If quarks had internal structure (contradicting current beliefs), what is the lowest
upper bound on their "radius" based on current experimental results?
If possible, I'd prefer to only consider experiments which probe protons and
neutrons (not other shorter-lived particles since their interpretations get biased
more by the standard model).
My only understanding is that this radius must be less than roughly 0.2 fm since
spacings are found to be 2 fm in high-energy proton scattering experiments. I
imagine "higher-energy scattering experiments" and "excited angular momentum
experiments" have probed this further, but am not familiar with any other results.
Or, is there some other reason why this radius must be zero? Honestly, with the
surprise of quarks 10,000x smaller than the electron cloud, it wouldn't be surprising
if we found some internal structure after another 10,000x zoom.
Best Answer
As I mentioned in another answer, what people actually report these days is not really an upper bound on the radius of a particle. Instead, what you'll find is a lower bound on what is sometimes called the "contact interaction scale" - the energy at which you start to see effects of interactions among the constituents of a quark, if they exist.
For example, the most recent information I can find is this paper from the CMS experiment. It presents lower bounds on the contact interaction scale ranging from $7.5\text{ TeV}$ to $14.5\text{ TeV}$, depending on which model of substructure you're looking at. (In order to extract a lower bound from the data you get out of the detector, you need to make some assumptions about what kind of substructure you might be looking for.) So roughly speaking, we're reasonably sure that the types of substructure considered in the paper do not have any effect at processes involving less than $7.5\text{ TeV}$ of energy.
You can convert these limits into distances using the formula $\lambda = h c/E$, which tells you the wavelength corresponding to a particle with that limiting energy. This is just a rough order-of-magnitude bound, but it's as close as you can get to declaring an upper bound on the quark's radius with the knowledge we have today. Based on the values in the paper, it's $1.6\times 10^{-19}\text{ m}$.