- \frac{\pi}{\omega_n}\right]\; ,
    \label{SCE_strong_SOC2}$$

which is reminiscent of Eq. (\[eq. SC\]) albeit expressed in the helical basis labelled by $\lambda$, rather than in the spin basis. For each helical band $\lambda$, the integral over momentum can be turned into an integral over $\xi$ using quasi-classical approach: $$\int_{-\infty}^{+\infty} \mathrm{d}p \rightarrow \frac{1}{v_\text{F}}\,\int_{-\infty}^{+\infty} \mathrm{d}\xi \;.$$ The integration over $\xi$ is then straightforward and the expression of the self-consistency equation is provided in Appendix \[app:SCE\].

\[subsec:diag\] Modulated phase in the presence of strong SOC
-------------------------------------------------------------

In the same way as Sec. \[subsec:DL\_weak\], we first investigate the emergence of the modulated phase. We consider small values of the Zeeman field $h_x,\, h_z \ll T_c$ and small values of $q$ such that $q \ll p_\text{F}$: $$\ln\left(\frac{T_c}{T_{c0}}\right) = 2\,\pi\,T_c\,\sum_{\omega_n \geq 0}\left[-\frac{h_z^2}{\omega_n^3} + \frac{3\,h_x^2\,h_z}{2\,\omega_n^3\,\alpha\,p_\text{F}^2}\,q - \frac{v_\text{F}^2}{4\,\omega_n^3}\,q^2\right] \;.
    \label{DL_strong_SOC}$$ The self-consistency relation Eq. (\[DL\_strong\_SOC\]) was obtained by considering only intraband pairing. In this case, we can notice that when $h_z = 0$, Eq. (\[DL\_strong\_SOC\]) leads to $T_c = T_{c0}$, which seems to indicate that superconductivity is not affected by the Zeeman field when this one is normal to the spin-orbit interaction. However, we have to keep in mind that taking interband correlations into account in Eq. (\[Trace\]) would modify 