On the measurement of the muon anomalous magnetic moment

Abstract: The ideas and formulas presented in the article will help to bring together the theoretical predictions for the anomalous magnetic moment of muon and the results of the "Muon g-2" experiment. In doing so, we are discussing the new effect exclusively within the Standard Model. In quantum physics a state with spin perpendicular to a magnetic field can be expressed as a superposition of energy eigenstates with spins parallel and antiparallel to the field: the resultant spin precession is due to the energy difference between the two eigenstates. If the state, like the muon, is unstable and can decay, it will have a natural energy spread. As a result the frequency of the spin precession can vary. For a constant magnetic field the measured spin precession velocity will be spread according to the Lorentzian distribution with width $\left(\gamma\tau\right)^{-1}$, for Lorentz gamma factor $\gamma=E/ m$, and particle lifetime $\tau$. Although the true mean and variance of a Lorentzian distribution is undefined, the latter can be estimated by the maximum likelihood method to be ${2 \over N (\gamma \tau)^2}$, twice that of a normal distribution. Thus, the statistical error on the anomalous magnetic moment in reality should turn out to be wider than with $\chi^2$ analysis of the experiment.