Model Independent Tests of the Hadronic Vacuum Polarization Contribution to the Muon $g$$-$$2$

The hadronic vacuum polarization (HVP) contributions to the muon $g$$-$$2$ are the crucial quantity to resolve whether new physics is present or not in the comparison between the Standard Model (SM) prediction and experimental measurements at Fermilab. They are commonly and historically determined via dispersion relations using a vast catalogue of experimentally measured, low-energy $e^+e^-\to \,\rm{hadrons}$ cross section data as input. These dispersive estimates result in a SM prediction that exhibits a muon $g$$-$$2$ discrepancy of more than $5\sigma$ when compared to experiment. However, recent lattice QCD evaluations of the HVP and a new hadronic cross section measurement from the CMD-3 experiment favor a no-new-physics scenario and, therefore, exhibit a common tension with the previous $e^+e^-\to \,\rm{hadrons}$ data. This study explores the current and future implications of these two scenarios on other observables that are also sensitive to the HVP contributions in the hope that they may provide independent tests of the current tensions observed in the muon $g$$-$$2$.