Feynman diagrams and the Wick contraction

[CAF123]

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, apparently 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. 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.

Should I write $$\langle p_3 p_4 | \phi_1^- \phi_2^- \phi_1^+ \phi_2^+ | p_1 p_2 \rangle = e^{-i (p_1-p_3) \cdot x_1} e^{-i (p_2-p_4) \cdot x_2} + e^{-i(p_2-p_3) \cdot x_1} e^{-i(p_1-p_4) \cdot x_2} + \text{perms}$$ where perms is sending $x_1 \leftrightarrow x_2$. Under the integration over $x_1$ and $x_2$ each perm matches exactly one of the terms written explicitly above so that is why we have factor of 2? Maybe? But using this logic does not explain the factor of 4 in the first term (as far as I can see).

Thanks!

 

[AndyW]

Hello, thank you for bringing up these questions. Let me clarify some points regarding the derivation of the $s,t,u$ diagrams from the second order term in the S matrix expansion.

Firstly, the factor of 4 in the first term and the factor of 2 in the second term come from the combinatorial prefactor that arises when expanding the product of four fields. This prefactor takes into account the different ways in which the fields can be ordered and is given by the number of permutations of four objects, which is 4! = 24. However, some of these permutations are equivalent, leading to a reduction in the combinatorial prefactor. In the first term, there are only 6 distinct permutations that contribute, leading to a factor of 4 in front of the integral. In the second term, there are only 3 distinct permutations, leading to a factor of 2.

Secondly, your expression for the matrix element is correct, but the reasoning for the factor of 2 is not quite accurate. The factor of 2 comes from the fact that we are summing over all possible ways in which the four fields can be arranged. This includes the case where the fields at $x_1$ and $x_2$ are swapped, which is taken into account by the "perms" term. However, we also need to consider the case where the fields at $x_1$ and $x_2$ are not swapped, which is why we have an additional factor of 2.

Finally, I want to point out that the derivation of the $s,t,u$ diagrams from the second order term in the S matrix expansion is not specific to this particular lagrangian. It is a general result that arises from the Feynman rules for scalar field theory. Therefore, it is not necessary to write out the full expression for the matrix element in order to understand where the $s,t,u$ diagrams come from. Instead, one can simply apply the Feynman rules to the lagrangian and see that the resulting diagrams correspond to the different channels in the scattering process.

I hope this clarifies your doubts. If you have any further questions, please do not hesitate to ask.