Quantum Field Theory Exam Q3: Draw Feynman Diagram | Advice Needed

[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.