This is the intention to formalize a theory of the weak interactions involving a dimensionless coupling constant and a particle working as a mediator of the weak interaction.
It is proposed to replace the Fermi current-current Lagrangian
by the Lagrangian
In this way, the couplings of the example are written as
Bearing in mind that and that there results that the (semi) weak
constant is dimensionless as it was requested. On the other hand, the
very short range of the weak interactions necessarily implies that the
intermediate boson must be massive. Remembering the expression for the
propagator of a vector boson with mass, it results for the
amplitude of the process of the example:
It must be noticed that for low momentum regions, that is to say,
This introduction shows that a model of weak interactions which bears a strong similitude with electrodinamics can be posed. Nevertheless, there are fundamental differences:
Due to these distinctive characteristics it can immediately be
observed that the theory of the intermediate boson of the weak
interactions leads to difficulties similar to the ones presented by
the model of point interaction. In other words, the higher order
perturbative corrections of the theory cannot be calculated due to
divergencies which continue to turn worse. The Lagrangian (117) has
therefore the same phenomenological status than the Fermi type. In
fact, the propagator of
It must be observed that in the equivalent contribution of the quantum electrodynamics pictured in Figure 15,
due to the fact that the photon has no mass, there appears
It is clear, then, that the renormalization of the theory with intermediate boson based on the Lagrangian (117) is hopeless. In other words, the previous discussion indicates that the limit imposed by the unitarity to the cross section of a given process, via the optical theorem, will surely be violated by the massive intermediate vector boson model.
It should be noticed that the contribution of the term in the propagator (122) is zero if the electron mass can be neglected due to the Dirac equation since these momenta multiply the lepton currents (see the amplitude (119)). Nevertheless the presence of a massive causes new difficulties. Consider the process (see Figure 16)
The difference is that there is no longitudinal polarized photon,
whereas W, being massive, implies both longitudinal and transverse
polarization states. In fact, the cross section for transverse
is well behaved
Therefore, the Lagrangian (117) has the same phenomenological status as the Fermi one. It gives rise to a non- renormalizable theory. On the contrary, QED is renormalizable. This important property is due to the fact that the photon is massless that in turn inhibits the longitudinal polarization state. This fact is directly related to the property of gauge invariance of electromagnetism.
In conclusion, it is necessary to implement gauge invariance in the weak interaction theory. This is not an easy task because there are clear differences with QED: while the photon is neutral and massless, the boson is charged and massive. Ability is needed.
It seems important to provide a cancellation mechanism between badly behaved amplitudes inspired in gauge invariant requierements.
Let us indicate the plausibility of these arguments analysing an example. We consider once more the annihilation reaction (125) that as it was established above, is ill behaved for . Using known (before 1970) particles, one can draw only the two graphs of Figure 17.
No cancellation is possible for the right electron state. Then, a new third graph is needed. One possibility is the one in Figure 6.4.
This graph includes a neutral weak boson that has to couple to both the right and the left states of the electron. This implies an electron neutral weak current and a precise relation between electromagnetism and neutral weak interactions just to guarantee the cancellation we were looking for. This scheme is correctly formalized by the model that we treat below.