Grand gauge-Higgs unification on $T^2/{\mathbb{Z}}_3$ via the diagonal embedding method

We study a novel six-dimensional gauge theory compactified on the $T^2/{\mathbb{Z}}_3$ orbifold utilizing the diagonal embedding method. The bulk gauge group is $G\times G\times G$, and the diagonal part $G^{\mathrm{diag}}$ remains manifest in the effective four-dimensional theory. Further spontaneous breaking of the gauge symmetry occurs through the dynamics of the zero modes of the extradimensional components of the gauge field. We apply this setup to the $SU(5)$ grand unified theory and examine the vacuum structure determined by the dynamics of the zero modes. We find bulk matter contents that radiatively induce spontaneous breakings of the unified symmetry $G^{\mathrm{diag}}\cong SU(5)$ down to $SU(3)\times SU(2)\times U(1)$ at the global minima of the one-loop effective potential for the zero modes. This spontaneous breaking provides notable features such as a realization of the doublet-triplet splitting without a fine tuning and a prediction of light adjoint fields.

Figure 1

The oblique coordinate system on $T^2/{\mathbb{Z}}_3$. The gray-shaded region is an independent domain of the torus $T^2$. The green-shaded region is a fundamental domain of the orbifold $T^2/{\mathbb{Z}}_3$. The small circles correspond to the fixed points.

Figure 2

The dual lattice ${\mathrm{\Lambda }}^{⟨p_t⟩}$ defined in Eq. (a12) for $p_t=0,1,-1$. The black dots show the point with $n_1=n_2=0$. As an example, we denote ${\mathit{N}}_{(\ell )}$ for $n_1=n_2=1$ by the yellow dots. Dots on the green, gray, and red-shaded regions belong to the sublattice ${\mathrm{\Lambda }}_{(1)}^{⟨p_t⟩},{\mathrm{\Lambda }}_{(2)}^{⟨p_t⟩}$, and ${\mathrm{\Lambda }}_{(3)}^{⟨p_t⟩}$ defined in Eq. (a14), respectively.