See L. Fermi, Atoms in the Family (1954, repr. 1988); biography by E. Segrè (1970).
Tree Feynman diagrams describe the interaction remarkably well. Unfortunately, loop diagrams cannot be calculated reliably because Fermi's interaction is not renormalizable. The solution is to replace the four-fermion contact interaction by a more complete theory (see UV completion) — an exchange of a W boson or a Z boson as explained in the electroweak theory. The electroweak theory is renormalizable.
Before the electroweak theory and the Standard Model were constructed, George Sudarshan and Robert Marshak, and also independently Richard Feynman and Murray Gell-Mann were able to determine the correct tensor structure (vector minus axial vector, V−A) of the four-fermion interaction.