= Solution
The canonical complex-scalar <kinetic term> is $\partial_\mu\widetilde\varphi^*\partial^\mu\widetilde\varphi$. Write $\widetilde\varphi=(a+ib)/\sqrt2$ 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:
$$
M_v^2=(m+gv_\pm)^2=m^2-2g\kappa=\Delta,
\qquad
\boxed{m_{\rm scalar}^2=|\Delta|,\qquad
m_{\rm scalar}=\sqrt{|m^2-2g\kappa|}.}
$$
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 $g=0,m\ne0$ this reduces to $|m|$; at a double critical point it is zero.
Back to article page
