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Spin-polarized normal-metal wires coupled to a superconductor can hostMajorana states at their ends. These decay into the bulk and are protected by aminigap in the spectrum. Previous studies have found that strong couplingbetween the superconductor and the normal-metal wire renormalizes theproperties of this low-energy phase. Here, we develop a semiclassicalscattering approach to explain these renormalization effects. We show that arenormalization of the propagation velocity in the normal wire originates fromdouble Andreev reflection processes at the superconductor interface and that itcontinues to exist in the absence of a proximity-induced minigap in thenormal-metal wire. We also show that the renormalization effects exist forarbitrary transparency of the normal-metal--superconductor interface, providedthe superconductor coherence length is sufficiently long in comparison to thethickness of the normal metal.

Renormalization effects in spin-polarized metallic wires proximitized by a superconductor: A scattering approach