condensed matter - Derivation of the gradient expansion of the Keldysh nonlinear sigma model for disorder metals

My confusion relates to Appendix C of this this paper although the same derivation is presented in many others. When deriving the gradient expansion of this term arrives at a term quadratic in the scalar potential, which takes the form $\mathrm{Tr}V^\alpha \Upsilon^{\alpha\beta} V^\beta$ where $\alpha,\beta \in \{\mathrm{c}l, \mathrm{q}\}$ are indices in the Keldysh matrix space, the trace is over space and time indices, and

$$\Upsilon^{\alpha\beta}(\omega) = - \frac{1}{2}\sum_{\mathbf{p}} \mathrm{Tr}_K \mathcal{G}(\mathbf{p}, \epsilon+\omega/2)\gamma^\alpha\mathcal{G}(\mathbf{p}, \epsilon-\omega/2)\gamma^\beta$$ with $\mathrm{Tr}_K$ the trace over Fermionic Keldysh indices, $\gamma^{cl} = \sigma^0$, $\gamma^q=\sigma^1$ are matrices in the Fermionic Keldysh space, and $\mathcal{G}$ is the disorder averaged Fermionic Green's function. In particular, we write the Fermionic Green's function as $$\hat{\mathcal{G}} = \frac{1}{2}\mathcal{G}^R \left(1 + \hat{\Lambda}\right) + \frac{1}{2}\mathcal{G}^A \left(1 - \hat{\Lambda}\right)$$ where $\mathcal{G}^{R(A)} = [\epsilon - \xi \pm \frac{i}{2\tau}]^{-1}$ is the retarded (advanced) disorder averaged Green's function and $\Lambda$ is the stationary saddle point solution of the non-linear sigma model $$\hat{\Lambda} = \begin{pmatrix} 1^R&2F\\ 0&-1^A \end{pmatrix}$$ c.f. Eq. 165. This expression is then evaluated to be $i\nu \hat{\sigma}_1^{\alpha\beta}$ where $\sigma_1$ is the first Pauli matrix in the bosonic Keldysh space and $\nu$ is the density of states at the Fermi surface.

So far so good. My confusion is the decision in Appendix C to only include the products $\mathcal{G}^R\mathcal{G}^R$ or $\mathcal{G}^A\mathcal{G}^A$ in the integral. Using the above parametrization of the Keldysh Green's function one can see that $\Upsilon^{\mathrm{cl},\mathrm{cl}}=0$ but each of the other components are not identically zero and contain $(\mathcal{G}^R)^2$, $(\mathcal{G}^A)^2$, and $\mathcal{G}^R\mathcal{G}^A$ terms. I don't follow the argument for why only the R-R terms should be included and in fact in the other parts of the gradient expansion it is instead the R-A terms which are kept and the R-R and A-A terms vanish due to the analytic structure of the Green's functions. Furthermore, it is not clear to me why the q-q component is zero. Am I missing something about the analytical structure of the integrands or is there a physical argument for this?