Calibrated correlation between heavy-quark masses and Hadronic Vacuum Polarization observables at the precision frontier
The theoretical prediction of the muon anomalous magnetic moment $a_\mu$ depends crucially on the Hadronic Vacuum Polarization (HVP), and the tension between its dispersive and lattice-QCD determinations remains unresolved. We show that part of this puzzle can be addressed in the heavy-quark sector, where both descriptions are theoretically clean, by recognizing that the heavy-quark mass and its contribution to $a_\mu$ are not independent quantities: both follow from integrals of the same hadronic spectral function, differing only in their integration kernel. Promoting this kernel to a free choice within the relativistic QCD Sum Rules used to determine heavy-quark masses, we break with the conventional notion of a single valid sum rule and instead determine the mass and its HVP contribution simultaneously, from a common, self-consistent framework. This intrinsic construction exploits the anticorrelation between the two quantities to sharpen the final uncertainty, and turns the residual disagreement between the perturbative and hadronic descriptions of the observable into a direct observable-specific diagnostic of residual theory/model dependence, including duality-violation and continuum-modeling effects, unavailable to a determination of the mass alone. We obtain $a_\mu^{\rm HVP_{c+b},LO} =(14.46(13)+0.3009(17))\times 10^{-10}$ at leading and $a_\mu^{\rm HVP_{c+b}, NLO_{a,b}} = ( -0.5738(95) - 0.01822(13) )\times 10^{-10}$ at next-to-leading order, for charm and bottom contributions, respectively. We compare our next-to-leading-order results with its first available lattice determination, finding good agreement in the charm sector. As a byproduct, we obtain $\hat m_c(\hat m_c)=1267.1(6.8)$ MeV and $\hat m_b(\hat m_b) = 4182.3(7.2)$ MeV, with unprecedented phenomenological precision.