For the phenomenological treatment of weak interactions we have the current-current effective Lagrangian (77). It is found that the known semileptonic and leptonic processes are well described by the matrix elements of the Lagrangian at first order of perturbation theory. Nevertheless, there are doubts about the behaviour of higher orders. That is to say, the renormalization properties (of theoretical interest) of the theory so formulated. It is found that the theory is not renormalizable because too many divergences appear. So it is valid to question where, at which energy scale, these divergences will turn unsustainable. That is to say where this model will stop having any value, even a phenomenological one.

A typical process to study is

(101) |

On the other hand, the optical theorem at high energies allows to write

(104) |

But in the process we are dealing with, one finds that in the angular
distribution only the waves are important, therefore

(105) |

(106) |

(107) |

called unitarity limit as it is provided by the optical theorem. In conclusion, from a certain energy of the order given by (110), the limit imposed by unitarity is violated and the effective model of four fermions point interaction lacks sense.

In order to obtain some inspiration to continue it is interesting to
compare the behaviour of a similar process but this time of the
quantum electrodynamics. For instance

(109) |

and the interaction terms are

The first order cross section of the process results

(111) |

(112) |

(113) |

(114) |

Summarizing, inspiration came: the necessity of searching for a theory of weak interactions involving a dimensionless coupling constant and a mediator of the interaction is clear.