A Goldilocks Higgs

Abstract: The Higgs could couple to a topological 4-form sector which yields a complex vacuum structure. In general such couplings could lead to direct CP violation in the Higgs sector. In many of the Higgs vacua electroweak symmetry is unbroken. In just as many it breaks when the 4-form flux is large enough. For a fixed value of flux, the symmetry breaking vacua have a smaller vacuum energy than the symmetric ones, where the difference is quantized because it is set by the $4$-form flux. This leads to the possibility that there is a value of the 4-form flux for any UV contributions to the Higgs {\it vev} that automatically cancels it down to the right value, $\sim$ TeV, if the 4-form charges are quantized in the units of the electroweak scale. This would still leave the cosmological constant which could be selected anthropically.