On systematic and GR effects on muon g − 2 experiments

We derive in full generality the equations that govern the time dependence of the energy E of the decay electrons in a muon g − 2 experiment. We include both electromagnetic and gravitational effects and we estimate possible systematics on the measurements of a ≡ (g − 2)/2, whose experimental uncertainty will soon reach ∆a/a ≈ 10−7 . In addition to the standard modulation of E when the motion is orthogonal to a constant magnetic field B, with angular frequency ωa = ea|B|/m, we study effects due to: (1) a non constant muon γ factor, in presence of electric fields E, (2) a correction due to a component of the muon velocity along B (the “pitch correction”), (3) corrections to the precession rate due to E fields, (4) non-trivial spacetime metrics. Oscillations along the radial and vertical directions of the muon lead to oscillations in E with a relative size of order 10−6 , for the BNL g − 2 experiment. We then find a subleading effect in the “pitch” correction, leading to a frequency shift of ∆ωa/ωa ≈ O(10−9 ) and subleading effects of about ∆ωa/ωa ≈ few × O(10−8–10−9 ) due to E fields. Finally we show that GR effects are dominated by the Coriolis force, due to the Earth rotation with angular frequency ωT , leading to a correction of about ∆ωa/ωa ≈ ωT /(γωa) ≈ O(10−12). A similar correction might be more appreciable for future electron g − 2 experiments, being of order ∆ωa/ωa,el ≈ ωT /(ωa,el) ≈ 7 × 10−13, compared to the present experimental uncertainty, ∆ael/ael ≈ 10−10, and forecasted to reach soon ∆ael/ael ≈ 10−1