$E_6$ grand-unification for a Dark Energy model

The unification of a recently proposed model for Dark Energy and Dark matter [1], based on an unbroken gauge group $SU(2)_Z$, with the Standard model (SM) is presented. The unifying group is $E_6$ and the symmetry breaking pattern takes the unusual route $E_6\to SU(2)_Z \otimes SU(6)\to SU(2)_Z \otimes SU(3)_c \otimes SU(3)_L \otimes U(1)_6\to SU(2)_Z \otimes SU(3)_c \otimes SU(2)_L \otimes U(1)_{3}\otimes U(1)_6\to SU(2)_Z \otimes SM$. We find that the SM couplings converge into $SU(6)$ at a mass scale $\ord(2 \times 10^{15}\gev)$, corresponding to a proton mean lifetime $\tau_p\sim 9 \times 10^{32}\textrm{yr}$, consistent with the actual lower bound, while the $E_6$ grand-unification occurs below the Planck scale. This scenario implies the existence of heavy mirror fermions with masses $\ord(250 \gev)$. \newline \newline [1] P. Q. Hung, hep-ph/0504060; \textit{ibid.}, Nucl. Phys. B {\textbf{747}}, 55 (2006).