S. G. Karshenboim, E. Yu. Korzinin, V. G. Ivanov
We consider a correction to energy levels in a pionic atom induced by the Uehling potential, i.e., by a free electron vacuum-polarization loop. The calculation is performed for circular states (l=n-1). The result is obtained in a closed analytic form as a function of $ Zα$ and the pion-to-electron mass ratio. Certain asymptotics of the result are also presented.
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The Uehling correction to the energy levels is presented in terms of the hypergeometric functions 2F1. This presentation allows to derived various asymptotics and approximations. Further applications of this method to other atomic characteristics are also considered.
We present the relativistic calculation of the hyperfine structure in pionic and kaonic atoms. A perturbation method has been applied to the Klein-Gordon equation to take into account the relativistic corrections. The perturbation operator has been obtained \textit{via} a multipole expansion of the nuclear electromagnetic potential. The hyperfine structure of pionic and kaonic atoms provide an additional term in the quantum electrodynamics calculation of the energy transition of these systems. Such a correction is required for a recent measurement of the pion mass.
We demonstrate the first step of a complete program, which consists in establishing an x-ray energy standard scale with the use of few-body atoms, in the few keV range. Light pionic and muonic atoms as well as one and two-electron ions from electron-cyclotron ion sources are used. The transition energies are calculable from quantum-electrodynamics, meaning that only a very limited subset need be measured and compared with theory, while providing a large number of standard lines. Here we show that circular transitions in pionic neon atoms, completely stripped from their electrons, reveal spectr…
The closed analytical expression for the Uehling potential is derived. The Uehling potential describes the lowest-order correction on vacuum polarisation in atomic and muon-atomic systems. We also derive the analytical formula for the interaction potential between two electrically charged point particles which includes correction to the vacuum polarisation, but has correct asymptotic behaviour at larger $r$. Our three-term analytical formula for the Uehling potential opens a new avenue in the study of the vacuum polarisation in light atomic systems.
Preliminary results of the strong interaction shift and width in pionic hydrogen ($πH$) using an X-ray spectrometer with spherically bent crystals and CCDs as X-ray detector are presented. In the experiment at the Paul Scherrer Institute three different $(np\to 1s)$ transitions in $πH$ were measured. Moreover the pion mass measurement using the $(5 \to 4)$ transitions in pionic nitrogen and muonic oxygen is presented
Effects of vacuum polarization modify the energy levels in atoms with an orbiting particle heavier than an electron. The dominant effect is due to the Uehling potential. In this paper we consider the relativistic corrections to the energy levels caused by the Uehling potential and in particular the fine structure in muonic and antiprotonic atoms. We derive general expressions and consider in detail specific regions of parameters which allow simple asymptotic expansion. We take into account the recoil effects and anomalous magnetic moment in the case of an antiproton as the orbiting particle.
We present a possibility of direct resonance production of pionic atoms (Coulomb bound states of a negative pion and a nucleus) with a rate of up to $\sim10^{10}$ per second using the gamma-ray beams from the Gamma Factory.
The collision-induced shift and broadening of selected dipole transition lines of pionic helium in gaseous helium at low temperatures up to $T=12$ K and pressure up to a few bar are calculated within variable phase function approach. We predict blue shift of the resonance frequencies of the $(n,l)=(16,15) \rightarrow (16,14) $ and $(16,15) \rightarrow (17,14)$ unfavored transitions and red shift for the favored transition $(17,16) \rightarrow (16,15)$. The result may be helpful in reducing the systematic error in proposed future experiments for determination of the negatively charged pion mass…
Probabilities of finding an antiparticle in an atom or ion containing a particle of spin 1/2 or spin 0 are determined. The spin 1/2 case was previously solved by Hans Bethe and his work is summarized. The spin 0 case is treated numerically for an arbitrary atomic number and analytically for small atomic numbers. The main tool for the spin 0 case is the Feshbach-Villars representation of the Klein-Gordon equation.
The relativistic recoil contributions to the Uehling corrections are revisited. We consider a controversy in recent calculations based on different approaches including Breit-type and Grotch-type calculations. We have found that calculations of those authors were in fact done in different gauges and in some of those gauges contributions the retardation and two-photon-exchange effects were missed. We have evaluated such effects and obtained a consistent result from those approaches. We present a correct expression for the Grotch-type approach which produces a correct gauge-invariant result. We …
Radiative corrections to the decay rate in the low-Z hydrogen-like muonic atoms are considered. The correction arises from the Uehling potential and it has the relative order of alpha. The numerical results are reported for the 2p-1s transition for the atom with the nuclear charge value up to Z=10.
The shift of atomic energy levels due to hadronic vacuum polarization is evaluated in a semiempirical way for hydrogenlike ions and for muonic hydrogen. A parametric hadronic polarization function obtained from experimental cross sections of $e^-e^+$ annihilation into hadrons is applied to derive an effective relativistic Uehling potential. The energy corrections originating from hadronic vacuum polarization are calculated for low-lying levels using analytical Dirac-Coulomb wave functions, as well as bound wave functions accounting for the finite nuclear size. Closed formulas for the hadronic …
A method for precise calculation of the energy corrections due to second order electric quadrupole interactions, as well as mixed electric quadrupole-vacuum polarization in the framework of the dynamic hyperfine structure in heavy muonic atoms is presented. For this, a multipole expansion of the Uehling potential is performed. The approach is applicable for an arbitrary nuclear electric charge distribution. By performing these calculations for muonic Rhenium and Uranium using a deformed Fermi distribution, it is shown that both corrections contribute on a level presumably visible in upcoming e…
Metastable pionic helium ($π{\rm He}^+$) is a three-body atom composed of a helium nucleus, an electron occupying the $1s$ ground state, and a negatively charged pion $π^-$ in a Rydberg state with principal- and orbital angular momentum quantum numbers of $n\sim \ell+1\sim 16$. We calculate the spin-independent energies of the $π{\rm ^3He}^+$ and $π{\rm ^4He}^+$ isotopes in the region $n=15$--19. These include relativistic and quantum electrodynamics corrections of orders $R_{\infty}α^2$ and $R_{\infty}α^3$ in atomic units, where $R_{\infty}$ and $α$ denote the Rydberg and fine structure const…
Using perturbed relativistic coupled-cluster (PRCC) theory we compute the ground state electric dipole polarizability, $α$, of doubly ionized alkaline earth metal ions $\rm{Mg}^{2+}$, $\rm{Ca}^{2+}$, $\rm{Sr}^{2+}$, $\rm{Ba}^{2+}$ and $\rm{Ra}^{2+}$. In the present work we use the Dirac-Coulomb-Breit atomic Hamiltonian and we also include the Uehling potential, which is the leading order term in the vacuum polarization corrections. We examine the correction to the orbital energies arising from the Uehling potential in the self-consistent field calculations as well as perturbatively. Our result…
Precision calculations of the fine and hyperfine structure of muonic atoms are performed in a relativistic approach and results for muonic 205 Bi, 147 Sm, and 89 Zr are presented. The hyperfine structure due to magnetic dipole and electric quadrupole splitting is calculated in first order perturbation theory, using extended nuclear charge and current distributions. The leading correction from quantum electrodynamics, namely vacuum polarization in Uehling approximation, is included as a potential directly in the Dirac equation. Also, an effective screening potential due to the surrounding elect…
Measurement of the charged pion mass using X-ray spectroscopy of exotic atoms
本研究通过同时测量1.4巴氮氧混合气体中π介子氮与μ介子氧原子的5g-4f跃迁，以μ介子谱线校准能量，测得负π介子质量为(139.57077±0.00018)MeV/c²，比现有世界平均值高4.2ppm。
Vacuum polarization corrections to hyperfine structure in many-electron atoms
本研究针对多电子原子超精细结构的真空极化修正开展理论计算，在Uehling近似与相对论框架下分析多体效应，明确不同原子态的修正近似条件与核心极化的关键作用。
We have developed a Fock-space relativistic coupled-cluster theory based method for the calculation of electric dipole polarizability of one-valence atoms and ions. We employ this method to compute the ground-state and spin-orbit coupled excited state electric dipole polarizability of Al and In. To check the quality of many-electron wavefunctions, we also compute the excitation energies of some low-lying states of Al and In. The effects of Breit interaction and QED corrections from the Uehling potential and the self-energy are included to improve the accuracy of $α$ further. Our recommended va…
High-resolution pionic-atom x-ray spectroscopy was performed with an x-ray spectrometer based on a 240-pixel array of superconducting transition-edge-sensor (TES) microcalorimeters at the piM1 beam line of the Paul Scherrer Institute. X-rays emitted by pionic carbon via the 4f->3d transition and the parallel 4d->3p transition were observed with a full-width-at-half-maximum energy resolution of 6.8 eV at 6.4 keV. Measured x-ray energies are consistent with calculated electromagnetic values which considered the strong-interaction effect assessed via the Seki-Masutani potential for the 3p energy …
Two-body $P$-state energies at $α^6$ order
本文研究了双体系统中nP态能级在α^6阶的修正，发现正电子对中存在此前未被注意到的修正，并对轻μ子原子的核电荷半径测量有重要意义。
We investigate quantum electrodynamic effects under the influence of an external, time-dependent electromagnetic field, which mediates dynamic modifications of the radiative corrections. Specifically, we consider the quantum electrodynamic vacuum-polarization tensor under the influence of two external background fields: a strong laser field and a nuclear Coulomb field. We calculate the charge and current densities induced by a nuclear Coulomb field in the presence of a laser field. We find the corresponding induced scalar and vector potentials. The induced potential, in first-order perturbatio…
Effective Field Theory for the Bound States and Scattering of a Heavy Charged Particle and a Neutral Atom
本研究提出诱导偶极有效场论（ID-EFT）描述重带电粒子与中性原子系统，通过接触相互作用重整化$1/r^4$势，验证了其在π氢散射相移及束缚态计算中的有效性。
Quantum calculation of the collision-induced line-shape effects in antiprotonic helium and the new accurate ab initio $\bar{p}$He$^{+}$-He potential energy surface
本文构建高精度从头算 $\ar{p}$He$^{+}$-He 势能面并进行量子散射计算，获得50个跃迁的压力展宽与频移系数，为反质子氦精密光谱及其他奇异原子研究提供理论基准。
Quantum electrodynamic corrections for molecules: Vacuum polarisation and electron self energy in a two-component relativistic framework
本文在双分量ZORA相对论框架中实现并评估了真空极化与电子自能QED修正，通过多种原子、分子及离子的能级与跃迁能计算，证明该方法高效且与四分量结果高度吻合。
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