O(10) kinks: Clash of symmetries on the brane and the gauge hierarchy problem

We study kink or domain wall solutions in O(10) Higgs models in the context of the ``clash of symmetries'' mechanism developed by Davidson, Toner, Volkas and Wali and, independently, by Pogosian and Vachaspati. We show that kink configurations employing Higgs fields in the 45 (the adjoint representation) of O(10) break up into three classes: those that at finite distances from the wall respect a U(5) subgroup of SO(10), and two others that respect the smaller subgroups U(3) x U(2) and U(4) x U(1). These smaller subgroups arise from the clash of symmetries mechanism: they are the intersections of two differently embedded U(5) subgroups of SO(10), the latter being the symmetries respected in asymptotia on opposite sides of the domain wall. The existence of the SO(10) -> U(3) x U(2) = SU(3) x SU(2) x U(1) x U(1)' = G_{SM} x U(1)' class advances the search for a realistic brane world model wherein some of the required symmetry breaking is achieved via the clash of symmetries rather than the conventional mechanism. At the centres of the walls, the unbroken symmetries are enhanced. In the U(3) x U(2) case, the symmetry is O(6) x U(2), which is closely related to the Pati-Salam-like SU(4) x SU(2) x U(1) group. If our universe is a brane located at the centre of such a wall, then we see the O(10) symmetry as being strongly broken to SU(4) x SU(2) x U(1). Interestingly, if the brane-world degrees of freedom enjoy a slight leakage off the wall, then an additional symmetry breakdown to U(3) x U(2) = G_{SM} x U(1)' is effectively induced on the brane. This provides a possible framework within which to address at least part of a gauge hierarchy problem: O(10) is strongly broken to SU(4) x SU(2) x U(1), then more weakly to G_{SM} x U(1)' depending on the amount of leakage off the brane. We also comment on kinks employing the 10 and 54 of O(10).