Dark Matter and Muon $g-2$ Anomaly via Scale Symmetry Breaking

The Standard Model (SM) without the Higgs mass term is scale invariant. Gildener and Weinberg generalized the scale invariant standard model (SISM) by including the multiplication of scalars in quartic forms. They pointed out that along the flat direction only one scalar -- called the scalon -- is classically massless and all other scalars are massive. Here we choose a SISM with one scalon and one heavy scalar and extend that further respecting the scale invariance by a vector-like lepton (VLL). By an appropriate choice of the flat direction, the heavy scalar enjoys the $\mathbb{Z}_2$ symmetry and is assumed as DM particle. The scalon connects the visible and dark sector via the Higgs-portal and by interacting with both the muon lepton and the VLL. The VLL is charged under $U(1)_Y$ and interacts with $\gamma/Z$ bosons. We show that the model correctly accounts for the observed dark matter (DM) relic abundance in the universe, while naturally evading the current and future bounds from direct detection (DD) experiments. Moreover, the model is capable to explain the $(g-2)_\mu$ anomaly observed in Fermilab. We also show a feature in SISM scenarios which is not present in other Higgs-portal models; despite having the Higgs-portal term $|H|^2 s^2$ ($s$ being the scalon) in SISM, the effective potential after the electroweak symmetry breaking lacks an important expected vertex $h s^2$. This property immediately forbids the tree-level invisible Higgs decay $h\to ss$ and the one-loop Higgs decay $h\to \mu^+ \mu^-$.