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We study analytically and numerically the renormalization of the $g$-factorin semiconducting Rashba nanowires (NWs), consisting of a normal andsuperconducting section. If the potential barrier between the sections is high,a quantum dot (QD) is formed in the normal section. For harmonic (hard-wall)confinement, the effective $g$-factor of all QD levels is suppressedexponentially (power-law) in the product of the spin-orbit interaction (SOI)wavevector and the QD length. If the barrier between the two sections isremoved, the $g$-factor of the emerging Andreev bound states is suppressed lessstrongly. In the strong SOI regime and if the chemical potential is tuned tothe SOI energy in both sections, the $g$-factor saturates to a universalconstant. Remarkably, the effective $g$-factor shows a pronounced peak at theSOI energy as function of the chemical potentials. In addition, if the SOI isuniform, the $g$-factor renormalization as a function of the chemical potentialis given by a universal dependence which is independent of the QD size. Thisprediction provides a powerful tool to determine experimentally whether the SOIin the whole NW is uniform and, moreover, gives direct access to the SOIstrengths of the NW via $g$-factor measurements. In addition, it allows one tofind the optimum position of the chemical potential for bringing the NW intothe topological phase at large magnetic fields.

Renormalization of quantum dot $g$-factor in superconducting Rashba nanowires