Modification of Predictions of Grand Unified Theories in the Presence of Spontaneous Compactification

There are speculations that grand unified theories (GUT's) may arise from higher-dimensional models of gravity in which the extra dimensions are spontaneously compactified. The GUT predictions for proton decay and ${{sin}^2\theta }_w$ can be significantly modified when nonrenormalizable interactions, scaled by inverse powers of the compactification scale $M_c$, are added to the standard GUT Lagrangian. For example, the decay rate for $p\to e^{+}{\pi }^0$ in minimal SU(5) can be lowered by one (two) orders of magnitude if $M_c$ is on the order of ${10}^{17}$ GeV. However, ${{sin}^2\theta }_w$ then decreases by 0.005 (0.01). The SO(10) model is also discussed.