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

We study analytically and numerically the renormalization of the $g$-factor in semiconducting Rashba nanowires (NWs), consisting of a normal and superconducting 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 suppressed exponentially (power-law) in the product of the spin-orbit interaction (SOI) wavevector and the QD length. If the barrier between the two sections is removed, the $g$-factor of the emerging Andreev bound states is suppressed less strongly. In the strong SOI regime and if the chemical potential is tuned to the SOI energy in both sections, the $g$-factor saturates to a universal constant. Remarkably, the effective $g$-factor shows a pronounced peak at the SOI energy as function of the chemical potentials. In addition, if the SOI is uniform, the $g$-factor renormalization as a function of the chemical potential is given by a universal dependence which is independent of the QD size. This prediction provides a powerful tool to determine experimentally whether the SOI in the whole NW is uniform and, moreover, gives direct access to the SOI strengths of the NW via $g$-factor measurements. In addition, it allows one to find the optimum position of the chemical potential for bringing the NW into the topological phase at large magnetic fields.