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 η = α ln(m_μ/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}