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In bound systems whose constituent particles are heavier than the electron, the dominant radiative correction to energy levels is given by light-fermion (electronic) vacuum polarization. In consequence, relativistic and recoil corrections to the one-loop vacuum-polarization correction are phenomenologically relevant. Here, we generalize the treatment, previously accomplished for systems with orbiting muons, to bound systems of constituents with more general spins: spin-0, spin- $1/2$, and spin-1. We discuss the application of our more general expressions to various systems of interest, including spinless systems (pionium), muonic hydrogen and deuterium, and devote special attention to the excited non- $S$states of deuteronium, the bound system of a deuteron and its antiparticle. The obtained energy corrections are of order ${\alpha }^5{m}_r$, where $\alpha$is the fine-structure constant and $m_r$is the reduced mass. 

We evaluate the energy levels of the deuteronium bound system, which consists of a deuteron and an antideuteron, with a special emphasis on states with nonvanishing orbital angular momenta. The excited atomic bound states of deuteronium constitute probes for the understanding of higher-order quantum electrodynamic corrections for spin-1 particles in a bound system where the typical field strength of the binding Coulomb field (at a distance of the generalized Bohr radius) exceeds Schwinger’s critical field strength. For states with nonvanishing angular momenta, effects due to the internal structure of the deuteron and virtual annihilation contributions are highly suppressed. Relevant transitions are found to be in a frequency range accessible by standard laser spectroscopic techniques. We evaluate the leading and next-to-leading energy corrections of orders ${\alpha }^3{m}_d$and ${\alpha }^4{m}_d$, where $\alpha$is the fine-structure constant and $m_d$is the deuteron mass, and also investigate internal-structure corrections: hadronic vacuum polarization, finite-size effects, and strong-interaction corrections. 

In exotic atomic systems with hadronic constituent particles, it is notoriously difficult to estimate the strong-interaction correction to energy levels. It is well known that, due to the strength of the nuclear interaction, the problem cannot be solved using Wigner–Brillouin perturbation theory alone. Recently, high-angular-momentum Rydberg states of exotic atomic systems with hadronic constituents have been identified as promising candidates in the search for new physics in the low-energy sector of the Standard Model. We thus derive a generalized Deser–Trueman formula for the induced energy shift for a general hydrogenic bound state with principal quantum number n and orbital angular momentum quantum number ℓ, and we find that the energy shift is given by the formula δE=2αn,ℓβℓ(ah/a0)2ℓ+1Eh/n3, where αn,0=1, αn,ℓ=∏s=1ℓ(s−2−n−2), βℓ=(2ℓ+1)/[(2ℓ+1)!!]2, Eh is the Hartree energy, ah is the hadronic radius and a0 is the generalized Bohr radius. The square of the double factorial, [(2ℓ+1)!!]2, in the denominator implies a drastic suppression of the effect for higher angular momenta. 

Three-loop electronic vacuum-polarization corrections due to irreducible diagrams are evaluated for two-body muonic ions with nuclear charge numbers $1\le Z\le 6$. The corrections are of order ${\alpha }^3(Z\alpha {)}^2{m}_r$, where $\alpha$is the fine-structure constant and $m_r$is the reduced mass. Numerically, the energy corrections are found to be of the same order of magnitude as the largest of the order ${\alpha }^2(Z\alpha {)}^6{m}_r$corrections, and are thus phenomenologically interesting. Our method of calculation eliminates numerical uncertainty encountered in other approaches. Published by the American Physical Society2025 

The recursive Neville algorithm allows one to calculate interpolating functions recursively. Upon a judicious choice of the abscissas used for the interpolation (and extrapolation), this algorithm leads to a method for convergence acceleration. For example, one can use the Neville algorithm in order to successively eliminate inverse powers of the upper limit of the summation from the partial sums of a given, slowly convergent input series. Here, we show that, for a particular choice of the abscissas used for the extrapolation, one can replace the recursive Neville scheme by a simple one-step transformation, while also obtaining access to subleading terms for the transformed series after convergence acceleration. The matrix-based, unified formulas allow one to estimate the rate of convergence of the partial sums of the input series to their limit. In particular, Bethe logarithms for hydrogen are calculated to 100 decimal digits. Generalizations of the method to series whose remainder terms can be expanded in terms of inverse factorial series, or series with half-integer powers, are also discussed. 

For an accurate theoretical description of muonic bound systems, it is crucial to consistently treat relativistic and recoil corrections to vacuum polarization. The one-loop vacuum-polarization effect is by far the dominant quantum electrodynamic (QED) energy correction for bound muons, being of order α(Zα)2mr, where α is the fine-structure constant, Z is the nuclear charge number, and mr is the reduced mass. Gauge invariance of the relativistic and recoil corrections to vacuum polarization of order α(Zα)4mr is investigated with respect to nonretarded and standard, renormalized variants of Coulomb gauge. The invariance is shown after including three-photon exchange diagrams. Our derivation is based on an adapted form of nonrelativistic quantum electrodynamics for bound muon systems (NRQEDμ), which is a version of NRQED where the hard scale is set at the muon mass instead of the electron mass. Updated values for the gauge-independent corrections for one-muon ions with nuclear charge numbers Z = 1, 2, 6 are presented. 

The calculation of higher-order binding corrections to bound systems is a fundamental problem of theoretical physics. For any nonrelativistic expansion, one needs the Foldy-Wouthuysen transformation, which disentangles the particle and the antiparticle degrees of freedom. This transformation is carried out here to eighth order in the momenta or to eighth order in the momentum operators, which is equivalent to the eighth order of the fine- structure constant. Matrix elements of the eighth-order terms are evaluated for F5/2 and F7/2 states in hydrogenlike ions and compared with the Dirac-Coulomb energy levels. 

We present a complete reevaluation of the irreducible two-loop vacuum-polarization correction to the photon propagator in quantum electrodynamics, i.e. with an electron-positron pair in the fermion propagators. The integration is carried out by reducing the integrations to a limited set of master integrals, which are calculated using integration-by-parts identities. Dimensional regularization is used in $D=4-2\epsilon$dimensions, and on-mass shell renormalization is employed. The one-loop effect is given to order $\epsilon$, to be combined with the $1/\epsilon$divergence of the two-loop amplitude. Master integrals are given. Final evaluations of two-loop energy shifts for $1S$, $2S$, and $2P$states are done analytically, and results are presented, with an emphasis on muonic hydrogen. For relativistic Dirac-Coulomb reference states, higher-order coefficients are obtained for the $Z\alpha$-expansion. We compare the results obtained to the existing literature. Published by the American Physical Society2024 

We discuss numerical aspects of instantons in two- and three-dimensional ${\varphi }^4$theories with an internal $O\left(N\right)$symmetry group, the so-called $N$-vector model. By combining asymptotic transseries expansions for large arguments with convergence acceleration techniques, we obtain high-precision values for certain integrals of the instanton that naturally occur in loop corrections around instanton configurations. Knowledge of these numerical properties is necessary in order to evaluate corrections to the large-order factorial growth of perturbation theory in ${\varphi }^4$theories. The results contribute to the understanding of the mathematical structures underlying the instanton configurations. Published by the American Physical Society2024 

We revisit the derivation of multipole contributions to the atom-wall interaction previously presented in Łach et al. [G. Łach, M. DeKieviet, and U. D. Jentschura, Phys. Rev. A 81, 052507 (2010)]. A careful reconsideration of the angular momentum decomposition of the second-, third-, and fourth-rank tensors composed of the derivatives of the electric-field modes leads to a modification for the results for the quadrupole, octupole, and hexadecupole contributions to the atom-wall interaction. Asymptotic results are given for the asymptotic long-range forms of the multipole terms, in both the short-range and long-range limits. Calculations are carried out for hydrogen and positronium in contact with α-quartz; a reanalysis of analytic models of the dielectric function of α-quartz is performed. Analytic results are provided for the multipole polarizabilities of hydrogen and positronium. The quadrupole correction is shown to be numerically significant for atom-surface interactions. The expansion into multipoles is shown to constitute a divergent, asymptotic series. Connections to van der Waals corrected density-functional theory and applications to physisorption are described