In order to compute the scattering probability that two particles of type 1 (associated to $\phi_1(x)$) which come from the far past with the momenta $p_1$ and $p_2$, to scatter and evolve into two particles of type 2 (associated to $\phi_2(x)$) with the momenta $p_3$ and $p_4$, I am going to apply the momentum space scattering rules, I'm just first of all trying to establish the number of distinct Feynman diagrams of $$\langle0|T( \phi_1(x_1)\phi_1(x_2)\phi_2(x_3)\phi_2(x_4))|0\rangle,$$ where the interacting Hamiltonian is $$H_\mathrm{int}= \frac g4 \phi_1^{2}(x)\phi_2^{2}(x)$$
QUESTION:
The solution lists these two Feynman diagrams only, see attachment 'dia 1'
But I am also getting this diagram, see attachment 'dia 2'
Where solid lines are associated to $\phi_1(x)$, dotted to $\phi_2(x)$, and $z$ and $w$ are the internal variables I am integrating over.
Is there a reason this should be excluded? I know that 'bubble/vacuum' diagrams are excluded - disconnected diagrams with no connection to external points, but I'm unsure here since it's connected...
On another note, my lecture notes describe the contribution from the $g^{0}$ term as 'trivial' since it describes non-interacting particles. so yes in this case it's $x_1, x_2$ and $x_3, x_4$ contracted since you can not contract different fields, however, what is meant by 'trivial' in this sense? Because I know that diagrams that have no connections to external points are not included since they are vacuum contributions and will just cancel out anyway, however here it is a diagram solely between external points, so whilst it doesn't describe any interaction, it still needs to be included right?