Consider a real scalar field described through the following lagrangian $$\mathcal L = \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi - \frac{1}{2}m^2 \phi^2 - \frac{g}{3!}\phi^3$$ The second order term in the S matrix expansion produces the diagrams in which we have a $2 \rightarrow 2$ scattering event and such diagrams are given by evaluating $$S^{(2)} = -\frac{9 g^2}{2 \cdot 6^2} \int d^4 x_1 \int d^4 x_2 i \Delta_F(x_1-x_2) (2 \langle p_3p_4| \phi_2^- \phi_2^- \phi_1^+ \phi_1^+ | p_2p_1\rangle + 4 \langle p_3p_4 | \phi_1^- \phi_2^- \phi_1^+ \phi_2^+| p_1p_2 \rangle) $$

My questions are: From this, apparantly it is clear that the $s,t,u$ diagrams can be obtained but I'm not sure how this is the case. I can see that in the first term, say, $p_1$ and $p_2$ are destroyed at $x_1$ and then $p_3$ and $p_4$ created at $x_2$ which would give the $s$ diagram and similarly depending on whether $\phi_2^+$ annihilates $p_1$ or $p_2$ at $x_2$ in the second term I can get the $t$ and $u$ channel diagrams. But there seems to be more diagrams?

There should also be a factor of $4$ alongside the first term there and a factor of $2$ for the second term (in order to cancel the combinatorial prefactor already present), but I don't see where this comes from


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