Bound-state QED Research Articles

A status report on experimental tests of QED in medium- Z systems

The electron beam ion trap has opened the door for experimental physics to perform critical tests of bound state QED in the medium- Z regime. The relatively small uncertainties associated with the theoretical predictions necessitate highly accurate experimental measurements of atomic transition energies. This work presents an overview of the issues and key concepts involved in the field. It also demonstrates several major steps taken to suppress systematic uncertainties that have limited past work and presents an error budget for current and future work.

Towards electronic g-factor measurements in medium-heavy hydrogen-like and lithium-like ions

Measurements of the anomalous magnetic moment of the electron bound in hydrogen-like ions with spinless nuclei have proven to be highly sensitive tests of corresponding calculations based on bound-state quantum electrodynamics. Measurements performed on H-like carbon 12C5+ and oxygen 16O7+ together with bound-state QED calculations on the same level of accuracy have achieved sensitivities around 0.25% of the QED bound state contributions to the calculated electronic g-factors of these ions. Currently, a similar experiment on hydrogen-like calcium 40Ca19+, lithium-like calcium 40Ca17+ and other medium-heavy ions is being prepared, which is capable of increasing this sensitivity on the experimental side by more than one order of magnitude. To that end, several novel experimental techniques have been developed. In the following, we will give a motivation for such a measurement, present the experimental requirements and technique and discuss the possible benefits.

PRECISION STUDY OF POSITRONIUM: TESTING BOUND STATE QED THEORY

As an unstable light pure leptonic system, positronium is a very specific probe atom to test bound state QED. In contrast to ordinary QED for free leptons, the bound state QED theory is not so well understood and bound state approaches deserve highly accurate tests. We present a brief overview of precision studies of positronium paying special attention to uncertainties of theory as well as comparison of theory and experiment. We also consider in detail advantages and disadvantages of positronium tests compared to other QED experiments.

Proton Zemach radius from measurements of the hyperfine splitting of hydrogen and muonic hydrogen

While measurements of the hyperfine structure of hydrogen-like atoms are traditionally regarded as test of bound-state QED, we assume that theoretical QED predictions are accurate and discuss the information about the electromagnetic structure of protons that could be extracted from the experimental values of the ground state hyperfine splitting in hydrogen and muonic hydrogen. Using recent theoretical results on the proton polarizability effects and the experimental hydrogen hyperfine splitting we obtain for the Zemach radius of the proton the value 1.040(16) fm. We compare it to the various theoretical estimates the uncertainty of which is shown to be larger that 0.016 fm. This point of view gives quite convincing arguments in support of projects to measure the hyperfine splitting of muonic hydrogen.

X-ray transitions studied for decelerated bare and H-like uranium ions at the ESR electron cooler

Here we report on X-ray spectra induced by spontaneous capture of free electrons into decelerated bare- and hydrogen-like uranium ions which we measured recently at the cooler section of the ESR storage ring. The most intense lines observed in spectra can be attributed to direct transition of electrons into the K shell of the projectile ions and to characteristic L→K (Lyα) transitions. Radiative recombination lines into the K shell of bare and H-like uranium can be exploited for measuring the two-electron contribution to the ground state binding energy in helium-like uranium. The goal is to probe for high-Z ions bound-state QED corrections which are of the order of α2. Besides the dominant characteristic L→K transitions, the strongly reduced Bremsstrahlung (due to the low cooler voltage applied to the decelerated ions) allowed us to observe for the very first time RR transitions into the L shell as well as the balmer radiation located at the low-energy part of the spectra.

Radiative corrections to hydrogenlike ions and heavy alkali-metal atoms in a magnetic field

A fully covariant scheme of renormalization is developed for the evaluation of radiative corrections to the energy levels of atoms and ions in an external magnetic field. Bound-state QED corrections to g factors are calculated for H-like ions in the ground state with nuclear charge numbers $1<~Z<~90$ and for the $\mathrm{ns}$ valence electrons in the atoms Cs, ${\mathrm{Ba}}^{+},$ and Fr. It is shown that these corrections should be taken into account in the comparison with experimental data for neutral atoms.

Renormalization of the two-photon vacuum polarization and the self energy vacuum polarization for a tightly bound electron

The renormalization method of Bogoljubov-Parasiuk-Hepp-Zimmermann (BPHZ) is used in order to derive the renormalized energy shift due to the gauge invariant Kallen-Sabry diagram of the two-photon vacuum polarization (VPVP) as well as the self energy vacuum polarization S(VP)E beyond the Uehling approximation. It is outlined, that no outer renormalization is required for the two-photon vacuum polarization and that only the inner renormalization has to be accomplished. It is shown that the so-called nongauge invariant spurious term is absent for a wide class of vacuum polarization (VP) diagrams if one applies the widely used spherical expansion of bound and free-electron propagator. This simplifies significantly calculations in bound state quantum electrodynamics. As one result of our paper the use of the BPHZ-approach in bound state QED is established.

Renormalization of the two-photon vacuum polarization and the self energy vacuum polarization for a tightly bound electron

The renormalization method of Bogoljubov-Parasiuk-Hepp-Zimmermann (BPHZ) is used in order to derive the renormalized energy shift due to the gauge invariant K\"all\'en-Sabry diagram of the two-photon vacuum polarization (VPVP) as well as the self energy vacuum polarization S(VP)E beyond the Uehling approximation. It is outlined, that no outer renormalization is required for the two-photon vacuum polarization and that only the inner renormalization has to b e accomplished. It is shown that the so-called nongauge invariant spurious term is absent for a wide class of vacuum polarization (VP) diagrams if one applies the widely used spherical expansion of bound and free-electron propagat or. This simplifies significantly calculations in bound state quantum electrodynamis . As one result of our paper the use of the BPHZ-approach in bound state QED is established.

QED procedure applied to the quasidegenerate fine-structure levels of He-like ions

A procedure for bound-state QED is presented, based upon a covariant form of the time-evolution operator. In contrast to the standard S-matrix formalism, our procedure is applicable also to states that are quasidegenerate. All (quasi)singularities, which appear when an intermediate state is (quasi)degenerate with the initial state, are eliminated. Our procedure is closely related to many-body perturbation theory (MBPT) and may open up possibilities to combine QED and MBPT in a more systematic fashion. The procedure is applied to the fine structure of the heliumlike neon and argon ions, and good agreement is obtained with recent experimental data.

Nuclear polarization in hydrogenlike heavy ions

The nuclear polarization (NP) correction is calculated for the ground states of the hydrogenlike ${}_{ 82}^{208}\mathrm{Pb}$ and ${}_{ 92}^{238}\mathrm{U}\mathrm{}$ ions. We take into account the transverse interaction, ignored in the previous studies, as well as the Coulomb interaction. The NP correction is formulated in the bound-state QED formalism with the ladder and cross diagrams of two-photon exchange between an electron and a nucleus. We take into account nuclear excitations of low-lying states and giant resonances with multipolarities of ${\ensuremath{\lambda}}^{\ensuremath{\pi}}{=0}^{+},$ ${1}^{\ensuremath{-}},$ ${2}^{+},$ and ${3}^{\ensuremath{-}},$ and describe them in a collective model. The NP corrections are obtained to be $\ensuremath{-}0.2$ and $\ensuremath{-}180.7$ meV, respectively, for ${}_{ 82}^{208}\mathrm{Pb}$ and ${}_{ 92}^{238}\mathrm{U}.$ The effect of the transverse interaction is found to be essential to the NP correction for electric dipole giant resonances, strongly canceling a contribution of the Coulomb interaction. Dependence of the NP correction on different nuclear excitations is investigated.

High-accuracy measurement of the magnetic moment anomaly of the electron bound in hydrogenlike carbon.

We present a new experimental value for the magnetic moment of the electron bound in hydrogenlike carbon (12C5+): g(exp) = 2.001 041 596 (5). This is the most precise determination of an atomic g(J) factor so far. The experiment was carried out on a single 12C5+ ion stored in a Penning trap. The high accuracy was made possible by spatially separating the induction of spin flips and the analysis of the spin direction. The current theoretical value amounts to g(th) = 2.001 041 591 (7). Together experiment and theory test the bound-state QED contributions to the g(J) factor of a bound electron to a precision of 1%.

Gjfactor of an electron bound in a hydrogenlike ion

We present a detailed theoretical evaluation for the ${g}_{j}$ factor of a bound electron in hydrogenlike ions up to $Z=94.$ All quantum electrodynamical corrections of order $(\ensuremath{\alpha}/\ensuremath{\pi})$ are evaluated in detail and various other contributions to the ${g}_{j}$ factor are computed and listed for 61 Z. A comparison with all existing experiments is carried out and excellent agreement is found. The present uncertainty in our calculations is discussed. It is not possible to improve this precision with only minor effort since two-photon bound-state QED terms are uncalculated up to now.

Density shift and broadening of transition lines in antiprotonic helium

The density shift and broadening of the transition lines of antiprotonic helium have been evaluated in the impact approximation using an interatomic potential calculated ab initio with the symmetry-adapted perturbation theory. The results help to remove an uncertainty of up to 10 ppm in the laser spectroscopy data on antiprotonic helium and are of importance in experimental tests of bound state QED and CPT invariance.

Estimates of the bound-state QED contributions to the g-factor of valence ns electrons in alkali metal atoms

The bound-state QED corrections to g-factors are partly calculated and partly estimated for the ns valence electrons in the atoms K, Rb, Cs, Ba + and Fr. It is shown that these corrections should be taken into account in the comparison with experimental data.

Counterterms for the second-order electron self-energy in bound-state QED

Counterterms for the second-order electron self-energy in bound-state QED

The second-order electron self-energy counterterms in bound state QED

The second-order electron self-energy counterterms are evaluated explicitly with the use of Pauli-Willars renormalization. These counterterms generalize the Feynman result for the first-order electron self-energy and can be useful for the future evaluation of the second-order Lamb shift for tightly bound electrons in highly charged ions.

Three-body corrections toO(α6)fine structure in helium

Three-body corrections of order {alpha}{sup 6}m{sup 2}c{sup 2}/M to the fine-structure splittings of helium and three-body terms of order {alpha}{sup 6}mc{sup 2} in lithium are derived and expressed in terms of expectation values of nonrelativistic operators. Corrections of order {alpha}{sup 6}{mu}c{sup 2} to non-S levels in two-body systems are also calculated. Perturbative calculation of one-body kinetic energy up to order 10{sup 6} of powers of {alpha} agrees with nonperturbatively numerical calculation within 19 significant figures. Such agreement provides a direct demonstration of the correctness of perturbation theory and of a simple technique developed previously [T. Zhang, Phys. Rev. A {bold 54}, 1252 (1996)] to calculate corrections arising from severely divergent nonrelativistic operators derived from the bound-state QED and relativistic kernels. {copyright} {ital 1997} {ital The American Physical Society}

Measurement of the gj factor of hydrogenic ions: a sensitive test of bound state QED

Thegj factor measurement of hydrogenic ions in the 1s ground state is with an expected accuracy of 10−7 a sensitive test of bound state QED. We expect to determine the deviations from the free electron value, caused by relativistic and radiative corrections, up to the orderα/4π(Zα)2 with an accuracy of 1%. As a first step, light ions like C5+ will be investigated. Later on, heavier hydrogenic ions up to U91+ will be examined using the accelerator facilities at GSI in Darmstadt.

Renormalization of the second-order electron self-energy for a tightly bound atomic electron: A detailed derivation.

A renormalized expression for the gauge invariant set of three second-order bound self-energy graphs is obtained. A nontrivial extra counterterm is found that is important for cancellation of infrared divergences. Also, a different type of infrared divergences specific to the bound-state QED is discussed and its cancellation is traced explicitly. The expression obtained is convenient for the application of the partial-wave renormalization technique. \textcopyright{} 1996 The American Physical Society.

Renormalization of the second-order self-energy for a tightly bound atomic electron

A renormalized expression for the gauge-invariant set of three second-order bound self-energy graphs is obtained. A non-trivial extra counter term is found that is important for the cancellation of infrared divergences. Also, a new type of infrared divergences specific to the bound-state QED is discussed and its cancellation is traced explicitly. The expression obtained is convenient for the application of the partial-wave renormalization technique.