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To discuss self energy, they introduced a vector field ##\phi = \{ \phi^a \}, a = 1, \cdots , N##. The action of the field is given by

[tex]

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)

[/tex]

The goal is to compute the perturbation expansion of the Green function

[tex]

G^{ab}(x-y)=\langle \phi^a (x) \phi^b (y)\rangle

[/tex]

using the self energy operator ##\Sigma_p##.

In momentum space, the Green function is given by

[tex]

G^{ab}_{p} = [(p^2+r- \hat \Sigma_p)^{-1}]^{ab},

[/tex]

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

[tex]

[\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'}}),

[/tex]

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.