Four-fermion interaction

A four-fermion interaction is a local operator containing four fermion fields. In four spacetime dimensions it has mass dimension six, so its coefficient has mass dimension minus two; writing it as the square of a current multiplied by a coefficient inside the square gives that coefficient mass dimension minus one.

For ${\mathcal{L}}_{\mathrm{int}}=g^2(\bar{\psi }\psi {)}^2/2$ with one species of Dirac field, the tree-level Feynman diagrams for $f(p_1)\bar{f}(p_2)\to f(p_3)\bar{f}(p_4)$ giveThe fermionic sign is the antisymmetry of the two external contractions. The initial spin average is one quarter of the spin sum. The two diagonal contributions are $g^4(t-4m^2{)}^2$ and $g^4(s-4m^2{)}^2$; the interference Dirac trace is $R=(s-4m^2{)}^2+(t-4m^2{)}^2-(u-4m^2{)}^2$, giving the displayed result after subtracting $g^{4}R/2$. Here the Mandelstam variables satisfy $s+t+u=4m^2$. At threshold this gives $16g^4{m}^4$; for zero mass it gives $g^4(s^2+t^2+u^2)/2$. The formula describes the leading scattering amplitude of an effective interaction and is not a statement of perturbative renormalizability.

Approximate a hadronic matrix element of a product of two currents by a product of separate current matrix elements. For a decay to two mesons, one factor can create a pion from the vacuum while the other gives a pseudoscalar-to-pseudoscalar form factor. Nonfactorizable quantum chromodynamics interactions are omitted; tree-level weak vertices do not alone justify this approximation.

The Fermi interaction is the low-energy four-fermion approximation to the weak charged current. Integrating out a W boson gives its coefficient $G_F/\sqrt{2}=g^2/(8M_W^2)$ at tree level.

For a tree-level left-handed three-body decay with one massive final fermion and two massless ones, the integrated width relative to the all-massless result includes $f(r)=1-8r+8r^3-r^4-12r^2\ln r$, where $r$ is the squared final-to-parent mass ratio. It satisfies $f(0)=1$ and $f(1)=0$. It is not a universal factor for channels with several massive daughters.

For a massive charged vector with interaction $-\kappa (W^{+}J+W^{-}{J}^{†})$, neglect derivatives at momenta small compared with its mass. Completing the square in $m_W^2{W}^{+}{W}^{-}$ leaves $-{\kappa }^2{J}^{†}J/m_W^2$. With $\kappa =g/(2\sqrt{2})$ this yields the Fermi constant relation $G_F/\sqrt{2}=g^2/(8m_W^2)$. Corrections contain extra derivatives and are suppressed by momentum squared over the heavy mass squared.

For a left-handed incoming neutrino, unpolarized electrons, and electron vector/axial couplings $c_V,c_A$, the four-fermion interaction gives $\overline{|\mathcal{M}{|}^2}=4G_F^2[(c_V+c_A{)}^2{s}^2+(c_V-c_A{)}^2{u}^2]$. Integrating massless two-body Lorentz-invariant phase space yields $\sigma =G_F^{2}s(c_V^2+c_V{c}_A+c_A^2)/(3\pi )$. The electron spin average is included, while no sterile neutrino helicity is averaged in.

Crossing a massless neutrino to an antineutrino reverses the antisymmetric part of its spin tensor. The $s^2$ and $u^2$ weights exchange, giving ${\sigma }_{\bar{\nu }}=G_F^{2}s[(c_V-c_A{)}^2+(c_V+c_A{)}^2/3]/(4\pi )$ for the same electron couplings. This is a low-energy contact statement rather than a replacement for a full weak-boson propagator.

Electron-neutrino scattering has a charged-current contribution absent for muon neutrinos. Fierz rearrangement combined with the external fermionic sign adds a left-electron current: $c_V\to c_V+1$, $c_A\to c_A+1$. The right-electron coupling stays unchanged; the left-electron coupling increases by two.

A Fierz rearrangement rewrites a product of fermion bilinears with one spinor pairing as a sum of products with another pairing. For left-handed vector currents, it lets a charged-current four-fermion operator be compared directly with a neutral-current operator.

The Fermi constant is the coefficient conventionally used for the low-energy charged-current weak interaction. Tree-level exchange of a $W$ boson gives $G_F/\sqrt{2}=g^2/(8M_W^2)$.