The non-resonant formation of $ddμ$ molecules in the loosely bound state in collisions of non-thermalized $dμ$ atoms with deuterium molecules D$_2$ has been considered. The process of such a type is possible only for collision energies exceeded the ionization potential of D$_2$. The calculated rates of $ddμ$ formation in the above-threshold energy region are about one order of magnitude higher than obtained earlier.
The role of epithermal non-resonant $μ$-molecule formation for the kinetics of $μ$CF processes in D$_2$ gas was studied. It was shown that the non-resonant $ddμ$ formation by $dμ$ atoms accelerated during the cascade can be directly observed in the neutron time spectra at very short initial times.
My previous calculations of the Lamb shift in muonic hydrogen are reviewed and compared with other work. In addition, numerical results for muonic deuterium are presented.
The deuteron charge radius is calculated from the measurement of the Lamb shift in muonic deuterium, taking into account the electron vacuum polarization correction to the nuclear-structure effects. This correction is unexpectedly large and gives a mean-square charge-radii difference $ r_d^2-r_p^2 = 3.81747(346)\, {\rm fm}^2 $, which is now consistent with that obtained from the ordinary H-D isotope shift in the 1$S$-2$S$ transition. This suggests that the long-standing discrepancy in the proton charge radius obtained from electronic and muonic systems is due to an underestimated uncertainty i…
Radiative deexcitation (RD) of the metastable 2S state of muonic protium and deuterium atoms has been observed. In muonic protium, we improve the precision on lifetime and population (formation probability) values for the short-lived μp(2S) component, and give an upper limit for RD of long-lived μp(2S) atoms. In muonic deuterium at 1 hPa, 3.1 +-0.3 % of all stopped muons form μd(2S) atoms. The short-lived 2S component has a population of 1.35 +0.57 -0.33 % and a lifetime of τ_short(μd) = 138 +32 -34 ns. We see evidence for RD of long-lived μd(2S) with a lifetime of τ_long(μd) = 1.15 +0.75 -0.5…
Quantum interference between energetically close states is theoretically investigated, with the state structure being observed via laser spectroscopy. In this work, we focus on hyperfine states of selected hydrogenic muonic isotopes, and on how quantum interference affects the measured Lamb shift. The process of photon excitation and subsequent photon decay is implemented within the framework of nonrelativistic second-order perturbation theory. Due to its experimental interest, calculations are performed for muonic hydrogen, deuterium, and helium-3. We restrict our analysis to the case of phot…
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_…
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 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 s…
The present knowledge of Lamb shift, fine- and hyperfine structure of the $\mathrm{2S}$ and $\mathrm{2P}$ states in muonic deuterium is reviewed in anticipation of the results of a first measurement of several $\mathrm{2S-2P}$ transition frequencies in muonic deuterium ($μ\mathrm{d}$). A term-by-term comparison of all available sources reveals reliable values and uncertainties of the QED and nuclear structure-dependent contributions to the Lamb shift, which are essential for a determination of the deuteron rms charge radius from $μ\mathrm{d}$. Apparent discrepancies between different sources a…
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 longitudinal structure function of the deuteron up through next-to-next-to-leading order in the framework of pionless effective field theory. We use these results to compute the two-photon polarizability contribution to Lamb shift in muonic deuterium, which can be utilized to extract the nuclear charge radius of the deuteron. We present analytical expressions order-by-order for the relevant transition matrix elements and the longitudinal structure function, and we give numerical results for the corresponding contributions to the Lamb shift. We also discuss the impact of relati…
Resonant scattering of muonic hydrogen atoms via back decay of molecular complex, a key process in the understanding of epithermal muonic molecular formation, is analyzed. The limitations of the effective rate approximation are discussed and the importance of the explicit treatment of the back decay is stressed. An expression of the energy distribution for the back-decayed atoms is given.
The formation and various disintegration modes of $2s$ states for muonic hydrogen and deuterium atoms at kinetic energies both above and below the $2s-2p$ threshold are studied. The cross sections of the collision-induced radiative quenching, elastic scattering, and Coulomb deexcitation of $(μ^- p)_{2s}$ and $(μ^- d)_{2s}$ atoms in collisions with ordinary hydrogen and deuterium atoms at collision energies below the $2s-2p$ threshold are calculated in the framework of the close-coupling approach. The basis set includes all open and closed channels corresponding to exotic-atom states with princ…
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)
In connection with recent and proposed experiments, and new theoretical results, my previous calculations of the Lamb shift in muonic hydrogen will be reviewed and compared with other work. In addition, numerical results for muonic deuterium and helium will be presented. Some previously neglected (but very small) effects are included.
Corrections to energy levels in light muonic atoms are investigated in order $α^2(Zα)^4m$. We pay attention to corrections which are specific for muonic atoms and include the electron vacuum polarization loop. In particular, we calculate relativistic and relativistic-recoil two-loop electron vacuum polarization contributions. The results are obtained for the levels with $n=1,2$ and in particular for the Lamb shift ($2p_{1/2}-2s_{1/2}$) and fine-structure intervals ($2p_{3/2}-2p_{1/2}$) in muonic hydrogen, deuterium, and muonic helium ions.
The ground bound states in the five-body muonic ions $a b μe_2$ (or $(a b μe_2)^{-}$), where $(a, b) = (p, d, t)$, are considered for the first time. As follows from accurate numerical computations of these five-body ions they are similar to the negatively charged hydrogen ion H$^{-}$ with a three-particle central quasi-nucleus $a b μ$ (or $(a b μ)^{+}$). These five-body ions play some role in the muon-catalyzed fusion of nuclear reaction in liquid deuterium and/or deuterium-tritium mixtures. We also discuss the bound states in the six-body $a b c μe_2$ quasi-molecules, where $(a, b, c) = (p, …
We provide a systematic assessment of the order-$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…
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.
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 relativistic recoil contributions to the Uehling corrections are revisited. We consider a controversy in recent calculations based on different approaches including Breit-type and Grotch-type calculations. We have found that calculations of those authors were in fact done in different gauges and in some of those gauges contributions the retardation and two-photon-exchange effects were missed. We have evaluated such effects and obtained a consistent result from those approaches. We present a correct expression for the Grotch-type approach which produces a correct gauge-invariant result. We …
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 measurement of the Lamb shift in muonic hydrogen and the subsequent emergence of the proton-radius puzzle have motivated an experimental campaign devoted to measuring the Lamb shift in other light muonic atoms, such as muonic deuterium and helium. For these systems it has been shown that two-photon exchange nuclear structure corrections are the largest source of uncertainty and consequently the bottleneck for exploiting the experimental precision to extract the nuclear charge radius. Utilizing techniques and methods developed to study electromagnetic reactions in light nuclei, recent calcu…
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.
The complete relativistic $O(α^2)$ nuclear structure correction to the energy levels of ordinary (electronic) and muonic hydrogen-like atoms is investigated. The elastic part of the nuclear structure correction is derived analytically. The resulting formula is valid for an arbitrary hydrogenic system and is much simpler than analogous expressions previously reported in the literature. The analytical result is verified by high-precision numerical calculations. The inelastic $O(α^2)$ nuclear structure correction is derived for the electronic and muonic deuterium atoms. The correction comes from …
The rate of $ddμ$ muonic molecule resonant formation in $dμ$ atom collision with a condensed deuterium target is expressed in terms of a single-particle response function. In particular, $ddμ$ formation in solid deuterium at low pressures is considered. Numerical calculations of the rate in the case of fcc polycrystalline deuterium at 3 K have been performed using the isotropic Debye model of solid. It is shown that the energy-dependent $ddμ$ formation rates in the solid differ strongly from those obtained for D$_2$ gaseous targets, even at high $dμ$ kinetic energies. Monte Carlo neutron spect…
The complete contribution of diagrams with the light-by-light scattering to the Lamb shift is found for muonic hydrogen, deuterium and helium ion. The results are obtained in the static muon approximation and a part of the paper is devoted to the verification of this approximation and analysis of its uncertainty.
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…
Recent calculations of nuclear structure corrections to the Lamb shift in light muonic atoms are based on an expansion in a parameter eta, where only terms up to second order are retained. The parameter eta can be shown to be proportional to the square root of the muon/proton mass ratio, so that it is small and the expansion is expected to converge. However, practical implementations show that the eta convergence may be slower than expected. In this work we probe the uncertainties due to this expansion using a different formalism, which is based on a multipole expansion of the longitudinal and…
We calculate the nuclear structure corrections to the Lamb shift in muonic deuterium by using state-of-the-art nucleon-nucleon potentials derived from chiral effective field theory. Our calculations complement previous theoretical work obtained from phenomenological potentials and the zero range approximation. The study of the chiral convergence order-by-order and the dependence on cutoff variations allows us to improve the estimates on the nuclear structure corrections and the theoretical uncertainty coming from nuclear potentials. This will enter the determination of the nuclear radius from …
Resonant formation of $dμt$ molecules in collisions of muonic tritium ($μt$) on D$_2$ was investigated using a beam of $μt$ atoms, demonstrating a new direct approach in muon catalyzed fusion studies. Strong epithermal resonances in $dμt$ formation were directly revealed for the first time. From the time-of-flight analysis of $2036\pm 116$ $dt$ fusion events, a formation rate consistent with $0.73\pm (0.16)_{meas} \pm (0.09)_{model}$ times the theoretical prediction was obtained. For the largest peak at a resonance energy of $0.423 \pm 0.037$ eV, this corresponds to a rate of $(7.1 \pm 1.8) \t…
