# Self Energy for N flavor phi^4 theory

1. Dec 23, 2014

### psi*psi

I am reading the fifth chapter on perturbation theory of Condensed Matter Field Theory by Altland and Simons. This question is about the section starting on page 223.

To discuss self energy, they introduced a vector field $\phi = \{ \phi^a \}, a = 1, \cdots , N$. The action of the field is given by
$$S[\phi] = \int d^dx (\frac{1}{2} \partial \phi \cdot \partial \phi + \frac{r}{2} \phi \cdot \phi + \frac{g}{4 N} (\phi \cdot \phi)^2)$$
The goal is to compute the perturbation expansion of the Green function
$$G^{ab}(x-y)=\langle \phi^a (x) \phi^b (y)\rangle$$
using the self energy operator $\Sigma_p$.

In momentum space, the Green function is given by
$$G^{ab}_{p} = [(p^2+r- \hat \Sigma_p)^{-1}]^{ab},$$
where the diagrams for $\Sigma_p$ is shown in the figure

The text claims that represented in terms of the Green functions, the first order contribution to the self-energy operator is given by
$$[\Sigma^{(1)}_{\mathbf{p}}]^{ab} = - \delta^{ab} \frac{g}{L^d} (\frac{1}{N} \sum_{\mathbf{p'}} G_{0,\mathbf{p'}} + \sum_{\mathbf{p'}} G_{0,\mathbf{p-p'}}),$$
where the first (second) term in the parenthesis corresponds to the first (second) diagram in the figure. I am having trouble reproducing this result. Specifically,
(1) Where does the overall minus sign come from?
(2) Since the interaction strength is given by $g/4N$, from the result, the first diagram has a contribution of $4$ and the second diagram has a contribution of $4N$. How do I get these factors?
(3) How do I derive the Feynman rules for these diagrams?

Thanks.

2. Dec 28, 2014