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Today, we consider quarks and electrons (leptons) as point-like or fundamental (structureless). Is there any way to indirectly probe quark/lepton substructure and guess if they are composite of something else and no point-like? If the compositeness scale of quark and leptons were different, any way to test it?

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There is ongoing research on elementary particle experiments on possible compositeness of elementary particles considered point particles in the standard model . See this review and links therein for methods and experiments and results.

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About the relative size of electrons and quarks, a 2016 paper analyzing the data from HERA estimated the upper limit on the quark radius to be at most $0.43 \times 10^{-18}$ meters. A more readable description of that paper is here.

The electron radius has been discussed here on Physics.SE before: Experimental boundaries for size of electron?. It has been quoted to be anywhere from at least as small as $10^{-21}$ meters in this not-so-widely-accepted 1988 paper and at least as big as the classical radius $2.817 940 3267 × 10^{-15} $ meters on pg 109 of the Particle Data Group's 2014 report (see PDF here). This paper also estimates a lower bound on the radius of the electron's "vacuum core" to be $4.9 \times 10^{-26}$.

Since the highest-resolution Scanning tunneling microscopes can still only distinguish features at the scale of a few picometers, there is no way to observationally confirm or deny the above numbers. There is also something called a "charge radius" which can be determined using spectroscopy, but even for the proton, which is likely to be bigger than quarks or electrons, the charge radius is at the center of one of the biggest open problems in physics right now: The proton radius puzzle.

The bottom line is that QCD theory places the radius of quarks at zero and the standard model places the radius of electrons at zero too. Any theory that predicts these radii to be any bigger would be considered a "beyond-standard-model theory" or fails at some scale (like classical mechanics does). There may be beyond-standard-model theories out there that predict sizes for electrons and/or quarks (one might be the Rishon model which all quarks and electrons are composed of three preons; one might be string theory in which rather than fundamental point-like 0-dimensional particles we have vibrating 1D strings), but such theories have not (yet) made a single experimentally confirmed prediction that can't already be predicted using the standard model or general relativity, so your question may just need to be revisited in the future :)

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  • $\begingroup$ To be mildly snarky, QCD theory shouldn't have much to say at all about the radius of the electron, which does not interact via the strong force. $\endgroup$ May 29, 2021 at 20:19
  • $\begingroup$ @JerrySchirmer It does about quarks though. I'll re-word that sentence to say SM rather than QCD :) $\endgroup$ May 29, 2021 at 20:22
  • $\begingroup$ @JerrySchirmer I disagree. I prefer to think of the electron’s “intrinsic radius” as a length scale where some new electron-specific interaction turns on (like how the nucleon radius is set by the Yukawa range of the light mesons). We have evidence against any such interaction at length- and energy-scales where QCD absolutely matters. $\endgroup$
    – rob
    May 30, 2021 at 2:51
  • $\begingroup$ Many e.g. the "~10^-21m" estimates come from g-factor argument, which is a nonsense: extrapolation by fitting parabola to two points(!) - you can "predict" whatever you want with such overfitting ( physics.stackexchange.com/questions/397022/… ). While these particles rather have no further parton structure, not to exceed 511keV mass with energy of electric field alone, this field needs to be deformed in fm-scale, size also suggested by electron-positron scattering. $\endgroup$
    – Jarek Duda
    May 30, 2021 at 7:02
  • $\begingroup$ I already said that the parabolic fit paper is "not-so-widely-accepted". $\endgroup$ May 30, 2021 at 16:07
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If we could isolate quarks, and then use particle accelerators to bombard them with each other, we most certainly could find something. But due to Confinement, we are unable to do so.

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