Does the Four Fermion Interaction Conserve Current Like Standard Models?

[quantumfireball]

four fermion interaction??

The four fermion interaction proposed by enrico fermi to learn weak interaction
postulates that four dirac fields interact via the interaction hamilton

J1$$^{}u$$*J2$$_{}u$$

where J1$$^{}u$$=phi_d(x)*y$$^{}u$$*phi_c(x)

but by question is this
is the bilinear form taken between wavefuctions asscoiated with different particles
conserve current just like $$\Psi$$
$$\bar{}$$*$$\gamma$$*$$\Psi$$

 

[javierR]

The Dirac Lagrangian for a massless fermion f has two symmetries each of U(1) type: "vector" and "axial" (or "chiral"). For the vector symmetry, there is an associated conserved current with gamma^{m} inbetween f-bar and f, with spacetime index m; and for the axial, there is instead a gamma^{m}*gamma^{5} inbetween. This also means that the currents with gamma^{m}*(1-gamma^{5}) and gamma^{m}(1+gamma^{5}) are conserved; these are chiral currents in that the (1 +or- gamma^{5}) are projections of general dirac spinors onto right and left chiral spinors.
Once you add a mass for the fermion, the chiral symmetry is ruined. For a massive fermion, chirality (and the axial current) is no longer a conserved quantity, so neither are the left and right chiral currents.
Take massless fermions again. If you have a new 4-fermion interaction among the fermions, you have to ask again about the symmetries and therefore which currents are conserved. The Fermi interaction for beta decay or quark interactions is
$\bar{f}_{1}\gamma^{m}(1-\gamma^{5})f_{2}\bar{f}_{3}\gamma_{m}(1-\gamma^{5})f_{4}$ where the numbered indices just label different types of fermions. (The projectors in parentheses mean you can rewite this term by replacing the Dirac fermions with left-chiral fermions $f_{L}$ and remove the projection operators in parentheses.) You can check this term preserves both vector and axial symmetries, and so also left-chiral currents and right-chiral currents are separately conserved (the interaction term, afterall, involves left-handed fermions only). Again, a mass term, or other types of interactions such as with QCD instantons, can break chiral symmetry.

 

[denian]

The four fermion interaction is a theoretical concept proposed by Enrico Fermi to explain the weak interaction between particles. It postulates that four Dirac fields interact through a specific interaction Hamiltonian. This interaction is described by the bilinear form between wavefunctions of different particles, similar to the conservation of current in the equation \Psi \bar{}*\gamma*\Psi. This concept has been studied extensively in particle physics and has contributed to our understanding of the fundamental forces in the universe. However, further research and experimentation are still needed to fully understand the implications and applications of the four fermion interaction.