Past exam of the gauge field theory course of the University of Cambridge / 2013 / Paper 1 / 3 / a / Solution

The operator $({\varphi }^{†}\varphi {)}^2$ has mass dimension four, so $[\lambda ]=0$. It is a power-counting renormalizable scalar quartic interaction. It describes scalar self-interactions, including the quartic term in the Higgs potential for the appropriate scalar representation. Its sign in the potential must be chosen for a stable large-field theory; dimension counting alone does not ensure stability.