A charged scalar particle is described by a complex field $\Phi(x) = \phi_{1}(x)+i\phi_{2}(x)$. Consider a Lagrangian of the $\lambda \Phi^{4}$-model whose potential in the Euclidean action is given by:
$V(\Phi)=\frac{m^{2}}{2}|\Phi|^{2}+\frac{\lambda}{4!}|\Phi|^{4} \ \ \ \ \ \ \ \ (1)$
with $|\Phi|^{2}=\Phi^{*}\Phi = \phi_{1}^{2} + \phi_{2}^{2}$
The condition for a minimum is given for $m^{2}<0$
$\frac{\partial V}{\partial \Phi} = m^{2} \Phi +\frac{\lambda}{3!}|\Phi|^{2}\Phi = 0 \ \ \ \ \ \ \ \ (2)$
How did the author get this equation?
The author seems to pass from a potential containing only scalar terms (1) to a derivative containing a vector term (2)?
What would be the second derivative of V?
I tried to express things in terms of real and imaginary parts and take derivatives separately with respect to each of them and combined but It did not seem to work
maybe there is a lack of understanding in my complex analysis.