Signals for CPT and Lorentz violation at the Planck scale may arise in hydrogen and antihydrogen spectroscopy. We show that certain 1S-2S and hyperfine transitions can exhibit theoretically detectable effects unsuppressed by any power of the fine-structure constant. [S0031-9007(99) PACS numbers: 11.30. Er, 12.20.Fv, 32.10.Fn, 32.80.Pj Experimental and theoretical studies of the spectrum of hydrogen (H) have historically been connected to several major advances in physics [1]. The recent production and observation of antihydrogen (H) [2,3] makes it plausible to consider a new class of spectroscopic measurements involving high-precision comparisons of the spectra of H and H [4]. The two-photon 1S-2S transition has received much attention because an eventual measurement of the line center to about 1 mHz, corresponding to a resolution of one part in 10 18 , appears feasible [5]. It has already been measured to 3.4 parts in 10 14 in a cold atomic beam of H [6] and to about 1 part in 10 12 in trapped H [7]. Proposed H spectroscopic investigations involve both beam and trapped-atom techniques [8,9].We consider here the use of spectroscopy of free or magnetically trapped H and H to search for CPT and Lorentz violation. The discrete symmetry CPT is an invariance of all local Lorentz-invariant quantum field theories of point particles [10], including the standard model and quantum electrodynamics (QED). However, the situation is less clear for a more fundamental theory combining the standard model with gravity, such as string theory, where spontaneous breaking of these symmetries may occur [11]. Low-energy manifestations would be suppressed by a power of the ratio of the low-energy scale to the Planck scale, so only a few exceptionally sensitive experiments are likely to detect them.In this paper, we show that effects of this type from the Planck scale can appear in H and H spectra at zeroth order in the fine-structure constant. Moreover, these effects are theoretically detectable not only in 1S-2S lines but also in hyperfine transitions.Our analysis is performed in the context of a standardmodel and QED extension [12] incorporating the idea of spontaneous CPT and Lorentz breaking at a more fundamental level. This quantum field theory appears at present to be the only existing candidate for a consistent extension of the standard model based on a microscopic theory of CPT and Lorentz violation. Desirable features such as energy-momentum conservation, gauge invariance, renormalizability, and microcausality are expected [12]. The theory has been applied to photon properties [12] [18]. To examine the spectra of free H and H, it suffices to perform a perturbative calculation in the context of relativistic quantum mechanics. In this approach, the unperturbed Hamiltonians and their eigenfunctions are the same for H and H. Moreover, all perturbative effects from conventional quantum electrodynamics are also identical for both systems. However, the CPT -and Lorentz-breaking couplings for the electron and positron can provide d...
Theories with spontaneous local Lorentz and diffeomorphism violation contain massless NambuGoldstone modes, which arise as field excitations in the minimum of the symmetry-breaking potential. If the shape of the potential also allows excitations above the minimum, then an alternative gravitational Higgs mechanism can occur in which massive modes involving the metric appear. The origin and basic properties of the massive modes are addressed in the general context involving an arbitrary tensor vacuum value. Special attention is given to the case of bumblebee models, which are gravitationally coupled vector theories with spontaneous local Lorentz and diffeomorphism violation. Mode expansions are presented in both local and spacetime frames, revealing the Nambu-Goldstone and massive modes via decomposition of the metric and bumblebee fields, and the associated symmetry properties and gauge fixing are discussed. The class of bumblebee models with kinetic terms of the Maxwell form is used as a focus for more detailed study. The nature of the associated conservation laws and the interpretation as a candidate alternative to Einstein-Maxwell theory are investigated. Explicit examples involving smooth and Lagrange-multiplier potentials are studied to illustrate features of the massive modes, including their origin, nature, dispersion laws, and effects on gravitational interactions. In the weak static limit, the massive mode and Lagrange-multiplier fields are found to modify the Newton and Coulomb potentials. The nature and implications of these modifications are examined.
A theoretical framework is introduced that describes possible CPT-violating effects in the context of quantum electrodynamics. Experiments comparing the anomalous magnetic moments of the electron and the positron can place tight limits on CPT violation. The conventional figure of merit adopted in these experiments, involving the difference between the corresponding g factors, is shown to provide a misleading measure of the precision of CPT limits. We introduce an alternative figure of merit, comparable to one commonly used in CPT tests with neutral mesons. To measure it, a straightforward extension of current experimental procedures is proposed. With current technology, a CPT bound better than about 1 part in 10 20 is attainable. [S0031-9007(97)03884-2] PACS numbers: 11.30. Er, 12.20.Fv, 13.40.Em, 14.60.Cd The CPT theorem [1] is a powerful result holding for local relativistic quantum field theories of point particles in flat spacetime. It states that such theories must be invariant under the combined operations of charge conjugation C, parity reversal P, and time reversal T. Among the implications of the theorem are the equality of particle and antiparticle masses and lifetimes.Invariance under CPT has been tested in a variety of experiments [2]. The tightest bound published to date arises from experiments with the neutral kaon system [3], where the CPT figure of merit(1) is known to be smaller than 2 parts in 10 18 . This remarkable precision is possible because neutral-kaon oscillations provide a natural interferometer with dimensionless sensitivity controlled by the mass difference between the physical K L and K S states: j͑m L 2 m S ͒͞m K j Ӎ 10 214 . The quoted precision for r K is thus attained via measurements with a precision of about 1 part in 10 4 .Atomic experiments have also confirmed CPT symmetry. High-precision comparisons of the anomalous magnetic moments of the electron and positron currently provide the most stringent bounds on CPT violation in lepton systems [4]. Denote the electron and positron g factors by g 2 and g 1 , respectively. Then, a conventional figure of merit used in these experiments is [2]which is known to be smaller than 2 parts in 10 12 . The experiments confine isolated single electrons or positrons in a Penning trap for the indefinite periods [4,5] and measure their cyclotron and anomaly frequencies to a precision of better than 1 part in 10 8 . These frequencies can be combined to determine g 2 2, which is of order 10 23 , and hence to yield the limit on r g .The figure of merit r g is poorer than r K by about 6 orders of magnitude, even though the experimental measurements involved in the g 2 2 experiments are about 4 orders of magnitude sharper. This discrepancy originates in the difference between the quantities entering the dimensionless figures of merit. One is a mass (energy) difference while the other is a coupling difference. Indeed, all CPT tests to date have looked for differences between particles and antiparticle masses, lifetimes, or couplings. An important ...
A theoretical analysis is performed of Penning-trap experiments testing CPT and Lorentz symmetry through measurements of anomalous magnetic moments and charge-to-mass ratios. Possible CPT and Lorentz violations arising from spontaneous symmetry breaking at a fundamental level are treated in the context of a general extension of the SU͑3͒ϫSU͑2͒ϫU͑1͒ standard model and its restriction to quantum electrodynamics. We describe signals that might appear in principle, introduce suitable figures of merit, and estimate CPT and Lorentz bounds attainable in present and future Penning-trap experiments. Experiments measuring anomaly frequencies are found to provide the sharpest tests of CPT symmetry. Bounds are attainable of approximately 10 Ϫ20 in the electron-positron case and of 10 Ϫ23 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 Ϫ18 in the electron-positron system and others at the level of 10 Ϫ21 in the protonantiproton system. In contrast, measurements comparing cyclotron frequencies are sensitive within the present theoretical framework to different kinds of Lorentz violation that preserve CPT. Constraints could be obtained on one figure of merit in the electron-positron system at the level of 10 Ϫ16 , on another in the proton-antiproton system at 10 Ϫ24 , and on a third at 10 Ϫ25 using comparisons of H Ϫ ions with antiprotons. ͓S0556-2821͑98͒04207-6͔
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