## The Higgs Mechanism

We have seen in Problem 30 that the four-Fermi interaction in good approximation can be written in terms of the exchange of a heavy vector particle. In lowest order we have resp. the diagrams in Figures 19.1(a) and 19.1(b). The first diagram comes from a four-fermion interaction term that can be written in terms of the product of two currents Jµ J µ, where Jµ = γµ. Here each fermion line typically carries its own flavour index, which was suppressed for simplicity. Figure 19.1(b) can be seen to effectively correspond to

− J˜ µ(−k) (

gµν − kµkν/M2 k2 − M2 + iε

) J˜ ν(k). (19.1)

At values of the exchanged momentum k2 M2, one will not see a difference between these two processes, provided the coupling constant for the fourFermi interactions [Figure 19.1(a)] is chosen suitably (see Problem 30). This is because for small k2, the propagator can be replaced by gµν/M2, which indeed converts eq. (19.1) to J µ Jµ/M2. It shows that the four-Fermi coupling constant is proportional to M−2, such that its weakness is explained by the heavy mass of the vector particle that mediates the interactions. Examples of four-Fermi interactions occur in the theory of β-decay, for example, the decay of a neutron into a proton, an electron and an antineutrino. In that case the current also contains a γ 5 (Problem 40).