CKM Matrix Parameters from the Exceptional Jordan Algebra

Abstract

We report a theoretical derivation of the Cabibbo–Kobayashi–Maskawa (CKM) matrix parameters and the accompanying mixing angles. These results are arrived at from the exceptional Jordan algebra applied to quark states, and from expressing flavor eigenstates (i.e., left chiral states) as a superposition of mass eigenstates (i.e., the right chiral states) weighted by the square root of mass. Flavor mixing for quarks is mediated by the square root mass eigenstates, and the mass ratios used are derived from earlier work from a left–right symmetric extension of the standard model. This permits a construction of the CKM matrix from first principles. There exist only four normed division algebras, and they can be listed as follows: the real numbers R, the complex numbers C, the quaternions H and the octonions O. The first three algebras are fairly well known; however, octonions as algebra are less studied. Recent research has pointed towards the importance of octonions in the study of high-energy physics. Clifford algebras and the standard model are being studied closely. The main advantage of this approach is that the spinor representations of the fundamental fermions can be constructed easily here as the left ideals of the algebra. Also, the action of various spin groups on these representations can also be studied easily. In this work, we build on some recent advances in the field and try to determine the CKM angles from an algebraic framework. We obtain the mixing angle values as θ12=11.093∘,θ13=0.172∘,θ23=4.054∘. In comparison, the corresponding experimentally measured values for these angles are 13.04∘±0.05∘,0.201∘±0.011∘,2.38∘±0.06∘. The agreement of theory with experiment is likely to improve when the running of quark masses is taken into account.

Quantum number conservation: a tool in the design and analysis of high energy experiments

“…Most recently, Pavšič [11] has given string theoretic arguments that Cl 8,8 is capable of providing a description of the elementary fermions and their interactions, but does not provide physical interpretations of the elements of this algebra. Some recent work [12,13] has employed more complex algebraic structures that 'go beyond' the SM. But, as these papers to build on the SM they fail to provide solutions to the fundamental problems, viz.…”

“…86-89). In the case of electrons there are the four distinct solutions e − ↑ , e − ↓ , e + ↑ , e + ↓ , each of which corresponds to one of the four combinations of eigenvalues μ 0 = ±1, μ 12 = ±i of the commuting elements γ 0 and γ 12 .…”