Complete theory of grand unification in five dimensions

A fully realistic unified theory is constructed, with $\mathrm{SU}(5)$ gauge symmetry and supersymmetry both broken by boundary conditions in a fifth dimension. Despite the resulting explicit breaking of $\mathrm{SU}(5)$ locally at a boundary of the dimension, when the size of the extra dimension is taken to be large precise predictions emerge for gauge coupling unification, ${\ensuremath{\alpha}}_{s}{(M}_{Z})=0.118\ifmmode\pm\else\textpm\fi{}0.003,$ and for Yukawa coupling unification, ${m}_{b}{(M}_{Z})=3.3\ifmmode\pm\else\textpm\fi{}0.2\mathrm{GeV}.$ The 5D theory is then valid over a large energy interval from the compactification scale, ${M}_{c}\ensuremath{\simeq}1\ifmmode\times\else\texttimes\fi{}{10}^{15}\mathrm{GeV},$ to the scale of strong coupling, ${M}_{s}\ensuremath{\simeq}1\ifmmode\times\else\texttimes\fi{}{10}^{17}\mathrm{GeV}.$ A complete understanding of the Higgs sector of the minimal supersymmetric standard model is given, with explanations for why the Higgs triplets are heavy, why the Higgs doublets are protected from a large tree-level mass, and why the $\ensuremath{\mu}$ and B parameters are naturally generated to be of order the supersymmetry breaking scale. All sources of proton decay from operators of dimension four and five are forbidden, while a new origin for baryon number violating dimension six operators is found to be important. The exchange of the superheavy gauge boson, with a brane-localized kinetic energy interaction, leads to ${\ensuremath{\tau}}_{p}\ensuremath{\approx}{10}^{34}\mathrm{yr},$ with several branching ratios determined in terms of a single mixing parameter. The theory is only realistic for an essentially unique choice of matter location in the fifth dimension: the ten-plets of the first two generations must lie in the bulk, with all other matter located on the $\mathrm{SU}(5)$ preserving boundary. Several aspects of flavor follow from this geometry: only the third generation possesses an $\mathrm{SU}(5)$ mass relation, and the lighter two generations have only small mixings with the heaviest generation except for neutrinos. The entire superpartner spectrum is predicted in terms of only two free parameters. The squark and slepton masses have sizes determined by their location in the fifth dimension, allowing a significant experimental test of the detailed structure of the extra dimension. Lepton flavor violation is found to be generically large in higher dimensional unified theories with nontrivial matter geometries, providing soft supersymmetry breaking operators are local up to the compactification scale. In our theory this forces a common location for all three neutrinos, predicting large neutrino mixing angles. Rates for $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\mu}}e\ensuremath{\gamma},$ $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\mu}}eee,$ $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\mu}}e$ conversion and $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\tau}}\ensuremath{\mu}\ensuremath{\gamma}$ are larger in our theory than in conventional 4D supersymmetric grand unified theories, and, once superpartner masses are measured, these rates are completely determined in terms of two leptonic mixing angles. Proposed experiments probing $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\mu}}e$ transitions will probe the entire interesting parameter space of our theory.