Extracting a Feynman diagram from a lagrangian?

[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?

 

[Orodruin]

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?

 

[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)

 

[Orodruin]

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.

 

[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.

 

[Orodruin]

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 ϕpi \phi_{pi} )?

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).

 

[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?

 

[fzero]

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?

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

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

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.

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?

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.

 

[Afanthomme]

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