The operator $(\phi^\dagger\phi)^2$ has <mass dimension> four, so $[\lambda]=0$. It is a \b[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.

Solution

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

 Solution (source code)