Charge radius of a proton as a function of its mass

[Cyril]

Hello !

I would like to know if the following relation, which numerically holds true according to the data available on wikipedia, can be analytically deduced from the current standard physics and model of a proton:

R = 4 x L x ( M / m )
=4 x 1.61619997e−35 x 2.1765113e−8 / 1.67262177774e−27 = 8.412368e-16 m which is correct according to wikipedia.
With R = charge radius of a proton, L = Planck length, M = Planck mass, m = Proton mass

As this relation is very simple and as it only uses fundamental constants such as Planck length and Planck mass, I would expect this relation to be found easily, maybe "by definition" or trivial equality. But I can't find this equality on the web in any standard physics document.

Can you tell me how to prove this? Is it a trivial equality?
(Please note I am not a physicist, but I have a scientific background)

Thank you very much!
Cyril

 

[mfb]

$$L\cdot M = \frac{\hbar}{c}$$
In natural units, your equation just reads R=4/m. The proportionality is expected from the uncertainty relation.
The proton charge radius measurements show a discrepancy of ~5% between regular and muonic hydrogen, that is not sufficient to see if the prefactor might be just a coincidence or something with a deeper meaning.

 

[Cyril]

Thank you very much for your answer mfb.

Could you detail a bit more how the proportionality is expected from the uncertainty relation?
If it is expected, what is the analytic expression of the proportionality factor?

 

[mfb]

If you want to confine something within a box of size 2r, its position uncertainty is smaller than r, which means its momentum uncertainty has to be larger than $\frac{\hbar}{2r}$. For r=0.8fm, this leads to ~117 MeV/c. That means the quarks are highly relativistic, and E≈pc, so the quarks have to be quite high-energetic.
This is a heuristic argument, it is one-dimensional and it does not take the strong interaction into account, but it shows the general idea. Distance and energy scales are always related in quantum mechanics.
I don't think there is an analytic prediction. Calculating the proton mass from scratch is very tricky even with numerical methods.