Two-loop renormalization of the electric charge in the Standard Model
Giuseppe Degrassi, Alessandro Vicini
TL;DR
This work computes the complete two-loop electroweak renormalization of the electric charge in the Standard Model using the Background Field Method, enabling Dyson summation for the full photon vacuum polarization. It defines the MSbar coupling $\hat{e}^{2}(\mu)$ and relates it to the on-shell charge via $\Delta\hat{\alpha}(m_Z^2)$, yielding a precise $\hat{\alpha}^{-1}(m_Z)\approx128.12\pm0.05$ for $\Delta\alpha^{(5)}_{had}(m_Z^2)=0.027572\pm0.000359$. The two-loop EW corrections contribute more than $10\times10^{-5}$ to $\Delta\hat{\alpha}(m_Z^2)$, with bosonic parts partially canceling top-quark effects, and they reduce the perturbative uncertainty in the determination of the running coupling. The results have important implications for electroweak precision fits and Higgs-mass constraints, and demonstrate the practical advantages of the Background Field Method for high-order gauge theory calculations.
Abstract
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 $\hat e(μ)$, defined in the $\ms$ renormalization scheme and present numerical results for different values of the scale $μ$. We find that the full 2-loop electroweak corrections contribute more than 10 parts in units $10^{-5}$ to the $Δ\hatα(\mz^2)$ parameter, obtaining $\hatα^{-1}(\mz)= 128.12 \pm 0.05$ for $Δα_{had}(\mz^2) =0.027572 \pm 0.000359$.