Both electrons and photons are defined in the SM as point-like particles. The electron does have rest mass while the photon is massless.

Now if I search for the upper limit on the size of the electron, I find a lot of questions, experiments, and even the official Particle Data Group list the upper limit on the electron radius (page 109).

The only official looking paper I have found on photon radius is this:

https://arxiv.org/ftp/arxiv/papers/1604/1604.03869.pdf

So far so good. Now if I try to search for the upper limit on the radius of the photon, I find a big empty nothing (official at least, and we do not rely on popsci articles). Experiments are usually for the photon mass (but not the radius). Why is that? Is it just the rest mass? Or is there another fundamental difference (EM charge maybe) that somehow makes a experiment able to be performed for the electron radius but not the photon radius? Most of the experiments and calculations for electrons talk about form factors, are these available for electrons only and not photons?

Question:

1. Why is there no official word on the upper limit of the photon radius (while there is for the radius of the electron)?
• The term "radius" is a bit ambiguous when applied to a photon. Any experiment that fires a single photon at a time will produce a radius estimate that is at least as big as the photon's wavelength. Jan 14 '20 at 22:50

If you think of "size" as meaning wavelength, then there is no upper bound on the size of a photon.

But usually in particle physics, when we talk about particles as having a size $$x$$, what we mean is that they're composites of other particles, which are bound together on distances at that scale. An example would be that a proton is a composite of quarks.

If photons were like this, then we would expect to see big errors in the predictions of QED when we did experiments with photons and charged particles having wavelengths of order $$x$$, and smaller errors even at wavelengths bigger than $$x$$. QED is actually incredibly accurate in predicting things like $$g-2$$ of the electron, so this constraint puts $$x$$ at many orders of magnitude less than the shortest wavelengths probed. So roughly speaking, that's your bound: less than the shortest wavelengths probed in particle physics experiments, and many orders of magnitude less than the shortest wavelengths probed in high-precision experiments.

If you want to make the bound more precise than that, then you're going to have to come up with a test theory that predicts something different from what QED predicts. One problem you're going to have constructing such a theory is that if it says a photon is a composite object with size $$x$$, then we would naively expect the energy of its constituents to be $$\gtrsim hc/x$$. If $$x$$ is, say, $$10^{-17}$$ meters, then this is $$10^{-8}$$ joules, which is equivalent to about $$10^{-25}$$ kg. But it's very difficult to make reasonable theories in which photons have mass, and empirical model-independent bounds on the mass of the photon are about $$10^{-47}$$ kg, or even lower if you use model-dependent experiments (Nieto 1992).

This is not to say that the naive bound of $$E\gtrsim hc/x$$ has to be right. E.g., the pion violates this bound.

By the way, you mention the classical electron radius in your question, but that's not a bound on the size of the electron. It's basically just the scale at which the classical model of the electron breaks down.

Reference

• thank you, can you please elaborate a little bit on "It's basically just the scale at which the classical model of the electron breaks down."? Jan 15 '20 at 2:21
• @ÁrpádSzendrei: You might want to ask a separate question on that. But basically if you integrate the energy density of the field of a point charge, you get infinity. This means that a point charge should have infinite mass, which is not what we observe. Therefore at small distances, the picture can't be classical. You can picture it in terms of a soup of virtual particles and antiparticles being created and annihilating.
– user4552
Jan 15 '20 at 2:34
• thank you so much Jan 15 '20 at 2:35

According to QM, all is made of waves. Then, one could divide everything into linear EM waves and circular (ring) EM waves, or particles. So, E= h/T could be used in the following way. h/E = T and T x c gives the wavelength or circumference of this ring wave. From there, divide by 2pi and get the radius of the ring wave...

Could be a fluke ... but, do this and you get 3.864258062x10-13m the electron radius of POELZ (2020)