Mass scales in grand unified theories

Abstract

Explicit expressions for the upper and lower bounds for all unification mass scales in allowed breakdowns of SU(5) and SO(10) are reviewed. All breakdown patterns of ${\mathrm{E}}_6$ through maximal subgroups are catalogued and bounds on the mass scales in such chains are derived. In all cases but one, the mass scale characterizing proton decay is at least as large as in SU(5). The dependence of the bounds on ${{sin}^2\theta }_W$ is examined and all bounds are found to be members of a single-parameter family whose significance is discussed. A weak-$\mathrm{S}{\mathrm{U}}_L\mathrm{}\left(2\right)\mathrm{}$ scale parameter ${\Lambda }_2$ is introduced in analogy with the strong scale ${\Lambda }_3$ and the unification masses are found to depend simply on ${\Lambda }_2$ and ${\Lambda }_3$ in a certain approximation.