Fermi's interaction should not be confused with Fermi contact interaction.
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.[1] 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.[2]
Fermi first introduced this coupling in his description of beta decay in 1933.[3] 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, of which the Fermi theory is the low-energy effective field theory.
According to Eugene Wigner, who together with Jordan introduced the Jordan–Wigner transformation, Fermi's paper on beta decay was his main contribution to the history of physics.[4]
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."[5] [6] 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.[7] 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.[8] [9] [10] [11] The paper did not appear at the time in a primary publication in English.[5] An English translation of the seminal paper was published in the American Journal of Physics in 1968.[11]
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 theory deals with three types of particles presumed to be in direct interaction: initially a “heavy particle” in the “neutron state” (
\rho=+1
\rho=-1
\psi=\sums\psisas,
where
\psi
\psis
as
s
as\Psi(N1,N2,\ldots,Ns,\ldots)=
N1+N2+ … +Ns-1 | |
(-1) |
(1-Ns)\Psi(N1,N2,\ldots,1-Ns,\ldots).
* | |
a | |
s |
s:
* | |
a | |
s |
\Psi(N1,N2,\ldots,Ns,\ldots)=
N1+N2+ … +Ns-1 | |
(-1) |
Ns\Psi(N1,N2,\ldots,1-Ns,\ldots).
Similarly,
\phi=\sum\sigma\phi\sigmab\sigma,
where
\phi
\phi\sigma
b\sigma
\sigma
b\sigma\Phi(M1,M2,\ldots,M\sigma,\ldots)=
M1+M2+ … +M\sigma-1 | |
(-1) |
(1-M\sigma)\Phi(M1,M2,\ldots,1-M\sigma,\ldots).
* | |
b | |
\sigma |
\sigma
\rho
\begin{pmatrix}1\\0\end{pmatrix}
represents a neutron, and
\begin{pmatrix}0\\1\end{pmatrix}
represents a proton (in the representation where
\rho
\sigmaz
The operators that change a heavy particle from a proton into a neutron and vice versa are respectively represented by
Q=\sigmax-i\sigmay=\begin{pmatrix}0&1\ 0&0\end{pmatrix}
and
Q*=\sigmax+i\sigmay=\begin{pmatrix}0&0\ 1&0\end{pmatrix}.
un
vn
n
The Hamiltonian is composed of three parts:
Hh.p.
Hl.p.
Hint.
Hh.p.=
1 | |
2 |
(1+\rho)N+
1 | |
2 |
(1-\rho)P,
where
N
P
\rho=1
Hh.p.=N
\rho=-1
Hh.p.=P
Hl.p.=\sumsHsNs+\sum\sigmaK\sigmaM\sigma,
where
Hs
sth
Ns
M\sigma
\sigmath
K\sigma
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 antineutrino), as well as a term for the inverse process; the Coulomb force between the electron and proton is ignored as irrelevant to the
\beta
Fermi proposes two possible values for
Hint.
Hint.=g\left[Q\psi(x)\phi(x)+Q*\psi*(x)\phi*(x)\right],
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
c
Hint.=g\left[Q\tilde{\psi}*\delta\phi+Q*\tilde{\psi}\delta\phi*\right],
where
\psi
\phi
\tilde{\psi}
\psi
\delta
\begin{pmatrix} 0&-1&0&0\\ 1&0&0&0\\ 0&0&0&1\\ 0&0&-1&0 \end{pmatrix}.
\rho,n,N1,N2,\ldots,M1,M2,\ldots,
\rho=\pm1
n
Ns
s
M\sigma
\sigma
Using the relativistic version of
Hint.
n
s
\sigma
m
s
\sigma
\rho=1,n,Ns=0,M\sigma=0 | |
H | |
\rho=-1,m,Ns=1,M\sigma=1 |
=\pmg\int
* | |
v | |
m |
un\tilde{\psi}s\delta
* | |
\phi | |
\sigma |
d\tau,
where the integral is taken over the entire configuration space of the heavy particles (except for
\rho
\pm
See also: Fermi's golden rule.
To calculate the lifetime of a neutron in a state
n
\psis
\phi\sigma
\rho=1,n,Ns=0,M\sigma=0 | |
H | |
\rho=-1,m,Ns=1,M\sigma=1 |
=\pmg\tilde{\psi}s\delta
* | |
\phi | |
\sigma |
\int
* | |
v | |
m |
und\tau,
where
\psis
\phi\sigma
According to Fermi's golden rule, the probability of this transition is
\rho=1,n,Ns=0,M\sigma=0 | |
\begin{align} \left|a | |
\rho=-1,m,Ns=1,M\sigma=1 |
\right|2&=
\rho=1,n,Ns=0,M\sigma=0 | |
\left|H | |
\rho=-1,m,Ns=1,M\sigma=1 |
x
| |||||
- |
1}{-W+Hs+
2 | |
K | |
\sigma}\right| |
\\ &=4
\rho=1,n,Ns=0,M\sigma=0 | |
\left|H | |
\rho=-1,m,Ns=1,M\sigma=1 |
\right|2 x
| ||||||||||
|
, \end{align}
where
W
Averaging over all positive-energy neutrino spin / momentum directions (where
\Omega-1
\left\langle
\rho=1,n,Ns=0,M\sigma=0 | |
\left|H | |
\rho=-1,m,Ns=1,M\sigma=1 |
\right|2\right\rangleavg=
g2 | |
4\Omega |
\left|\int
* | |
v | |
m |
und\tau\right|2\left(\tilde{\psi}s\psis-
\muc2 | |
K\sigma |
\tilde{\psi}s\beta\psis\right),
where
\mu
\beta
Noting that the transition probability has a sharp maximum for values of
p\sigma
-W+Hs+K\sigma=0
t | 8\pi3g2 |
h4 |
x \left|\int
* | |
v | |
m |
und\tau\right|2
| |||||||
v\sigma |
\left(\tilde{\psi}s\psis-
\muc2 | |
K\sigma |
\tilde{\psi}s\beta\psis\right),
where
p\sigma
K\sigma
-W+Hs+K\sigma=0
Fermi makes three remarks about this function:
K\sigma>\muc2
\beta
Hs\leqW-\muc2
Hs>mc2
\beta
W\geq(m+\mu)c2
* | |
Q | |
mn |
=\int
* | |
v | |
m |
und\tau
in the transition probability is normally of magnitude 1, but in special circumstances it vanishes; this leads to (approximate) selection rules for
\beta
See main article: article and Forbidden transition.
As noted above, when the inner product
* | |
Q | |
mn |
un
vm
If the description of the nucleus in terms of the individual quantum states of the protons and neutrons is accurate to a good approximation,
* | |
Q | |
mn |
un
vm
Shortly after Fermi's paper appeared, Werner Heisenberg noted in a letter to Wolfgang Pauli[12] 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
Const. | |
r5 |
The following year, Hideki Yukawa picked up on this idea,[14] 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.[15]
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
\sigma ≈
2 | |
G | |
\rmF |
E2
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.[16]
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.[17] [18]
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.[19] [20]
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).[21] In modern terms, the "reduced Fermi constant", that is, the constant in natural units is[22]
| |||||
G | = | ||||
\rmF |
\sqrt{2 | |
v=\left(\sqrt{2}
0\right) | |
G | |
\rmF |
-1/2\simeq246.22 rm{GeV}
More directly, approximately (tree level for the standard model),
0\simeq | |
G | |
\rmF |
\pi\alpha | |
\sqrt{2 |
~
2 | |
M | |
\rmW |
(1-
2 | |
M | |
\rmW |
2 | |
/M | |
\rmZ |
)}.
This can be further simplified in terms of the Weinberg angle using the relation between the W and Z bosons with
M | ||||
|
0\simeq | |
G | |
\rmF |
\pi\alpha | |
\sqrt{2 |
~
2 | |
M | |
\rmZ |
\cos2\theta\rm\sin2\theta\rm