Topology of the Electroweak Vacua

Abstract

In the Standard Model, the electroweak symmetry is broken by a complex, $SU(2)$-doublet Higgs field and the vacuum manifold $SU(2)\times U(1)/U(1)$ has the topology of a 3-sphere. We remark that there exist theoretical alternatives that are locally isomorphic, but in which the vacuum manifold is homeomorphic to an arbitrary non-trivial principal $U(1)$-bundle over a 2-sphere. These alternatives have non-trivial fundamental group and thus feature topologically-stable electroweak strings. An alternative based on the manifold $\mathbb{R}P^3$ (with fundamental group $\mathbb{Z}/2$) allows custodial protection of gauge boson masses and their couplings to fermions, but, in common with all alternatives to $S^3$, has a problem with fermion masses.