Unification of gauge couplings and the Higgs mass in vectorlike particle theories extended into NMSSM

The minimal supersymmetry standard model (MSSM) extended with vectorlike theories have long been discussed. If we extend the vectorlike MSSM theory into NMSSM and let the vectorlike particles couple with the singlet (S), we find out a natural way to generate the vectorlike particle masses near 1 TeV through the breaking of the $Z_3$ group. Compared with the $\text{MSSM}+\text{vectorlike}$ models, vectorlike models extended into NMSSM contain more Yukawa couplings and can help us adjust the renormalization group (RG) trajectories of the gauge couplings in order to unify the intersections. They can also help press down the gauge RG-$\beta$ functions for a $5+\overline{5}+10+\overline{10}$ model, in order for the RG trajectories of the gauge couplings to unify before the Landau pole. We also discuss the Higgs mass contributed from the vectorlike sectors in this case.

Figure 1

The coupling constants near $Q=1\mathrm{TeV}$ in the boundary condition that $\lambda (Q_{\mathrm{GUT}})=\kappa (Q_{\mathrm{GUT}})=3$ and ${\lambda }_D(Q_{\mathrm{GUT}})={\lambda }_L(Q_{\mathrm{GUT}})$. The GUT scale is defined as $Q_{\mathrm{GUT}}=1.81\times 10^{16}\mathrm{GeV}$, and different values of ${\lambda }_D(Q_{\mathrm{GUT}})={\lambda }_L(Q_{\mathrm{GUT}})$ are taken into the RG input parameters.

Figure 2

Running couplings in $5+\overline{5}$ theory. The left panel shows the trajectories of the gauge couplings. Notice that they converge into one point accurately. The right panel shows the corresponding Yukawa couplings. It is the Yukawa couplings that contribute into the gauge RG-$\beta$ functions so that they can converge into one point at the scale $Q=1.81\times 10^{16}\mathrm{GeV}$.

Figure 3

The coupling constant trajectories of $10+\overline{10}$ theory. The left panel shows the gauge couplings, and the right panel shows the Yukawa couplings.

Figure 4

The two-loop trajectories of three gauge-coupling constants under $5+\overline{5}+10+\overline{10}+\mathrm{NMSSM}$ theory. All Yukawa couplings are closed and the trajectories are all run from $Q=1\mathrm{TeV}$.

Figure 5

The coupling constant trajectories of $5+\overline{5}+10+\overline{10}$ theory. The left panel show the gauge couplings, and the right panel shows the Yukawa couplings.