  We present a theoretical study of Coulomb exchange interaction for electrons
confined in a cylindrical quantum wire and subject to a Rashba-type spin-orbit
coupling with radial electric field. The effect of spin splitting on the
single-particle band dispersions, the quasiparticle effective mass, and the
system's total exchange energy per particle are discussed. Exchange interaction
generally suppresses the quasiparticle effective mass in the lowest nanowire
subband, and a finite spin splitting is found to significantly increase the
magnitude of the quasiparticle-mass suppression (by upto 15\% in the
experimentally relevant parameter regime). In contrast, spin-orbit coupling
causes a modest (1\%-level) reduction of the magnitude of the exchange energy
per particle. Our results shed new light on the interplay of spin-orbit
coupling and Coulomb interaction in quantum-confined systems, including those
that are expected to host exotic quasiparticle excitations.
