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 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.

## Contents

- History of initial rejection and later publication
- Electron state
- Neutrino state
- Heavy particle state
- Hamiltonian
- Matrix elements
- Transition probability
- Forbidden transitions
- Influence
- Later developments
- Fermi coupling constant
- References

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 "because it contained speculations too remote from reality to be of interest to the reader." *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.

## Electron state

## Neutrino state

Similarly,

## Heavy particle state

The operators that change a heavy particle from a proton into a neutron and vice versa are respectively represented by

and

## Hamiltonian

The Hamiltonian is composed of three parts:

where

where

The interaction part must contain a term representing the transformation of a proton into a neutron along with the emission of an electron and a neutrino (now known to be an anti-neutrino), as well as a term for the inverse process; the Coulomb force between the electron and proton is ignored as irrelevant to the

Fermi proposes two possible values for

and subsequently a version assuming that the light particles are four-component Dirac spinors, but that speed of the heavy particles is small relative to

where

## Matrix elements

The state of the system is taken to be given by the tuple

Using the relativistic version of

where the integral is taken over the entire configuration space of the heavy particles (except for

## Transition probability

To calculate the lifetime of a neutron in a state

where

According to Fermi's golden rule, the probability of this transition is

where

Averaging over all positive-energy neutrino spin / momentum directions (where

Noting that the transition probability has a sharp maximum for values of

Fermi makes three remarks about this function:

- Since the neutrino states are considered to be free,
K σ > μ c 2 β -spectrum isH s ≤ W − μ c 2 - Since for the electrons
H s > m c 2 β -decay to occur, the proton–neutron energy difference must beW ≥ ( m + μ ) c 2 - The factor
Q m n ∗ = ∫ v m ∗ u n d τ in the transition probability is normally of magnitude 1, but in special circumstances it vanishes; this leads to (approximate) selection rules forβ -decay.

## Forbidden transitions

As noted above, when the inner product

If the description of the nucleus in terms of the individual quantum states of the protons and neutrons is good,

## Influence

Shortly after Fermi's paper appeared, Werner Heisenberg noted in a letter to Wolfgang Pauli 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

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, or probability of interaction, grows as the square of the energy
*G*_{F} is the Fermi coupling 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 (UV completion)—an exchange of a W or Z boson as explained in the electroweak theory.

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.

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.

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 coupling 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 *G*_{F} (when neglecting the muon mass against the mass of the W boson). In modern terms:

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

More directly, approximately (tree level for the standard model),