String scale gauge coupling unification with vector-like exotics and noncanonical U(1)_y normalization

We use a new approach to study string scale gauge coupling unification systematically, allowing both the possibility of noncanonical U(1)_Y normalization and the existence of vector-like particles whose quantum numbers are the same as those of the Standard Model (SM) fermions and their Hermitian conjugates and the SM adjoint particles. We first give all the independent sets (Yi) of particles that can be employed to achieve SU(3)_C and SU(2)_L string scale gauge coupling unification and calculate their masses. Second, for a noncanonical U(1)_Y normalization, we obtain string scale SU(3)_C × SU(2)_L × U(1) _Y gauge coupling unification by choosing suitable U(1)_Y normalizations for each of the Yi sets. Alternatively, for the canonical U(1)_Y normalization, we achieve string scale gauge coupling unification by considering suitable combinations of the Yi sets or by introducing additional independent sets (Zi), that do not affect the SU(3) _O × SU(2)_L unification at tree level, and then choosing suitable combinations, one from the Yi sets and one from the Zi sets. We also briefly discuss string scale gauge coupling unification in models with higher Kac-Moody levels for SU(2)_L or SU(3)_C.