SO(10)-Grand Unification and Fermion Masses

In the state of the art the Standard Model is the best gauge theory
describing interactions among elementary particles. It comprises
all of the fundamental interactions in nature except gravitation.
Its predictions have been experimentally tested to a high level of
accuracy. However, it is not considered to be the fundamental
theory of gauge interactions. It contains a lot of arbitrary
parameters. It can not predict the fermion masses and fails to
explain the smallness of neutrino masses which have been observed
by recent experiments. It contains no gauge bosons that can mediate
nucleon decays via baryon and lepton number violating process,
which are needed to explain the baryon asymmetry in our universe.
Furthermore, CP violation has to be introduced into the CKM and MNS
matrices by hand. The shortcomings of the Standard Model can be
solved in the framework of grand unified gauge theories (GUTs)
which have greater degrees of freedom. GUT's which have truly one
coupling constant are based on gauge groups that contain the
Standard Model as a subgroup. There are a limited number of such
gauge groups. SO(10) is a fully symmetric gauge group that has two
outstanding features: It unifies all the known gauge interactions
under a single coupling strength and classifies all the known
fermions of a family under a single spinor. In this work, we will
study SO(10) grand unification in its full extent by using
different explicit matrix representations which exhibit the
structure of SO(10) in a very transparent way. Our approach
consists mainly of two stages: We will derive the explicit
expressions of the mass-eigenvalues and mass-eigenstates of the
physical gauge bosons from a mass squared-matrix that contains all
the information about the mixing parameters among the gauge fields
and the phases which are sources for CP violation. In the light of
this analysis, we will derive the explicit expressions for the
interaction Lagrangians of the charged currents, the neutral
currents and the charged and colored currents in SO(10). We will
present explicit expressions of the vector and axial-vector
couplings of the two neutral currents in SO(10). We will show how
the baryon, lepton and baryon minus lepton number violating
processes and their explicit CP violating phases are accommodated
in the SO(10) theory. The Higgs potential that we use to implement
in the Higgs mechanism will be constructed in a most general
fashion through a careful study of the Higgs fields of SO(10),
where we give special emphasis on illustrating the explicit matrix
representation of these Higgs fields. The potential part of the
Higgs Lagrangian will give us the properties of the minimum of the
vacuum, and the kinetic part will give us the mass-squared matrix
of the gauge bosons via spontaneous symmetry breakdown. The same
Higgs multiplets will be coupled to fermions through a democratic
Yukawa matrix. Thereby, we will derive explicit expressions for the
fermion masses of the third family including Majorana and Dirac
masses for neutrinos. We will introduce a flavor-eigenbasis for
neutrinos and find the mass-eigenstates and mass-eigenvalues of the
neutrinos. Explicit expressions for CP violation in the neutrino
sector will be obtained. In the second stage of our work, we will
evaluate all the above mentioned quantities. We will compare our
results with those of the Standard Model like the W and Z masses
and the vector and axial-vector coupling of the NC current and the
fermion masses of the third family. In addition, we will present
the values of the physical quantities that are not present in the
Standard Model like the masses of new gauge bosons, the vector and
axial-vector couplings of a new NC current, the masses of a light
left-handed and a heavier right-neutrino, the values of various
mixing parameters and CP phases etc. The input values required for
these evaluations will be acquired mainly from two sources: First,
we will determine the vacuum expectation values and the coupling
strengths of gauge interactions given by the SO(10) theory in so
far as possible through studying the mass scales in SO(10) in the
framework of coupling unification. Complementarily, we will
determine the vacuum expectation values and their phases by
adjusting them to the masses of the known gauge bosons and fermions
below the Fermi scale which are accurately measured and known. We
will be able to predict more than 67 parameters with an input of 7
vacuum expectation values, 5 angles, 1 gauge coupling and 1 Yukawa
coupling.