In Grand Unified Theories (GUTs) from orbifold and various string
constructions the generic vector-like particles do not need to form complete
SU(5) or SO(10) representations. To realize them concretely, we present
orbifold SU(5) models, orbifold SO(10) models where the gauge symmetry can be
broken down to flipped SU(5) X U(1)_X or Pati-Salam SU(4)_C X SU(2)_L X SU(2)_R
gauge symmetries, and F-theory SU(5) models. Interestingly, these vector-like
particles can be at the TeV-scale so that the lightest CP-even Higgs boson mass
can be lifted, or play the messenger fields in the Gauge Mediated Supersymmetry
Breaking (GMSB). Considering GMSB, ultraviolet insensitive Anomaly Mediated
Supersymmetry Breaking (AMSB), and the deflected AMSB, we study the general
gaugino mass relations and their indices, which are valid from the GUT scale to
the electroweak scale at one loop, in the SU(5) models, the flipped SU(5) X
U(1)_X models, and the Pati-Salam SU(4)_C X SU(2)_L X SU(2)_R models. In the
deflected AMSB, we also define the new indices for the gaugino mass relations,
and calculate them as well. Using these gaugino mass relations and their
indices, we may probe the messenger fields at intermediate scale in the GMSB
and deflected AMSB, determine the supersymmetry breaking mediation mechanisms,
and distinguish the four-dimensional GUTs, orbifold GUTs, and F-theory GUTs.Comment: RevTex4, 45 pages, 15 tables, version to appear in JHE
