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 $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ 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 ${10}^{\ensuremath{-}20}$ in the electron-positron case and of ${10}^{\ensuremath{-}23}$ 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 ${10}^{\ensuremath{-}18}$ in the electron-positron system and others at the level of ${10}^{\ensuremath{-}21}$ 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 $\mathrm{CPT}.$ Constraints could be obtained on one figure of merit in the electron-positron system at the level of ${10}^{\ensuremath{-}16},$ on another in the proton-antiproton system at ${10}^{\ensuremath{-}24},$ and on a third at ${10}^{\ensuremath{-}25}$ using comparisons of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ ions with antiprotons.