Two-loop Self-energy Correction Research Articles

Combined two-loop self-energy corrections at finite and zero temperatures

Combined two-loop self-energy corrections at finite and zero temperatures

Two-photon atomic level widths at finite temperatures

The thermal two-photon level broadening of the excited energy levels in the hydrogen and H-like helium is evaluated via the imaginary part of thermal two-loop self-energy correction for bound electron. All the derivations are presented in the framework of rigorous quantum electrodynamic theory at finite temperatures and are applicable for the H-like ions. On this basis, we found a contribution to the level broadening induced by the blackbody radiation which is fundamentally different from the usual line broadening caused by the stimulated two-photon decay and the Raman scattering of thermal photons. Numerical calculations of the two-loop thermal correction to the two-photon width for the $2s$ state in hydrogen and singly ionized helium atoms show that the effect could significantly exceed the higher-order relativistic and radiative QED corrections commonly included in the calculations. In addition, the thermal two-loop self-energy correction significantly exceeds the "ordinary" stimulated one-photon depopulation rate at the relevant laboratory temperatures. In this work, detailed analysis and the corresponding comparison of the effect with the existing laboratory measurements in H-like ions are carried out.

Theory of the two-loop self-energy correction to the g factor in nonperturbative Coulomb fields

Two-loop self-energy corrections to the bound-electron $g$ factor are investigated theoretically to all orders in the nuclear binding strength parameter $Z\alpha$. The separation of divergences is performed by dimensional regularization, and the contributing diagrams are regrouped into specific categories to yield finite results. We evaluate numerically the loop-after-loop terms, and the remaining diagrams by treating the Coulomb interaction in the electron propagators up to first order. The results show that such two-loop terms are mandatory to take into account for projected near-future stringent tests of quantum electrodynamics and for the determination of fundamental constants through the $g$ factor.

Two-loop self-energy in the Lamb shift of the ground and excited states of hydrogenlike ions

The two-loop self-energy correction to the Lamb shift of hydrogen-like ions is calculated for the $1s$, $2s$, and $2p_{1/2}$ states and nuclear charge numbers $Z = 30$-$100$. The calculation is performed to all orders in the nuclear binding strength parameter $Z\alpha$. As compared to previous calculations of this correction, numerical accuracy is improved by an order of magnitude and the region of the nuclear charges is extended. An analysis of the $Z$-dependence of the obtained results demonstrates their consistency with the known $Z\alpha$-expansion coefficients.

The two-loop self-energy: diagrams in the coordinate-momentum representation

The paper reports a technique of evaluation of Feynman diagrams in the mixed coordinate-momentum representation. The technique is employed for a recalculation of the two-loop self-energy correction for the ground state of hydrogen-like ions with the nuclear charge numbers Z=10-30. The numerical accuracy is considerably improved as compared to the previous calculations. The higher-order (in Z\alpha) remainder function is inferred from the numerical results and extrapolated towards Z=0 and 1. The extrapolated value for hydrogen is consistent (but still not in perfect agreement) with the analytical result obtained within the perturbative approach.

Two-loop self-energy for the ground state of medium-Zhydrogenlike ions

The two-loop self-energy correction to the ground state Lamb shift is calculated for hydrogen-like ions with the nuclear charge Z=10-30 without any expansion in the binding field of the nucleus. A calculational technique is reported for treatment of Feynman diagrams in the mixed coordinate-momentum representation, which yields significant improvement in numerical accuracy as compared to previous results. An extrapolation of the all-order numerical data towards Z=1 yields a result for the higher-order remainder function for hydrogen. The previously reported disagreement between the all-order and the perturbative approaches is reduced to the marginal agreement.

Two-loop QED corrections in few-electron ions

Results of a calculation valid to all orders in the nuclear-strength parameter Zα are presented for the two-loop Lamb shift, notably for the two-loop self-energy correction, for the ground and first excited states of ions with the nuclear charge numbers Z = 60–100. A detailed comparison of the all-order calculation with earlier investigations based on an expansion in the parameter Zα is given. The calculation removes the largest theoretical uncertainty for the 2pj–2s transition energies in heavy Li-like ions and is important for interpretation of experimental data.PACS Nos.: 31.30.Jv, 31.30.–i, 31.10.+z

Nonperturbative Calculation of the Two-Loop Lamb Shift in Li-Like Ions

A calculation valid to all orders in the nuclear-strength parameter is presented for the two-loop Lamb shift, notably for the two-loop self-energy correction, to the 2p-2s transition energies in heavy Li-like ions. The calculation removes the largest theoretical uncertainty for these transitions and yields the first experimental identification of two-loop QED effects in the region of the strong binding field.

Two-loop self-energy correction in a strong Coulomb nuclear field

The two-loop self-energy correction to the ground-state energy levels of hydrogen-like ions with nuclear charges Z≥10 is calculated without the Zα expansion, where α is the fine-structure constant. The data obtained are compared with the results of analytical calculations within the Zα expansion; significant disagreement with the analytical results of order α2(Zα)6 has been found. Extrapolation is used to obtain the most accurate value for the two-loop self-energy correction for the 1s state in hydrogen.

Two-loop self-energy correction to the ground-state Lamb shift in H-like ions

The two-loop self-energy correction is evaluated to all orders in Zα for the ground-state Lamb shift of H-like ions with Z⩾20, where Z is the nuclear charge number and α is the fine structure constant. The results obtained are compared with analytical calculations performed within the Zα-expansion.

Two-loop self-energy contribution to the Lamb shift in H-like ions

The two-loop self-energy correction is evaluated to all orders in $Z\ensuremath{\alpha}$ for the ground-state Lamb shift of H-like ions with $Z\ensuremath{\geqslant}10$, where $Z$ is the nuclear charge number and $\ensuremath{\alpha}$ is the fine-structure constant. The results obtained are compared with the analytical values for the $Z\ensuremath{\alpha}$-expansion coefficients. An extrapolation of the all-order numerical results to $Z=1$ is presented and implications of our calculation for the hydrogen Lamb shift are discussed.

Some recent advances in bound-state quantum electrodynamics

We discuss recent progress in various problems related to bound-state quantum electrodynamics: the bound-electron g factor, two-loop self-energy corrections, and the laser-dressed Lamb shift. The progress relies on various advances in the bound-state formalism, including ideas inspired by effective field theories such as nonrelativistic quantum electrodynamics. Radiative corrections in dynamical processes represent a promising field for further investigations. PACS Nos.: 31.15.–p, 12.20.Ds

Evaluation of the two-loop self-energy correction to the ground state energy of H-like ions to all orders in Z$\alpha$

We present an evaluation of the complete gauge-invariant set of the two-loop self-energy diagrams carried out to all orders in the parameter \(Z\alpha\). The calculation is performed for the ground state of H-like ions with \(Z\ge 40\). As a result, we significantly improve the accuracy of theoretical values for the ground-state Lamb shift in high-Z H-like ions. We provide a compilation of various contributions to the ground-state energy of H-like ions and estimate uncertainties of the corresponding theoretical predictions.

Double-logarithmic two-loop self-energy corrections to the Lamb shift

Self-energy corrections involving logarithms of the parameter $Z\ensuremath{\alpha}$ can often be derived within a simplified approach, avoiding calculational difficulties typical of the problematic nonlogarithmic corrections (as customary in bound-state quantum electrodynamics, we denote by Z the nuclear charge number, and by $\ensuremath{\alpha}$ the fine-structure constant). For some logarithmic corrections, it is sufficient to consider internal properties of the electron characterized by form factors. We provide a detailed derivation of related self-energy ``potentials'' that give rise to the logarithmic corrections; these potentials are local in coordinate space. We focus on the double-logarithmic two-loop coefficient ${B}_{62}$ for P states and states with higher angular momenta in hydrogenlike systems. We complement the discussion by a systematic derivation of ${B}_{62}$ based on nonrelativistic quantum electrodynamics. In particular, we find that an additional double logarithm generated by the loop-after-loop diagram cancels when the entire gauge-invariant set of two-loop self-energy diagrams is considered. This double logarithm is not contained in the effective-potential approach.

Aproblematic set of two-loop self-energy corrections

We investigate a specific set of two-loop self-energycorrections involving squared decay rates and point out thattheir interpretation is highly problematic. The correctionscannot be interpreted as radiative energy shifts in the usualsense. Some of the problematic corrections find a naturalinterpretation as radiative nonresonant corrections to thenatural line shape. They cannot uniquely be associated with oneand only one atomic level. While the problematic corrections arerather tiny when expressed in units of frequency (a few hertzfor hydrogenic P levels) and do not affect the reliability ofquantum electrodynamics at the current level of experimentalaccuracy, they may be of importance for future experiments. Theproblems are connected with the limitations of the so-calledasymptotic-state approximation, which means that atomic in- andout-states in the S-matrix are assumed to have an infinitelifetime.

Two-loop self-energy corrections to the fine structure

We investigate two-loop higher-order binding corrections to the fine structure, which contribute to the spin-dependent part of the Lamb shift. Our calculation focuses on the so-called ``two-loop self-energy'' involving two virtual closed photon loops. For bound states, this correction has proven to be notoriously difficult to evaluate. The calculation of the binding corrections to the bound-state two-loop self-energy is simplified by a separate treatment of hard and soft virtual photons. The two photon-energy scales are matched at the end of the calculation. We explain the significance of the mathematical methods employed in the calculation in a more general context, and present results for the fine-structure difference of the two-loop self-energy through the order of $\alpha^8$.

Two-loop self-energy correction in H-like ions

A part of the two-loop self-energy correction, the so-called P term, is evaluated numerically for the $1s$ state to all orders in $Z\ensuremath{\alpha}.$ Our calculation, combined with the previous investigation [S. Mallampalli and J. Sapirstein, Phys. Rev. A 57, 1548 (1998)], yields the total answer for the two-loop self-energy correction in H-like uranium and bismuth. As a result, the major uncertainty is eliminated from the theoretical prediction for the Lamb shift in these systems. The total value of the ground-state Lamb shift in H-like uranium is found to be 463.95(50) eV.