Finite nuclear size (FNS) effects on the magnetic-dipole hyperfine splitting in muonic hydrogenlike ions are investigated within a fully relativistic Dirac framework. The FNS contribution is quantified through the correction factor $δ$, defined by $ΔE_{\mathrm{ext}} = ΔE_{\mathrm{point}}(1 - δ)$, where $ΔE_{\mathrm{ext}}$ is evaluated using Dirac wavefunctions computed for an extended nuclear charge distribution.
Two nuclear models are considered: a homogeneously charged sphere and a two-parameter Fermi distribution. Bound-state energies and radial wavefunctions are obtained using a numerical iterative solver, while a semi-analytic matching scheme provides reference values and initial seeds. We present a systematic dataset of $δ$ values for the $1s$, $2s$, and $2p_{1/2}$ states over a wide range of nuclear charge numbers $Z$. Nuclear-model dependence is quantified, including uncertainties induced by the nuclear radius in the uniform-sphere model.
The results show that $δ$ increases monotonically with $Z$ and exhibits clear state dependence, with reduced magnitude for the $2p_{1/2}$ state relative to $s$ states. A pronounced sensitivity to the nuclear charge distribution is observed, highlighting the importance of realistic nuclear modeling in precision hyperfine studies of muonic atoms.
The hyperfine splitting in hydrogenlike $^{209}$Bi, $^{203}$Tl, and $^{205}$Tl is calculated with the nuclear magnetization determined from experimental data on the hyperfine splitting in the corresponding muonic atoms. The single-particle and configuration-mixing nuclear models are considered. The QED corrections are taken into account for both electronic and muonic atoms. The obtained results are compared with other calculations and with experiment.
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.
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…
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
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…
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.
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 splittings are determined for all five bound $S(L = 0)-$states in the three symmetric muonic molecular ions: $ppμ, ddμ$ and $ttμ$. The expectation values of all interparticle delta-functions used in our calculations have been determined in recent highly accurate computations.
Nuclear structure corrections of orders $Zα\, E_F$ and $(Zα)^2 E_F$ are calculated for the hyperfine splitting of the muonic deuterium. The obtained results disagree with previous calculations and lead to a $5\,σ$ disagreement with the current experimental value of the $2S$ hyperfine splitting in $μ$D.
The recoil, vacuum polarization and electron vertex corrections of first and second orders in the fine structure constant $α$ and the ratio of electron to muon and electron to α-particle masses are calculated in the hyperfine splitting of the $1s^{(e)}_{1/2}2s^{(μ)}_{1/2}$ state of muonic helium atom (μe ^4_2He) on the basis of a perturbation theory. We obtain total result for the muonically excited state hyperfine splitting $Δν^{hfs}=4295.66$ MHz which improves previous calculations due to the account of new corrections and more accurate treatment of the electron vertex contribution.
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 make precise calculation of hyperfine structure of $S$-states in muonic ions of lithium, beryllium and boron in quantum electrodynamics. Corrections of orders $α^5$ and $α^6$ due to the vacuum polarization, nuclear structure and recoil in first and second orders of perturbation theory are taken into account. We obtain estimates of the total values of hyperfine splittings which can be used for a comparison with future experimental data.
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.
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 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.
Non-relativisitic second-order corrections to the wave function at origin in muonic and exotic atoms are considered. The corrections are due to the electronic vacuum polarization. Such corrections are of interest due to various effective approaches, which take into account QED and hadronic effects. The wave function at origin plays a key role in the calculation of the pionium lifetime, various finite nuclear size effects and the hyperfine splitting. The results are obtained for the $1s$ and $2s$ states in pionic and muonic hydrogen and deuterium and in pionium, a bound system of $π^+$ and $π^-…
Laser Excitation of Muonic 1S Hydrogen Hyperfine Transition: Effects of Multi-pass Cell Interference
NOVA School of Science and Technology, NOVA University Lisbon（新大学科学技术学院，新里斯本大学）；PSI Center for Neutron and Muon Sciences（瑞士保罗谢勒研究所中子与μ子科学中心）；LIBPhys-UC, Department of Physics, University of Coimbra（科英布拉大学物理系LIBPhys-UC）；Institute for Particle Physics and Astrophysics, ETH Zurich（苏黎世联邦理工学院粒子物理与天体物理研究所）
The calculation of hadronic polarizability contribution of the nucleus to hyperfine structure of muonic hydrogen and helium is carried out within the unitary isobar model and experimental data on the polarized structure functions of deep inelastic lepton-proton and lepton-deuteron scattering. The calculation of virtual absorption cross sections of transversely and longitudinally polarized photons by nucleons in the resonance region is performed in the framework of the program MAID.
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 calculate hyperfine structure intervals $ΔE^{hfs}(2P_{1/2})$ and $ΔE^{hfs}(2P_{3/2})$ for P-states in muonic ions of lithium, beryllium and boron. To construct the particle interaction operator in momentum space we use the tensor method of projection operators on states with definite quantum numbers of total atomic momentum F and total muon momentum j. We take into account vacuum polarization, relativistic, quadruple and structure corrections of orders $α^4$, $α^5$ and $α^6$. The obtained numerical values of hyperfine splittings can be used for a comparison with future experimental data of …
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…
The hyperfine structures of the ground states in the ${}^{3}$He$^{2+} μ^{-} e^{-}$ and ${}^{4}$He$^{2+} μ^{-} e^{-}$ helium-muonic atoms are investigated with the use of highly accurate variational wave functions. The differences between corresponding levels of hyperfine structure (i.e. hyperfine splittings) are determined to very high numerical accuracy. In particular, we have found that the hyperfine structure splitting in the ground state of the ${}^{4}$He$^{2+} μ^{-} e^{-}$ atom is $Δν\approx 4464.55454(3)$ $MHz$, while analogous splitting for the ${}^{3}$He$^{2+}μ^{-}e^{-}$ atom is $Δν\ap…
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.
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.
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.
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 …
On the basis of the perturbation theory in the fine structure constant $α$ and the ratio of the electron to muon masses we calculate one-loop vacuum polarization and electron vertex corrections and the nuclear structure corrections to the hyperfine splitting of the ground state of muonic helium atom $(μ e \ ^3_2He)$. We obtain total result for the ground state hyperfine splitting $Δν^{hfs}=4166.471$ MHz which improves the previous calculation of Lakdawala and Mohr due to the account of new corrections of orders $α^5$ and $α^6$. The remaining difference between our theoretical result and experi…
The hyperfine splitting of the ground and low-energy $3/2^+(7.8 \pm 0.5$ eV) levels in the $^{229}$Th nucleus in muonic atom ($μ^-_{1S_{1/2}}{}^{229}$Th$)^*$ has been calculated considering the distribution of the nuclear magnetization in the framework of collective nuclear model with the wave functions of the Nilsson model for the unpaired neutron. It is shown that (a) the hyperfine splitting of the $3/2^+$ isomeric state is anomalously weak, and the reduction of hyperfine structure in comparison with the model of a point nuclear magnetic dipole is close to 100\%, (b) partial inversion of lev…
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…
Probing a Fifth Force in Muonic Atoms through Lamb Shifts and Hyperfine Structure
Hebei University（河北大学）；College of Physics Science and Technology, Hebei University（河北大学物理科学与技术学院）；Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所）；Nanjing University（南京大学）；School of Physics, Nanjing University（南京大学物理学院）；RIKEN Nishina Center, RIKEN（理化学研究所西村中心）；University of Chinese Academy of Sciences（中国科学院大学）；School of Nuclear Science and Technology, University of Chinese Academy of Sciences（中国科学院大学核科学与技术学院）；Southern Center for Nuclear Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所南方核科学理论中心）
Weak interaction contributions to hyperfine splitting and Lamb shift in light electronic and muonic atoms are calculated. We notice that correction to hyperfine splitting turns into zero for deuterium. Weak correction to the Lamb shift in hydrogen is additionally suppressed in comparison with other cases by a small factor $(1-4\sin^2θ_W)$.
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 …
The finite-nuclear size correction to the fine structure of muonic atoms are considered. The procedure for the analytical calculation of the energies and wave functions has been derived in a homogeneously charged sphere nuclear charge distribution approximation. The finite-nuclear size effect was calculated in a first few orders of the perturbation energy, with the accurate estimations of the convergence. Finally, we present energies of the low-lying electronic and muonic states with the finite-nuclear size correction, calculated analytically, for ${}^{185}_{75}\text{Re}$ ion.
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…
High-precision laser spectroscopy of atomic energy levels enables the measurement of nuclear properties. Sensitivity to these properties is particularly enhanced in muonic atoms which are bound systems of a muon and a nucleus. Exemplary is the measurement of the proton charge radius from muonic hydrogen performed by the CREMA collaboration which resulted in an order of magnitude more precise charge radius as extracted from other methods but at a variance of 7 standard deviations. Here, we summarize the role of muonic atoms for the extraction of nuclear charge radii, we present the status of th…
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 …
The present knowledge of Lamb shift, fine-, and hyperfine structure of the 2S and 2P states in muonic helium-3 ions is reviewed in anticipation of the results of a first measurement of several $\mathrm{2S\rightarrow2P}$ transition frequencies in the muonic helium-3 ion, $\mathrm{μ^3He^+}$. This ion is the bound state of a single negative muon $μ^-$ and a bare helium-3 nucleus (helion), $\mathrm{^3He^{++}}$. A term-by-term comparison of all available sources, including new, updated, and so far unpublished calculations, reveals reliable values and uncertainties of the QED and nuclear structure-d…
A model for description of nonresonant formation of the muonic molecules in collisions of the muonic hydrogen atoms with the hydrogenic molecules has been developed with taking into account the internal motion of all nuclei. It has been shown that such a motion leads to a significant smearing of the calculated energy-dependent formation rates at low collision energies. In particular, this effect is strong in the $ddμ$ and $dtμ$ formation. An appreciable isotopic effect in the case of nonresonant $ddμ$ formation in $dμ$ collisions with the molecules D$_2$ and HD has been found. All these effect…
Two recent experimental observations at the ATOMKI Institute of the Hungarian Academy of Sciences (regarding the angular emission pattern of electron-positron pairs from nuclear transitions from excited states in 8Be and 4He) indicate the possible existence of a particle of a rest mass energy of roughly 17 MeV. The so-called X17 particle constitutes a virtual state in the process, preceding the emission of the electron-positron pair. Based on the symmetry of the nuclear transitions ($1^+$ to $0^+$ and $0^-$ to $0^+$), the X17 could either be a vector, or a pseudoscalar particle. Here, we calcu…
The diffusion radius of the 1S muonic hydrogen atoms in gaseous H_2 targets with various deuterium admixtures has been determined for temperatures T=30 and 300 K. The Monte Carlo calculations have been performed using the partial differential crosssections for $pμ$ and $dμ$ atom scattering from the molecules H$_2$, HD and D$_2$. These cross sections include hyperfine transitions in the muonic atoms, the muon exchange between the nuclei $p$ and $d$, and rotational-vibrational transitions in the target molecules. The Monte Carlo results have been used for preparing the time-projection chamber fo…
The first fully relativistic, rigorous QED calculations of the self-energy correction to the fine-structure levels of heavy muonic atoms are reported. We discuss nuclear model and parameter dependence for this contribution as well as numerical convergence issues. The presented results show sizable disagreement with previously reported estimations, including ones used for the determination of the nuclear root-mean-square radii, and underline the importance of rigorous QED calculations for the theoretical prediction of the spectra of muonic atoms.
We evaluate the two-photon exchange corrections to the Lamb shift and hyperfine splitting of S states in electronic hydrogen relying on modern experimental data and present the two-photon exchange on a neutron inside the electronic and muonic atoms. These results are relevant for the precise extraction of the isotope shift as well as in the analysis of the ground state hyperfine splitting in usual and muonic hydrogen.
Nuclear structure corrections to energy levels of light muonic atoms are derived with particular attention to the nuclear mass dependence. The obtained result for the 2P-2S transition of 1.717(19) meV serves for determination of the nuclear charge radius from the spectroscopic measurement in muonic deuterium.
Based on the results of accurate variational calculations we demonstrate stability of the five-body negatively charged ions $a^{+} b^{+} μ^{-} e^{-} e^{-}$. Each of these five-body ions contains two electrons $e^{-}$, one negatively charged muon $μ^{-}$ and two nuclei of the hydrogen isotopes $a, b = (p, d, t)$. The bound state properties of these five-body ions, including their hyperfine structure, are briefly discussed. We also investigate the hyperfine structure of the ground states of the four-body muonic quasi-atoms $a^{+} b^{+} μ^{-} e^{-}$. In particular, we determine the hyperfine stru…
Precision spectroscopy of light muonic atoms provides unique information about the atomic and nuclear structure of these systems and thus represents a way to access fundamental interactions, properties and constants. One application comprises the determination of absolute nuclear charge radii with unprecedented accuracy from measurements of the 2S$\,$-$\,$2P Lamb shift. Here, we review recent results of nuclear charge radii extracted from muonic hydrogen and helium spectroscopy and present experiment proposals to access light muonic atoms with $Z \geq 3$. In addition, our approaches towards a …
High-harmonic generation from hydrogenlike muonic atoms exposed to ultraintense high-frequency laser fields is calculated. Systems of low nuclear charge number Z are considered where a nonrelativistic description applies. By comparing the radiative response for different isotopes we demonstrate characteristic signatures of the finite nuclear mass and size in the harmonic spectra. In particular, for Z>1, an effective muon charge appears in the Schrödinger equation for the relative particle motion, which influences the position of the harmonic cutoff. Cutoff energies in the MeV domain can be ach…
High-harmonic generation from muonic atoms exposed to intense laser fields is considered. Our particular interest lies in effects arising from the finite nuclear mass and size. We numerically perform a fully quantum mechanical treatment of the muon-nucleus dynamics by employing modified soft-core and hard-core potentials. It is shown that the position of the high-energy cutoff of the harmonic spectrum depends on the nuclear mass, while the height of the spectral plateau is sensitive to the nuclear radius. We also demonstrate that $γ$-ray harmonics can be generated from muonic atoms in ultrastr…
