The Proton Radius Problem

The hydrogen atom is one of the primary examples studied in a typical introductory quantum mechanics course. Recent measurements indicate that this simple system may still have surprises for us. Could this be a hint of new physics? This post is based on the following papers:

“Muonic hydrogen and MeV forces” by D. Tucker-Smith and I. Yavin [1011.4922], Phys. Rev. D83 (2011) 101702

“Proton size anomaly” by V. Barger, C. Chiang, W. Keung, D. Marfatia [1011.3519], Phys. Rev. Lett. 106 (2011) 153001

“The Size of the Proton” by Pohl et al. in Nature 466 (2010) 213

Quantum mechanically, the proton is an object whose electric charge is smeared out over a small region. Experiments that scatter electrons off protons can probe this spatial extent and recent measurements indicate an effective proton charge radius of 0.877(7) femtometers.

Electron scattering experiments see a proton charge radius of 0.88 fm.
Electron scattering experiments measure a particular proton radius. (Image by the author.)

Muons are heavy copies of electrons and can similarly form muonic hydrogen: an atom formed from a proton and a muon. Because the muons are heavier, they exist closer to the nucleus and are more sensitive to the extent of the proton charge: the effective Coulomb force is reduced as one dips into the charge distribution in the same way that the gravitational force decreases as one digs towards the center of the Earth.

By ‘tickling’ the muon into a higher energy level with a laser and then measuring the resulting X-ray emission, one can deduce the proton radius. Since lasers can be tuned to very precise frequencies, one can make a very precise measurement of the Lamb shift in the muonic hydrogen energy levels. This, in turn, can be converted into a measurement of the proton radius because the energy levels are sensitive to the overlap of the muon and proton probability distributions. Intuitively, when the muon is inside the proton charge radius, it experiences a weaker Coulomb potential due to screening.

The big surprise is that the muonic hydrogen measurement gives a radius of 0.842(7) femtometers, this is over five standard deviations smaller than the expected result based on regular hydrogen!

Measurements of the proton charge radius from the lamb shift of muonic hydrogen (a proton--muon bound state) are smaller than that from electron scattering.
Measurements of the proton charge radius from the Lamb shift of muonic hydrogen are smaller than that from electron scattering by five standard deviations. (Image by the author)

This discrepancy remains an open question despite several proposed solutions based on more precise theoretical calculations to relate the Lamb shift to the proton radius. One optimistic approach is to entertain the possibility that this is an indicator of new fundamental physics, such as a heretofore undiscovered force that tugs on the muon and electron differently. It turns out that these types of models are difficult to construct. One of the main constraints is actually nearly 40 years old and comes from the effect of such a new force on neutron–lead scattering.

Meanwhile, a new set of experiments to probe the proton radius anomaly are already underway. One of these is the Muon-Proton Scattering Experiment (MUSE); this would directly probe if the origin of the discrepancy came from the two different proton radius measurements described above: scattering for electrons versus spectroscopy for muons.

Further reading:

  • 1301.0905: a recent review covering theoretical and experimental aspects of the proton radius problem
  • The Proton Radius Problem,” J. Bernauer and R. Pohl in Scientific American, Feb. 2014. [paywall]
  • 1303.2160: a summary of the upcoming MUSE experiment to test muon-proton scattering