I am reading Kamenev's book "Field theory of non-equilibrium systems" and I have some questions about his approach to understanding disordered fermionic systems in chapter 14. I understand the arguments of section 14.2 and how he arrives at the corrections to the fermionic propagator in equation 14.24: $$ \Lambda_\epsilon^{R/A} = \frac{i}{\pi\nu} \sum_{\vec k} \frac{1}{\epsilon-\xi_{\vec k} + \frac{i}{\Lambda_\epsilon^{R/A}}}.$$ Here $R/A$ stand for retarded and advanced. This is essentially the self-consistent Born approximation: letting $G_0^{R/A}$ be the bare green functions, we find the corrected green function $G$ is given by equation 14.29 $(G_0^{-1} + (i\Lambda/2\tau) )G = 1$. So far so good.
The part I get confused at is section 14.7 in which he treat the time-reversal-invariant case. Here he says we need to double the fields to account for the 'time-reversal space.' He then proceeds to have a paragraph describing how one might go about modifying the calculation he did previously to account for this doubling of fields. What I'm wondering is can someone be a little more explicit and derive the generalized version of the self-consistent Born approximation (reproduced above)? Do we get the same factor of $\Lambda$ to correct the fermionic green function? Or is it modified in some capacity?
