On the quantification of GR effects in muon g-2 experiments

Recently, Morishima, Futamase and Shimizu published a series of manuscripts, putting forward arguments, based on a post-Newtonian approximative calculation, that there can be a sizable general relativistic (GR) correction in the experimental determination of the muon magnetic moment, i.e., in muon g-2 experiments. In response, other authors argued that the effect must be much smaller than claimed. Further authors argued that the effect exactly cancels. Also the known formulae for de Sitter and Lense-Thirring effect do not apply due to the non-geodesic motion. All this indicates that it is difficult to estimate from first principles the influence of GR corrections in the problem of spin propagation. Therefore, in this paper we present a full general relativistic calculation in order to quantify this effect. The used methodology is the purely differential geometrical tool of Fermi-Walker transport over a Schwarzschild background. This is compared to the Minkowski limit in order to quantify the GR corrections. The correction turns out to be of first order in terms of the Schwarzschild radius over Earth radius, and is suppressed by the squared ratio of the storage ring radius to the Earth radius, for ultrarelativistic particles. The calculated effect can be basically attributed to the contribution of GR to the Thomas precession, which appears since the muons are forced to move on a non-geodesic trajectory. Our calculation, however, does not include the Larmor precession, which is present in the real experiment, only the Thomas precession of the gyroscopic motion which is of purely kinematic origin. This estimation yields a negligible relative systematic error, as a preliminary result.