Two-loop renormalization of the electric charge in the standard model

We discuss the renormalization of the electric charge at the two-loop level in the standard model of the electroweak interactions. We explicitly calculate the expression of the complete on-shell two-loop counterterm using the background field method and discuss the advantages of this computational approach. We consider the related quantity e²(μ), defined in the MS̄ renormalization scheme and present numerical results for different values of the scale {μ}. We find that the full two-loop electroweak corrections contribute more than ten parts in units 10^-5 to the Δ^α(mz²) parameter, obtaining ^α^-1(mz)=128.12±0.05 for Δαhad⁽⁵⁾(mz²)=0.027572±0.000359.