thumb|360px| decay in an [[atomic nucleus (the accompanying antineutrino is omitted).
The inset shows beta decay of a free neutron. In both processes, the intermediate emission of a virtual boson (which then decays to electron and antineutrino) is not shown.]]
In particle physics, Fermi's interaction (also the Fermi theory of beta decay or the Fermi four-fermion interaction) is an explanation of the beta decay, proposed by Enrico Fermi in 1933. The theory posits four fermions directly interacting with one another (at one vertex of the associated Feynman diagram). This interaction explains beta decay of a neutron by direct coupling of a neutron with an electron, a neutrino (later determined to be an antineutrino) and a proton.
Fermi first introduced this coupling in his description of beta decay in 1933. The Fermi interaction was the precursor to the theory for the weak interaction where the interaction between the proton–neutron and electron–antineutrino is mediated by a virtual W<sup>−</sup> boson, of which the Fermi theory is the low-energy effective field theory.
According to Eugene Wigner, who together with Pascual Jordan introduced the Jordan–Wigner transformation, Fermi's paper on beta decay was his main contribution to the history of physics.
History of initial rejection and later publication
Fermi first submitted his "tentative" theory of beta decay to the prestigious science journal Nature, which rejected it "because it contained speculations too remote from reality to be of interest to the reader." It has been argued that Nature later admitted the rejection to be one of the great editorial blunders in its history, but Fermi's biographer David N. Schwartz has objected that this is both unproven and unlikely. Fermi then submitted revised versions of the paper to Italian and German publications, which accepted and published them in those languages in 1933 and 1934. The paper did not appear at the time in a primary publication in English. that the emission and absorption of neutrinos and electrons in the nucleus should, at the second order of perturbation theory, lead to an attraction between protons and neutrons, analogously to how the emission and absorption of photons leads to the electromagnetic force. He found that the force would be of the form <math>\frac{\text{Const.{r^5}</math>, but noted that contemporary experimental data led to a value that was too small by a factor of a million.
The following year, Hideki Yukawa picked up on this idea, but in his theory the neutrinos and electrons were replaced by a new hypothetical particle with a rest mass approximately 200 times heavier than the electron.
Later developments
Fermi's four-fermion theory describes the weak interaction remarkably well. Unfortunately, the calculated cross-section, the probability of the interaction multiplied by the common interaction area, grows as the square of the energy <math> \sigma \approx G_{\rm F}^2 E^2 </math>. Since this cross section grows without bound, the theory is not valid at energies much higher than about 100 GeV. Here is the Fermi constant, which denotes the strength of the interaction. This eventually led to the replacement of the four-fermion contact interaction by a more complete theory (ultraviolet completion)—an exchange of a W or Z boson as explained in the electroweak theory.
x110px|thumb|right|Fermi's interaction showing the 4-point fermion vector current, coupled under Fermi's Coupling Constant . Fermi's Theory was the first theoretical effort in describing nuclear decay rates for β decay.
The interaction could also explain muon decay via a coupling of a muon, electron-antineutrino, muon-neutrino and electron, with the same fundamental strength of the interaction. This hypothesis was put forward by Semyon Gershtein and Yakov Zeldovich and is known as the vector current conservation hypothesis.
In the original theory, Fermi assumed that the form of interaction is a contact coupling of two vector currents. Subsequently, it was pointed out by T.-D. Lee and C. N. Yang that nothing prevented the appearance of an axial, parity violating current, and this was confirmed by experiments carried out by Chien-Shiung Wu.
The inclusion of parity violation in Fermi's interaction was done by George Gamow and Edward Teller in the so-called Gamow–Teller transitions which described Fermi's interaction in terms of parity-violating "allowed" decays and parity-conserving "superallowed" decays in terms of anti-parallel and parallel electron and neutrino spin states respectively. Before the advent of the electroweak theory and the Standard Model, 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, ) of the four-fermion interaction.
Fermi constant
The most precise experimental determination of the Fermi constant comes from measurements of the muon lifetime, which is inversely proportional to the square of (when neglecting the muon mass against the mass of the W boson). In modern terms, the "reduced Fermi constant", that is, the constant in natural units is
:<math>G_{\rm F}^0=\frac{G_{\rm F{(\hbar c)^3}=\frac{\sqrt{2{8}\frac{g^{2{M_{\rm W}^{2} c^4}=1.1663787(6)\times10^{-5} \; \textrm{GeV}^{-2} \approx 4.5437957\times10^{14} \; \textrm{J}^{-2}\ .</math>
Here, is the coupling constant of the weak interaction, and is the mass of the W boson, which mediates the decay in question.
In the Standard Model, the Fermi constant is related to the Higgs vacuum expectation value
:<math>v = \left(\sqrt{2} \, G_{\rm F}^0\right)^{-1/2} \simeq 246.22 \; \textrm{GeV}</math>.
More directly, approximately (tree level for the standard model),
:<math>
G_{\rm F}^0\simeq \frac {\pi \alpha}{\sqrt{2}~ M_{\rm W}^2 (1- M^2_{\rm W}/M^2_{\rm Z} )}.
</math>
This can be further simplified in terms of the Weinberg angle using the relation between the W and Z bosons with <math>M_\text{Z}=\frac{M_\text{W{\cos\theta_\text{W</math>, so that
:<math>
G_{\rm F}^0\simeq \frac {\pi \alpha}{\sqrt{2}~ M_{\rm Z}^{2}\cos^{2}\theta_{\rm W}\sin^{2}\theta_{\rm W.
</math>