SO(10) grand unified theory in generalized covariant derivative formalism

The SO(10) grand unified theory is reformulated in a new field theory with a unified field strength for both the gauge and Higgs fields, and the fine-tuning problem and the condition for the symmetry breakings are investigated. The unified field strength for the gauge and Higgs fields, which takes values in the Dirac algebra, is defined by means of the commutator of the generalized covariant derivative for a multi-spinor field describing all families of quarks and leptons. The bosonic Lagrangian is constructed from a general sum of quadratic invariants of the field strength. Among the Yukawa coupling constants and other parameters related to the Higgs field self-couplings, there exist additional relations that are interpreted as initial conditions for renormalization equations. The grand unified symmetry is broken down to the low energy symmetry by the 210-, 45-, 126- and 10-dimensional Higgs fields. The Higgs potential turns out to have a discrete symmetry among Higgs fields. This symmetry makes it subtle and difficult to solve the fine-tuning problem, which requires accurate adjustments of the parameters included in the generalized covariant derivative. It is shown that the 45-dimensional Higgs field plays an essential role to break the discrete symmetry and to solve the fine-tuning problem.