Challenging the standard model by high-precision comparisons of the fundamental properties of protons and antiprotons

The BASE collaboration investigates the fundamental properties of protons and antiprotons, such as charge-to-mass ratios and magnetic moments, using advanced cryogenic Penning trap systems. In recent years, we performed the most precise measurement of the magnetic moments of both the proton and the antiproton, and conducted the most precise comparison of the proton-to-antiproton charge-to-mass ratio. In addition, we have set the most stringent constraint on directly measured antiproton lifetime, based on a unique reservoir trap technique. Our matter/antimatter comparison experiments provide stringent tests of the fundamental charge–parity–time invariance, which is one of the fundamental symmetries of the standard model of particle physics. This article reviews the recent achievements of BASE and gives an outlook to our physics programme in the ELENA era. This article is part of the Theo Murphy meeting issue ‘Antiproton physics in the ELENA era’.


Introduction
As a consequence of the combined charge, parity and time reversal (CPT) invariance [1]

Materials and methods
(a) Penning trap BASE uses an advanced Penning trap system to investigate and compare the fundamental properties of single trapped protons and antiprotons. In a Penning trap, a homogeneous magnetic field B in the axial direction (z) is superimposed with an electrostatic quadrupole potential φ(z, ), being the radial coordinate. In such a configuration of static fields, a charged particle describes a trajectory which is a superposition of three independent harmonic oscillator motions. The modified cyclotron motion at frequency ν + and the magnetron motion at frequency ν − are perpendicular to the magnetic field axis, and the axial motion at frequency ν z oscillates along the magnetic field lines. An invariance theorem ν 2 + + ν 2 z + ν 2 − = ν c relates the trap eigenfrequencies to the free cyclotron frequency ν c = (qB)/(2π m), which is defined by the fundamental properties, charge q and mass m, of the trapped particle [20]. If the particle carries spin, the associated magnetic moment precesses in the magnetic field of the trap with the Larmor frequency ν L = (g p,p /2) ν c , where g p,p /2 is the magnetic moment μ p,p of the trapped proton/antiproton in units of the nuclear magneton μ N .
For protons p and antiprotonsp, the Penning trap therefore provides access to two types of measurements: (i) charge-to-mass ratio comparisons, by measuring the cyclotron frequencies ν c,p and ν c,p , ν c,p ν c,p = R = (q p /m p ) (qp/mp) , and (ii) determinations of magnetic moments g p,p /2 = ±μ p,p /μ N by measuring ν L and ν c .
The determination of ν c is based on image current detection, which is nowadays a straightforward standard method relying on highly sensitive superconducting tuned circuits [21], while the measurement of the Larmor frequency ν L relies on the technically challenging application of the continuous Stern-Gerlach effect [3].

(b) Continuous Stern-Gerlach effect
Unlike in cyclotron frequency measurements, the Larmor precession of the particle's spin is not accompanied by a shift of charge. Therefore, it cannot be detected by direct image current methods. To access the Larmor precession, we superimpose a magnetic bottle B(z) = B 0 + B 2 z 2 on the Penning trap ( figure 1a). This adds a spin-dependent magnetic axial potential to the axial electrostatic trapping potential. Accordingly, the axial oscillation frequency becomes a function of the spin eigenstate, This method has been applied with great success in measurements of electron/positron magnetic moments [3]. The difficulty in applying these methods to the proton/antiproton is related to thẽ μ/m scaling of the axial frequency jump ν z,SF [22]. For protons and antiprotons, this ratio μ/m is about 10 6 times smaller than for the electron/positron. To tackle this challenge, we impose on our trap a magnetic bottle with a strength of B 2 ≈ 300 000 T m −2 . In this extreme magnetic environment, a proton/antiproton spin transition induces an axial frequency shift of only ν z,SF ≈ 170 mHz out of typically ν z = 680 kHz.
Once an apparatus has been designed which enables the detection of the axial frequency shifts induced by spin transitions, the spin-flip probability as a function of an applied radio-frequency (rf) can be measured [22]. The result leads to a well-understood line-shape [23], as shown in figure 1b, from which ν L is extracted. The same principle can be applied to measure the cyclotron frequency ν c [18].
The development of the methods which enable the detection of these tiny axial frequency shifts in the presence of the ultra-strong B 2 required several years of dedicated research and development work and is documented in a sequence of publications in which we have demonstrated the first observation of spin flips with a single trapped proton [22] as well as the first high-fidelity detection of single-proton spin transitions [24]. Related work by another effort has been published in [11].   The above-explained principle to determine the g-factor of a single proton/antiproton in a Penning trap with a superimposed magnetic bottle is limited by the bottle itself, which produces a considerable line-width. Quantum transitions in the magnetron oscillator mode limit the determination of ν L and ν c to the ppm level [18]. To overcome this limitation, Häffner et al. [25] introduced the double Penning trap technique. This method has been developed to enhance measurement precision in determinations of the g-factor of the electron bound to highly charged ions. BASE adapted this method and successfully developed a double Penning trap instrument [14] with 30-fold improved sensitivity with respect to magnetic moments (figure 2) [22].
In the double-trap method, a homogeneous precision Penning trap (PT) is added to the trap assembly. In this PT, the magnetic field is about 100 000 times more homogeneous than in the trap with the superimposed magnetic bottle (analysis trap (AT)). In a double-trap measurement sequence, the spin state is identified in the AT; afterwards the particle is transported to the precision trap, where its cyclotron frequency ν c is measured while spin transitions are induced by applying a magnetic rf drive at ν rf . Subsequently, the particle is transported back to the AT where its spin state is analysed. By repeating this sequence many times at different drive frequencies, the . Multi-trap assembly as used in the proton-to-antiproton charge-to-mass ratio comparison, the first high-precision Penning trap experiment in which the 'fast shuttling' method was applied [13]. The image also shows the reservoir trap, which enables antiproton experiments, independent of accelerator up-times.
spin-flip probability as a function of ν rf /ν c is obtained. Compared to single-trap measurements, the homogeneous magnetic field in the PT drastically narrows the width of the g-factor resonance, typically enabling measurements on the ppb level [14]. The application of the double-trap method requires high-fidelity spin state detection, which means that the spin state of the particle can be clearly determined within two axial frequency measurements, constituting the major challenge in such measurements [24].
(ii) Charge-to-mass ratio measurement in a triple trap Classical mass spectrometers usually use single measurement traps. In frequency-ratio measurements, the particles are loaded from external or in situ sources; this re-loading/ preparation procedure typically takes several tens of minutes. An elegant update to this classical method has been introduced by Gabrielse et al., which compared cyclotron frequencies of antiprotons and negatively charged hydrogen ions [10]. In this experiment, both particles were stored in the same trap, the measurement particle in the centre and the second particle on a large orbit. Including particle swapping, a single frequency-ratio measurement typically required 4 h. In [13], we have applied a novel three-trap method, which uses two park traps and one measurement trap, as illustrated in figure 3. Here we prepare a particle in the centre of the measurement trap and another one in a park electrode. Shuttling of this particle configuration along the magnetic field axis enables particle exchange within typically 10 s. By combining this fast shuttling technique with sideband measurement methods to determine the cyclotron frequency, we were able to perform one single frequency-ratio measurement in 240 s, which corresponds to a 60-fold improved ratio sampling rate, compared to previous proton/antiproton charge-to-mass ratio comparisons.  we demonstrated antiproton trapping for about 405 days. Figure 4 shows the content of the reservoir trap as a function of time; steps are related to documented extractions. None of the steps is caused by an antiproton decay or annihilation with background gas. By combining all available reservoir trap datasets, we extract a lower antiproton lifetime limit of τp > 10.2 a [17]. Compared to other CPT tests, antiproton lifetime measurements test the conservation of the baryon number.

Physics
In 1999, the APEX collaboration set limits on several antiproton decay channels in a storage ring [26]. Cosmic ray studies have set antiproton lifetime constraints based on model-dependent antiproton production mechanisms in the cosmos [27]. By contrast, our measurement sets the best directly measured limits on the full half-life of the antiproton. It is sensitive to potentially invisible decay channels which are favoured by some grand unified theories [28,29]. BASE plans to further push the limit in τp by improving the trapping rate by a factor of 10-100.

(b) Proton-to-antiproton charge-to-mass ratio
With the fast shuttling technique which has been described above, we performed in the 2014 antiproton run about 6500 cyclotron-frequency-ratio comparisons of antiprotons and negatively charged hydrogen ions [13]. Using hydrogen ions instead of protons drastically reduces systematics in the frequency-ratio comparisons [10,13]. The hydrogen ion serves down to the sub-ppt level as a perfect, negatively charged, proxy of the proton, its mass being Here the first correction term is the electron-to-proton mass ratio, the second a polarizability shift; the last two corrections account for the binding energies of the electrons. This leads to a theoretically expected cyclotron frequency ratio ν c,p /ν c,p = R =  which is within 1.2 standard errors (68% confidence level) consistent with CPT invariance. The high sampling rate in our charge-to-mass ratio measurements also enables the search for sidereal variations. Such variations are possible by Lorentz-violating extensions of the standard model. By performing a lock-in analysis to the dataset shown in figure 5b, we obtain the output shown in figure 5c, which sets to any Lorentz-violating diurnal effect a limit dR/R(t diurnal ) < 0.72 ppb. Given the assumption that CPT invariance holds, the above measurement allows for a test of the weak equivalence principle (WEP). In this case, the proton/antiproton cyclotron frequencies are, in the absence of a gravitational field, identical by definition. If a gravitational field is being introduced while the WEP is broken to some extent, the proton and the antiproton will experience a different gravitational red-shift. The combined value (CPT) QM constrains |α g − 1| < 6.9 × 10 −7 , the parameter |α g − 1| representing a possible gravitational anomaly acting on antimatter [30]. Note that we follow here the previously published literature, which uses the gravitational potential of the local galactic supercluster, an assumption which has been controversially discussed. To set direct constraints on the gravitational red-shift, which do not require additional assumptions, e.g. by performing charge-to-mass ratio measurements at different altitude, a fractional resolution on the 0.01 ppt scale would be required. The AEgIS, ALPHA-g and GBAR experiments at CERN are planning to investigate the WEP using antihydrogen. Their expected results will provide important information to disentangle the limits on WEP and CPT violation from proton/antiproton cyclotron frequency comparisons.

(c) Proton and antiproton magnetic moments
Several years of research and development work were required to implement the methods to compare the magnetic moments of the proton and the antiproton. In this context, we have reported on the first observation of spin flips with a single trapped proton [22], the first   [24] and the first successful demonstration of the double Penning trap technique [31]. These achievements culminated in the first direct highprecision measurement of the proton magnetic moment [14] in units of the nuclear magneton. From the recorded double-trap g-factor resonance, we obtain g 2 p = 2.792 847 350 (9).
This result constitutes the most precise measurement of the proton magnetic moment to date, agrees with the previous best value which was extracted from hydrogen maser spectroscopy [32], but is about three times more precise. One of the major goals of the BASE collaboration is the application of the double Penning trap technique to measure the magnetic moment of the antiproton at a comparable level of precision. The implementation of this measurement in the noisy environment of an accelerator hall is a highly challenging task. In a first step towards this major goal, we have performed a single Penning trap measurement of the antiproton magnetic moment, relying on frequency measurements in the strong magnetic bottle. In that measurement campaign, we have recorded in total 12 cyclotron resonances, interleaved by Larmor resonance scans [18]. The resulting g-factor which was obtained from this measurement campaign, has a fractional precision of 0.8 ppm at 95% confidence level. This measurement improves the fractional precision of the previous best value [11] by a factor of six, and is consistent with CPT invariance: (CPT) μ = (g/2)p (g/2) p = 0.999 999 70 (82), limited by the measurement precision of the antiproton magnetic moment.
With the successful application of the double Penning trap technique to measure the magnetic moment of the antiproton, an at least 100-fold improved CPT test with proton/antiproton magnetic moments will become possible. Very recently, a major step towards this challenging goal has been achieved. For the first time, we have observed single antiproton spin transitions [19]. A representative result on the unambiguous detection of such antiproton spin flips is shown in figure 6.
In the experiment, the axial frequency of a single antiproton is measured in the magnetic bottle of the analysis trap. After each frequency measurement, a resonant rf drive is applied  Table 1. SME coefficients which are constrained by the BASE-Mainz and the BASE-CERN experiment. Second column: previous constraints as published in [35]. Third column: updated constraints published in [21].

Discussion
Several models exist which provide a framework to interpret and compare sensitivities of tests of CPT invariance [33], the most prominent of which is the standard model extension (SME) [34]. This carefully constructed model adds CPT-odd and Lorentz-violating contributions to the standard model; the shape of the considered corrections is inspired by effective field theory approaches applied to string theory. A comprehensive paper by Ding & Kostelecký [35] describes the derivation of SME coefficients based on experiments located at different coordinates and different alignment with respect to the rotational axis of the Earth's laboratory frame. In Penning traps, the SME interactions modify the energy levels of the trapped particle, which results in a CPT-odd frequency difference for quantum transitions involving a spin flip. By combining the BASE-Mainz data [14] with the BASE-CERN data [18] we are able to constrain energy difference modifications of the quantum-level structure resulting in limits on six combinations of SME coefficients. Table 1 summarizes previous upper bounds together with the updated numbers published in [18]. The recent BASE magnetic moment measurement improves the upper limits of the leading coefficient by a factor of 10-20.

BASE physics in the ELENA era
The major goal of BASE in the ELENA era is the further improvement of measurement precision in (CPT) QM and (CPT) μ . The next logical step to improve (CPT) μ is the application of the double Penning trap technique to measure the magnetic moment of the antiproton. Major steps towards this goal have been achieved [19,21]. We anticipate that in first measurements after successful implementation of the double-trap method fractional precisions on the ppb level will be reached, with the potential to be further improved by at least a factor of 10. Highly stabilized superconducting trap magnets together with elegant phase-sensitive detection techniques [36] applied to measure the proton/antiproton oscillation frequencies will allow further reduction of the resonance line-widths.
The time budget of current magnetic moment measurements is limited by selective resistive cooling which is necessary to achieve single spin-flip resolution [14,19,22,24]. The implementation of sympathetic cooling of antiprotons by coupling the particles to laser-cooled Be + ions using a common endcap method [37] or by direct Coulomb coupling in a micro-fabricated Penning trap [38], which are currently being prepared by collaboration members of BASE [37,38], will enable measurements at improved data-sampling rate. We anticipate that by combining all these techniques magnetic moment measurements at the 10 ppt level will become possible, in the long term.
Our 2014 (CPT) QM figure has reached limits imposed by the accelerator; another systematic error arose from systematic frequency shifts caused by residual inhomogeneities of the traps magnetic field [13]. To further improve (CPT) QM we have developed a self-shielding solenoid system with an improved shielding factor, reduced the magnetic field inhomogeneity of the trap and implemented tuneable axial detection systems which enable measurements in a static electric field [39] to further reduce systematic limitations. By combining these technical upgrades, we expect to improve the current best proton-to-antiproton charge-to-mass ratio comparison by at least a factor of 5. In addition, it is planned to implement the Pritchard two-particle phase method [40]. This technique makes use of simultaneous cyclotron frequency measurements using co-trapped particles on phase-locked magnetron orbits and is insensitive to first-order external magnetic field fluctuations.

Conclusion
Since the approval in 2013, the BASE collaboration has carried out the most precise comparison of the proton-to-antiproton charge-to-mass ratio [13], the most precise measurements of the proton [14] and antiproton magnetic moments [18], and has set the most stringent limit on directly measured antiproton lifetime [17]. Very recently, we resolved single spin flips with a single trapped antiproton [19] which is a major step towards an at least 100-fold improved measurement of μp. We anticipate that in the ELENA era the uncertainty in both numbers (CPT) QM and (CPT) μ can be improved to the level of 10 ppt and 1 ppt, respectively.
Data accessibility. All data presented in this paper will be provided on reasonable request. Authors' contributions. This short review summarizes work of the BASE collaboration; author contributions to the presented datasets are described in the referenced original papers. The paper has been written by S.U. and edited and improved by all co-authors.
Competing interests. We have no competing interests.