MAP, MAC, and vortex-rings configurations in the Weinberg–Salam model

We report on the presence of new axially symmetric monopoles, antimonopoles and vortex-rings solutions of the SU(2)×U(1) Weinberg–Salam model of electromagnetic and weak interactions. When the ϕϕ-winding number n=1n=1, and 2, the configurations are monopole–antimonopole pair (MAP) and monopole–antimonopole chain (MAC) with poles of alternating sign magnetic charge arranged along the zz-axis. Vortex-rings start to appear from the MAP and MAC configurations when the winding number n=3n=3. The MAP configurations possess zero net magnetic charge whereas the MAC configurations possess net magnetic charge of 4πn/e4πn/e.In the MAP configurations, the monopole–antimonopole pair is bounded by the Z0Z0 field flux string and there is an electromagnetic current loop encircling it. The monopole and antimonopole possess magnetic charges ±4πnesin2θW respectively. In the MAC configurations there is no string connecting the monopole and the adjacent antimonopole and they possess magnetic charges ±4πne respectively. The MAC configurations possess infinite total energy and zero magnetic dipole moment whereas the MAP configurations which are actually sphalerons possess finite total energy and magnetic dipole moment. The configurations were investigated for varying values of Higgs self-coupling constant 0≤λ≤400≤λ≤40 at Weinberg angle θW=π4.