Lamb Shift In Hydrogen Research Articles

Proton radius puzzle and quantum gravity at the Fermi scale

We show how the “proton radius puzzle” emerging from the measurement of the Lamb shift in muonic hydrogen may be solved by means of a binding energy contribution due to an effective Yukawian gravitational potential related to charged weak interactions. The residual discrepancy from the experimental result should be mainly attributable to the need for the experimental determination of the gravitational radius of the proton. The absence of an analogous contribution in the Lamb shift of electronic hydrogen should imply the existence of generation-dependent interactions, corroborating previous proposals. Muonic hydrogen plays a crucial role to test possible scenarios for a gravitoweak unification, with weak interactions seen as manifestations of quantum gravity effects at the Fermi scale.

Muonic-hydrogen Lamb shift: Dispersing the nucleon-excitation uncertainty with a finite-energy sum rule

We assess the two-photon exchange contribution to the Lamb shift in muonic hydrogen with forward dispersion relations. The subtraction constant $\bar T(0,Q^2)$ that is necessary for a dispersive evaluation of the forward doubly-virtual Compton amplitude, through a finite energy sum rule, is related to the fixed J=0 pole generalized to the case of virtual photons. We evaluated this sum rule using excellent virtual photoabsorption data that are available. We find that the "proton polarizability correction" to the Lamb shift in muonic hydrogen is $-(40\pm5)\mu$eV. We conclude that nucleon structure-dependent uncertainty by itself is unlikely to resolve the large (300$\mu$eV) discrepancy between direct measurement of the Lamb shift in $\mu H$ and expectations based on conventional Hydrogen measurements.

Nonidentical protons

We have calculated the proton charge radius by assuming that the real proton radius is not unique and the radii are randomly distributed in a certain range. This is performed by averaging the elastic electron-proton differential cross section over the form factor cutoff. By using a dipole form factor and fitting the middle value of the cutoff to the low-${Q}^{2}$ Mainz data, we found the lowest ${\ensuremath{\chi}}^{2}/N$ for a cutoff $\ensuremath{\Lambda}=0.8203\ifmmode\pm\else\textpm\fi{}0.0003$ GeV, which corresponds to a proton charge radius ${r}_{E}=0.8333\ifmmode\pm\else\textpm\fi{}0.0004$ fm. The result is compatible with the recent precision measurement of the Lamb shift in muonic hydrogen as well as recent calculations using more sophisticated techniques. Our result indicates that the relative variation of the form factor cutoff should be around 21.5$%$. Based on this result we have investigated effects of the nucleon radius variation on the symmetric nuclear matter (SNM) and the neutron star matter (NSM) by considering the excluded volume effect in our calculation. The mass-radius relation of a neutron star is found to be sensitive to this variation. The nucleon effective mass in the SNM and the equation of state of both the SNM and the NSM exhibit a similar sensitivity.

Theory of the 2S–2P Lamb shift and 2S hyperfine splitting in muonic hydrogen

The 7σ discrepancy between the proton rms charge radius from muonic hydrogen and the CODATA-2010 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; see Pohl et al. (2010) [9] and Antognini et al. (2013) [71].

Proton polarizability contribution: Muonic hydrogen Lamb shift and elastic scattering

The uncertainty in the contribution to the Lamb shift in muonic hydrogen, ΔEsubt arising from proton polarizability effects in the two-photon exchange diagram at large virtual photon momenta is shown large enough to account for the proton radius puzzle. This is because ΔEsubt is determined by an integrand that falls very slowly with very large virtual photon momenta. We evaluate the necessary integral using a set of chosen form factors and also a dimensional regularization procedure which makes explicit the need for a low energy constant. The consequences of our two-photon exchange interaction for low-energy elastic lepton–proton scattering are evaluated and could be observable in a planned low energy lepton–proton scattering experiment planned to run at PSI.

Proton polarisability contribution to the Lamb shift in muonic hydrogen at fourth order in chiral perturbation theory

We calculate the amplitude T1 for forward doubly virtual Compton scattering in heavy-baryon chiral perturbation theory, to fourth order in the chiral expansion and with the leading contribution of the \( \gamma\) N \( \Delta\) form factor. This provides a model-independent expression for the amplitude in the low-momentum region, which is the dominant one for its contribution to the Lamb shift. It allows us to significantly reduce the theoretical uncertainty in the proton polarisability contributions to the Lamb shift in muonic hydrogen. We also stress the importance of consistency between the definitions of the Born and structure parts of the amplitude. Our result leaves no room for any effect large enough to explain the discrepancy between proton charge radii as determined from muonic and normal hydrogen.

Weak-interaction contributions to hyperfine splitting and Lamb shift in light muonic atoms

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 $(1-4\sin^2\theta_W)$.

Brane world regularization of point particle classical self-energy

Physical effects in brane worlds models emerge by the incorporation of field modes coming from extra dimensions with the usual four dimensional ones. Such effects can be tested with well established experiments to set bounds on the parameters of the brane models. In this work we extend a previous result which gave finite electromagnetic potentials and self energies for a source looking pointlike to an observer sitting in a 4D Minkowski subspace of a single brane of a Randall-Sundrum spacetime including compact dimensions, and along which the source stretches uniformly. We show that a scalar particle produces a nonsingular static potential, possess a finite self-energy and that technically its analysis is very similar to the electrostatic case. As for the latter, we use the deviations from the Coulomb potential to set bounds on the anti de Sitter radius of the brane model on the basis of two experiments, namely, one of the Cavendish type and other being the scattering of electrons by Helium atoms. We found these are less stringent than others previously obtained using the Lamb shift in Hydrogen.

Lamb shift in light muonic atoms — Revisited

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.

Model Independent Analysis of Proton Structure for Hydrogenic Bound States

Proton structure effects in hydrogenic bound states are analyzed using nonrelativistic QED effective field theory. Implications for the Lamb shift in muonic hydrogen are discussed. Model-dependent assumptions in previous analyses are isolated, and sensitivity to poorly constrained hadronic structure in the two-photon exchange contribution is identified.

High-precision measurement of the proton elastic form factor ratio [formula omitted] at low [formula omitted

We report a new, high-precision measurement of the proton elastic form factor ratio μpGE/GM for the four-momentum transfer squared Q2=0.3–0.7(GeV/c)2. The measurement was performed at Jefferson Lab (JLab) in Hall A using recoil polarimetry. With a total uncertainty of approximately 1%, the new data clearly show that the deviation of the ratio μpGE/GM from unity observed in previous polarization measurements at high Q2 continues down to the lowest Q2 value of this measurement. The updated global fit that includes the new results yields an electric (magnetic) form factor roughly 2% smaller (1% larger) than the previous global fit in this Q2 range. We obtain new extractions of the proton electric and magnetic radii, which are 〈rE2〉1/2=0.875±0.010 fm and 〈rM2〉1/2=0.867±0.020 fm. The charge radius is consistent with other recent extractions based on the electron–proton interaction, including the atomic hydrogen Lamb shift measurements, which suggests a missing correction in the comparison of measurements of the proton charge radius using electron probes and the recent extraction from the muonic hydrogen Lamb shift.

Muonic hydrogen and MeV forces

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 point of view. One can nevertheless make predictions for energy shifts in muonic deuterium, muonic helium, and true muonium under the assumption that the new particle couples dominantly to muons and protons.

Proton-structure corrections to hyperfine splitting in muonic hydrogen

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 uncertainty in the hyperfine splitting. Additionally, hyperfine splittings have an impact on the muonic hydrogen Lamb shift and proton radius measurement, however the correction we advocate has a small effect there.

Proton Size Anomaly

A measurement of the Lamb shift in muonic hydrogen yields a charge radius of the proton that is smaller than the CODATA value by about 5 standard deviations. We explore the possibility that new scalar, pseudoscalar, vector, and tensor flavor-conserving nonuniversal interactions may be responsible for the discrepancy. We consider exotic particles that, among leptons, couple preferentially to muons and mediate an attractive nucleon-muon interaction. We find that the many constraints from low energy data disfavor new spin-0, spin-1, and spin-2 particles as an explanation.

Going from classical to quantum description of bound charged particles II: Implications for the atomic physics

This paper is the continuation of the analysis of bound quantum systems started in part I (A.L. Kholmetskii, T. Yarman and O.V. Missevitch, Going from classical to quantum description of bound charged particles. I: Basic concepts and assertions), which is based on a novel approach to the transition from classical to quantum description of electrically bound charges, involving the requirement of energy-momentum conservation for the bound electromagnetic (EM) field, when the EM radiation is forbidden. It has been shown that the modified expression for the energy levels of hydrogenic atoms within such a pure bound field theory (PBFT) provides the same gross and fine structure of energy levels, like in the standard theory. At the same time, at the scale of hyperfine interactions, our approach, in general, does evoke some important corrections to the energy levels. Part of such corrections, like the spin-spin splitting in the hydrogen atom, is less than the present theoretical/experimental uncertainties in the evaluation of hyperfine contributions into the atomic levels. But the most interesting result is the appearance of a number of significant corrections (the 1S -2S interval and 1S spin-spin interval in positronium, 1S and 2S -2P Lamb shift in light hydrogenic atoms), which improve considerably the convergence between theoretical predictions and experimental results. In particular, the corrected 1S -2S interval and 1S spin-spin splitting in positronium practically eliminate the existing up-to-date discrepancy between theoretical and experimental data. The re-estimated classic 2S -2P Lamb shift as well as ground-state Lamb shift in the hydrogen atom lead to the proton charge radius r p = 0.841(6) fm (from 2S -2P Lamb shift), and r p = 0.846(22) fm (from 1S Lamb shift), which perfectly agrees with the latest estimation of proton size via the measurement of 2S -2P Lamb shift in muonic hydrogen, i.e. r p = 0.84184(67) fm. Finally, we consider the decay of bound muons in meso-atoms and achieve a quantitative agreement between experimental data and the results obtained through our approach.

Third Zemach moment of the proton

Modern electron scattering experiments have determined the proton electric form factor ${G}_{\mathit{Ep}}({Q}^{2})$ to high precision. We utilize this data, represented by the different empirical form-factor parametrizations, to compute the third Zemach moment of the proton charge distribution. We find that existing data rule out a value of the third Zemach moment large enough to explain the current puzzle with the proton charge radius, determined from the Lamb shift in muonic hydrogen. This is in contrast to the recent paper of De R\'ujula. We also demonstrate that the size of the third Zemach moment is largely governed by the fourth moment of the conventional charge distributions $\ensuremath{\langle}{r}^{4}\ensuremath{\rangle}$, which enables us to obtain a rigorous upper bound on the magnitude of the third Zemach moment of the proton.

Nuclear-size correction to the Lamb shift of one-electron atoms

The nuclear-size effect on the one-loop self-energy and vacuum polarization is evaluated for the $1s$, $2s$, $3s$, $2{p}_{1/2}$, and $2{p}_{3/2}$ states of hydrogen-like ions. The calculation is performed to all orders in the nuclear binding strength parameter $Z\ensuremath{\alpha}$. Detailed comparison is made with previous all-order calculations and calculations based on the expansion in the parameter $Z\ensuremath{\alpha}$. Extrapolation of the all-order numerical results obtained toward $Z=1$ provides results for the radiative nuclear-size effect on the hydrogen Lamb shift.

The RMS charge radius of the proton and Zemach moments

On the basis of recent precise measurements of the electric form factor of the proton, the Zemach moments, needed as input parameters for the determination of the proton rms radius from the measurement of the Lamb shift in muonic hydrogen, are calculated. It turns out that the new moments give an uncertainty as large as the presently stated error of the recent Lamb shift measurement of Pohl et al. De Rújula's idea of a large Zemach moment in order to reconcile the five standard deviation discrepancy between the muonic Lamb shift determination and the result of electronic experiments is shown to be in clear contradiction with experiment. Alternative explanations are touched upon.

The Lamb shift in muonic hydrogenThis paper was presented at the International Conference on Precision Physics of Simple Atomic Systems, held at École de Physique, les Houches, France, 30 May – 4 June, 2010.

The long quest for a measurement of the Lamb shift in muonic hydrogen is over. Last year we measured the 2S 1/2F=1 –2P 3/2F=2 energy splitting (Pohl et al., Nature, 466, 213 (2010)) in μp with an experimental accuracy of 15 ppm, twice better than our proposed goal. Using current QED calculations of the fine, hyperfine, QED, and finite size contributions, we obtain a root-mean-square proton charge radius of rp = 0.841 84 (67) fm. This value is 10 times more precise, but 5 standard deviations smaller, than the 2006 CODATA value of rp. The origin of this discrepancy is not known. Our measurement, together with precise measurements of the 1S–2S transition in regular hydrogen and deuterium, gives improved values of the Rydberg constant, R∞ = 10 973 731.568 160 (16) m–1 and the rms charge radius of the deuteron rd = 2.128 09 (31) fm.

Spectroscopy as a test of Coulomb’s law: A probe of the hidden sector

High precision spectroscopy can provide a sensitive tool to test Coulomb's law on atomic length scales. This can then be used to constrain particles such as extra ``hidden'' photons or minicharged particles that are predicted in many extensions of the standard model, and which cause small deviations from Coulomb's law. In this paper we use a variety of transitions in atomic hydrogen, hydrogenic ions, and exotic atoms to probe Coulomb's law. This extends the region of pure Coulomb's law tests to larger masses. For hidden photons and minicharged particles this region is already tested by other astrophysical and laboratory probes. However, future tests of true muonium and muonic atoms are likely to probe new parameter space and therefore have good discovery potential for new physics. Finally, we investigate whether the discrepancy between the theoretical calculation of the $2{s}_{1/2}^{F=1}\ensuremath{-}2{p}_{3/2}^{F=2}$ transition in muonic hydrogen and its recent experimental measurement at PSI can be explained by the existence of a hidden photon. This explanation is ruled out by measurements of the Lamb shift in ordinary hydrogen.