In the past decade several new results have been published that impact the testing of QED in positronium. These are comparison between theory and experiment for the decay rates of ortho- (1³S₁) and para-positronium (11S0) and for the energy splittings between the hyperfine states (1³S₁-1¹S₀), the n = 2 fine structure (2³S₁-2³P_J, J = 0, 1, 2; 2³S₁-2¹P₁), and the two-photon transition between the n = 1 and n = 2 levels (1³S₁-2³S₁). In the early part of the decade all the experimental values for these quantities were known to higher precision than the corresponding theoretical values. In the last few years this trend has been almost completely reversed by theoretical advances. In the early part of the decade the only clear discrepancy between theory and experiment was in the decay rate of ortho-positronium. That discrepancy has persisted, with additional controversy introduced by further experimental and theoretical results. In light of recent theoretical calculations, the hyperfine splitting too, shows hints of a discrepancy. 

The thermalization of positronium in gases has systematic effects in some of the measurements of both the ortho-positronium decay rate and the hyperfine splitting. These systematics will be discussed in detail in a separate poster. Future prospects for improvements in each of these measurements will be discussed.

Work of the Michigan positron group is supported by NSF grant PHY 97-31861 and the University of Michigan.

The muonium atom consits of two leptons from two different generations, a positve muon and an electron^1,2. The absence of any known internal structure for these particles allows to calculate level energies to very high accuracy within the framework of bound state Quantum Electrodynamics (QED). In th case of the hyperfine structure the agreement between theory and experiment is substantially better than for natural hydrogen where the yet not well known   structure of the proton and dynamics of its charge carrying constituents prevent more stringent conclusions. 

In two recent experiments the hyperfine structure interval and the Zeeman effect in the ground state³ and the 1s-2s energy splitting⁴ were measured with microwave respectively Doppler-free two-photon laser spectroscopy. The results were in good agreement with theoretical predictions which include the dominant QED part and contributions from strong and weak interactions.  Most accurate values for the muon magnetic moment, the muon mass, the muon electron charge ratio and a precise number for fine structure constant were extracted from the measurements^2,3,4.

Muonium has proven to be an ideal system for testing QED and  fundamental symmetries in physics⁵ as well as for providing accurate values of fundamental constants in the past. It could continue to do so in the future. Improvements in accuracy can be expected with higher numbers of muonium atoms as they are expected to be available from future high flux muon sources such as the PRISM facility of the Japanese Hadron Facility, the  Oak Ridge spallation neutron source or the front end of a muon collider^6,7. An additional boost in accuracy can then be gained from new techniques like using cw lasers in the case of the 1s-2s experiment⁵.

References:

1. V.W. Hughes and G. zu Putlitz, in: Quantum Electrodynamics, ed. T. Kinoshita, World Scientific, p. 822 (1990).
2. K. Jungmann, in: Muon Science, eds. S.L. Lee, S.H. Kilcoyne and R. Cywinsky, Inst. of Physics Publ., p. 405 (1999).
3. W. Liu et al., Phys. Rev. Lett. 82, 711 (1999).
4. V. Meyer et al., Phys. Rev. Lett. 84, 1136 (2000).
5. L. Willmann et al., Phys. Rev. Lett. 82, 49 (1999).
6. K. Jungmann, in: Proceedings of the HISMUS99 Workshop, ed. Y. Kuno, World Scientific, in print (2000).
7. M.G. Boshier et al., Comm. At. Mol. Phys. 33, 17 (1996).

Order  Corrections to the Decay Rate of Orthopositronium

G.S .Adkins¹, R. Fell², and J. Sapirstein³

¹ Department of Physics and Astroomy, Franklin & Marshall Colege, P.O. Box 3003, Lancaster, PA, 17604, USA, E­mail: g_adkins@acad.fandm.edu
² Brandeis University, Waltham, MA, 01742 USA
³ Department of Physics, University of Nore Dame, Nore Dame, IN, 46556 USA

The discrepancy between theory and experiment for the decay rate of orthopositronium has long been one of the outstanding problems in precision QED. The calculated decay rate is [1-3] 

where the lowest order contributionis  [4]. The one-loop correction is known to be A = -10.286606(10) [5]. The result of the present calculationis a value for the two-loop correction B. 

Our calculation was done in the context of Nonrelativistic Quantum Electrodynamics (also knownas NRQED) [6] following the approach outlined by Labelle, Lepage, and Magnea [7].This method allows the high­energy part of the calculation to be treatedas an on­shell scattering process. The high­energy calculationis part of a "matching" procedure in which a set of nonrelativistic interaction operators is defined. These operators are used to work out the bound­state aspects of the problem. Our calculation of the high­energy process followed by a bound­state calculation using the effective interaction operators allowed us to complete the determination of B [8]. 

Acknowledgments. The work of GA was partially supported by NSF grants PHY­9711991 and PHY­9722074, and that of JS by PHY­9870017. Useful conversations with P. Labelle, G.P. Lepage, and R. Hill are acknowledged. 

References:

1. W.E. Caswell, G.P. Lepage, and J. Sapirstein, Phys. Rev. Lett. 38, 488 (1977). 
2. W.E. Caswell and G.P. Lepage, Phys. Rev. A 20, 36 (1979). 
3. S.G. Karshenboim, Zh. Eksp. Teor. Fiz. 103, 1105 (1993) [JETP 76, 541 (1993)]. 
4. A. Ore and J.L. Powell, Phys. Rev. 75, 1696 (1949). 
5. G.S. Adkins, Phys. Rev. Lett. 76, 4903 (1996). 
6. W.E. Caswell and G.P. Lepage, Phys. Lett. 167B , 437(1986). 
7. P. Labelle, G.P. Lepage, and U. Magnea, Phys. Rev. Lett. 72, 2006 (1994). 
8. G.S. Adkins, R. Fell, and J. Sapirstein, hep­ph/0003028. 

Following a suggestion of A. Kostelecký et al. [1], we are evaluating a test of CPT and Lorentz invariance from the microwave spectroscopy of muonium [2]. Precise measurements have been reported for the transition frequencies  and  for ground state muonium in a magnetic field H of 1.7 T. These frequencies depend on both the hyperfine interaction and Zeeman effect. Hamiltonian terms beyond the standard model which violate CPT and Lorentz invariance would contribute  and . The nonstandard theory indicates that  and  should oscillate with the earth's sidereal frequency and indeed  and  would be anticorrelated. 

We are analyzing our muonium data and expect to report results at the Hydrogen II Conference. 

References:

1. D. Calladay and V.A. Kostelecký, Phys. Rev. D 55, 6760 (1997) ; R. Bluhm, V.A. Kostelecký and C.D. Lane, CPT and Lorentz Tests with Muons, submitted to Phys. Rev. Lett. 
2. W. Liu et al., Phys. Rev. Lett 82, 711 (1999). 

Positronium, an elementary atom which consists of electron and positron, provides a unique laboratory to test the theory of weakly bound states in QED.  Because of the small value of the electron mass, the uncertainties due to strong interactions are not important at the current level of theoretical and experimental precision. This provides a unique opportunity to confront high precision experiments that study various properties of positronium with theoretical predictions. 

Recently, we have applied dimensionally regularized non-relativistic QED to positronium spectroscopy and the decay width of parapositronium. An analytic expressions for  corrections to the ground state positronium hyperfine splitting [1] and  singlet energy level shifts [2] have been obtained. We have also derived  corrections to positronium energy levels [3]. For the most precisely measured quantity, the positronium ground state hyperfine splitting, we obtain [2,3] 203392(1) MHz, which differs from experimental results by about three standard deviations.

Another interesting quantity is the parapositronium decay width into two photons. Since a rather precise measurement of the parapositronium decay rate is available, the theoretical result becomes of considerable interest. We obtain [4,5]   which agrees very well with experimental result. 

References:

1. A. Czarnecki, K. Melnikov and A. Yelkhovsky, Phys. Rev. Lett. 82, 311 (1999).
2. A. Czarnecki, K. Melnikov and A. Yelkhovsky,  Phys. Rev. A 59, 4316 (1999).
3. K. Melnikov and A. Yelkhovsky, Phys. Lett. B 458, 143 (1999).
4. A. Czarnecki, K. Melnikov and A. Yelkhovsky, Phys. Rev. Lett. 83, 1135 (1999).
5. A. Czarnecki, K. Melnikov and A. Yelkhovsky, Phys. Rev. A 61, 052502 (2000).

Positronium (Ps) is the bound state of an electron and its antiparticle the positron. Both components are structureless and pointlike leptons, thus avoiding the difficulties encountered with the proton structure in hydrogen. The advantage, compared with muonium, is the absence of an additional free parameter like the muon mass. Moreover Ps is an eigenstate of the charge conjugation operator, which opens new channels of real and virtual annihilation. For these reasons Ps is an ideal test object for bound state QED. Ps is completely described by only two parameters: the Rydberg constant Ry and the fine structure constant . QCD effects and the weak interaction play no role at the present state of accuracy.

The energy levels of Ps have been completely calculated up to the order . The theoretical uncertainty results only from uncalculated higher order terms and is estimated to be 1 MHz for the ground state n = 1. The experimental accuracy is comparable and there is moderate agreement with theory within 3 standard deviations. In the excited state n = 2 the intervals between the P-levels can be calculated to an accuracy of 10 kHz. Here the experiments need the full width of 3 standard deviations (3 × 1 MHz ) to come into agreement with theory.

The annihilation rate of triplet Ps in the ground state is completely calculated up to the order . The contribution from higher orders is only estimated and introduces a relative theoretical uncertainty of 2×10-4. The experimental situation is controversial: a measurement at the University of Tokyo is in agreement with theory, whereas the results from the University of Ann Arbor disagree with theory by more than 4 standard deviations.

The annihilation rate of singlet Ps in the in the ground state has been completely calculated up to the order . The estimated higher order contributions introduce a relative theoretical uncertainty of 1×10-4. The experiment has comparable accuracy and is in good agreement with theory.

The project of Low Energy Particle Toroidal Accumulator (LEPTA) [1], which is under construction in the JINR now, is dedicated to the creation of small positron storage ring with electron cooling of positrons circulating in the ring. The potential of this device is the generation of intense streams of electron-positron bound states, known as positronium, and - together with low energy antiprotons - for the synthesis of antihydrogen atoms in copious numbers. 

The focusing system with longitudinal magnetic field and electron cooling of positrons are essential features of the LEPTA. The single turn injection of positrons is performed by special kicker coil. At the first stage of the LEPTA operation we plan to use a positron source on the base of radioactive isotope ²²Na. Nearest prototype of the injection system is the positron trap of the ATHENA project. Special septum coils and centrifugal drift of the electrons are used for superposition and separation of the cooling electron beam and the circulating positron one. The positronium is generated in collisions of positrons with free electrons of the cooling electron beam that have velocities very close to the positron ones. This permits obtaining a high positronium flux with small angular and velocity spreads of the atoms and provides a significant advantage for proposed arrangements of experiments, so-called positronium-in-flight set-ups [2], as compared with traditional approaches in which positronium is generated in targets. In particular, the precision in measuring positronium parameters can be enhanced by several orders of magnitude. Moreover, some experiments, that are unrealistic within traditional schemes, becomes feasible with the proposed facility. The ring circumference is about 18 m, magnetic field value is 400 G, positron energy is of the order of 10 keV. Expected angular and relative energy spreads of positronium flux are 2×10^-3 and (1 - 5)×10^-4 correspondingly. At 10⁹ positrons circulating in the ring the flux value is about 10⁴ atoms per second.

Presently the design of the storage ring and the elaboration of the technology of the ring elements manufacturing are completed. The vacuum chamber of the ring was constructed and tested. Solenoid of electron cooling system was constructed, tested and adjusted. Other general elements of the magnetic system are under construction. In very beginning of the LEPTA ring operation the following problems have to be experimentally investigated: dynamics of circulating beam; measurements of the friction force components due to electron cooling of positrons, investigation of the equilibrium state after finishing of the cooling process; measurements of the e⁺e^- recombination rate. Their solution will give a base for detail elaboration of the first physical experiments with positronium in-flight, namely o-Ps life-time measurements and precise comparison of positron and electron electric charges. 

This work is supported by Grant RFBR 99-02-17716.

References:

1. Yu.V. Korotaev, I.N. Meshkov, S.V. Mironov, A.O. Sidorin, and E. Syresin, 6^th European Particle Accelerator Conference, Stockholm, 1998, p. 853.
2. I.N. Meshkov, Fiz. El.Ch.A.Yad. 28 , 495 (1997); Phys. Part. Nucl. 28, 198 (1997).

Positronium (Ps) is an excellent system to study QED, with gases often being the formation medium used. At an initial energy of several eV when formed in gases, Ps then collides with the gas atoms and approaches thermal equilibrium. It has been recently found¹ that the rate of thermalization is significantly slower than previously believed. Corrections used to remove gas related collisional effects in high-precision experiments must include the non-thermal nature of the Ps population. Examples of affected experiments include orthopositronium (o-Ps) vacuum decay rate measurements² () and ground state singlet-triplet splitting³ (). The latter contains gas pressure (Stark) shifts and the former gas collisional quenching effects ().

To experimentally investigate the effect of thermalization on , a measurement of the temperature dependence⁴ of  in the gases used in Ref. 2 (isobutane, neopentane, Ne, and N₂) was made. It was found that  increases linearly with temperature rather than remaining constant as was previously assumed. In light of the previous two experiments, a systematic reanalysis of   was then performed. The 1989 data have been refitted to an elastic thermalization model in which the effective thermalization rate of positronium near room temperature is a freely fitted parameter. The corrections remove the observed overdispersive nature of the data and results in a correction downward of about . This brings the data in good agreement with a measurement of  in vacuum⁵. Both measurements remain in disagreement with the QED theoretical value⁶ and another measurement using low-density SiO₂ powders⁷.

Measurements^3,8 of , which are each in  disagreement with recent theory, may also suffer from systematic effects due to thermalization. Data in Ref. 3 are acquired at sufficiently low gas densities where it cannot be assumed that the o-Ps was thermalized. However, there are no data on the temperature dependence of the   pressure shift and thus the exact impact of thermalization on  cannot be determined at this time. Implications of this effect will be considered.

This research is supported by NSF Grant PHY-9731861 and the University of Michigan.

References:

1. M. Skalsey, R.K. Bithell, J.J. Engbrecht, R.S. Vallery, and D.W. Gidley, Phys. Rev. Lett. 80, 3727 (1998). 
2. C.I. Westbrook, D.W. Gidley, R.S. Conti, and A. Rich, Phys. Rev. A 40, 5489 (1989).
3. M.W. Ritter, P.O. Egan, V.W. Hughes, and K.A. Woodle, Phys. Rev. A. 30, 1331 (1984).
4. R.S. Vallery, A.E. Leanhardt, M. Skalsey, and D.W. Gidley, accepted for publication in J. Phys. B.
5. J.S. Nico, D.W. Gidley, A. Rich, and Zitzewitz, Phys. Rev. Lett. 65, 1344 (1990).
6. G.S. Adkins, R.N. Fell, and J. Sapirstein, submitted to Phys. Rev. Lett..
7. S. Asai , S. Orito, and Hinohara, Phys. Lett. B 357, 475 (1995).
8. A.P.Mills, Jr., and G.H. Bearman, Phys. Rev. Lett. 34, 246 (1975). 

We present highly accurate ab initio theoretical study to simulate ionization probabilities and line profiles for a two-step 3-photon resonant ionization process, , of the ground state of muonium (or any hydrogen-like atom). The 1S-2S transition offers unique opportunities for ultra-high precision spectroscopy due to the narrow natural line width  of 2S­state. In hydrogen atom, for example, the smallness of   KHz has already allowed the quality factor  Hz be achieved in the measurement of the 1S-2S energy separation [1]. Experimentally, the 1S-2S transition can be induced Doppler-free by absorbing two photons from two identical counter-propagating laser beams. These can be generated by either asufficiently powerful pulsed laser (as with muonium [2]) or a continuous laser (as with hydrogen [1]). In the new 1S-2S in muonim recently finalized at the Rutherford Appleton Laboratory, the use of intense pulsed laser source to induce above 3-photon process has enabled the 1S-2S energy interval to be determined to the 9.8 MHz accuracy [2]. 

In the present work we report new results of our simulations intended to account for most important contributions to the energy intervals between atomic level involved in the above resonant 3-photon process, which arise due to a number of systematic spurious effects that make their appearance whenever an atom is subject to laser radiation of arbitrary intensity. The relevance and strong motivation of this study are discussed in more detail in [2,3]. Our present model constitutes a significant improvement over our obsolete scheme [3] which was lately used in analyzing experimental 1S to the model on equal footing. Without imposing any restriction on the strength of the laser field, the model developed is currently capable of accounting for its arbitrary spatial and temporary inhomogeneities, non-zero ionization rates of intermediate atomic relay levels, along with their Stark shifts and appropriate exact fine structure contributions. In addition, it allows for the second order Doppler shifts as well, i.e. takes into account the movement of atoms in a media to order (v/c)². This enables the model to be efficiently employed for a highly accurate analysis of either 2- or 3-photon resonant phenomena with few-particle systems, which are induced by either pulsed or continuous sources of laser radiation. Although technically difficult to construct, high-power continuous CW lasers offer nowadays a number of inviting and unique opportunities for ultra-high precision spectroscopy with simplest atoms, especially for those containing -muons. This makes the current work of relevance and use for both current and future highly precise studies with various fundamental bounded quantum systems. 

References:

1. A. Huber, Udem Th., B. Gross, J. Reichert, M. Kourogi et al., Phys. Rev. Lett. 80, 468 (1998). 
2. V. Meyer, S.N. Bagaev, P.E.G. Baird, P. Bakule, M.G. Boshier et al. Phys. Rev. Lett. 84, 1136 (2000). 
3. V. Yakhontov, R. Santra and K. Jungmann, J. Phys. B: Atom. Mol. Opt. Phys. 32, 1615 (1999).