Ultra-weak sector, Higgs boson mass, and the dilaton

The Higgs boson mass may arise from a portal coupling to a singlet field $\sigma$ which has a very large VEV $f \gg m_\text{Higgs}$. This requires a sector of "ultra-weak" couplings $\zeta_i$, where $\zeta_i \lesssim m_\text{Higgs}^2 / f^2$. Ultra-weak couplings are technically naturally small due to a custodial shift symmetry of $\sigma$ in the $\zeta_i \rightarrow 0$ limit. The singlet field $\sigma$ has properties similar to a pseudo-dilaton. We engineer explicit breaking of scale invariance in the ultra-weak sector via a Coleman-Weinberg potential, which requires hierarchies amongst the ultra-weak couplings.