Leptonic systems

Positronium, being a readily available purely leptonic system and also a particle-antiparticle pair, has attracted considerable experimental interest over the years as a testing ground for the existence of exotic particles or couplings. The latter may perhaps manifest themselves in the decay properties of positronium, so that attempts have been made to observe forbidden modes. In particular, the longstanding discrepancy between the Michigan experimental value for oAo-ps and the results from QED calculations, described in subsection 7.1.1, has acted as a spur to such investigations. 

Positronium (e+e ) is a purely leptonic system, free of nuclear structure effects, but suffers from reduced corrections in the worst possible case of equal masses. This makes the system difficult to treat, since quantum electrodynamical calculations start from an infinite nuclear mass and treat reduced mass effects as a perturbation. 

QED can be considered to be one of the most precisely tested theories in physics at present. It provides an extremely accurate description of systems such as hydrogen and helium atoms, as well as for bound-leptonic systems, for example, positronium and muonium. Remarkable agreement between theory and experiment has been achieved with respect to the determination of the hyperfine structure and the Lamb shift. The same holds true for the electronic and muonic g-factors. The free-electron g-factor is determined at present as... 

Although measurements of the ground state hfs splitting and the n=2 Lamb shift have been made, the analogous two-photon laser experiment (ls-2s) to that in hydrogen has only recently been comtemplated because of developments in the production of slow muonium atomsl. It should be noted that both positronium and muonium are pure leptonic systems and therefore do not suffer from any uncertainty in nuclear size in the case of hydrogen the present error in the proton size determination is 4 (see equation (5)). 

It is inevitable that work of this kind faces limits as far as determination of numerical values for the constants and also as precision tests of fundamental theory. These limits may be seen to arise in two ways. First of all, the hydrogenic energy levels and their splittings are sensitive to some extent to the inner structure of proton. Secondly, since interesting electrodynamic effects scale with some power of Za, measurements of great precision are required. In the case of the former problem, one is attracted to the notion of the spectroscopy of purely leptonic systems, while the latter invites consideration of the single electron spectra of high Z ions. 

Second Quantized Description of a System of Noninteracting Spin Particles.—All the spin particles discovered thus far in nature have the property that particles and antiparticles are distinct from one another. In fact there operates in nature conservation laws (besides charge conservation) which prevent such a particle from turning into its antiparticle. These laws operate independently for light particles (leptons) and heavy particles (baryons). For the light fermions, i.e., the leptons neutrinos, muons, and electrons, the conservation law is that of leptons, requiring that the number of leptons minus the number of antileptons is conserved in any process. For the baryons (nucleons, A, E, and S hyperons) the conservation law is the... 

One of our main interests is to describe quark matter at the interior of a compact star since this is one of the possibilities to find color superconducting matter in nature. It is therefore important to consider electrically and color neutral2 matter in /3-equilibrium. In addition to the quarks we also allow for the presence of leptons, especially electrons muons. As we consider stars older than a few minutes, when neutrinos can freely leave the system, lepton number is not conserved. The conditions for charge neutrality read... 

The Total Bary on Number Remains Constant. A baryon is a nucleon (proton or neutron) or any panicle heavier than those that can he considered to have an atomic mass number A — I. Some mesons have a mass greater than the proton, but they have a mass number. 4 - 0. so they are not barvuns. In computing the number or barvons present in a system, each baryon counts I each ami baryon counts — I and leptons and mesons count 0. 

For slow collisions the transitions occur from the continuum state of the J — 0 (spherical) symmetry to the bound state of the J = 1 symmetry, and are induced by the electric dipole moment of the H — H system. This dipole moment is the expectation value of the dipole moment operator for all four particles with respect to the ground state leptonic wave function, D = where r, denote the positions of... 

It is interesting to note that a similar radiative association process is not possible for the two hydrogen atoms. On the symmetry grounds the dipole moment of the H — H system (which is inversion symmetric) vanishes. In that case the nuclear dipole moment is identically 0 and the electronic dipole moments induced in the two approaching atoms have opposite orientations and cancel each other. For the H — H system (which lacks the inversion symmetry) the dipole moment (in the adiabatic and non-relativistic approximation) is finite. In that case the hadronic moment is e R and the induced leptonic moments of H and H have the same orientations and add together to a non vanishing dipole moment (which tends to 0 in the limit of infinite separation R between the atoms). 

Abstract. Muonium is a hydrogen-like system which in many respects may be viewed as an ideal atom. Due to the close confinement of the bound state of the two pointlike leptons it can serve as a test object for Quantum Electrodynamics. The nature of the muon as a heavy copy of the electron can be verified. Furthermore, searches for additional, yet unknown interactions between leptons can be carried out. Recently completed experimental projects cover the ground state hyperfine structure, the ls-2s energy interval, a search for spontaneous conversion of muonium into antimuonium and a test of CPT and Lorentz invariance. Precision experiments allow the extraction of accurate values for the electromagnetic fine structure constant, the muon magnetic moment and the muon mass. Most stringent limits on speculative models beyond the standard theory have been set. 

We will review here experimental tests of quantum electrodynamics (QED) and relativistic bound-state formalism in the positron-electron (e+,e ) system, positronium (Ps). Ps is an attractive atom for such tests because it is purely leptonic (i.e. without the complicating effects of nuclear structure as in normal atoms), and because the e and e+ are antiparticles, and thus the unique effects of annihilation (decay into photons) on the real and imaginary (related to decay) energy levels of Ps can be tested to high precision. In addition, positronium constitutes an equal-mass, two-body system in which recoil effects are very important. 

The purely leptonic hydrogen atom, muonium, consists of a positive muon and an electron. It is the ideal atom, free of the nuclear structure effects of H, D and T and also of the difficult, reduced mass corrections of positronium. An American-Japanese group has observed the 1S-2S transition in muonium to a precision somewhat better than a part in 107. [10] Because there were very few atoms available, the statistical errors precluded an accurate measurement. The "ultimate" value of this system is very great, being limited by the natural width of the 1S-2S line of 72 kHz, set by the 2.2 nsec lifetime of the muon. 

Positrons have the charge of protons and the mass of electrons. Quantum chemists who take an interest in mixed electron-positron systems immediately recognize some interesting consequences of this transparent observation. For example, (a) the familiar Born-Oppenheimer approximation cannot be used for positrons, but rather positrons must be treated as distinguishable electrons (b) electron-positron correlation is more important, pair by pair, than correlation between leptons of like charge and (c) there are always core electrons (except for the simplest systems), but positrons congregate in the valence region or beyond. 

The QMC method is ideally suited for mixed systems because electron-positron correlation, which is difficult to treat with Cl methods, is automatically treated correctly. Systems of up to a bit more than ten leptons are routinely treated. Effective core potential methods can be used to extend the method to larger systems. Expectation values of local operators for the distribution k 2 are calculated by straightforward sampling procedures, but nonlocal operators, such as those for the annihilation rate, are problematic and are under active investigation [12],... 

All the particles in Table 10.1 have spin. Quantum mechanical calculations and experimental observations have shown that each particle has a fixed spin energy which is determined by the spin quantum number s s = h for leptons and nucleons). Particles of non-integral spin are csWeA fermions because they obey the statistical rules devised by Fermi and Dirac, which state that two such particles cannot exist in the same closed system (nucleus or electron shell) having all quantum numbers the same (referred to as the Pauli principle). Fermions can be created and destroyed only in conjunction with an anti-particle of the same class. For example if an electron is emitted in 3-decay it must be accompanied by the creation of an anti-neutrino. Conversely, if a positron — which is an anti-electron — is emitted in the ]3-decay, it is accompanied by the creation of a neutrino. 

As is apparent from equation (9), in order to calculate the cross-section for leptonic annihilation in flight we need the scattering wave function in the initial channel of the colliding system,. The latter may be obtained by means of the formalism developed in our previous work [3,5]. 

We solve the hydrogen-antihydrogen scattering problem in a distorted wave approximation based on the separation of the leptonic and hadronic motions as implied by the Born-Oppenheimer approximation. The total wave function of the system is written as... 

It is instructive and helpful for assessing the correctness of the numerical apparatus to investigate the asymptotic behavior of the leptonic coalescence density P R). In the limit R 00 the ground-state HH system dissociates into separated H(1 s) and H( 1 s) atoms. In this case the electron and positron densities are spatially separated and do not overlap. Necessarily, lim/ oo P R) = 0. 

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