Reconciling Grand Unification with Strings by Anisotropic Compactifications

We analyze gauge coupling unification in the context of heterotic strings on anisotropic orbifolds. This construction is very much analogous to effective 5 dimensional orbifold GUT field theories. Our analysis assumes three fundamental scales, the string scale, $\text{mstring}$, a compactification scale, $\text{mc}$, and a mass scale for some of the vector-like exotics, $\text{mex}$; the other exotics are assumed to get mass at $\text{mstring}$. In the particular models analyzed, we show that gauge coupling unification is not possible with $\text{mex}=\text{mc}$ and in fact we require $\text{mex}\ll \text{mc}\sim 3\times {10}^{16}$ GeV. We find that about 10% of the parameter space has a proton lifetime (from dimension 6 gauge exchange) ${10}^{33}\mathrm{y}\mathrm{r}\lesssim \tau (p\to {\pi }^0{e}^{+})\lesssim {10}^{36}\mathrm{y}\mathrm{r}$. The other 80% of the parameter space gives proton lifetimes below Super-K bounds. The next generation of proton decay experiments should be sensitive to the remaining parameter space.