@Sestoft:2018; @shabani:2016].

\[subsec:helical\_basis\] Hamiltonian in the helical basis
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We still consider the same model Eq. (\[H\]) for the wire, but we are going to use a different basis to facilitate the study of the large spin-orbit coupling regime. In the absence of superconductivity, the Hamiltonian describing the wire reads : $$\hat{h}_\text{N} = \left(\xi + \alpha\,p\,\sigma_z\right)\,\tau_z + \vec{h}\cdot\vec{\sigma} \, ,$$ where $\sigma_{x,y,z}$ and $\tau_{x,y,z}$ are the Pauli matrices acting in spin and Nambu space respectively.

Using the unitary transformation $$\hat{h}_\text{N} \rightarrow \mathcal{U}^\dagger\,\hat{h}_\text{N}\,\mathcal{U} = \hat{h}_\text{hel} \, ,$$ with $ \mathcal{U}$ given explicitly in Appendix \[app:hamiltonian\], the normal state Hamiltonian can be written in the helical basis as : $$\hat{h}_\text{hel} = \xi\,\tau_z - \frac{\lambda}{2}\left[\left(\varepsilon_+ + \varepsilon_-\right)\,\tau_0 + \left(\varepsilon_+ - \varepsilon_-\right)\,\tau_z\right]\;,
    \label{hamiltonian_diag}$$ where $\lambda = \pm$ labels the energy bands, and $\varepsilon_\pm = \sqrt{h_x^2 + \left(h_z \pm \alpha\,p\right)^2}$. Matrix $\tau_0$ is the identity matrix acting in Nambu space.

This allows to solve the equations for the normal state Green’s functions for each helical band independently : $$\left(\text{i}\,\omega_n - \hat{h}_\text{hel}\right)\hat{G}_0 = \mathbb{1} \, ,$$ which leads to the solutions in Nambu-helical space: $$\hat{G}_0 = \begin{pmatrix}
    G_\lambda\left(p,\, \omega_n\right) & 0\\
    0 & -G_\lambda\left(-p,\, -\omega_n\right)
    \end{pmatrix} \;,$$ where $$G_\lambda\left(p,\, \omega_n\right) = \frac{1}{\text{i}\,\omega_n - \xi + \lambda\,\sqrt{h_x^2 