Standard model with spontaneously broken quantum scale invariance

We explore the possibility that scale symmetry is a quantum symmetry that is broken only spontaneously and apply this idea to the standard model. We compute the quantum corrections to the potential of the Higgs field ($\varphi$) in the classically scale-invariant version of the standard model ($m_{\varphi }=0$ at tree level) extended by the dilaton ($\sigma$). The tree-level potential of $\varphi$ and $\sigma$, dictated by scale invariance, may contain nonpolynomial effective operators, e.g., ${\varphi }^6/{\sigma }^2$, ${\varphi }^8/{\sigma }^4$, ${\varphi }^{10}/{\sigma }^6$, etc. The one-loop scalar potential is scale invariant, since the loop calculations manifestly preserve the scale symmetry, with the dimensional regularization subtraction scale $\mu$ generated spontaneously by the dilaton vacuum expectation value $\mu \sim ⟨\sigma ⟩$. The Callan-Symanzik equation of the potential is verified in the presence of the gauge, Yukawa, and the nonpolynomial operators. The couplings of the nonpolynomial operators have nonzero beta functions that we can actually compute from the quantum potential. At the quantum level, the Higgs mass is protected by spontaneously broken scale symmetry, even though the theory is nonrenormalizable. We compare the one-loop potential to its counterpart computed in the “traditional” dimensional regularization scheme that breaks scale symmetry explicitly ($\mu =\text{constant}$) in the presence at the tree level of the nonpolynomial operators.