1) Composite Operator Insertions

How do we calculate diagrams with insertions of composite operators?

Iain:

You can think of an operator insertion as just an additional Feynman rule, where you insert that operator once. The idea behind the Composite Operator Renormalization homework problem is that you can make this very explicit, by simply thinking about the composite operator as an additional term that you add to the Lagrangian. If you do not want to think about it this way, but rather simply as an operator, then the review by Buras (see entry for Chapter 5 on the syllabus) goes through this in detail.

Sam:

 This is partially addressed in MQ3 as well, quoted below, which uses inspiration from Peskin and Schroeder as well as Collins' book.

``Doing the source procedure for composite fields is actually a bit tricky, so I hope that one of this problem can help elucidate the procedure. First, we add \(\int d^4 x J_{\phi^2}(x) \phi_0^2(x)\) to the action.

I think about this as simply adding an extra Feynman rule: instead of trying to do the path integral exactly, imagine that \(J(x)\) behaves like a field.

 In this naive picture, we have a new Feynman rule where the legs of a diagram can include \(J\)s, and we have a 3-point vertex which couples two \(\phi\)s to a \(J\).

 In momentum space, this is a bit easier for me to think about, and the Feynman rule must satisfy momentum conservation as usual. 

 Since \(\int d^4 x J_{\phi^2}(x) \phi_0^2(x) \int \frac{d^4 k}{(2\pi)^4}  \widetilde{J_{\phi^2}}(-p) \widetilde{\phi_0^2}(p)\), we can find the result of inserting \(\phi^2(x)\) by taking functional derivatives with respect to \(\widetilde{J_{\phi^2}}(-p)\) to get insertions of \(\widetilde{\phi_0^2}(p)\), setting \(J_{\phi^2} = 0\), and then Fourier transforming to find the result of inserting \(\phi^2(x)\).''