$\mathrm{CPT}$ and Lorentz tests in Penning traps

A theoretical analysis is performed of Penning-trap experiments testing $\mathrm{CPT}$ and Lorentz symmetry through measurements of anomalous magnetic moments and charge-to-mass ratios. Possible $\mathrm{CPT}$ and Lorentz violations arising from spontaneous symmetry breaking at a fundamental level are treated in the context of a general extension of the $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{U}\left(1\right)\mathrm{}$ standard model and its restriction to quantum electrodynamics. We describe signals that might appear in principle, introduce suitable figures of merit, and estimate $\mathrm{CPT}$ and Lorentz bounds attainable in present and future Penning-trap experiments. Experiments measuring anomaly frequencies are found to provide the sharpest tests of $\mathrm{CPT}$ symmetry. Bounds are attainable of approximately ${10}^{-20}$ in the electron-positron case and of ${10}^{-23}$ for a suggested experiment with protons and antiprotons. Searches for diurnal frequency variations in these experiments could also limit certain types of Lorentz violation to the level of ${10}^{-18}$ in the electron-positron system and others at the level of ${10}^{-21}$ in the proton-antiproton system. In contrast, measurements comparing cyclotron frequencies are sensitive within the present theoretical framework to different kinds of Lorentz violation that preserve $\mathrm{CPT}.\mathrm{}$ Constraints could be obtained on one figure of merit in the electron-positron system at the level of ${10}^{-16},$ on another in the proton-antiproton system at ${10}^{-24},$ and on a third at ${10}^{-25}$ using comparisons of ${\mathrm{H}}^{\mathrm{-}}$ ions with antiprotons.