Classically conformal $\mathrm{U}(1{)}^{\prime }$ extended standard model and Higgs vacuum stability

We consider the minimal $\mathrm{U}(1{)}^{\prime }$ extension of the standard model (SM) with conformal invariance at the classical level, where in addition to the SM particle contents, three generations of right-handed neutrinos and a $\mathrm{U}(1{)}^{\prime }$ Higgs field are introduced. In the presence of the three right-handed neutrinos, which are responsible for the seesaw mechanism, this model is free from all the gauge and gravitational anomalies. The $\mathrm{U}(1{)}^{\prime }$ gauge symmetry is radiatively broken via the Coleman-Weinberg mechanism, by which the $\mathrm{U}(1{)}^{\prime }$ gauge boson ($Z^{\prime }$ boson) mass as well as the Majorana mass for the right-handed neutrinos are generated. The radiative $\mathrm{U}(1{)}^{\prime }$ symmetry breaking also induces a negative mass squared for the SM Higgs doublet to trigger the electroweak symmetry breaking. In this context, we investigate a possibility to solve the SM Higgs vacuum instability problem. The model includes only three free parameters ($\mathrm{U}(1{)}^{\prime }$ charge of the SM Higgs doublet, $\mathrm{U}(1{)}^{\prime }$ gauge coupling and $Z^{\prime }$ boson mass), for which we perform parameter scan, and identify a parameter region resolving the SM Higgs vacuum instability. We also examine naturalness of the model. The heavy states associated with the $\mathrm{U}(1{)}^{\prime }$ symmetry breaking contribute to the SM Higgs self-energy. We find an upper bound on $Z^{\prime }$ boson mass, $m_{Z^{\prime }}\lesssim 6\mathrm{TeV}$, in order to avoid a fine-tuning severer than 10% level. The $Z^{\prime }$ boson in this mass range can be discovered at the LHC Run-2 in the near future.

Figure 1

(a) The evolutions of the Higgs quartic coupling ${\lambda }_H$ (solid line) for the inputs $m_t=173.34\mathrm{GeV}$ and $m_h=125.03\mathrm{GeV}$, along with the SM case (dashed line). (b) The RG evolutions of ${\lambda }_{\mathrm{\Phi }}$ (solid line) and ${\lambda }_{\text{mix}}$ (dashed line). Here, we have taken $x_H=2$, $v_{\varphi }=19\mathrm{TeV}$ and $g^{\prime }(v_{\varphi })=0.09$.

Figure 2

(a) The result of parameter scan for $x_H$ and $g^{\prime }$ with a fixed $v_{\varphi }=19\mathrm{TeV}$, shown in $(m_{Z^{\prime }},x_H)$-plane with $m_{Z^{\prime }}=\sqrt{(x_{\mathrm{\Phi }}{g}^{\prime }{v}_{\varphi }{)}^2+(x_H{g}^{\prime }{v}_h{)}^2}\simeq x_{\mathrm{\Phi }}{g}^{\prime }{v}_{\varphi }$. As a reference, horizontal lines are depicted for $x_H=2$, 0 [$\mathrm{U}(1{)}_{B-L}$ case], $-16/41$ [orthogonal case], and $-1$ [$\mathrm{U}(1{)}_R$ case]. (b) Same as (a), but parameter scan for $x_H$ and $v_{\varphi }$ with a fixed $g^{\prime }=0.09$.

Figure 3

(a) The result of parameter scan for $v_{\varphi }$ and $g^{\prime }$ with a fixed $x_H=2$, shown in $(m_{Z^{\prime }},{\alpha }_{g^{\prime }})$-plane. (b) The allowed region at the TeV scale in (a) is magnified, along with the LEP bound (dashed line) and the LHC bound (solid line) from direct search for $Z^{\prime }$ boson resonance. The region on the left side of the lines are excluded. Here, the naturalness bounds for 10% (dotted line) and 30% (dashed-dotted line) fine-tuning levels are also depicted.

Figure 4

(a) Same as Fig. 3, but for $x_H=-2.5$. (b) Same as Fig. 3, but for $x_H=-2.5$