# Feynman diagrams in the Yukawa theory

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?

• Open any introductory textbook on QFT, e.g. Peskin & Schroeder, and it will have the Feynman rules for the Yukawa theory worked out. Form there it should be a pice of cake to write all (connected) diagrams of order 4. Feynman diagrams were after all invented not to go through the rigmarole you suggest. Commented May 27, 2021 at 17:03

## 2 Answers

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 $$O(y^4)$$.

• Thanks for your answer. I'm still quite lost. Why are you talking only of diagrams like $\phi \phi \rightarrow \phi \phi$? What about diagrams like $\phi \rightarrow \phi$ or $\phi \rightarrow \phi \phi \phi$ or the same configurations with dirac fields instead of scalar fields? Commented May 27, 2021 at 16:55
• I thought you mentioned the 4-point correlator right? So $\phi \to \phi$ is 2-point. $\phi \to \phi \phi \phi$ is $\phi \phi \to \phi \phi$. What matters is the number of external legs (4). Commented May 27, 2021 at 17:14
• Oh I see. So with these conditions then, the only diagramm which I can think of is the one which I added to my question. Could there be more? Commented May 27, 2021 at 18:20
• Precisely, that's the only possibility! Note that you can still swap the external legs between themselves. There's three configurations in total (s,t,u channels). Commented May 28, 2021 at 12:12
• thank you very much for your help! Commented May 28, 2021 at 12:34

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.