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I am finding some difficulties understanding the following problem. I have the following logarithm for which I have to identify branch cuts:

$$\lim_{\epsilon\rightarrow0}\ln{\frac{(p+2p_F)^2+\epsilon^2}{(p-2p_F)^2+\epsilon^2}}.$$

The branch points are the zeros of the argument of the logarithm, i.e.:

$p = \pm 2p_F \pm i\epsilon$

However if I look at Fetter-Walecka (pp. 178) or in the image I am attaching in this question, I cannot understand why the branch cuts are defined in the upper half-line along the rays:

$$\pm2p_F+i\epsilon+is\quad 0\le s < \infty.$$

Could you help me please?

The integral I have to solve is:

$$I = \int_{-\infty}^{+\infty}dp\frac{pe^{ipr}}{p^2+(½)q_{TF}^2*(1+g(p))}$$

where $$g(p)\propto\lim_{\epsilon\rightarrow0}\ln{\frac{(p+2p_F)^2+\epsilon^2}{(p-2p_F)^2+\epsilon^2}}.$$

enter image description here

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Here we will only answer OP's question about branch cuts. Replace the complex logarithm $${\rm Ln}{\frac{(p+2p_F)^2+\epsilon^2}{(p-2p_F)^2+\epsilon^2}}$$ with $${\rm Ln}[(p+2p_F+i\epsilon)(p+2p_F-i\epsilon)] - {\rm Ln}[(p-2p_F+i\epsilon)(p-2p_F-i\epsilon)] $$ for simplicity. The branch cuts in Fig A.1 (and their mirrors below the real $p$-axis) then corresponds precisely to the standard choice of branch cut for the complex logarithm $z\mapsto {\rm Ln}(z)$ along the negative real $z$-axis. One can of course move the branch cuts as long as they don't intersect the integration contour (=the real $p$-axis).

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