Post-Newtonian effects of Dirac particle in curved spacetime - III : the muon g-2 in the Earth's gravity

The general relativistic effects to the anomalous magnetic moment of muons moving in the Earth's gravitational field have been examined. The Dirac equation generalized to include the general relativity suggests the magnetic moment of fermions measured on the ground level is influenced by the Earth's gravitational field as $\mu_{\rm m}^{\rm eff} \!\simeq\! (1\!+\!3\phi/c^2)\,\mu_{\rm m}$, where $\mu_{\rm m}$ is the magnetic moment in the flat spacetime and $\phi\!=\!-{G M}/{r}$ is the Earth's gravitational potential. It implies that the muon anomalous magnetic moment measured on the Earth $a_\mu\!\equiv\!{\rm g}_\mu/2\!-\!1$ contains the gravitational correction of $\left\vert a_\mu\right\vert\simeq 2.1\!\times\! 10^{-9}$ in addition to the quantum radiative corrections. The gravitationally induced anomaly may affect the comparison between the theoretical and experimental values recently reported: $a_{\mu({\rm EXP})}-a_{\mu({\rm SM})}=28.8\,\,(8.0)\times10^{-10}\,\,(3.6\,\sigma)$. In this paper, the comparison between the theory and the experiment is examined by considering the influence of the spacetime curvature to the measurement on the muon ${\rm g}_\mu\!\!-\!2$ experiment using the storage ring on the basis of the general relativity up to the post-Newtonian order of $O(1/c^2)$.