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Opus 200: How Big is the Proton?

by John G. Cramer

Alternate View Column AV-200
Keywords:  proton radius, electron-hydrogen spectroscopy, muon-hydrogen spectroscopy, electron scattering, 
Published in the May-June-2019 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted 01/21/2019 and is copyrighted ©2019 by John G. Cramer.
All rights reserved. No part may be reproduced in any form without
the explicit permission of the auth

This column represents a milestone.  It is the 200th Alternate View column that I have written for Analog.  In 1984 I was on a sabbatical at the Hahn-Meitner Institute in West Berlin, in the days before the Wall came down.    I received a letter from the then-Analog-Editor Stanley Schmidt saying that Jerry Pournelle no longer wanted to do his end of the Alternate View columns in Analog.  Stan asked me if I was interested in replacing Jerry, taking on the responsibility of writing a 2,000 word AV column every two months.  Earlier in my career as a physicist I had decided to make a run at learning to explain physics at the popular level, and I had earlier written several science-fact pieces for Analog, but I was not at all sure that I could do a regular column.  Was 2,000 words really enough space to explain complicated physics ideas?  Would I miss the audience with too much complexity?   What if I couldn't find anything to write about?  Anyhow, after thinking about it for a while I decided to do it, and the rest is history.  So here is my 200th try at explaining a current physics development to you, my loyal readers:

As we now understand it, the proton, the main form of matter in the universe and the nucleus of a hydrogen atom, is not really a fundamental particle.  Rather, it is a composite system consisting of three fundamental particles, two up quarks and one down quark, bound together by the strong force so tightly that the three-body system gives the appearance of a single particle.

The properties of a proton are mass, charge, spin, magnetic moment, and charge radius.    The latter is the radius of a sphere of distributed charge that would produce the same electric field as a proton, both inside and outside.  Experimental physicists have used three ways of measuring the charge radius of the proton: (a) deducing it from the energy levels of the hydrogen atom, i.e. the way an electron orbits a proton (from a considerable distance), (b) deducing it from the way a negative mu lepton (muon) orbits a proton (which is from a much smaller distance), and (c) deducing it by scattering energetic electrons from a proton target.  All three methods have measured a quantity called rp (see below), which is conventionally assumed to be the charge radius of the proton. The problem is that as of 2010, these three methods have given at least two different and inconsistent answers.  Electron-hydrogen spectroscopy reports rp = 0.8768(67) fm (where 1 fm = 10-15 m).  On the other hand, with somewhat more accuracy muon-hydrogen spectroscopy reports rp = 0.84184(67) fm.  The results from electron scattering have greater uncertainties than the spectroscopy values, but give rp = 0.8775(5) fm, which tends to agree with the electron-hydrogen spectroscopy value.

Thus, there is a discrepancy of about 4% in estimates of the proton charge radius given by these trusted methods.  What's going on?  Is there some unsuspected difference in the interaction of protons with electrons vs. with muons that is not included in the Standard Model?  This 4% difference is large enough to represent a serious problem for the physics community, and it has proved difficult to understand.  It has become known as "The Proton Radius Puzzle".

In order to understand the puzzle in more detail, let's examine the measurement methods one at a time, starting with electron scattering, which is similar to Rutherford 's scattering experiments that first established the existence of the atomic nucleus.  Here, a hydrogen gas or solid target is placed in a beam of accelerated electrons, and the probability of electrons scattering in a particular direction with a particular final energy is carefully measured by accumulating a large number of such events.  Because the only significant interaction between an electron and a proton in this situation is the electromagnetic interaction, such scattering data probes the charge structure of the proton.  These data, particularly the results at small momentum transfers between electron and proton, are analyzed to extract the charge radius of the proton.   The accuracy of the radius estimate is limited by counting statistics and by systematic errors.  Data in which the momentum transferred in the scattering process is very small are best at probing the charge distribution, but these are difficult to measure with high statistics.

Next we'll consider muon-hydrogen spectroscopy.  The muon is a fundamental leptonic particle very similar to the electron, except that the muon has a mass that is 207 times larger.  While the electron is stable, the muon is not, and it decays to an electron and two neutrinos with a half life of 2.2 microseconds.  Muons are produced in high-energy particle accelerators by first creating pi mesons (pions) in nuclear collisions and then allowing the pions to decay into muons.  Careful beam handling can slow the muons from such pion decays and deliver them to an experiment with very low kinetic energies.  The muons can then be allowed to be captured by a hydrogen target to form muonic hydrogen atoms, i.e., positively-charged protons orbited by negatively-charged muons.

Since the muon is 207 times more massive than the electron, the electron-like orbits formed by a muon as it orbits a proton in muonic hydrogen are 207 times closer to the proton than an electron's orbits would be.  Quantum mechanics tells us that at such reduced orbital radii, the muon has a high probability of being found inside the proton, where the electric force is lower.  This leads to an easily-observed shift in the orbital energy levels of muonic hydrogen.  For this reason, careful measurement of the energies of photons produced by orbital quantum jumps in muonic hydrogen provide very direct information about the size of the proton.

Finally, let's consider electron-hydrogen spectroscopy.  As in the case of the muon, when an electron orbits a proton its location, as represented by its wave function, is spread out over the entire atom, and a small part of the wave function extends to the proton and intrudes inside.  This small part of the electron present in the proton's interior, much smaller than in the muon case, causes a small but detectable shift in the characteristics energies of the electron's orbits.  Because orbit jumps in the hydrogen atom make photons with energies that can be measured very precisely, this energy shift becomes accessible to measurement and can be used to deduce the proton's charge radius.  As mentioned above, the value of rp given by this analysis is 0.8768(67) fm.

The eminent theoretical physicist Robert G. Sachs (1916-1999), a founder and Director of Argonne National Laboratory in  Illinois and the author of the 1953 standard physics textbook Nuclear Theory (a book that I studied as a graduate student), played a key role in connecting the spectroscopic information to the proton charge radius.  Sachs defined the electric form factor GE(Q2), which represents the probability that, at a given momentum transfer Q, the proton can absorb that momentum while still remaining in its ground state.  The variable GE(Q2) is important because it can be readily extracted from the analysis of spectroscopy or scattering data.

Sachs went on to provide a calculation showing that -6G'E(0) = rp2, where rp is a quantity with the dimensions of distance and the prime (') indicates differentiation with respect to Q2.  Sachs asserted that this rp quantity was the radius of a three-dimensional diffuse sphere of charge representing the proton, i.e., the proton charge radius.   In other words, according to Sachs the magnitude of the slope of the electric form factor as it approaches zero momentum transfer (Q=0) is equal to 1/6 of the proton's charge radius squared.  This Sachs relation has been used in almost all of the electron-hydrogen spectroscopy and electron scattering analyses in extracting the proton's charge radius.  As stated above, the value of rp given by such analyses of electron-hydrogen spectroscopy is 0.8768(69) fm, about 4% larger than the muon value.  This is the core of the puzzle.

From my perspective, the Sachs relation looks fishy because of Heisenberg's uncertainty principle.  Setting the momentum to precisely zero (with no uncertainty) both before and after a proton's interaction with a photon should lead to infinite uncertainty in any related position variable, yet it is used to extract the charge radius of the proton with no uncertainty.   Indeed, my University of Washington colleague, Prof. Gerald A. Miller, has identified the Sachs relation as the source of the discrepancy underlying the Proton Radius Puzzle, because it ignores the effects of special relativity.  The quantity rp is not the proton's charge radius.  It is a definition of a distance-related quantity that can readily be extracted from spectroscopy data, but it bears only a rather complicated relation with the true charge radius of the proton.

We now understand (as no one did when Sachs published his calculation) that the proton is not a static sphere of diffuse electric charge.   Instead, it is a strong-force "box" containing three point-like charged quarks.  The individual quarks have rest masses that are two orders of magnitude smaller than the rest mass is the proton they form.  This huge mass difference between the proton and its components arises from Heisenberg's uncertainty principle.  The quarks, tightly constrained in position by the strong-force interaction between them, must have a large momentum uncertainty.  The quarks therefore violently rattle around in their little box, and their large kinetic energies, called Fermi motion, give the proton most of its mass.  Further, because the quarks' kinetic energies are so much greater that their rest energies, they are highly relativistic particles, and so relativistic effects and Lorentz invariance cannot be ignored.

Miller has reconsidered the electron plus 3-quark-proton problem using relativistic quantum mechanics and has shown that the Sachs relation is simply a definition of rp and cannot be taken directly as an estimate of the proton's charge radius.  A correct relativistically invariant derivation of the relation between orbital energy shifts and the charge distribution of the quarks within the proton shows that one cannot even treat the proton as a static sphere of charge in three dimensions, as Sachs did.  Miller's treatment shows clearly that the quantity being measured in spectroscopy and scattering experiments is the same: it is the slope of GE. This resolves some conceptual issues involving the Proton Radius Puzzle. The actual value of the proton charge radius will be determined by future measurements.  The quantity rp is a convenient variable that is expected to be different when leptons of different masses interact with a proton, and it is not the proton charge radius.

The moral here is that relativity cannot be neglected, even in a simple system like the hydrogen atom.  Einstein is not to be ignored.

John G. Cramer's 2016 nonfiction book (Amazon gives it 5 stars) describing his transactional interpretation of quantum mechanics, The Quantum Handshake - Entanglement, Nonlocality, and Transactions, (Springer, January-2016) is available online as a hardcover or eBook at: or

SF Novels by John Cramer: Printed editions of John's hard SF novels Twistor and Einstein's Bridge are available from Amazon at and His new novel, Fermi's Question may be coming soon.

Alternate View Columns Online: Electronic reprints of 212 or more "The Alternate View" columns by John G. Cramer published in Analog between 1984 and the present are currently available online at: .


The Proton Radius Puzzle:

"Proton radius puzzle", Wikipedia,

"Defining the Proton Radius: a Unified Treatment", Gerald A. Miller, to be published in Phys. Rev. C,

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