+ \left(h_z + \alpha\,p\right)^2}} \;.$$

![image](Fig_2c.png)

Let us now consider the system in the presence of superconductivity in the limit $\Delta_0 \ll T_c$. In order to take into account the possibility of a modulated superconducting order parameter $\Delta_0 e^{iqx}$, the self-consistency equation reads $$\Delta_0 = \frac{|\gamma|}{4}\,T\,\sum_{\omega_n} \int_{-\infty}^{+\infty} \text{Tr}\left(\hat{G}^S\left(\mathcal{U}^{(q)}\right)^\dagger\tau_x\,\mathcal{U}^{(q)}\right) \frac{\mathrm{d}p}{2\,\pi} \;,
    \label{SCE_strong_SOC}$$ where $\mathcal{U}^{(q)}$ is the “momentum shifted” version (Appendix \[app:Adding\_SC\]) of the unitary matrix connecting the original spin basis and the helical basis. The matrix $\hat{G}^S$ is the Green’s function in the presence of superconductivity. At first order in $\Delta_0$: $$\hat{G}^S = \hat{G}_0^{(q)} - \Delta_0\hat{G}_0^{(q)}\left(\mathcal{U}^{(q)}\right)^\dagger\tau_x\,\mathcal{U}^{(q)}\,\hat{G}_0^{(q)} \;.
    \label{Gq}$$ The label $S$ in $\hat{G}^{S}$ denotes the presence of superconductivity, and the superscript $q$ in $\hat{G}_0^{(q)}$ and $\mathcal{U}^{(q)}$ refer to the “shifted versions” of $\hat{G}_0$ and $\mathcal{U}$ defined in the appendix B. The Green’s function $\hat{G}^S$ contains both intraband and interband terms (see Appendix \[app:SCE\]). In the regime of strong spin-orbit interaction, $\alpha\,p_\text{F} \gg h_x,\,h_z,\,T_c$ and $E_\text{so} = \frac{1}{2}\,m\,\alpha^2 \ll \mu$, the interband terms are negligible and it is sufficient to consider only intraband pairing. It is convenient to eliminate $\gamma$, and rewrite the self-consistency relation Eq. (\[SCE\_strong\_SOC\]) as :

$$\ln\left(\frac{T_c}{T_{c0}}\right) = T_c\,\sum_{\omega_n \geq 0,\,\lambda} \left[\frac{v_\text{F}}{2}\,\text{Re}\left(\int_{-\infty}^{+\infty} G_\lambda\left(p + \frac{q}{2},\, \omega_n\right)\,G_\lambda\left(-p + \frac{q}{2},\, -\omega_n\right)\mathrm{d}p\right) 