Vacuum Polarization Correction to the Proton Charge Radius Puzzle: A Lepton-Mass Dependent Effective Metric Approach

The persistent discrepancy between the proton charge radius measured from muonic hydrogen (rₚ 0. 84 fm) and electronic hydrogen (rₚ 0. 88 fm) —the "proton radius puzzle"—suggests a potential violation of lepton universality or an incomplete understanding of vacuum structure at short distances. In this Letter, we propose a mechanism of nonlinear vacuum metric response. By treating the vacuum as a dielectric-like medium with a finite elastic modulus coupled to electromagnetic energy density, we derive a conformal scaling factor for the local metric. The contraction of the effective metric is analytically derived as = (m_/mₑ), where the coupling factor arises from the summation over vacuum polarization modes. This formula yields a radius shrinkage of 3. 8906\%, which agrees with the experimental discrepancy of 3. 9115\% to within 0. 02\%. We further predict a radius of 0. 823 fm for tauonic hydrogen. abstract