Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 45 / 2 / f / Solution

The canonical complex-scalar kinetic term is ${\partial }_{\mu }{\tilde{\phi }}^{\ast }{\partial }^{\mu }\tilde{\phi }$. Write $\tilde{\phi }=(a+ib)/\sqrt{2}$ so that $a,b$ each have real canonical normalization. The quadratic potential is $|M_v{|}^2(a^2+b^2)/2$, giving equal squared masses for both real modes:The first equality is an algebraic square of the generally complex mass parameter; the physical squared mass is its modulus squared. In particular it is not a negative squared mass when real parameters have $\Delta <0$: the vacuum then lies off the original real slice. The constant $\alpha$ does not enter. For $g=0,m\ne 0$ this reduces to $|m|$; at a double critical point it is zero.