CKM and PMNS mixing matrices from discrete subgroups of SU(2)

Abstract

Remaining within the realm of the Standard Model(SM) local gauge group, this first principles derivation of both the PMNS and CKM matrices utilizes quaternion generators of the three discrete (i.e., finite) binary rotational subgroups of SU(2) called [3,3,2], [4,3,2], and [5,3,2] for three lepton families in R3 and four related discrete binary rotational subgroups [3,3,3], [4,3,3], [3,4,3], and [5,3,3] represented by four quark families in R4. The traditional 3x3 CKM matrix is extracted as a submatrix of the 4x4 CKM4 matrix. If these two additional quarks b' and t' of a 4th quark family exist, there is the possibility that the SM lagrangian may apply all the way down to the Planck scale. There are then numerous other important consequences. The Weinberg angle is derived using these same quaternion generators, and the triangle anomaly cancellation is satisfied even though there is an obvious mismatch of three lepton families to four quark families. In a discrete space, one can also use these generators to derive a unique connection from the electroweak local gauge group SU(2)L x U(1)Y acting in R4 to the discrete group Weyl E8 in R8. By considering Lorentz transformations in discrete (3,1)-D spacetime, one obtains another Weyl E8 discrete symmetry group in R8, so that the combined symmetry is Weyl E8 x Weyl E8 = "discrete" SO(9,1) in 10-D spacetime. This unique connection is in direct contrast to the 10500 possible connections for superstring theory!