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. 2018 Feb 19;376(2116):20170276. doi: 10.1098/rsta.2017.0276

Prospects for testing Lorentz and CPT symmetry with antiprotons

Arnaldo J Vargas 1,✉
PMCID: PMC5829178  PMID: 29459417

Abstract

A brief overview of the prospects of testing Lorentz and CPT symmetry with antimatter experiments is presented. The models discussed are applicable to atomic spectroscopy experiments, Penning-trap experiments and gravitational tests. Comments about the sensitivity of the most recent antimatter experiments to the models reviewed here are included.

This article is part of the Theo Murphy meeting issue ‘Antiproton physics in the ELENA era’.

Keywords: Lorentz violation, CPT violation, antimatter

1. Introduction

The CPT theorem states that any Lorentz-invariant local quantum field theory is CPT-invariant. The standard model of particle physics satisfies the premises of the theorem and so CPT (charge, parity and time reversal) symmetry is generally considered a symmetry of Nature. The huge success of the standard model has justified the idea that we can predict the behaviour of antimatter by studying the behaviour of matter. For example, the standard model allows us to predict the absorption and emission spectra of antimatter from studying how ordinary matter emits and absorbs light.

Most of these predictions about the behaviour of antimatter remain unverified experimentally due to the low abundance of antimatter in the Universe and the technical difficulties in manipulating and trapping antimatter. It could be argued that the low abundance of antimatter observed in the Universe indicates that there is something fundamentally different about matter and antimatter beyond the amount of charge-conjugation violation allowed in the standard model of particle physics. This not only puts into question if CPT symmetry is really a symmetry of Nature but also if the weak equivalence principle (WEP) holds for antimatter. For these reasons it is important to test the predictions of the standard model and general relativity with antimatter experiments.

Other theoretical motivations for testing CPT symmetry are related to the question of how to describe gravity at the quantum level. One of the challenges of formulating a quantum theory of gravity is the lack of relevant experimental data that could distinguish between different quantum gravity candidate theories. Contrary to the situation with the other fundamental interactions, the gravitational interaction between elementary particles cannot be probed directly via scattering experiments, as even the best accelerators in the world are incapable of accelerating particles to energies where the gravitational effects are expected to be relevant. Conceivable methods for collecting experimental data that could distinguish between the candidate theories have been suggested, such as looking for low-energy exotic signals of these candidate theories. Examples of these exotic signals are CPT and Lorentz violation, which have been suggested as a plausible low-energy signal for candidate theories such as string theory [1,2], non-commutative field theory [3–8], loop quantum gravity [9], granular space–time models [10] and multiverse scenarios [11].

This article discusses several models for Lorentz and CPT violation that are of relevance to collaborations, such as Antihydrogen Laser Physics Apparatus (ALPHA), Atomic Spectroscopy and Collisions Using Slow Antiprotons (ASACUSA), the Antihydrogen Trap (ATRAP), Baryon Antibaryon Symmetry Experiment (BASE), Antihydrogen Experiment: Gravity, Interferometry, Spectroscopy (AEGIS) and Gravitational Behaviour of Antihydrogen at Rest (GBAR). The experimental scenarios considered in these models include spectroscopy experiments with antimatter, gravitational tests with antimatter and Penning-trap experiments with antiprotons.

2. Standard-model extension

The standard-model extension (SME) is an effective field theory that was conceived to facilitate the systematic search for Lorentz violation [12–14]. The SME extends Einstein’s theory of general relativity and the standard model by adding to the Lagrange density all the operators that break Lorentz symmetry and that can be obtained from conventional field operators. The SME can be understood as a framework for the development of test models for experimental tests of Lorentz symmetry. Models for Lorentz violation obtained from the SME exist for spectroscopy experiments [15–19], spin-precession experiments [17,19–21], neutrino oscillation experiments [22], accelerator experiments [23], short-range gravity experiments [24] and WEP tests [25], among others.

Because of the deep connection between CPT and Lorentz symmetry, the SME contains any realistic test model for CPT violation. The reason for this is that the SME includes all the terms that break Lorentz symmetry, and any local interacting quantum field theory that breaks CPT symmetry also breaks Lorentz symmetry [26]. This suggests that experimental groups around the world that are conducting experiments to test CPT symmetry can use the SME as a framework to quantify the results of their experiments as bounds on controlling coefficients for CPT violation. The bounds on the controlling coefficients for CPT and Lorentz violation are collected annually in the ‘Data tables for Lorentz and CPT violation’ [27]. By looking at the tables, the experimental groups will be able to compare the sensitivity of their tests of CPT symmetry to the sensitivities reached by other groups around the world. Furthermore, as explained in this article, testing Lorentz symmetry with antimatter experiments is fundamental in any systematic search for Lorentz violation. This is another motivation to use the SME as the framework for creating the test models for antimatter experiments.

The controlling coefficients for Lorentz violation can be interpreted as the components of background fields [12,13]. In general, each experiment will be sensitive only to some of the components of the background fields, and only by using the results of several experiments can a bound on the size of the Lorentz-violating background fields be obtained. As the components of the background fields can change from reference frame to reference frame, it is essential to report all the measurements of the controlling coefficients for Lorentz and CPT violation in the same reference frame. The reference frame used to report the results for Lorentz and CPT violation in the data tables is the canonical Sun-centred frame [27,28]. The orientation of the frame is fixed by requiring that the Sun is at rest in this frame with the X-axis pointing in the direction of the vernal equinox, and the Z-axis parallel to the Earth’s axis of rotation.

3. Spectroscopy experiments with antimatter

The antimatter collaborations ALPHA [29], ASACUSA [30] and ATRAP [31] have performed or are planning to perform spectroscopy experiments with antimatter. The first transitions to be targeted by these collaborations are the hyperfine transition of the ground state and the 1S–2S transition. The general expressions for the dominant modifications to the spectrum of antihydrogen due to CPT- and Lorentz-violating operators were obtained in [15].

As the transitions to be studied are between nS1/2 states, it is enough to discuss the Lorentz-violating energy shift of the nS1/2 states. For hydrogen, the energy shift in the presence of a weak magnetic field is given by

3. 3.1

with the quantum number mF being the eigenvalue of the projection of the total angular momentum in the direction of the applied magnetic field and w the flavour index of the Lorentz-violating operator, with w=e corresponding to the Lorentz-violating operators in the electron sector of the SME and w=p in the proton sector. The symbols Sw, Inline graphic, Inline graphic and Inline graphic represent combinations of coefficients for Lorentz and CPT violation. For example,

3. 3.2

where α is the fine-structure constant, me the mass of the electron and n the principal quantum number. The explicit expressions for the other combinations of coefficients in equation (3.1) can be found in [15]. The controlling coefficients for Lorentz violation awNRkjm and cwNRkjm are known as the effective non-relativistic coefficients and are defined in [32]. The coefficient awNRkjm also controls CPT-violating effects.

The energy shift for antihydrogen has the same structure as the shift for hydrogen except that the effective combinations of coefficients are different, and this is represented by the use of an asterisk (*) on the symbols representing the linear combinations of coefficients. Using this notation, the energy shift for antihydrogen is given by

3. 3.3

The explicit expression for the symbol S*w is given by

3. 3.4

Note that the only difference between S*w and Sw is the sign in front of the awNRk00 coefficients, which are the coefficients controlling the CPT violation.

As a general rule, low-energy matter experiments are sensitive to the combination of coefficients Inline graphic and antimatter experiments to the combination Inline graphic. This difference between the linear combinations of coefficients in matter experiments compared to antimatter experiments is one of the main motivations to test Lorentz symmetry with antimatter. For example, it is possible to construct a model for Lorentz violation, where Inline graphic, suppressing the signals for Lorentz violation in ordinary matter experiments and enhancing them in antimatter experiments.

In equations (3.1) and (3.3), the unit vector Inline graphic is the direction of the applied magnetic field and β is the instantaneous velocity of the laboratory frame relative to the Sun-centred frame. The instantaneous velocity of the laboratory frame β can be approximated as

3. 3.5

where β⊕ is the Earth’s instantaneous orbital velocity with respect to the Sun and βL the instantaneous tangential velocity relative to the centre of mass of the Earth. Approximating the orbits as circular orbits, we can express them as

3. 3.6

and

3. 3.7

In these expressions Ω⊕≃2π/(365.26 d) is the Earth’s orbital angular frequency, η≃23.4° is the angle between the Equator and the orbital plane of the Earth, β⊕≃10−4 is the Earth’s orbital speed, T is the time as measured in the Sun-centred frame, T⊕ is the sidereal time and ω⊕≃2π/(23 h 56 m) is the Earth’s sidereal frequency.

The direction of the magnetic field Inline graphic can be expressed as

3. 3.8

where ϑ is the angle between the magnetic field and the rotational axis of the Earth. The angle ϕ is the azimuthal angle of the magnetic field in the Sun-centred frame at T⊕=0.

The ASACUSA collaboration will measure the hyperfine-Zeeman transitions of the ground state of hydrogen in a low magnetic field. In the hyperfine-Zeeman regime, the Lorentz-violating frequency shift to the transition frequency between the sublevels of the 1S1/2 state is given by

3. 3.9

where ΔmF is the change in mF during the transition. The first thing to notice from this expression is that there is no frequency shift if ΔmF=0 and that means that at leading order the sigma transition is insensitive to Lorentz and CPT violation. Because of the time dependence of β and Inline graphic, the frequency shift (3.9) changes with sidereal and annual times. For instance, the terms Inline graphic and Inline graphic can produce sidereal variations, with the former producing variations in the first harmonic of the sidereal frequency and the latter in the first and second harmonics of the sidereal frequency. The term Inline graphic can produce annual variations of the frequency. Also there is a contribution to equation (3.9) that is given by

3. 3.10

that does not introduce any time variation of the frequency.

Annual and sidereal variation studies of the transition frequencies in antihydrogen are sensitive to CPT- and Lorentz-violating effects that cannot be studied by simply comparing the value of the frequency obtained in a hydrogen experiment to the one obtained in an antihydrogen experiment. The coefficients that can be constrained by comparing the transition in hydrogen with the one in antihydrogen are only those shown in equation (3.10). Note also that the combination of coefficients that will be constrained by the experiment also depends on the direction of the magnetic field.

The ALPHA collaboration measured the hyperfine splitting in a strong magnetic field [33]. The frequency shift is given by

3. 3.11

where Δsw represents the spin flips of the antiproton and the positron. The ALPHA collaboration measured the frequency of the transition Inline graphic that corresponds to the spin flip of the positron between two states with the spin of the antiproton parallel to the applied magnetic field and the transition Inline graphic that is the spin flip between the states with the spin of the antiproton antiparallel to the magnetic field. The Lorentz-violating shifts to both transition frequencies are identical and they are given by

3. 3.12

Because only the spin state of the positron is affected in the transitions, they are only sensitive to the electron coefficients.

The difference between the positron spin-flip transitions Inline graphic and Inline graphic is equal to the hyperfine splitting of the ground state; however, because the leading-order Lorentz-violating shifts to the transitions are identical, they cancel out. For this reason, the recent comparison of the hyperfine transition of hydrogen and antihydrogen by the ALPHA collaboration is not sensitive to the leading-order CPT effects. The result reported by ALPHA could be, in principle, sensitive to CPT-violating effects that are considerably more suppressed that the ones considered in [15]. In spite of this possibility, the crucial point is that the transitions Inline graphic and Inline graphic individually are more sensitive to CPT violation than the difference between them.

Another transition that has been measured by ALPHA is the 1S–2S transition in antihydrogen [34]. Contrarily to the hyperfine splitting of the ground state, the 1S–2S transition is sensitive to the S*w and V *w combinations of coefficients. For example, the contribution from the S*w term to the Lorentz-violating frequency shift of the 1S–2S transition is given by

3. 3.13

As in the case of equation (3.10), the terms in equation (3.13) are the terms that can be tested by comparing hydrogen and antihydrogen. For instance, a measurement of the 1S–2S transition in antihydrogen with a sensitivity of 100 kHz could lead to bounds on the awNR200 coefficients of the order of 10−7 GeV−1. That implies that the recent measurement of the 1S–2S transition in antihydrogen [34] can be used to impose competitive bounds on coefficients for Lorentz and CPT violation.

As mentioned before, comparing the transitions in hydrogen with the transitions in antihydrogen is only sensitive to a small number of the CPT-violating operators that can be studied in antihydrogen experiments. Searching for the sidereal and annual variations of the spectroscopy frequencies generated by the term β⋅V*w is a way to bound CPT-violating effects that are inaccessible just by comparing hydrogen and antihydrogen transitions.

As mentioned in [15], transitions involving states with higher angular momentum in general are sensitive to coefficients for CPT and Lorentz violation that cannot be studied in transitions that only involve nS1/2 states. For example, the proposed experiment to measure the Lamb shift of antihydrogen [35] could be sensitive to CPT-violating effects that cannot be studied using the 1S–2S transition or the hyperfine transition of the ground state.

Another interesting point is that, as demonstrated in [15], spectroscopy experiments with atoms with heavier nucleus can enhance the sensitivity to proton Lorentz-violating operators by around eight orders of magnitude. That means that CPT tests with antideuterium or antihelium will be significantly more sensitive to some CPT-violating operators than antihydrogen experiments.

4. Spin-precession experiments with antiprotons

Models for testing Lorentz symmetry in proton spin-precession experiments already exist in the literature [20,21]. In the most recent of these models [21], the frequency shift to the anomalous frequency due to Lorentz violation takes the form

4. 4.1

where Inline graphic and Inline graphic are linear combinations of proton coefficients for Lorentz violation defined in [21]. As was the case in the previous section, the vector B represents the applied magnetic field that in this case is the magnetic field of the Penning trap. The conventions used in expressing equation (4.1) are different than the conventions used in [21], where the original expression was derived. The expression (4.1) was adapted to match the notation used in the previous section of this article.

The terms Inline graphic and Inline graphic can produce sidereal variations with the first harmonic of the sidereal frequency and the term Inline graphic can also produce variations with the second harmonic of the sidereal frequency. Comparing equations (4.1) and (3.1) the reader can note that corrections that are proportional to the relative velocity between the laboratory frame and the Sun-centred frame are absent in equation (4.1). These effects are generally suppressed due to the small relative velocity between the two frames and for that reason they were not considered in [21]. Despite the lack of any model for these higher-order corrections, it is possible to deduce some of the signals that will emerge from them, such as sidereal variations with the third harmonic of the sidereal frequency and annual variations.

As discussed in the previous section, the models for Lorentz and CPT violation in antimatter experiments have the same structure as in matter experiments, with the difference that the combinations of coefficients that can be studied in antimatter experiments is independent of the combinations of coefficients that can be studied in matter experiments. In Penning-trap experiments, the combinations of coefficients that can be studied with antimatter experiments are represented by the effective coefficients Inline graphic and Inline graphic. Using this notation, the corrections to the anomalous frequency of the antiproton due to Lorentz violation can be expressed as

4. 4.2

Recently, the ATRAP [36] and the BASE [37] collaborations have measured the g-factor of the antiproton. Comparing these values with the best measurement of the g-factor of the proton [38] is a way to constrain the terms in equations (4.1) and (4.2) that are independent of the sidereal time. The contribution to equation (4.2) that is independent of the sidereal time can take the form

4. 4.3

in the case that the applied magnetic field is pointing radially outwards from the centre of the Earth and the experiment is located at the colatitude χ. The equivalent expression to equation (4.1) can be obtained by replacing Inline graphic and Inline graphic in equation (4.3). Bounds on the coefficients for Lorentz and CPT violation were obtained in [21] by comparing the g-factor of the proton and antiproton and afterwards the bounds were improved by the BASE collaboration [37]. Similar to the discussion of the previous section, it is important to understand that sidereal variation studies of the anomalous frequency of the antiproton are sensitive to CPT-violating effects that were not constrained by comparing the g-factor of the proton and the antiproton.

5. Gravitational tests with antimatter

Certain models for Lorentz and CPT violation allow for a breaking of the WEP, including the possibility that antimatter and matter experience different gravitational accelerations [25]. For instance, one of the models derived from the SME in [25] that breaks the WEP is given by the Lagrangian

5. 5.1

where Inline graphic is the effective inertial mass of the test particle, mTg its effective gravitational mass, v its velocity and r its position relative to a source with effective gravitational mass mSg.

The interesting feature of the model is that the effective inertial and gravitational masses can depend on the composition of the test particle and the source. For the particular case that the test particle is a hydrogen atom, the effective masses are given by

5. 5.2

where mH is the conventional mass of hydrogen and mw is the mass of the corresponding subatomic particle. The constant α in this expression is not the fine-structure constant but instead is a constant that is related to the fluctuations of the Lorentz-violating background fields around their vacuum expectation values that incidentally are represented by the coefficients Inline graphic and Inline graphic [25].

In the case that we consider antihydrogen as our test particle instead of hydrogen, the expression for the effective inertial mass will remain the same and the expression for the effective gravitational mass will be given by

5. 5.3

with Inline graphic being the mass of antihydrogen. Again the only difference between the models for the matter and antimatter experiments is the sign in front of the coefficients associated with CPT-violating effects.

The isotropic ‘parachute’ model (IPM) [25] is the model obtained if we assume that

5. 5.4

Under this assumption, the effective gravitational and inertial masses of hydrogen will be identical. On the other hand, the model predicts that the gravitational and inertial masses of antihydrogen will be different. In this model, any deviation of the WEP in ordinary matter experiments is suppressed but such suppression is not present in antimatter experiments [25]. Some of the predictions of the model are that the gravitational accelerations of hydrogen and antihydrogen are different, the gravitational accelerations of hydrogen and deuterium are the same, and the gravitational accelerations of antihydrogen and antideuterium are different.

Indirect limits on an anomalous response of antimatter to gravity can be found in the literature [25]. These indirect bounds on the difference between the gravitational acceleration of matter and that of antimatter are based on specific models. For example, most of the arguments found in the literature are ineffective to limit the kind of anomalous gravitational response of antimatter predicted by the IPM [25].

The model (5.2) only considers the contributions from isotropic Lorentz-violating operators in the Sun-centred frame. Allowing contributions from anisotropic operators will introduce annual and sidereal variations to the gravitational responses of matter and antimatter [25]. This means that the gravitational acceleration of antimatter might change at different times of the day and at different times of the year. For this reason, the collaborations that are planning to measure or have measured the gravitational acceleration of antimatter, such as AEGIS [39], GBAR [40] and ALPHA [41], should perform analyses looking for sidereal and annual variations of their data if they want to study all the CPT-violating effects that can be accessed in their experiments.

Data accessibility

This article has no additional data.

Competing interests

I declare I have no competing interests.

Funding

This work was supported in part by the US Department of Energy under grant number DE-SC0010120 and by the Indiana University Center for Spacetime Symmetries.

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