Fermi's interaction

Not to be confused with Fermi contact interaction.

β− 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 W− boson (which then decays to electron and antineutrino) is not shown.

In particle physics, Fermi's interaction (also the Fermi theory of beta decay) 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.
For example, this interaction explains beta decay of a neutron by direct coupling of a neutron with:

1. an electron,
2. an antineutrino and
3. 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⁻ boson.

History of initial rejection and later publication

Fermi first submitted his "tentative" theory of beta decay to the famous science journal Nature, which rejected it for being "too speculative." Nature later admitted the rejection to be one of the great editorial blunders in its history. Fermi then submitted the paper to Italian and German publications, which accepted and published it in 1933 in those languages, but it did not appear at the time in a primary publication in English (Nature finally belatedly republished Fermi's report on beta decay in English on January 16, 1939).

Fermi found the initial rejection of the paper so troubling that he decided to take some time off from theoretical physics, and do only experimental physics. This would lead shortly to his famous work with activation of nuclei with slow neutrons.

The nature of the interaction

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 Gershtein and Zeldovich and is known as the Vector Current Conservation hypothesis.

Fermi's four-fermion theory describes the weak interaction remarkably well. Unfortunately, the calculated cross-section grows as the square of the energy $\sigma \approx G_{\rm F}^2 E^2$, making it unlikely that the theory is valid at energies much higher than about 100 GeV. The solution is to replace the four-fermion contact interaction by a more complete theory (UV completion)—an exchange of a W or Z boson as explained in the electroweak theory.

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 Lee and Yang that nothing prevented the appearance of an axial, parity violating current, and this was confirmed by experiments carried out by Chien-Shiung Wu.

Fermi's interaction showing the 4-point fermion vector current, coupled under Fermi's Coupling Constant GF. Fermi's Theory was the first theoretical effort in describing nuclear decay rates for Beta-Decay.

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, V − A) of the four-fermion interaction.

Fermi constant

The strength of Fermi's interaction is given by the Fermi coupling constant G_F. The most precise experimental determination of the Fermi constant comes from measurements of the muon lifetime, which is inversely proportional to the square of G_F (when neglecting the muon mass against the mass of the W boson). In modern terms:

$\frac{G_{\rm F}}{(\hbar c)^3}=\frac{\sqrt{2}}{8}\frac{g^{2}}{m_{\rm W}^{2}}=1.16637(1)\times10^{-5} \; \textrm{GeV}^{-2} \ .$

Here g is the coupling constant of the weak interaction, and m_W is the mass of the W boson which mediates the decay in question.

In the Standard Model, Fermi's constant is related to the Higgs vacuum expectation value $v = (\sqrt{2}G_{\rm F})^{-1/2} \simeq 246.22 \; \textrm{GeV}$