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.