A new calculation of the electron’s self-energy improves determination of fundamental constants

When quantum electrodynamics, the quantum field theory of electrons and photons, was being developed after World War II, one of the major challenges for theorists was calculating a value for the Lamb shift, the energy of a photon resulting from an electron transitioning from one hydrogen hyperfine energy level to another.

The effect was first detected by Willis Lamb and Robert Retherford in 1947, with the emitted photon having a frequency of 1,000 megahertz, corresponding to a photon wavelength of 30 cm and an energy of 4 millionths of an electronvolt—right on the lower edge of the microwave spectrum. It came when the one electron of the hydrogen atom transitioned from the 2P_1/2 energy level to the 2S_1/2 level. (The leftmost number is the principal quantum number, much like the discrete but increasing circular orbits of the Bohr atom.)

Conventional quantum mechanics didn’t have such transitions, and Dirac’s relativistic Schrödinger equation (naturally called the Dirac equation) did not have such a hyperfine transition either, because the shift is a consequence of interactions with the vacuum, and Dirac’s vacuum was a “sea” that did not interact with real particles.

As theorists worked to produce a workable theory of quantum electrodynamics (QED), predicting the Lamb shift was an excellent challenge as the QED calculation contained the prominent thorns of the theory, such as divergent integrals at both low and high energies and singularity points.

On Lamb’s 65th birthday in 1978, Freeman Dyson said to him, “Those years, when the Lamb shift was the central theme of physics, were golden years for all the physicists of my generation. You were the first to see that this tiny shift, so elusive and hard to measure, would clarify our thinking about particles and fields.”

Precisely predicting the Lamb shift, as well as the anomalous magnetic moment of the electron, has been a challenge for theorists of every generation since. The theoretically predicted value for the shift allows the fine-structure constant to be measured with an uncertainty of less than one part in a million.

Now, a new step in the evolution of the Lamb shift calculation has been published in Physical Review Letters by a group of three scientists from the Max Planck Institute for Nuclear Physics in Germany. To be exact, they calculated the “two-loop” electron self-energy.

Self-energy is the energy a particle (here, an electron) has as a result of changes that it causes in its environment. For example, the electron in a hydrogen atom attracts the proton that is the nucleus, so the effective distance between them changes.

QED has a prescription to calculate the self-energy, and it’s easiest via Feynman diagrams. “Two-loops” refers to the Feynman diagrams that describe this quantum process—two virtual photons from the quantum vacuum that influence the electron’s behavior. They pop in from the vacuum, stay a shorter time than is set by the Heisenberg Uncertainty Principle, then are absorbed by the 1S electron state, which has spin 1/2.

Accounting for the two-loop self-energy is one of only three mathematical terms that describe the Lamb shift, but it constitutes a major problem that most influences the result for the Lamb energy shift.

Lead author Vladimir Yerokhin and his colleagues determined an enhanced precision for it from numerical calculations. Importantly, they calculated the two-loop correction to all orders in an important parameter, Zα that represents the interaction with the nucleus. (Z is the atomic number of the nucleus. The atom still has only one electron, but a nucleus bigger than hydrogen’s is included for generality. α is the fine structure constant.)

Although it was computationally challenging, the trio produced a significant improvement on previous two-loop calculations of the electron self-energy that reduces the 1S–2S Lamb shift in hydrogen by a frequency difference of 2.5 kHz and reduces its theoretical uncertainty. In particular, this reduces the value of the Rydberg constant by one part in a trillion.

Introduced by the Swedish spectroscopist Johannes Rydberg in 1890, this number appears in simple equations for the spectral lines of hydrogen. The Rydberg constant is a fundamental constant that is one of the most precisely known constants in physics, containing 12 significant figures with, previously, a relative uncertainty of about two parts in a trillion.

Overall, they write, “the calculational approach developed in this Letter allowed us to improve the numerical accuracy of this effect by more than an order of magnitude and extend calculations to lower nuclear charges [Z] than previously possible.” This, in turn, has consequences for the Rydberg constant.

Their methodology also has consequences for other celebrated QED calculations: other two-loop corrections to the Lamb shift, and especially to the two-loop QED effects for the anomalous magnetic moment of the electron and the muon, also called their “g-factors.” A great deal of experimental effort is currently being put into precisely determining the muon’s g-factor, such as the Muon g-2 experiment at Fermilab, as it could point the way to physics beyond the standard model.

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