Krzysztof Pachucki

Muonic hydrogen and the proton radius puzzle

The extremely precise extraction of the proton radius obtained 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 discrepancy is the proton radius puzzle. In this review, we explain the origins of the puzzle and the reasons for believing it to be very significant. We identify various possible solutions of the puzzle and discuss future research needed to resolve the puzzle.

Theoretical Determination of the Dissociation Energy of Molecular Hydrogen

The dissociation energy of molecular hydrogen is determined theoretically with a careful estimation of error bars by including nonadiabatic, relativistic, and quantum electrodynamics (QED) corrections. The relativistic and QED corrections were obtained at the adiabatic level of theory by including all contributions of the order α(2) and α(3) as well as the major (one-loop) α(4) term, where α is the fine-structure constant. The computed α(0), α(2), α(3), and α(4) components of the dissociation energy of the H2 isotopomer are 36 118.7978(2), -0.5319(3), -0.1948(2), and -0.0016(8) cm(-1), respectively, while their sum amounts to 36 118.0695(10) cm(-1), where the total uncertainty includes the estimated size (±0.0004 cm(-1)) of the neglected relativistic nonadiabatic/recoil corrections. The obtained theoretical value of the dissociation energy is in excellent agreement with the most recent experimental determination 36 118.0696(4) cm(-1) [J. Liu et al. J. Chem. Phys. 2009, 130, 174 306]. This agreement would have been impossible without inclusion of several subtle QED contributions which have not been considered, thus far, for molecules. A similarly good agreement is observed for the leading vibrational and rotational energy differences. For the D2 molecule we observe, however, a small disagreement between our value 36 748.3633(9) cm(-1) and the experimental result 36 748.343(10) cm(-1) obtained in a somewhat older and less precise experiment [Y. P. Zhang et al. Phys. Rev. Lett. 2004, 92, 203003]. The reason of this discrepancy is not known.

Nonadiabatic corrections to rovibrational levels of H2

The leading nonadiabatic corrections to rovibrational levels of a diatomic molecule are expressed in terms of three functions of internuclear distance: corrections to the adiabatic potential, the effective nuclear mass, and the effective moment of inertia. The resulting radial Schrodinger equation for nuclear motion is solved numerically yielding accurate nonadiabatic energies for all rovibrational levels of the H(2) molecule. Results for states with J < or = 10 are in excellent agreement with previous calculations by Wolniewicz, and for states with J > 10 are new.

Quantum Electrodynamics Effects in Rovibrational Spectra of Molecular Hydrogen.

The dissociation energies from all rovibrational levels of H2 and D2 in the ground electronic state are calculated with high accuracy by including relativistic and quantum electrodynamics (QED) effects in the nonadiabatic treatment of the nuclear motion. For D2, the obtained energies have theoretical uncertainties of 0.001 cm(-1). For H2, similar uncertainties are for the lowest levels, while for the higher ones the uncertainty increases to 0.005 cm(-1). Very good agreement with recent high-resolution measurements of the rotational v = 0 levels of H2, including states with large angular momentum J, is achieved. This agreement would not have been possible without accurate evaluation of the relativistic and QED contributions and may be viewed as the first observation of the QED effects, mainly the electron self-energy, in a molecular spectrum. For several electric quadrupole transitions, we still observe certain disagreement with experimental results, which remains to be explained.

PROTON STRUCTURE EFFECTS IN MUONIC HYDROGEN

Proton structure effects, including finite size, polarizability, and self-energy are considered and their influence on energy levels of muonic hydrogen is recalculated. A theoretical prediction for the Lamb shift is presented together with improved values of all known QED contributions for muonic hydrogen. This would allow for the substantial improvement of the proton charge radius determination.

Fundamental Vibration of Molecular Hydrogen

The fundamental ground tone vibration of H(2), HD, and D(2) is determined to an accuracy of 2×10(-4) cm(-1) from Doppler-free laser spectroscopy in the collisionless environment of a molecular beam. This rotationless vibrational splitting is derived from the combination difference between electronic excitation from the X(1)Σ(g)(+), v=0, and v=1 levels to a common EF(1)Σ(g)(+), v=0 level. Agreement within 1σ between the experimental result and a full ab initio calculation provides a stringent test of quantum electrodynamics in a chemically bound system.

Complete two-loop correction to the bound-electron g factor

Within a systematic approach based on dimensionally regularized nonrelativistic quantum electrodynamics, we derive a complete result for the two-loop correction to order ({alpha}/{pi}){sup 2}(Z{alpha}){sup 4} for the g factor of an electron bound in an nS state of a hydrogenlike ion. The results obtained significantly improve the accuracy of the theoretical predictions for the hydrogenlike carbon and oxygen ions and influence the value of the electron mass inferred from g-factor measurements.

Bounds on fifth forces from precision measurements on molecules

ioncanbeinterpretedintermsofconstraintsonpossiblefifth-forceinteractions.Wherethehydrogen atom is a probe for yet unknown lepton-hadron interactions, and the helium atom is sensitiveforlepton-lepton interactions, molecules open the domain to search for additional long-range hadron-hadronforces. First principles calculations in the framework of quantum electrodynamics have now advanced tothe level that hydrogen molecules and hydrogen molecular ions have become calculable systems, makingthemasearchgroundforfifthforces.Followingaphenomenologicaltreatmentofunknownhadron-hadroninteractions written in terms of a Yukawapotential of the form V

Born-Oppenheimer potential for HeH+

The Born-Oppenheimer potential for the {sup 1}{Sigma}{sub g}{sup +} state of H{sub 2} is obtained in the range 0.1-20 a.u., using analytic formulas and recursion relations for two-center two-electron integrals with exponential functions. For small distances, the James-Coolidge basis is used, while for large distances, the Heitler-London functions with arbitrary polynomial in electron variables are used. In the whole range of internuclear distance, about 10{sup -15} precision is achieved; as an example, at the equilibrium distance r=1.4011 a.u., the Born-Oppenheimer potential amounts to -1.174 475 931 400 216 7(3). Results for the exchange energy verify the formula of Herring and Flicker [Phys. Rev. A 134, 362 (1964)] for the large-internuclear-distance asymptotics. The presented analytic approach to Slater integrals opens a window for high-precision calculations in an arbitrary diatomic molecule.

A problematic set of two-loop self-energy corrections

We investigate a specific set of two-loop self-energy corrections involving squared decay rates and point out that their interpretation is highly problematic. The corrections cannot be interpreted as radiative energy shifts in the usual sense. Some of the problematic corrections find a natural interpretation as radiative nonresonant corrections to the natural line shape. They cannot uniquely be associated with one and only one atomic level. While the problematic corrections are rather tiny when expressed in units of frequency (a few hertz for hydrogenic P levels) and do not affect the reliability of quantum electrodynamics at the current level of experimental accuracy, they may be of importance for future experiments. The problems are connected with the limitations of the so-called asymptotic-state approximation, which means that atomic in- and out-states in the S-matrix are assumed to have an infinite lifetime.