The Muon $(g-2)$ Spin Equations, the Magic $\gamma$, What's small and what's not

Abstract

We review the spin equations for the muon in the 1.45~T muon storage ring at Brookhaven National Laboratory, which has subsequently been relocated to Fermilab. Muons are stored in a uniform 1.45~T magnetic field, and vertical focusing is provided by four sets of electrostatic quadrupoles placed symmetrically around the storage ring. The storage ring is operated at the "magic $\gamma = 29.3$" so that the effect of the motional magnetic field cancels for muons at the magic momentum. We point out the relative sizes of the various terms in the spin equations, and show that for experiments that use the magic $\gamma$ and electric quadrupole focusing to store the muon beam, any proposed effect that multiplies either the motional magnetic field $\vec \beta \times \vec E$ or the muon pitching motion $\vec \beta \cdot \vec B$ term, will be smaller by three or more orders of magnitude, relative to the spin precession from the storage ring magnetic field. We use a recently proposed General Relativity correction as an example, to demonstrate the smallness of any such contribution, and point out that the revised preprint from these authors still contains a conceptual error, that significantly overestimates the magnitude of their proposed correction. We have prepared this document in the hope that future authors will find it useful, should they wish to propose corrections from some additional term added to the Thomas equation, Eq. 13, below. Our goal is to clarify how the experiment is done, and how the small corrections due to the presence of the radial electric field and the vertical pitching motion of themuons (betatron motion) in the storage ring are taken into account.