Since we know that both have positive and negative muons contribute to the mean muon lifetime, it is also important to have some sort of quantitative understanding of the ratio, \(\rho\) of muons to antimuons is. We can calculate \(\rho\) via Equation 5. \[\rho=-\frac{\tau^{+}}{\tau^{-}}(\frac{\tau^{-}-\tau_{obs}}{\tau^{+}-\tau_{obs}})=\frac{N^{+}}{N^{-}}\] Where \(\tau^{+}\) is the known value of the mean lifetime for an antimuon,\(2.19 \mu\textrm{seconds}\), \(\tau^{-}\) is the known value of the mean lifetime for a muon, \(2.03 \mu\textrm{seconds}\), and \(\tau_{obs}\) is the value \(2.04\pm0.04 \mu\textrm{seconds}\), taken from the best fit of our data. Thus the calculated ratio is approximately \(0.072\pm.001\). This value is quite small, which confirms our previous hypothesis that the number of muon decays detected was significantly greater than the number of antimuon decays detected.
The Fermi Coupling Constant is the strength of the weak force, and can be calculated from the observed muon lifetime, \(\tau_{obs}\). The relationship between the muon lifetime and the Fermi Coupling Constant is shown below, \[\tau=\frac{192\pi^{3}\hslash^{7}}{G_{f}^{2}m^{5}c^{4}}\] In order to find the Fermi Coupling Constant we can rework the equation such that, \[G_{f}=\sqrt[]{\frac{192\pi^{3}\hslash^{7}}{\tau m^{5}c^{4}}}\] From Equation 7 it was found the the Fermi Coupling Constant was \(1.179\pm0.004 GeV^{-2}\), which is close to the accepted value \(1.166 GeV^{-2}\), but does match within experimental uncertainty. The discrepancy is not particularly problematic, since \(\tau_{obs}\) was an average of the mean muon lifetime and the mean antimuon lifetime. Therefore \(\tau_{obs}\) is not expected to match the mean muon lifetime in a vacuum. Since the Fermi Coupling constant is calculated using the mean muon lifetime in a vacuum, it would make sense that the value observed and the accepted value did not match within uncertainty.