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 for the high-precision measurement of the nuclear $μ^{-}$ capture in the ground-state $pμ$ atom, which is now underway at the Paul Scherrer Institute.
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 Lamb shift in muonic hydrogen continues to be a subject of experimental and theoretical investigation. Here my older work on the subject is updated to provide a complementary calculation of the energies of the 2p-2s transitions in muonic hydrogen.
The proton structure effects, including finite size, polarizability and self-energy is considered and their influence on energy levels of muonic hydrogen is recalculated. A new theoretical prediction for the Lamb shift is presented together with improved values of all known QED contributions.
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 differential cross sections for low-energy muonic hydrogen atom scattering in solid molecular H$_2$, D$_2$ and T$_2$ targets under low pressure have been calculated for various temperatures. The polycrystalline fcc and hcp structure of the solid hydrogenic targets are considered. The Bragg and phonon scattering processes are described using the Debye model of a solid. The calculated cross sections are used for Monte Carlo simulations of the muonic atom slowing down in these targets. They have been successfully applied for a description of the production of the muonic atom beams in the mult…
The differential cross sections for low-energy muonic hydrogen atom scattering in liquid hydrogenic targets have been calculated in the incoherent approximation using the Van Hove response function. A simple model of liquids and the available experimental parameters have been employed for a description of the diffusive and vibrational modes in these targets. At collision energies below about 10 meV, the obtained cross sections are very different from the analogous cross sections for scattering in hydrogenic gases.
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^-…
We report our investigations of X rays emitted during the muonic cascade in hydrogen employing charge coupled devices as X-ray detectors. The density dependence of the relative X-ray yields for the muonic hydrogen lines (K_alpha, K_beta, K_gamma) has been measured at densities between 0.00115 and 0.97 of liquid hydrogen density. In this density region collisional processes dominate the cascade down to low energy levels. A comparison with recent calculations is given in order to demonstrate the influence of Coulomb deexcitation.
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
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…
In the comment we discuss an additional term to the $α^3$ correction to the Lamb shift, which was missed in the calculation by Kinoshita and Nio (Phys. Rev. Lett. 82, 3240 (1999)).
We perform dynamical calculations of direct muon-transfer rates from thermalized muonic hydrogen isotopes to bare nuclei C$^{6+}$ and O$^{8+}$. For these three-body charge-transfer reactions with Coulomb interaction in the final state we use two-component integro-differential Faddeev-Hahn-type equations in configuration space using close-coupling approximation. To take into account the high polarizability of the muonic hydrogen due to the large charge of the incident nuclei, a polarization potential is included in the elastic channel. The large final-state Coulomb interaction is explicitly inc…
The "proton radius puzzle" was recently solved by reducing the four-standard deviation discrepancy between the results for electronic hydrogen ($H$) and muonic hydrogen ($μH$) atoms to $3.3$ value. The value of the root-mean-square radius of the proton ($r_p$), extracted from experiments on measuring the one-photon $2s-4p$ transition and the Lamb shift in hydrogen, is now $0.8335(95)$ fm, that is in good agreement with the muonic hydrogen experiments, $0.84087(39)$ fm. Even so, these values deviate significantly from the CODATA value, which is determined as the average using the results for va…
The differential cross sections for low-energy muonic hydrogen atom scattering from hydrogenic molecules are directly expressed by the corresponding amplitudes for muonic atom scattering from hydrogen-isotope nuclei. The energy and angular dependence of these three-body amplitudes is thus taken naturally into account in scattering from molecules, without involving any pseudopotentials. Effects of the internal motion of nuclei inside the target molecules are included for every initial rotational-vibrational state. These effects are very significant as the considered three-body amplitudes often …
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…
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.
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_…
We report the first experimental determination of the collision-energy dependence of the muon transfer rate from the ground state of muonic hydrogen to oxygen at near-thermal energies. A sharp increase by nearly an order of magnitude in the energy range 0 - 70 meV was found that is not observed in other gases. The results set a reliable reference for quantum-mechanical calculations of low-energy processes with exotic atoms, and provide firm ground for the measurement of the hyperfine splitting in muonic hydrogen and the determination of the Zemach radius of the proton by the FAMU collaboration…
The kinetic energy distribution of ground state muonic hydrogen atoms mu-p(1S) is determined from time-of-flight spectra measured at 4, 16, and 64 hPa H2 room-temperature gas. A 0.9 keV-component is discovered and attributed to radiationless deexcitation of long-lived mu-p(2S) atoms in collisions with H2 molecules. The analysis reveals a relative population of about 1%, and a pressure-dependent lifetime (e.g. (30.4 +21.4 -9.7) ns at 64 hPa) of the long-lived mu-p(2S) population, equivalent to a 2S-quench rate in mu-p(2S) + H2 collisions of (4.4 +2.1 -1.8) 10^11 s^-1 at liquid hydrogen density.
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 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 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…
We determine the third Zemach moment of hydrogen (<r^3>_(2)) using only the world data on elastic electron-proton scattering. This moment dominates the O (Z alpha)^5 hadronic correction to the Lamb shift in muonic atoms. The resulting moment, <r^3 >_(2) = 2.71(13) fm^3, is somewhat larger than previously inferred values based on models. The contribution of that moment to the muonic hydrogen 2S level is -0.0247(12) meV.
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.
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 first measurement of the temperature dependence of the muon transfer rate from muonic hydrogen to oxygen was performed by the FAMU collaboration in 2016. The results provide evidence that the transfer rate rises with the temperature in the range 104-300 K. This paper presents the results of the experiment done in 2018 to extend the measurements towards lower (70 K) and higher (336 K) temperatures. The 2018 results confirm the temperature dependence of the muon transfer rate observed in 2016 and sets firm ground for comparison with the theoretical predictions.
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.
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.
Metastable ${2S}$ muonic-hydrogen atoms undergo collisional ${2S}$-quenching, with rates which depend strongly on whether the $μp$ kinetic energy is above or below the ${2S}\to {2P}$ energy threshold. Above threshold, collisional ${2S} \to {2P}$ excitation followed by fast radiative ${2P} \to {1S}$ deexcitation is allowed. The corresponding short-lived $μp ({2S})$ component was measured at 0.6 hPa $\mathrm{H}_2$ room temperature gas pressure, with lifetime $τ_{2S}^\mathrm{short} = 165 ^{+38}_{-29}$ ns (i.e., $λ_{2S}^\mathrm{quench} = 7.9 ^{+1.8}_{-1.6} \times 10^{12} \mathrm{s}^{-1}$ at liquid…
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 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…
Recently the muonic hydrogen lamb shift has been measured with unprecedented accuracy, allowing for a precise determination of the proton radius. This determination is 5 sigma away from the previous CODATA value obtained from (mainly) the hydrogen lamb shift and the electron-proton scattering. Within an effective field theory formalism, I will define the proton radius and briefly review some aspects of the theoretical prediction for the muonic hydrogen lamb shift, studying both the pure QED-like computation and the hadronic effects.
The Lamb shift in muonic hydrogen was measured some time ago to a high accuracy. The theoretical prediction of this value is very sensitive to the proton-finite-size effects. The proton radius extracted from muonic hydrogen is in contradiction with the results extracted from elastic electron-proton scattering. That creates a certain problem for the interpretation of the results from the muonic hydrogen Lamb shift. For the latter we need also to take into account the two-photon-exchange contribution with the proton finite size involved. The only way to describe it relies on the data from the sc…
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…
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…
Studies of muonic hydrogen atoms and molecules have been performed traditionally in bulk targets of gas, liquid or solid. At TRIUMF, Canada's meson facility, we have developed a new type of target system using multilayer thin films of solid hydrogen, which provides a beam of muonic hydrogen atoms in vacuum. Using the time-of-flight of the muonic atoms, the energy-dependent information of muonic reactions are obtained in direct manner. We discuss some unique measurements enabled by the new technique, with emphasis on processes relevant to muon catalyzed fusion.
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.
Recently a high-precision measurement of the Lamb shift in muonic hydrogen has been performed. An accurate value of the proton charge radius can be extracted from this datum with a high accuracy. To do that a sufficient accuracy should be achieved also on the theoretical side, including an appropriate treatment of higher-order proton-structure effects. Here we consider a higher-order contribution of the finite size of the proton to the Lamb shift in muonic hydrogen. Only model-dependent results for this correction have been known up to date. Meantime, the involved models are not consistent eit…
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 …
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 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.
To date the magnetic radius of the proton has been determined only by means of electron-proton scattering, which is not free of controversies. Any existing atomic determinations are irrelevant because they are strongly model-dependent. We consider a so-called Zemach contribution to the hyperfine interval in ordinary and muonic hydrogen and derive a self-consistent model-independent value of the magnetic radius of the proton. More accurately, we constrain not a value of the magnetic radius by itself, but its certain combination with the electric-charge radius of the proton, namely, R_E^2+R_M^2.…
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…
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…
Radiative corrections to the decay rate in the low-Z hydrogen-like muonic atoms are considered. The correction arises from the Uehling potential and it has the relative order of alpha. The numerical results are reported for the 2p-1s transition for the atom with the nuclear charge value up to Z=10.
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 …
We demonstrate that criticisms concerning our work nucl-th/9704043, made in the recent preprint hep-ph/9903352, are unfounded. The improvements over our results claimed in hep-ph/9903352 are in fact spurious, being based mainly on misunderstandings and on dubious theoretical models.
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.
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 …
We illustrate how to apply modern effective field theory techniques and dimensional regularization to factorise the various scales which appear in QED bound states at finite temperature. We focus here on the muonic hydrogen atom. Vacuum polarization effects make the physics of this atom at finite temperature very close to that of heavy quarkonium states. We comment on the implications of our results for these states in the quark gluon plasma. In particular, we estimate the effects of a finite charm quark mass in the dissociation temperature of bottomonium.
The extremely precise extraction of the proton radius by Pohl et al. from the measured energy difference between the 2P and 2S states of muonic hydrogen disagrees significantly with that extracted from electronic hydrogen or elastic electron-proton scattering. This is the proton radius puzzle. The origins of the puzzle and the reasons for believing it to be very significant are explained. Various possible solutions of the puzzle are identified, and future work needed to resolve the puzzle is discussed.
