I'm currently studying how to deduce Feynman rules for general theories, and I've managed to deduce them for $\phi^3$ and $\phi^4$ theories. Up to this point I've considered the same field for all cases, and deduced the Feynman rules by expanding the interacting term in the correlator and using Wick's theorem to make the contractions.
My question is, if we consider a interacting theory for two different fields, how could we deduce the Feynman rules from Wick's theorem. Consider, for example, the decay of a particle given by the interacting lagrangian term
$$L_\mathrm{int}=-\lambda\Phi\phi^2$$
We see directly it only has vertices with 3 lines. If I expand the exponential that usually comes from this:
$$\exp(-i\lambda\int d^4y\ \Phi\phi^2)=1+(i\lambda)\int d^4y\Phi_y\phi_y^3+(i\lambda)^2\int d^4y_1d^4y_4\ \Phi_{y1}\phi_{y1}^3\Phi_{y2}\phi_{y2}^3+...$$ $$$$
However, I'm not sure which terms I should write for the correlators. For simplicity, let's consider the 2 particle correlator:
$$\langle \Omega |T[\phi_1\phi_2]|\Omega\rangle=\lim_{T\rightarrow \infty}\frac{\langle 0 |T[\phi_1\phi_2 \exp(-i\lambda\int d^4y\ \Phi\phi^2)]|0\rangle}{\langle 0 |\exp(-i\lambda\int d^4y\ \Phi\phi^2)|0\rangle}$$
Let's focus on the numerator, which could either be:
$$N(x_1,x_2)=\langle 0 | \phi_{x1}\phi_{x2}\exp(-i\lambda\int d^4y\ \Phi\phi^2) | 0 \rangle$$
Or could it be:
$$N(x_1,x_2)=\langle 0 | \phi_{x1}\Phi_{x2}\exp(-i\lambda\int d^4y\ \Phi\phi^2) | 0 \rangle$$
Or even it could be:
$$N(x_1,x_2)=\langle 0 | \Phi_{x1}\Phi_{x2}\exp(-i\lambda\int d^4y\ \Phi\phi^2) | 0 \rangle$$
Which one should I consider for the expansion?
