  In Grand Unified Theories (GUTs), the Standard Model (SM) gauge couplings
need not be unified at the GUT scale due to the high-dimensional operators.
Considering gravity mediated supersymmetry breaking, we study for the first
time the generic gauge coupling relations at the GUT scale, and the general
gaugino mass relations which are valid from the GUT scale to the electroweak
scale at one loop. We define the index k for these relations, which can be
calculated in GUTs and can be determined at the Large Hadron Collider and the
future International Linear Collider. Thus, we give a concrete definition of
the GUT scale in these theories, and suggest a new way to test general GUTs at
future experiments. We also discuss five special scenarios with interesting
possibilities. With our generic formulae, we present all the GUT-scale gauge
coupling relations and all the gaugino mass relations in the SU(5) and SO(10)
models, and calculate the corresponding indices k. Especially, the index k is
5/3 in the traditional SU(5) and SO(10) models that have been studied
extensively so far. Furthermore, we discuss the field theory realization of the
U(1) flux effects on the SM gauge kinetic functions in F-theory GUTs, and
calculate their indices k as well.
