Complete 2-Loop Quantum Electrodynamic Contributions to the Muon Lifetime in the Fermi Model

Abstract

The complete 2-loop quantum electrodynamic corrections to the muon lifetime are calculated in the Fermi theory. The exact result for the effects of virtual and real photons, virtual electrons, muons as well as ${e}^{+}{e}^{\ensuremath{-}}$ pair creation, is $\ensuremath{\Delta}{\ensuremath{\Gamma}}_{\mathrm{QED}}^{(2)}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\Gamma}}_{0}(\frac{\ensuremath{\alpha}}{\ensuremath{\pi}}{)}^{2}[\frac{156815}{5184}\ensuremath{-}\frac{1036}{27}\ensuremath{\zeta}(2)\ensuremath{-}\frac{895}{36}\ensuremath{\zeta}(3)+\frac{67}{8}\ensuremath{\zeta}(4)+53\ensuremath{\zeta}(2)\mathrm{ln}(2)]\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\Gamma}}_{0}(\frac{\ensuremath{\alpha}}{\ensuremath{\pi}}{)}^{2}6.743$, where ${\ensuremath{\Gamma}}_{0}$ is the tree-level width. The theoretical error in the value of the Fermi coupling constant ${G}_{F}$ is now rendered negligible compared to the experimental uncertainty coming from the measurement of the muon lifetime. The overall error in ${G}_{F}$ is then roughly halved, giving ${G}_{F}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(1.16637\ifmmode\pm\else\textpm\fi{}0.00001)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}{\mathrm{GeV}}^{\ensuremath{-}2}$.