Structure of cosmic strings for a gauged B-L symmetry and two Higgs fields

In the Standard Model, the difference between the baryon number $B$ and the lepton number $L$ is conserved, which represents a symmetry that is global and exact, and therefore unnatural. We turn it into a naturally exact, local symmetry by coupling the quantum number $B-L$ to an Abelian gauge field. Gauge anomalies are cancelled by adding right-handed neutrinos $ν_R$, which are not sterile in this case. Therefore the usual Majorana term is forbidden by gauge symmetry, but we arrange for a $ν_R$-mass by adding a Higgs-type 1-component complex scalar field. Thus we arrive at a modest, well-motivated extension of the Standard Model. In this framework, we investigate the field equations in the extended gauge-Higgs sector, involving both Higgs fields and the non-standard U(1) gauge field. They are given by a set of coupled, non-linear differential equations, which we solve numerically, focusing on the structures of cosmic strings. For a variety of parameters, we identify the string profiles and energy densities, assuming appropriate boundary conditions. In particular, these profiles depend on the winding number of each of the Higgs fields. As an amazing peculiarity, we discover -- for high winding numbers -- ``overshooting'' and ``co-axial'' solutions; in the latter case, the profile of the standard Higgs field changes its sign near the core of the cosmic string.