We investigate analytically and numerically the effects of disorder on the helical edge of a two-dimensional topological insulator in the presence of a Zeeman field and superconductivity. Employing bosonization and a renormalization-group analysis, we study how impurity potentials modify charge- and spin-density wave correlations as well as superconducting pair correlations. Our results reveal that the Zeeman field controls the competition: in the attractive regime, it amplifies the superconducting gap, while in the repulsive regime, it stabilizes impurity effects by keeping the system longer in the relevant regime for disorder. We also find that disorder induces logarithmic suppression of density-wave correlations, while at the same time introducing positive logarithmic corrections that enhance superconducting pairing correlations and contribute to their stability. These effects directly modify the scaling of spin conductance, providing experimentally accessible signatures of the interplay between disorder and superconductivity in topological edge channels.

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• Charge density waves
• Pair density wave
• Spin density waves
• 1-dimensional systems
• Topological superconductors
• Luttinger liquid model
• Single impurity model