Tests of CPT and Lorentz symmetry from muon anomalous magnetic dipole moment

We derive the relativistic factor for splitting of the $g$-factors of a fermion and its anti-fermion partner, which is important for placing constraints on dimension-5, $CPT$-odd and Lorentz-invariance-violating interactions from experiments performed in a cyclotron. From existing data, we extract limits (1$\sigma$) on the coupling strengths of the temporal component, $f^0$, of a background field (including the field amplitude), which is responsible for such $g$-factor splitting, with an electron, proton, and muon: $\mathrm{\mid }{f}_e^0\mathrm{\mid }<2.3\times {10}^{-12}{\mu }_{\text{B}}$, $|f^0_p|< 4 \times 10^{-9} ~\mu_{$\text{B}$}$,and$|f^0_\mu|< 8 \times 10^{-11} ~\mu_{$\text{B}$}$, respectively, in the laboratory frame. From existing data, we also extract limits on the coupling strengths of the spatial components, $d^{\perp }$, of related dimension-5 interactions of a background field with an electron, proton, neutron, and muon: $| {d}e^{\perp} | \lesssim 10^{-9} ~\mu{$\text{B}$}$,$| {d}p^{\perp} | \lesssim 10^{-9} ~\mu{$\text{B}$}$,$| {d}n^{\perp} | \lesssim 10^{-10} ~\mu{$\text{B}$}$,and$| {d}\mu^{\perp} | \lesssim 10^{-9} ~\mu{$\text{B}$}$, respectively, in the laboratory frame.