Executive Summary : | This proposal aims to explore the signatures of Left-Right scenarios arising from breaking the exceptional group $E_6$, which is closely connected to string theory. The E_6 group has $SU(3)^3$ subgroup, with one remaining unbroken and identified as strong force group $SU(3)_c$. The other two $SU(3)$ further break into $SU(2)_L \times SU(2)_H \times U(1)_X$, embedding the electroweak group $SU(2)_L \times U(1)_Y$. There are three different ways to embed $SU(2)_H$ into $SU(3)_H$: the usual left-right symmetric model (LRSM), Alternative left-right model (ALRSM), and an inert doublet model. The first scenario involves tree-level flavor violations due to the Higgs sector, making the additional Higgs and gauge bosons heavy and unlikely to reach the LHC. The second variant of conventional Left-Right models, which bring remarkable changes in the physical spectrum and possess interesting features, can produce compelling signatures testable at the LHC and ILC. The top quark Flavour Changing Neutral Current (FCNC) interactions, which are forbidden within the Standard Model (SM) due to the GIM Mechanism, are considered. The Branching Ratio for these decays in the SM is $BR(t\to (Z/\gamma/g)u) \sim 10^-17}$, with current experimental constraints at $\sim 10^-4}$. The projected reach of these BR's at the high luminosity LHC is $2.5-5.5 \times 10^-5}$. Colliders with electron beams have multiple advantages in probing these interactions. |