Feynman diagrams in the Yukawa theory

Question

I'm trying to find out the best method to find the Feynman diagrams which appear in the Yukawa theory $L_{int} = -y \bar{\psi} \psi \phi $ in the 4 point Greens function $g^{(4)}$ with external scalar fields up to $O(y^4)$.

Is there any better/faster way than to calculate the generating functional $Z[J,\eta,\bar{\eta}]$ and then differentiate the functional 4 times and set the sources to zero? In other words,

$$g^{(4)} = \frac{\delta Z[J,\eta,\bar{\eta}]}{i \delta J \, i \delta \eta \, i \delta \bar{\eta} } \bigg| _{J=\eta=\bar{\eta}=0}$$ with $$Z[J,\eta,\bar{\eta}]= \Omega \, exp\left( \,i \int d^4x \,(-y) \left(-\frac{\delta}{i \delta \eta(x) } \right) \left(\frac{\delta}{i \delta \bar{\eta}(x) } \right)\left(\frac{\delta}{i \delta J(x) } \right) \,\right) Z_0[J,\eta,\bar{\eta}]$$ and $$Z_0[J,\eta,\bar{\eta}] = exp\left( \int d^4z\int d^4z'\left[ \frac{i}{2}J(z)i\Delta_F(z-z')iJ(z') + i\bar{\eta}(z)iS_F(z-z')i\eta(z') \right] \right)$$ with $\Omega$ being the normalisation constant.

So is there a way to guess the diagrams without going through this calculation?

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Answer 1

The moment you have the Feynman rules you don't need to consider the generating functional anymore. It's now rather a combinatorial problem. In your case, to find the diagrams of $\phi \phi \to \phi \phi$ up to order $O(y^4)$ you need to find all the possible ways to connect the four $\phi$ external lines with the four vertices. Ignoring the vacuum bubbles (the disconnected diagrams) you get the scattering amplitude at order $O(y^4)$.

Answer 2

you have incorrectly defined the 4-point green function since you have a third order variational derivative; by definition it must be fourth order - two for scalar sources and two for fermionics Review the pion nucleon interaction model in Ryder chapter 6 it can help you.