\left(\xi + \alpha\,p\right) - h_z\right]\left[\text{i}\,\omega_n \pm \left(\xi - \alpha\,p\right) + h_z\right] - h_x^2\; .
\label{D}$$

In the limit $\Delta_0 \rightarrow 0$, the Gor’kov Green’s function $F^\dagger$ can be written as : $$F^{\dagger} = -\Delta_0\,g_0(p)\,\text{i}\,\sigma_y G_0(p + q)\; .
\label{F}$$ Finally, using Eqs. (\[G0\], \[g0\], \[D\]) to compute the off-diagonal entry $F_{\downarrow\uparrow}^{\dagger}$ of Eq. (\[F\]), one obtains :

$$\frac{F_{\downarrow\uparrow}^{\dagger}}{\Delta_0} = -\frac{\left(\text{i}\,\omega + \xi_p - \alpha\,p + h_z\right)\,
\left(i\,\omega - \xi_{p+q}+\alpha\left(p+q\right) + h_z\right) + h_x^2}{\left[\left(\text{i}\,\omega + \xi_p + \alpha\,p - h_z\right)\,
\left(\text{i}\,\omega + \xi_p - \alpha\,p + h_z\right) - h_x^2\right]\left[\left(\text{i}\,\omega - \xi_{p+q}-\alpha\left(p+q\right) - h_z\right)\,
\left(\text{i}\,\omega - \xi_{p+q}+\alpha\left(p+q\right) + h_z\right) - h_x^2\right]}\;.
\label{eq. Fdu-single}$$

This expression Eq. (\[eq. Fdu-single\]) is the solution of the linearized Gor’kov equations for the anomalous Green’s function. At this stage, no assumptions are made upon the Zeeman fields ($h_x$ and $h_z$) and spin-orbit coupling $\alpha$ in the above equation, the only assumption being the limit $\Delta_0 \rightarrow 0$.

\[subsec:DL\_weak\] Self-consistency relation and critical temperature
----------------------------------------------------------------------

The self-consistency equation for the superconducting order parameter, $\Delta(x)= |\gamma| F_{\downarrow\uparrow}^{\dagger}(x,x,\tau=0) $, reads in momentum and Matsubara frequency representation : $$\Delta_0 = |\gamma|\,T\,\displaystyle\sum_{\omega_n} \int_{-\infty}^{+\infty}F_{\downarrow\uparrow}^{\dagger}(p)\frac{\mathrm{d}p}{2\,\pi}\;,
\label{SCE1}$$ where $\gamma$ is the attractive electron-electron BCS effective coupling constant. Near the critical temperature, Eq. (\[SCE1\]) can be 