The muonic hydrogen experiment measuring the $2P-2S$ transition energy [R. Pohl {\em et al.}, Nature {\bf 466}, 213 (2010)] is significantly discrepant with theoretical predictions based on quantum electrodynamics. A possible approach to resolve this conundrum is to compare experimental values with theoretical predictions in another system, muonic deuterium $μ$D. The only correction which might be questioned in $μ$D is that due to the deuteron polarizability. We investigate this effect in detail and observe cancellation with the elastic contribution. The total value obtained for the deuteron structure correction in the $2P-2S$ transition is 1.680(16) meV.
The reduced-mass dependence of relativistic and radiative effects in simple muonic bound systems is investigated. The spin-dependent nuclear recoil correction of order (Zalpha)^4 mu^3/m_N^2 is evaluated for muonic hydrogen and deuterium, and muonic helium ions (mu is the reduced mass and m_N is the nuclear mass). Relativistic corrections to vacuum polarization of order alpha (Zalpha)^4 mu are calculated, with a full account of the reduced-mass dependence. The results shift theoretical predictions. The radiative-recoil correction to vacuum polarization of order alpha (Zα)^5 ln^2(Zalpha) mu^2/m_…
Contributions to the energy levels in light muonic atoms and, in particular, to the Lamb shift fall into a few well-distinguished classes. The related diagrams are calculated using different approaches. In particular, there is a specific kind of non-relativistic contributions. Here we consider such corrections to the Lamb shift in order $α^5m_μ$. These contributions are due to free vacuum polarization loops as well as to various effects of light-by-light scattering. The closed loop in the related diagrams is an electronic one, which allows a non-relativistic consideration of the muon. Both kin…
We calculate the corrections of orders alpha^3, alpha^4 and alpha^5 to the Lamb shift of the 1S and 2S energy levels of muonic hydrogen (mu p) and muonic deuterium (mu d). The nuclear structure effects are taken into account in terms of the proton r_p and deuteron r_d charge radii for the one-photon interaction and by means of the proton and deuteron electromagnetic form factors in the case of one-loop amplitudes. The obtained numerical value of the isotope shift (mu d) - (mu p) for the splitting (1S-2S) 101003.3495 meV can be considered as a reliable estimation for corresponding experiment wi…
Hadronic deuteron polarizability correction to the Lamb shift of muonic deuterium is calculated on the basis of unitary isobar model and modern experimental data on the structure functions of deep inelastic lepton-deuteron scattering and their parameterizations in the resonance and nonresonance regions.
On the basis of quasipotential approach to the bound state problem in quantum electrodynamics we calculate hyperfine structure intervals Delta E^{hfs}(2P_{1/2}) and Delta E^{hfs}(2P_{3/2}) for P-states in muonic deuterium. The tensor method of projection operators for the calculation of the hyperfine structure of P-states with definite quantum numbers of total atomic momentum F and total muon momentum j in muonic deuterium is formulated. We take into account vacuum polarization, relativistic, quadrupole and structure corrections of orders alpha^4, alpha^5 and alpha^6. The obtained numerical va…
On the basis of quasipotential approach in quantum electrodynamics we calculate vacuum polarization and quadrupole corrections in first and second orders of perturbation theory in hyperfine structure of P-states in muonic deuterium. All corrections are presented in integral form and evaluated analytically and numerically. The obtained results can be used for the improvement of the transition frequencies between levels 2P and 2S.
The one-loop nuclear structure corrections of order (Z alpha)^5 to the Lamb shift and hyperfine splitting of the deuterium are calculated. The contribution of the deuteron structure effects to the isotope shift (ep)-(ed), (mu p)-(mu d) in the interval (1S - 2S) is obtained on the basis of modern experimental data on the deuteron electromagnetic form factors. The comparison with the similar contributions to the Lamb shift for electronic and muonic hydrogen shows, that the relative contribution due to the nucleus structure increases when passing from the hydrogen to the deuterium.
A recent letter [1] investigated the effects of deuteron polarizability in the mu D atom. The importance of understanding this atom has been heightened because of the possible light it could shed on the proton radius puzzle. Here we confirm, clarify, and emphasize the significance of the result [1] that the effects of the third Zemach moment are canceled by part of the deuteron polarizability effects. 1. K. Pachucki, Phys. Rev. Lett. 106, 193007 (2011)
We present new investigation of the Lamb shift (2P_{1/2}-2S_{1/2}) in muonic deuterium (mu d) atom using the three-dimensional quasipotential method in quantum electrodynamics. The vacuum polarization, nuclear structure and recoil effects are calculated with the account of contributions of orders alpha^3, alpha^4, alpha^5 and alpha^6. The results are compared with earlier performed calculations. The obtained numerical value of the Lamb shift 202.4139 meV can be considered as a reliable estimate for the comparison with forthcoming experimental data.
On the basis of quasipotential method in quantum electrodynamics we calculate corrections of order $α^5$ and $α^6$ to hyperfine structure of S-wave energy levels of muonic deuterium. Relativistic corrections, effects of vacuum polarization in first, second and third orders of perturbation theory, nuclear structure and recoil corrections are taken into account. The obtained numerical values of hyperfine splitting $ΔE^{hfs}(1S)=50.2814$ meV (1S state) and $ΔE^{hfs}(2S)=6.2804$ meV (2S state) represent reliable estimate for a comparison with forthcoming experimental data of CREMA collaboration. T…
We provide a systematic assessment of the order-$α^5$ nuclear contributions to the Lamb shift of muonic deuterium, including the accompanying radiative corrections due to vacuum polarization, up to next-to-next-to-next-to-leading order (N3LO) within the pionless effective field theory (EFT). We also evaluate higher-order corrections due to the single-nucleon structure, which are expected to be the most important corrections beyond N3LO. We find a correlation between the deuteron charge and Friar radii, which can be useful to judge the quality of charge form factor parametrisations. We refine t…
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 $(1-4\sin^2θ_W)$.
From the comparison between the experimental and theoretical results for the energy difference between 2s1/2 and 2p1/2 states in hydrogen and hydrogen-like ions with different values of Z, we estimate bounds for the presence of an extra fractional dimension. These bounds are higher when we work with muonic hydrogen and muonic deuterium, which open the possibility that future measurements of the Lamb shift in muonic atoms with one electron and higher values of Z may show evidence of the extra dimension.
We present our studies of the forward unpolarised doubly-virtual Compton scattering (VVCS) off the deuteron and the closely related two-photon-exchange ($2γ$-exchange) corrections to the Lamb shift of muonic deuterium. The deuteron VVCS amplitude is calculated in the framework of pionless effective field theory, up to next-to-next-to-next-to-leading order (N3LO) for the longitudinal and next-to-leading order (NLO) for the transverse amplitude. The charge elastic form factor of the deuteron, obtained from the residue of the longitudinal VVCS amplitude, is used to extract the value of the single…
We consider the two-photon exchange contribution to the $2P-2S$ Lamb shift in muonic deuterium in the framework of forward dispersion relations. The dispersion integrals are evaluated using experimental data on elastic deuteron form factors and inelastic electron-deuteron scattering, both in the quasielastic and hadronic range. The subtraction constant that is required to ensure convergence of the dispersion relation for the forward Compton amplitude $T_1(ν,Q^2)$ is related to the deuteron magnetic polarizability $β(Q^2)$. Based on phenomenological information, we obtain for the Lamb shift $ΔE…
Leading order ($α^4$) finite size corrections in muonic deuterium are evaluated within a few body formalism for the $μ^- p n$ system in muonic deuterium and found to be sensitive to the input of the deuteron wave function. We show that this sensitivity, taken along with the precise deuteron charge radius determined from muonic atom spectroscopy can be used to determine the elusive deuteron D-state probability, $P_D$, for a given model of the nucleon-nucleon (NN) potential. The radius calculated with a $P_D$ of 4.3\% in the chiral NN models and about 5.7\% in the high precision NN potentials is…
We present a study of the two-photon-exchange ($2γ$-exchange) corrections to the $S$-levels in muonic ($μ$D) and ordinary (D) deuterium within the pionless effective field theory (pionless EFT). Our calculation proceeds up to next-to-next-to-next-to-leading order (N3LO) in the pionless EFT expansion. The only unknown low-energy constant entering the calculation at this order corresponds to the coupling of a longitudinal photon to the nucleon-nucleon system. To minimise its correlation with the deuteron charge radius, it is extracted using the information about the hydrogen-deuterium isotope sh…
We explore the possibility that a new interaction between muons and protons is responsible for the discrepancy between the CODATA value of the proton radius and the value deduced from the measurement of the Lamb shift in muonic hydrogen. We show that a new force carrier with roughly MeV mass can account for the observed energy shift as well as the discrepancy in the muon anomalous magnetic moment. However, measurements in other systems constrain the couplings to electrons and neutrons to be suppressed relative to the couplings to muons and protons, which seems challenging from a theoretical po…
The current discrepancy of theory and experiment observed recently in muonic hydrogen necessitates a reinvestigation of all corrections to contribute to the Lamb shift in muonic hydrogen muH, muonic deuterium muD, the muonic 3He ion, as well as in the muonic 4He ion. Here, we choose a semi-analytic approach and evaluate a number of higher-order corrections to vacuum polarization (VP) semi-analytically, while remaining integrals over the spectral density of VP are performed numerically. We obtain semi-analytic results for the second-order correction, and for the relativistic correction to VP. T…
On the basis of quasipotential method in quantum electrodynamics we calculate nuclear finite size radiative corrections of order $α(Z α)^5$ to the Lamb shift in muonic hydrogen and helium. To construct the interaction potential of particles, which gives the necessary contributions to the energy spectrum, we use the method of projection operators to states with a definite spin. Separate analytic expressions for the contributions of the muon self-energy, the muon vertex operator and the amplitude with spanning photon are obtained. We present also numerical results for these contributions using m…
The radiative correction to the nuclear charge radius contribution to the Lamb shift of order $α(Zα)^5m_r^3<r^2>$ is calculated. In view of the recent high precision experimental data, this theoretical correction produces a significant contribution to the hydrogen-deuterium isotopic shift.
Bound muons (also pions, kaons, etc) increase the fission barrier and produce some stabilizing effects for highly charged nuclei. If the binding energy of the muon exceeds $mc^2$, it becomes stable. The $1s$ state of a muon inside an exotic nucleus with atomic number $A= 5Z/2$ and such large charge $Z$ that the $1s$ energy $E$ is in the range $0\le E<-mc^2$ is considered.
On the basis of quasipotential method in quantum electrodynamics we calculate nuclear finite size radiative corrections of order $α(Zα)^5$ to the hyperfine structure of S-wave energy levels in muonic hydrogen and muonic deuterium. For the construction of the particle interaction operator we employ the projection operators on the particle bound states with definite spins. The calculation is performed in the infrared safe Fried-Yennie gauge. Modern experimental data on the electromagnetic form factors of the proton and deuteron are used.
Recent progress in laser and x-ray spectroscopy of muonic atoms offers promising long-term possibilities at the intersection of atomic, nuclear and particle physics. In muonic hydrogen, laser spectroscopy measurements will determine the ground-state hyperfine splitting (HFS) and additionally improve the Lamb shift by a factor of 5. Precision spectroscopy with cryogenic microcalorimeters has the potential to significantly improve the charge radii of the light nuclei in the $Z=3-8$ range. Complementary progress in precision should be achieved on the theory of nucleon- and nuclear-structure effec…
We present a precise calculation of the Lamb shift $(2P_{1/2}-2S_{1/2})$ in muonic ions $(μ^6_3Li)^{2+},~(μ^7_3Li)^{2+}$, $(μ^9_4Be)^{3+},~(μ^{10}_4Be)^{3+}$, $(μ^{10}_5B)^{4+},~(μ^{11}_5B)^{4+}$. The contributions of orders $α^3\divα^6$ to the vacuum polarization, nuclear structure and recoil, relativistic effects are taken into account. Our numerical results are consistent with previous calculations and improve them due to account of new corrections. The obtained results can be used for the comparison with future experimental data, and extraction more accurate values of nuclear charge radii.
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.
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…
We determine the Zemach moments of hydrogen and deuterium for the first time using only the world data on elastic electron-proton and electron-deuteron scattering. Such moments are required for the calculation of the nuclear corrections to the hyperfine structure of these hydrogenic atoms. We compare the resulting HFS predictions to the available high-precision data and provide an estimate of the size of the nuclear polarization corrections necessary to produce agreement between experimental HFS and theoretical calculations.
Transition rates for coherent muon-electron conversion in muonic atoms, mu+N --> e+N, are computed for various types of muon number violating interactions. Attention is paid to relativistic atomic effects, Coulomb distortion, finite nuclear size, and nucleon distributions. Discrepancies with previously published results are pointed out and explained. Results are presented for several elements of current and future experimental interest.
The energy interval $ (3S-1S) $ in muonic hydrogen is calculated on the basis of quasipotential approach in quantum electrodynamics. We take into account different corrections of orders $α^3÷α^6 $, which are determined by relativistic effects, the effects of vacuum polarization, nuclear structure and recoil, as well as combined corrections including the above. Nuclear structure effects are expressed in terms of the charge radius of the proton in the case of one-photon interaction and the proton electromagnetic form factors in the case of two-photon exchange interaction. The value of the energy…
Corrections of orders alpha^5, alpha^6 are calculated in the hyperfine splitting of the 2S state in the muonic hydrogen. The nuclear structure effects are taken into account in the one- and two-loop Feynman amplitudes by means of the proton electromagnetic form factors. Total numerical value of the 2S state hyperfine splitting 22.8148 meV in the (μp) can be considered as reliable estimation for the corresponding experiment with the accuracy 10^{-5}. The value of the Sternheim's hyperfine splitting interval [8ΔE^{HFS}(2S)-ΔE^{HFS}(1S)] is obtained with the accuracy 10^{-6}.
Nuclear reactions induced by muon neutrino with energies in the range from 0.2 to 1.5 GeV in Monte Carlo calculations framework in the intra-nuclear cascade model are studied. This study was done by comparison between the available experimental data and theoretical values of total cross section, and the energy distribution of emitted lepton energy in the reaction muon neutrino-nucleus, using the targets: $^{12}$C, $^{16}$O, $^{27}$Al, $^{40}$Ar, $^{56}$Fe and $^{208}$Pb. A phenomenological toy model of primary neutrino-nucleon interaction gives a good agreement of our theoretical inclusive neu…
Corrections of orders alpha^5 and alpha^6 are calculated in the hyperfine splittings of 1S and 2S - energy levels in the ion of muonic helium. The electron vacuum polarization effects, the nuclear structure corrections and recoil corrections are taken into account. The obtained numerical values of the hyperfine splittings -1334.56 meV (1S state), -166.62 meV (2S state) can be considered as a reliable estimate for the comparison with the future experimental data. The hyperfine splitting interval Delta_{12}=(8 Delta E^{hfs}(2S)- Delta E^{hfs}(1S)) = 1.64 meV can be used for the check of quantum …
Corrections of orders alpha^5, alpha^6 are calculated in the hyperfine splitting of the muonic hydrogen ground state. The nuclear structure effects are taken into account in the one- and two-loop Feynman amplitudes by means of the proton electromagnetic form factors. The modification of the hyperfine splitting part of the Breit potential due to the electron vacuum polarization is considered. Total numerical value of the 1S state hyperfine splitting 182.638 meV in the (mu p) can play the role of proper estimation for the corresponding experiment with the accuracy 30 ppm.
Laser spectroscopy of muonic atoms has been recently used to probe properties of light nuclei with unprecedented precision. We introduce nuclear effects in hydrogen-like atoms, nucleon structure quantities (form factors, structure functions, polarizabilities) and their effects in the Lamb shift and hyperfine splitting (HFS) of muonic hydrogen ($μ$H). Updated theory predictions for the Lamb shift and HFS in $μ$H are presented. We review the challenges of the ongoing effort to measure the ground-state HFS in $μ$H and its impact on our understanding of the nucleon spin structure. To narrow down t…
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 $(μe ^4_2He)$. We obtain total result for the ground state hyperfine splitting $Δν^{hfs}=4465.526$ MHz which improves the previous calculation of Lakdawala and Mohr due to the account of new corrections. The remaining difference between the theoretical result and experimental value of the hyperfine…
In view of the recently observed discrepancy of theory and experiment for muonic hydrogen [R. Pohl et al., Nature vol. 466, p. 213 (2010)], we reexamine the theory on which the quantum electrodynamic (QED) predictions are based. In particular, we update the theory of the 2P-2S Lamb shift, by calculating the self-energy of the bound muon in the full Coulomb+vacuum polarization (Uehling) potential. We also investigate the relativistic two-body corrections to the vacuum polarization shift, and we analyze the influence of the shape of the nuclear charge distribution on the proton radius determinat…
A specific combination of $s$-state Lamb shift $ΔE_L(1s_{1/2}) - n^3 ΔE_L(ns_{1/2})$ is considered. Its value is calculated both in the hydrogen and deuterium atoms for $n$ up to 12. The result inludes all correction which can contribute $1 kHz$ and particulary: one-loop self energy and vacuum polarization, two-loop contribution. Nuclear finite-size corrections for the isotopic difference of the combination are also evaluated.
The EMC effect in deuterium and helium-3 is studied using a convolution formalism that allows isolating the impact of high-momentum nucleons in short-ranged correlated (SRC) pairs. We assume that the modification of the structure function of bound nucleons is given by a universal (i.e. nucleus independent) function of their virtuality, and find that the effect of such modifications is dominated by nucleons in SRC pairs. This SRC-dominance of nucleon modifications is observed despite the fact that the bulk of the nuclear inelastic scattering cross-section comes from interacting with low-momentu…
Koike et al. proposed the charged lepton flavor violation (CLFV) decay of the muonic atom $μ^-e^-\rightarrow e^-e^-$ as one of the promising processes to search for new physics beyond the standard model. It was found that the attractive interaction of leptons with the nucleus enhances the transition rate of the $μ^-e^-\rightarrow e^-e^-$ process. We report on our improved analysis of this process by taking into account the distortion of the out-going electrons in the nuclear Coulomb potential and the relativistic treatment of the muon and the electrons. As results, we found significant enhance…
On the basis of perturbation theory in fine structure constant alpha and the ratio of electron to muon masses we calculate one-loop vacuum polarization, electron vertex corrections, nuclear structure and recoil corrections to hyperfine splitting of the ground state in muonic lithium ions $(μ e\ ^6_3Li)^+$ and $(μ e\ ^7_3Li)^+$. We obtain total results for the ground state small hyperfine splittings in $(μ e\ ^6_3Li)^+$ $Δν_1=21572.16$ MHz and $Δν_2=14152.56$ MHz and in $(μ e\ ^7_3Li)^+$ $Δν_1=21733.06$ MHz and $Δν_2=13994.35$ MHz which can be considered as a reliable estimate for a comparison …
We present a novel method for extracting the proton radius from elastic electron-proton ($ep$) scattering data. The approach is based on interpolation via continued fractions augmented by statistical sampling and avoids any assumptions on the form of function used for the representation of data and subsequent extrapolation onto $Q^2\simeq 0$. Applying the method to extant modern $e p$ data sets, we find that all results are mutually consistent and, combining them, arrive at $r_p=0.847(8)\,$fm. This result compares favourably with values obtained from contemporary measurements of the Lamb shift…
Up to now, all charge radius measurements of the proton and deuteron assumed uniform spheroidal charge distribution. We investigate the nuclear deformation effects on these charge radius measurements by assuming a uniform prolate charge distribution for the proton and deuteron. We solve the energy levels of the corresponding muonic and electric atoms with such deformed nucleus and present how the purely quadruple deformation of proton and deuteron affects their Lamb shifts. The numerical results suggest that the deformation of proton and deuteron leads to that the charge radius extracted from …
We report the results of a new global QCD analysis, which includes deep-inelastic $e/μ$ scattering data off proton and deuterium, as well as Drell-Yan lepton pair production in proton-proton and proton-deuterium collisions and $W^\pm/Z$ boson production data from $pp$ and $p \bar p$ collisions at the LHC and Tevatron. Nuclear effects in the deuteron are treated in terms of a nuclear convolution approach with bound off-shell nucleons within a weak binding approximation. The off-shell correction is controlled by a universal function of the Bjorken variable $x$ describing the modification of part…
We consider deuterium compressed to higher than atomic, but lower than nuclear densities. At such densities deuterium is a superconducting quantum liquid. Generically, two superconducting phases compete, a "ferromagnetic" and a "nematic" one. We provide a power counting argument suggesting that the dominant interactions in the deuteron liquid are perturbative (but screened) Coulomb interactions. At very high densities the ground state is determined by very small nuclear interaction effects that probably favor the ferromagnetic phase. At lower densities the symmetry of the theory is effectively…
An Electron-Ion Collider (EIC) with suitable forward detection capabilities would enable a unique experimental program of deep-inelastic scattering (DIS) from polarized light nuclei (deuterium 2H, helium 3He) with spectator nucleon tagging. Such measurements promise significant advances in several key areas of nuclear physics and QCD: (a) neutron spin structure, by using polarized deuterium and eliminating nuclear effects through on-shell extrapolation in the spectator proton momentum; (b) quark/gluon structure of the bound nucleon at x > 0.1 and the dynamical mechanisms acting on it, by measu…
