Drawing a Feynman diagram with limited vertex types

[latentcorpse]

Consider q3 in this exam:
http://www.maths.cam.ac.uk/postgrad/mathiii/pastpapers/2006/Paper48.pdf

I reckon I can manage a good part of the rest of the question. Unfortunately, I cannot manage the very first bit (drawing the Feynman diagram) and this is preventing me from continuing!

I have attached my best attempt so far...

The problem is in completing the loop. We are only given those to types of vertices and I just cannot get the lines to join up - I end having to add more and more lines and it just gets more and more complicated!

Any advice?

 

[grey_earl]

Don't worry too much about the distinction between $$\psi$$ and $$\bar\psi$$, since one is the other when changing the direction of time. You can just close the loop without inserting anything more. The Fermion running in the loop is a $$\psi$$, but from every interaction vertex there is one coming and one going, so you have a $$\psi \bar\psi$$ at every vertex.

 

[latentcorpse]

Don't worry too much about the distinction between $$\psi$$ and $$\bar\psi$$, since one is the other when changing the direction of time. You can just close the loop without inserting anything more. The Fermion running in the loop is a $$\psi$$, but from every interaction vertex there is one coming and one going, so you have a $$\psi \bar\psi$$ at every vertex.

I don't get it. If I close the loop as it is just now using a $\psi$ then that bottom right hand vertex has a $\psi,\psi,\Phi$ at it. Which we clearly can't have because of the interaction terms in the lagrangian!

Can you explain how this works in a bit more detail please?

Thanks!

 

[grey_earl]

In reality you don't have $$\psi$$ and $$\bar\psi$$ at any vertex, you have a $$\psi$$ entering and a $$\psi$$ leaving. But since the $$\psi$$ entering "comes from the future", from the point of view of this vertex it is a $$\bar\psi$$. Take your first vertex, the $$\Phi\psi\bar\psi$$ one. Assuming time advances from left to right, the upper $$\psi$$ leaves, advances in time, and so your naming and your arrow are correct. The lower, the one you baptised $$\bar\psi$$, needs the arrow in the other direction, since it is a $$\bar\psi$$ leaving from the vertex. If you insist on your arrow direction, however, you must baptise it $$\psi$$, since a $$\bar\psi$$ coming from the future (as the arrow shows) is a $$\psi$$ going into the future.

For the other vertices, the reasoning is the same, if you want one arrow coming in and one leaving (for the $$\psi$$'s), they are both $$\psi$$'s, but if you want to write $$\psi$$ and $$\bar\psi$$, both arrows are leaving.

Note that for the $$\Phi$$ and $$\phi$$ this doesn't make any difference, because they are their own antiparticles.