Fermi's interaction

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. |url-access=subscription

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, for which he received a Nobel Prize in Physics in 1938.

The "tentativo"

Definitions

The theory deals with three types of particles presumed to be in direct interaction: initially a "heavy particle" in the "neutron state" (\rho=+1), which then transitions into its "proton state" (\rho = -1) with the emission of an electron and a neutrino.

Electron state

:\psi = \sum_s \psi_s a_s,

where \psi is the single-electron wavefunction, \psi_s are its stationary states.

a_s is the operator which annihilates an electron in state s which acts on the Fock space as

:a_s \Psi(N_1, N_2, \ldots, N_s, \ldots) = (-1)^{N_1 + N_2 + \cdots + N_s - 1} (1 - N_s) \Psi(N_1, N_2, \ldots, 1 - N_s, \ldots).

a_s^* is the creation operator for electron state s:

:a_s^* \Psi(N_1, N_2, \ldots, N_s, \ldots) = (-1)^{N_1 + N_2 + \cdots + N_s - 1} N_s \Psi(N_1, N_2, \ldots, 1 - N_s, \ldots).

Neutrino state

Similarly,

:\phi = \sum_\sigma \phi_\sigma b_\sigma,

where \phi is the single-neutrino wavefunction, and \phi_\sigma are its stationary states.

b_\sigma is the operator which annihilates a neutrino in state \sigma which acts on the Fock space as

:b_\sigma \Phi(M_1, M_2, \ldots, M_\sigma, \ldots) = (-1)^{M_1 + M_2 + \cdots + M_\sigma - 1} (1 - M_\sigma) \Phi(M_1, M_2, \ldots, 1 - M_\sigma, \ldots).

b_\sigma^* is the creation operator for neutrino state \sigma.

Heavy particle state

\rho is the operator introduced by Heisenberg (later generalized into isospin) that acts on a heavy particle state, which has eigenvalue +1 when the particle is a neutron, and −1 if the particle is a proton. Therefore, heavy particle states will be represented by two-row column vectors, where

:\begin{pmatrix}1\0\end{pmatrix}

represents a neutron, and

:\begin{pmatrix}0\1\end{pmatrix}

represents a proton (in the representation where \rho is the usual \sigma_z spin matrix).

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

:Q = \sigma_x - i \sigma_y = \begin{pmatrix}0 & 1\ 0 & 0\end{pmatrix}

and

:Q^* = \sigma_x + i \sigma_y = \begin{pmatrix}0 & 0\ 1 & 0\end{pmatrix}.

u_n(x) resp. v_n(x) are the eigenfunctions for the heavy particle in the nth stationary state when \rho = +1 (neutron) resp. \rho = -1 (proton), where x is the position of the heavy particle.

Hamiltonian

The Hamiltonian is composed of three parts: H_\text{h.p.}, representing the energy of the free heavy particles, H_\text{l.p.}, representing the energy of the free light particles, and a part giving the interaction H_\text{int.}.

:H_\text{h.p.} = \frac{1}{2}(1 + \rho)N + \frac{1}{2}(1 - \rho)P,

where N and P are the energy operators of the neutron and proton respectively, so that if \rho = 1, H_\text{h.p.} = N, and if \rho = -1, H_\text{h.p.} = P.

:H_\text{l.p.} = \sum_s H_s N_s + \sum_\sigma K_\sigma M_\sigma,

where H_s is the energy of the electron in the s^\text{th} state in the nucleus's Coulomb field, and N_s is the number of electrons in that state; M_\sigma is the number of neutrinos in the \sigma^\text{th} state, and K_\sigma energy of each such neutrino (assumed to be in a free, plane wave state).

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-decay process.

Fermi proposes two possible values for H_\text{int.}: first, a non-relativistic version which ignores spin:

:H_\text{int.} = 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 and that the interaction terms analogous to the electromagnetic vector potential can be ignored:

:H_\text{int.} = g \left[ Q \tilde{\psi}^* \delta \phi + Q^* \tilde{\psi} \delta \phi^* \right],

where \psi and \phi are now four-component Dirac spinors, \tilde{\psi} represents the Hermitian conjugate of \psi, and \delta is a matrix

:\begin{pmatrix} 0 & -1 & 0 & 0\ 1 & 0 & 0 & 0\ 0 & 0 & 0 & 1\ 0 & 0 & -1 & 0 \end{pmatrix}.

Matrix elements

The state of the system is taken to be given by the tuple \rho, n, N_1, N_2, \ldots, M_1, M_2, \ldots, where \rho = \pm 1 specifies whether the heavy particle is a neutron or proton, n is the quantum state of the heavy particle, N_s is the number of electrons in state s and M_\sigma is the number of neutrinos in state \sigma.

Using the relativistic version of H_\text{int.}, Fermi gives the matrix element between the state with a neutron in state n and no electrons resp. neutrinos present in state s resp. \sigma , and the state with a proton in state m and an electron and a neutrino present in states s and \sigma as

:H^{\rho=1, n, N_s=0, M_\sigma=0}{\rho=-1,m,N_s=1,M\sigma=1} = \pm g \int v_m^* u_n \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). The \pm is determined by whether the total number of light particles is odd (−) or even (+).

Transition probability

To calculate the lifetime of a neutron in a state n according to the usual quantum perturbation theory, the above matrix elements must be summed over all unoccupied electron and neutrino states. This is simplified by assuming that the electron and neutrino eigenfunctions \psi_s and \phi_\sigma are constant within the nucleus (i.e., their Compton wavelength is much larger than the size of the nucleus). This leads to

:H^{\rho=1, n, N_s=0, M_\sigma=0}{\rho=-1,m,N_s=1,M\sigma=1} = \pm g \tilde{\psi}s \delta \phi\sigma^* \int v_m^* u_n d\tau,

where \psi_s and \phi_\sigma are now evaluated at the position of the nucleus.

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

:\begin{align} \left|a^{\rho=1, n, N_s=0, M_\sigma=0}{\rho=-1,m,N_s=1,M\sigma=1}\right|^2 &= \left|H^{\rho=1, n, N_s=0, M_\sigma=0}{\rho=-1,m,N_s=1,M\sigma=1} \times \frac{\exp{\frac{2\pi i}{h} (-W + H_s + K_\sigma) t} - 1}{-W + H_s + K_\sigma}\right|^2 \ &= 4 \left|H^{\rho=1, n, N_s=0, M_\sigma=0}{\rho=-1,m,N_s=1,M\sigma=1}\right|^2 \times \frac{\sin^2\left(\frac{\pi t}{h}(-W + H_s + K_\sigma)\right)}{(-W + H_s + K_\sigma)^2}, \end{align}

where W is the difference in the energy of the proton and neutron states.

Averaging over all positive-energy neutrino spin / momentum directions (where \Omega^{-1} is the density of neutrino states, eventually taken to infinity), we obtain

: \left\langle \left|H^{\rho=1, n, N_s=0, M_\sigma=0}{\rho=-1,m,N_s=1,M\sigma=1}\right|^2 \right \rangle_\text{avg} = \frac{g^2}{4\Omega} \left|\int v_m^* u_n d\tau\right|^2 \left( \tilde{\psi}s \psi_s - \frac{\mu c^2}{K\sigma} \tilde{\psi}_s \beta \psi_s\right),

where \mu is the rest mass of the neutrino and \beta is the Dirac matrix.

Noting that the transition probability has a sharp maximum for values of p_\sigma for which -W + H_s + K_\sigma = 0, this simplifies to

: t\frac{8\pi^3 g^2}{h^4} \times \left| \int v_m^* u_n d\tau \right|^2 \frac{p_\sigma^2}{v_\sigma}\left(\tilde{\psi}s \psi_s - \frac{\mu c^2}{K\sigma} \tilde{\psi}_s \beta \psi_s\right),

where p_\sigma and K_\sigma is the values for which -W + H_s + K_\sigma = 0.

Fermi makes three remarks about this function:

• Since the neutrino states are considered to be free, K_\sigma \mu c^2 and thus the upper limit on the continuous \beta-spectrum is H_s \leq W - \mu c^2.
• Since for the electrons H_s mc^2, in order for \beta-decay to occur, the proton–neutron energy difference must be W \geq (m + \mu)c^2
• The factor

:in the transition probability is normally of magnitude 1, but in special circumstances it vanishes; this leads to (approximate) selection rules for \beta-decay.

Forbidden transitions

Main article: Forbidden transition

As noted above, when the inner product Q_{mn}^* between the heavy particle states u_n and v_m vanishes, the associated transition is "forbidden" (or, rather, much less likely than in cases where it is closer to 1).

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}^* vanishes unless the neutron state u_n and the proton state v_m have the same angular momentum; otherwise, the total angular momentum of the entire nucleus before and after the decay must be used.

Influence

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

The following year, Hideki Yukawa picked up on this idea,

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 \sigma \approx G_{\rm F}^2 E^2 . Since this cross section grows without bound, the theory is not valid at energies much higher than about 100 GeV. Here GF 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.

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.{{cite journal

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. |doi-access=free |doi-access=free

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 most precise experimental determination of the Fermi constant comes from measurements of the muon lifetime, which is inversely proportional to the square of GF (when neglecting the muon mass against the mass of the W boson). |author2-link=MuLan |display-authors=etal |pmid=17678280|s2cid=3255120 :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}\ . 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, the Fermi constant is related to the Higgs vacuum expectation value :v = \left(\sqrt{2} , G_{\rm F}^0\right)^{-1/2} \simeq 246.22 ; \textrm{GeV}. |article-number=053008

More directly, approximately (tree level for the standard model), : G_{\rm F}^0\simeq \frac {\pi \alpha}{\sqrt{2}~ M_{\rm W}^2 (1- M^2_{\rm W}/M^2_{\rm Z} )}.

This can be further simplified in terms of the Weinberg angle using the relation between the W and Z bosons with M_\text{Z}=\frac{M_\text{W}}{\cos\theta_\text{W}}, so that : G_{\rm F}^0\simeq \frac {\pi \alpha}{\sqrt{2}~ M_{\rm Z}^{2}\cos^{2}\theta_{\rm W}\sin^{2}\theta_{\rm W}}.

References

References

1. (2012). "Fermi's β-decay Theory". Asia Pacific Physics Newsletter.
2. Fermi, Enrico. (2004). "Fermi Remembered". University of Chicago Press.
3. Close, Frank. (23 February 2012). "Neutrino". Oxford University Press.
4. (1986). "Inward Bound". Oxford University Press.
5. Schwartz, David N.. (2017). "The Last Man Who Knew Everything. The Life and Times of Enrico Fermi, Father of the Nuclear Age". Basic Books.
6. Brown, Laurie M. (1996). "The Origin of the Concept of Nuclear Forces". Institute of Physics Publishing.
7. (1958). "Theory of the Fermi interaction". Physical Review.
8. (1958). "Chirality invariance and the universal Fermi interaction". Physical Review.
9. (June 2015). "CODATA Value: Fermi coupling constant". US [[National Institute of Standards and Technology]].