Sphalerons at finite mixing angle

The classical sphaleron is constructed for the full SU(2)\ifmmode\times\else\texttimes\fi{}U(1) electroweak theory. Unlike the sphaleron of the SU(2) Yang-Mills-Higgs theory, it is not spherically symmetric due to the coupling to the U(1) field. It is symmetric only under rotations around the z axis and parity reflections. The mixing angle ${\mathrm{\ensuremath{\theta}}}_{\mathit{W}}$ is varied over the full range 0\ensuremath{\le}${\mathrm{\ensuremath{\theta}}}_{\mathit{W}}$\ensuremath{\le}\ensuremath{\pi}/2. When the mixing angle is increased the energy of the sphaleron decreases, and the energy density changes its shape from a sphere at ${\mathrm{\ensuremath{\theta}}}_{\mathit{W}}$=0 to a very elongated spheroid at large values of the mixing angle. At the physical value of the mixing angle, however, the electroweak sphaleron differs only a little from the spherical sphaleron.

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