and model-independent method to fit and reliably extrapolate the future MUonE experimental data, extracting $a_\mu^{\mathrm{HVP,\,LO}}$ with a conservative but competitive uncertainty, using no, or very limited, external information. The method relies on fundamental analytic properties of the two-point correlator underlying $a_\mu^{\mathrm{HVP,\,LO}}$ and provides lower and upper bounds for the result for $a_\mu^{\mathrm{HVP,\,LO}}$. We demonstrate the reliability of the method using toy data sets generated from a model for $\Delta \alpha_{\rm had}(t)$ reflecting the expected statistics of the MUonE experiment.