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The claim is $r_p m_p = 4 L_0 M_0 = 4\hbar/c$, where $r_p$ and $m_p$ are the proton's charge radius and mass, and $L_0$ and $M_0$ are the Planck length and mass.

Using the muon measurement $r_p=0.84087 \times 10^{-15}$ m, $m_p=1.6726219 \times 10^{-27}$ kg, $r_p m_p = 1.40646 \times 10^{−42}$ kg m.

$4\hbar/c = 1.40706915 \times 10^{-42}$ kg m, agreement to $0.04\%$. That seems pretty incredible.

Another way to put it is as a "prediction" of $r_p = \dfrac{4\hbar}{m_p c} = .84124$ fm.

With the error bars, the muon measurement is $.84087 \pm .00039$ fm, so call it min $.84048$ fm, max $.84126$ fm. The "prediction" is very close to the max.

The CODATA value for the proton's charge radius (electron measurement) is $.8751 \pm .0061 $ fm, close but not so incredibly close as the muon measurement.

Coincidence?

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – David Z May 5 '16 at 13:38
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    $\begingroup$ Voting to leave open because the comments left by CuriousOne (now moved to chat) would make a perfectly good answer. I wish people here were able to distinguish between "science has ultimately determined that it cannot answer this question" and "novice scientists should be disallowed from asking this question." They are two completely different things. $\endgroup$ – Nathaniel May 6 '16 at 12:36
  • $\begingroup$ If anything, the mass should be divided rather than multiplied by the radius, similarly to Newton's law of universal gravitation. Conversely, Planck's constant should be multiplied rather than divided by the speed of light, similarly to the definition of the momentum. $\endgroup$ – Lucian Mar 23 '18 at 16:14
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In one sense, this should not be a coincidence, because the mass and the charge radius are actually determined by more fundamental quantities (quark masses, strong force coupling constant). So given that those quantities have their particular values, in theory it inexorably determines what "mass times charge radius" is going to be.

What should be a coincidence is the appearance of the number 4, when "mass times charge radius" is expressed in natural units. However, since charge radius depends on the distribution of charge inside the proton, it's not inconceivable that, e.g., the distribution of "partons" (quarks and gluons), the distribution of momentum among them, etc., follows some principle which really does imply that the answer is "4 + small corrections". At least, I can't think of a simple reason this should not be so.

For me the deepest fact is that fundamental theory should predict something for this quantity. That prediction might be 4, or it might be e^1.37, but it must be something close to 4.

Update: A comment in chat by @Rococo allows @dandb's observation to be expressed in a very crisp way:

The charge radius of the proton (in muonic hydrogen) is almost exactly four times the reduced Compton wavelength of the proton.

Update 2017: Via P.R. Silva (eqn 6), I have run across a heuristic model of the nucleon in which M = 4/R (in natural units). Here R is the radius of the bag in the "bag model". See Xiangdong Ji, "Mass of the hadron", slide 20. I have not found where this argument originates, but a remark in a 1994 paper by Ji (see paragraph beginning "In the chiral limit...", on the final page) hints at it.

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  • $\begingroup$ What do you mean by "something close to 4"? The same order of magnitude? Or to the nearest whole number? Of course fundamental theory should predict that the product will be of the same order of magnitude as h/c, but isn't that because the OP has chosen appropriate units (Planck length, Planck mass) for measuring length and mass on this scale? Should we be surprised that length of shoes is of the order of magnitude of 1 foot? $\endgroup$ – sammy gerbil May 5 '16 at 11:46
  • $\begingroup$ All I mean is that the empirical value (in these units) is close to 4, so the theoretical prediction had better be close to 4 too. This is true whether or not "4" plays a role in the theoretical calculation. $\endgroup$ – Mitchell Porter May 5 '16 at 12:17
  • $\begingroup$ The 4 is of course a guess, like the whole thing. But it wouldn't be the first time a 4 popped up in a formula. E.g. the ratio of the horizon surface area of a black hole to its entropy is the similar looking $4L_0^2/k$, made out of Boltzmann's constant and the Planck area. $\endgroup$ – dandb May 5 '16 at 12:33
  • $\begingroup$ Thank you so much for your answer. Mr. Porter. I think you are getting at the roots of my exasperation. The coupling constants and quark masses are indeed more fundamental than the proton size & mass. But nothing's more fundamental than $\hbar$ and $c$. It's a sad state of affairs when my feeble attempt to derive particle properties from fundamental constants is laughed off as numerology, instead of lamenting how much we've accepted that we've failed at this task and these quantities will forever remain parameters in our theories. $\endgroup$ – dandb May 5 '16 at 13:36
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    $\begingroup$ @dandb it is unfair to think that theorists are not working on the mass of the proton from basics arxiv.org/abs/0906.0126 . It is just hard and challenging $\endgroup$ – anna v May 5 '16 at 14:06
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Another attractive feature of this conjecture is that it is similar to another conjecture related to hadrons that is known to be true: that the spin of a hadron is equal to the sum of the spins of the quarks in the hadron (which come in discrete half integer increments), even though non-quark partons in the hadron have non-zero spins that "magically" cancel out in the total for reasons that are not well understood (i.e. the "proton spin crisis")). Until we understand why this is the case for spin in hadrons, we can't rule out that this conjecture is exactly true for related reasons.

I'd also note that the in lots of other areas of the Standard Model (e.g. some of the more obscure relationships between electroweak constants in the Standard Model), there are lots of known exact relationships between Standard Model, so it wouldn't be a priori unreasonable to wonder if there was such a relationship here, particularly given that charge radius is an electroweak phenomenon. I don't think that there is any known Standard Model constant relationship that can explain why this conjecture should be true, but the precision of the the conjectured relationship is sufficiently great that it isn't unreasonable to entertain the possibility that it is exactly or exactly subject to small corrections, true.

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