Extrapolation of the Zalpha-Expansion and Two--Loop Bound--State Energy Shifts

Quantum electrodynamic (QED) effects that shift the binding energies of hydrogenic energy levels have been expressed in terms of a semi-analytic expansion in powers of Zalpha and ln[(Zalpha)^{-2}], where Z is the nuclear charge number and alpha is the fine-structure constant. For many QED effects, numerical data are available in the domain of high Z where the Zalpha expansion fails. In this Letter, we demonstrate that it is possible, within certain limits of accuracy, to extrapolate the Zalpha-expansion from the low-Z to the high-Z domain. We also review two-loop self-energy effects and provide an estimate for the problematic nonlogarithmic coefficient B_60.Comment: 10 pages, LaTeX, Phys. Lett. B, in pres

Calculation of the Characteristic Functions of Anharmonic Oscillators

The energy levels of quantum systems are determined by quantization conditions. For one-dimensional anharmonic oscillators, one can transform the Schrodinger equation into a Riccati form, i.e., in terms of the logarithmic derivative of the wave function. A perturbative expansion of the logarithmic derivative of the wave function can easily be obtained. The Bohr-Sommerfeld quantization condition can be expressed in terms of a contour integral around the poles of the logarithmic derivative. Its functional form is B_m(E,g) = n + 1/2, where B is a characteristic function of the anharmonic oscillator of degree m, E is the resonance energy, and g is the coupling constant. A recursive scheme can be devised which facilitates the evaluation of higher-order Wentzel-Kramers-Brioullin (WKB) approximants. The WKB expansion of the logarithmic derivative of the wave function has a cut in the tunneling region. The contour integral about the tunneling region yields the instanton action plus corrections, summarized in a second characteristic function A_m(E,g). The evaluation of A_m(E,g) by the method of asymptotic matching is discussed for the case of the cubic oscillator of degree m=3.Comment: 11 pages, LaTeX; three further typographical errors correcte

A Problematic Set of Two-Loop Self-Energy Corrections

We investigate a specific set of two-loop self-energy corrections involving squared decay rates and point out that their interpretation is highly problematic. The corrections cannot be interpreted as radiative energy shifts in the usual sense. Some of the problematic corrections find a natural interpretation as radiative nonresonant corrections to the natural line shape. They cannot uniquely be associated with one and only one atomic level. While the problematic corrections are rather tiny when expressed in units of frequency (a few Hertz for hydrogenic P levels) and do not affect the reliability of quantum electrodynamics at the current level of experimental accuracy, they may be of importance for future experiments. The problems are connected with the limitations of the so-called asymptotic-state approximation which means that atomic in- and out-states in the S-matrix are assumed to have an infinite lifetime.Comment: 12 pages, 3 figures (New J. Phys., in press, submitted 28th May

Non-uniform convergence of two-photon decay rates for excited atomic states

Two-photon decay rates in simple atoms such as hydrogenlike systems represent rather interesting fundamental problems in atomic physics. The sum of the energies of the two emitted photons has to fulfill an energy conservation condition, the decay takes place via intermediate virtual states, and the total decay rate is obtained after an integration over the energy of one of the emitted photons. Here, we investigate cases with a virtual state having an energy intermediate between the initial and the final state of the decay process, and we show that due to non-uniform convergence, only a careful treatment of the singularities infinitesimally displaced from the photon integration contour leads to consistent and convergent results.Comment: 3 pages; LaTe

Light Sea Fermions in Electron-Proton and Muon-Proton Interactions

The proton radius conundrum [R. Pohl et al., Nature vol.466, p.213 (2010) and A. Antognini et al., Science vol.339, p.417 (2013)] highlights the need to revisit any conceivable sources of electron-muon nonuniversality in lepton-proton interactions within the Standard Model. Superficially, a number of perturbative processes could appear to lead to such a nonunversality. One of these is a coupling of the scattered electron into an electronic as opposed to a muonic vacuum polarization loop in the photon exchange of two valence quarks, which is present only for electron projectiles as opposed to muon projectiles. However, we can show that this effect actually is part of the radiative correction to the proton's polarizability contribution to the Lamb shift, equivalent to a radiative correction to double scattering. We conclude that any conceivable genuine nonuniversality must be connected with a nonperturbative feature of the proton's structure, e.g., with the possible presence of light sea fermions as constituent components of the proton. If we assume an average of roughly 0.7*10^(-7) light sea positrons per valence quark, then we can show that virtual electron-positron annihiliation processes lead to an extra term in the electron-proton versus muon-proton interaction, which has the right sign and magnitude to explain the proton radius discrepancy.Comment: 6 pages; RevTeX; published in Physical Review A in 2013; as compare to the journal version, we have added a note at the end of the paper which pertains to the (new) Ref. [42]; otherwise unchange

Gravitational Correction to Vacuum Polarization

We consider the gravitational correction to (electronic) vacuum polarization in the presence of a gravitational background field. The Dirac propagators for the virtual fermions are modified to include the leading gravitational correction (potential term) which corresponds to a coordinate-dependent fermion mass. The mass term is assumed to be uniform over a length scale commensurate with the virtual electron-positron pair. The on-mass shell renormalization condition ensures that the gravitational correction vanishes on the mass shell of the photon, i.e., the speed of light is unaffected by the quantum field theoretical loop correction, in full agreement with the equivalence principle. Nontrivial corrections are obtained for off-shell, virtual photons. We compare our findings to other works on generalized Lorentz transformations and combined quantum-electrodynamic gravitational corrections to the speed of light which have recently appeared in the literature.Comment: 9 pages; RevTeX; typographical errors corrected and references adde

Techniques in Analytic Lamb Shift Calculations

Quantum electrodynamics has been the first theory to emerge from the ideas of regularization and renormalization, and the coupling of the fermions to the virtual excitations of the electromagnetic field. Today, bound-state quantum electrodynamics provides us with accurate theoretical predictions for the transition energies relevant to simple atomic systems, and steady theoretical progress relies on advances in calculational techniques, as well as numerical algorithms. In this brief review, we discuss one particular aspect connected with the recent progress: the evaluation of relativistic corrections to the one-loop bound-state self-energy in a hydrogenlike ion of low nuclear charge number, for excited non-S states, up to the order of alpha (Zalpha)^6 in units of the electron mass. A few details of calculations formerly reported in the literature are discussed, and results for 6F, 7F, 6G and 7G states are given.Comment: 16 pages, LaTe

Separation of Transitions with Two Quantum Jumps from Cascades

We consider the general scenario of an excited level |i> of a quantum system that can decay via two channels: (i) via a single-quantum jump to an intermediate, resonant level |bar m>, followed by a second single-quantum jump to a final level |f>, and (ii) via a two-quantum transition to a final level |f>. Cascade processes |i> -> |bar m> -> | f> and two-quantum transitions |i> -> |m> -> |f> compete (in the latter case, |m> can be both a nonresonant as well as a resonant level). General expressions are derived within second-order time-dependent perturbation theory, and the cascade contribution is identified. When the one-quantum decay rates of the virtual states are included into the complex resonance energies that enter the propagator denominator, it is found that the second-order decay rate contains the one-quantum decay rate of the initial state as a lower-order term. For atomic transitions, this implies that the differential-in-energy two-photon transition rate with complex resonance energies in the propagator denominators can be used to good accuracy even in the vicinity of resonance poles.Comment: 9 pages; RevTe

Dirac Hamiltonian with Imaginary Mass and Induced Helicity-Dependence by Indefinite Metric

It is of general theoretical interest to investigate the properties of superluminal matter wave equations for spin one-half particles. One can either enforce superluminal propagation by an explicit substitution of the real mass term for an imaginary mass, or one can use a matrix representation of the imaginary unit that multiplies the mass term. The latter leads to the tachyonic Dirac equation, while the equation obtained by the substitution m->i*m in the Dirac equation is naturally referred to as the imaginary-mass Dirac equation. Both the tachyonic as well as the imaginary-mass Dirac Hamiltonians commute with the helicity operator. Both Hamiltonians are pseudo-Hermitian and also possess additional modified pseudo-Hermitian properties, leading to constraints on the resonance eigenvalues. Here, by an explicit calculation, we show that specific sum rules over the spectrum hold for the wave functions corresponding to the well-defined real energy eigenvalues and complex resonance and anti-resonance energies. In the quantized imaginary-mass Dirac field, one-particle states of right-handed helicity acquire a negative norm ("indefinite metric") and can be excluded from the physical spectrum by a Gupta--Bleuler type condition.Comment: 8 pages; RevTeX; published in J.Mod.Phy

Muonic bound systems, virtual particles and proton radius

The proton radius puzzle questions the self-consistency of theory and experiment in light muonic and electronic bound systems. Here, we summarize the current status of virtual particle models as well as Lorentz-violating models that have been proposed in order to explain the discrepancy. Highly charged one-electron ions and muonic bound systems have been used as probes of the strongest electromagnetic fields achievable in the laboratory. The average electric field seen by a muon orbiting a proton is comparable to hydrogenlike Uranium and, notably, larger than the electric field in the most advanced strong-laser facilities. Effective interactions due to virtual annihilation inside the proton (lepton pairs) and process-dependent corrections (nonresonant effects) are discussed as possible explanations of the proton size puzzle. The need for more experimental data on related transitions is emphasized.Comment: 11 pages; RevTe