Model-independent extrapolation of MUonE data with Padé and D-Log approximants

The MUonE experiment is designed to extract the hadronic contribution to the electromagnetic coupling in the space-like region, $\mathrm{\Delta }{\alpha }_{\mathrm{h}\mathrm{a}\mathrm{d}}(t)$, from elastic $e\mu$ scattering. The leading order hadronic vacuum polarization contribution to the muon $g-2$, $a_{\mu }^{\mathrm{H}\mathrm{V}\mathrm{P},\mathrm{L}\mathrm{O}}$, can then be obtained from a weighted integral over $\mathrm{\Delta }{\alpha }_{\mathrm{h}\mathrm{a}\mathrm{d}}(t)$. This, however, requires knowledge of $\mathrm{\Delta }{\alpha }_{\mathrm{h}\mathrm{a}\mathrm{d}}(t)$ in the whole domain of integration, which cannot be achieved by experiment. In this work, we propose to use Padé and D-Log Padé approximants as a systematic and model-independent method to fit and reliably extrapolate the future MUonE experimental data, extracting $a_{\mu }^{\mathrm{H}\mathrm{V}\mathrm{P},\mathrm{L}\mathrm{O}}$ 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 $a_{\mu }^{\mathrm{H}\mathrm{V}\mathrm{P},\mathrm{L}\mathrm{O}}$ and provides lower and upper bounds for the result for $a_{\mu }^{\mathrm{H}\mathrm{V}\mathrm{P},\mathrm{L}\mathrm{O}}$. We demonstrate the reliability of the method using toy data sets generated from a model for $\mathrm{\Delta }{\alpha }_{\mathrm{h}\mathrm{a}\mathrm{d}}(t)$ reflecting the expected statistics of the MUonE experiment.