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# Fermi's interaction

In particle physics, Fermi's interaction also known as Fermi coupling, is an old explanation of the weak force, proposed by Enrico Fermi, in which four fermions directly interact with one another at one vertex.[1] For example, this interaction explains beta decay of a neutron by direct coupling of a neutron (two down quarks and an up quark) with an electron, antineutrino and a proton (two up quarks and a down quark). The interaction could also explain muon decay via a coupling of a muon, electron-antineutrino, muon-antineutrino and electron. Fermi first introduced this coupling in his description of beta decay in 1933.[2]

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 or Z boson as explained in the electroweak theory. The electroweak theory is renormalizable.

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 GF. The most precise experimental determination of the Fermi constant comes from measurements of the muon lifetime, which is inversely proportional to the square of GF.[3] In modern terms:[2]

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

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

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

References

^ Feynman, R.P. (1962). Theory of Fundamental Processes. W.A. Benjamin. Chapters 6&7

^ a b Griffiths, David (2009). Introduction to Elementary Particles. pp. 314–315. ISBN 978-3-527-40601-2.

^ D. Chitwood et al., Phys.Rev.Lett. 99 (2007) 032001

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