Cosmological Constant in a Model with Superstring-Inspired E₆ Unification and Shadow Theta-Particles

We have developed a concept of parallel existence of the ordinary (O) and mirror (M), or shadow (Sh) worlds. In the first part of the paper we consider a mirror world with broken mirror parity and the breaking $E_6\to SU(3)^3$ in both worlds. We show that in this case the evolutions of coupling constants in the O- and M-worlds are not identical, having different parameters for similar evolutions. $E_6$ unification, inspired by superstring theory, restores the broken mirror parity at the scale $\sim 10^{18}$ GeV. With the aim to explain the tiny cosmological constant, in the second part we consider the breakings: $E_6 \to SO(10)\times U(1)_Z$ -- in the O-world, and $E'_6 \to SU(6)'\times SU(2)'_{\theta}$ -- in the Sh-world. We assume the existence of shadow $\theta$-particles and the low energy symmetry group $SU(3)'_C\times SU(2)'_L\times SU(2)'_{\theta}\times U(1)'_Y$ in the shadow world, instead of the Standard Model. The additional non-Abelian $SU(2)'_{\theta}$ group with massless gauge fields, "theton", has a macroscopic confinement radius $1/\Lambda'_{\theta}$. The assumption that $\Lambda'_{\theta}\approx 2.3\cdot 10^{-3}$ eV explains the tiny cosmological constant given by recent astrophysical measurements. In this way the present work opens the possibility to specify a grand unification group, such as $E_6$, from Cosmology.