A Notable Coincidence Related To The Proton Mass And Charge Radius

There is a functional relationship between the mass of the proton and the charge radius of the proton that is consistent with experimental measurements of those quantities, that doesn't have an obvious cause.

The simple proton mass and charge radius relationship

From @dandb at Physics Stack Exchange on May 5, 2016 (ten years ago). This can also be stated another way:

The charge radius of the proton is almost exactly four times the reduced Compton wavelength of the proton.

The reduced Planck's constant h-bar, is Planck's constant divided by 2π. So, it could also be stated as r = 2h/πmc, for Planck's constant h, the proton charge radius r, and the proton mass m.

This relationship is consistent with experimental measurements made to 0.05% precision

The uncertainty in the "predicted" value of the charge radius of the proton from this relationship, which is 0.84124 to five significant digits, is negligible, because the speed of light (c) and the reduced Planck's constant (h-bar) are quantities used to define SI units of measurement which are thus known "exactly" in terms of SI units of measurement, and the mass of the proton is known to the exquisite precision of about one part per hundred billion. See the Particle Data Group table of physical constants.

The Particle Data Group world average value is currently 0.8409(4) fm, i.e. a one sigma range of 0.8405 to 0.8413 This is a relative uncertainty of 0.048% (i.e. about one part per two thousand).

The PDG value is also consistent with a February 11, 2026 measurement of the charge radius of the proton with a relative uncertainty of 0.18% published in the prestigious peer reviewed journal Nature found it to be rp = 0.8406(15) fm, i.e. a one sigma range of 0.8392 to 0.8421 fm. 

So, the conjectured relationship is consistent with the experimentally measured value of the charge radius of the proton. 

At the time that this Physics Stack Exchange post was written, there was a discrepancy between the electron measurement of the proton charge radius and the muon measurement of the proton charge radius, but that has since been resolved. The muon measurement was found to be correct, and the electron measurement was found to have been incorrect due to experimental measurement errors not fully reflected in the stated uncertainty of the measurement.

This "prediction" is also notable because it is a testable hypothesis. As measurements of the proton charge radius grow more precise, we can find out if the experimentally measured value continues to be consistent with this prediction.

For example, if this hypothesis is merely numerology with no deeper meaning, it would be highly likely that it would grow less consistent with the experimental measurement if the experimental measurement's precision were increased by a factor of ten.

Analysis

Since the charge radius of the proton and the mass of the proton are both, in principle, derived quantities in the Standard Model, that this isn't actually a "coincidence" so much as it is a simple relationship arising from Standard Model physics whose source isn't trivially obvious.

The reason that it isn't trivially obvious is that the calculation of the mass and charge radius of the proton in the Standard Model are primarily functions at leading order of (1) the QCD coupling constant (which describes the strength of the "strong force"), (2) the mass of the up quark, (3) the mass of the down quark, and (4) the electromagnetic coupling constant. Yet, none of these experimentally measured physical constants have a functional relationship to Planck's constant or the speed of light.

There are comparatively minor contributions to these quantities that tweak their value beyond the leading order values from the masses of the other quarks (especially the strange quark), the weak force coupling constant, the W boson mass, and the CKM matrix elements (especially the  two elements of the nine elements in the matrix involving up-down quark transitions and up-strange quark transitions).

The stack exchange thread linked above contains some speculations as to why this is true, but they are only speculations.

The reason that this relationship is surprising is that there is no known functional relationship between the reduced Planck's constant or the speed of light, and the other experimentally measured determinants of the proton mass and the proton charge radius (such as the Standard Model coupling constants, the quark masses, and the CKM matrix elements).

One possibility, which is to some extent the default one, is that this numerical coincidence of these two values has no deep meaning or connection and doesn't point to anything. In other words, this relationship just happens to hold for one hadron out of hundreds, for one of a large set of possible combinations of other physical constants that have no actually physical relationship to each other.

Another reason that this could be true is that the contributions of the experimentally measured constants cancel out in the combination of the proton mass and the proton charge radius, since the same experimentally measured constants enter into both calculations.

If true, this would suggest that should be a way of calculating the proton charge radius from first principles that more transparently and obviously reveals this cancelation.

This would be very interesting, would provide us to a deeper understand of the Standard Model and hadron physics. 

It would also suggest that this relationship ought be to generalizable in some way to the relationship between hadron mass and hadron charge radius for many hadrons (hadrons are composite particles made up of quark and/or gluons bound by the strong force of the Standard Model).

A calculation in this form would also have practical use, because the first principles Standard Model calculation of the proton mass has less than one part per thousand precision (vastly less than the precision of the experimentally measured value). And, in general, this would provide a quick and easy way to calculate hadron charge radii (which are no more precise than first principles calculations of hadron masses using current methods) which could then be compared to experimental measurements of hadron charge radii.

A third possibility, which would be even more grand, is that the values of the physical constants of the Standard Model that go into calculating the mass and charge radius of the proton actually have some deep functional connection to Planck's constant and the speed of light that has not previously been recognized or hypothesized.