# Extracting a feynman diagram from a lagrangian?

1. Jul 24, 2015

### Afanthomme

Hi everyone, sorry if this is not the right place to post that question but i'm new to this forum, i'll delete if necessary.

I am currently trying to learn QFT from Matthew Schwartz's "Quantum field theory and the standard model", quite clear during the first chapters, but i have been completely lost by the chapter about the feynman rules (can be found here : http://isites.harvard.edu/fs/docs/icb.topic1097985.files/I-7-Feynman.pdf )

I understand the first examples and the 2 derivations, but once he starts adding the derivative coupling (p.17) i'm completely lost : i don't understand how he gets the diagrams.
Is it that he decides to study 2->2 scattering of the first 2 fields through the third given the interaction?

If this is the case, why are there 4 diagrams instead of 1 after the integration by parts?

And one last question : why is there only the p2 momentum in the first amplitude? Since the two particles enter the first vertice (and exit the second), shouldn't they both give a factor?

2. Jul 24, 2015

### Orodruin

Staff Emeritus
It is a bit unclear exactly what you are having trouble with. Are you having trouble in deriving the Feynman rules or using them to obtain the available Feynman diagrams?

3. Jul 24, 2015

### Afanthomme

it's mostly the second problem, i really don't understand how to link a physical situation to the diagrams.
I also have a problem with the momentum factors due to particles entering/leaving a vertex (but i think this might become more understandable if i understand how to get the diagrams)

4. Jul 24, 2015

### Orodruin

Staff Emeritus
In order to get the diagrams you simply draw all allowed diagrams with the correct in and out states (to a given order in perturbation theory). Start by deriving the Feynman rules and then use these to draw the diagrams.

5. Jul 24, 2015

### Afanthomme

Well i'm not really sure that helps me...
When you say "allowed diagrams with the correct in and out states", you mean all diagrams with a given number of vertex that link the external points ( for example $\phi_{pi}$)?

Edit : in fact i understood what bothered me with the derivative.

Last edited: Jul 24, 2015
6. Jul 24, 2015

### Orodruin

Staff Emeritus
Yes. But you must also make sure that the external lines are for the correct fields ... You cannot make a Feynman diagram with an outgoing $\phi_1$ if your outgoing field should be a $\phi_2$ (unless there is also an outgoing $\phi_1$ of course, but you get my meaning).

7. Jul 24, 2015

### Afanthomme

Ok i get that !
One last question: when asked to compute "the amplitude for a $\phi->\phi \phi$ decay" without further precision, how do we fix the external points?
Do we have to integrate over all possible momenta configurations?

8. Jul 24, 2015

### fzero

Yes, he chose $1+2\rightarrow 1+2$, but you could work the amplitude for $2+3 \rightarrow 2+3$ for example.

You have two interaction terms in the Lagrangian. There are 2 choices for which to use at each vertex, hence $2\cdot 2 = 4$ possibilities in all.

I think you mentioned that you worked this out. In case it's not clear, it's because there is no derivative on the $\phi_1$ factor and we've specified that there is a $1+2$ in the initial state.

9. Jul 24, 2015

### Afanthomme

Thank you very much, this really made it much clearer !