How did physicists run tests to try to study the internal structure of an electrons? How did physicists run tests to try to study the internal structure of an electrons? How specifically did they run tests to try and study it? I've looked around online and I can't seem to find any studies done. If anyone could explain this to me or point me in the right direction it would be most appreciated.
 A: Physicists used accelerators such as the Stanford Linear Collider to study high-energy collisions between electrons and positrons. These experiments found that neither particle has internal constituents at the length scales that the accelerator could probe. They behave as fundamental, non-composite particles, as the Standard Model of particle physics assumes.
However, they are not as simple as the point particles that the Dirac equation describes. Quantum field theory makes them more complicated objects, “dressed” with virtual particles due to their electromagnetic and weak interactions.
The International Linear Collider is a proposed electron-positron collider, 30-50 kilometers long, that would use higher energies to probe smaller length scales.
Since electrons and positron do not, apparently, have constituents, their collisions are “clean”. There is no question about what is colliding with what, so the results are easier to analyze.
A: The electron is a spin 1/2 particle. This severely limits the possible structure it can have. If we ignore the weak interaction, and only consider its charge, then it talks to the rest of the world through a photon vertex, e.g.:

The figure shows a presumably structureless electron on the left (coupling through the Dirac matrices, $\gamma_{\mu}$, and a well know spin 1/2 particle with structure on the right). The coupling (in elastic scattering) must have the form:
$$ \Gamma_{\mu} = \gamma_{\mu}F_1(Q^2) + \frac{i\sigma_{\mu\nu}}{2M}q^{\nu}F_2(Q^2)$$
where $q^{\mu}$ is the 4-momentum of the virtual photon and $Q^2=-q^{\mu}q_{\mu}$ describes the (inverse) length scale (squared) at which the photon probes the target.
The Dirac and Pauli form factors ($F_1(Q^2)$ and $F_2(Q^2)$) parameterize the target structure as a function of length scale.
At low $Q^2$, or large distance scales, the form factors must be:
$$ F_1(0) = 1 $$
(which means all the charge is contained in the region probed by the photon), and
$$ F_2(0) = \kappa $$
where $$\kappa=0$$ is the first order anomalous magnetic moment of the target (not the QED higher order moment referenced in G Smith's answer).
Of course, the electron doesn't have a $\kappa$, because it has no structure, so it's not even "a thing", but if it did, it would be $0$.
The above diagram leads to a scattering cross section that looks like:
$$\frac{d\sigma}{d\Omega}=
\big(\frac{d\sigma}{d\Omega}\big)_{\rm Mott} \times
\frac{\epsilon G^2_E(Q^2) + \tau G^2_M(Q^2)}{\epsilon(1+\tau)} $$
(Here $G_E$ and $G_M$ are the Sachs electric and magnetic form factors, which are linear functions of the Dirac and Pauli form factors. The $\epsilon$ and $\tau$ are kinematic factors related to the photon polarization and energy. All the is the subject of Robert Hofstadter's 1961 Nobel Prize).
The Mott cross section is the cross section for structureless spin 1/2 scattering. If there is structure, then the Mott cross section is modified by form factors that parameterize the structure. For example, $F_1(Q^2)$ is the Fourier transform of the radial charge distribution.
Moreover, if there is structure, you can eventually break the target apart. This leads to more functions that parameterize the internal structure. For nucleons, it is called Deep Inelastic Scattering, and is the subject of the 1990 Nobel Prize.
All this formalism exists for the neutron and proton, because they have structure, and none exists for the electron because it (so far) dose not.
Had the electron shown structure, then it probably would have been probed by Compton scattering, where the point-like $\gamma e^-$ vertex would have been modified by structure functions.
