Symmetry factors (Srednicki, figure 9.11)

[kexue]

In Srednicki's QFT book on page 63, figure 9.11, the diagram in the middle of the second row is a Feynman diagram with four external lines, two vertices, one internal line and one loop placed on one external line. It has symmetry factor 4.

Does the symmetry facor stand for the 4 possibilities to place the loop on all four external lines respectively, so that these four diagrams would be equivalent?

thanks

 

[Avodyne]

No. There is a 2 from switching the top & bottom dark circles on the right, and a 2 from switching the two internal propagators in the loop.

 

[kexue]

Ok. But then the next question, why does the book say that in figure 9.11 we see all connected diagrams with E=4 and V=4? What, for example, with the three other possible diagrams where the loop is placed on one of the three external lines? Where is that accounted for?

thank you

 

[LAHLH]

Ok. But then the next question, why does the book say that in figure 9.11 we see all connected diagrams with E=4 and V=4? What, for example, with the three other possible diagrams where the loop is placed on one of the three external lines? Where is that accounted for?

We've already counted up those diagrams in the procedure outlined by Srednicki at the bottom of p60, were he talks about rearranging functional derivatives 3! ways, rearranging vertices themselves V! etc...

The point in the symmetry factors is to cancel off any duplicates we may have got by this naive counting, i.e. on the diagram in question swapping the two external props is equivalent to interaching the two func derivs from the right hand vertex, so we'd have naivley counted two diagrams that are exactly the same in our first guess at the number of diagrams, so we need a 1/2 to be introduced to start with.

Swapping the whole loop to another external propagator however is not something we need to consider in the sym factor, as to get the loop onto another arm would mean swapping the two vertices, whilst simultaneously swapping the internal prop and swapping the other external props around. Their is no other way to duplicate this by another set of our moves (we've counted correctly), no way we have overcounted etc. Compare this to get the upper right leg to be the bottom leg, we could either swap the propagators, OR we could swap the functional derivs on the right vertex.

As for why he hasn't drawn the other diagrams, with the loop on other legs, that is just because the diagrams are all equivalent, because they rotate into each other (yes, this is symmetry of the diagram, but a subtely different kind to symmetry factors)

 

[kexue]

Cool! Thank you very much, Avodyne and LAHLH!