content_image/416353.jpg content_image/416354.jpg content_image/416355.jpg content_image/416362.jpg content_image/416357.jpg content_image/416358.jpg content_image/416359.jpg content_image/416356.jpg content_image/416360.jpg content_image/416348.jpg content_image/416361.jpg content_image/416363.jpg content_image/416364.jpg
After carrying out the angular integrals we obtain 

<img src='content_image/416353.jpg'>

<img src='content_image/416354.jpg'>

<img src='content_image/416355.jpg'>

where 〈 L$_{2}$ 〉 is the angle integrated loop L$_{2}$ , which is ob- tain from (A12) via the substitution x 2 = 1 / 3. Using the transformation dpp 2 = m $^{∗}$p$_{F}$ d ( p $^{2}$/ 2 m $^{∗}$) = m $^{∗}$p$_{F}$ dξ$_{p}$, and the relation ϵ$_{p}$dϵ$_{p}$ = ξ$_{p}$dξ$_{p}$ we obtain 

To make further progress we need to separate the real and imaginary parts of the integrals. We shall first compute the imaginary parts. They are extracted with the help of the identity 

<img src='content_image/416362.jpg'>

where P denotes the principal value and δ ( n $^{)}$( D ) is the n -th derivative of the delta function. 

<img src='content_image/416357.jpg'>

<img src='content_image/416358.jpg'>

<img src='content_image/416359.jpg'>

For positive ω the integral (B1) gives [21] 

<img src='content_image/416356.jpg'>

Consider the integral (B2). First, note that the term ∝ D − 1 vanishes, since the delta function enforces ω 2 = 4 ϵ 2 $_{p}$. After dropping this term we are left with the integral 

<img src='content_image/416360.jpg'>

This and similar integrals, which contain derivatives of the delta function, are computed via the formula 

<img src='content_image/416348.jpg'>

The result of integration is 

<img src='content_image/416361.jpg'>

In the integral (B3) we again omit terms ∝ D $^{1}$, since their prefactors are zero after integration. After inserting x 2 = 1 / 3 in the remainder we obtain 

<img src='content_image/416363.jpg'>

Applying Eq. (B13) we obtain 

<img src='content_image/416364.jpg'>

Finally, adding the three integrals (B11), (B14), and (B16) we obtain Eq. (42) of the main text. 

