# Integration of Dirac propagator

I've got confused with Dirac (or Spinor) Propagator. Everywhere in books I have examples of integration Klein-Gordon propagators, which are quite easy. But I don't understand how to integrate 4-component function.

My second question - how to deal with a mix of Dirac and KG propagators in one equation? I know that if we have many scalar vertices we have to regularize our integral. And examples are shown in lot of books - Peskin, Maggiore, many more. But I couldn't find case with Dirac propagators.

My equations were with: $$S_a(x_1-x)D(x-y)S_b(x-y)S_a(y-z)S_b(z-y_1)D(z-y_2)$$

Where $x_1$ were position of initial particle (it's 1->2 decay process) and $y_i$ are positions of final particles. $x,y,z$ coordinates are for internal vertices. We started from a-fermion particle (that's why there is $S_a$ propagator) and we ended with b-fermion and $\phi$ - scalar particles. Interaction Hamiltonian contained only these 3 particles.

If there is required, I can even post Feynman diagram here for process.

EDIT: Here are the two Feynman Diagrams for decay process, in third order of perturbation. Described problem was for the first one:

Ah: our theory is $\mathcal{H}_I=k\varphi\bar{\psi_b}\psi_a$

• The process is essentially identical to dealing with scalar fields. The only difference being that now you are integrating over matrices. But a matrix integral just means an integral for each component(though you can do them all at once) so it isn't any harder then the scalar case. If you want more help then I'd recommend posting the diagram and the expression you are having difficulty simplifying. – JeffDror May 21 '14 at 18:20
• Link to image broken... – jinawee Jun 19 '14 at 22:39