Theoretical investigations of the hyperfine structure of the hydrogen molecular ion (one electron and two protons) are discussed. The nuclear spin-rotation interaction has been found to be of the same sign as in the hydrogen molecule and the hyperfine transition frequencies can be accurately predicted. With measurements of the hyperfine structure of the deuterium molecular ion (or of HD+) it should be possible to obtain a value of the deuteron quadrupole moment that could be compared with the values obtained from the deuterium molecule and from nuclear theory.
The one-loop self-energy correction to the hyperfine structure splitting of the 1s and 2s states of hydrogenlike ions is calculated both for the point and finite nucleus. The results of the calculation are combined with other corrections to find the ground state hyperfine splitting in lithiumlike ^{209}Bi^{80+} and ^{165}Ho^{64+}.
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 have applied an optical method to the measurement of the 2S hyperfine interval in atomic hydrogen. The interval has been measured by means of two-photon spectroscopy of the 1S-2S transition on a hydrogen atomic beam shielded from external magnetic fields. The measured value of the 2S hyperfine interval is equal to 177 556 860(15) Hz and represents the most precise measurement of this interval to date. The theoretical evaluation of the specific combination of 1S and 2S hyperfine intervals D_21 is in moderately good agreement with the value for D_21 deduced from our measurement.
The 7 standard deviations between the proton rms charge radius from muonic hydrogen and the CODATA-10 value from hydrogen spectroscopy and electron-scattering has caused considerable discussions. Here, we review the theory of the 2S-2P Lamb shift and 2S hyperfine splitting in muonic hydrogen combining the published contributions and theoretical approaches. The prediction of these quantities is necessary for the determination of both proton charge and Zemach radii from the two 2S-2P transition frequencies measured in muonic hydrogen.
The current status of the determination of corrections to the hyperfine splitting of the ground state in hydrogen is considered. Improved calculations are provided taking into account the most recent value for the proton charge radius. Comparing experimental data with predictions for the hyperfine splitting, the Zemach radius of the proton is deduced to be $1.045(16)$ fm. Employing exponential parametrizations for the electromagnetic form factors we determine the magnetic radius of the proton to be $0.778(29)$ fm. Both values are compared with the corresponding ones derived from the data obtai…
The usefulness of study of hyperfine splitting in the hydrogen atom is limited on a level of 10 ppm by our knowledge of the proton structure. One way to go beyond 10 ppm is to study a specific difference of the hyperfine structure intervals 8 Delta nu_2 - Delta nu_1. Nuclear effects for are not important this difference and it is of use to study higher-order QED corrections.
The largest contributions to the $n=2$ Lamb-shift, fine structure interval and $2s$ hyperfine structure of muonic hydrogen are calculated by exact numerical evaluations of the Dirac equation, rather than by a perturbation expansion in powers of $1/c$, in the framework of non-relativistic quantum electrodynamics. Previous calculations and the validity of the perturbation expansion for light elements are confirmed. The dependence of the various effects on the nuclear size and model are studied
The ground state hyperfine splitting values of high Z hydrogenlike ions are calculated. The relativistic, nuclear and QED corrections are taken into account. The nuclear magnetization distribution correction (the Bohr-Weisskopf effect) is evaluated within the single particle model with the g_{S}-factor chosen to yield the observed nuclear moment. An additional contribution caused by the nuclear spin-orbit interaction is included in the calculation of the Bohr-Weisskopf effect. It is found that the theoretical value of the wavelength of the transition between the hyperfine splitting components …
We consider a simplified model of the optical multi-pass cavity that is being currently developed by the FAMU collaboration for the measurement of the hyperfine splitting in the ground state of muonic hydrogen and of the Zemach radius of the proton. The model is focused on the time distribution of the events of laser-stimulated hyperfine transitions in the muonuc atom and may be helpful in the preliminary design of the FAMU experimental set-up and, more generally, in the optimization of multi-pass optical cavities for experiments with pulsed lasers.
NRQED approach to the fine and hyperfine structure corrections of order m$α$ 6 and m$α$ 6 (m/M)-Application to the hydrogen atom The NRQED approach is applied to the calculation of relativistic corrections to the fine and hyperfine structure of hydrogenlike atoms at orders m$α$ 6 and m$α$ 6 (m/M). Results are found to be in agreement with those of the relativistic theory. This confirms that the derived NRQED effective potentials are correct, and may be used for studying more complex atoms or molecules. Furthermore, we verify the equivalence between different forms of the NRQED Lagrangian used …
An optical measurement of the 2S hyperfine interval in atomic hydrogen using two-photon spectroscopy of the 1S-2S transition gives a value of 177 556 834.3(6.7) Hz. The uncertainty is 2.4 times smaller than achieved by our group in 2003 and more than 4 times smaller than for any independent radio-frequency measurement. The specific combination of the 2S and 1S hyperfine intervals predicted by QED theory $D_{21}=8 f_{\rm HFS}({2S}) - f_{\rm HFS}({1S})=48 953(3)$ Hz is in good agreement with the value of 48 923(54) Hz obtained from this experiment.
The antihydrogen programme of the ASACUSA collaboration at the antiproton decelerator of CERN focuses on Rabi-type measurements of the ground-state hyperfine splitting of antihydrogen for a test of the combined Charge-Parity-Time symmetry. The spectroscopy apparatus consists of a microwave cavity to drive hyperfine transitions and a superconducting sextupole magnet for quantum state analysis via Stern-Gerlach separation. However, the small production rates of antihydrogen forestall comprehensive performance studies on the spectroscopy apparatus. For this purpose a hydrogen source and detector …
We revisit the $m α^6 (m/M)$ order corrections to the hyperfine splitting in the H$_2^+$ ion, and find a hitherto unrecognized second-order relativistic contribution associated with the vibrational motion of the nuclei. Inclusion of this correction term produces theoretical predictions which are in excellent agreement with experimental data [K. B. Jefferts, Phys.\ Rev.\ Lett.\ \textbf{23}, 1476 (1969)], thereby concluding a nearly fifty years long theoretical quest to explain the experimental results within their 1-ppm error. The agreement between theory and experiment corroborates the proton …
The purpose of this article is twofold. First, we attempt to give a brief overview of the different application areas of quantum electrodynamics (QED). These include fundamental physics (prediction of atomic energy levels), where the atom may be exposed to additional external fields (hyperfine splitting and g factor). We also mention QED processes in highly intense laser fields, and more applied areas like Casimir and Casimir--Polder interactions. Both the unifying aspects as well as the differences in the the theoretical treatment required by these application areas (such as the treatment of …
The hyperfine structure (hfs) and the g factor of a bound electron are caused by external magnetic fields. For the hfs, the magnetic field is due to the nuclear spin. A uniform-in-space and constant-in-time magnetic field is used to probe the bound-electron g factor. The self-energy corrections to these effects are more difficult to evaluate than those to the Lamb shift. Here, we describe a numerical approach for both effects in the notoriously problematic regime of hydrogen-like bound systems with low nuclear charge numbers. The calculation is nonperturbative in the binding Coulomb field. Acc…
This study addresses the effect of the magnetic hyperfine interaction on the relativistic H1s wave functions. These are used to calculate the electric, magnetic, and confinement force densities acting on the 1s electron. The magnetic field couples Dirac equations for different angular momenta. These are solved numerically for the hyperfine singlet and triplet, as well as for a classical magnetic dipole. In the singlet ground state the hyperfine interaction shifts the electron density toward the proton. A similar shift is found for the classical dipole, and an opposite shift for the triplet. Th…
We find a new correction to the hydrogen atom ground state hyperfine energy levels splitting in magnetic field. It can be interpreted as magnetic focusing of the wave function at the origin. The effect might be within the reach of experiment.
We present tables for the bound-state energies for atomic hydrogen. The tabulated energies include the hyperfine structure, and thus this work extends the work of Rev. Mod. Phys. {\bf 84}, 1527 (2012), which excludes hyperfine structure. The tabulation includes corrections of the hyperfine structure due to the anomalous moment of the electron, due to the finite mass of the proton, and due to off-diagonal matrix elements of the hyperfine Hamiltonian. These corrections are treated incorrectly in most other works. Simple formulas valid for all quantum numbers are presented for the hyperfine corre…
We study the proton-hydrogen collision in the energy range from 0 to 5 K where the hyperfine structure of the hydrogen atom is important. The proper multichannel treatment of the hyperfine structure is found to be crucial at cold temperatures compared to the elastic approximation used for higher temperatures. Both elastic and hyperfine-changing inelastic processes are investigated using the newly developed multichannel quantum-defect theory (MQDT) as well as the traditional coupled-channel numerical method. Cross section results from the two methods coincide well throughout the energy range. T…
The ASACUSA collaboration is developing a polarized beam of antihydrogen atoms to precisely determine the ground-state hyperfine structure for studies of CPT and Lorentz invariance. Using a beam of ordinary hydrogen, measurements of both the $σ$ and $π$-transition have been performed, investigating orientation-dependent SME coefficients. Furthermore a first hyperfine experiment with a beam of deuterium is being prepared.
The fully relativistic theory of the Zeeman splitting of the $1s$ hyperfine structure levels in hydrogenlike ions is considered for the magnetic field magnitude in the range from 1 to 10 T. The second-order corrections to the Breit -- Rabi formula are calculated and discussed. The results can be used for a precise determination of nuclear magnetic moments from $g$ factor experiments.
We consider the most accurate tests of bound state QED theory of the hyperfine splitting in two-body atoms related to the HFS interval of the 1s state in muonium and positronium and the 2s state in hydrogen, deuterium and the helium-3 ion. We summarize their QED theory and pay special attention to involved effects of strong interactions and to recent optical measurements of the 2s HFS interval in hydrogen and deuterium. We present results for specific ratios of the 1s-2s frequencies in hydrogen and deuterium which happen to be among the most accurately measured and calculated quantities.
The prospects of tests of CPT symmetry using precision spectroscopy of antihydrogen are discussed with special emphasis on the ground-state hyperfine structure, a measurement of which is the aim of the ASACUSA collaboration at the AD/ELENA facility of CERN. Ongoing parallel experiments using hyperfine spectroscopy of hydrogen and deuterium aiming at studying Lorentz invariance by determining coefficients of the Standard Model Extension framework are described.
Antihydrogen, the lightest atom consisting purely of antimatter, is an ideal laboratory to study the CPT symmetry by comparison to hydrogen. With respect to absolute precision, transitions within the ground-state hyperfine structure (GS-HFS) are most appealing by virtue of their small energy separation. ASACUSA proposed employing a beam of cold antihydrogen atoms in a Rabi-type experiment to determine the GS-HFS in a field-free region. Here we present a measurement of the zero-field hydrogen GS-HFS using the spectroscopy apparatus of ASACUSA's antihydrogen experiment. The measured value of $ν_…
The (3p - 1s) X-ray transition to the muonic hydrogen ground state was measured with a high resolution crystal spectrometer. A Doppler effect broadening of the X-ray line was established which could be attributed to different Coulomb de-excitation steps preceding the measured transition. The assumption of a statistical population of the hyperfine levels of the muonic hydrogen ground state was directly confirmed by the experiment and measured values for the hyperfine splitting can be reported. The results allow a decisive test of advanced cascade model calculations and establish a method to ext…
The contribution from the axial-vector meson exchange to the potential of the muon-proton interaction in muonic hydrogen induced by anomalous axial-vector meson coupling to two photon state is calculated. It is shown that such contribution to the hyperfine splitting in muonic hydrogen is large and important for a comparison with precise experimental data. In the light of our result, the proton radius "puzzle" is discussed.
We have investigated the correction to the hyperfine structure of heavy multicharged ions, which is connected with the nuclear-polarization effect caused by the unpaired bound electron. Numerical calculations are performed for hydrogenlike ions taking into account the dominant collective nuclear excitations. The correction defines the ultimate limit of precision in accurate theoretical predictions of the hyperfine-structure splittings.
We present the leading-order prediction of baryon chiral perturbation theory for the proton polarizability contribution to the 2S hyperfine splitting in muonic hydrogen, and compare with the results of dispersive calculations.
Relativistic self-consistent-field calculations of the radial hyperfine integrals have been performed in the 4d-shell element ions. The comparison with available experimental results gives an estimate of configuration interaction effects in the hyperfine interaction in these ions. The results can also be used to derive nuclear moments from laserspectroscopic measurements of radioactive isotopes. PACS Numbers: 31.15.Ne, 31.30.Gs, 32.10.Fn
The ASACUSA collaboration at the Antiproton Decelerator of CERN aims at a precise measurement of the antihydrogen ground-state hyperfine structure as a test of the fundamental CPT symmetry. A beam of antihydrogen atoms is formed in a CUSP trap, undergoes Rabi-type spectroscopy and is detected downstream in a dedicated antihydrogen detector. In parallel measurements using a polarized hydrogen beam are being performed to commission the spectroscopy apparatus and to perform measurements of parameters of the Standard Model Extension (SME). The current status of antihydrogen spectroscopy is reviewe…
We present a precise determination of the polarizability and other proton structure dependent contributions to the hydrogen hyperfine splitting, based heavily on the most recent published data on proton spin dependent structure functions from the EG1 experiment at the Jefferson Laboratory. As a result, the total calculated hyperfine splitting now has a standard deviation slightly under 1 part-per-million, and is about 1 standard deviation away from the measured value. We also present results for muonic hydrogen hyperfine splitting, taking care to ensure the compatibility of the recoil and pola…
By solving the two-body Schordinger equation in a very high precise nonperturbative numerical (NPnum) way, we reexamine the contributions of fine, hyperfine structure splittings of muonic hydrogen based on the Breit potential. The comparison of our results with those by the first order perturbative theory ($^{1st}$PT) in the literature shows, when the structure of proton is considered, the differences between the results by the $^{1st}$PT and NPnum methods are small for the fine and hyperfine splitting of $2P$ state, while are about $-0.009$ meV and $0.08$ meV for the $F=1$ and total hyperfine…
We show that the non-spherical charge distribution of the proton manifests itself in hydrogen hyperfine splitting as an increase (in absolute value) of the proton Zemach radius and polarization contributions.
We present the derivation of the formulas for the proton structure-dependent terms in the hyperfine splitting of muonic hydrogen. We use compatible conventions throughout the calculations to derive a consistent set of formulas that reconcile differences between our results and some specific terms in earlier work. Convention conversion corrections are explicitly presented, which reduce the calculated hyperfine splitting by about 46 ppm. We also note that using only modern fits to the proton elastic form factors gives a smaller than historical spread of Zemach radii and leads to a reduced uncert…
We present a high-precision calculation of the recoil--finite-size correction to the hyperfine splitting (HFS) in muonic and electronic hydrogen based on nucleon electromagnetic form factors obtained from dispersion theory. This will help guide the upcoming searches of the HFS transition in muonic hydrogen, and will allow a precise determination of the polarizability and Zemach radius contributions when this transition is found.
State-dependent quantum electrodynamic corrections are evaluated for the hyperfine splitting of nS states for arbitrary principal quantum number n. The calculations comprise both the self-energy and the vacuum-polarization correction of order alpha(Z alpha)^2 E_F and the recoil correction of order (Z alpha)^2 (m/M) E_F. Higher-order corrections are summarized and partly reevaluated as well. Accurate predictions for hydrogen hyperfine splitting intervals of nS states with n=2,...,8 are presented. The results obtained are important due to steady progress in hydrogen spectroscopy for transitions …
The mα^6(m/M) order corrections to the hyperfine splitting in the H_2^+ ion are calculated. That allows to reduce uncertainty in the frequency intervals between hyperfine sublevels of a given rovibrational state to about 10 ppm. Results are in good agreement with the high precision experiment carried out by Jefferts in 1969.
We provide an accurate evaluation of the two-photon exchange correction to the hyperfine splitting of S energy levels in muonic hydrogen exploiting the corresponding measurements in electronic hydrogen. The proton structure uncertainty in the calculation of $α^5$ contribution is sizably reduced.
Hyperfine structures of the triplet $n^3S-$states in the four-electron Be-atom(s) and Be-like ions are considered. It is shown that to determine the hyperfine structure splitting in such atomic systems one needs to know the triplet electron density at the central atomic nucleus $ρ_T(0)$. We have developed the procedure which allows allows one to determine such an electron density $ρ_T(0)$ for arbitrary four-electron atoms and ions.
The interaction of two excited hydrogen atoms in metastable states constitutes a theoretically interesting problem because of the quasi-degenerate 2P_{1/2} levels which are removed from the 2S states only by the Lamb shift. The total Hamiltonian of the system is composed of the van der Waals Hamiltonian, the Lamb shift and the hyperfine effects. The van der Waals shift becomes commensurate with the 2S-2P_{3/2} fine-structure splitting only for close approach (R < 100 a_0, where a_0 is the Bohr radius) and one may thus restrict the discussion to the levels with n=2 and J=1/2 to good approximati…
Relativistic and QED corrections are calculated for the hyperfine splitting (hfs) in the $2S_{1/2}$ ground state of $^{9}$Be$^+$ ions with an exact account for electronic correlations. The achieved accuracy is sufficient to determine the finite nuclear size effects from the comparison to the experimental hfs value. The obtained results establish the ground to determine the neutron halo in $^{11}$Be.
In this paper we study the influence of electron screening on the Bohr-Weisskopf (BW) effect in many-electron atoms. The BW effect gives the finite-nucleus magnetization contribution to the hyperfine structure. Relativistic atomic many-body calculations are performed for s and p_1/2 states of several systems of interest for studies of atomic parity violation and time-reversal-violating electric dipole moments -- Rb, Cs, Fr, Ba+, Ra+, and Tl. For s states, electron screening effects are small, and the relative BW correction for hydrogenlike ions and neutral atoms is approximately the same. We r…
The hyperfine structure (HFS) of a bound electron is modified by the self-interaction of the electron with its own radiation field. This effect is known as the self-energy correction. In this work, we discuss the evaluation of higher-order self-energy corrections to the HFS of bound P states. These are expressed in a semi-analytic expansion involving powers of Zalpha and ln(Zalpha), where Z is the nuclear charge number and alpha is the fine-structure constant. We find that the correction of relative order alpha (Zalpha)^2 involves only a single logarithm ln(Zalpha) for P_1/2 states [but no ter…
We reevaluate the Zemach, recoil and polarizability corrections to the hyperfine splitting in muonic hydrogen expressing them through the low-energy proton structure constants and obtain the precise values of the Zemach radius and two-photon exchange (TPE) contribution. The uncertainty of TPE correction to S energy levels in muonic hydrogen of 105 ppm exceeds the ppm accuracy level of the forthcoming 1S hyperfine splitting measurements at PSI, J-PARC and RIKEN-RAL.
In the framework of the quasipotential method in quantum electrodynamics we calculate corrections to the nuclear structure proportional to $r_N^2$ from two-photon exchange amplitudes in the fine and hyperfine structure of P-states in muonic hydrogen, as well as the photon-photon interaction amplitudes, leading to the exchange of the axial vector meson. In constructing the quasipotential of the muon-nucleus interaction, we use the method of projection operators on states of two particles with a definite spin and total angular momentum. Analytical calculation of the matrix elements is performed …
A complete effective Hamiltonian for relativistic corrections at orders $mα^6$ and $mα^6(m/M)$ in a one-electron molecular system is derived from the NRQED Lagrangian. It includes spin-independent corrections to the energy levels and spin-spin scalar interactions contributing to the hyperfine splitting, both of which had been studied previously. In addition, corrections to electron spin-orbit and spin-spin tensor interactions are newly obtained. This allows improving the hyperfine structure theory in the hydrogen molecular ions. Improved values of the spin-orbit hyperfine coefficient are calcu…
We investigate the influence of the spatial extent of the proton magnetization and charge densities on the 2S hyperfine splitting in muonic hydrogen. The use of a non-perturbative relativistic Dirac approach leads to corrections of 15% to values obtained from the perturbative treatment encapsulated by the Zemach radius, which surpass the next-leading order contribution in the perturbation series by an order of magnitude.
The CREMA collaboration is pursuing a measurement of the ground-state hyperfine splitting (HFS) in muonic hydrogen ($μ$p) with 1 ppm accuracy by means of pulsed laser spectroscopy to determine the two-photon-exchange contribution with $2\times10^{-4}$ relative accuracy. In the proposed experiment, the $μ$p atom undergoes a laser excitation from the singlet hyperfine state to the triplet hyperfine state, {then} is quenched back to the singlet state by an inelastic collision with a H$_2$ molecule. The resulting increase of kinetic energy after the collisional deexcitation is used as a signature …
We study the hydrogen and muonic hydrogen within an effective field theory framework. We perform the matching between heavy baryon effective theory coupled to photons and leptons and the relevant effective field theory at atomic scales. This matching can be performed in a perturbative expansion in alpha, 1/m_p and the chiral counting. We then compute the O(m_{l_i}^3 alpha^5/m_p^2 x logarithms) contribution (including the leading chiral logarithms) to the Hyperfine splitting and compare with experiment. They can explain about 2/3 of the difference between experiment and the pure QED prediction …
Energy levels, hyperfine interaction constants, and Landé $g_J$-factors are reported for n=2 states in beryllium-, boron-, carbon-, and nitrogen-like ions from relativistic configuration interaction calculations. Valence, core-valence, and core-core correlation effects are taken into account through single and double-excitations from multireference expansions to increasing sets of active orbitals. A systematic comparison of the calculated hyperfine interaction constants is made with values from the available literature.
We obtain a model-independent prediction for the two-photon exchange contribution to the hyperfine splitting in muonic hydrogen. We use the relation of the Wilson coefficients of the spin-dependent dimension-six four-fermion operator of NRQED applied to the electron-proton and to the muon-proton sectors. Their difference can be reliably computed using chiral perturbation theory, whereas the Wilson coefficient of the electron-proton sector can be determined from the hyperfine splitting in hydrogen. This allows us to give a precise model-independent determination of the Wilson coefficient for th…
The hyperfine structure of the long-lived $5D_{3/2}$ and $5D_{5/2}$ levels of Ba$^+$ ion is analyzed. A procedure for extracting relatively unexplored nuclear magnetic moments $Ω$ is presented. The relevant electronic matrix elements are computed in the framework of the ab initio relativistic many-body perturbation theory. Both the first- and the second-order (in the hyperfine interaction) corrections to the energy levels are analyzed. It is shown that a simultaneous measurement of the hyperfine structure of the entire $5D_J$ fine-structure manifold allows one to extract $Ω$ without contaminat…
We report on the status of the in-beam hyperfine-structure measurements on ground-state antihydrogen by ASACUSA and on recent results obtained in supporting measurements from hydrogen. The $σ_1$ and $π_1$ transitions can now be investigated, which is beneficial from both theoretical and experimental perspectives. We discuss systematic effects from resonance interference originating from the chosen field geometries in the interaction region, and how their impact can be managed by appropriate data-taking or design concepts.
