An Electroweak-like Theory from Four Fermion Interactions

An electroweak-like theory of a broken chiral symmetry that is constructed by the collective modes of fermion pairs from four fermion interactions of one lepton generation is presented. The products of Dirac spinors lead to the separation of the two chiral fermions to couple respectively with two different kinds of polarization states. Because of a broken vacuum, a fermion and an anti-fermion out of the four pair up to form vector bosons, which behave like gauge bosons, such as $W^\pm$, $Z$ and $\gamma$ in a group structure of $SU(2)_L\times U(1)_Y$. The pairing of spinors only allows left-handed fermions to interact with charged bosons to secure the gauge invariance, while, as desired, $Z$-like bosons mediate different weak forces for two chiral fermions and $\gamma$-like bosons interact freely with fermions.