Vacuum stability and triviality analyses of the renormalizable coloron model

The renormalizable coloron model is built around a minimally extended color gauge group, which is spontaneously broken to QCD. The formalism introduces massive color-octet vector bosons (colorons), as well as several new scalars and fermions associated with the symmetry breaking sector. In this paper, we examine vacuum stability and triviality conditions within the context of the renormalizable coloron model up to a cutoff energy scale of 100 TeV, by computing the $\beta$-functions of all relevant couplings and determining their running behavior as a function of the renormalization scale. We constrain the parameter space of the theory for four separate scenarios based on differing fermionic content, and demonstrate that the vectorial scenarios are less constrained by vacuum stability and triviality bounds than the chiral scenarios. Our results are summarized in exclusion plots for the separate scenarios, with previous bounds on the model overlaid for comparison. We find that a 100 TeV hadron collider could explore the entire allowed parameter space of the chiral models very effectively.

Figure 1

The $\sin \chi -m_s$ exclusion plots with no spectator fermions, $N_Q=0$, for a singlet VEV $v_s=3\mathrm{TeV}$. The panels display benchmark values of $m_{G_H},M_C\in \{2.5,3.5\}\mathrm{TeV}$, so that one can observe the impact of having each mass lying below or above $v_s$. The constraints from considering the global minimum of the potential coinciding with the VEVs (long-dashed green line), unitarity (dot-dashed blue line), electroweak precision tests (dotted red line), direct Higgs coupling measurements by the LHC (short-dashed black line), and vacuum stability and triviality up to 100 TeV (solid brown line) are exhibited. The enumerated curves correspond to several values of $n\equiv m_{\mathcal{A}}/m_{G_H}$ between 0 and 1. All colored regions are excluded.

Figure 2

The $\sin \chi -m_s$ exclusion plots with no spectator fermions, $N_Q=0$, for singlet VEVs $v_s=5,10\mathrm{TeV}$, and $m_{G_H},M_C\sim v_s$. Once more, several $n\equiv m_{\mathcal{A}}/m_{G_H}$ curves between 0 and 1 are displayed. All colored regions are excluded. Note that as $v_s$ increases, the unitarity bounds (corresponding, from bottom to top, to $n=0,0.5,1$) restrict $\sin \chi$ more strongly, the direct Higgs constraints restrict $\sin \chi$ less strongly, and the regions allowed by electroweak precision tests, vacuum stability, and triviality become both more constrained in $\sin \chi$ and less constrained in $m_s$. Once more, the lower bounds (corresponding, from bottom to top, to $n=0.5,1$) are due to demanding that the global minimum of the potential coincides with the VEVs. (See the caption of Fig. 1 for further details.)

Figure 3

The running of the scalar quartic couplings (top and middle rows) and the gauge and Yukawa couplings (bottom panel) as a function of the renormalization scale $\mu$, for $N_Q=0$, $v_s=3\mathrm{TeV}$, $m_{G_H}=M_C=3.5\mathrm{TeV}$ (bottom-right panel of Fig. 1). The panels in the top and middle rows correspond to different regions of the $\sin \chi -m_s$ parameter space: the top-left panel illustrates a point in the allowed region for a heavy pseudoscalar, the top-right panel corresponds to a point excluded because $\sin \chi$ is too large, and the middle-left panel corresponds to a point excluded because $m_s$ is too large. The middle-right panel shows a point that is allowed for a heavy pseudoscalar (as in the top-left panel) but becomes excluded when the pseudoscalar is light. The vertical solid line indicates the imposed 100 TeV cutoff, and a Landau pole is indicated by a coupling that exceeds $4\pi$; in the top-right panel, ${\lambda }_h$ suffers a Landau pole below 100 TeV, while in the middle row it is ${\lambda }_s^{\prime }$ that does so. The flow of the gauge and Yukawa couplings (bottom panel) is the same in all panels, since they do not depend on the mixing angle or the (pseudo)scalar masses. Note that the Higgs quartic coupling, ${\lambda }_h$, receives a shift at the threshold $v_s$, whereas the strong coupling $g_s$ is replaced by the two extended gauge group couplings, $g_{s_1}$ and $g_{s_2}$, at the same threshold.

Figure 4

The $\sin \chi -m_s$ exclusion plots with one spectator fermion generation, $N_Q=1$, for $v_s=m_{G_H}=M_C=3\mathrm{TeV}$. The panels correspond to two representative values of the spectator mass, $M_Q=1,2\mathrm{TeV}$, and various $n\equiv m_{\mathcal{A}}/m_{G_H}$ curves between 0 and 1 (together with their corresponding unitarity and the global-minimum constraints from bottom to top) are displayed. Note that in the right panel an additional $m_s$ lower bound is developed for a heavy pseudoscalar ($n=1$). All colored regions are excluded. (See the caption of Fig. 1 for further details.)

Figure 5

The running of the scalar quartic couplings (top and middle rows) and the gauge and Yukawa couplings (bottom panel) as a function of the renormalization scale $\mu$, for $N_Q=1$, $v_s=m_{G_H}=M_C=3\mathrm{TeV}$, and $M_Q=2\mathrm{TeV}$ (right panel of Fig. 4). The panels in the top and middle rows correspond to different regions of the $\sin \chi -m_s$ parameter space: the top-left panel illustrates a point in the allowed region for a light pseudoscalar, the top-right panel corresponds to a point excluded because $\sin \chi$ is too large (${\lambda }_h$ suffers a Landau pole), and the middle-left panel corresponds to a point excluded because $m_s$ is too large (${\lambda }_s^{\prime }$ suffers a Landau pole). The middle-right panel demonstrates a point which is allowed for a light pseudoscalar but becomes excluded for a heavy pseudoscalar, where ${\kappa }_s$ becomes negative and therefore destabilizes the potential. (See the caption of Fig. 3 for further details.)

Figure 6

The $\sin \chi -m_s$ exclusion plots with three spectator fermion generation, $N_Q=3$, for $v_s=m_{G_H}=M_C=3\mathrm{TeV}$, and the spectator mass $M_Q=1\mathrm{TeV}$. Various $n\equiv m_{\mathcal{A}}/m_{G_H}$ curves between 0 and 1 (together with their corresponding unitarity and the global-minimum constraints from bottom to top) are displayed. All colored regions are excluded. (See the caption of Fig. 1 for further details.)

Figure 7

The $\sin \chi -m_s$ exclusion plots with one Dirac spectator fermion generation, $N_Q^{\text{Dirac}}=1$, for three benchmark values $v_s=3,5,10\mathrm{TeV}$, and the corresponding values for the vector and scalar color-octet masses. Several curves within the allowed range of the pseudoscalar masses ($n=0-1$) are presented. A universal spectator mass $M_Q=1\mathrm{TeV}$ is selected for illustration. All colored regions are excluded. (See the caption of Fig. 1 for details.)