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Homonuclear molecules have emerged as a crucial component in the pursuit of frequency standards, offering a promising avenue for the discovery of new physics phenomena that transcend the standard model. They also provide a unique approach to constraining variations in fundamental constants over time, thereby complementing the capabilities of atomic clocks. A notable challenge faced by molecular and single atomic quantum systems is the management of blackbody radiation (BBR), which introduces significant systematic errors and is challenging to regulate effectively. To address this issue, we perform {\it ab-initio} quantum chemical calculations to accurately determine the potential energy curve and the polarizability tensor for the ground state of the N$_2^+$ molecular ion, one of the most promising candidates for searching for variation of $m_{e}/m_{p}$ and creating frequency standards. We then calculate the BBR shifts affecting the vibrational levels of the ground electronic X$^2\Sigma_g^+$ state, marking a substantial contribution towards the precise experimental measurements.

In the present work, quasi-molecular compounds consisting of one antiproton ($\bar{p}$) and one hydrogen-like ion are investigated: $\mathrm{He}^{+} - \bar{p}$, $\mathrm{Li}^{2+} - \bar{p}$, $\mathrm{C}^{5+} - \bar{p}$, $\mathrm{S}^{15+} - \bar{p}$, $\mathrm{Kr}^{35+} - \bar{p}$, $\mathrm{Ho}^{66+} - \bar{p}$, $\mathrm{Re}^{74+} - \bar{p}$, $\mathrm{U}^{91+} - \bar{p}$. For the calculations, the Dirac equation with two-center potential is solved numerically using the dual-kinetically balanced finite-basis-set method adapted to systems with axial symmetry (A-DKB). Adiabatic potential curves are constructed for the ground state of the above quasi-molecular compounds in the framework of the A-DKB approach. Calculations were also performed for the case of an external magnetic field (the field is taken into account non-perturbatively). Zeeman shifts of the quasi-molecular terms are obtained for a homogeneous magnetic field with a strength of the laboratory order (up to 100 Tesla) directed along the axis of the molecule., Comment: 14 pages, 4 figures

This paper discusses some features of the spectral line profile theory used in the treatment of measured atomic transitions. It is shown that going beyond the established linear approximation for the spectral line contour in the case of its nonresonant extension, the potential for a more accurate extraction of atomic characteristics from experimental data arises. Using the example of the Lyman-$\alpha$ (Ly$_\alpha$) transition in hydrogen, a simple analysis of the observed spectral line distorted by a possible interfering transitions is given. In particular, the results obtained in the present work clearly demonstrate that the processing of the same experimental data at different settings can provide an accurate determination of the transition frequency, the centre of gravity as well as the hyperfine splitting of the ground state in hydrogen-like atomic systems. The latter is especially important for setting up precision spectroscopic experiments on the antihydrogen atom.

Modern resonant spectroscopic experiments to measure transition frequencies in atoms have reached a level where a meticulous description of all aspects of the processes under study has become obligatory. The precision achieved in the experiments of A. Beyer, et al., Science 358, 79 (2017), has led to the fact that the determination of the transition frequency based on measured data is significantly refined by theoretical treatment of the observed spectral line profile. As it was predicted theoretically, a great impact of effects arising beyond the resonance approximation was found experimentally. These findings marked the beginning of the upcoming epoch in the resonant atomic spectroscopy when many commonly understood ideas became invalid. For example, the atomic transition may be characterized by several different but equally acceptable frequencies. Furthermore, we show that the picture becomes even more complicated when the observed spectral line profile is "identified" with one of the processes - emission or absorption. Precise determination of the transition frequency requires a description of the absorption line profile inseparable from the emission process and vice versa. The theoretical aspects discussed in this work provide prerequisites for more accurate and yet simpler experiments than those reported in Science 358, 79 (2017). Implementing the new physics expected in atomic resonance spectroscopy in the near future beyond the resonance approximation is unfeasible without resolving these issues., Comment: 20 pages, 4 figures, 4 tables

In this study, we reexamine the long-range interaction between two atoms placed in an equilibrium thermal radiation environment. Employing the formalism of quantum electrodynamics at finite temperatures, we derive an expression for the thermal correction to the interaction potential and explore various asymptotic behaviors. The numerical calculations of temperature-dependent dispersion coefficients for both the ground and highly excited states of the hydrogen atom are performed. We proceed from the first principles of the theory to derive the dipole-dipole interaction at finite temperature. The analysis presented in this work reveals that the expressions established earlier in the context of phenomenological extrapolation from zero- to finite-temperature scenarios exhibit disparate asymptotic behavior and lead to overestimated results to those of the rigorous quantum electrodynamics approach.

The electronic spectra of light one-electron quasi-molecular compounds H-H$^+$, He$^+$-He$^2+$ and He$^+$-H$^+$ are analyzed. To this end, the two-center Dirac equation is solved by the dual-kinetically balanced finite-basis-set method for axially symmetric systems termed as A-DKB. This method allows a complete relativistic consideration of these systems at fixed internuclear distances. A comparison of the obtained results with the nonrelativistic and relativistic calculations presented in the literature is performed. The advantages and disadvantages of the approach are discussed in details.

In this paper we investigate higher-order corrections to the energies of bound states in hydrogen subjected to the external blackbody radiation field. In particular, within the framework of thermal quantum electrodynamics and $S$-matrix approach we analyze combined type of two-loop self-energy corrections, including one zero-vacuum and one loop at finite temperature. By utilizing the method of dimensional regularization, we derive closed analytical expressions for the energy shifts of atomic levels. Our numerical calculations demonstrate that even at room temperature these corrections can be significant for excited states, reaching the magnitude of the thermal induced Stark contribution.

In this paper, we consider a one-loop thermal correction to the $g$-factors of the ground and low-lying excited states in neutral alkaline-like atomic systems. As an example, we carried out calculations for Rb and Cs atoms, where optical measurements of transition frequencies set the metrological level of accuracy. With the development of atomic systems as the most accurate tool for spectroscopic experiments, it has become increasingly important to improve the accuracy of measuring the electron $g$-factor, which is currently at ~$10^{-12}$. The rapid progress in this field in recent years requires analysis not only of various relativistic, QED and other effects, but also theoretical studies of phenomena stimulated by the external thermal environment. We can expect that the thermal contribution to the $g$-factor calculated in our work may be of interest in the near future.

The paper discusses the line profile asymmetry of the photon scattering process that arises naturally in quantum electrodynamics (QED). Based on precision spectroscopic experiments conducted on hydrogen atoms, we focus our attention on the two-photon $1s-2s$ transition. As one of the most precisely determined transition frequencies, it is a key pillar of optical frequency standards and is used in determining fundamental physical constants, testing physical principles, and searching constraints on new fundamental interactions. The results obtained in this work show the need to take into account the natural line profile asymmetry in precision spectroscopic experiments.

In this work, we consider the thermal correction to the hyperfine interaction in hydrogen, deuterium, and the $^3$He$^+$ ion. This correction is effectively described by one-loop Feynman graphs in the framework of the quantum electrodynamics theory for bound states at a finite temperature. A simple analysis shows the importance of the obtained results for future prospects for measuring hyperfine splitting. In addition, the application for testing the time variation of fundamental constants is briefly discussed.

We study the influence of constant magnetic field combined with a field induced by the external thermal environment on the atomic decay rates. The importance of radiative corrections, including magnetic interaction, is demonstrated for hydrogen and hydrogen-like ions with low nuclear charge values $Z$. Based on the quantum electrodynamics description, the principal possibility of determining the $g$-factor by observing fluorescence is shown. The considered effects can be used in precision spectroscopic experiments and astrophysical studies., Comment: 8 figures, 3 Tables, 12 pages

It is well-known that the 4-rotation in the 4-dimensional space-time is equivalent to the CPT-transformation (C is the charge conjugation, P is the space inversion and T is the time-reversal). The standard definition of the T-reversal includes the change of the sign of time variable and replacement of the initial state of the particle (system of particles) by the final state and vice versa. Since the time-reversal operation changes the state of a particle, the particle's wave function cannot be the eigenfunction of the corresponding operator with a certain eigenvalue, as in the case of space parity. Unlike the CPT-transformation, the separate P, T, or C transformations cannot be reduced to any 4-rotation. The extended Lorentz group incorporates all the separate C, P, or T transformations which do not bring the time axis out of the corresponding light cone. The latter restriction is included in the standard definition of the time-reversal. In the present communication, we ignore this restriction. This allows to introduce the "time arrow" operator and characterize every particle by the new quantum number - "time arrow" value. The wave functions of all particles are eigenfunctions of this operator with eigenvalues equal to "time arrow" values. The particles with the "time arrow" values opposite to the "time arrow" value in our universe form another universe (anti-universe). The existence of anti-universe can be confirmed, in principle, by laboratory (atomic) experiments. The anti-universe may be also considered as a candidate to the role of dark matter.

In this review, we have investigated the asymmetry of the line profile in precision one- and two-photon spectroscopy of hydrogen and helium atoms within the framework of a rigorous QED approach. A detailed analysis of the angular correlations of the quantum interference effect has been carried out using various examples. Nonresonant effects are also considered in relation to some astrophysical problems. In particular, a rigorous QED derivation of the nonresonant extension for the Lorentz line profile is given using the Ly$_{\alpha}$ transition as an example; such a QED derivation has been lacking in the literature.

The influence of the blackbody radiation field on the $g$-factor of light hydrogenlike ions is considered within the framework of quantum electrodynamics at finite temperature for bound states. One-loop thermal corrections are examined for a wide range of temperatures. The numerical results for $1s$, $2s$, $2p_{1/2}$, and $2p_{3/2}$ states are presented. It is shown that for excited states finite temperature corrections to the bound-electron $g$-factor are close to the level of current experimental uncertainty even at room temperatures and can be discerned within the measurements anticipated in the near future.

Thermal corrections, including relativistic effects, for the positronium atom are discussed. The theoretical description of thermal corrections is carried out within the framework of relativistic quantum electrodynamics. As a result, thermal corrections to atomic energy levels with a fine and hyperfine structure and to the probabilities of annihilation of a positronium atom placed in a thermal environment (blackbody radiation) are taken into account. Numerical results are discussed throughout the paper in view of modern experiments and theoretical searches for verification of fundamental interactions., Comment: 10 pages, 2 figures, 2 tables

In the present paper, the correction due to the thermal interaction of two charges to the recombination and ionization processes for the hydrogen atom is considered. The evaluation is based on a rigorous quantum electrodynamic (QED) approach within the framework of perturbation theory. The lowest-order radiative correction to the recombination/ionization cross-section is examined for a wide range of temperatures corresponding to laboratory and astrophysical conditions. The found thermal contribution is discussed both for specific states and for the total recombination and ionization coefficients., Comment: 7 pages, 2 figures, 4 Tables

In the present paper, we study nonresonant corrections for experimental measurements of the transition frequencies in the helium atom. Having attracted more attention, such effects can make a significant contribution to experiments based on one- and two-photon atomic spectroscopy. The quantum interference effects in the measurements of $2^3S_1-n^3D_1$ ($ n=3,\,4,\,5 $) transition frequencies based on Doppler-free two-photon spectroscopy, are considered as a possible source of current discrepancy between the experimental and theoretical data. We demonstrate that line profile asymmetry caused by the quantum interference of fine sub-levels of the $ ^3D_{J} $ $ (J=1,2,3) $ state can reach tenths of a megahertz for different experimental conditions. Thus, previously unaccounted nonresonant corrections should be taken in next-generation experimental measurements of transitions frequencies in helium. However, they could not completely eliminate the current imbalance in the study of helium spectra and the question is still open.

In the present paper, we consider nonresonant corrections to $ 2s-ns/nd $ transition frequencies in hydrogen for the experiments based on two-photon spectroscopy. A detailed study of angular correlations of quantum interference effects within the framework of rigorous quantum electrodynamics is given. Closed expressions for the resonant two-photon scattering cross sections on an atomic level with dependence on all atomic quantum numbers including fine and hyperfine structure are derived. These expressions are applied for the description of experiments based on two-photon spectroscopy with fixing of incident (outgoing) photon polarizations. We demonstrate that for certain experimental geometry nonresonant corrections could be significant for the determination of Rydberg constant and proton charge radius.

This paper discusses relativistic corrections to the thermal Coulomb potential for simple atomic systems. The theoretical description of the revealed thermal corrections is carried out within the framework of relativistic quantum electrodymamics (QED). As a result, thermal corrections to the fine and hyperfine strucutres of atomic levels are introduced. The theory presented in this paper is based on the assumption that the atom is placed in a thermal environment created by the blackbody radiation (BBR). The numerical results allow us to expect hteir significance for modern experiments and testing the fundamental interactions., Comment: 8 pages, 3 Tables, 1 figure

The "proton radius puzzle" was recently solved by reducing the four-standard deviation discrepancy between the results for electronic hydrogen ($H$) and muonic hydrogen ($\mu H$) atoms to $3.3$ value. The value of the root-mean-square radius of the proton ($r_p$), extracted from experiments on measuring the one-photon $2s-4p$ transition and the Lamb shift in hydrogen, is now $0.8335(95)$ fm, that is in good agreement with the muonic hydrogen experiments, $0.84087(39)$ fm. Even so, these values deviate significantly from the CODATA value, which is determined as the average using the results for various spectral lines including two-photon transitions in the hydrogen atom. The solution of the proton radius puzzle was realized by taking into account the influence of interference effect in one-photon scattering processes. The importance of interfering effects in atomic frequencies measurements gives an impetus to the study of experiments based on two-photon spectroscopy with the suchlike thoroughness. It is shown here that the effect of interfering pathways for two-photon $2s-nd$ transitions in a hydrogen atom is also significant in determining the proton charge radius and Rydberg constant.

Thermal corrections to the one-photon spontaneous and induced transition probabilities for hydrogen and hydrogen-like ions are evaluated. The found thermal corrections are given by the vertex Feynman graph, where the vertex represents the thermal interaction between the bound electron and the nucleus. All derivations of thermal corrections to bound-bound transitions for an atom exposed to blackbody radiation (BBR) are made in a fully relativistic approach within the framework of the adiabatic $S$-matrix formalism. It is found that the vertex-type radiative corrections to the transition rates can be at the level of a few percent to corresponding spontaneous rates for highly excited states in the hydrogen atom. A comprehensive analysis of the vertex-type thermal corrections for hydrogen-like atomic systems is presented., Comment: 10 pages, 2 figures, 5 tables

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.

In this paper the new type of thermal corrections for the helium and helium-like atomic systems are introduced. These are thermal one-photon exchange between the bound electrons and nucleus as well as between the bound electrons induced by the blackbody radiation (BBR). All the derivations are given within the rigorous QED theory. It is shown that these thermal corrections are the same order in powers of $\alpha$ (fine structure constant) as the well-known BBR-induced Stark shift but the different behaviour in temperature. The numerical results presented in this paper make possible to expect their significance for modern experiments and testing the fundamental interactions in helium., Comment: 5 pages, 2 figures, 3 tables

Within the framework of QED theory at finite temperature the thermal radiative corrections to spontaneous one-photon transition rates in hydrogen atom are investigated. The radiative one-loop self-energy corrections are described in the thermal case. Closed analytical expressions are derived and their numerical calculations for the spontaneous decay rate of the $ 2p $ state are carried out. Dominance of thermal radiative corrections to spontaneous Ly$_{\alpha} $ decay rate over ordinary induced transition rate up to temperatures $ T < 6000 $ K is demonstrated.

We discuss field theories appearing as a result of applying field transformations with derivatives (differential field transformations, DFT) to a known theory. We begin with some simple examples of DFTs to see the basic properties of the procedure. In this process the dynamics of the theory might either change or conserve. After that we concentrate on the theories of gravity which appear as a result of various DFT applied to general relativity, namely the mimetic gravity and Regge-Teitelboim embedding theory. We review main results related to the extension of dynamics in these theories, as well as the possibility to write down the action of a theory after DFT as the action of the original theory before DFT plus an additional term. Such a term usually contains some constraints with Lagrange multipliers and can be interpreted as an action of additional matter, which might be of use in cosmological applications, e.g. for the explanation of the effects of dark matter., Comment: 18 pages

We demonstrate that the present superaccurate measurements of transition processes between atomic states in hydrogen atom reached the limit of accuracy when transition frequency cannot be defined anymore in a unique way. This was predicted earlier and is due to the necessity to include the nonresonant corrections in the description of resonant processes. The observed spectral line profile becomes asymmetric, and it becomes impossible to extract the value of transition frequency from this profile in a unique way. Nonresonant corrections depend on the type of experiment and on the experimental arrangement. However, the line profile itself for any resonant process can be defined with any desired level of accuracy. A popular trend in modern search for atomic frequency standards and atomic clocks is the search for transitions where the nonresonant corrections are negligible. In this paper we present closed expressions for the resonant photon scattering cross sections on an atomic level with dependence on all atomic quantum numbers including fine and hyperfine structure. These expressions are given for different types of experiments with fixing of the incident (outgoing) photon propagation directions and incident (outgoing) photon polarization. Using these expressions we demonstrate that the transition frequencies in particular cases cannot be derived uniquely if the accuracy of measurement reaches the level quoted in A. Beyer {\it et al}. Our interpretation of the results of this experiment is alternative to the interpretation given by A. Beyer {\it et al}.

In present paper the Quantum Electrodynamics theory at finite temperatures for the bound states is presented. To describe the thermal effects arising in a heat bath the Hadamard form of thermal photon propagator is employed. This form permits the simple introduction of thermal gauges in a way similar to the 'ordinary' Feynman propagator and, therefore, the gauge invariance can be proved for all the considered effects. Moreover, contrary to the 'standard' form of thermal photon propagator, the Hadamard expression has a well defined analytical properties. However, this thermal photon propagator contains the divergent contribution which requires the introduction of regularization procedure within the framework of constructed theory. The method of regularization in conjunction with the physical interpretation is given in the paper. Correctness of regularization procedure is confirmed also by the gauge invariance of final results and coincidence of the results (on the example of self-energy correction) for two different forms of photon propagator. On the basis of constructed theory the thermal Coulomb potential and its asymptotics at the large distances are found. Finally, we discuss in details the thermal effects of lowest order in the fine structure constant and temperature. Such effects are presented by the thermal one-photon exchange between bound electron and nucleus, thermal one-loop self-energy, thermal vacuum polarization, recoil corrections and correction on the finite size of the nucleus. Introduction of the regularization allows us do not apply the renormalization procedure. To confirm this we describe also the thermal vertex (with one, two and three vertices) corrections within the adiabatic $S$-matrix formalism. Finally, the influence of thermal effects on the determination of proton radius and Rydberg constant is discussed in the paper., Comment: 32 pages, 10 figures, 6 Tables

We study the problem of construction of global isometric embedding for spherically symmetric black holes with negative cosmological constant in various dimensions. Firstly, we show that there is no such embedding for 4D RN-AdS black hole in 6D flat ambient space, completing the classification which we started earlier. Then we construct an explicit embedding of non-spinning BTZ black hole in 6D flat ambient space. Using this embedding as an anzats, we then construct a global explicit embedding of (d)-dimensional Schwarzschild-AdS black hole in a flat (d+3)-dimensional ambient space., Comment: 15 pages

The process of recombination for the hydrogen atom in the heat bath creating the blackbody radiation is descibed within the frameworks of quantum electrodynamics. For this purpose the self-energy for unbound electron in the field of the nucleus is considered. The imaginary part of this self-energy is directly connected with the recombination cross-section. The same procedure is applied to the hydrogen atom in the field of blackbody radiation. This leads to the new thermal correction to the process of recombination for the hydrogen atom in the heat bath. This correction takes into account the finite lifetimes of atomic levels and appears to be important for special astrophysical studies., Comment: 8 pages, 2 figures, 2 tables

This paper concerns the most intriguing question of modern atomic physics: determination of the proton root-mean-square (rms) charge radius ($r_p$). This problem was announced by the difference in $r_p$ values extracted from the experiments on measurement of transition frequencies in hydrogen ($H$) atom and measurement of Lamb shift in muonic hydrogen ($\mu H$) atom. The both experiments give the possibility for the very accurate comparison of values evaluated theoretically and experimentally. As a result of this analysis the value of proton charge radius extracted from the $\mu H$ experiment was found on $4\%$ smaller than the value given by the hydrogenic experiments. For a decade a lot of theoretical efforts were devoted to the search of 'new physics' on the basis of such deviation. In this paper we reproduce the analysis on determination of proton charge radius in the hydrogen atom., Comment: 5 pages, 1 figure, 2 tables

The paper analyzes the absorption line profile at 21 cm for the hydrogen atom in the interstellar medium. The hydrogen atom is treated as a three-level system illuminated by a powerful light source at neighboring resonances corresponding to the hyperfine splitting of the ground state and Ly$_\alpha$ transition. The field acting upon the resonances gives rise to physical processes, which can be explained as interfering pathways between different transitions. The paper considers particular cases when the 21 cm line profile is substantially modified by the Ly$_\alpha$ transition. A correction to the optical depth is introduced as a result of theory. It is shown that the correction can be considerable and should be taken into account when determining the column density of hydrogen atoms in the interstellar medium. The paper also deals with the effects of none-Doppler broadening and frequency shift., Comment: 13 pages, 3 figures, 2 tables

The precise calculations of blackbody radiation (BBR)-induced Stark shifts and depopulation rates for low-lying states of helium atom with the use of variational approach are presented. An effect of the BBR-induced induced Stark-mixing of energy levels is considered. It is shown that this effect leads to a significant reduction of lifetimes of helium excited states. As a consequence the influence of Stark-mixing effect on the decay rates of metastable states in helium is discussed in context of formation processes of the cosmic microwave background.

An effect of atomic line broadening induced by the blackbody radiation is discussed. The level mixing effect and anti-Stokes Raman scattering are compared. It is shown that the mixing effect gives the most significant contribution to the line broadening and it is indicated how to distinguish these two effects in laboratory experiments. The influence of the level mixing on the recombination history of primordial plasma is also discussed.