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Antimatter and Gravity (WAG 2013) International Journal of Modern Physics: Conference Series Vol. 30 (2014) 1460260 (9 pages) c The Author DOI: 10.1142/S2010194514602609
Constraints on gravitational properties of antimatter from cyclotron-frequency measurements
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Michael H. Holzscheiter Department of Physics & Astronomy, University of New Mexico 1919 Lomas Blvd, Albuquerque, NM 87131, USA
[email protected] Received 7 February 2014 Revised 17 March 2014 Published 7 May 2014 A fundamental question in physics that has yet to be addressed experimentally is whether particles of antimatter, such as the antiproton or positron, obey the weak equivalence principle (WEP). Several theoretical arguments have been put forward arguing limits for possible violations of WEP. No direct ‘classical’ gravitational experiment, the measurement of the free fall of an antiparticle, has been performed to date to determine if a particle of antimatter would experience a force in the gravitational potential of a normal matter body that is different from normal gravity. 30 years ago we proposed a free fall experiment using protons and antiprotons, modeled after the experiment to measure the gravitational acceleration of a free electron. At that time we gave consideration to yet another possible observation of gravitational differences between matter and antimatter based on the gravitational red shift of clocks. I will recall the original arguments and make a number of comments pertaining to the technical problems and other issues that prevented the execution of the antiproton free fall measurement. Note that a different gravitational force on antimatter in the gravitational field of matter would not constitute a violation of CPT, as this is only concerned with the gravitational acceleration of antimatter in the gravitational field of an antimatter body. Keywords: Gravity; antimatter; clocks.
1. Introduction A fundamental question in physics that has recently attracted the interest of several experimental collaborations at CERN1−3 and elsewhere4 is whether particles of antimatter, such as the antiproton, the positron, or an antihydrogen atom obey the weak equivalence principle. In contrast to a proposal submitted to CERN and approved for running at LEAR as PS2005 in 1985, all current experiments under development plan to use a neutral antimatter particle, the antihydrogen atom. This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 3.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited. 1460260-1
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This is based on the expectation that stray electric fields would make it impossible to perform a measurement on the antiproton. Our original proposal coincided with a very active period in physics looking for deviation from normal gravity for matter particles on many different length scales and stringent limits were given for such “fifth force” effects.6 At the same time arguments by Morrison7 and Schiff8 were given attempting to rule out the notion of “antigravity”. Good9 interpreted the lack of observation of regeneration of KL into KS as an indication against antigravity. An excellent review of these arguments, and a much deeper insight into the theoretical background behind these discussions as can be presented in the limited space of this report, has been given by M. Nieto and T. Goldman.10 Meanwhile, during the time of development of the PS200 apparatus, highprecision experimental results on the equality of the inertial masses (or more precisely the equality of the charge–to-mass ratios) of protons and antiprotons had become available from an experiment comparing the cyclotron frequencies of the particles in the same magnetic field.11 Since then these measurements have been improved by nearly 3 orders of magnitude.12 These results are usually regarded as very sensitive tests of CPT symmetry.13 However, assuming exact CPT symmetry, they can also provide tests of the weak equivalence principle for a gravitational coupling to the energy of positrons and antiprotons, using certain assumptions for the coupling of gravity to energy and mass. This possibility arises because the frequencies in question constitute local ”clocks” and as such are subject to a gravitational red shift, which may be formulated as a test of weak equivalence for their energy content.
2. Einstein’s “Dumb Waiter” Experiment For this, we will consider a modified version of Einstein’s “Gedanken Experiment” described in his seminal paper on the influence of gravitation on the propagation of light.14 There he showed that the gravitational red shift can be directly derived from conservation of energy in the gravitational field. Consider some local clock based on the photon frequency in the transition A∗ → A + γ between system A and its excited state A∗ . In a uniform gravitational field, A(A∗) experiences gravitational acceleration g(g ∗ ) associated with inertial mass m(m∗ ), respectively. The relative rates of two such clocks at different heights in the gravitational field may be compared by exchanging photons.14 We start with system A located at some height h1 and system A∗ above it at height h2 , with h2 − h1 = l. Next, A and A∗ are interchanged, releasing an amount of energy Eout = (m∗ g ∗ − mg)l.
(1)
System A∗ , now located at the lower height h1 , is allowed to decay to its ground state A by emission of a photon of frequency ω1 with ω1 = (m∗ − m)c2 , which defines the local clock frequency. This photon is allowed to propagate up to height 1460260-2
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h2 , where its frequency will have suffered a generalized red shift to some value ω2 , according to the local clock at h2 , ω2 = ω1 (1 − gR l/c2 )
(2)
where the parameter gR has the dimension of an acceleration. In order to excite the system A at the upper level h2 and recover the initial configuration, the energy carried by the photon must be augmented by an amount
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Ein = (m∗ − m)gR l.
(3)
Conservation of energy requires Ein = Eout , and hence gR =
∆(clock weight) m∗ g ∗ − mg = . ∗ m −m ∆(clock mass)
(4)
Thus, the gravitational red shift is a test of weak equivalence for the energy content of the clock, and the conventional red shift, for which gR = g, only arises if weak equivalence, g ∗ = g, is obeyed. 3. Gravity on Antimatter The notion of precision spectroscopy on anti-atoms was at the time of our original paper still a futuristic idea (and to some extend still is today) and we therefore considered in place of some internal transition to an excited state the cyclotron frequency of an antiproton (proton) trapped in the field configuration of a Penning trap, which, aside from minor corrections due to the presence of the electric fields, is given simply by ωc =
e × B, m
(5)
with e and m the charge and inertial mass for the particle under consideration, and B the strength of the external magnetic field. If CPT symmetry is assumed to be exact, the particle and antiparticle cyclotronfrequency clocks will have identical rates at “infinity” (beyond the range of any equivalence-principle-violating interaction). However, from the above argument, if the proton or electron respectively respect the weak equivalence principle, and we assume that any violation of equivalence for the antiproton (positron) occurs through an anomalous coupling strength of gravity to its energy, the antiproton’s (positron’s) cyclotron frequency will red shift by an amount different from the proton (electron) when they are lowered to the same height in a gravitational field from “infinity”, resulting in a measurable frequency difference. To transform this general argument into a quantitative and measurable statement we consider a phenomenological model for a violation of the weak equivalence for antiprotons or positrons. Near the surface of the Earth, protons, electrons, 1460260-3
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and electromagnetism experience conventional gravity through the interaction Lagrangian L = hµν T µν
(6)
where T µν is the energy-momentum tensor of the particles and electromagnetism. The (weak) tensor gravitational field is
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hµν = (2U/c2 )diag(1, 1, 1, 1)
(7)
where U is the Newtonian gravitational potential. A violation of the weak equivalence for antiprotons or positrons can now be introduced by modifying the above Lagrangian to the form L = αhµν T µν
(8)
where α is an adjustable coupling parameter. Protons and electrons experience a gravitational acceleration g, antiparticles experience an anomalous acceleration g = αg. To obtain the effect of gravity on the particle’s cyclotron frequency terms (6) and (8) need to be added to the action of a single particle with charge e and mass m in an electromagnetic field, which we have shown13 to lead to the replacements for mass, magnetic field, and ultimately the cyclotron frequency of the forms m → m(1 − 3U/c2 ) B → B(1 − 3U/c2 )
(9)
ωc → ωc (1 + U/c2 ). For the cyclotron frequency ωc of an antiproton or positron (9) is replaced by ωc → ω c (1 + [3α − 2]U/c2 ),
(10)
leading to a difference of the proton and antiproton clock frequency of (ω c − ωc )/ωc = 3(α − 1)U/c2
(11)
at the same height in the gravitational potential. The dependency of the frequency difference on the gravitational potential leads to the observation that one no longer can change the value of the potential by adding a constant, as is implied by the Newtonian field equations, and the choice of “infinity” i.e. “gravity free space” becomes a critical issue. Good9 suggested to set U = 0 at infinity, to make the frequency difference disappear in the absence of gravity. A somewhat different way to approach this problem is in giving a finite mass and thus a final range to the field. Upper limits for the mass of the graviton have been given by Goldhaber and Nieto15 and more recently by S. S. Gershtein 1460260-4
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et al.16 which inspires us to chose the potential U in equation 11 to be the potential of the local supercluster: |U/c2 | ≈ 3 × 10−5 .
(12)
The most precise comparison on antiproton/proton cyclotron frequencies available in the literature is from the measurement by G. Gabrielse et al.12 , constraining the ratio of antiproton and proton cyclotron frequencies at the Earth’s surface to
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ωc = 0.99999999991(9), and therefore ωc |α − 1| < 1 × 10−6 .
(13) (14)
This leads us to the conclusion that the gravitational acceleration of the antiproton could differ from that of the proton at the most by 1 × 10−6 (under the assumption that the gravitational force is mediated by a single tensor field). Other scenarios are discussed in Hughes et al.13 , including the possibility to mediate gravity by the usual tensor gravitational field coupling to the antimatter article with its normal strength, / but then having in addition a new hypothetical tensor field hµν of finite range coupling to antimatter only. U has then the form of a Yukawa potential and its value depends on the range of this new interaction. This would weaken the constraints given above, i.e. using the potential of the local galaxy would lower the limit of |α-1|to 10−4 , but limits would still be in the range below a percent. The reader is referred to the article by M. Jankoviak, which is appearing in the same volume. But without a definite knowledge on possible forms of tensor or scalar couplings or new fields coupling only to one but not to the other particle, any direct measurement in the percent range, as proposed by the currently proposed experiments, will be an extremely valuable contribution to this complex field. 4. Comments on Measurements of Gravitational Acceleration of Charged Particles The proposed experiment PS200 was based on the description of a measurement of the net vertical component of the force experienced by freely falling electrons in a vacuum enclosed by a copper tube for shielding of any stray electric field by F.C. Witteborn and W.M. Fairbank in the late 1960’s.18,19 The observed result of geff = 0.09 g was interpreted as a complete cancellation of the force of gravity on the electron by an electrical field due to the compression of the free electron gas in the wall of the shielding tube. This result was heavily contested and many arguments were given why other external forces that were predicted significantly stronger than either the force of gravity or the force caused by the electrical field due to electron gas compression were not detected by the set-up. Specifically the “patch-effect” and the gravitational compression of the lattice in the shield wall were heavily debated. Ultimately the proof of the experiment should have been given by repeating the same measurement using positrons, which then should have experienced a net force 1460260-5
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geff = 2.0 g. But due to a lack of sources of low energy positrons at the time of these measurements such an experiment could not be performed. To defend the measurements and to discuss all critical arguments put forward by other groups, 10 years after their original announcement of their result from the electron measurement, Witteborn and Fairbank published a more detailed paper20 on the experimental method, offering a deeper insight into the methodology of the experiment and the many tests performed. The following paragraphs summarize the key findings and serve as an explanation that the PS200 collaboration had (and still has today) all reasons to be convinced that a comparative measurement between protons and antiprotons should have been entirely feasible. After all, protons and antiprotons have the same charge as electrons or positrons, but 1836 times the inertial mass, in principle making the experiment roughly 2000 times easier. In short the experiment consisted of launching short bursts of electrons from a cold cathode located at the bottom of a long vertical copper tube in a vacuum chamber with walls cooled to 4 K. A vertical magnetic field constrained the electrons to move along the axis of the vertical tube to be detected by an electron multiplier at the upper end of the tube. Each electron resulted in a short electrical pulse from the electron multiplier that was stored in a multichannel analyzer. Due to the thermal distribution emitted from the cathode this resulted in a time-of-flight (TOF) distribution with a cut-off signaling the energy below which the particles would turn around due to gravity or other forces before reaching the detector. Known forces could be applied to the electrons inside the tube by running currents axially through the walls of the tube in order to influence the TOF distributions. Studying the TOF distributions for several such known forces allowed then determining the intrinsic forces present in the tube. The results of the experiment has led to the suggestion that the 4.2 K enclosure surrounding the free falling electrons was free of electrical stray fields larger than about 10−11 V/m except for a uniform field that cancelled the force of gravity on electrons to within 9%. This is in total contradiction to the expectations that the variations of the work function at the surface of the drift tube, due to random crystal orientation (called patch effect), would cause potential variations at a distance of 1 cm from the surface of about 10−3 eV. A possible explanation given was that the low temperature of the drift tube would lead to the absorption of hydrogen and helium atoms from the rest gas during the initial cool down. The high mobility of these gases at 4.2 K would allow them to move to a site of minimum energy, resulting in a surface potential with spatial variations on the atomic scale. Such a potential variation would be much more likely to produce a uniform field a few centimeter away from the surface. Following this suspicion that the low temperature might be responsible in some way for the absence of the large fields, Lockhart, Witteborn, and Fairbank20,21 repeated the experiment at temperatures ranging from 4.2 to 300 K, finding a sharp transition in the force on electrons at 4.5 K. At 4.2 K the results were consistent with the results of the original experiments while at temperatures above 4.5 K the 1460260-6
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results of the time-of-flight measurements on electrons were in agreement with the expectations from the predicted electric fields (patch effect and lattice compression). A detailed discussion of the different effects and possible explanations for not observing these in the Witteborn-Fairbank experiment can be found in the review paper by Darling et al.22 As far as the author is aware off, this controversy has never been resolved, and as mentioned before, the ultimate experimental proof, the repetition of the experiment with positrons has never been attempted, despite the fact that low energy positron sources now are available. Armed with the detailed knowledge from the Witteborn-Fairbank analysis and having been able to recruit both F.C. Witteborn and T. Darling to the PS2005 collaboration, we were well equipped for the task at hand. In 1986 we proposed measuring the acceleration of antiprotons in the Earth’s gravitational field by launching antiprotons from a thermal distribution at 4 K in groups of approximately 100 particles upwards against the force of gravity and measuring their time-of-flight (TOF) for a 1 m flight path. Due to the distribution of initial energies, a time of flight distribution would be observed. This TOF distribution will exhibit a cut-off representing the minimum kinetic energy necessary to reach the detector at the top of the experiment. This cut-off time is independent of the particle’s inertial mass and of the detailed shape of the thermal distribution the particles are launched from, and is a direct measure of g for the particles under study. We planned to compare this cut-off time, and thereby g, of negative hydrogen ions and antiprotons to a level of accuracy of better than 0.1%. Two critical components were developed in parallel. At CERN experiment PS200 developed an elongated Penning trap of about 30 cm length, and based on the energy degradation of a 5 MeV antiproton pulse by a thin foil at the entrance of the trap demonstrated trapping and electron cooling of 1 million antiprotons from a single shot from the Low Energy Antiproton Ring (LEAR).23 Ejection of the bunch from the trap was demonstrated24 and preliminary studies of stray fields above conducting surfaces were in progress.25 In parallel a superconducting magnet with field variations of less than 1 × 10−6 in a cylindrical volume of 1 m length and 2 cm diameter was developed and constructed and initially used at Los Alamos for initial tests of field homogeneity and launch procedures for ions from an ECR ion source before it was transferred to CERN. The ion source did get replaced by a small Penning trap and the system was continuing its preparation for the experiment. Before the two systems could be integrated into a full experiment the LEAR facility was shut down, and the source of antiprotons seized to exist. Fortunately a new antiproton source, the antiproton decelerator (AD), could be developed based on the original antiproton target and the antiproton accumulator ring. This was up and running a few years later, but the main interest of the community by then had shifted to the neutral antihydrogen atom for precision spectroscopy and first discussions of measuring gravity on neutral antihydrogen started to appear. Meanwhile a number of high profile experiments at the AD are now ready to move forward with gravity studies on the neutral particle. Considering the complexity of these 1460260-7
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experiments only time will tell if using antihydrogen will make a measurement of gravity on antimatter easier, or if a set of problems were simply exchanged for a different combination of experimental issues. Recently, Hohensee et al.26 set stringent limits on EEP and Lorentz invariant violation within the phenomenological framework of the standard model extension (SME) using data from precise dysprosium (Dy) spectroscopy. No matter what theoretical restrictions can be placed on the difference in gravitational acceleration of matter and antimatter, the lack of a complete theoretical description of gravity in the context of quantum physics still calls for an experimental test of the difference of gravity on antimatter and matter, but we can expect that this question will remain unanswered until experiments can reach a precision well below the 1% level. 5. Conclusions Ultimately it was the time that was running out too fast and the experimental development was stopped when the LEAR accelerator at CERN was shut down in 1996. At that time PS200 had achieved significant progress on trapping and cooling of antiprotons and in parallel had started a test experiment searching for the effects of gravity on heavy ions with the goal of reducing the mass of the particles stepwise until reaching the level of the proton or H− ion. Acknowledgments The author is indebted to many members of the PS200 collaboration during the years of planning, designing, and constructing parts of the experimental apparatus. He also expresses his gratitude to the organizers of the WAG 2013 Conference in BERN to invite him to participate, allowing him to learn that this is still an important and active field of physics and acknowledges receipt of travel support, which allowed him to step back in time and participate in an inspiring meeting on an important topic in physics he has left behind nearly 20 years ago. References 1. A. Kellerbauer et al., Nucl. Instr. Methods B 266, 351 (2008). 2. The Alpha Collaboration and A. E. Charman, Nat. Commun. DOI: 10.1038/ncomms2787 (2013). 3. P. Perez and Y. Sacquin, Class. Quantum Grav. 29, 184008 (2012). 4. A. Cronin et al., Antimatter Gravity Experiment (AGE) at Fermilab, Letter of Intent (2009) 5. Beverini et al., CERN/PSCC/86-2 (1986). 6. E. Adelberger, G., S. W. Stubbs, B. R. Heckel, Y. Su, H. E. Swanson, G. Smith, J. H. Gundlach and W. F. Rogers, Phys. Rev. D 42, 3267 (1990). 7. P. Morrison, Richtmyer memorial address of the American Association of Physics Teachers (1958). 8. L. I. Schiff, Phys. Rev. Lett. 1, 254 (1958). 1460260-8
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9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
M. L. Good, Phys. Rev. 121, 311 (1961). M. M. Nieto and T. Goldman, Phys. Rep. 205, 221 (1991). G. Gabrielse et al., Phys. Rev. Lett. 63, 1360 (1989). G. Gabrielse et al., Phys. Rev. Lett. 82, 3198 (1999). R. J. Hughes and M. H. Holzscheiter, Phys. Rev. Lett. 66, 854 (1991). A. Einstein, Ann. Phys. 35, 898 (1911). A. S. Goldhaber and M. M. Nieto, Phys. Rev. D 9, 1119 (1974). S. S. Gershtein, A. A. Logunov and M. A. Mestvirishvili, arXiv:hep-th/9711147v1 (1997). F. C. Witteborn and W. M. Fairbank, Phys. Rev. Lett. 19, 3198 (1967). F. C. Witteborn and W. M. Fairbank, Nature 220, 436 (1981). F. C. Witteborn and W. M. Fairbank, Rev. Sci. Instrum. 48, 1 (1977). J. M. Lockhart, F. C. Witteborn and W. M. Fairbank, Phys. Rev. Lett. 67 (1991). J. M. Lockhart, “Experimental evidence for a temperature-dependent surface shielding effect inside a copper tube” PhD thesis Stanford University (1976). T. W. Darling, F. Rossi, G. I. Opat and G. F. Moorhead, Rev. Mod. Phys. 54, 237 (1992). M. H. Holzscheiter et al., Phys. Lett. A 214, 279 (1996). X. Feng et al., Hyperfine Int. 100, 103 (1996). G. Testera, Hyperfine Int. 109, 333 (1997). M.A. Hohensee et al., Phys. Rev. Lett. 111, 050401 (2013).
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