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.